diff --git a/build.rs b/build.rs index 23e1178..29521ab 100644 --- a/build.rs +++ b/build.rs @@ -41,6 +41,19 @@ mod musl_reference_tests { "rem_pio2.rs", "rem_pio2_large.rs", "rem_pio2f.rs", + "remquo.rs", + "remquof.rs", + "lgamma.rs", // lgamma passed, lgamma_r has more than 1 result + "lgammaf.rs", // lgammaf passed, lgammaf_r has more than 1 result + "frexp.rs", // more than 1 result + "frexpf.rs", // more than 1 result + "sincos.rs", // more than 1 result + "sincosf.rs", // more than 1 result + "modf.rs", // more than 1 result + "modff.rs", // more than 1 result + "asinef.rs", // not exists + "jn.rs", // passed, but very slow + "jnf.rs", // passed, but very slow ]; struct Function { @@ -78,12 +91,9 @@ mod musl_reference_tests { let contents = fs::read_to_string(file).unwrap(); let mut functions = contents.lines().filter(|f| f.starts_with("pub fn")); - let function_to_test = functions.next().unwrap(); - if functions.next().is_some() { - panic!("more than one function in"); + while let Some(function_to_test) = functions.next() { + math.push(parse(function_to_test)); } - - math.push(parse(function_to_test)); } // Generate a bunch of random inputs for each function. This will @@ -330,7 +340,7 @@ mod musl_reference_tests { src.push_str(match function.ret { Ty::F32 => "if _eqf(output, f32::from_bits(*expected as u32)).is_ok() { continue }", Ty::F64 => "if _eq(output, f64::from_bits(*expected as u64)).is_ok() { continue }", - Ty::I32 => "if output as i64 == expected { continue }", + Ty::I32 => "if output as i64 == *expected { continue }", Ty::Bool => unreachable!(), }); diff --git a/ci/run-docker.sh b/ci/run-docker.sh old mode 100755 new mode 100644 diff --git a/ci/run.sh b/ci/run.sh old mode 100755 new mode 100644 diff --git a/src/math/acosh.rs b/src/math/acosh.rs index 3494e34..95dc57d 100644 --- a/src/math/acosh.rs +++ b/src/math/acosh.rs @@ -1,22 +1,22 @@ -use super::{log, log1p, sqrt}; - -const LN2: f64 = 0.693147180559945309417232121458176568; /* 0x3fe62e42, 0xfefa39ef*/ - -/* acosh(x) = log(x + sqrt(x*x-1)) */ -pub fn acosh(x: f64) -> f64 { - let u = x.to_bits(); - let e = ((u >> 52) as usize) & 0x7ff; - - /* x < 1 domain error is handled in the called functions */ - - if e < 0x3ff + 1 { - /* |x| < 2, up to 2ulp error in [1,1.125] */ - return log1p(x-1.0+sqrt((x-1.0)*(x-1.0)+2.0*(x-1.0))); - } - if e < 0x3ff + 26 { - /* |x| < 0x1p26 */ - return log(2.0*x-1.0/(x+sqrt(x*x-1.0))); - } - /* |x| >= 0x1p26 or nan */ - return log(x) + LN2; -} +use super::{log, log1p, sqrt}; + +const LN2: f64 = 0.693147180559945309417232121458176568; /* 0x3fe62e42, 0xfefa39ef*/ + +/* acosh(x) = log(x + sqrt(x*x-1)) */ +pub fn acosh(x: f64) -> f64 { + let u = x.to_bits(); + let e = ((u >> 52) as usize) & 0x7ff; + + /* x < 1 domain error is handled in the called functions */ + + if e < 0x3ff + 1 { + /* |x| < 2, up to 2ulp error in [1,1.125] */ + return log1p(x - 1.0 + sqrt((x - 1.0) * (x - 1.0) + 2.0 * (x - 1.0))); + } + if e < 0x3ff + 26 { + /* |x| < 0x1p26 */ + return log(2.0 * x - 1.0 / (x + sqrt(x * x - 1.0))); + } + /* |x| >= 0x1p26 or nan */ + return log(x) + LN2; +} diff --git a/src/math/acoshf.rs b/src/math/acoshf.rs index 1e298a9..f50a003 100644 --- a/src/math/acoshf.rs +++ b/src/math/acoshf.rs @@ -1,21 +1,21 @@ -use super::{log1pf, logf, sqrtf}; - -const LN2: f32 = 0.693147180559945309417232121458176568; - -/* acosh(x) = log(x + sqrt(x*x-1)) */ -pub fn acoshf(x: f32) -> f32 { - let u = x.to_bits(); - let a = u & 0x7fffffff; - - if a < 0x3f800000+(1<<23) { - /* |x| < 2, invalid if x < 1 or nan */ - /* up to 2ulp error in [1,1.125] */ - return log1pf(x-1.0 + sqrtf((x-1.0)*(x-1.0)+2.0*(x-1.0))); - } - if a < 0x3f800000+(12<<23) { - /* |x| < 0x1p12 */ - return logf(2.0*x - 1.0/(x+sqrtf(x*x-1.0))); - } - /* x >= 0x1p12 */ - return logf(x) + LN2; -} +use super::{log1pf, logf, sqrtf}; + +const LN2: f32 = 0.693147180559945309417232121458176568; + +/* acosh(x) = log(x + sqrt(x*x-1)) */ +pub fn acoshf(x: f32) -> f32 { + let u = x.to_bits(); + let a = u & 0x7fffffff; + + if a < 0x3f800000 + (1 << 23) { + /* |x| < 2, invalid if x < 1 or nan */ + /* up to 2ulp error in [1,1.125] */ + return log1pf(x - 1.0 + sqrtf((x - 1.0) * (x - 1.0) + 2.0 * (x - 1.0))); + } + if a < 0x3f800000 + (12 << 23) { + /* |x| < 0x1p12 */ + return logf(2.0 * x - 1.0 / (x + sqrtf(x * x - 1.0))); + } + /* x >= 0x1p12 */ + return logf(x) + LN2; +} diff --git a/src/math/asinef.rs b/src/math/asinef.rs index d2cd826..cd1428b 100644 --- a/src/math/asinef.rs +++ b/src/math/asinef.rs @@ -1,95 +1,93 @@ -/* @(#)z_asinef.c 1.0 98/08/13 */ -/****************************************************************** - * The following routines are coded directly from the algorithms - * and coefficients given in "Software Manual for the Elementary - * Functions" by William J. Cody, Jr. and William Waite, Prentice - * Hall, 1980. - ******************************************************************/ -/****************************************************************** - * Arcsine - * - * Input: - * x - floating point value - * acosine - indicates acos calculation - * - * Output: - * Arcsine of x. - * - * Description: - * This routine calculates arcsine / arccosine. - * - *****************************************************************/ - -use super::{fabsf, sqrtf}; - -const P: [f32; 2] = [ 0.933935835, -0.504400557 ]; -const Q: [f32; 2] = [ 0.560363004e+1, -0.554846723e+1 ]; -const A: [f32; 2] = [ 0.0, 0.785398163 ]; -const B: [f32; 2] = [ 1.570796326, 0.785398163 ]; -const Z_ROOTEPS_F: f32 = 1.7263349182589107e-4; - -pub fn asinef(x: f32, acosine: usize) -> f32 -{ - let flag: usize; - let i: usize; - let mut branch: bool = false; - let g: f32; - let mut res: f32 = 0.0; - let mut y: f32; - - /* Check for special values. */ - //i = numtestf (x); - if x.is_nan() || x.is_infinite() { - force_eval!(x); - return x; - } - - y = fabsf(x); - flag = acosine; - - if y > 0.5 { - i = 1 - flag; - - /* Check for range error. */ - if y > 1.0 { - return 0.0 / 0.0; - } - - g = (1.0 - y) / 2.0; - y = -2.0 * sqrtf(g); - branch = true; - } else { - i = flag; - if y < Z_ROOTEPS_F { - res = y; - g = 0.0; // pleasing the uninitialized variable - } else { - g = y * y; - } - } - - if y >= Z_ROOTEPS_F || branch { - /* Calculate the Taylor series. */ - let p = (P[1] * g + P[0]) * g; - let q = (g + Q[1]) * g + Q[0]; - let r = p / q; - - res = y + y * r; - } - - /* Calculate asine or acose. */ - if flag == 0 { - res = (A[i] + res) + A[i]; - if x < 0.0 { - res = -res; - } - } else { - if x < 0.0 { - res = (B[i] + res) + B[i]; - } else { - res = (A[i] - res) + A[i]; - } - } - - return res; -} +/* @(#)z_asinef.c 1.0 98/08/13 */ +/****************************************************************** + * The following routines are coded directly from the algorithms + * and coefficients given in "Software Manual for the Elementary + * Functions" by William J. Cody, Jr. and William Waite, Prentice + * Hall, 1980. + ******************************************************************/ +/****************************************************************** + * Arcsine + * + * Input: + * x - floating point value + * acosine - indicates acos calculation + * + * Output: + * Arcsine of x. + * + * Description: + * This routine calculates arcsine / arccosine. + * + *****************************************************************/ + +use super::{fabsf, sqrtf}; + +const P: [f32; 2] = [ 0.933935835, -0.504400557 ]; +const Q: [f32; 2] = [ 0.560363004e+1, -0.554846723e+1 ]; +const A: [f32; 2] = [ 0.0, 0.785398163 ]; +const B: [f32; 2] = [ 1.570796326, 0.785398163 ]; +const Z_ROOTEPS_F: f32 = 1.7263349182589107e-4; + +pub fn asinef(x: f32, acosine: bool) -> f32 { + let i: usize; + let mut branch: bool = false; + let g: f32; + let mut res: f32 = 0.0; + let mut y: f32; + + /* Check for special values. */ + //i = numtestf (x); + if x.is_nan() || x.is_infinite() { + force_eval!(x); + return x; + } + + y = fabsf(x); + let flag = acosine; + + if y > 0.5 { + i = (!flag) as usize; + + /* Check for range error. */ + if y > 1.0 { + return 0.0 / 0.0; + } + + g = (1.0 - y) / 2.0; + y = -2.0 * sqrtf(g); + branch = true; + } else { + i = flag; + if y < Z_ROOTEPS_F { + res = y; + g = 0.0; // pleasing the uninitialized variable + } else { + g = y * y; + } + } + + if y >= Z_ROOTEPS_F || branch { + /* Calculate the Taylor series. */ + let p = (P[1] * g + P[0]) * g; + let q = (g + Q[1]) * g + Q[0]; + let r = p / q; + + res = y + y * r; + } + + /* Calculate asine or acose. */ + if flag == 0 { + res = (A[i] + res) + A[i]; + if x < 0.0 { + res = -res; + } + } else { + if x < 0.0 { + res = (B[i] + res) + B[i]; + } else { + res = (A[i] - res) + A[i]; + } + } + + return res; +} diff --git a/src/math/asinh.rs b/src/math/asinh.rs index 09e8945..b29093b 100644 --- a/src/math/asinh.rs +++ b/src/math/asinh.rs @@ -1,35 +1,35 @@ -use super::{log, log1p, sqrt}; - -const LN2: f64 = 0.693147180559945309417232121458176568; /* 0x3fe62e42, 0xfefa39ef*/ - -/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */ -pub fn asinh(mut x: f64) -> f64 { - let mut u = x.to_bits(); - let e = ((u >> 52) as usize) & 0x7ff; - let sign = (u >> 63) != 0; - - /* |x| */ - u &= (!0) >> 1; - x = f64::from_bits(u); - - if e >= 0x3ff + 26 { - /* |x| >= 0x1p26 or inf or nan */ - x = log(x) + LN2; - } else if e >= 0x3ff + 1 { - /* |x| >= 2 */ - x = log(2.0*x + 1.0/(sqrt(x*x+1.0)+x)); - } else if e >= 0x3ff - 26 { - /* |x| >= 0x1p-26, up to 1.6ulp error in [0.125,0.5] */ - x = log1p(x + x*x/(sqrt(x*x+1.0)+1.0)); - } else { - /* |x| < 0x1p-26, raise inexact if x != 0 */ - let x1p120 = f64::from_bits(0x4770000000000000); - force_eval!(x + x1p120); - } - - if sign { - -x - } else { - x - } -} +use super::{log, log1p, sqrt}; + +const LN2: f64 = 0.693147180559945309417232121458176568; /* 0x3fe62e42, 0xfefa39ef*/ + +/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */ +pub fn asinh(mut x: f64) -> f64 { + let mut u = x.to_bits(); + let e = ((u >> 52) as usize) & 0x7ff; + let sign = (u >> 63) != 0; + + /* |x| */ + u &= (!0) >> 1; + x = f64::from_bits(u); + + if e >= 0x3ff + 26 { + /* |x| >= 0x1p26 or inf or nan */ + x = log(x) + LN2; + } else if e >= 0x3ff + 1 { + /* |x| >= 2 */ + x = log(2.0 * x + 1.0 / (sqrt(x * x + 1.0) + x)); + } else if e >= 0x3ff - 26 { + /* |x| >= 0x1p-26, up to 1.6ulp error in [0.125,0.5] */ + x = log1p(x + x * x / (sqrt(x * x + 1.0) + 1.0)); + } else { + /* |x| < 0x1p-26, raise inexact if x != 0 */ + let x1p120 = f64::from_bits(0x4770000000000000); + force_eval!(x + x1p120); + } + + if sign { + -x + } else { + x + } +} diff --git a/src/math/asinhf.rs b/src/math/asinhf.rs index 236916d..9812433 100644 --- a/src/math/asinhf.rs +++ b/src/math/asinhf.rs @@ -1,34 +1,34 @@ -use super::{logf, log1pf, sqrtf}; - -const LN2: f32 = 0.693147180559945309417232121458176568; - -/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */ -pub fn asinhf(mut x: f32) -> f32 { - let u = x.to_bits(); - let i = u & 0x7fffffff; - let sign = (u >> 31) != 0; - - /* |x| */ - x = f32::from_bits(i); - - if i >= 0x3f800000 + (12<<23) { - /* |x| >= 0x1p12 or inf or nan */ - x = logf(x) + LN2; - } else if i >= 0x3f800000 + (1<<23) { - /* |x| >= 2 */ - x = logf(2.0*x + 1.0/(sqrtf(x*x+1.0)+x)); - } else if i >= 0x3f800000 - (12<<23) { - /* |x| >= 0x1p-12, up to 1.6ulp error in [0.125,0.5] */ - x = log1pf(x + x*x/(sqrtf(x*x+1.0)+1.0)); - } else { - /* |x| < 0x1p-12, raise inexact if x!=0 */ - let x1p120 = f32::from_bits(0x7b800000); - force_eval!(x + x1p120); - } - - if sign { - -x - } else { - x - } -} +use super::{log1pf, logf, sqrtf}; + +const LN2: f32 = 0.693147180559945309417232121458176568; + +/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */ +pub fn asinhf(mut x: f32) -> f32 { + let u = x.to_bits(); + let i = u & 0x7fffffff; + let sign = (u >> 31) != 0; + + /* |x| */ + x = f32::from_bits(i); + + if i >= 0x3f800000 + (12 << 23) { + /* |x| >= 0x1p12 or inf or nan */ + x = logf(x) + LN2; + } else if i >= 0x3f800000 + (1 << 23) { + /* |x| >= 2 */ + x = logf(2.0 * x + 1.0 / (sqrtf(x * x + 1.0) + x)); + } else if i >= 0x3f800000 - (12 << 23) { + /* |x| >= 0x1p-12, up to 1.6ulp error in [0.125,0.5] */ + x = log1pf(x + x * x / (sqrtf(x * x + 1.0) + 1.0)); + } else { + /* |x| < 0x1p-12, raise inexact if x!=0 */ + let x1p120 = f32::from_bits(0x7b800000); + force_eval!(x + x1p120); + } + + if sign { + -x + } else { + x + } +} diff --git a/src/math/atan2.rs b/src/math/atan2.rs index 7ab6360..313bec4 100644 --- a/src/math/atan2.rs +++ b/src/math/atan2.rs @@ -53,7 +53,7 @@ pub fn atan2(y: f64, x: f64) -> f64 { let lx = x.to_bits() as u32; let mut iy = (y.to_bits() >> 32) as u32; let ly = y.to_bits() as u32; - if ((ix - 0x3ff00000) | lx) == 0 { + if ((ix.wrapping_sub(0x3ff00000)) | lx) == 0 { /* x = 1.0 */ return atan(y); } diff --git a/src/math/atanh.rs b/src/math/atanh.rs index ea44480..2833715 100644 --- a/src/math/atanh.rs +++ b/src/math/atanh.rs @@ -1,33 +1,32 @@ -use super::{log1p}; - -/* atanh(x) = log((1+x)/(1-x))/2 = log1p(2x/(1-x))/2 ~= x + x^3/3 + o(x^5) */ -pub fn atanh(mut x: f64) -> f64 { - let mut u = x.to_bits(); - let e = ((u >> 52) as usize) & 0x7ff; - let sign = (u >> 63) != 0; - - /* |x| */ - u &= 0x7fffffff; - x = f64::from_bits(u); - - if e < 0x3ff - 1 { - if e < 0x3ff - 32 { - /* handle underflow */ - if e == 0 { - force_eval!(x as f32); - } - } else { - /* |x| < 0.5, up to 1.7ulp error */ - x = 0.5*log1p(2.0*x + 2.0*x*x/(1.0-x)); - } - } else { - /* avoid overflow */ - x = 0.5*log1p(2.0*(x/(1.0-x))); - } - - if sign { - -x - } else { - x - } -} +use super::log1p; + +/* atanh(x) = log((1+x)/(1-x))/2 = log1p(2x/(1-x))/2 ~= x + x^3/3 + o(x^5) */ +pub fn atanh(x: f64) -> f64 { + let u = x.to_bits(); + let e = ((u >> 52) as usize) & 0x7ff; + let sign = (u >> 63) != 0; + + /* |x| */ + let mut y = f64::from_bits(u & 0x7fff_ffff_ffff_ffff); + + if e < 0x3ff - 1 { + if e < 0x3ff - 32 { + /* handle underflow */ + if e == 0 { + force_eval!(y as f32); + } + } else { + /* |x| < 0.5, up to 1.7ulp error */ + y = 0.5 * log1p(2.0 * y + 2.0 * y * y / (1.0 - y)); + } + } else { + /* avoid overflow */ + y = 0.5 * log1p(2.0 * (y / (1.0 - y))); + } + + if sign { + -y + } else { + y + } +} diff --git a/src/math/atanhf.rs b/src/math/atanhf.rs index 77d451b..709a955 100644 --- a/src/math/atanhf.rs +++ b/src/math/atanhf.rs @@ -1,32 +1,32 @@ -use super::{log1pf}; - -/* atanh(x) = log((1+x)/(1-x))/2 = log1p(2x/(1-x))/2 ~= x + x^3/3 + o(x^5) */ -pub fn atanhf(mut x: f32) -> f32 { - let mut u = x.to_bits(); - let sign = (u >> 31) != 0; - - /* |x| */ - u &= 0x7fffffff; - x = f32::from_bits(u); - - if u < 0x3f800000 - (1<<23) { - if u < 0x3f800000 - (32<<23) { - /* handle underflow */ - if u < (1<<23) { - force_eval!((x*x) as f32); - } - } else { - /* |x| < 0.5, up to 1.7ulp error */ - x = 0.5*log1pf(2.0*x + 2.0*x*x/(1.0-x)); - } - } else { - /* avoid overflow */ - x = 0.5*log1pf(2.0*(x/(1.0-x))); - } - - if sign { - -x - } else { - x - } -} +use super::log1pf; + +/* atanh(x) = log((1+x)/(1-x))/2 = log1p(2x/(1-x))/2 ~= x + x^3/3 + o(x^5) */ +pub fn atanhf(mut x: f32) -> f32 { + let mut u = x.to_bits(); + let sign = (u >> 31) != 0; + + /* |x| */ + u &= 0x7fffffff; + x = f32::from_bits(u); + + if u < 0x3f800000 - (1 << 23) { + if u < 0x3f800000 - (32 << 23) { + /* handle underflow */ + if u < (1 << 23) { + force_eval!((x * x) as f32); + } + } else { + /* |x| < 0.5, up to 1.7ulp error */ + x = 0.5 * log1pf(2.0 * x + 2.0 * x * x / (1.0 - x)); + } + } else { + /* avoid overflow */ + x = 0.5 * log1pf(2.0 * (x / (1.0 - x))); + } + + if sign { + -x + } else { + x + } +} diff --git a/src/math/ceilf.rs b/src/math/ceilf.rs index 88f9ecc..0be53c5 100644 --- a/src/math/ceilf.rs +++ b/src/math/ceilf.rs @@ -12,7 +12,7 @@ pub fn ceilf(x: f32) -> f32 { } } let mut ui = x.to_bits(); - let e = (((ui >> 23) & 0xff) - 0x7f) as i32; + let e = (((ui >> 23) & 0xff).wrapping_sub(0x7f)) as i32; if e >= 23 { return x; diff --git a/src/math/copysign.rs b/src/math/copysign.rs index 74b761e..9c5362a 100644 --- a/src/math/copysign.rs +++ b/src/math/copysign.rs @@ -1,7 +1,7 @@ -pub fn copysign(x: f64, y: f64) -> f64 { - let mut ux = x.to_bits(); - let uy = y.to_bits(); - ux &= (!0) >> 1; - ux |= uy & (1<<63); - f64::from_bits(ux) -} +pub fn copysign(x: f64, y: f64) -> f64 { + let mut ux = x.to_bits(); + let uy = y.to_bits(); + ux &= (!0) >> 1; + ux |= uy & (1 << 63); + f64::from_bits(ux) +} diff --git a/src/math/copysignf.rs b/src/math/copysignf.rs index a0a814b..b42fd39 100644 --- a/src/math/copysignf.rs +++ b/src/math/copysignf.rs @@ -1,7 +1,7 @@ -pub fn copysignf(x: f32, y: f32) -> f32 { - let mut ux = x.to_bits(); - let uy = y.to_bits(); - ux &= 0x7fffffff; - ux |= uy & 0x80000000; - f32::from_bits(ux) -} +pub fn copysignf(x: f32, y: f32) -> f32 { + let mut ux = x.to_bits(); + let uy = y.to_bits(); + ux &= 0x7fffffff; + ux |= uy & 0x80000000; + f32::from_bits(ux) +} diff --git a/src/math/erf.rs b/src/math/erf.rs index b3ad2ce..d53a4c8 100644 --- a/src/math/erf.rs +++ b/src/math/erf.rs @@ -1,297 +1,306 @@ -use super::{exp, fabs, get_high_word, with_set_low_word}; -/* origin: FreeBSD /usr/src/lib/msun/src/s_erf.c */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ -/* double erf(double x) - * double erfc(double x) - * x - * 2 |\ - * erf(x) = --------- | exp(-t*t)dt - * sqrt(pi) \| - * 0 - * - * erfc(x) = 1-erf(x) - * Note that - * erf(-x) = -erf(x) - * erfc(-x) = 2 - erfc(x) - * - * Method: - * 1. For |x| in [0, 0.84375] - * erf(x) = x + x*R(x^2) - * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] - * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] - * where R = P/Q where P is an odd poly of degree 8 and - * Q is an odd poly of degree 10. - * -57.90 - * | R - (erf(x)-x)/x | <= 2 - * - * - * Remark. The formula is derived by noting - * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) - * and that - * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 - * is close to one. The interval is chosen because the fix - * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is - * near 0.6174), and by some experiment, 0.84375 is chosen to - * guarantee the error is less than one ulp for erf. - * - * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and - * c = 0.84506291151 rounded to single (24 bits) - * erf(x) = sign(x) * (c + P1(s)/Q1(s)) - * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 - * 1+(c+P1(s)/Q1(s)) if x < 0 - * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 - * Remark: here we use the taylor series expansion at x=1. - * erf(1+s) = erf(1) + s*Poly(s) - * = 0.845.. + P1(s)/Q1(s) - * That is, we use rational approximation to approximate - * erf(1+s) - (c = (single)0.84506291151) - * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] - * where - * P1(s) = degree 6 poly in s - * Q1(s) = degree 6 poly in s - * - * 3. For x in [1.25,1/0.35(~2.857143)], - * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) - * erf(x) = 1 - erfc(x) - * where - * R1(z) = degree 7 poly in z, (z=1/x^2) - * S1(z) = degree 8 poly in z - * - * 4. For x in [1/0.35,28] - * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 - * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6 x >= 28 - * erf(x) = sign(x) *(1 - tiny) (raise inexact) - * erfc(x) = tiny*tiny (raise underflow) if x > 0 - * = 2 - tiny if x<0 - * - * 7. Special case: - * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, - * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, - * erfc/erf(NaN) is NaN - */ - -const ERX: f64 = 8.45062911510467529297e-01; /* 0x3FEB0AC1, 0x60000000 */ -/* - * Coefficients for approximation to erf on [0,0.84375] - */ -const EFX8: f64 = 1.02703333676410069053e+00; /* 0x3FF06EBA, 0x8214DB69 */ -const PP0: f64 = 1.28379167095512558561e-01; /* 0x3FC06EBA, 0x8214DB68 */ -const PP1: f64 = -3.25042107247001499370e-01; /* 0xBFD4CD7D, 0x691CB913 */ -const PP2: f64 = -2.84817495755985104766e-02; /* 0xBF9D2A51, 0xDBD7194F */ -const PP3: f64 = -5.77027029648944159157e-03; /* 0xBF77A291, 0x236668E4 */ -const PP4: f64 = -2.37630166566501626084e-05; /* 0xBEF8EAD6, 0x120016AC */ -const QQ1: f64 = 3.97917223959155352819e-01; /* 0x3FD97779, 0xCDDADC09 */ -const QQ2: f64 = 6.50222499887672944485e-02; /* 0x3FB0A54C, 0x5536CEBA */ -const QQ3: f64 = 5.08130628187576562776e-03; /* 0x3F74D022, 0xC4D36B0F */ -const QQ4: f64 = 1.32494738004321644526e-04; /* 0x3F215DC9, 0x221C1A10 */ -const QQ5: f64 = -3.96022827877536812320e-06; /* 0xBED09C43, 0x42A26120 */ -/* - * Coefficients for approximation to erf in [0.84375,1.25] - */ -const PA0: f64 = -2.36211856075265944077e-03; /* 0xBF6359B8, 0xBEF77538 */ -const PA1: f64 = 4.14856118683748331666e-01; /* 0x3FDA8D00, 0xAD92B34D */ -const PA2: f64 = -3.72207876035701323847e-01; /* 0xBFD7D240, 0xFBB8C3F1 */ -const PA3: f64 = 3.18346619901161753674e-01; /* 0x3FD45FCA, 0x805120E4 */ -const PA4: f64 = -1.10894694282396677476e-01; /* 0xBFBC6398, 0x3D3E28EC */ -const PA5: f64 = 3.54783043256182359371e-02; /* 0x3FA22A36, 0x599795EB */ -const PA6: f64 = -2.16637559486879084300e-03; /* 0xBF61BF38, 0x0A96073F */ -const QA1: f64 = 1.06420880400844228286e-01; /* 0x3FBB3E66, 0x18EEE323 */ -const QA2: f64 = 5.40397917702171048937e-01; /* 0x3FE14AF0, 0x92EB6F33 */ -const QA3: f64 = 7.18286544141962662868e-02; /* 0x3FB2635C, 0xD99FE9A7 */ -const QA4: f64 = 1.26171219808761642112e-01; /* 0x3FC02660, 0xE763351F */ -const QA5: f64 = 1.36370839120290507362e-02; /* 0x3F8BEDC2, 0x6B51DD1C */ -const QA6: f64 = 1.19844998467991074170e-02; /* 0x3F888B54, 0x5735151D */ -/* - * Coefficients for approximation to erfc in [1.25,1/0.35] - */ -const RA0: f64 = -9.86494403484714822705e-03; /* 0xBF843412, 0x600D6435 */ -const RA1: f64 = -6.93858572707181764372e-01; /* 0xBFE63416, 0xE4BA7360 */ -const RA2: f64 = -1.05586262253232909814e+01; /* 0xC0251E04, 0x41B0E726 */ -const RA3: f64 = -6.23753324503260060396e+01; /* 0xC04F300A, 0xE4CBA38D */ -const RA4: f64 = -1.62396669462573470355e+02; /* 0xC0644CB1, 0x84282266 */ -const RA5: f64 = -1.84605092906711035994e+02; /* 0xC067135C, 0xEBCCABB2 */ -const RA6: f64 = -8.12874355063065934246e+01; /* 0xC0545265, 0x57E4D2F2 */ -const RA7: f64 = -9.81432934416914548592e+00; /* 0xC023A0EF, 0xC69AC25C */ -const SA1: f64 = 1.96512716674392571292e+01; /* 0x4033A6B9, 0xBD707687 */ -const SA2: f64 = 1.37657754143519042600e+02; /* 0x4061350C, 0x526AE721 */ -const SA3: f64 = 4.34565877475229228821e+02; /* 0x407B290D, 0xD58A1A71 */ -const SA4: f64 = 6.45387271733267880336e+02; /* 0x40842B19, 0x21EC2868 */ -const SA5: f64 = 4.29008140027567833386e+02; /* 0x407AD021, 0x57700314 */ -const SA6: f64 = 1.08635005541779435134e+02; /* 0x405B28A3, 0xEE48AE2C */ -const SA7: f64 = 6.57024977031928170135e+00; /* 0x401A47EF, 0x8E484A93 */ -const SA8: f64 = -6.04244152148580987438e-02; /* 0xBFAEEFF2, 0xEE749A62 */ -/* - * Coefficients for approximation to erfc in [1/.35,28] - */ -const RB0: f64 = -9.86494292470009928597e-03; /* 0xBF843412, 0x39E86F4A */ -const RB1: f64 = -7.99283237680523006574e-01; /* 0xBFE993BA, 0x70C285DE */ -const RB2: f64 = -1.77579549177547519889e+01; /* 0xC031C209, 0x555F995A */ -const RB3: f64 = -1.60636384855821916062e+02; /* 0xC064145D, 0x43C5ED98 */ -const RB4: f64 = -6.37566443368389627722e+02; /* 0xC083EC88, 0x1375F228 */ -const RB5: f64 = -1.02509513161107724954e+03; /* 0xC0900461, 0x6A2E5992 */ -const RB6: f64 = -4.83519191608651397019e+02; /* 0xC07E384E, 0x9BDC383F */ -const SB1: f64 = 3.03380607434824582924e+01; /* 0x403E568B, 0x261D5190 */ -const SB2: f64 = 3.25792512996573918826e+02; /* 0x40745CAE, 0x221B9F0A */ -const SB3: f64 = 1.53672958608443695994e+03; /* 0x409802EB, 0x189D5118 */ -const SB4: f64 = 3.19985821950859553908e+03; /* 0x40A8FFB7, 0x688C246A */ -const SB5: f64 = 2.55305040643316442583e+03; /* 0x40A3F219, 0xCEDF3BE6 */ -const SB6: f64 = 4.74528541206955367215e+02; /* 0x407DA874, 0xE79FE763 */ -const SB7: f64 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ - -fn erfc1(x: f64) -> f64 { - let s: f64; - let p: f64; - let q: f64; - - s = fabs(x) - 1.0; - p = PA0+s*(PA1+s*(PA2+s*(PA3+s*(PA4+s*(PA5+s*PA6))))); - q = 1.0+s*(QA1+s*(QA2+s*(QA3+s*(QA4+s*(QA5+s*QA6))))); - - 1.0 - ERX - p/q -} - -fn erfc2(ix: u32, mut x: f64) -> f64 { - let s: f64; - let r: f64; - let big_s: f64; - let z: f64; - - if ix < 0x3ff40000 { /* |x| < 1.25 */ - return erfc1(x); - } - - x = fabs(x); - s = 1.0/(x*x); - if ix < 0x4006db6d { /* |x| < 1/.35 ~ 2.85714 */ - r = RA0+s*(RA1+s*(RA2+s*(RA3+s*(RA4+s*( - RA5+s*(RA6+s*RA7)))))); - big_s = 1.0+s*(SA1+s*(SA2+s*(SA3+s*(SA4+s*( - SA5+s*(SA6+s*(SA7+s*SA8))))))); - } else { /* |x| > 1/.35 */ - r = RB0+s*(RB1+s*(RB2+s*(RB3+s*(RB4+s*( - RB5+s*RB6))))); - big_s = 1.0+s*(SB1+s*(SB2+s*(SB3+s*(SB4+s*( - SB5+s*(SB6+s*SB7)))))); - } - z = with_set_low_word(x, 0); - - exp(-z*z-0.5625)*exp((z-x)*(z+x)+r/big_s)/x -} - -pub fn erf(x: f64) -> f64 { - let r: f64; - let s: f64; - let z: f64; - let y: f64; - let mut ix: u32; - let sign: usize; - - ix = get_high_word(x); - sign = (ix>>31) as usize; - ix &= 0x7fffffff; - if ix >= 0x7ff00000 { - /* erf(nan)=nan, erf(+-inf)=+-1 */ - return 1.0-2.0*(sign as f64) + 1.0/x; - } - if ix < 0x3feb0000 { /* |x| < 0.84375 */ - if ix < 0x3e300000 { /* |x| < 2**-28 */ - /* avoid underflow */ - return 0.125*(8.0*x + EFX8*x); - } - z = x*x; - r = PP0+z*(PP1+z*(PP2+z*(PP3+z*PP4))); - s = 1.0+z*(QQ1+z*(QQ2+z*(QQ3+z*(QQ4+z*QQ5)))); - y = r/s; - return x + x*y; - } - if ix < 0x40180000 { /* 0.84375 <= |x| < 6 */ - y = 1.0 - erfc2(ix,x); - } else { - let x1p_1022 = f64::from_bits(0x0010000000000000); - y = 1.0 - x1p_1022; - } - - if sign != 0 { - -y - } else { - y - } -} - -pub fn erfc(x: f64) -> f64 { - let r: f64; - let s: f64; - let z: f64; - let y: f64; - let mut ix: u32; - let sign: usize; - - ix = get_high_word(x); - sign = (ix>>31) as usize; - ix &= 0x7fffffff; - if ix >= 0x7ff00000 { - /* erfc(nan)=nan, erfc(+-inf)=0,2 */ - return 2.0*(sign as f64) + 1.0/x; - } - if ix < 0x3feb0000 { /* |x| < 0.84375 */ - if ix < 0x3c700000 { /* |x| < 2**-56 */ - return 1.0 - x; - } - z = x*x; - r = PP0+z*(PP1+z*(PP2+z*(PP3+z*PP4))); - s = 1.0+z*(QQ1+z*(QQ2+z*(QQ3+z*(QQ4+z*QQ5)))); - y = r/s; - if sign != 0 || ix < 0x3fd00000 { /* x < 1/4 */ - return 1.0 - (x+x*y); - } - return 0.5 - (x - 0.5 + x*y); - } - if ix < 0x403c0000 { /* 0.84375 <= |x| < 28 */ - if sign != 0 { - return 2.0 - erfc2(ix,x); - } else { - return erfc2(ix,x); - } - } - - let x1p_1022 = f64::from_bits(0x0010000000000000); - if sign != 0 { - 2.0 - x1p_1022 - } else { - x1p_1022*x1p_1022 - } -} +use super::{exp, fabs, get_high_word, with_set_low_word}; +/* origin: FreeBSD /usr/src/lib/msun/src/s_erf.c */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ +/* double erf(double x) + * double erfc(double x) + * x + * 2 |\ + * erf(x) = --------- | exp(-t*t)dt + * sqrt(pi) \| + * 0 + * + * erfc(x) = 1-erf(x) + * Note that + * erf(-x) = -erf(x) + * erfc(-x) = 2 - erfc(x) + * + * Method: + * 1. For |x| in [0, 0.84375] + * erf(x) = x + x*R(x^2) + * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] + * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] + * where R = P/Q where P is an odd poly of degree 8 and + * Q is an odd poly of degree 10. + * -57.90 + * | R - (erf(x)-x)/x | <= 2 + * + * + * Remark. The formula is derived by noting + * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) + * and that + * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 + * is close to one. The interval is chosen because the fix + * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is + * near 0.6174), and by some experiment, 0.84375 is chosen to + * guarantee the error is less than one ulp for erf. + * + * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and + * c = 0.84506291151 rounded to single (24 bits) + * erf(x) = sign(x) * (c + P1(s)/Q1(s)) + * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 + * 1+(c+P1(s)/Q1(s)) if x < 0 + * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 + * Remark: here we use the taylor series expansion at x=1. + * erf(1+s) = erf(1) + s*Poly(s) + * = 0.845.. + P1(s)/Q1(s) + * That is, we use rational approximation to approximate + * erf(1+s) - (c = (single)0.84506291151) + * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] + * where + * P1(s) = degree 6 poly in s + * Q1(s) = degree 6 poly in s + * + * 3. For x in [1.25,1/0.35(~2.857143)], + * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) + * erf(x) = 1 - erfc(x) + * where + * R1(z) = degree 7 poly in z, (z=1/x^2) + * S1(z) = degree 8 poly in z + * + * 4. For x in [1/0.35,28] + * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 + * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6 x >= 28 + * erf(x) = sign(x) *(1 - tiny) (raise inexact) + * erfc(x) = tiny*tiny (raise underflow) if x > 0 + * = 2 - tiny if x<0 + * + * 7. Special case: + * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, + * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, + * erfc/erf(NaN) is NaN + */ + +const ERX: f64 = 8.45062911510467529297e-01; /* 0x3FEB0AC1, 0x60000000 */ +/* + * Coefficients for approximation to erf on [0,0.84375] + */ +const EFX8: f64 = 1.02703333676410069053e+00; /* 0x3FF06EBA, 0x8214DB69 */ +const PP0: f64 = 1.28379167095512558561e-01; /* 0x3FC06EBA, 0x8214DB68 */ +const PP1: f64 = -3.25042107247001499370e-01; /* 0xBFD4CD7D, 0x691CB913 */ +const PP2: f64 = -2.84817495755985104766e-02; /* 0xBF9D2A51, 0xDBD7194F */ +const PP3: f64 = -5.77027029648944159157e-03; /* 0xBF77A291, 0x236668E4 */ +const PP4: f64 = -2.37630166566501626084e-05; /* 0xBEF8EAD6, 0x120016AC */ +const QQ1: f64 = 3.97917223959155352819e-01; /* 0x3FD97779, 0xCDDADC09 */ +const QQ2: f64 = 6.50222499887672944485e-02; /* 0x3FB0A54C, 0x5536CEBA */ +const QQ3: f64 = 5.08130628187576562776e-03; /* 0x3F74D022, 0xC4D36B0F */ +const QQ4: f64 = 1.32494738004321644526e-04; /* 0x3F215DC9, 0x221C1A10 */ +const QQ5: f64 = -3.96022827877536812320e-06; /* 0xBED09C43, 0x42A26120 */ +/* + * Coefficients for approximation to erf in [0.84375,1.25] + */ +const PA0: f64 = -2.36211856075265944077e-03; /* 0xBF6359B8, 0xBEF77538 */ +const PA1: f64 = 4.14856118683748331666e-01; /* 0x3FDA8D00, 0xAD92B34D */ +const PA2: f64 = -3.72207876035701323847e-01; /* 0xBFD7D240, 0xFBB8C3F1 */ +const PA3: f64 = 3.18346619901161753674e-01; /* 0x3FD45FCA, 0x805120E4 */ +const PA4: f64 = -1.10894694282396677476e-01; /* 0xBFBC6398, 0x3D3E28EC */ +const PA5: f64 = 3.54783043256182359371e-02; /* 0x3FA22A36, 0x599795EB */ +const PA6: f64 = -2.16637559486879084300e-03; /* 0xBF61BF38, 0x0A96073F */ +const QA1: f64 = 1.06420880400844228286e-01; /* 0x3FBB3E66, 0x18EEE323 */ +const QA2: f64 = 5.40397917702171048937e-01; /* 0x3FE14AF0, 0x92EB6F33 */ +const QA3: f64 = 7.18286544141962662868e-02; /* 0x3FB2635C, 0xD99FE9A7 */ +const QA4: f64 = 1.26171219808761642112e-01; /* 0x3FC02660, 0xE763351F */ +const QA5: f64 = 1.36370839120290507362e-02; /* 0x3F8BEDC2, 0x6B51DD1C */ +const QA6: f64 = 1.19844998467991074170e-02; /* 0x3F888B54, 0x5735151D */ +/* + * Coefficients for approximation to erfc in [1.25,1/0.35] + */ +const RA0: f64 = -9.86494403484714822705e-03; /* 0xBF843412, 0x600D6435 */ +const RA1: f64 = -6.93858572707181764372e-01; /* 0xBFE63416, 0xE4BA7360 */ +const RA2: f64 = -1.05586262253232909814e+01; /* 0xC0251E04, 0x41B0E726 */ +const RA3: f64 = -6.23753324503260060396e+01; /* 0xC04F300A, 0xE4CBA38D */ +const RA4: f64 = -1.62396669462573470355e+02; /* 0xC0644CB1, 0x84282266 */ +const RA5: f64 = -1.84605092906711035994e+02; /* 0xC067135C, 0xEBCCABB2 */ +const RA6: f64 = -8.12874355063065934246e+01; /* 0xC0545265, 0x57E4D2F2 */ +const RA7: f64 = -9.81432934416914548592e+00; /* 0xC023A0EF, 0xC69AC25C */ +const SA1: f64 = 1.96512716674392571292e+01; /* 0x4033A6B9, 0xBD707687 */ +const SA2: f64 = 1.37657754143519042600e+02; /* 0x4061350C, 0x526AE721 */ +const SA3: f64 = 4.34565877475229228821e+02; /* 0x407B290D, 0xD58A1A71 */ +const SA4: f64 = 6.45387271733267880336e+02; /* 0x40842B19, 0x21EC2868 */ +const SA5: f64 = 4.29008140027567833386e+02; /* 0x407AD021, 0x57700314 */ +const SA6: f64 = 1.08635005541779435134e+02; /* 0x405B28A3, 0xEE48AE2C */ +const SA7: f64 = 6.57024977031928170135e+00; /* 0x401A47EF, 0x8E484A93 */ +const SA8: f64 = -6.04244152148580987438e-02; /* 0xBFAEEFF2, 0xEE749A62 */ +/* + * Coefficients for approximation to erfc in [1/.35,28] + */ +const RB0: f64 = -9.86494292470009928597e-03; /* 0xBF843412, 0x39E86F4A */ +const RB1: f64 = -7.99283237680523006574e-01; /* 0xBFE993BA, 0x70C285DE */ +const RB2: f64 = -1.77579549177547519889e+01; /* 0xC031C209, 0x555F995A */ +const RB3: f64 = -1.60636384855821916062e+02; /* 0xC064145D, 0x43C5ED98 */ +const RB4: f64 = -6.37566443368389627722e+02; /* 0xC083EC88, 0x1375F228 */ +const RB5: f64 = -1.02509513161107724954e+03; /* 0xC0900461, 0x6A2E5992 */ +const RB6: f64 = -4.83519191608651397019e+02; /* 0xC07E384E, 0x9BDC383F */ +const SB1: f64 = 3.03380607434824582924e+01; /* 0x403E568B, 0x261D5190 */ +const SB2: f64 = 3.25792512996573918826e+02; /* 0x40745CAE, 0x221B9F0A */ +const SB3: f64 = 1.53672958608443695994e+03; /* 0x409802EB, 0x189D5118 */ +const SB4: f64 = 3.19985821950859553908e+03; /* 0x40A8FFB7, 0x688C246A */ +const SB5: f64 = 2.55305040643316442583e+03; /* 0x40A3F219, 0xCEDF3BE6 */ +const SB6: f64 = 4.74528541206955367215e+02; /* 0x407DA874, 0xE79FE763 */ +const SB7: f64 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ + +fn erfc1(x: f64) -> f64 { + let s: f64; + let p: f64; + let q: f64; + + s = fabs(x) - 1.0; + p = PA0 + s * (PA1 + s * (PA2 + s * (PA3 + s * (PA4 + s * (PA5 + s * PA6))))); + q = 1.0 + s * (QA1 + s * (QA2 + s * (QA3 + s * (QA4 + s * (QA5 + s * QA6))))); + + 1.0 - ERX - p / q +} + +fn erfc2(ix: u32, mut x: f64) -> f64 { + let s: f64; + let r: f64; + let big_s: f64; + let z: f64; + + if ix < 0x3ff40000 { + /* |x| < 1.25 */ + return erfc1(x); + } + + x = fabs(x); + s = 1.0 / (x * x); + if ix < 0x4006db6d { + /* |x| < 1/.35 ~ 2.85714 */ + r = RA0 + s * (RA1 + s * (RA2 + s * (RA3 + s * (RA4 + s * (RA5 + s * (RA6 + s * RA7)))))); + big_s = 1.0 + + s * (SA1 + + s * (SA2 + s * (SA3 + s * (SA4 + s * (SA5 + s * (SA6 + s * (SA7 + s * SA8))))))); + } else { + /* |x| > 1/.35 */ + r = RB0 + s * (RB1 + s * (RB2 + s * (RB3 + s * (RB4 + s * (RB5 + s * RB6))))); + big_s = + 1.0 + s * (SB1 + s * (SB2 + s * (SB3 + s * (SB4 + s * (SB5 + s * (SB6 + s * SB7)))))); + } + z = with_set_low_word(x, 0); + + exp(-z * z - 0.5625) * exp((z - x) * (z + x) + r / big_s) / x +} + +pub fn erf(x: f64) -> f64 { + let r: f64; + let s: f64; + let z: f64; + let y: f64; + let mut ix: u32; + let sign: usize; + + ix = get_high_word(x); + sign = (ix >> 31) as usize; + ix &= 0x7fffffff; + if ix >= 0x7ff00000 { + /* erf(nan)=nan, erf(+-inf)=+-1 */ + return 1.0 - 2.0 * (sign as f64) + 1.0 / x; + } + if ix < 0x3feb0000 { + /* |x| < 0.84375 */ + if ix < 0x3e300000 { + /* |x| < 2**-28 */ + /* avoid underflow */ + return 0.125 * (8.0 * x + EFX8 * x); + } + z = x * x; + r = PP0 + z * (PP1 + z * (PP2 + z * (PP3 + z * PP4))); + s = 1.0 + z * (QQ1 + z * (QQ2 + z * (QQ3 + z * (QQ4 + z * QQ5)))); + y = r / s; + return x + x * y; + } + if ix < 0x40180000 { + /* 0.84375 <= |x| < 6 */ + y = 1.0 - erfc2(ix, x); + } else { + let x1p_1022 = f64::from_bits(0x0010000000000000); + y = 1.0 - x1p_1022; + } + + if sign != 0 { + -y + } else { + y + } +} + +pub fn erfc(x: f64) -> f64 { + let r: f64; + let s: f64; + let z: f64; + let y: f64; + let mut ix: u32; + let sign: usize; + + ix = get_high_word(x); + sign = (ix >> 31) as usize; + ix &= 0x7fffffff; + if ix >= 0x7ff00000 { + /* erfc(nan)=nan, erfc(+-inf)=0,2 */ + return 2.0 * (sign as f64) + 1.0 / x; + } + if ix < 0x3feb0000 { + /* |x| < 0.84375 */ + if ix < 0x3c700000 { + /* |x| < 2**-56 */ + return 1.0 - x; + } + z = x * x; + r = PP0 + z * (PP1 + z * (PP2 + z * (PP3 + z * PP4))); + s = 1.0 + z * (QQ1 + z * (QQ2 + z * (QQ3 + z * (QQ4 + z * QQ5)))); + y = r / s; + if sign != 0 || ix < 0x3fd00000 { + /* x < 1/4 */ + return 1.0 - (x + x * y); + } + return 0.5 - (x - 0.5 + x * y); + } + if ix < 0x403c0000 { + /* 0.84375 <= |x| < 28 */ + if sign != 0 { + return 2.0 - erfc2(ix, x); + } else { + return erfc2(ix, x); + } + } + + let x1p_1022 = f64::from_bits(0x0010000000000000); + if sign != 0 { + 2.0 - x1p_1022 + } else { + x1p_1022 * x1p_1022 + } +} diff --git a/src/math/erff.rs b/src/math/erff.rs index 0aaa897..ef67c33 100644 --- a/src/math/erff.rs +++ b/src/math/erff.rs @@ -1,210 +1,218 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/s_erff.c */ -/* - * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. - */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -use super::{expf, fabsf}; - -const ERX: f32 = 8.4506291151e-01; /* 0x3f58560b */ -/* - * Coefficients for approximation to erf on [0,0.84375] - */ -const EFX8: f32 = 1.0270333290e+00; /* 0x3f8375d4 */ -const PP0: f32 = 1.2837916613e-01; /* 0x3e0375d4 */ -const PP1: f32 = -3.2504209876e-01; /* 0xbea66beb */ -const PP2: f32 = -2.8481749818e-02; /* 0xbce9528f */ -const PP3: f32 = -5.7702702470e-03; /* 0xbbbd1489 */ -const PP4: f32 = -2.3763017452e-05; /* 0xb7c756b1 */ -const QQ1: f32 = 3.9791721106e-01; /* 0x3ecbbbce */ -const QQ2: f32 = 6.5022252500e-02; /* 0x3d852a63 */ -const QQ3: f32 = 5.0813062117e-03; /* 0x3ba68116 */ -const QQ4: f32 = 1.3249473704e-04; /* 0x390aee49 */ -const QQ5: f32 = -3.9602282413e-06; /* 0xb684e21a */ -/* - * Coefficients for approximation to erf in [0.84375,1.25] - */ -const PA0: f32 = -2.3621185683e-03; /* 0xbb1acdc6 */ -const PA1: f32 = 4.1485610604e-01; /* 0x3ed46805 */ -const PA2: f32 = -3.7220788002e-01; /* 0xbebe9208 */ -const PA3: f32 = 3.1834661961e-01; /* 0x3ea2fe54 */ -const PA4: f32 = -1.1089469492e-01; /* 0xbde31cc2 */ -const PA5: f32 = 3.5478305072e-02; /* 0x3d1151b3 */ -const PA6: f32 = -2.1663755178e-03; /* 0xbb0df9c0 */ -const QA1: f32 = 1.0642088205e-01; /* 0x3dd9f331 */ -const QA2: f32 = 5.4039794207e-01; /* 0x3f0a5785 */ -const QA3: f32 = 7.1828655899e-02; /* 0x3d931ae7 */ -const QA4: f32 = 1.2617121637e-01; /* 0x3e013307 */ -const QA5: f32 = 1.3637083583e-02; /* 0x3c5f6e13 */ -const QA6: f32 = 1.1984500103e-02; /* 0x3c445aa3 */ -/* - * Coefficients for approximation to erfc in [1.25,1/0.35] - */ -const RA0: f32 = -9.8649440333e-03; /* 0xbc21a093 */ -const RA1: f32 = -6.9385856390e-01; /* 0xbf31a0b7 */ -const RA2: f32 = -1.0558626175e+01; /* 0xc128f022 */ -const RA3: f32 = -6.2375331879e+01; /* 0xc2798057 */ -const RA4: f32 = -1.6239666748e+02; /* 0xc322658c */ -const RA5: f32 = -1.8460508728e+02; /* 0xc3389ae7 */ -const RA6: f32 = -8.1287437439e+01; /* 0xc2a2932b */ -const RA7: f32 = -9.8143291473e+00; /* 0xc11d077e */ -const SA1: f32 = 1.9651271820e+01; /* 0x419d35ce */ -const SA2: f32 = 1.3765776062e+02; /* 0x4309a863 */ -const SA3: f32 = 4.3456588745e+02; /* 0x43d9486f */ -const SA4: f32 = 6.4538726807e+02; /* 0x442158c9 */ -const SA5: f32 = 4.2900814819e+02; /* 0x43d6810b */ -const SA6: f32 = 1.0863500214e+02; /* 0x42d9451f */ -const SA7: f32 = 6.5702495575e+00; /* 0x40d23f7c */ -const SA8: f32 = -6.0424413532e-02; /* 0xbd777f97 */ -/* - * Coefficients for approximation to erfc in [1/.35,28] - */ -const RB0: f32 = -9.8649431020e-03; /* 0xbc21a092 */ -const RB1: f32 = -7.9928326607e-01; /* 0xbf4c9dd4 */ -const RB2: f32 = -1.7757955551e+01; /* 0xc18e104b */ -const RB3: f32 = -1.6063638306e+02; /* 0xc320a2ea */ -const RB4: f32 = -6.3756646729e+02; /* 0xc41f6441 */ -const RB5: f32 = -1.0250950928e+03; /* 0xc480230b */ -const RB6: f32 = -4.8351919556e+02; /* 0xc3f1c275 */ -const SB1: f32 = 3.0338060379e+01; /* 0x41f2b459 */ -const SB2: f32 = 3.2579251099e+02; /* 0x43a2e571 */ -const SB3: f32 = 1.5367296143e+03; /* 0x44c01759 */ -const SB4: f32 = 3.1998581543e+03; /* 0x4547fdbb */ -const SB5: f32 = 2.5530502930e+03; /* 0x451f90ce */ -const SB6: f32 = 4.7452853394e+02; /* 0x43ed43a7 */ -const SB7: f32 = -2.2440952301e+01; /* 0xc1b38712 */ - -fn erfc1(x: f32) -> f32 { - let s: f32; - let p: f32; - let q: f32; - - s = fabsf(x) - 1.0; - p = PA0+s*(PA1+s*(PA2+s*(PA3+s*(PA4+s*(PA5+s*PA6))))); - q = 1.0+s*(QA1+s*(QA2+s*(QA3+s*(QA4+s*(QA5+s*QA6))))); - return 1.0 - ERX - p/q; -} - -fn erfc2(mut ix: u32, mut x: f32) -> f32 { - let s: f32; - let r: f32; - let big_s: f32; - let z: f32; - - if ix < 0x3fa00000 { /* |x| < 1.25 */ - return erfc1(x); - } - - x = fabsf(x); - s = 1.0/(x*x); - if ix < 0x4036db6d { /* |x| < 1/0.35 */ - r = RA0+s*(RA1+s*(RA2+s*(RA3+s*(RA4+s*( - RA5+s*(RA6+s*RA7)))))); - big_s = 1.0+s*(SA1+s*(SA2+s*(SA3+s*(SA4+s*( - SA5+s*(SA6+s*(SA7+s*SA8))))))); - } else { /* |x| >= 1/0.35 */ - r = RB0+s*(RB1+s*(RB2+s*(RB3+s*(RB4+s*( - RB5+s*RB6))))); - big_s = 1.0+s*(SB1+s*(SB2+s*(SB3+s*(SB4+s*( - SB5+s*(SB6+s*SB7)))))); - } - ix = x.to_bits(); - z = f32::from_bits(ix&0xffffe000); - - expf(-z*z - 0.5625) * expf((z-x)*(z+x) + r/big_s)/x -} - -pub fn erff(x: f32) -> f32 -{ - let r: f32; - let s: f32; - let z: f32; - let y: f32; - let mut ix: u32; - let sign: usize; - - ix = x.to_bits(); - sign = (ix>>31) as usize; - ix &= 0x7fffffff; - if ix >= 0x7f800000 { - /* erf(nan)=nan, erf(+-inf)=+-1 */ - return 1.0-2.0*(sign as f32) + 1.0/x; - } - if ix < 0x3f580000 { /* |x| < 0.84375 */ - if ix < 0x31800000 { /* |x| < 2**-28 */ - /*avoid underflow */ - return 0.125*(8.0*x + EFX8*x); - } - z = x*x; - r = PP0+z*(PP1+z*(PP2+z*(PP3+z*PP4))); - s = 1.0+z*(QQ1+z*(QQ2+z*(QQ3+z*(QQ4+z*QQ5)))); - y = r/s; - return x + x*y; - } - if ix < 0x40c00000 { /* |x| < 6 */ - y = 1.0 - erfc2(ix,x); - } else { - let x1p_120 = f32::from_bits(0x03800000); - y = 1.0 - x1p_120; - } - - if sign != 0 { - -y - } else { - y - } -} - -pub fn erfcf(x: f32) -> f32 { - let r: f32; - let s: f32; - let z: f32; - let y: f32; - let mut ix: u32; - let sign: usize; - - ix = x.to_bits(); - sign = (ix>>31) as usize; - ix &= 0x7fffffff; - if ix >= 0x7f800000 { - /* erfc(nan)=nan, erfc(+-inf)=0,2 */ - return 2.0*(sign as f32) + 1.0/x; - } - - if ix < 0x3f580000 { /* |x| < 0.84375 */ - if ix < 0x23800000 { /* |x| < 2**-56 */ - return 1.0 - x; - } - z = x*x; - r = PP0+z*(PP1+z*(PP2+z*(PP3+z*PP4))); - s = 1.0+z*(QQ1+z*(QQ2+z*(QQ3+z*(QQ4+z*QQ5)))); - y = r/s; - if sign != 0 || ix < 0x3e800000 { /* x < 1/4 */ - return 1.0 - (x+x*y); - } - return 0.5 - (x - 0.5 + x*y); - } - if ix < 0x41e00000 { /* |x| < 28 */ - if sign != 0 { - return 2.0 - erfc2(ix, x); - } else { - return erfc2(ix, x); - } - } - - let x1p_120 = f32::from_bits(0x03800000); - if sign != 0 { - 2.0 - x1p_120 - } else { - x1p_120*x1p_120 - } -} +/* origin: FreeBSD /usr/src/lib/msun/src/s_erff.c */ +/* + * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. + */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +use super::{expf, fabsf}; + +const ERX: f32 = 8.4506291151e-01; /* 0x3f58560b */ +/* + * Coefficients for approximation to erf on [0,0.84375] + */ +const EFX8: f32 = 1.0270333290e+00; /* 0x3f8375d4 */ +const PP0: f32 = 1.2837916613e-01; /* 0x3e0375d4 */ +const PP1: f32 = -3.2504209876e-01; /* 0xbea66beb */ +const PP2: f32 = -2.8481749818e-02; /* 0xbce9528f */ +const PP3: f32 = -5.7702702470e-03; /* 0xbbbd1489 */ +const PP4: f32 = -2.3763017452e-05; /* 0xb7c756b1 */ +const QQ1: f32 = 3.9791721106e-01; /* 0x3ecbbbce */ +const QQ2: f32 = 6.5022252500e-02; /* 0x3d852a63 */ +const QQ3: f32 = 5.0813062117e-03; /* 0x3ba68116 */ +const QQ4: f32 = 1.3249473704e-04; /* 0x390aee49 */ +const QQ5: f32 = -3.9602282413e-06; /* 0xb684e21a */ +/* + * Coefficients for approximation to erf in [0.84375,1.25] + */ +const PA0: f32 = -2.3621185683e-03; /* 0xbb1acdc6 */ +const PA1: f32 = 4.1485610604e-01; /* 0x3ed46805 */ +const PA2: f32 = -3.7220788002e-01; /* 0xbebe9208 */ +const PA3: f32 = 3.1834661961e-01; /* 0x3ea2fe54 */ +const PA4: f32 = -1.1089469492e-01; /* 0xbde31cc2 */ +const PA5: f32 = 3.5478305072e-02; /* 0x3d1151b3 */ +const PA6: f32 = -2.1663755178e-03; /* 0xbb0df9c0 */ +const QA1: f32 = 1.0642088205e-01; /* 0x3dd9f331 */ +const QA2: f32 = 5.4039794207e-01; /* 0x3f0a5785 */ +const QA3: f32 = 7.1828655899e-02; /* 0x3d931ae7 */ +const QA4: f32 = 1.2617121637e-01; /* 0x3e013307 */ +const QA5: f32 = 1.3637083583e-02; /* 0x3c5f6e13 */ +const QA6: f32 = 1.1984500103e-02; /* 0x3c445aa3 */ +/* + * Coefficients for approximation to erfc in [1.25,1/0.35] + */ +const RA0: f32 = -9.8649440333e-03; /* 0xbc21a093 */ +const RA1: f32 = -6.9385856390e-01; /* 0xbf31a0b7 */ +const RA2: f32 = -1.0558626175e+01; /* 0xc128f022 */ +const RA3: f32 = -6.2375331879e+01; /* 0xc2798057 */ +const RA4: f32 = -1.6239666748e+02; /* 0xc322658c */ +const RA5: f32 = -1.8460508728e+02; /* 0xc3389ae7 */ +const RA6: f32 = -8.1287437439e+01; /* 0xc2a2932b */ +const RA7: f32 = -9.8143291473e+00; /* 0xc11d077e */ +const SA1: f32 = 1.9651271820e+01; /* 0x419d35ce */ +const SA2: f32 = 1.3765776062e+02; /* 0x4309a863 */ +const SA3: f32 = 4.3456588745e+02; /* 0x43d9486f */ +const SA4: f32 = 6.4538726807e+02; /* 0x442158c9 */ +const SA5: f32 = 4.2900814819e+02; /* 0x43d6810b */ +const SA6: f32 = 1.0863500214e+02; /* 0x42d9451f */ +const SA7: f32 = 6.5702495575e+00; /* 0x40d23f7c */ +const SA8: f32 = -6.0424413532e-02; /* 0xbd777f97 */ +/* + * Coefficients for approximation to erfc in [1/.35,28] + */ +const RB0: f32 = -9.8649431020e-03; /* 0xbc21a092 */ +const RB1: f32 = -7.9928326607e-01; /* 0xbf4c9dd4 */ +const RB2: f32 = -1.7757955551e+01; /* 0xc18e104b */ +const RB3: f32 = -1.6063638306e+02; /* 0xc320a2ea */ +const RB4: f32 = -6.3756646729e+02; /* 0xc41f6441 */ +const RB5: f32 = -1.0250950928e+03; /* 0xc480230b */ +const RB6: f32 = -4.8351919556e+02; /* 0xc3f1c275 */ +const SB1: f32 = 3.0338060379e+01; /* 0x41f2b459 */ +const SB2: f32 = 3.2579251099e+02; /* 0x43a2e571 */ +const SB3: f32 = 1.5367296143e+03; /* 0x44c01759 */ +const SB4: f32 = 3.1998581543e+03; /* 0x4547fdbb */ +const SB5: f32 = 2.5530502930e+03; /* 0x451f90ce */ +const SB6: f32 = 4.7452853394e+02; /* 0x43ed43a7 */ +const SB7: f32 = -2.2440952301e+01; /* 0xc1b38712 */ + +fn erfc1(x: f32) -> f32 { + let s: f32; + let p: f32; + let q: f32; + + s = fabsf(x) - 1.0; + p = PA0 + s * (PA1 + s * (PA2 + s * (PA3 + s * (PA4 + s * (PA5 + s * PA6))))); + q = 1.0 + s * (QA1 + s * (QA2 + s * (QA3 + s * (QA4 + s * (QA5 + s * QA6))))); + return 1.0 - ERX - p / q; +} + +fn erfc2(mut ix: u32, mut x: f32) -> f32 { + let s: f32; + let r: f32; + let big_s: f32; + let z: f32; + + if ix < 0x3fa00000 { + /* |x| < 1.25 */ + return erfc1(x); + } + + x = fabsf(x); + s = 1.0 / (x * x); + if ix < 0x4036db6d { + /* |x| < 1/0.35 */ + r = RA0 + s * (RA1 + s * (RA2 + s * (RA3 + s * (RA4 + s * (RA5 + s * (RA6 + s * RA7)))))); + big_s = 1.0 + + s * (SA1 + + s * (SA2 + s * (SA3 + s * (SA4 + s * (SA5 + s * (SA6 + s * (SA7 + s * SA8))))))); + } else { + /* |x| >= 1/0.35 */ + r = RB0 + s * (RB1 + s * (RB2 + s * (RB3 + s * (RB4 + s * (RB5 + s * RB6))))); + big_s = + 1.0 + s * (SB1 + s * (SB2 + s * (SB3 + s * (SB4 + s * (SB5 + s * (SB6 + s * SB7)))))); + } + ix = x.to_bits(); + z = f32::from_bits(ix & 0xffffe000); + + expf(-z * z - 0.5625) * expf((z - x) * (z + x) + r / big_s) / x +} + +pub fn erff(x: f32) -> f32 { + let r: f32; + let s: f32; + let z: f32; + let y: f32; + let mut ix: u32; + let sign: usize; + + ix = x.to_bits(); + sign = (ix >> 31) as usize; + ix &= 0x7fffffff; + if ix >= 0x7f800000 { + /* erf(nan)=nan, erf(+-inf)=+-1 */ + return 1.0 - 2.0 * (sign as f32) + 1.0 / x; + } + if ix < 0x3f580000 { + /* |x| < 0.84375 */ + if ix < 0x31800000 { + /* |x| < 2**-28 */ + /*avoid underflow */ + return 0.125 * (8.0 * x + EFX8 * x); + } + z = x * x; + r = PP0 + z * (PP1 + z * (PP2 + z * (PP3 + z * PP4))); + s = 1.0 + z * (QQ1 + z * (QQ2 + z * (QQ3 + z * (QQ4 + z * QQ5)))); + y = r / s; + return x + x * y; + } + if ix < 0x40c00000 { + /* |x| < 6 */ + y = 1.0 - erfc2(ix, x); + } else { + let x1p_120 = f32::from_bits(0x03800000); + y = 1.0 - x1p_120; + } + + if sign != 0 { + -y + } else { + y + } +} + +pub fn erfcf(x: f32) -> f32 { + let r: f32; + let s: f32; + let z: f32; + let y: f32; + let mut ix: u32; + let sign: usize; + + ix = x.to_bits(); + sign = (ix >> 31) as usize; + ix &= 0x7fffffff; + if ix >= 0x7f800000 { + /* erfc(nan)=nan, erfc(+-inf)=0,2 */ + return 2.0 * (sign as f32) + 1.0 / x; + } + + if ix < 0x3f580000 { + /* |x| < 0.84375 */ + if ix < 0x23800000 { + /* |x| < 2**-56 */ + return 1.0 - x; + } + z = x * x; + r = PP0 + z * (PP1 + z * (PP2 + z * (PP3 + z * PP4))); + s = 1.0 + z * (QQ1 + z * (QQ2 + z * (QQ3 + z * (QQ4 + z * QQ5)))); + y = r / s; + if sign != 0 || ix < 0x3e800000 { + /* x < 1/4 */ + return 1.0 - (x + x * y); + } + return 0.5 - (x - 0.5 + x * y); + } + if ix < 0x41e00000 { + /* |x| < 28 */ + if sign != 0 { + return 2.0 - erfc2(ix, x); + } else { + return erfc2(ix, x); + } + } + + let x1p_120 = f32::from_bits(0x03800000); + if sign != 0 { + 2.0 - x1p_120 + } else { + x1p_120 * x1p_120 + } +} diff --git a/src/math/exp10.rs b/src/math/exp10.rs index d12fa0b..9537f76 100644 --- a/src/math/exp10.rs +++ b/src/math/exp10.rs @@ -1,24 +1,21 @@ -use super::{exp2, modf, pow}; - -const LN10: f64 = 3.32192809488736234787031942948939; -const P10: &[f64] = &[ - 1e-15, 1e-14, 1e-13, 1e-12, 1e-11, 1e-10, - 1e-9, 1e-8, 1e-7, 1e-6, 1e-5, 1e-4, 1e-3, 1e-2, 1e-1, - 1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9, - 1e10, 1e11, 1e12, 1e13, 1e14, 1e15 -]; - -pub fn exp10(x: f64) -> f64 -{ - let (mut y, n) = modf(x); - let u: u64 = n.to_bits(); - /* fabs(n) < 16 without raising invalid on nan */ - if (u>>52 & 0x7ff) < 0x3ff+4 { - if y == 0.0 { - return P10[((n as isize) + 15) as usize]; - } - y = exp2(LN10 * y); - return y * P10[((n as isize) + 15) as usize]; - } - return pow(10.0, x); -} +use super::{exp2, modf, pow}; + +const LN10: f64 = 3.32192809488736234787031942948939; +const P10: &[f64] = &[ + 1e-15, 1e-14, 1e-13, 1e-12, 1e-11, 1e-10, 1e-9, 1e-8, 1e-7, 1e-6, 1e-5, 1e-4, 1e-3, 1e-2, 1e-1, + 1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9, 1e10, 1e11, 1e12, 1e13, 1e14, 1e15, +]; + +pub fn exp10(x: f64) -> f64 { + let (mut y, n) = modf(x); + let u: u64 = n.to_bits(); + /* fabs(n) < 16 without raising invalid on nan */ + if (u >> 52 & 0x7ff) < 0x3ff + 4 { + if y == 0.0 { + return P10[((n as isize) + 15) as usize]; + } + y = exp2(LN10 * y); + return y * P10[((n as isize) + 15) as usize]; + } + return pow(10.0, x); +} diff --git a/src/math/exp10f.rs b/src/math/exp10f.rs index 8fb88a5..d45fff3 100644 --- a/src/math/exp10f.rs +++ b/src/math/exp10f.rs @@ -1,22 +1,21 @@ -use super::{exp2, exp2f, modff}; - -const LN10_F32: f32 = 3.32192809488736234787031942948939; -const LN10_F64: f64 = 3.32192809488736234787031942948939; -const P10: &[f32] = &[ - 1e-7, 1e-6, 1e-5, 1e-4, 1e-3, 1e-2, 1e-1, - 1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7 -]; - -pub fn exp10f(x: f32) -> f32 { - let (mut y, n) = modff(x); - let u = n.to_bits(); - /* fabsf(n) < 8 without raising invalid on nan */ - if (u>>23 & 0xff) < 0x7f+3 { - if y == 0.0 { - return P10[((n as isize) + 7) as usize] - } - y = exp2f(LN10_F32 * y); - return y * P10[((n as isize) + 7) as usize]; - } - return exp2(LN10_F64 * (x as f64)) as f32; -} +use super::{exp2, exp2f, modff}; + +const LN10_F32: f32 = 3.32192809488736234787031942948939; +const LN10_F64: f64 = 3.32192809488736234787031942948939; +const P10: &[f32] = &[ + 1e-7, 1e-6, 1e-5, 1e-4, 1e-3, 1e-2, 1e-1, 1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, +]; + +pub fn exp10f(x: f32) -> f32 { + let (mut y, n) = modff(x); + let u = n.to_bits(); + /* fabsf(n) < 8 without raising invalid on nan */ + if (u >> 23 & 0xff) < 0x7f + 3 { + if y == 0.0 { + return P10[((n as isize) + 7) as usize]; + } + y = exp2f(LN10_F32 * y); + return y * P10[((n as isize) + 7) as usize]; + } + return exp2(LN10_F64 * (x as f64)) as f32; +} diff --git a/src/math/fma.rs b/src/math/fma.rs index 21c854c..6b06248 100644 --- a/src/math/fma.rs +++ b/src/math/fma.rs @@ -43,7 +43,7 @@ fn mul(x: u64, y: u64) -> (u64, u64) { t1 = xlo * ylo; t2 = xlo * yhi + xhi * ylo; t3 = xhi * yhi; - let lo = t1 + (t2 << 32); + let lo = t1.wrapping_add(t2 << 32); let hi = t3 + (t2 >> 32) + (t1 > lo) as u64; (hi, lo) } @@ -116,7 +116,7 @@ pub fn fma(x: f64, y: f64, z: f64) -> f64 { let mut nonzero: i32 = 1; if samesign { /* r += z */ - rlo += zlo; + rlo = rlo.wrapping_add(zlo); rhi += zhi + (rlo < zlo) as u64; } else { /* r -= z */ diff --git a/src/math/fmod.rs b/src/math/fmod.rs index ecc9b39..2cdd8a9 100644 --- a/src/math/fmod.rs +++ b/src/math/fmod.rs @@ -46,7 +46,7 @@ pub fn fmod(x: f64, y: f64) -> f64 { /* x mod y */ while ex > ey { - i = uxi - uyi; + i = uxi.wrapping_sub(uyi); if i >> 63 == 0 { if i == 0 { return 0.0 * x; @@ -56,7 +56,7 @@ pub fn fmod(x: f64, y: f64) -> f64 { uxi <<= 1; ex -= 1; } - i = uxi - uyi; + i = uxi.wrapping_sub(uyi); if i >> 63 == 0 { if i == 0 { return 0.0 * x; diff --git a/src/math/fmodf.rs b/src/math/fmodf.rs index 98f51f4..3e6779a 100644 --- a/src/math/fmodf.rs +++ b/src/math/fmodf.rs @@ -52,7 +52,7 @@ pub fn fmodf(x: f32, y: f32) -> f32 { /* x mod y */ while ex > ey { - i = uxi - uyi; + i = uxi.wrapping_sub(uyi); if i >> 31 == 0 { if i == 0 { return 0.0 * x; @@ -64,7 +64,7 @@ pub fn fmodf(x: f32, y: f32) -> f32 { ex -= 1; } - i = uxi - uyi; + i = uxi.wrapping_sub(uyi); if i >> 31 == 0 { if i == 0 { return 0.0 * x; diff --git a/src/math/frexp.rs b/src/math/frexp.rs index 45733a3..badad78 100644 --- a/src/math/frexp.rs +++ b/src/math/frexp.rs @@ -1,20 +1,20 @@ -pub fn frexp(x: f64) -> (f64, isize) { - let mut y = x.to_bits(); - let ee = ((y>>52) & 0x7ff) as isize; - - if ee == 0 { - if x != 0.0 { - let x1p64 = f64::from_bits(0x43f0000000000000); - let (x, e) = frexp(x*x1p64); - return (x, e - 64); - } - return (x, 0); - } else if ee == 0x7ff { - return (x, 0); - } - - let e = ee - 0x3fe; - y &= 0x800fffffffffffff; - y |= 0x3fe0000000000000; - return (f64::from_bits(y), e); -} +pub fn frexp(x: f64) -> (f64, i32) { + let mut y = x.to_bits(); + let ee = ((y >> 52) & 0x7ff) as i32; + + if ee == 0 { + if x != 0.0 { + let x1p64 = f64::from_bits(0x43f0000000000000); + let (x, e) = frexp(x * x1p64); + return (x, e - 64); + } + return (x, 0); + } else if ee == 0x7ff { + return (x, 0); + } + + let e = ee - 0x3fe; + y &= 0x800fffffffffffff; + y |= 0x3fe0000000000000; + return (f64::from_bits(y), e); +} diff --git a/src/math/frexpf.rs b/src/math/frexpf.rs index 1c9dae0..2919c0a 100644 --- a/src/math/frexpf.rs +++ b/src/math/frexpf.rs @@ -1,21 +1,21 @@ -pub fn frexpf(x: f32) -> (f32, isize) { - let mut y = x.to_bits(); - let ee: isize = ((y>>23) & 0xff) as isize; - - if ee == 0 { - if x != 0.0 { - let x1p64 = f32::from_bits(0x5f800000); - let (x, e) = frexpf(x*x1p64); - return (x, e - 64); - } else { - return (x, 0); - } - } else if ee == 0xff { - return (x, 0); - } - - let e = ee - 0x7e; - y &= 0x807fffff; - y |= 0x3f000000; - return (f32::from_bits(y), e); -} +pub fn frexpf(x: f32) -> (f32, i32) { + let mut y = x.to_bits(); + let ee: i32 = ((y >> 23) & 0xff) as i32; + + if ee == 0 { + if x != 0.0 { + let x1p64 = f32::from_bits(0x5f800000); + let (x, e) = frexpf(x * x1p64); + return (x, e - 64); + } else { + return (x, 0); + } + } else if ee == 0xff { + return (x, 0); + } + + let e = ee - 0x7e; + y &= 0x807fffff; + y |= 0x3f000000; + (f32::from_bits(y), e) +} diff --git a/src/math/ilogb.rs b/src/math/ilogb.rs index 78fe030..8a1289c 100644 --- a/src/math/ilogb.rs +++ b/src/math/ilogb.rs @@ -1,31 +1,31 @@ -const FP_ILOGBNAN: isize = -1 - (((!0) >> 1)); -const FP_ILOGB0: isize = FP_ILOGBNAN; - -pub fn ilogb(x: f64) -> isize { - let mut i: u64 = x.to_bits(); - let e = ((i>>52) & 0x7ff) as isize; - - if e == 0 { - i <<= 12; - if i == 0 { - force_eval!(0.0/0.0); - return FP_ILOGB0; - } - /* subnormal x */ - let mut e = -0x3ff; - while (i>>63) == 0 { - e -= 1; - i <<= 1; - } - return e; - } - if e == 0x7ff { - force_eval!(0.0/0.0); - if (i<<12) != 0 { - return FP_ILOGBNAN; - } else { - return isize::max_value(); - } - } - return e - 0x3ff; -} +const FP_ILOGBNAN: i32 = -1 - ((!0) >> 1); +const FP_ILOGB0: i32 = FP_ILOGBNAN; + +pub fn ilogb(x: f64) -> i32 { + let mut i: u64 = x.to_bits(); + let e = ((i >> 52) & 0x7ff) as i32; + + if e == 0 { + i <<= 12; + if i == 0 { + force_eval!(0.0 / 0.0); + return FP_ILOGB0; + } + /* subnormal x */ + let mut e = -0x3ff; + while (i >> 63) == 0 { + e -= 1; + i <<= 1; + } + return e; + } + if e == 0x7ff { + force_eval!(0.0 / 0.0); + if (i << 12) != 0 { + return FP_ILOGBNAN; + } else { + return i32::max_value(); + } + } + return e - 0x3ff; +} diff --git a/src/math/ilogbf.rs b/src/math/ilogbf.rs index 9ca1c36..1bf4670 100644 --- a/src/math/ilogbf.rs +++ b/src/math/ilogbf.rs @@ -1,31 +1,31 @@ -const FP_ILOGBNAN: isize = -1 - (((!0) >> 1)); -const FP_ILOGB0: isize = FP_ILOGBNAN; - -pub fn ilogbf(x: f32) -> isize { - let mut i = x.to_bits(); - let e = ((i>>23) & 0xff) as isize; - - if e == 0 { - i <<= 9; - if i == 0 { - force_eval!(0.0/0.0); - return FP_ILOGB0; - } - /* subnormal x */ - let mut e = -0x7f; - while (i>>31) == 0 { - e -= 1; - i <<= 1; - } - return e; - } - if e == 0xff { - force_eval!(0.0/0.0); - if (i<<9) != 0 { - return FP_ILOGBNAN; - } else { - return isize::max_value(); - } - } - return e - 0x7f; -} +const FP_ILOGBNAN: i32 = -1 - ((!0) >> 1); +const FP_ILOGB0: i32 = FP_ILOGBNAN; + +pub fn ilogbf(x: f32) -> i32 { + let mut i = x.to_bits(); + let e = ((i >> 23) & 0xff) as i32; + + if e == 0 { + i <<= 9; + if i == 0 { + force_eval!(0.0 / 0.0); + return FP_ILOGB0; + } + /* subnormal x */ + let mut e = -0x7f; + while (i >> 31) == 0 { + e -= 1; + i <<= 1; + } + return e; + } + if e == 0xff { + force_eval!(0.0 / 0.0); + if (i << 9) != 0 { + return FP_ILOGBNAN; + } else { + return i32::max_value(); + } + } + return e - 0x7f; +} diff --git a/src/math/j0.rs b/src/math/j0.rs index 02625b0..c4258cc 100644 --- a/src/math/j0.rs +++ b/src/math/j0.rs @@ -1,392 +1,422 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/e_j0.c */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunSoft, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ -/* j0(x), y0(x) - * Bessel function of the first and second kinds of order zero. - * Method -- j0(x): - * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ... - * 2. Reduce x to |x| since j0(x)=j0(-x), and - * for x in (0,2) - * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x; - * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 ) - * for x in (2,inf) - * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) - * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) - * as follow: - * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) - * = 1/sqrt(2) * (cos(x) + sin(x)) - * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4) - * = 1/sqrt(2) * (sin(x) - cos(x)) - * (To avoid cancellation, use - * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) - * to compute the worse one.) - * - * 3 Special cases - * j0(nan)= nan - * j0(0) = 1 - * j0(inf) = 0 - * - * Method -- y0(x): - * 1. For x<2. - * Since - * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...) - * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function. - * We use the following function to approximate y0, - * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2 - * where - * U(z) = u00 + u01*z + ... + u06*z^6 - * V(z) = 1 + v01*z + ... + v04*z^4 - * with absolute approximation error bounded by 2**-72. - * Note: For tiny x, U/V = u0 and j0(x)~1, hence - * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27) - * 2. For x>=2. - * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0)) - * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) - * by the method mentioned above. - * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0. - */ - -use super::{cos, get_low_word, get_high_word, fabs, log, sin, sqrt}; -const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */ -const TPI: f64 = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */ - -/* common method when |x|>=2 */ -fn common(ix: u32, x: f64, y0: bool) -> f64 { - let s: f64; - let mut c: f64; - let mut ss: f64; - let mut cc: f64; - let z: f64; - - /* - * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x-pi/4)-q0(x)*sin(x-pi/4)) - * y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x-pi/4)+q0(x)*cos(x-pi/4)) - * - * sin(x-pi/4) = (sin(x) - cos(x))/sqrt(2) - * cos(x-pi/4) = (sin(x) + cos(x))/sqrt(2) - * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) - */ - s = sin(x); - c = cos(x); - if y0 { - c = -c; - } - cc = s+c; - /* avoid overflow in 2*x, big ulp error when x>=0x1p1023 */ - if ix < 0x7fe00000 { - ss = s-c; - z = -cos(2.0*x); - if s*c < 0.0 { - cc = z/ss; - } else { - ss = z/cc; - } - if ix < 0x48000000 { - if y0 { - ss = -ss; - } - cc = pzero(x)*cc-qzero(x)*ss; - } - } - return INVSQRTPI*cc/sqrt(x); -} - -/* R0/S0 on [0, 2.00] */ -const R02: f64 = 1.56249999999999947958e-02; /* 0x3F8FFFFF, 0xFFFFFFFD */ -const R03: f64 = -1.89979294238854721751e-04; /* 0xBF28E6A5, 0xB61AC6E9 */ -const R04: f64 = 1.82954049532700665670e-06; /* 0x3EBEB1D1, 0x0C503919 */ -const R05: f64 = -4.61832688532103189199e-09; /* 0xBE33D5E7, 0x73D63FCE */ -const S01: f64 = 1.56191029464890010492e-02; /* 0x3F8FFCE8, 0x82C8C2A4 */ -const S02: f64 = 1.16926784663337450260e-04; /* 0x3F1EA6D2, 0xDD57DBF4 */ -const S03: f64 = 5.13546550207318111446e-07; /* 0x3EA13B54, 0xCE84D5A9 */ -const S04: f64 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */ - -pub fn j0(mut x: f64) -> f64 -{ - let z: f64; - let r: f64; - let s: f64; - let mut ix: u32; - - ix = get_high_word(x); - ix &= 0x7fffffff; - - /* j0(+-inf)=0, j0(nan)=nan */ - if ix >= 0x7ff00000 { - return 1.0/(x*x); - } - x = fabs(x); - - if ix >= 0x40000000 { /* |x| >= 2 */ - /* large ulp error near zeros: 2.4, 5.52, 8.6537,.. */ - return common(ix,x,false); - } - - /* 1 - x*x/4 + x*x*R(x^2)/S(x^2) */ - if ix >= 0x3f200000 { /* |x| >= 2**-13 */ - /* up to 4ulp error close to 2 */ - z = x*x; - r = z*(R02+z*(R03+z*(R04+z*R05))); - s = 1.0+z*(S01+z*(S02+z*(S03+z*S04))); - return (1.0+x/2.0)*(1.0-x/2.0) + z*(r/s); - } - - /* 1 - x*x/4 */ - /* prevent underflow */ - /* inexact should be raised when x!=0, this is not done correctly */ - if ix >= 0x38000000 { /* |x| >= 2**-127 */ - x = 0.25*x*x; - } - return 1.0 - x; -} - -const U00: f64 = -7.38042951086872317523e-02; /* 0xBFB2E4D6, 0x99CBD01F */ -const U01: f64 = 1.76666452509181115538e-01; /* 0x3FC69D01, 0x9DE9E3FC */ -const U02: f64 = -1.38185671945596898896e-02; /* 0xBF8C4CE8, 0xB16CFA97 */ -const U03: f64 = 3.47453432093683650238e-04; /* 0x3F36C54D, 0x20B29B6B */ -const U04: f64 = -3.81407053724364161125e-06; /* 0xBECFFEA7, 0x73D25CAD */ -const U05: f64 = 1.95590137035022920206e-08; /* 0x3E550057, 0x3B4EABD4 */ -const U06: f64 = -3.98205194132103398453e-11; /* 0xBDC5E43D, 0x693FB3C8 */ -const V01: f64 = 1.27304834834123699328e-02; /* 0x3F8A1270, 0x91C9C71A */ -const V02: f64 = 7.60068627350353253702e-05; /* 0x3F13ECBB, 0xF578C6C1 */ -const V03: f64 = 2.59150851840457805467e-07; /* 0x3E91642D, 0x7FF202FD */ -const V04: f64 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */ - -pub fn y0(x: f64) -> f64 -{ - let z: f64; - let u: f64; - let v: f64; - let ix: u32; - let lx: u32; - - ix = get_high_word(x); - lx = get_low_word(x); - - /* y0(nan)=nan, y0(<0)=nan, y0(0)=-inf, y0(inf)=0 */ - if ((ix<<1) | lx) == 0 { - return -1.0/0.0; - } - if (ix>>31) != 0 { - return 0.0/0.0; - } - if ix >= 0x7ff00000 { - return 1.0/x; - } - - if ix >= 0x40000000 { /* x >= 2 */ - /* large ulp errors near zeros: 3.958, 7.086,.. */ - return common(ix,x,true); - } - - /* U(x^2)/V(x^2) + (2/pi)*j0(x)*log(x) */ - if ix >= 0x3e400000 { /* x >= 2**-27 */ - /* large ulp error near the first zero, x ~= 0.89 */ - z = x*x; - u = U00+z*(U01+z*(U02+z*(U03+z*(U04+z*(U05+z*U06))))); - v = 1.0+z*(V01+z*(V02+z*(V03+z*V04))); - return u/v + TPI*(j0(x)*log(x)); - } - return U00 + TPI*log(x); -} - -/* The asymptotic expansions of pzero is - * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. - * For x >= 2, We approximate pzero by - * pzero(x) = 1 + (R/S) - * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 - * S = 1 + pS0*s^2 + ... + pS4*s^10 - * and - * | pzero(x)-1-R/S | <= 2 ** ( -60.26) - */ -const PR8: [f64; 6] = [ /* for x in [inf, 8]=1/[0,0.125] */ - 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ - -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */ - -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */ - -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */ - -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */ - -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */ -]; -const PS8: [f64; 5] = [ - 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */ - 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */ - 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */ - 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */ - 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */ -]; - -const PR5: [f64; 6] = [ /* for x in [8,4.5454]=1/[0.125,0.22001] */ - -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */ - -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */ - -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */ - -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */ - -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */ - -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */ -]; -const PS5: [f64; 5] = [ - 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */ - 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */ - 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */ - 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */ - 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */ -]; - -const PR3: [f64; 6] = [/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ - -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */ - -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */ - -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */ - -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */ - -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */ - -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */ -]; -const PS3: [f64; 5] = [ - 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */ - 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */ - 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */ - 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */ - 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */ -]; - -const PR2: [f64; 6] = [/* for x in [2.8570,2]=1/[0.3499,0.5] */ - -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */ - -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */ - -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */ - -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */ - -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */ - -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */ -]; -const PS2: [f64; 5] = [ - 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */ - 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */ - 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */ - 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */ - 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */ -]; - -fn pzero(x: f64) -> f64 -{ - let p: &[f64; 6]; - let q: &[f64; 5]; - let z: f64; - let r: f64; - let s: f64; - let mut ix: u32; - - ix = get_high_word(x); - ix &= 0x7fffffff; - if ix >= 0x40200000 {p = &PR8; q = &PS8;} - else if ix >= 0x40122E8B {p = &PR5; q = &PS5;} - else if ix >= 0x4006DB6D {p = &PR3; q = &PS3;} - else /*ix >= 0x40000000*/{p = &PR2; q = &PS2;} - z = 1.0/(x*x); - r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); - s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); - return 1.0 + r/s; -} - - -/* For x >= 8, the asymptotic expansions of qzero is - * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. - * We approximate pzero by - * qzero(x) = s*(-1.25 + (R/S)) - * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 - * S = 1 + qS0*s^2 + ... + qS5*s^12 - * and - * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) - */ -const QR8: [f64; 6] = [ /* for x in [inf, 8]=1/[0,0.125] */ - 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ - 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */ - 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */ - 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */ - 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */ - 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */ -]; -const QS8: [f64; 6] = [ - 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */ - 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */ - 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */ - 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */ - 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */ - -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */ -]; - -const QR5: [f64; 6] = [ /* for x in [8,4.5454]=1/[0.125,0.22001] */ - 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */ - 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */ - 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */ - 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */ - 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */ - 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */ -]; -const QS5: [f64; 6] = [ - 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */ - 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */ - 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */ - 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */ - 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */ - -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */ -]; - -const QR3: [f64; 6] = [/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ - 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */ - 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */ - 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */ - 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */ - 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */ - 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */ -]; -const QS3: [f64; 6] = [ - 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */ - 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */ - 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */ - 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */ - 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */ - -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */ -]; - -const QR2: [f64; 6] = [/* for x in [2.8570,2]=1/[0.3499,0.5] */ - 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */ - 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */ - 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */ - 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */ - 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */ - 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */ -]; -const QS2: [f64; 6] = [ - 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */ - 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */ - 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */ - 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */ - 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */ - -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */ -]; - -fn qzero(x: f64) -> f64 -{ - let p: &[f64; 6]; - let q: &[f64; 6]; - let s: f64; - let r: f64; - let z: f64; - let mut ix: u32; - - ix = get_high_word(x); - ix &= 0x7fffffff; - if ix >= 0x40200000 {p = &QR8; q = &QS8;} - else if ix >= 0x40122E8B {p = &QR5; q = &QS5;} - else if ix >= 0x4006DB6D {p = &QR3; q = &QS3;} - else /*ix >= 0x40000000*/{p = &QR2; q = &QS2;} - z = 1.0/(x*x); - r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); - s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); - return (-0.125 + r/s)/x; -} +/* origin: FreeBSD /usr/src/lib/msun/src/e_j0.c */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ +/* j0(x), y0(x) + * Bessel function of the first and second kinds of order zero. + * Method -- j0(x): + * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ... + * 2. Reduce x to |x| since j0(x)=j0(-x), and + * for x in (0,2) + * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x; + * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 ) + * for x in (2,inf) + * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) + * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) + * as follow: + * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) + * = 1/sqrt(2) * (cos(x) + sin(x)) + * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4) + * = 1/sqrt(2) * (sin(x) - cos(x)) + * (To avoid cancellation, use + * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) + * to compute the worse one.) + * + * 3 Special cases + * j0(nan)= nan + * j0(0) = 1 + * j0(inf) = 0 + * + * Method -- y0(x): + * 1. For x<2. + * Since + * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...) + * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function. + * We use the following function to approximate y0, + * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2 + * where + * U(z) = u00 + u01*z + ... + u06*z^6 + * V(z) = 1 + v01*z + ... + v04*z^4 + * with absolute approximation error bounded by 2**-72. + * Note: For tiny x, U/V = u0 and j0(x)~1, hence + * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27) + * 2. For x>=2. + * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0)) + * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) + * by the method mentioned above. + * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0. + */ + +use super::{cos, fabs, get_high_word, get_low_word, log, sin, sqrt}; +const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */ +const TPI: f64 = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */ + +/* common method when |x|>=2 */ +fn common(ix: u32, x: f64, y0: bool) -> f64 { + let s: f64; + let mut c: f64; + let mut ss: f64; + let mut cc: f64; + let z: f64; + + /* + * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x-pi/4)-q0(x)*sin(x-pi/4)) + * y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x-pi/4)+q0(x)*cos(x-pi/4)) + * + * sin(x-pi/4) = (sin(x) - cos(x))/sqrt(2) + * cos(x-pi/4) = (sin(x) + cos(x))/sqrt(2) + * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) + */ + s = sin(x); + c = cos(x); + if y0 { + c = -c; + } + cc = s + c; + /* avoid overflow in 2*x, big ulp error when x>=0x1p1023 */ + if ix < 0x7fe00000 { + ss = s - c; + z = -cos(2.0 * x); + if s * c < 0.0 { + cc = z / ss; + } else { + ss = z / cc; + } + if ix < 0x48000000 { + if y0 { + ss = -ss; + } + cc = pzero(x) * cc - qzero(x) * ss; + } + } + return INVSQRTPI * cc / sqrt(x); +} + +/* R0/S0 on [0, 2.00] */ +const R02: f64 = 1.56249999999999947958e-02; /* 0x3F8FFFFF, 0xFFFFFFFD */ +const R03: f64 = -1.89979294238854721751e-04; /* 0xBF28E6A5, 0xB61AC6E9 */ +const R04: f64 = 1.82954049532700665670e-06; /* 0x3EBEB1D1, 0x0C503919 */ +const R05: f64 = -4.61832688532103189199e-09; /* 0xBE33D5E7, 0x73D63FCE */ +const S01: f64 = 1.56191029464890010492e-02; /* 0x3F8FFCE8, 0x82C8C2A4 */ +const S02: f64 = 1.16926784663337450260e-04; /* 0x3F1EA6D2, 0xDD57DBF4 */ +const S03: f64 = 5.13546550207318111446e-07; /* 0x3EA13B54, 0xCE84D5A9 */ +const S04: f64 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */ + +pub fn j0(mut x: f64) -> f64 { + let z: f64; + let r: f64; + let s: f64; + let mut ix: u32; + + ix = get_high_word(x); + ix &= 0x7fffffff; + + /* j0(+-inf)=0, j0(nan)=nan */ + if ix >= 0x7ff00000 { + return 1.0 / (x * x); + } + x = fabs(x); + + if ix >= 0x40000000 { + /* |x| >= 2 */ + /* large ulp error near zeros: 2.4, 5.52, 8.6537,.. */ + return common(ix, x, false); + } + + /* 1 - x*x/4 + x*x*R(x^2)/S(x^2) */ + if ix >= 0x3f200000 { + /* |x| >= 2**-13 */ + /* up to 4ulp error close to 2 */ + z = x * x; + r = z * (R02 + z * (R03 + z * (R04 + z * R05))); + s = 1.0 + z * (S01 + z * (S02 + z * (S03 + z * S04))); + return (1.0 + x / 2.0) * (1.0 - x / 2.0) + z * (r / s); + } + + /* 1 - x*x/4 */ + /* prevent underflow */ + /* inexact should be raised when x!=0, this is not done correctly */ + if ix >= 0x38000000 { + /* |x| >= 2**-127 */ + x = 0.25 * x * x; + } + return 1.0 - x; +} + +const U00: f64 = -7.38042951086872317523e-02; /* 0xBFB2E4D6, 0x99CBD01F */ +const U01: f64 = 1.76666452509181115538e-01; /* 0x3FC69D01, 0x9DE9E3FC */ +const U02: f64 = -1.38185671945596898896e-02; /* 0xBF8C4CE8, 0xB16CFA97 */ +const U03: f64 = 3.47453432093683650238e-04; /* 0x3F36C54D, 0x20B29B6B */ +const U04: f64 = -3.81407053724364161125e-06; /* 0xBECFFEA7, 0x73D25CAD */ +const U05: f64 = 1.95590137035022920206e-08; /* 0x3E550057, 0x3B4EABD4 */ +const U06: f64 = -3.98205194132103398453e-11; /* 0xBDC5E43D, 0x693FB3C8 */ +const V01: f64 = 1.27304834834123699328e-02; /* 0x3F8A1270, 0x91C9C71A */ +const V02: f64 = 7.60068627350353253702e-05; /* 0x3F13ECBB, 0xF578C6C1 */ +const V03: f64 = 2.59150851840457805467e-07; /* 0x3E91642D, 0x7FF202FD */ +const V04: f64 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */ + +pub fn y0(x: f64) -> f64 { + let z: f64; + let u: f64; + let v: f64; + let ix: u32; + let lx: u32; + + ix = get_high_word(x); + lx = get_low_word(x); + + /* y0(nan)=nan, y0(<0)=nan, y0(0)=-inf, y0(inf)=0 */ + if ((ix << 1) | lx) == 0 { + return -1.0 / 0.0; + } + if (ix >> 31) != 0 { + return 0.0 / 0.0; + } + if ix >= 0x7ff00000 { + return 1.0 / x; + } + + if ix >= 0x40000000 { + /* x >= 2 */ + /* large ulp errors near zeros: 3.958, 7.086,.. */ + return common(ix, x, true); + } + + /* U(x^2)/V(x^2) + (2/pi)*j0(x)*log(x) */ + if ix >= 0x3e400000 { + /* x >= 2**-27 */ + /* large ulp error near the first zero, x ~= 0.89 */ + z = x * x; + u = U00 + z * (U01 + z * (U02 + z * (U03 + z * (U04 + z * (U05 + z * U06))))); + v = 1.0 + z * (V01 + z * (V02 + z * (V03 + z * V04))); + return u / v + TPI * (j0(x) * log(x)); + } + return U00 + TPI * log(x); +} + +/* The asymptotic expansions of pzero is + * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. + * For x >= 2, We approximate pzero by + * pzero(x) = 1 + (R/S) + * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 + * S = 1 + pS0*s^2 + ... + pS4*s^10 + * and + * | pzero(x)-1-R/S | <= 2 ** ( -60.26) + */ +const PR8: [f64; 6] = [ + /* for x in [inf, 8]=1/[0,0.125] */ + 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ + -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */ + -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */ + -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */ + -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */ + -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */ +]; +const PS8: [f64; 5] = [ + 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */ + 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */ + 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */ + 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */ + 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */ +]; + +const PR5: [f64; 6] = [ + /* for x in [8,4.5454]=1/[0.125,0.22001] */ + -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */ + -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */ + -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */ + -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */ + -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */ + -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */ +]; +const PS5: [f64; 5] = [ + 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */ + 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */ + 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */ + 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */ + 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */ +]; + +const PR3: [f64; 6] = [ + /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ + -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */ + -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */ + -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */ + -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */ + -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */ + -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */ +]; +const PS3: [f64; 5] = [ + 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */ + 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */ + 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */ + 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */ + 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */ +]; + +const PR2: [f64; 6] = [ + /* for x in [2.8570,2]=1/[0.3499,0.5] */ + -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */ + -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */ + -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */ + -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */ + -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */ + -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */ +]; +const PS2: [f64; 5] = [ + 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */ + 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */ + 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */ + 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */ + 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */ +]; + +fn pzero(x: f64) -> f64 { + let p: &[f64; 6]; + let q: &[f64; 5]; + let z: f64; + let r: f64; + let s: f64; + let mut ix: u32; + + ix = get_high_word(x); + ix &= 0x7fffffff; + if ix >= 0x40200000 { + p = &PR8; + q = &PS8; + } else if ix >= 0x40122E8B { + p = &PR5; + q = &PS5; + } else if ix >= 0x4006DB6D { + p = &PR3; + q = &PS3; + } else + /*ix >= 0x40000000*/ + { + p = &PR2; + q = &PS2; + } + z = 1.0 / (x * x); + r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5])))); + s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4])))); + return 1.0 + r / s; +} + +/* For x >= 8, the asymptotic expansions of qzero is + * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. + * We approximate pzero by + * qzero(x) = s*(-1.25 + (R/S)) + * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 + * S = 1 + qS0*s^2 + ... + qS5*s^12 + * and + * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) + */ +const QR8: [f64; 6] = [ + /* for x in [inf, 8]=1/[0,0.125] */ + 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ + 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */ + 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */ + 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */ + 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */ + 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */ +]; +const QS8: [f64; 6] = [ + 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */ + 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */ + 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */ + 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */ + 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */ + -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */ +]; + +const QR5: [f64; 6] = [ + /* for x in [8,4.5454]=1/[0.125,0.22001] */ + 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */ + 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */ + 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */ + 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */ + 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */ + 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */ +]; +const QS5: [f64; 6] = [ + 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */ + 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */ + 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */ + 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */ + 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */ + -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */ +]; + +const QR3: [f64; 6] = [ + /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ + 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */ + 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */ + 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */ + 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */ + 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */ + 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */ +]; +const QS3: [f64; 6] = [ + 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */ + 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */ + 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */ + 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */ + 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */ + -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */ +]; + +const QR2: [f64; 6] = [ + /* for x in [2.8570,2]=1/[0.3499,0.5] */ + 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */ + 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */ + 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */ + 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */ + 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */ + 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */ +]; +const QS2: [f64; 6] = [ + 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */ + 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */ + 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */ + 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */ + 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */ + -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */ +]; + +fn qzero(x: f64) -> f64 { + let p: &[f64; 6]; + let q: &[f64; 6]; + let s: f64; + let r: f64; + let z: f64; + let mut ix: u32; + + ix = get_high_word(x); + ix &= 0x7fffffff; + if ix >= 0x40200000 { + p = &QR8; + q = &QS8; + } else if ix >= 0x40122E8B { + p = &QR5; + q = &QS5; + } else if ix >= 0x4006DB6D { + p = &QR3; + q = &QS3; + } else + /*ix >= 0x40000000*/ + { + p = &QR2; + q = &QS2; + } + z = 1.0 / (x * x); + r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5])))); + s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5]))))); + return (-0.125 + r / s) / x; +} diff --git a/src/math/j0f.rs b/src/math/j0f.rs index e2faed0..91c03db 100644 --- a/src/math/j0f.rs +++ b/src/math/j0f.rs @@ -1,330 +1,359 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/e_j0f.c */ -/* - * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. - */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -use super::{cosf, fabsf, logf, sinf, sqrtf}; - -const INVSQRTPI: f32 = 5.6418961287e-01; /* 0x3f106ebb */ -const TPI: f32 = 6.3661974669e-01; /* 0x3f22f983 */ - -fn common(ix: u32, x: f32, y0: bool) -> f32 -{ - let z: f32; - let s: f32; - let mut c: f32; - let mut ss: f32; - let mut cc: f32; - /* - * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) - * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) - */ - s = sinf(x); - c = cosf(x); - if y0 { - c = -c; - } - cc = s+c; - if ix < 0x7f000000 { - ss = s-c; - z = -cosf(2.0*x); - if s*c < 0.0 { - cc = z/ss; - } else { - ss = z/cc; - } - if ix < 0x58800000 { - if y0 { - ss = -ss; - } - cc = pzerof(x)*cc-qzerof(x)*ss; - } - } - return INVSQRTPI*cc/sqrtf(x); -} - -/* R0/S0 on [0, 2.00] */ -const R02: f32 = 1.5625000000e-02; /* 0x3c800000 */ -const R03: f32 = -1.8997929874e-04; /* 0xb947352e */ -const R04: f32 = 1.8295404516e-06; /* 0x35f58e88 */ -const R05: f32 = -4.6183270541e-09; /* 0xb19eaf3c */ -const S01: f32 = 1.5619102865e-02; /* 0x3c7fe744 */ -const S02: f32 = 1.1692678527e-04; /* 0x38f53697 */ -const S03: f32 = 5.1354652442e-07; /* 0x3509daa6 */ -const S04: f32 = 1.1661400734e-09; /* 0x30a045e8 */ - -pub fn j0f(mut x: f32) -> f32 -{ - let z: f32; - let r: f32; - let s: f32; - let mut ix: u32; - - ix = x.to_bits(); - ix &= 0x7fffffff; - if ix >= 0x7f800000 { - return 1.0/(x*x); - } - x = fabsf(x); - - if ix >= 0x40000000 { /* |x| >= 2 */ - /* large ulp error near zeros */ - return common(ix, x, false); - } - if ix >= 0x3a000000 { /* |x| >= 2**-11 */ - /* up to 4ulp error near 2 */ - z = x*x; - r = z*(R02+z*(R03+z*(R04+z*R05))); - s = 1.0+z*(S01+z*(S02+z*(S03+z*S04))); - return (1.0+x/2.0)*(1.0-x/2.0) + z*(r/s); - } - if ix >= 0x21800000 { /* |x| >= 2**-60 */ - x = 0.25*x*x; - } - return 1.0 - x; -} - -const U00: f32 = -7.3804296553e-02; /* 0xbd9726b5 */ -const U01: f32 = 1.7666645348e-01; /* 0x3e34e80d */ -const U02: f32 = -1.3818567619e-02; /* 0xbc626746 */ -const U03: f32 = 3.4745343146e-04; /* 0x39b62a69 */ -const U04: f32 = -3.8140706238e-06; /* 0xb67ff53c */ -const U05: f32 = 1.9559013964e-08; /* 0x32a802ba */ -const U06: f32 = -3.9820518410e-11; /* 0xae2f21eb */ -const V01: f32 = 1.2730483897e-02; /* 0x3c509385 */ -const V02: f32 = 7.6006865129e-05; /* 0x389f65e0 */ -const V03: f32 = 2.5915085189e-07; /* 0x348b216c */ -const V04: f32 = 4.4111031494e-10; /* 0x2ff280c2 */ - -pub fn y0f(x: f32) -> f32 -{ - let z: f32; - let u: f32; - let v: f32; - let ix: u32; - - ix = x.to_bits(); - if (ix & 0x7fffffff) == 0 { - return -1.0/0.0; - } - if (ix>>31) !=0 { - return 0.0/0.0; - } - if ix >= 0x7f800000 { - return 1.0/x; - } - if ix >= 0x40000000 { /* |x| >= 2.0 */ - /* large ulp error near zeros */ - return common(ix,x,true); - } - if ix >= 0x39000000 { /* x >= 2**-13 */ - /* large ulp error at x ~= 0.89 */ - z = x*x; - u = U00+z*(U01+z*(U02+z*(U03+z*(U04+z*(U05+z*U06))))); - v = 1.0+z*(V01+z*(V02+z*(V03+z*V04))); - return u/v + TPI*(j0f(x)*logf(x)); - } - return U00 + TPI*logf(x); -} - -/* The asymptotic expansions of pzero is - * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. - * For x >= 2, We approximate pzero by - * pzero(x) = 1 + (R/S) - * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 - * S = 1 + pS0*s^2 + ... + pS4*s^10 - * and - * | pzero(x)-1-R/S | <= 2 ** ( -60.26) - */ -const PR8: [f32; 6] = [ /* for x in [inf, 8]=1/[0,0.125] */ - 0.0000000000e+00, /* 0x00000000 */ - -7.0312500000e-02, /* 0xbd900000 */ - -8.0816707611e+00, /* 0xc1014e86 */ - -2.5706311035e+02, /* 0xc3808814 */ - -2.4852163086e+03, /* 0xc51b5376 */ - -5.2530439453e+03, /* 0xc5a4285a */ -]; -const PS8: [f32; 5] = [ - 1.1653436279e+02, /* 0x42e91198 */ - 3.8337448730e+03, /* 0x456f9beb */ - 4.0597855469e+04, /* 0x471e95db */ - 1.1675296875e+05, /* 0x47e4087c */ - 4.7627726562e+04, /* 0x473a0bba */ -]; -const PR5: [f32; 6] = [ /* for x in [8,4.5454]=1/[0.125,0.22001] */ - -1.1412546255e-11, /* 0xad48c58a */ - -7.0312492549e-02, /* 0xbd8fffff */ - -4.1596107483e+00, /* 0xc0851b88 */ - -6.7674766541e+01, /* 0xc287597b */ - -3.3123129272e+02, /* 0xc3a59d9b */ - -3.4643338013e+02, /* 0xc3ad3779 */ -]; -const PS5: [f32; 5] = [ - 6.0753936768e+01, /* 0x42730408 */ - 1.0512523193e+03, /* 0x44836813 */ - 5.9789707031e+03, /* 0x45bad7c4 */ - 9.6254453125e+03, /* 0x461665c8 */ - 2.4060581055e+03, /* 0x451660ee */ -]; - -const PR3: [f32; 6] = [/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ - -2.5470459075e-09, /* 0xb12f081b */ - -7.0311963558e-02, /* 0xbd8fffb8 */ - -2.4090321064e+00, /* 0xc01a2d95 */ - -2.1965976715e+01, /* 0xc1afba52 */ - -5.8079170227e+01, /* 0xc2685112 */ - -3.1447946548e+01, /* 0xc1fb9565 */ -]; -const PS3: [f32; 5] = [ - 3.5856033325e+01, /* 0x420f6c94 */ - 3.6151397705e+02, /* 0x43b4c1ca */ - 1.1936077881e+03, /* 0x44953373 */ - 1.1279968262e+03, /* 0x448cffe6 */ - 1.7358093262e+02, /* 0x432d94b8 */ -]; - -const PR2: [f32; 6] = [/* for x in [2.8570,2]=1/[0.3499,0.5] */ - -8.8753431271e-08, /* 0xb3be98b7 */ - -7.0303097367e-02, /* 0xbd8ffb12 */ - -1.4507384300e+00, /* 0xbfb9b1cc */ - -7.6356959343e+00, /* 0xc0f4579f */ - -1.1193166733e+01, /* 0xc1331736 */ - -3.2336456776e+00, /* 0xc04ef40d */ -]; -const PS2: [f32; 5] = [ - 2.2220300674e+01, /* 0x41b1c32d */ - 1.3620678711e+02, /* 0x430834f0 */ - 2.7047027588e+02, /* 0x43873c32 */ - 1.5387539673e+02, /* 0x4319e01a */ - 1.4657617569e+01, /* 0x416a859a */ -]; - -fn pzerof(x: f32) -> f32 -{ - let p: &[f32; 6]; - let q: &[f32; 5]; - let z: f32; - let r: f32; - let s: f32; - let mut ix: u32; - - ix = x.to_bits(); - ix &= 0x7fffffff; - if ix >= 0x41000000 {p = &PR8; q = &PS8;} - else if ix >= 0x409173eb {p = &PR5; q = &PS5;} - else if ix >= 0x4036d917 {p = &PR3; q = &PS3;} - else /*ix >= 0x40000000*/{p = &PR2; q = &PS2;} - z = 1.0/(x*x); - r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); - s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); - return 1.0 + r/s; -} - - -/* For x >= 8, the asymptotic expansions of qzero is - * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. - * We approximate pzero by - * qzero(x) = s*(-1.25 + (R/S)) - * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 - * S = 1 + qS0*s^2 + ... + qS5*s^12 - * and - * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) - */ -const QR8: [f32; 6] = [ /* for x in [inf, 8]=1/[0,0.125] */ - 0.0000000000e+00, /* 0x00000000 */ - 7.3242187500e-02, /* 0x3d960000 */ - 1.1768206596e+01, /* 0x413c4a93 */ - 5.5767340088e+02, /* 0x440b6b19 */ - 8.8591972656e+03, /* 0x460a6cca */ - 3.7014625000e+04, /* 0x471096a0 */ -]; -const QS8: [f32; 6] = [ - 1.6377603149e+02, /* 0x4323c6aa */ - 8.0983447266e+03, /* 0x45fd12c2 */ - 1.4253829688e+05, /* 0x480b3293 */ - 8.0330925000e+05, /* 0x49441ed4 */ - 8.4050156250e+05, /* 0x494d3359 */ - -3.4389928125e+05, /* 0xc8a7eb69 */ -]; - -const QR5: [f32; 6] = [ /* for x in [8,4.5454]=1/[0.125,0.22001] */ - 1.8408595828e-11, /* 0x2da1ec79 */ - 7.3242180049e-02, /* 0x3d95ffff */ - 5.8356351852e+00, /* 0x40babd86 */ - 1.3511157227e+02, /* 0x43071c90 */ - 1.0272437744e+03, /* 0x448067cd */ - 1.9899779053e+03, /* 0x44f8bf4b */ -]; -const QS5: [f32; 6] = [ - 8.2776611328e+01, /* 0x42a58da0 */ - 2.0778142090e+03, /* 0x4501dd07 */ - 1.8847289062e+04, /* 0x46933e94 */ - 5.6751113281e+04, /* 0x475daf1d */ - 3.5976753906e+04, /* 0x470c88c1 */ - -5.3543427734e+03, /* 0xc5a752be */ -]; - -const QR3: [f32; 6] = [/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ - 4.3774099900e-09, /* 0x3196681b */ - 7.3241114616e-02, /* 0x3d95ff70 */ - 3.3442313671e+00, /* 0x405607e3 */ - 4.2621845245e+01, /* 0x422a7cc5 */ - 1.7080809021e+02, /* 0x432acedf */ - 1.6673394775e+02, /* 0x4326bbe4 */ -]; -const QS3: [f32; 6] = [ - 4.8758872986e+01, /* 0x42430916 */ - 7.0968920898e+02, /* 0x44316c1c */ - 3.7041481934e+03, /* 0x4567825f */ - 6.4604252930e+03, /* 0x45c9e367 */ - 2.5163337402e+03, /* 0x451d4557 */ - -1.4924745178e+02, /* 0xc3153f59 */ -]; - -const QR2: [f32; 6] = [/* for x in [2.8570,2]=1/[0.3499,0.5] */ - 1.5044444979e-07, /* 0x342189db */ - 7.3223426938e-02, /* 0x3d95f62a */ - 1.9981917143e+00, /* 0x3fffc4bf */ - 1.4495602608e+01, /* 0x4167edfd */ - 3.1666231155e+01, /* 0x41fd5471 */ - 1.6252708435e+01, /* 0x4182058c */ -]; -const QS2: [f32; 6] = [ - 3.0365585327e+01, /* 0x41f2ecb8 */ - 2.6934811401e+02, /* 0x4386ac8f */ - 8.4478375244e+02, /* 0x44533229 */ - 8.8293585205e+02, /* 0x445cbbe5 */ - 2.1266638184e+02, /* 0x4354aa98 */ - -5.3109550476e+00, /* 0xc0a9f358 */ -]; - -fn qzerof(x: f32) -> f32 -{ - let p: &[f32; 6]; - let q: &[f32; 6]; - let s: f32; - let r: f32; - let z: f32; - let mut ix: u32; - - ix = x.to_bits(); - ix &= 0x7fffffff; - if ix >= 0x41000000 {p = &QR8; q = &QS8;} - else if ix >= 0x409173eb {p = &QR5; q = &QS5;} - else if ix >= 0x4036d917 {p = &QR3; q = &QS3;} - else /*ix >= 0x40000000*/{p = &QR2; q = &QS2;} - z = 1.0/(x*x); - r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); - s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); - return (-0.125 + r/s)/x; -} +/* origin: FreeBSD /usr/src/lib/msun/src/e_j0f.c */ +/* + * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. + */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +use super::{cosf, fabsf, logf, sinf, sqrtf}; + +const INVSQRTPI: f32 = 5.6418961287e-01; /* 0x3f106ebb */ +const TPI: f32 = 6.3661974669e-01; /* 0x3f22f983 */ + +fn common(ix: u32, x: f32, y0: bool) -> f32 { + let z: f32; + let s: f32; + let mut c: f32; + let mut ss: f32; + let mut cc: f32; + /* + * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) + * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) + */ + s = sinf(x); + c = cosf(x); + if y0 { + c = -c; + } + cc = s + c; + if ix < 0x7f000000 { + ss = s - c; + z = -cosf(2.0 * x); + if s * c < 0.0 { + cc = z / ss; + } else { + ss = z / cc; + } + if ix < 0x58800000 { + if y0 { + ss = -ss; + } + cc = pzerof(x) * cc - qzerof(x) * ss; + } + } + return INVSQRTPI * cc / sqrtf(x); +} + +/* R0/S0 on [0, 2.00] */ +const R02: f32 = 1.5625000000e-02; /* 0x3c800000 */ +const R03: f32 = -1.8997929874e-04; /* 0xb947352e */ +const R04: f32 = 1.8295404516e-06; /* 0x35f58e88 */ +const R05: f32 = -4.6183270541e-09; /* 0xb19eaf3c */ +const S01: f32 = 1.5619102865e-02; /* 0x3c7fe744 */ +const S02: f32 = 1.1692678527e-04; /* 0x38f53697 */ +const S03: f32 = 5.1354652442e-07; /* 0x3509daa6 */ +const S04: f32 = 1.1661400734e-09; /* 0x30a045e8 */ + +pub fn j0f(mut x: f32) -> f32 { + let z: f32; + let r: f32; + let s: f32; + let mut ix: u32; + + ix = x.to_bits(); + ix &= 0x7fffffff; + if ix >= 0x7f800000 { + return 1.0 / (x * x); + } + x = fabsf(x); + + if ix >= 0x40000000 { + /* |x| >= 2 */ + /* large ulp error near zeros */ + return common(ix, x, false); + } + if ix >= 0x3a000000 { + /* |x| >= 2**-11 */ + /* up to 4ulp error near 2 */ + z = x * x; + r = z * (R02 + z * (R03 + z * (R04 + z * R05))); + s = 1.0 + z * (S01 + z * (S02 + z * (S03 + z * S04))); + return (1.0 + x / 2.0) * (1.0 - x / 2.0) + z * (r / s); + } + if ix >= 0x21800000 { + /* |x| >= 2**-60 */ + x = 0.25 * x * x; + } + return 1.0 - x; +} + +const U00: f32 = -7.3804296553e-02; /* 0xbd9726b5 */ +const U01: f32 = 1.7666645348e-01; /* 0x3e34e80d */ +const U02: f32 = -1.3818567619e-02; /* 0xbc626746 */ +const U03: f32 = 3.4745343146e-04; /* 0x39b62a69 */ +const U04: f32 = -3.8140706238e-06; /* 0xb67ff53c */ +const U05: f32 = 1.9559013964e-08; /* 0x32a802ba */ +const U06: f32 = -3.9820518410e-11; /* 0xae2f21eb */ +const V01: f32 = 1.2730483897e-02; /* 0x3c509385 */ +const V02: f32 = 7.6006865129e-05; /* 0x389f65e0 */ +const V03: f32 = 2.5915085189e-07; /* 0x348b216c */ +const V04: f32 = 4.4111031494e-10; /* 0x2ff280c2 */ + +pub fn y0f(x: f32) -> f32 { + let z: f32; + let u: f32; + let v: f32; + let ix: u32; + + ix = x.to_bits(); + if (ix & 0x7fffffff) == 0 { + return -1.0 / 0.0; + } + if (ix >> 31) != 0 { + return 0.0 / 0.0; + } + if ix >= 0x7f800000 { + return 1.0 / x; + } + if ix >= 0x40000000 { + /* |x| >= 2.0 */ + /* large ulp error near zeros */ + return common(ix, x, true); + } + if ix >= 0x39000000 { + /* x >= 2**-13 */ + /* large ulp error at x ~= 0.89 */ + z = x * x; + u = U00 + z * (U01 + z * (U02 + z * (U03 + z * (U04 + z * (U05 + z * U06))))); + v = 1.0 + z * (V01 + z * (V02 + z * (V03 + z * V04))); + return u / v + TPI * (j0f(x) * logf(x)); + } + return U00 + TPI * logf(x); +} + +/* The asymptotic expansions of pzero is + * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. + * For x >= 2, We approximate pzero by + * pzero(x) = 1 + (R/S) + * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 + * S = 1 + pS0*s^2 + ... + pS4*s^10 + * and + * | pzero(x)-1-R/S | <= 2 ** ( -60.26) + */ +const PR8: [f32; 6] = [ + /* for x in [inf, 8]=1/[0,0.125] */ + 0.0000000000e+00, /* 0x00000000 */ + -7.0312500000e-02, /* 0xbd900000 */ + -8.0816707611e+00, /* 0xc1014e86 */ + -2.5706311035e+02, /* 0xc3808814 */ + -2.4852163086e+03, /* 0xc51b5376 */ + -5.2530439453e+03, /* 0xc5a4285a */ +]; +const PS8: [f32; 5] = [ + 1.1653436279e+02, /* 0x42e91198 */ + 3.8337448730e+03, /* 0x456f9beb */ + 4.0597855469e+04, /* 0x471e95db */ + 1.1675296875e+05, /* 0x47e4087c */ + 4.7627726562e+04, /* 0x473a0bba */ +]; +const PR5: [f32; 6] = [ + /* for x in [8,4.5454]=1/[0.125,0.22001] */ + -1.1412546255e-11, /* 0xad48c58a */ + -7.0312492549e-02, /* 0xbd8fffff */ + -4.1596107483e+00, /* 0xc0851b88 */ + -6.7674766541e+01, /* 0xc287597b */ + -3.3123129272e+02, /* 0xc3a59d9b */ + -3.4643338013e+02, /* 0xc3ad3779 */ +]; +const PS5: [f32; 5] = [ + 6.0753936768e+01, /* 0x42730408 */ + 1.0512523193e+03, /* 0x44836813 */ + 5.9789707031e+03, /* 0x45bad7c4 */ + 9.6254453125e+03, /* 0x461665c8 */ + 2.4060581055e+03, /* 0x451660ee */ +]; + +const PR3: [f32; 6] = [ + /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ + -2.5470459075e-09, /* 0xb12f081b */ + -7.0311963558e-02, /* 0xbd8fffb8 */ + -2.4090321064e+00, /* 0xc01a2d95 */ + -2.1965976715e+01, /* 0xc1afba52 */ + -5.8079170227e+01, /* 0xc2685112 */ + -3.1447946548e+01, /* 0xc1fb9565 */ +]; +const PS3: [f32; 5] = [ + 3.5856033325e+01, /* 0x420f6c94 */ + 3.6151397705e+02, /* 0x43b4c1ca */ + 1.1936077881e+03, /* 0x44953373 */ + 1.1279968262e+03, /* 0x448cffe6 */ + 1.7358093262e+02, /* 0x432d94b8 */ +]; + +const PR2: [f32; 6] = [ + /* for x in [2.8570,2]=1/[0.3499,0.5] */ + -8.8753431271e-08, /* 0xb3be98b7 */ + -7.0303097367e-02, /* 0xbd8ffb12 */ + -1.4507384300e+00, /* 0xbfb9b1cc */ + -7.6356959343e+00, /* 0xc0f4579f */ + -1.1193166733e+01, /* 0xc1331736 */ + -3.2336456776e+00, /* 0xc04ef40d */ +]; +const PS2: [f32; 5] = [ + 2.2220300674e+01, /* 0x41b1c32d */ + 1.3620678711e+02, /* 0x430834f0 */ + 2.7047027588e+02, /* 0x43873c32 */ + 1.5387539673e+02, /* 0x4319e01a */ + 1.4657617569e+01, /* 0x416a859a */ +]; + +fn pzerof(x: f32) -> f32 { + let p: &[f32; 6]; + let q: &[f32; 5]; + let z: f32; + let r: f32; + let s: f32; + let mut ix: u32; + + ix = x.to_bits(); + ix &= 0x7fffffff; + if ix >= 0x41000000 { + p = &PR8; + q = &PS8; + } else if ix >= 0x409173eb { + p = &PR5; + q = &PS5; + } else if ix >= 0x4036d917 { + p = &PR3; + q = &PS3; + } else + /*ix >= 0x40000000*/ + { + p = &PR2; + q = &PS2; + } + z = 1.0 / (x * x); + r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5])))); + s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4])))); + return 1.0 + r / s; +} + +/* For x >= 8, the asymptotic expansions of qzero is + * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. + * We approximate pzero by + * qzero(x) = s*(-1.25 + (R/S)) + * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 + * S = 1 + qS0*s^2 + ... + qS5*s^12 + * and + * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) + */ +const QR8: [f32; 6] = [ + /* for x in [inf, 8]=1/[0,0.125] */ + 0.0000000000e+00, /* 0x00000000 */ + 7.3242187500e-02, /* 0x3d960000 */ + 1.1768206596e+01, /* 0x413c4a93 */ + 5.5767340088e+02, /* 0x440b6b19 */ + 8.8591972656e+03, /* 0x460a6cca */ + 3.7014625000e+04, /* 0x471096a0 */ +]; +const QS8: [f32; 6] = [ + 1.6377603149e+02, /* 0x4323c6aa */ + 8.0983447266e+03, /* 0x45fd12c2 */ + 1.4253829688e+05, /* 0x480b3293 */ + 8.0330925000e+05, /* 0x49441ed4 */ + 8.4050156250e+05, /* 0x494d3359 */ + -3.4389928125e+05, /* 0xc8a7eb69 */ +]; + +const QR5: [f32; 6] = [ + /* for x in [8,4.5454]=1/[0.125,0.22001] */ + 1.8408595828e-11, /* 0x2da1ec79 */ + 7.3242180049e-02, /* 0x3d95ffff */ + 5.8356351852e+00, /* 0x40babd86 */ + 1.3511157227e+02, /* 0x43071c90 */ + 1.0272437744e+03, /* 0x448067cd */ + 1.9899779053e+03, /* 0x44f8bf4b */ +]; +const QS5: [f32; 6] = [ + 8.2776611328e+01, /* 0x42a58da0 */ + 2.0778142090e+03, /* 0x4501dd07 */ + 1.8847289062e+04, /* 0x46933e94 */ + 5.6751113281e+04, /* 0x475daf1d */ + 3.5976753906e+04, /* 0x470c88c1 */ + -5.3543427734e+03, /* 0xc5a752be */ +]; + +const QR3: [f32; 6] = [ + /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ + 4.3774099900e-09, /* 0x3196681b */ + 7.3241114616e-02, /* 0x3d95ff70 */ + 3.3442313671e+00, /* 0x405607e3 */ + 4.2621845245e+01, /* 0x422a7cc5 */ + 1.7080809021e+02, /* 0x432acedf */ + 1.6673394775e+02, /* 0x4326bbe4 */ +]; +const QS3: [f32; 6] = [ + 4.8758872986e+01, /* 0x42430916 */ + 7.0968920898e+02, /* 0x44316c1c */ + 3.7041481934e+03, /* 0x4567825f */ + 6.4604252930e+03, /* 0x45c9e367 */ + 2.5163337402e+03, /* 0x451d4557 */ + -1.4924745178e+02, /* 0xc3153f59 */ +]; + +const QR2: [f32; 6] = [ + /* for x in [2.8570,2]=1/[0.3499,0.5] */ + 1.5044444979e-07, /* 0x342189db */ + 7.3223426938e-02, /* 0x3d95f62a */ + 1.9981917143e+00, /* 0x3fffc4bf */ + 1.4495602608e+01, /* 0x4167edfd */ + 3.1666231155e+01, /* 0x41fd5471 */ + 1.6252708435e+01, /* 0x4182058c */ +]; +const QS2: [f32; 6] = [ + 3.0365585327e+01, /* 0x41f2ecb8 */ + 2.6934811401e+02, /* 0x4386ac8f */ + 8.4478375244e+02, /* 0x44533229 */ + 8.8293585205e+02, /* 0x445cbbe5 */ + 2.1266638184e+02, /* 0x4354aa98 */ + -5.3109550476e+00, /* 0xc0a9f358 */ +]; + +fn qzerof(x: f32) -> f32 { + let p: &[f32; 6]; + let q: &[f32; 6]; + let s: f32; + let r: f32; + let z: f32; + let mut ix: u32; + + ix = x.to_bits(); + ix &= 0x7fffffff; + if ix >= 0x41000000 { + p = &QR8; + q = &QS8; + } else if ix >= 0x409173eb { + p = &QR5; + q = &QS5; + } else if ix >= 0x4036d917 { + p = &QR3; + q = &QS3; + } else + /*ix >= 0x40000000*/ + { + p = &QR2; + q = &QS2; + } + z = 1.0 / (x * x); + r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5])))); + s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5]))))); + return (-0.125 + r / s) / x; +} diff --git a/src/math/j1.rs b/src/math/j1.rs index 92289a6..02a65ca 100644 --- a/src/math/j1.rs +++ b/src/math/j1.rs @@ -1,387 +1,414 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/e_j1.c */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunSoft, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ -/* j1(x), y1(x) - * Bessel function of the first and second kinds of order zero. - * Method -- j1(x): - * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ... - * 2. Reduce x to |x| since j1(x)=-j1(-x), and - * for x in (0,2) - * j1(x) = x/2 + x*z*R0/S0, where z = x*x; - * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 ) - * for x in (2,inf) - * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1)) - * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) - * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) - * as follow: - * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) - * = 1/sqrt(2) * (sin(x) - cos(x)) - * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) - * = -1/sqrt(2) * (sin(x) + cos(x)) - * (To avoid cancellation, use - * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) - * to compute the worse one.) - * - * 3 Special cases - * j1(nan)= nan - * j1(0) = 0 - * j1(inf) = 0 - * - * Method -- y1(x): - * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN - * 2. For x<2. - * Since - * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...) - * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function. - * We use the following function to approximate y1, - * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2 - * where for x in [0,2] (abs err less than 2**-65.89) - * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4 - * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5 - * Note: For tiny x, 1/x dominate y1 and hence - * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54) - * 3. For x>=2. - * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) - * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) - * by method mentioned above. - */ - -use super::{cos, get_high_word, get_low_word, fabs, log, sin, sqrt}; - -const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */ -const TPI: f64 = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */ - -fn common(ix: u32, x: f64, y1: bool, sign: bool) -> f64 -{ - let z: f64; - let mut s: f64; - let c: f64; - let mut ss: f64; - let mut cc: f64; - - /* - * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x-3pi/4)-q1(x)*sin(x-3pi/4)) - * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x-3pi/4)+q1(x)*cos(x-3pi/4)) - * - * sin(x-3pi/4) = -(sin(x) + cos(x))/sqrt(2) - * cos(x-3pi/4) = (sin(x) - cos(x))/sqrt(2) - * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) - */ - s = sin(x); - if y1 { - s = -s; - } - c = cos(x); - cc = s-c; - if ix < 0x7fe00000 { - /* avoid overflow in 2*x */ - ss = -s-c; - z = cos(2.0*x); - if s*c > 0.0 { - cc = z/ss; - } else { - ss = z/cc; - } - if ix < 0x48000000 { - if y1 { - ss = -ss; - } - cc = pone(x)*cc-qone(x)*ss; - } - } - if sign { - cc = -cc; - } - return INVSQRTPI*cc/sqrt(x); -} - -/* R0/S0 on [0,2] */ -const R00: f64 = -6.25000000000000000000e-02; /* 0xBFB00000, 0x00000000 */ -const R01: f64 = 1.40705666955189706048e-03; /* 0x3F570D9F, 0x98472C61 */ -const R02: f64 = -1.59955631084035597520e-05; /* 0xBEF0C5C6, 0xBA169668 */ -const R03: f64 = 4.96727999609584448412e-08; /* 0x3E6AAAFA, 0x46CA0BD9 */ -const S01: f64 = 1.91537599538363460805e-02; /* 0x3F939D0B, 0x12637E53 */ -const S02: f64 = 1.85946785588630915560e-04; /* 0x3F285F56, 0xB9CDF664 */ -const S03: f64 = 1.17718464042623683263e-06; /* 0x3EB3BFF8, 0x333F8498 */ -const S04: f64 = 5.04636257076217042715e-09; /* 0x3E35AC88, 0xC97DFF2C */ -const S05: f64 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */ - -pub fn j1(x: f64) -> f64 -{ - let mut z: f64; - let r: f64; - let s: f64; - let mut ix: u32; - let sign: bool; - - ix = get_high_word(x); - sign = (ix>>31) != 0; - ix &= 0x7fffffff; - if ix >= 0x7ff00000 { - return 1.0/(x*x); - } - if ix >= 0x40000000 { /* |x| >= 2 */ - return common(ix, fabs(x), false, sign); - } - if ix >= 0x38000000 { /* |x| >= 2**-127 */ - z = x*x; - r = z*(R00+z*(R01+z*(R02+z*R03))); - s = 1.0+z*(S01+z*(S02+z*(S03+z*(S04+z*S05)))); - z = r/s; - } else { - /* avoid underflow, raise inexact if x!=0 */ - z = x; - } - return (0.5 + z)*x; -} - -const U0: [f64; 5] = [ - -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */ - 5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */ - -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */ - 2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */ - -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */ -]; -const V0: [f64; 5] = [ - 1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */ - 2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */ - 1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */ - 6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */ - 1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */ -]; - -pub fn y1(x: f64) -> f64 -{ - let z: f64; - let u: f64; - let v: f64; - let ix: u32; - let lx: u32; - - ix = get_high_word(x); - lx = get_low_word(x); - - /* y1(nan)=nan, y1(<0)=nan, y1(0)=-inf, y1(inf)=0 */ - if (ix<<1 | lx) == 0 { - return -1.0/0.0; - } - if (ix>>31) != 0 { - return 0.0/0.0; - } - if ix >= 0x7ff00000 { - return 1.0/x; - } - - if ix >= 0x40000000 { /* x >= 2 */ - return common(ix, x, true, false); - } - if ix < 0x3c900000 { /* x < 2**-54 */ - return -TPI/x; - } - z = x*x; - u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); - v = 1.0+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); - return x*(u/v) + TPI*(j1(x)*log(x)-1.0/x); -} - -/* For x >= 8, the asymptotic expansions of pone is - * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. - * We approximate pone by - * pone(x) = 1 + (R/S) - * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 - * S = 1 + ps0*s^2 + ... + ps4*s^10 - * and - * | pone(x)-1-R/S | <= 2 ** ( -60.06) - */ - -const PR8: [f64; 6] = [ /* for x in [inf, 8]=1/[0,0.125] */ - 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ - 1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */ - 1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */ - 4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */ - 3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */ - 7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */ -]; -const PS8: [f64; 5] = [ - 1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */ - 3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */ - 3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */ - 9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */ - 3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */ -]; - -const PR5: [f64; 6] = [ /* for x in [8,4.5454]=1/[0.125,0.22001] */ - 1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */ - 1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */ - 6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */ - 1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */ - 5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */ - 5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */ -]; -const PS5: [f64; 5] = [ - 5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */ - 9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */ - 5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */ - 7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */ - 1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */ -]; - -const PR3: [f64; 6] = [ - 3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */ - 1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */ - 3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */ - 3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */ - 9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */ - 4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */ -]; -const PS3: [f64; 5] = [ - 3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */ - 3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */ - 1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */ - 8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */ - 1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */ -]; - -const PR2: [f64; 6] = [/* for x in [2.8570,2]=1/[0.3499,0.5] */ - 1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */ - 1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */ - 2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */ - 1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */ - 1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */ - 5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */ -]; -const PS2: [f64; 5] = [ - 2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */ - 1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */ - 2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */ - 1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */ - 8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */ -]; - -fn pone(x: f64) -> f64 -{ - let p: &[f64; 6]; - let q: &[f64; 5]; - let z: f64; - let r: f64; - let s: f64; - let mut ix: u32; - - ix = get_high_word(x); - ix &= 0x7fffffff; - if ix >= 0x40200000 {p = &PR8; q = &PS8;} - else if ix >= 0x40122E8B {p = &PR5; q = &PS5;} - else if ix >= 0x4006DB6D {p = &PR3; q = &PS3;} - else /*ix >= 0x40000000*/{p = &PR2; q = &PS2;} - z = 1.0/(x*x); - r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); - s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); - return 1.0+ r/s; -} - -/* For x >= 8, the asymptotic expansions of qone is - * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. - * We approximate pone by - * qone(x) = s*(0.375 + (R/S)) - * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 - * S = 1 + qs1*s^2 + ... + qs6*s^12 - * and - * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) - */ - -const QR8: [f64; 6] = [ /* for x in [inf, 8]=1/[0,0.125] */ - 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ - -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */ - -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */ - -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */ - -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */ - -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */ -]; -const QS8: [f64; 6] = [ - 1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */ - 7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */ - 1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */ - 7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */ - 6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */ - -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */ -]; - -const QR5: [f64; 6] = [ /* for x in [8,4.5454]=1/[0.125,0.22001] */ - -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */ - -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */ - -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */ - -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */ - -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */ - -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */ -]; -const QS5: [f64; 6] = [ - 8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */ - 1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */ - 1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */ - 4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */ - 2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */ - -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */ -]; - -const QR3: [f64; 6] = [ - -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */ - -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */ - -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */ - -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */ - -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */ - -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */ -]; -const QS3: [f64; 6] = [ - 4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */ - 6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */ - 3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */ - 5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */ - 1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */ - -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */ -]; - -const QR2: [f64; 6] = [/* for x in [2.8570,2]=1/[0.3499,0.5] */ - -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */ - -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */ - -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */ - -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */ - -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */ - -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */ -]; -const QS2: [f64; 6] = [ - 2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */ - 2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */ - 7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */ - 7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */ - 1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */ - -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */ -]; - -fn qone(x: f64) -> f64 -{ - let p: &[f64; 6]; - let q: &[f64; 6]; - let s: f64; - let r: f64; - let z: f64; - let mut ix: u32; - - ix = get_high_word(x); - ix &= 0x7fffffff; - if ix >= 0x40200000 {p = &QR8; q = &QS8;} - else if ix >= 0x40122E8B {p = &QR5; q = &QS5;} - else if ix >= 0x4006DB6D {p = &QR3; q = &QS3;} - else /*ix >= 0x40000000*/{p = &QR2; q = &QS2;} - z = 1.0/(x*x); - r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); - s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); - return (0.375 + r/s)/x; -} +/* origin: FreeBSD /usr/src/lib/msun/src/e_j1.c */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ +/* j1(x), y1(x) + * Bessel function of the first and second kinds of order zero. + * Method -- j1(x): + * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ... + * 2. Reduce x to |x| since j1(x)=-j1(-x), and + * for x in (0,2) + * j1(x) = x/2 + x*z*R0/S0, where z = x*x; + * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 ) + * for x in (2,inf) + * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1)) + * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) + * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) + * as follow: + * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) + * = 1/sqrt(2) * (sin(x) - cos(x)) + * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) + * = -1/sqrt(2) * (sin(x) + cos(x)) + * (To avoid cancellation, use + * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) + * to compute the worse one.) + * + * 3 Special cases + * j1(nan)= nan + * j1(0) = 0 + * j1(inf) = 0 + * + * Method -- y1(x): + * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN + * 2. For x<2. + * Since + * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...) + * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function. + * We use the following function to approximate y1, + * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2 + * where for x in [0,2] (abs err less than 2**-65.89) + * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4 + * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5 + * Note: For tiny x, 1/x dominate y1 and hence + * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54) + * 3. For x>=2. + * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) + * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) + * by method mentioned above. + */ + +use super::{cos, fabs, get_high_word, get_low_word, log, sin, sqrt}; + +const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */ +const TPI: f64 = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */ + +fn common(ix: u32, x: f64, y1: bool, sign: bool) -> f64 { + let z: f64; + let mut s: f64; + let c: f64; + let mut ss: f64; + let mut cc: f64; + + /* + * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x-3pi/4)-q1(x)*sin(x-3pi/4)) + * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x-3pi/4)+q1(x)*cos(x-3pi/4)) + * + * sin(x-3pi/4) = -(sin(x) + cos(x))/sqrt(2) + * cos(x-3pi/4) = (sin(x) - cos(x))/sqrt(2) + * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) + */ + s = sin(x); + if y1 { + s = -s; + } + c = cos(x); + cc = s - c; + if ix < 0x7fe00000 { + /* avoid overflow in 2*x */ + ss = -s - c; + z = cos(2.0 * x); + if s * c > 0.0 { + cc = z / ss; + } else { + ss = z / cc; + } + if ix < 0x48000000 { + if y1 { + ss = -ss; + } + cc = pone(x) * cc - qone(x) * ss; + } + } + if sign { + cc = -cc; + } + return INVSQRTPI * cc / sqrt(x); +} + +/* R0/S0 on [0,2] */ +const R00: f64 = -6.25000000000000000000e-02; /* 0xBFB00000, 0x00000000 */ +const R01: f64 = 1.40705666955189706048e-03; /* 0x3F570D9F, 0x98472C61 */ +const R02: f64 = -1.59955631084035597520e-05; /* 0xBEF0C5C6, 0xBA169668 */ +const R03: f64 = 4.96727999609584448412e-08; /* 0x3E6AAAFA, 0x46CA0BD9 */ +const S01: f64 = 1.91537599538363460805e-02; /* 0x3F939D0B, 0x12637E53 */ +const S02: f64 = 1.85946785588630915560e-04; /* 0x3F285F56, 0xB9CDF664 */ +const S03: f64 = 1.17718464042623683263e-06; /* 0x3EB3BFF8, 0x333F8498 */ +const S04: f64 = 5.04636257076217042715e-09; /* 0x3E35AC88, 0xC97DFF2C */ +const S05: f64 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */ + +pub fn j1(x: f64) -> f64 { + let mut z: f64; + let r: f64; + let s: f64; + let mut ix: u32; + let sign: bool; + + ix = get_high_word(x); + sign = (ix >> 31) != 0; + ix &= 0x7fffffff; + if ix >= 0x7ff00000 { + return 1.0 / (x * x); + } + if ix >= 0x40000000 { + /* |x| >= 2 */ + return common(ix, fabs(x), false, sign); + } + if ix >= 0x38000000 { + /* |x| >= 2**-127 */ + z = x * x; + r = z * (R00 + z * (R01 + z * (R02 + z * R03))); + s = 1.0 + z * (S01 + z * (S02 + z * (S03 + z * (S04 + z * S05)))); + z = r / s; + } else { + /* avoid underflow, raise inexact if x!=0 */ + z = x; + } + return (0.5 + z) * x; +} + +const U0: [f64; 5] = [ + -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */ + 5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */ + -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */ + 2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */ + -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */ +]; +const V0: [f64; 5] = [ + 1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */ + 2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */ + 1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */ + 6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */ + 1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */ +]; + +pub fn y1(x: f64) -> f64 { + let z: f64; + let u: f64; + let v: f64; + let ix: u32; + let lx: u32; + + ix = get_high_word(x); + lx = get_low_word(x); + + /* y1(nan)=nan, y1(<0)=nan, y1(0)=-inf, y1(inf)=0 */ + if (ix << 1 | lx) == 0 { + return -1.0 / 0.0; + } + if (ix >> 31) != 0 { + return 0.0 / 0.0; + } + if ix >= 0x7ff00000 { + return 1.0 / x; + } + + if ix >= 0x40000000 { + /* x >= 2 */ + return common(ix, x, true, false); + } + if ix < 0x3c900000 { + /* x < 2**-54 */ + return -TPI / x; + } + z = x * x; + u = U0[0] + z * (U0[1] + z * (U0[2] + z * (U0[3] + z * U0[4]))); + v = 1.0 + z * (V0[0] + z * (V0[1] + z * (V0[2] + z * (V0[3] + z * V0[4])))); + return x * (u / v) + TPI * (j1(x) * log(x) - 1.0 / x); +} + +/* For x >= 8, the asymptotic expansions of pone is + * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. + * We approximate pone by + * pone(x) = 1 + (R/S) + * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 + * S = 1 + ps0*s^2 + ... + ps4*s^10 + * and + * | pone(x)-1-R/S | <= 2 ** ( -60.06) + */ + +const PR8: [f64; 6] = [ + /* for x in [inf, 8]=1/[0,0.125] */ + 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ + 1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */ + 1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */ + 4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */ + 3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */ + 7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */ +]; +const PS8: [f64; 5] = [ + 1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */ + 3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */ + 3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */ + 9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */ + 3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */ +]; + +const PR5: [f64; 6] = [ + /* for x in [8,4.5454]=1/[0.125,0.22001] */ + 1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */ + 1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */ + 6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */ + 1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */ + 5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */ + 5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */ +]; +const PS5: [f64; 5] = [ + 5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */ + 9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */ + 5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */ + 7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */ + 1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */ +]; + +const PR3: [f64; 6] = [ + 3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */ + 1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */ + 3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */ + 3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */ + 9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */ + 4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */ +]; +const PS3: [f64; 5] = [ + 3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */ + 3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */ + 1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */ + 8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */ + 1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */ +]; + +const PR2: [f64; 6] = [ + /* for x in [2.8570,2]=1/[0.3499,0.5] */ + 1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */ + 1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */ + 2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */ + 1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */ + 1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */ + 5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */ +]; +const PS2: [f64; 5] = [ + 2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */ + 1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */ + 2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */ + 1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */ + 8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */ +]; + +fn pone(x: f64) -> f64 { + let p: &[f64; 6]; + let q: &[f64; 5]; + let z: f64; + let r: f64; + let s: f64; + let mut ix: u32; + + ix = get_high_word(x); + ix &= 0x7fffffff; + if ix >= 0x40200000 { + p = &PR8; + q = &PS8; + } else if ix >= 0x40122E8B { + p = &PR5; + q = &PS5; + } else if ix >= 0x4006DB6D { + p = &PR3; + q = &PS3; + } else + /*ix >= 0x40000000*/ + { + p = &PR2; + q = &PS2; + } + z = 1.0 / (x * x); + r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5])))); + s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4])))); + return 1.0 + r / s; +} + +/* For x >= 8, the asymptotic expansions of qone is + * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. + * We approximate pone by + * qone(x) = s*(0.375 + (R/S)) + * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 + * S = 1 + qs1*s^2 + ... + qs6*s^12 + * and + * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) + */ + +const QR8: [f64; 6] = [ + /* for x in [inf, 8]=1/[0,0.125] */ + 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ + -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */ + -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */ + -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */ + -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */ + -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */ +]; +const QS8: [f64; 6] = [ + 1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */ + 7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */ + 1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */ + 7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */ + 6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */ + -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */ +]; + +const QR5: [f64; 6] = [ + /* for x in [8,4.5454]=1/[0.125,0.22001] */ + -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */ + -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */ + -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */ + -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */ + -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */ + -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */ +]; +const QS5: [f64; 6] = [ + 8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */ + 1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */ + 1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */ + 4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */ + 2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */ + -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */ +]; + +const QR3: [f64; 6] = [ + -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */ + -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */ + -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */ + -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */ + -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */ + -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */ +]; +const QS3: [f64; 6] = [ + 4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */ + 6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */ + 3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */ + 5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */ + 1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */ + -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */ +]; + +const QR2: [f64; 6] = [ + /* for x in [2.8570,2]=1/[0.3499,0.5] */ + -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */ + -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */ + -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */ + -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */ + -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */ + -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */ +]; +const QS2: [f64; 6] = [ + 2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */ + 2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */ + 7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */ + 7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */ + 1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */ + -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */ +]; + +fn qone(x: f64) -> f64 { + let p: &[f64; 6]; + let q: &[f64; 6]; + let s: f64; + let r: f64; + let z: f64; + let mut ix: u32; + + ix = get_high_word(x); + ix &= 0x7fffffff; + if ix >= 0x40200000 { + p = &QR8; + q = &QS8; + } else if ix >= 0x40122E8B { + p = &QR5; + q = &QS5; + } else if ix >= 0x4006DB6D { + p = &QR3; + q = &QS3; + } else + /*ix >= 0x40000000*/ + { + p = &QR2; + q = &QS2; + } + z = 1.0 / (x * x); + r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5])))); + s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5]))))); + return (0.375 + r / s) / x; +} diff --git a/src/math/j1f.rs b/src/math/j1f.rs index 7cf9c45..83ac1ac 100644 --- a/src/math/j1f.rs +++ b/src/math/j1f.rs @@ -1,331 +1,358 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/e_j1f.c */ -/* - * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. - */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -use super::{cosf, fabsf, logf, sinf, sqrtf}; - -const INVSQRTPI: f32 = 5.6418961287e-01; /* 0x3f106ebb */ -const TPI: f32 = 6.3661974669e-01; /* 0x3f22f983 */ - -fn common(ix: u32, x: f32, y1: bool, sign: bool) -> f32 -{ - let z: f64; - let mut s: f64; - let c: f64; - let mut ss: f64; - let mut cc: f64; - - s = sinf(x) as f64; - if y1 { - s = -s; - } - c = cosf(x) as f64; - cc = s-c; - if ix < 0x7f000000 { - ss = -s-c; - z = cosf(2.0*x) as f64; - if s*c > 0.0 { - cc = z/ss; - } else { - ss = z/cc; - } - if ix < 0x58800000 { - if y1 { - ss = -ss; - } - cc = (ponef(x) as f64)*cc-(qonef(x) as f64)*ss; - } - } - if sign { - cc = -cc; - } - return INVSQRTPI*(cc as f32)/sqrtf(x); -} - -/* R0/S0 on [0,2] */ -const R00: f32 = -6.2500000000e-02; /* 0xbd800000 */ -const R01: f32 = 1.4070566976e-03; /* 0x3ab86cfd */ -const R02: f32 = -1.5995563444e-05; /* 0xb7862e36 */ -const R03: f32 = 4.9672799207e-08; /* 0x335557d2 */ -const S01: f32 = 1.9153760746e-02; /* 0x3c9ce859 */ -const S02: f32 = 1.8594678841e-04; /* 0x3942fab6 */ -const S03: f32 = 1.1771846857e-06; /* 0x359dffc2 */ -const S04: f32 = 5.0463624390e-09; /* 0x31ad6446 */ -const S05: f32 = 1.2354227016e-11; /* 0x2d59567e */ - -pub fn j1f(x: f32) -> f32 -{ - let mut z: f32; - let r: f32; - let s: f32; - let mut ix: u32; - let sign: bool; - - ix = x.to_bits(); - sign = (ix>>31) != 0; - ix &= 0x7fffffff; - if ix >= 0x7f800000 { - return 1.0/(x*x); - } - if ix >= 0x40000000 { /* |x| >= 2 */ - return common(ix, fabsf(x), false, sign); - } - if ix >= 0x39000000 { /* |x| >= 2**-13 */ - z = x*x; - r = z*(R00+z*(R01+z*(R02+z*R03))); - s = 1.0+z*(S01+z*(S02+z*(S03+z*(S04+z*S05)))); - z = 0.5 + r/s; - } else { - z = 0.5; - } - return z*x; -} - -const U0: [f32; 5] = [ - -1.9605709612e-01, /* 0xbe48c331 */ - 5.0443872809e-02, /* 0x3d4e9e3c */ - -1.9125689287e-03, /* 0xbafaaf2a */ - 2.3525259166e-05, /* 0x37c5581c */ - -9.1909917899e-08, /* 0xb3c56003 */ -]; -const V0: [f32; 5] = [ - 1.9916731864e-02, /* 0x3ca3286a */ - 2.0255257550e-04, /* 0x3954644b */ - 1.3560879779e-06, /* 0x35b602d4 */ - 6.2274145840e-09, /* 0x31d5f8eb */ - 1.6655924903e-11, /* 0x2d9281cf */ -]; - -pub fn y1f(x: f32) -> f32 -{ - let z: f32; - let u: f32; - let v: f32; - let ix: u32; - - ix = x.to_bits(); - if (ix & 0x7fffffff) == 0 { - return -1.0/0.0; - } - if (ix>>31) != 0{ - return 0.0/0.0; - } - if ix >= 0x7f800000 { - return 1.0/x; - } - if ix >= 0x40000000 { /* |x| >= 2.0 */ - return common(ix,x,true,false); - } - if ix < 0x33000000 { /* x < 2**-25 */ - return -TPI/x; - } - z = x*x; - u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); - v = 1.0+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); - return x*(u/v) + TPI*(j1f(x)*logf(x)-1.0/x); -} - -/* For x >= 8, the asymptotic expansions of pone is - * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. - * We approximate pone by - * pone(x) = 1 + (R/S) - * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 - * S = 1 + ps0*s^2 + ... + ps4*s^10 - * and - * | pone(x)-1-R/S | <= 2 ** ( -60.06) - */ - -const PR8: [f32; 6] = [ /* for x in [inf, 8]=1/[0,0.125] */ - 0.0000000000e+00, /* 0x00000000 */ - 1.1718750000e-01, /* 0x3df00000 */ - 1.3239480972e+01, /* 0x4153d4ea */ - 4.1205184937e+02, /* 0x43ce06a3 */ - 3.8747453613e+03, /* 0x45722bed */ - 7.9144794922e+03, /* 0x45f753d6 */ -]; -const PS8: [f32; 5] = [ - 1.1420736694e+02, /* 0x42e46a2c */ - 3.6509309082e+03, /* 0x45642ee5 */ - 3.6956207031e+04, /* 0x47105c35 */ - 9.7602796875e+04, /* 0x47bea166 */ - 3.0804271484e+04, /* 0x46f0a88b */ -]; - -const PR5: [f32; 6] = [ /* for x in [8,4.5454]=1/[0.125,0.22001] */ - 1.3199052094e-11, /* 0x2d68333f */ - 1.1718749255e-01, /* 0x3defffff */ - 6.8027510643e+00, /* 0x40d9b023 */ - 1.0830818176e+02, /* 0x42d89dca */ - 5.1763616943e+02, /* 0x440168b7 */ - 5.2871520996e+02, /* 0x44042dc6 */ -]; -const PS5: [f32; 5] = [ - 5.9280597687e+01, /* 0x426d1f55 */ - 9.9140142822e+02, /* 0x4477d9b1 */ - 5.3532670898e+03, /* 0x45a74a23 */ - 7.8446904297e+03, /* 0x45f52586 */ - 1.5040468750e+03, /* 0x44bc0180 */ -]; - -const PR3: [f32; 6] = [ - 3.0250391081e-09, /* 0x314fe10d */ - 1.1718686670e-01, /* 0x3defffab */ - 3.9329774380e+00, /* 0x407bb5e7 */ - 3.5119403839e+01, /* 0x420c7a45 */ - 9.1055007935e+01, /* 0x42b61c2a */ - 4.8559066772e+01, /* 0x42423c7c */ -]; -const PS3: [f32; 5] = [ - 3.4791309357e+01, /* 0x420b2a4d */ - 3.3676245117e+02, /* 0x43a86198 */ - 1.0468714600e+03, /* 0x4482dbe3 */ - 8.9081134033e+02, /* 0x445eb3ed */ - 1.0378793335e+02, /* 0x42cf936c */ -]; - -const PR2: [f32; 6] = [/* for x in [2.8570,2]=1/[0.3499,0.5] */ - 1.0771083225e-07, /* 0x33e74ea8 */ - 1.1717621982e-01, /* 0x3deffa16 */ - 2.3685150146e+00, /* 0x401795c0 */ - 1.2242610931e+01, /* 0x4143e1bc */ - 1.7693971634e+01, /* 0x418d8d41 */ - 5.0735230446e+00, /* 0x40a25a4d */ -]; -const PS2: [f32; 5] = [ - 2.1436485291e+01, /* 0x41ab7dec */ - 1.2529022980e+02, /* 0x42fa9499 */ - 2.3227647400e+02, /* 0x436846c7 */ - 1.1767937469e+02, /* 0x42eb5bd7 */ - 8.3646392822e+00, /* 0x4105d590 */ -]; - -fn ponef(x: f32) -> f32 -{ - let p: &[f32; 6]; - let q: &[f32; 5]; - let z: f32; - let r: f32; - let s: f32; - let mut ix: u32; - - ix = x.to_bits(); - ix &= 0x7fffffff; - if ix >= 0x41000000 {p = &PR8; q = &PS8;} - else if ix >= 0x409173eb {p = &PR5; q = &PS5;} - else if ix >= 0x4036d917 {p = &PR3; q = &PS3;} - else /*ix >= 0x40000000*/{p = &PR2; q = &PS2;} - z = 1.0/(x*x); - r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); - s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); - return 1.0 + r/s; -} - -/* For x >= 8, the asymptotic expansions of qone is - * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. - * We approximate pone by - * qone(x) = s*(0.375 + (R/S)) - * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 - * S = 1 + qs1*s^2 + ... + qs6*s^12 - * and - * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) - */ - -const QR8: [f32; 6] = [ /* for x in [inf, 8]=1/[0,0.125] */ - 0.0000000000e+00, /* 0x00000000 */ - -1.0253906250e-01, /* 0xbdd20000 */ - -1.6271753311e+01, /* 0xc1822c8d */ - -7.5960174561e+02, /* 0xc43de683 */ - -1.1849806641e+04, /* 0xc639273a */ - -4.8438511719e+04, /* 0xc73d3683 */ -]; -const QS8: [f32; 6] = [ - 1.6139537048e+02, /* 0x43216537 */ - 7.8253862305e+03, /* 0x45f48b17 */ - 1.3387534375e+05, /* 0x4802bcd6 */ - 7.1965775000e+05, /* 0x492fb29c */ - 6.6660125000e+05, /* 0x4922be94 */ - -2.9449025000e+05, /* 0xc88fcb48 */ -]; - -const QR5: [f32; 6] = [ /* for x in [8,4.5454]=1/[0.125,0.22001] */ - -2.0897993405e-11, /* 0xadb7d219 */ - -1.0253904760e-01, /* 0xbdd1fffe */ - -8.0564479828e+00, /* 0xc100e736 */ - -1.8366960144e+02, /* 0xc337ab6b */ - -1.3731937256e+03, /* 0xc4aba633 */ - -2.6124443359e+03, /* 0xc523471c */ -]; -const QS5: [f32; 6] = [ - 8.1276550293e+01, /* 0x42a28d98 */ - 1.9917987061e+03, /* 0x44f8f98f */ - 1.7468484375e+04, /* 0x468878f8 */ - 4.9851425781e+04, /* 0x4742bb6d */ - 2.7948074219e+04, /* 0x46da5826 */ - -4.7191835938e+03, /* 0xc5937978 */ -]; - -const QR3: [f32; 6] = [ - -5.0783124372e-09, /* 0xb1ae7d4f */ - -1.0253783315e-01, /* 0xbdd1ff5b */ - -4.6101160049e+00, /* 0xc0938612 */ - -5.7847221375e+01, /* 0xc267638e */ - -2.2824453735e+02, /* 0xc3643e9a */ - -2.1921012878e+02, /* 0xc35b35cb */ -]; -const QS3: [f32; 6] = [ - 4.7665153503e+01, /* 0x423ea91e */ - 6.7386511230e+02, /* 0x4428775e */ - 3.3801528320e+03, /* 0x45534272 */ - 5.5477290039e+03, /* 0x45ad5dd5 */ - 1.9031191406e+03, /* 0x44ede3d0 */ - -1.3520118713e+02, /* 0xc3073381 */ -]; - -const QR2: [f32; 6] = [ /* for x in [2.8570,2]=1/[0.3499,0.5] */ - -1.7838172539e-07, /* 0xb43f8932 */ - -1.0251704603e-01, /* 0xbdd1f475 */ - -2.7522056103e+00, /* 0xc0302423 */ - -1.9663616180e+01, /* 0xc19d4f16 */ - -4.2325313568e+01, /* 0xc2294d1f */ - -2.1371921539e+01, /* 0xc1aaf9b2 */ -]; -const QS2: [f32; 6] = [ - 2.9533363342e+01, /* 0x41ec4454 */ - 2.5298155212e+02, /* 0x437cfb47 */ - 7.5750280762e+02, /* 0x443d602e */ - 7.3939318848e+02, /* 0x4438d92a */ - 1.5594900513e+02, /* 0x431bf2f2 */ - -4.9594988823e+00, /* 0xc09eb437 */ -]; - -fn qonef(x: f32) -> f32 -{ - let p: &[f32; 6]; - let q: &[f32; 6]; - let s: f32; - let r: f32; - let z: f32; - let mut ix: u32; - - ix = x.to_bits(); - ix &= 0x7fffffff; - if ix >= 0x41000000 {p = &QR8; q = &QS8;} - else if ix >= 0x409173eb {p = &QR5; q = &QS5;} - else if ix >= 0x4036d917 {p = &QR3; q = &QS3;} - else /*ix >= 0x40000000*/{p = &QR2; q = &QS2;} - z = 1.0/(x*x); - r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); - s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); - return (0.375 + r/s)/x; -} +/* origin: FreeBSD /usr/src/lib/msun/src/e_j1f.c */ +/* + * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. + */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +use super::{cosf, fabsf, logf, sinf, sqrtf}; + +const INVSQRTPI: f32 = 5.6418961287e-01; /* 0x3f106ebb */ +const TPI: f32 = 6.3661974669e-01; /* 0x3f22f983 */ + +fn common(ix: u32, x: f32, y1: bool, sign: bool) -> f32 { + let z: f64; + let mut s: f64; + let c: f64; + let mut ss: f64; + let mut cc: f64; + + s = sinf(x) as f64; + if y1 { + s = -s; + } + c = cosf(x) as f64; + cc = s - c; + if ix < 0x7f000000 { + ss = -s - c; + z = cosf(2.0 * x) as f64; + if s * c > 0.0 { + cc = z / ss; + } else { + ss = z / cc; + } + if ix < 0x58800000 { + if y1 { + ss = -ss; + } + cc = (ponef(x) as f64) * cc - (qonef(x) as f64) * ss; + } + } + if sign { + cc = -cc; + } + return INVSQRTPI * (cc as f32) / sqrtf(x); +} + +/* R0/S0 on [0,2] */ +const R00: f32 = -6.2500000000e-02; /* 0xbd800000 */ +const R01: f32 = 1.4070566976e-03; /* 0x3ab86cfd */ +const R02: f32 = -1.5995563444e-05; /* 0xb7862e36 */ +const R03: f32 = 4.9672799207e-08; /* 0x335557d2 */ +const S01: f32 = 1.9153760746e-02; /* 0x3c9ce859 */ +const S02: f32 = 1.8594678841e-04; /* 0x3942fab6 */ +const S03: f32 = 1.1771846857e-06; /* 0x359dffc2 */ +const S04: f32 = 5.0463624390e-09; /* 0x31ad6446 */ +const S05: f32 = 1.2354227016e-11; /* 0x2d59567e */ + +pub fn j1f(x: f32) -> f32 { + let mut z: f32; + let r: f32; + let s: f32; + let mut ix: u32; + let sign: bool; + + ix = x.to_bits(); + sign = (ix >> 31) != 0; + ix &= 0x7fffffff; + if ix >= 0x7f800000 { + return 1.0 / (x * x); + } + if ix >= 0x40000000 { + /* |x| >= 2 */ + return common(ix, fabsf(x), false, sign); + } + if ix >= 0x39000000 { + /* |x| >= 2**-13 */ + z = x * x; + r = z * (R00 + z * (R01 + z * (R02 + z * R03))); + s = 1.0 + z * (S01 + z * (S02 + z * (S03 + z * (S04 + z * S05)))); + z = 0.5 + r / s; + } else { + z = 0.5; + } + return z * x; +} + +const U0: [f32; 5] = [ + -1.9605709612e-01, /* 0xbe48c331 */ + 5.0443872809e-02, /* 0x3d4e9e3c */ + -1.9125689287e-03, /* 0xbafaaf2a */ + 2.3525259166e-05, /* 0x37c5581c */ + -9.1909917899e-08, /* 0xb3c56003 */ +]; +const V0: [f32; 5] = [ + 1.9916731864e-02, /* 0x3ca3286a */ + 2.0255257550e-04, /* 0x3954644b */ + 1.3560879779e-06, /* 0x35b602d4 */ + 6.2274145840e-09, /* 0x31d5f8eb */ + 1.6655924903e-11, /* 0x2d9281cf */ +]; + +pub fn y1f(x: f32) -> f32 { + let z: f32; + let u: f32; + let v: f32; + let ix: u32; + + ix = x.to_bits(); + if (ix & 0x7fffffff) == 0 { + return -1.0 / 0.0; + } + if (ix >> 31) != 0 { + return 0.0 / 0.0; + } + if ix >= 0x7f800000 { + return 1.0 / x; + } + if ix >= 0x40000000 { + /* |x| >= 2.0 */ + return common(ix, x, true, false); + } + if ix < 0x33000000 { + /* x < 2**-25 */ + return -TPI / x; + } + z = x * x; + u = U0[0] + z * (U0[1] + z * (U0[2] + z * (U0[3] + z * U0[4]))); + v = 1.0 + z * (V0[0] + z * (V0[1] + z * (V0[2] + z * (V0[3] + z * V0[4])))); + return x * (u / v) + TPI * (j1f(x) * logf(x) - 1.0 / x); +} + +/* For x >= 8, the asymptotic expansions of pone is + * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. + * We approximate pone by + * pone(x) = 1 + (R/S) + * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 + * S = 1 + ps0*s^2 + ... + ps4*s^10 + * and + * | pone(x)-1-R/S | <= 2 ** ( -60.06) + */ + +const PR8: [f32; 6] = [ + /* for x in [inf, 8]=1/[0,0.125] */ + 0.0000000000e+00, /* 0x00000000 */ + 1.1718750000e-01, /* 0x3df00000 */ + 1.3239480972e+01, /* 0x4153d4ea */ + 4.1205184937e+02, /* 0x43ce06a3 */ + 3.8747453613e+03, /* 0x45722bed */ + 7.9144794922e+03, /* 0x45f753d6 */ +]; +const PS8: [f32; 5] = [ + 1.1420736694e+02, /* 0x42e46a2c */ + 3.6509309082e+03, /* 0x45642ee5 */ + 3.6956207031e+04, /* 0x47105c35 */ + 9.7602796875e+04, /* 0x47bea166 */ + 3.0804271484e+04, /* 0x46f0a88b */ +]; + +const PR5: [f32; 6] = [ + /* for x in [8,4.5454]=1/[0.125,0.22001] */ + 1.3199052094e-11, /* 0x2d68333f */ + 1.1718749255e-01, /* 0x3defffff */ + 6.8027510643e+00, /* 0x40d9b023 */ + 1.0830818176e+02, /* 0x42d89dca */ + 5.1763616943e+02, /* 0x440168b7 */ + 5.2871520996e+02, /* 0x44042dc6 */ +]; +const PS5: [f32; 5] = [ + 5.9280597687e+01, /* 0x426d1f55 */ + 9.9140142822e+02, /* 0x4477d9b1 */ + 5.3532670898e+03, /* 0x45a74a23 */ + 7.8446904297e+03, /* 0x45f52586 */ + 1.5040468750e+03, /* 0x44bc0180 */ +]; + +const PR3: [f32; 6] = [ + 3.0250391081e-09, /* 0x314fe10d */ + 1.1718686670e-01, /* 0x3defffab */ + 3.9329774380e+00, /* 0x407bb5e7 */ + 3.5119403839e+01, /* 0x420c7a45 */ + 9.1055007935e+01, /* 0x42b61c2a */ + 4.8559066772e+01, /* 0x42423c7c */ +]; +const PS3: [f32; 5] = [ + 3.4791309357e+01, /* 0x420b2a4d */ + 3.3676245117e+02, /* 0x43a86198 */ + 1.0468714600e+03, /* 0x4482dbe3 */ + 8.9081134033e+02, /* 0x445eb3ed */ + 1.0378793335e+02, /* 0x42cf936c */ +]; + +const PR2: [f32; 6] = [ + /* for x in [2.8570,2]=1/[0.3499,0.5] */ + 1.0771083225e-07, /* 0x33e74ea8 */ + 1.1717621982e-01, /* 0x3deffa16 */ + 2.3685150146e+00, /* 0x401795c0 */ + 1.2242610931e+01, /* 0x4143e1bc */ + 1.7693971634e+01, /* 0x418d8d41 */ + 5.0735230446e+00, /* 0x40a25a4d */ +]; +const PS2: [f32; 5] = [ + 2.1436485291e+01, /* 0x41ab7dec */ + 1.2529022980e+02, /* 0x42fa9499 */ + 2.3227647400e+02, /* 0x436846c7 */ + 1.1767937469e+02, /* 0x42eb5bd7 */ + 8.3646392822e+00, /* 0x4105d590 */ +]; + +fn ponef(x: f32) -> f32 { + let p: &[f32; 6]; + let q: &[f32; 5]; + let z: f32; + let r: f32; + let s: f32; + let mut ix: u32; + + ix = x.to_bits(); + ix &= 0x7fffffff; + if ix >= 0x41000000 { + p = &PR8; + q = &PS8; + } else if ix >= 0x409173eb { + p = &PR5; + q = &PS5; + } else if ix >= 0x4036d917 { + p = &PR3; + q = &PS3; + } else + /*ix >= 0x40000000*/ + { + p = &PR2; + q = &PS2; + } + z = 1.0 / (x * x); + r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5])))); + s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4])))); + return 1.0 + r / s; +} + +/* For x >= 8, the asymptotic expansions of qone is + * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. + * We approximate pone by + * qone(x) = s*(0.375 + (R/S)) + * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 + * S = 1 + qs1*s^2 + ... + qs6*s^12 + * and + * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) + */ + +const QR8: [f32; 6] = [ + /* for x in [inf, 8]=1/[0,0.125] */ + 0.0000000000e+00, /* 0x00000000 */ + -1.0253906250e-01, /* 0xbdd20000 */ + -1.6271753311e+01, /* 0xc1822c8d */ + -7.5960174561e+02, /* 0xc43de683 */ + -1.1849806641e+04, /* 0xc639273a */ + -4.8438511719e+04, /* 0xc73d3683 */ +]; +const QS8: [f32; 6] = [ + 1.6139537048e+02, /* 0x43216537 */ + 7.8253862305e+03, /* 0x45f48b17 */ + 1.3387534375e+05, /* 0x4802bcd6 */ + 7.1965775000e+05, /* 0x492fb29c */ + 6.6660125000e+05, /* 0x4922be94 */ + -2.9449025000e+05, /* 0xc88fcb48 */ +]; + +const QR5: [f32; 6] = [ + /* for x in [8,4.5454]=1/[0.125,0.22001] */ + -2.0897993405e-11, /* 0xadb7d219 */ + -1.0253904760e-01, /* 0xbdd1fffe */ + -8.0564479828e+00, /* 0xc100e736 */ + -1.8366960144e+02, /* 0xc337ab6b */ + -1.3731937256e+03, /* 0xc4aba633 */ + -2.6124443359e+03, /* 0xc523471c */ +]; +const QS5: [f32; 6] = [ + 8.1276550293e+01, /* 0x42a28d98 */ + 1.9917987061e+03, /* 0x44f8f98f */ + 1.7468484375e+04, /* 0x468878f8 */ + 4.9851425781e+04, /* 0x4742bb6d */ + 2.7948074219e+04, /* 0x46da5826 */ + -4.7191835938e+03, /* 0xc5937978 */ +]; + +const QR3: [f32; 6] = [ + -5.0783124372e-09, /* 0xb1ae7d4f */ + -1.0253783315e-01, /* 0xbdd1ff5b */ + -4.6101160049e+00, /* 0xc0938612 */ + -5.7847221375e+01, /* 0xc267638e */ + -2.2824453735e+02, /* 0xc3643e9a */ + -2.1921012878e+02, /* 0xc35b35cb */ +]; +const QS3: [f32; 6] = [ + 4.7665153503e+01, /* 0x423ea91e */ + 6.7386511230e+02, /* 0x4428775e */ + 3.3801528320e+03, /* 0x45534272 */ + 5.5477290039e+03, /* 0x45ad5dd5 */ + 1.9031191406e+03, /* 0x44ede3d0 */ + -1.3520118713e+02, /* 0xc3073381 */ +]; + +const QR2: [f32; 6] = [ + /* for x in [2.8570,2]=1/[0.3499,0.5] */ + -1.7838172539e-07, /* 0xb43f8932 */ + -1.0251704603e-01, /* 0xbdd1f475 */ + -2.7522056103e+00, /* 0xc0302423 */ + -1.9663616180e+01, /* 0xc19d4f16 */ + -4.2325313568e+01, /* 0xc2294d1f */ + -2.1371921539e+01, /* 0xc1aaf9b2 */ +]; +const QS2: [f32; 6] = [ + 2.9533363342e+01, /* 0x41ec4454 */ + 2.5298155212e+02, /* 0x437cfb47 */ + 7.5750280762e+02, /* 0x443d602e */ + 7.3939318848e+02, /* 0x4438d92a */ + 1.5594900513e+02, /* 0x431bf2f2 */ + -4.9594988823e+00, /* 0xc09eb437 */ +]; + +fn qonef(x: f32) -> f32 { + let p: &[f32; 6]; + let q: &[f32; 6]; + let s: f32; + let r: f32; + let z: f32; + let mut ix: u32; + + ix = x.to_bits(); + ix &= 0x7fffffff; + if ix >= 0x41000000 { + p = &QR8; + q = &QS8; + } else if ix >= 0x409173eb { + p = &QR5; + q = &QS5; + } else if ix >= 0x4036d917 { + p = &QR3; + q = &QS3; + } else + /*ix >= 0x40000000*/ + { + p = &QR2; + q = &QS2; + } + z = 1.0 / (x * x); + r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5])))); + s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5]))))); + return (0.375 + r / s) / x; +} diff --git a/src/math/jn.rs b/src/math/jn.rs index 7f7c06f..70c9802 100644 --- a/src/math/jn.rs +++ b/src/math/jn.rs @@ -1,338 +1,343 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/e_jn.c */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunSoft, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ -/* - * jn(n, x), yn(n, x) - * floating point Bessel's function of the 1st and 2nd kind - * of order n - * - * Special cases: - * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; - * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. - * Note 2. About jn(n,x), yn(n,x) - * For n=0, j0(x) is called, - * for n=1, j1(x) is called, - * for n<=x, forward recursion is used starting - * from values of j0(x) and j1(x). - * for n>x, a continued fraction approximation to - * j(n,x)/j(n-1,x) is evaluated and then backward - * recursion is used starting from a supposed value - * for j(n,x). The resulting value of j(0,x) is - * compared with the actual value to correct the - * supposed value of j(n,x). - * - * yn(n,x) is similar in all respects, except - * that forward recursion is used for all - * values of n>1. - */ - -use super::{cos, fabs, get_high_word, get_low_word, log, j0, j1, sin, sqrt, y0, y1}; - -const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */ - -pub fn jn(n: isize, mut x: f64) -> f64 -{ - let mut ix: u32; - let lx: u32; - let nm1: isize; - let mut i: isize; - let mut sign: bool; - let mut a: f64; - let mut b: f64; - let mut temp: f64; - - ix = get_high_word(x); - lx = get_low_word(x); - sign = (ix>>31) != 0; - ix &= 0x7fffffff; - - // -lx == !lx + 1 - if (ix | (lx|(!lx+1))>>31) > 0x7ff00000 { /* nan */ - return x; - } - - /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) - * Thus, J(-n,x) = J(n,-x) - */ - /* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */ - if n == 0 { - return j0(x); - } - if n < 0 { - nm1 = -(n+1); - x = -x; - sign = !sign; - } else { - nm1 = n-1; - } - if nm1 == 0 { - return j1(x); - } - - sign &= (n & 1) != 0; /* even n: 0, odd n: signbit(x) */ - x = fabs(x); - if (ix|lx) == 0 || ix == 0x7ff00000 { /* if x is 0 or inf */ - b = 0.0; - } else if (nm1 as f64) < x { - /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ - if ix >= 0x52d00000 { /* x > 2**302 */ - /* (x >> n**2) - * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) - * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) - * Let s=sin(x), c=cos(x), - * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then - * - * n sin(xn)*sqt2 cos(xn)*sqt2 - * ---------------------------------- - * 0 s-c c+s - * 1 -s-c -c+s - * 2 -s+c -c-s - * 3 s+c c-s - */ - temp = match nm1&3 { - 0 => -cos(x)+sin(x), - 1 => -cos(x)-sin(x), - 2 => cos(x)-sin(x), - 3 | _ => cos(x)+sin(x), - }; - b = INVSQRTPI*temp/sqrt(x); - } else { - a = j0(x); - b = j1(x); - i = 0; - while i < nm1 { - i += 1; - temp = b; - b = b*(2.0*(i as f64)/x) - a; /* avoid underflow */ - a = temp; - } - } - } else { - if ix < 0x3e100000 { /* x < 2**-29 */ - /* x is tiny, return the first Taylor expansion of J(n,x) - * J(n,x) = 1/n!*(x/2)^n - ... - */ - if nm1 > 32 { /* underflow */ - b = 0.0; - } else { - temp = x*0.5; - b = temp; - a = 1.0; - i = 2; - while i <= nm1 + 1 { - a *= i as f64; /* a = n! */ - b *= temp; /* b = (x/2)^n */ - i += 1; - } - b = b/a; - } - } else { - /* use backward recurrence */ - /* x x^2 x^2 - * J(n,x)/J(n-1,x) = ---- ------ ------ ..... - * 2n - 2(n+1) - 2(n+2) - * - * 1 1 1 - * (for large x) = ---- ------ ------ ..... - * 2n 2(n+1) 2(n+2) - * -- - ------ - ------ - - * x x x - * - * Let w = 2n/x and h=2/x, then the above quotient - * is equal to the continued fraction: - * 1 - * = ----------------------- - * 1 - * w - ----------------- - * 1 - * w+h - --------- - * w+2h - ... - * - * To determine how many terms needed, let - * Q(0) = w, Q(1) = w(w+h) - 1, - * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), - * When Q(k) > 1e4 good for single - * When Q(k) > 1e9 good for double - * When Q(k) > 1e17 good for quadruple - */ - /* determine k */ - let mut t: f64; - let mut q0: f64; - let mut q1: f64; - let mut w: f64; - let h: f64; - let mut z: f64; - let mut tmp: f64; - let nf: f64; - - let mut k: isize; - - nf = (nm1 as f64) + 1.0; - w = 2.0*nf/x; - h = 2.0/x; - z = w+h; - q0 = w; - q1 = w*z - 1.0; - k = 1; - while q1 < 1.0e9 { - k += 1; - z += h; - tmp = z*q1 - q0; - q0 = q1; - q1 = tmp; - } - t = 0.0; - i = k; - while i >= 0 { - t = 1.0/(2.0*((i as f64)+nf)/x - t); - i -= 1; - } - a = t; - b = 1.0; - /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) - * Hence, if n*(log(2n/x)) > ... - * single 8.8722839355e+01 - * double 7.09782712893383973096e+02 - * long double 1.1356523406294143949491931077970765006170e+04 - * then recurrent value may overflow and the result is - * likely underflow to zero - */ - tmp = nf*log(fabs(w)); - if tmp < 7.09782712893383973096e+02 { - i = nm1; - while i > 0 { - temp = b; - b = b*(2.0*(i as f64))/x - a; - a = temp; - i -= 1; - } - } else { - i = nm1; - while i > 0 { - temp = b; - b = b*(2.0*(i as f64))/x - a; - a = temp; - /* scale b to avoid spurious overflow */ - let x1p500 = f64::from_bits(0x5f30000000000000); // 0x1p500 == 2^500 - if b > x1p500 { - a /= b; - t /= b; - b = 1.0; - } - i -= 1; - } - } - z = j0(x); - w = j1(x); - if fabs(z) >= fabs(w) { - b = t*z/b; - } else { - b = t*w/a; - } - } - } - - if sign { - -b - } else { - b - } -} - - -pub fn yn(n: isize, x: f64) -> f64 -{ - let mut ix: u32; - let lx: u32; - let mut ib: u32; - let nm1: isize; - let mut sign: bool; - let mut i: isize; - let mut a: f64; - let mut b: f64; - let mut temp: f64; - - ix = get_high_word(x); - lx = get_low_word(x); - sign = (ix>>31) != 0; - ix &= 0x7fffffff; - - // -lx == !lx + 1 - if (ix | (lx|(!lx+1))>>31) > 0x7ff00000 { /* nan */ - return x; - } - if sign && (ix|lx) != 0 { /* x < 0 */ - return 0.0/0.0; - } - if ix == 0x7ff00000 { - return 0.0; - } - - if n == 0 { - return y0(x); - } - if n < 0 { - nm1 = -(n+1); - sign = (n&1) != 0; - } else { - nm1 = n-1; - sign = false; - } - if nm1 == 0 { - if sign { - return -y1(x); - } else { - return y1(x); - } - } - - if ix >= 0x52d00000 { /* x > 2**302 */ - /* (x >> n**2) - * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) - * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) - * Let s=sin(x), c=cos(x), - * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then - * - * n sin(xn)*sqt2 cos(xn)*sqt2 - * ---------------------------------- - * 0 s-c c+s - * 1 -s-c -c+s - * 2 -s+c -c-s - * 3 s+c c-s - */ - temp = match nm1&3 { - 0 => -sin(x)-cos(x), - 1 => -sin(x)+cos(x), - 2 => sin(x)+cos(x), - 3 | _ => sin(x)-cos(x), - }; - b = INVSQRTPI*temp/sqrt(x); - } else { - a = y0(x); - b = y1(x); - /* quit if b is -inf */ - ib = get_high_word(b); - i = 0; - while i < nm1 && ib != 0xfff00000 { - i += 1; - temp = b; - b = (2.0*(i as f64)/x)*b - a; - ib = get_high_word(b); - a = temp; - } - } - - if sign { - -b - } else { - b - } -} +/* origin: FreeBSD /usr/src/lib/msun/src/e_jn.c */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ +/* + * jn(n, x), yn(n, x) + * floating point Bessel's function of the 1st and 2nd kind + * of order n + * + * Special cases: + * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; + * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. + * Note 2. About jn(n,x), yn(n,x) + * For n=0, j0(x) is called, + * for n=1, j1(x) is called, + * for n<=x, forward recursion is used starting + * from values of j0(x) and j1(x). + * for n>x, a continued fraction approximation to + * j(n,x)/j(n-1,x) is evaluated and then backward + * recursion is used starting from a supposed value + * for j(n,x). The resulting value of j(0,x) is + * compared with the actual value to correct the + * supposed value of j(n,x). + * + * yn(n,x) is similar in all respects, except + * that forward recursion is used for all + * values of n>1. + */ + +use super::{cos, fabs, get_high_word, get_low_word, j0, j1, log, sin, sqrt, y0, y1}; + +const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */ + +pub fn jn(n: i32, mut x: f64) -> f64 { + let mut ix: u32; + let lx: u32; + let nm1: i32; + let mut i: i32; + let mut sign: bool; + let mut a: f64; + let mut b: f64; + let mut temp: f64; + + ix = get_high_word(x); + lx = get_low_word(x); + sign = (ix >> 31) != 0; + ix &= 0x7fffffff; + + // -lx == !lx + 1 + if (ix | (lx | (!lx + 1)) >> 31) > 0x7ff00000 { + /* nan */ + return x; + } + + /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) + * Thus, J(-n,x) = J(n,-x) + */ + /* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */ + if n == 0 { + return j0(x); + } + if n < 0 { + nm1 = -(n + 1); + x = -x; + sign = !sign; + } else { + nm1 = n - 1; + } + if nm1 == 0 { + return j1(x); + } + + sign &= (n & 1) != 0; /* even n: 0, odd n: signbit(x) */ + x = fabs(x); + if (ix | lx) == 0 || ix == 0x7ff00000 { + /* if x is 0 or inf */ + b = 0.0; + } else if (nm1 as f64) < x { + /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ + if ix >= 0x52d00000 { + /* x > 2**302 */ + /* (x >> n**2) + * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Let s=sin(x), c=cos(x), + * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then + * + * n sin(xn)*sqt2 cos(xn)*sqt2 + * ---------------------------------- + * 0 s-c c+s + * 1 -s-c -c+s + * 2 -s+c -c-s + * 3 s+c c-s + */ + temp = match nm1 & 3 { + 0 => -cos(x) + sin(x), + 1 => -cos(x) - sin(x), + 2 => cos(x) - sin(x), + 3 | _ => cos(x) + sin(x), + }; + b = INVSQRTPI * temp / sqrt(x); + } else { + a = j0(x); + b = j1(x); + i = 0; + while i < nm1 { + i += 1; + temp = b; + b = b * (2.0 * (i as f64) / x) - a; /* avoid underflow */ + a = temp; + } + } + } else { + if ix < 0x3e100000 { + /* x < 2**-29 */ + /* x is tiny, return the first Taylor expansion of J(n,x) + * J(n,x) = 1/n!*(x/2)^n - ... + */ + if nm1 > 32 { + /* underflow */ + b = 0.0; + } else { + temp = x * 0.5; + b = temp; + a = 1.0; + i = 2; + while i <= nm1 + 1 { + a *= i as f64; /* a = n! */ + b *= temp; /* b = (x/2)^n */ + i += 1; + } + b = b / a; + } + } else { + /* use backward recurrence */ + /* x x^2 x^2 + * J(n,x)/J(n-1,x) = ---- ------ ------ ..... + * 2n - 2(n+1) - 2(n+2) + * + * 1 1 1 + * (for large x) = ---- ------ ------ ..... + * 2n 2(n+1) 2(n+2) + * -- - ------ - ------ - + * x x x + * + * Let w = 2n/x and h=2/x, then the above quotient + * is equal to the continued fraction: + * 1 + * = ----------------------- + * 1 + * w - ----------------- + * 1 + * w+h - --------- + * w+2h - ... + * + * To determine how many terms needed, let + * Q(0) = w, Q(1) = w(w+h) - 1, + * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), + * When Q(k) > 1e4 good for single + * When Q(k) > 1e9 good for double + * When Q(k) > 1e17 good for quadruple + */ + /* determine k */ + let mut t: f64; + let mut q0: f64; + let mut q1: f64; + let mut w: f64; + let h: f64; + let mut z: f64; + let mut tmp: f64; + let nf: f64; + + let mut k: i32; + + nf = (nm1 as f64) + 1.0; + w = 2.0 * nf / x; + h = 2.0 / x; + z = w + h; + q0 = w; + q1 = w * z - 1.0; + k = 1; + while q1 < 1.0e9 { + k += 1; + z += h; + tmp = z * q1 - q0; + q0 = q1; + q1 = tmp; + } + t = 0.0; + i = k; + while i >= 0 { + t = 1.0 / (2.0 * ((i as f64) + nf) / x - t); + i -= 1; + } + a = t; + b = 1.0; + /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) + * Hence, if n*(log(2n/x)) > ... + * single 8.8722839355e+01 + * double 7.09782712893383973096e+02 + * long double 1.1356523406294143949491931077970765006170e+04 + * then recurrent value may overflow and the result is + * likely underflow to zero + */ + tmp = nf * log(fabs(w)); + if tmp < 7.09782712893383973096e+02 { + i = nm1; + while i > 0 { + temp = b; + b = b * (2.0 * (i as f64)) / x - a; + a = temp; + i -= 1; + } + } else { + i = nm1; + while i > 0 { + temp = b; + b = b * (2.0 * (i as f64)) / x - a; + a = temp; + /* scale b to avoid spurious overflow */ + let x1p500 = f64::from_bits(0x5f30000000000000); // 0x1p500 == 2^500 + if b > x1p500 { + a /= b; + t /= b; + b = 1.0; + } + i -= 1; + } + } + z = j0(x); + w = j1(x); + if fabs(z) >= fabs(w) { + b = t * z / b; + } else { + b = t * w / a; + } + } + } + + if sign { + -b + } else { + b + } +} + +pub fn yn(n: i32, x: f64) -> f64 { + let mut ix: u32; + let lx: u32; + let mut ib: u32; + let nm1: i32; + let mut sign: bool; + let mut i: i32; + let mut a: f64; + let mut b: f64; + let mut temp: f64; + + ix = get_high_word(x); + lx = get_low_word(x); + sign = (ix >> 31) != 0; + ix &= 0x7fffffff; + + // -lx == !lx + 1 + if (ix | (lx | (!lx + 1)) >> 31) > 0x7ff00000 { + /* nan */ + return x; + } + if sign && (ix | lx) != 0 { + /* x < 0 */ + return 0.0 / 0.0; + } + if ix == 0x7ff00000 { + return 0.0; + } + + if n == 0 { + return y0(x); + } + if n < 0 { + nm1 = -(n + 1); + sign = (n & 1) != 0; + } else { + nm1 = n - 1; + sign = false; + } + if nm1 == 0 { + if sign { + return -y1(x); + } else { + return y1(x); + } + } + + if ix >= 0x52d00000 { + /* x > 2**302 */ + /* (x >> n**2) + * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Let s=sin(x), c=cos(x), + * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then + * + * n sin(xn)*sqt2 cos(xn)*sqt2 + * ---------------------------------- + * 0 s-c c+s + * 1 -s-c -c+s + * 2 -s+c -c-s + * 3 s+c c-s + */ + temp = match nm1 & 3 { + 0 => -sin(x) - cos(x), + 1 => -sin(x) + cos(x), + 2 => sin(x) + cos(x), + 3 | _ => sin(x) - cos(x), + }; + b = INVSQRTPI * temp / sqrt(x); + } else { + a = y0(x); + b = y1(x); + /* quit if b is -inf */ + ib = get_high_word(b); + i = 0; + while i < nm1 && ib != 0xfff00000 { + i += 1; + temp = b; + b = (2.0 * (i as f64) / x) * b - a; + ib = get_high_word(b); + a = temp; + } + } + + if sign { + -b + } else { + b + } +} diff --git a/src/math/jnf.rs b/src/math/jnf.rs index 4cd848a..360f62e 100644 --- a/src/math/jnf.rs +++ b/src/math/jnf.rs @@ -1,255 +1,259 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/e_jnf.c */ -/* - * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. - */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -use super::{fabsf, j0f, j1f, logf, y0f, y1f}; - -pub fn jnf(n: isize, mut x: f32) -> f32 -{ - let mut ix: u32; - let mut nm1: isize; - let mut sign: bool; - let mut i: isize; - let mut a: f32; - let mut b: f32; - let mut temp: f32; - - ix = x.to_bits(); - sign = (ix>>31) != 0; - ix &= 0x7fffffff; - if ix > 0x7f800000 { /* nan */ - return x; - } - - /* J(-n,x) = J(n,-x), use |n|-1 to avoid overflow in -n */ - if n == 0 { - return j0f(x); - } - if n < 0 { - nm1 = -(n+1); - x = -x; - sign = !sign; - } else { - nm1 = n-1; - } - if nm1 == 0 { - return j1f(x); - } - - sign &= (n&1) != 0; /* even n: 0, odd n: signbit(x) */ - x = fabsf(x); - if ix == 0 || ix == 0x7f800000 { /* if x is 0 or inf */ - b = 0.0; - } else if (nm1 as f32) < x { - /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ - a = j0f(x); - b = j1f(x); - i = 0; - while i < nm1 { - i += 1; - temp = b; - b = b*(2.0*(i as f32)/x) - a; - a = temp; - } - } else { - if ix < 0x35800000 { /* x < 2**-20 */ - /* x is tiny, return the first Taylor expansion of J(n,x) - * J(n,x) = 1/n!*(x/2)^n - ... - */ - if nm1 > 8 { /* underflow */ - nm1 = 8; - } - temp = 0.5 * x; - b = temp; - a = 1.0; - i = 2; - while i <= nm1 + 1 { - a *= i as f32; /* a = n! */ - b *= temp; /* b = (x/2)^n */ - i += 1; - } - b = b/a; - } else { - /* use backward recurrence */ - /* x x^2 x^2 - * J(n,x)/J(n-1,x) = ---- ------ ------ ..... - * 2n - 2(n+1) - 2(n+2) - * - * 1 1 1 - * (for large x) = ---- ------ ------ ..... - * 2n 2(n+1) 2(n+2) - * -- - ------ - ------ - - * x x x - * - * Let w = 2n/x and h=2/x, then the above quotient - * is equal to the continued fraction: - * 1 - * = ----------------------- - * 1 - * w - ----------------- - * 1 - * w+h - --------- - * w+2h - ... - * - * To determine how many terms needed, let - * Q(0) = w, Q(1) = w(w+h) - 1, - * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), - * When Q(k) > 1e4 good for single - * When Q(k) > 1e9 good for double - * When Q(k) > 1e17 good for quadruple - */ - /* determine k */ - let mut t: f32; - let mut q0: f32; - let mut q1: f32; - let mut w: f32; - let h: f32; - let mut z: f32; - let mut tmp: f32; - let nf: f32; - let mut k: isize; - - nf = (nm1 as f32)+1.0; - w = 2.0*(nf as f32)/x; - h = 2.0/x; - z = w+h; - q0 = w; - q1 = w*z - 1.0; - k = 1; - while q1 < 1.0e4 { - k += 1; - z += h; - tmp = z*q1 - q0; - q0 = q1; - q1 = tmp; - } - t = 0.0; - i = k; - while i >= 0 { - t = 1.0/(2.0*((i as f32)+nf)/x-t); - i -= 1; - } - a = t; - b = 1.0; - /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) - * Hence, if n*(log(2n/x)) > ... - * single 8.8722839355e+01 - * double 7.09782712893383973096e+02 - * long double 1.1356523406294143949491931077970765006170e+04 - * then recurrent value may overflow and the result is - * likely underflow to zero - */ - tmp = nf*logf(fabsf(w)); - if tmp < 88.721679688 { - i = nm1; - while i > 0 { - temp = b; - b = 2.0*(i as f32)*b/x - a; - a = temp; - i -= 1; - } - } else { - i = nm1; - while i > 0 { - temp = b; - b = 2.0*(i as f32)*b/x - a; - a = temp; - /* scale b to avoid spurious overflow */ - let x1p60 = f32::from_bits(0x5d800000); // 0x1p60 == 2^60 - if b > x1p60 { - a /= b; - t /= b; - b = 1.0; - } - i -= 1; - } - } - z = j0f(x); - w = j1f(x); - if fabsf(z) >= fabsf(w) { - b = t*z/b; - } else { - b = t*w/a; - } - } - } - - if sign { - -b - } else { - b - } -} - -pub fn ynf(n: isize, x: f32) -> f32 -{ - let mut ix: u32; - let mut ib: u32; - let nm1: isize; - let mut sign: bool; - let mut i: isize; - let mut a: f32; - let mut b: f32; - let mut temp: f32; - - ix = x.to_bits(); - sign = (ix>>31) != 0; - ix &= 0x7fffffff; - if ix > 0x7f800000 { /* nan */ - return x; - } - if sign && ix != 0 { /* x < 0 */ - return 0.0/0.0; - } - if ix == 0x7f800000 { - return 0.0; - } - - if n == 0 { - return y0f(x); - } - if n < 0 { - nm1 = -(n+1); - sign = (n&1) != 0; - } else { - nm1 = n-1; - sign = false; - } - if nm1 == 0 { - if sign { - return -y1f(x); - } else { - return y1f(x); - } - } - - a = y0f(x); - b = y1f(x); - /* quit if b is -inf */ - ib = b.to_bits(); - i = 0; - while i < nm1 && ib != 0xff800000 { - i += 1; - temp = b; - b = (2.0*(i as f32)/x)*b - a; - ib = b.to_bits(); - a = temp; - } - - if sign { - -b - } else { - b - } -} +/* origin: FreeBSD /usr/src/lib/msun/src/e_jnf.c */ +/* + * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. + */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +use super::{fabsf, j0f, j1f, logf, y0f, y1f}; + +pub fn jnf(n: i32, mut x: f32) -> f32 { + let mut ix: u32; + let mut nm1: i32; + let mut sign: bool; + let mut i: i32; + let mut a: f32; + let mut b: f32; + let mut temp: f32; + + ix = x.to_bits(); + sign = (ix >> 31) != 0; + ix &= 0x7fffffff; + if ix > 0x7f800000 { + /* nan */ + return x; + } + + /* J(-n,x) = J(n,-x), use |n|-1 to avoid overflow in -n */ + if n == 0 { + return j0f(x); + } + if n < 0 { + nm1 = -(n + 1); + x = -x; + sign = !sign; + } else { + nm1 = n - 1; + } + if nm1 == 0 { + return j1f(x); + } + + sign &= (n & 1) != 0; /* even n: 0, odd n: signbit(x) */ + x = fabsf(x); + if ix == 0 || ix == 0x7f800000 { + /* if x is 0 or inf */ + b = 0.0; + } else if (nm1 as f32) < x { + /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ + a = j0f(x); + b = j1f(x); + i = 0; + while i < nm1 { + i += 1; + temp = b; + b = b * (2.0 * (i as f32) / x) - a; + a = temp; + } + } else { + if ix < 0x35800000 { + /* x < 2**-20 */ + /* x is tiny, return the first Taylor expansion of J(n,x) + * J(n,x) = 1/n!*(x/2)^n - ... + */ + if nm1 > 8 { + /* underflow */ + nm1 = 8; + } + temp = 0.5 * x; + b = temp; + a = 1.0; + i = 2; + while i <= nm1 + 1 { + a *= i as f32; /* a = n! */ + b *= temp; /* b = (x/2)^n */ + i += 1; + } + b = b / a; + } else { + /* use backward recurrence */ + /* x x^2 x^2 + * J(n,x)/J(n-1,x) = ---- ------ ------ ..... + * 2n - 2(n+1) - 2(n+2) + * + * 1 1 1 + * (for large x) = ---- ------ ------ ..... + * 2n 2(n+1) 2(n+2) + * -- - ------ - ------ - + * x x x + * + * Let w = 2n/x and h=2/x, then the above quotient + * is equal to the continued fraction: + * 1 + * = ----------------------- + * 1 + * w - ----------------- + * 1 + * w+h - --------- + * w+2h - ... + * + * To determine how many terms needed, let + * Q(0) = w, Q(1) = w(w+h) - 1, + * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), + * When Q(k) > 1e4 good for single + * When Q(k) > 1e9 good for double + * When Q(k) > 1e17 good for quadruple + */ + /* determine k */ + let mut t: f32; + let mut q0: f32; + let mut q1: f32; + let mut w: f32; + let h: f32; + let mut z: f32; + let mut tmp: f32; + let nf: f32; + let mut k: i32; + + nf = (nm1 as f32) + 1.0; + w = 2.0 * (nf as f32) / x; + h = 2.0 / x; + z = w + h; + q0 = w; + q1 = w * z - 1.0; + k = 1; + while q1 < 1.0e4 { + k += 1; + z += h; + tmp = z * q1 - q0; + q0 = q1; + q1 = tmp; + } + t = 0.0; + i = k; + while i >= 0 { + t = 1.0 / (2.0 * ((i as f32) + nf) / x - t); + i -= 1; + } + a = t; + b = 1.0; + /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) + * Hence, if n*(log(2n/x)) > ... + * single 8.8722839355e+01 + * double 7.09782712893383973096e+02 + * long double 1.1356523406294143949491931077970765006170e+04 + * then recurrent value may overflow and the result is + * likely underflow to zero + */ + tmp = nf * logf(fabsf(w)); + if tmp < 88.721679688 { + i = nm1; + while i > 0 { + temp = b; + b = 2.0 * (i as f32) * b / x - a; + a = temp; + i -= 1; + } + } else { + i = nm1; + while i > 0 { + temp = b; + b = 2.0 * (i as f32) * b / x - a; + a = temp; + /* scale b to avoid spurious overflow */ + let x1p60 = f32::from_bits(0x5d800000); // 0x1p60 == 2^60 + if b > x1p60 { + a /= b; + t /= b; + b = 1.0; + } + i -= 1; + } + } + z = j0f(x); + w = j1f(x); + if fabsf(z) >= fabsf(w) { + b = t * z / b; + } else { + b = t * w / a; + } + } + } + + if sign { + -b + } else { + b + } +} + +pub fn ynf(n: i32, x: f32) -> f32 { + let mut ix: u32; + let mut ib: u32; + let nm1: i32; + let mut sign: bool; + let mut i: i32; + let mut a: f32; + let mut b: f32; + let mut temp: f32; + + ix = x.to_bits(); + sign = (ix >> 31) != 0; + ix &= 0x7fffffff; + if ix > 0x7f800000 { + /* nan */ + return x; + } + if sign && ix != 0 { + /* x < 0 */ + return 0.0 / 0.0; + } + if ix == 0x7f800000 { + return 0.0; + } + + if n == 0 { + return y0f(x); + } + if n < 0 { + nm1 = -(n + 1); + sign = (n & 1) != 0; + } else { + nm1 = n - 1; + sign = false; + } + if nm1 == 0 { + if sign { + return -y1f(x); + } else { + return y1f(x); + } + } + + a = y0f(x); + b = y1f(x); + /* quit if b is -inf */ + ib = b.to_bits(); + i = 0; + while i < nm1 && ib != 0xff800000 { + i += 1; + temp = b; + b = (2.0 * (i as f32) / x) * b - a; + ib = b.to_bits(); + a = temp; + } + + if sign { + -b + } else { + b + } +} diff --git a/src/math/lgamma.rs b/src/math/lgamma.rs index 35b2526..b1a321e 100644 --- a/src/math/lgamma.rs +++ b/src/math/lgamma.rs @@ -1,309 +1,323 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunSoft, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - * - */ -/* lgamma_r(x, signgamp) - * Reentrant version of the logarithm of the Gamma function - * with user provide pointer for the sign of Gamma(x). - * - * Method: - * 1. Argument Reduction for 0 < x <= 8 - * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may - * reduce x to a number in [1.5,2.5] by - * lgamma(1+s) = log(s) + lgamma(s) - * for example, - * lgamma(7.3) = log(6.3) + lgamma(6.3) - * = log(6.3*5.3) + lgamma(5.3) - * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) - * 2. Polynomial approximation of lgamma around its - * minimun ymin=1.461632144968362245 to maintain monotonicity. - * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use - * Let z = x-ymin; - * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) - * where - * poly(z) is a 14 degree polynomial. - * 2. Rational approximation in the primary interval [2,3] - * We use the following approximation: - * s = x-2.0; - * lgamma(x) = 0.5*s + s*P(s)/Q(s) - * with accuracy - * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 - * Our algorithms are based on the following observation - * - * zeta(2)-1 2 zeta(3)-1 3 - * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... - * 2 3 - * - * where Euler = 0.5771... is the Euler constant, which is very - * close to 0.5. - * - * 3. For x>=8, we have - * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... - * (better formula: - * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) - * Let z = 1/x, then we approximation - * f(z) = lgamma(x) - (x-0.5)(log(x)-1) - * by - * 3 5 11 - * w = w0 + w1*z + w2*z + w3*z + ... + w6*z - * where - * |w - f(z)| < 2**-58.74 - * - * 4. For negative x, since (G is gamma function) - * -x*G(-x)*G(x) = PI/sin(PI*x), - * we have - * G(x) = PI/(sin(PI*x)*(-x)*G(-x)) - * since G(-x) is positive, sign(G(x)) = sign(sin(PI*x)) for x<0 - * Hence, for x<0, signgam = sign(sin(PI*x)) and - * lgamma(x) = log(|Gamma(x)|) - * = log(PI/(|x*sin(PI*x)|)) - lgamma(-x); - * Note: one should avoid compute PI*(-x) directly in the - * computation of sin(PI*(-x)). - * - * 5. Special Cases - * lgamma(2+s) ~ s*(1-Euler) for tiny s - * lgamma(1) = lgamma(2) = 0 - * lgamma(x) ~ -log(|x|) for tiny x - * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero - * lgamma(inf) = inf - * lgamma(-inf) = inf (bug for bug compatible with C99!?) - * - */ - -use super::{floor, k_cos, k_sin, log}; - -const PI: f64 = 3.14159265358979311600e+00; /* 0x400921FB, 0x54442D18 */ -const A0: f64 = 7.72156649015328655494e-02; /* 0x3FB3C467, 0xE37DB0C8 */ -const A1: f64 = 3.22467033424113591611e-01; /* 0x3FD4A34C, 0xC4A60FAD */ -const A2: f64 = 6.73523010531292681824e-02; /* 0x3FB13E00, 0x1A5562A7 */ -const A3: f64 = 2.05808084325167332806e-02; /* 0x3F951322, 0xAC92547B */ -const A4: f64 = 7.38555086081402883957e-03; /* 0x3F7E404F, 0xB68FEFE8 */ -const A5: f64 = 2.89051383673415629091e-03; /* 0x3F67ADD8, 0xCCB7926B */ -const A6: f64 = 1.19270763183362067845e-03; /* 0x3F538A94, 0x116F3F5D */ -const A7: f64 = 5.10069792153511336608e-04; /* 0x3F40B6C6, 0x89B99C00 */ -const A8: f64 = 2.20862790713908385557e-04; /* 0x3F2CF2EC, 0xED10E54D */ -const A9: f64 = 1.08011567247583939954e-04; /* 0x3F1C5088, 0x987DFB07 */ -const A10: f64 = 2.52144565451257326939e-05; /* 0x3EFA7074, 0x428CFA52 */ -const A11: f64 = 4.48640949618915160150e-05; /* 0x3F07858E, 0x90A45837 */ -const TC: f64 = 1.46163214496836224576e+00; /* 0x3FF762D8, 0x6356BE3F */ -const TF: f64 = -1.21486290535849611461e-01; /* 0xBFBF19B9, 0xBCC38A42 */ -/* tt = -(tail of TF) */ -const TT: f64 = -3.63867699703950536541e-18; /* 0xBC50C7CA, 0xA48A971F */ -const T0: f64 = 4.83836122723810047042e-01; /* 0x3FDEF72B, 0xC8EE38A2 */ -const T1: f64 = -1.47587722994593911752e-01; /* 0xBFC2E427, 0x8DC6C509 */ -const T2: f64 = 6.46249402391333854778e-02; /* 0x3FB08B42, 0x94D5419B */ -const T3: f64 = -3.27885410759859649565e-02; /* 0xBFA0C9A8, 0xDF35B713 */ -const T4: f64 = 1.79706750811820387126e-02; /* 0x3F9266E7, 0x970AF9EC */ -const T5: f64 = -1.03142241298341437450e-02; /* 0xBF851F9F, 0xBA91EC6A */ -const T6: f64 = 6.10053870246291332635e-03; /* 0x3F78FCE0, 0xE370E344 */ -const T7: f64 = -3.68452016781138256760e-03; /* 0xBF6E2EFF, 0xB3E914D7 */ -const T8: f64 = 2.25964780900612472250e-03; /* 0x3F6282D3, 0x2E15C915 */ -const T9: f64 = -1.40346469989232843813e-03; /* 0xBF56FE8E, 0xBF2D1AF1 */ -const T10: f64 = 8.81081882437654011382e-04; /* 0x3F4CDF0C, 0xEF61A8E9 */ -const T11: f64 = -5.38595305356740546715e-04; /* 0xBF41A610, 0x9C73E0EC */ -const T12: f64 = 3.15632070903625950361e-04; /* 0x3F34AF6D, 0x6C0EBBF7 */ -const T13: f64 = -3.12754168375120860518e-04; /* 0xBF347F24, 0xECC38C38 */ -const T14: f64 = 3.35529192635519073543e-04; /* 0x3F35FD3E, 0xE8C2D3F4 */ -const U0: f64 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */ -const U1: f64 = 6.32827064025093366517e-01; /* 0x3FE4401E, 0x8B005DFF */ -const U2: f64 = 1.45492250137234768737e+00; /* 0x3FF7475C, 0xD119BD6F */ -const U3: f64 = 9.77717527963372745603e-01; /* 0x3FEF4976, 0x44EA8450 */ -const U4: f64 = 2.28963728064692451092e-01; /* 0x3FCD4EAE, 0xF6010924 */ -const U5: f64 = 1.33810918536787660377e-02; /* 0x3F8B678B, 0xBF2BAB09 */ -const V1: f64 = 2.45597793713041134822e+00; /* 0x4003A5D7, 0xC2BD619C */ -const V2: f64 = 2.12848976379893395361e+00; /* 0x40010725, 0xA42B18F5 */ -const V3: f64 = 7.69285150456672783825e-01; /* 0x3FE89DFB, 0xE45050AF */ -const V4: f64 = 1.04222645593369134254e-01; /* 0x3FBAAE55, 0xD6537C88 */ -const V5: f64 = 3.21709242282423911810e-03; /* 0x3F6A5ABB, 0x57D0CF61 */ -const S0: f64 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */ -const S1: f64 = 2.14982415960608852501e-01; /* 0x3FCB848B, 0x36E20878 */ -const S2: f64 = 3.25778796408930981787e-01; /* 0x3FD4D98F, 0x4F139F59 */ -const S3: f64 = 1.46350472652464452805e-01; /* 0x3FC2BB9C, 0xBEE5F2F7 */ -const S4: f64 = 2.66422703033638609560e-02; /* 0x3F9B481C, 0x7E939961 */ -const S5: f64 = 1.84028451407337715652e-03; /* 0x3F5E26B6, 0x7368F239 */ -const S6: f64 = 3.19475326584100867617e-05; /* 0x3F00BFEC, 0xDD17E945 */ -const R1: f64 = 1.39200533467621045958e+00; /* 0x3FF645A7, 0x62C4AB74 */ -const R2: f64 = 7.21935547567138069525e-01; /* 0x3FE71A18, 0x93D3DCDC */ -const R3: f64 = 1.71933865632803078993e-01; /* 0x3FC601ED, 0xCCFBDF27 */ -const R4: f64 = 1.86459191715652901344e-02; /* 0x3F9317EA, 0x742ED475 */ -const R5: f64 = 7.77942496381893596434e-04; /* 0x3F497DDA, 0xCA41A95B */ -const R6: f64 = 7.32668430744625636189e-06; /* 0x3EDEBAF7, 0xA5B38140 */ -const W0: f64 = 4.18938533204672725052e-01; /* 0x3FDACFE3, 0x90C97D69 */ -const W1: f64 = 8.33333333333329678849e-02; /* 0x3FB55555, 0x5555553B */ -const W2: f64 = -2.77777777728775536470e-03; /* 0xBF66C16C, 0x16B02E5C */ -const W3: f64 = 7.93650558643019558500e-04; /* 0x3F4A019F, 0x98CF38B6 */ -const W4: f64 = -5.95187557450339963135e-04; /* 0xBF4380CB, 0x8C0FE741 */ -const W5: f64 = 8.36339918996282139126e-04; /* 0x3F4B67BA, 0x4CDAD5D1 */ -const W6: f64 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ - -/* sin(PI*x) assuming x > 2^-100, if sin(PI*x)==0 the sign is arbitrary */ -fn sin_pi(mut x: f64) -> f64 -{ - let mut n: isize; - - /* spurious inexact if odd int */ - x = 2.0*(x*0.5 - floor(x*0.5)); /* x mod 2.0 */ - - n = (x*4.0) as isize; - n = (n+1)/2; - x -= (n as f64)*0.5; - x *= PI; - - match n { - 1 => k_cos(x, 0.0), - 2 => k_sin(-x, 0.0, 0), - 3 => -k_cos(x, 0.0), - 0|_ => k_sin(x, 0.0, 0), - } -} - -pub fn lgamma(x: f64) -> f64 { - lgamma_r(x).0 -} - -pub fn lgamma_r(mut x: f64) -> (f64, isize) -{ - let u: u64 = x.to_bits(); - let mut t: f64; - let y: f64; - let mut z: f64; - let nadj: f64; - let p: f64; - let p1: f64; - let p2: f64; - let p3: f64; - let q: f64; - let mut r: f64; - let w: f64; - let ix: u32; - let sign: bool; - let i: isize; - let mut signgam: isize; - - /* purge off +-inf, NaN, +-0, tiny and negative arguments */ - signgam = 1; - sign = (u>>63) != 0; - ix = ((u>>32) as u32) & 0x7fffffff; - if ix >= 0x7ff00000 { - return (x*x, signgam); - } - if ix < (0x3ff-70)<<20 { /* |x|<2**-70, return -log(|x|) */ - if sign { - x = -x; - signgam = -1; - } - return (-log(x), signgam); - } - if sign { - x = -x; - t = sin_pi(x); - if t == 0.0 { /* -integer */ - return (1.0/(x-x), signgam); - } - if t > 0.0 { - signgam = -1; - } else { - t = -t; - } - nadj = log(PI/(t*x)); - } else { - nadj = 0.0; - } - - /* purge off 1 and 2 */ - if (ix == 0x3ff00000 || ix == 0x40000000) && (u & 0xffffffff) == 0 { - r = 0.0; - } - /* for x < 2.0 */ - else if ix < 0x40000000 { - if ix <= 0x3feccccc { /* lgamma(x) = lgamma(x+1)-log(x) */ - r = -log(x); - if ix >= 0x3FE76944 { - y = 1.0 - x; - i = 0; - } else if ix >= 0x3FCDA661 { - y = x - (TC-1.0); - i = 1; - } else { - y = x; - i = 2; - } - } else { - r = 0.0; - if ix >= 0x3FFBB4C3 { /* [1.7316,2] */ - y = 2.0 - x; - i = 0; - } else if ix >= 0x3FF3B4C4 { /* [1.23,1.73] */ - y = x - TC; - i = 1; - } else { - y = x - 1.0; - i = 2; - } - } - match i { - 0 => { - z = y*y; - p1 = A0+z*(A2+z*(A4+z*(A6+z*(A8+z*A10)))); - p2 = z*(A1+z*(A3+z*(A5+z*(A7+z*(A9+z*A11))))); - p = y*p1+p2; - r += p-0.5*y; - } - 1 => { - z = y*y; - w = z*y; - p1 = T0+w*(T3+w*(T6+w*(T9 +w*T12))); /* parallel comp */ - p2 = T1+w*(T4+w*(T7+w*(T10+w*T13))); - p3 = T2+w*(T5+w*(T8+w*(T11+w*T14))); - p = z*p1-(TT-w*(p2+y*p3)); - r += TF + p; - } - 2 => { - p1 = y*(U0+y*(U1+y*(U2+y*(U3+y*(U4+y*U5))))); - p2 = 1.0+y*(V1+y*(V2+y*(V3+y*(V4+y*V5)))); - r += -0.5*y + p1/p2; - } - #[cfg(feature = "checked")] - _ => unreachable!(), - #[cfg(not(feature = "checked"))] - _ => {} - } - } else if ix < 0x40200000 { /* x < 8.0 */ - i = x as isize; - y = x - (i as f64); - p = y*(S0+y*(S1+y*(S2+y*(S3+y*(S4+y*(S5+y*S6)))))); - q = 1.0+y*(R1+y*(R2+y*(R3+y*(R4+y*(R5+y*R6))))); - r = 0.5*y+p/q; - z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */ - // TODO: In C, this was implemented using switch jumps with fallthrough. - // Does this implementation have performance problems? - if i >= 7 { z *= y + 6.0; } - if i >= 6 { z *= y + 5.0; } - if i >= 5 { z *= y + 4.0; } - if i >= 4 { z *= y + 3.0; } - if i >= 3 { - z *= y + 2.0; - r += log(z); - } - } else if ix < 0x43900000 { /* 8.0 <= x < 2**58 */ - t = log(x); - z = 1.0/x; - y = z*z; - w = W0+z*(W1+y*(W2+y*(W3+y*(W4+y*(W5+y*W6))))); - r = (x-0.5)*(t-1.0)+w; - } else { /* 2**58 <= x <= inf */ - r = x*(log(x)-1.0); - } - if sign { - r = nadj - r; - } - return (r, signgam); -} +/* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + * + */ +/* lgamma_r(x, signgamp) + * Reentrant version of the logarithm of the Gamma function + * with user provide pointer for the sign of Gamma(x). + * + * Method: + * 1. Argument Reduction for 0 < x <= 8 + * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may + * reduce x to a number in [1.5,2.5] by + * lgamma(1+s) = log(s) + lgamma(s) + * for example, + * lgamma(7.3) = log(6.3) + lgamma(6.3) + * = log(6.3*5.3) + lgamma(5.3) + * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) + * 2. Polynomial approximation of lgamma around its + * minimun ymin=1.461632144968362245 to maintain monotonicity. + * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use + * Let z = x-ymin; + * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) + * where + * poly(z) is a 14 degree polynomial. + * 2. Rational approximation in the primary interval [2,3] + * We use the following approximation: + * s = x-2.0; + * lgamma(x) = 0.5*s + s*P(s)/Q(s) + * with accuracy + * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 + * Our algorithms are based on the following observation + * + * zeta(2)-1 2 zeta(3)-1 3 + * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... + * 2 3 + * + * where Euler = 0.5771... is the Euler constant, which is very + * close to 0.5. + * + * 3. For x>=8, we have + * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... + * (better formula: + * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) + * Let z = 1/x, then we approximation + * f(z) = lgamma(x) - (x-0.5)(log(x)-1) + * by + * 3 5 11 + * w = w0 + w1*z + w2*z + w3*z + ... + w6*z + * where + * |w - f(z)| < 2**-58.74 + * + * 4. For negative x, since (G is gamma function) + * -x*G(-x)*G(x) = PI/sin(PI*x), + * we have + * G(x) = PI/(sin(PI*x)*(-x)*G(-x)) + * since G(-x) is positive, sign(G(x)) = sign(sin(PI*x)) for x<0 + * Hence, for x<0, signgam = sign(sin(PI*x)) and + * lgamma(x) = log(|Gamma(x)|) + * = log(PI/(|x*sin(PI*x)|)) - lgamma(-x); + * Note: one should avoid compute PI*(-x) directly in the + * computation of sin(PI*(-x)). + * + * 5. Special Cases + * lgamma(2+s) ~ s*(1-Euler) for tiny s + * lgamma(1) = lgamma(2) = 0 + * lgamma(x) ~ -log(|x|) for tiny x + * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero + * lgamma(inf) = inf + * lgamma(-inf) = inf (bug for bug compatible with C99!?) + * + */ + +use super::{floor, k_cos, k_sin, log}; + +const PI: f64 = 3.14159265358979311600e+00; /* 0x400921FB, 0x54442D18 */ +const A0: f64 = 7.72156649015328655494e-02; /* 0x3FB3C467, 0xE37DB0C8 */ +const A1: f64 = 3.22467033424113591611e-01; /* 0x3FD4A34C, 0xC4A60FAD */ +const A2: f64 = 6.73523010531292681824e-02; /* 0x3FB13E00, 0x1A5562A7 */ +const A3: f64 = 2.05808084325167332806e-02; /* 0x3F951322, 0xAC92547B */ +const A4: f64 = 7.38555086081402883957e-03; /* 0x3F7E404F, 0xB68FEFE8 */ +const A5: f64 = 2.89051383673415629091e-03; /* 0x3F67ADD8, 0xCCB7926B */ +const A6: f64 = 1.19270763183362067845e-03; /* 0x3F538A94, 0x116F3F5D */ +const A7: f64 = 5.10069792153511336608e-04; /* 0x3F40B6C6, 0x89B99C00 */ +const A8: f64 = 2.20862790713908385557e-04; /* 0x3F2CF2EC, 0xED10E54D */ +const A9: f64 = 1.08011567247583939954e-04; /* 0x3F1C5088, 0x987DFB07 */ +const A10: f64 = 2.52144565451257326939e-05; /* 0x3EFA7074, 0x428CFA52 */ +const A11: f64 = 4.48640949618915160150e-05; /* 0x3F07858E, 0x90A45837 */ +const TC: f64 = 1.46163214496836224576e+00; /* 0x3FF762D8, 0x6356BE3F */ +const TF: f64 = -1.21486290535849611461e-01; /* 0xBFBF19B9, 0xBCC38A42 */ +/* tt = -(tail of TF) */ +const TT: f64 = -3.63867699703950536541e-18; /* 0xBC50C7CA, 0xA48A971F */ +const T0: f64 = 4.83836122723810047042e-01; /* 0x3FDEF72B, 0xC8EE38A2 */ +const T1: f64 = -1.47587722994593911752e-01; /* 0xBFC2E427, 0x8DC6C509 */ +const T2: f64 = 6.46249402391333854778e-02; /* 0x3FB08B42, 0x94D5419B */ +const T3: f64 = -3.27885410759859649565e-02; /* 0xBFA0C9A8, 0xDF35B713 */ +const T4: f64 = 1.79706750811820387126e-02; /* 0x3F9266E7, 0x970AF9EC */ +const T5: f64 = -1.03142241298341437450e-02; /* 0xBF851F9F, 0xBA91EC6A */ +const T6: f64 = 6.10053870246291332635e-03; /* 0x3F78FCE0, 0xE370E344 */ +const T7: f64 = -3.68452016781138256760e-03; /* 0xBF6E2EFF, 0xB3E914D7 */ +const T8: f64 = 2.25964780900612472250e-03; /* 0x3F6282D3, 0x2E15C915 */ +const T9: f64 = -1.40346469989232843813e-03; /* 0xBF56FE8E, 0xBF2D1AF1 */ +const T10: f64 = 8.81081882437654011382e-04; /* 0x3F4CDF0C, 0xEF61A8E9 */ +const T11: f64 = -5.38595305356740546715e-04; /* 0xBF41A610, 0x9C73E0EC */ +const T12: f64 = 3.15632070903625950361e-04; /* 0x3F34AF6D, 0x6C0EBBF7 */ +const T13: f64 = -3.12754168375120860518e-04; /* 0xBF347F24, 0xECC38C38 */ +const T14: f64 = 3.35529192635519073543e-04; /* 0x3F35FD3E, 0xE8C2D3F4 */ +const U0: f64 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */ +const U1: f64 = 6.32827064025093366517e-01; /* 0x3FE4401E, 0x8B005DFF */ +const U2: f64 = 1.45492250137234768737e+00; /* 0x3FF7475C, 0xD119BD6F */ +const U3: f64 = 9.77717527963372745603e-01; /* 0x3FEF4976, 0x44EA8450 */ +const U4: f64 = 2.28963728064692451092e-01; /* 0x3FCD4EAE, 0xF6010924 */ +const U5: f64 = 1.33810918536787660377e-02; /* 0x3F8B678B, 0xBF2BAB09 */ +const V1: f64 = 2.45597793713041134822e+00; /* 0x4003A5D7, 0xC2BD619C */ +const V2: f64 = 2.12848976379893395361e+00; /* 0x40010725, 0xA42B18F5 */ +const V3: f64 = 7.69285150456672783825e-01; /* 0x3FE89DFB, 0xE45050AF */ +const V4: f64 = 1.04222645593369134254e-01; /* 0x3FBAAE55, 0xD6537C88 */ +const V5: f64 = 3.21709242282423911810e-03; /* 0x3F6A5ABB, 0x57D0CF61 */ +const S0: f64 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */ +const S1: f64 = 2.14982415960608852501e-01; /* 0x3FCB848B, 0x36E20878 */ +const S2: f64 = 3.25778796408930981787e-01; /* 0x3FD4D98F, 0x4F139F59 */ +const S3: f64 = 1.46350472652464452805e-01; /* 0x3FC2BB9C, 0xBEE5F2F7 */ +const S4: f64 = 2.66422703033638609560e-02; /* 0x3F9B481C, 0x7E939961 */ +const S5: f64 = 1.84028451407337715652e-03; /* 0x3F5E26B6, 0x7368F239 */ +const S6: f64 = 3.19475326584100867617e-05; /* 0x3F00BFEC, 0xDD17E945 */ +const R1: f64 = 1.39200533467621045958e+00; /* 0x3FF645A7, 0x62C4AB74 */ +const R2: f64 = 7.21935547567138069525e-01; /* 0x3FE71A18, 0x93D3DCDC */ +const R3: f64 = 1.71933865632803078993e-01; /* 0x3FC601ED, 0xCCFBDF27 */ +const R4: f64 = 1.86459191715652901344e-02; /* 0x3F9317EA, 0x742ED475 */ +const R5: f64 = 7.77942496381893596434e-04; /* 0x3F497DDA, 0xCA41A95B */ +const R6: f64 = 7.32668430744625636189e-06; /* 0x3EDEBAF7, 0xA5B38140 */ +const W0: f64 = 4.18938533204672725052e-01; /* 0x3FDACFE3, 0x90C97D69 */ +const W1: f64 = 8.33333333333329678849e-02; /* 0x3FB55555, 0x5555553B */ +const W2: f64 = -2.77777777728775536470e-03; /* 0xBF66C16C, 0x16B02E5C */ +const W3: f64 = 7.93650558643019558500e-04; /* 0x3F4A019F, 0x98CF38B6 */ +const W4: f64 = -5.95187557450339963135e-04; /* 0xBF4380CB, 0x8C0FE741 */ +const W5: f64 = 8.36339918996282139126e-04; /* 0x3F4B67BA, 0x4CDAD5D1 */ +const W6: f64 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ + +/* sin(PI*x) assuming x > 2^-100, if sin(PI*x)==0 the sign is arbitrary */ +fn sin_pi(mut x: f64) -> f64 { + let mut n: i32; + + /* spurious inexact if odd int */ + x = 2.0 * (x * 0.5 - floor(x * 0.5)); /* x mod 2.0 */ + + n = (x * 4.0) as i32; + n = (n + 1) / 2; + x -= (n as f64) * 0.5; + x *= PI; + + match n { + 1 => k_cos(x, 0.0), + 2 => k_sin(-x, 0.0, 0), + 3 => -k_cos(x, 0.0), + 0 | _ => k_sin(x, 0.0, 0), + } +} + +pub fn lgamma(x: f64) -> f64 { + lgamma_r(x).0 +} + +pub fn lgamma_r(mut x: f64) -> (f64, i32) { + let u: u64 = x.to_bits(); + let mut t: f64; + let y: f64; + let mut z: f64; + let nadj: f64; + let p: f64; + let p1: f64; + let p2: f64; + let p3: f64; + let q: f64; + let mut r: f64; + let w: f64; + let ix: u32; + let sign: bool; + let i: i32; + let mut signgam: i32; + + /* purge off +-inf, NaN, +-0, tiny and negative arguments */ + signgam = 1; + sign = (u >> 63) != 0; + ix = ((u >> 32) as u32) & 0x7fffffff; + if ix >= 0x7ff00000 { + return (x * x, signgam); + } + if ix < (0x3ff - 70) << 20 { + /* |x|<2**-70, return -log(|x|) */ + if sign { + x = -x; + signgam = -1; + } + return (-log(x), signgam); + } + if sign { + x = -x; + t = sin_pi(x); + if t == 0.0 { + /* -integer */ + return (1.0 / (x - x), signgam); + } + if t > 0.0 { + signgam = -1; + } else { + t = -t; + } + nadj = log(PI / (t * x)); + } else { + nadj = 0.0; + } + + /* purge off 1 and 2 */ + if (ix == 0x3ff00000 || ix == 0x40000000) && (u & 0xffffffff) == 0 { + r = 0.0; + } + /* for x < 2.0 */ + else if ix < 0x40000000 { + if ix <= 0x3feccccc { + /* lgamma(x) = lgamma(x+1)-log(x) */ + r = -log(x); + if ix >= 0x3FE76944 { + y = 1.0 - x; + i = 0; + } else if ix >= 0x3FCDA661 { + y = x - (TC - 1.0); + i = 1; + } else { + y = x; + i = 2; + } + } else { + r = 0.0; + if ix >= 0x3FFBB4C3 { + /* [1.7316,2] */ + y = 2.0 - x; + i = 0; + } else if ix >= 0x3FF3B4C4 { + /* [1.23,1.73] */ + y = x - TC; + i = 1; + } else { + y = x - 1.0; + i = 2; + } + } + match i { + 0 => { + z = y * y; + p1 = A0 + z * (A2 + z * (A4 + z * (A6 + z * (A8 + z * A10)))); + p2 = z * (A1 + z * (A3 + z * (A5 + z * (A7 + z * (A9 + z * A11))))); + p = y * p1 + p2; + r += p - 0.5 * y; + } + 1 => { + z = y * y; + w = z * y; + p1 = T0 + w * (T3 + w * (T6 + w * (T9 + w * T12))); /* parallel comp */ + p2 = T1 + w * (T4 + w * (T7 + w * (T10 + w * T13))); + p3 = T2 + w * (T5 + w * (T8 + w * (T11 + w * T14))); + p = z * p1 - (TT - w * (p2 + y * p3)); + r += TF + p; + } + 2 => { + p1 = y * (U0 + y * (U1 + y * (U2 + y * (U3 + y * (U4 + y * U5))))); + p2 = 1.0 + y * (V1 + y * (V2 + y * (V3 + y * (V4 + y * V5)))); + r += -0.5 * y + p1 / p2; + } + #[cfg(feature = "checked")] + _ => unreachable!(), + #[cfg(not(feature = "checked"))] + _ => {} + } + } else if ix < 0x40200000 { + /* x < 8.0 */ + i = x as i32; + y = x - (i as f64); + p = y * (S0 + y * (S1 + y * (S2 + y * (S3 + y * (S4 + y * (S5 + y * S6)))))); + q = 1.0 + y * (R1 + y * (R2 + y * (R3 + y * (R4 + y * (R5 + y * R6))))); + r = 0.5 * y + p / q; + z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */ + // TODO: In C, this was implemented using switch jumps with fallthrough. + // Does this implementation have performance problems? + if i >= 7 { + z *= y + 6.0; + } + if i >= 6 { + z *= y + 5.0; + } + if i >= 5 { + z *= y + 4.0; + } + if i >= 4 { + z *= y + 3.0; + } + if i >= 3 { + z *= y + 2.0; + r += log(z); + } + } else if ix < 0x43900000 { + /* 8.0 <= x < 2**58 */ + t = log(x); + z = 1.0 / x; + y = z * z; + w = W0 + z * (W1 + y * (W2 + y * (W3 + y * (W4 + y * (W5 + y * W6))))); + r = (x - 0.5) * (t - 1.0) + w; + } else { + /* 2**58 <= x <= inf */ + r = x * (log(x) - 1.0); + } + if sign { + r = nadj - r; + } + return (r, signgam); +} diff --git a/src/math/lgammaf.rs b/src/math/lgammaf.rs index 60effa3..8fe8060 100644 --- a/src/math/lgammaf.rs +++ b/src/math/lgammaf.rs @@ -1,244 +1,258 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/e_lgammaf_r.c */ -/* - * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. - */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -use super::{floorf, k_cosf, k_sinf, logf}; - -const PI: f32 = 3.1415927410e+00; /* 0x40490fdb */ -const A0: f32 = 7.7215664089e-02; /* 0x3d9e233f */ -const A1: f32 = 3.2246702909e-01; /* 0x3ea51a66 */ -const A2: f32 = 6.7352302372e-02; /* 0x3d89f001 */ -const A3: f32 = 2.0580807701e-02; /* 0x3ca89915 */ -const A4: f32 = 7.3855509982e-03; /* 0x3bf2027e */ -const A5: f32 = 2.8905137442e-03; /* 0x3b3d6ec6 */ -const A6: f32 = 1.1927076848e-03; /* 0x3a9c54a1 */ -const A7: f32 = 5.1006977446e-04; /* 0x3a05b634 */ -const A8: f32 = 2.2086278477e-04; /* 0x39679767 */ -const A9: f32 = 1.0801156895e-04; /* 0x38e28445 */ -const A10: f32 = 2.5214456400e-05; /* 0x37d383a2 */ -const A11: f32 = 4.4864096708e-05; /* 0x383c2c75 */ -const TC: f32 = 1.4616321325e+00; /* 0x3fbb16c3 */ -const TF: f32 = -1.2148628384e-01; /* 0xbdf8cdcd */ -/* TT = -(tail of TF) */ -const TT: f32 = 6.6971006518e-09; /* 0x31e61c52 */ -const T0: f32 = 4.8383611441e-01; /* 0x3ef7b95e */ -const T1: f32 = -1.4758771658e-01; /* 0xbe17213c */ -const T2: f32 = 6.4624942839e-02; /* 0x3d845a15 */ -const T3: f32 = -3.2788541168e-02; /* 0xbd064d47 */ -const T4: f32 = 1.7970675603e-02; /* 0x3c93373d */ -const T5: f32 = -1.0314224288e-02; /* 0xbc28fcfe */ -const T6: f32 = 6.1005386524e-03; /* 0x3bc7e707 */ -const T7: f32 = -3.6845202558e-03; /* 0xbb7177fe */ -const T8: f32 = 2.2596477065e-03; /* 0x3b141699 */ -const T9: f32 = -1.4034647029e-03; /* 0xbab7f476 */ -const T10: f32 = 8.8108185446e-04; /* 0x3a66f867 */ -const T11: f32 = -5.3859531181e-04; /* 0xba0d3085 */ -const T12: f32 = 3.1563205994e-04; /* 0x39a57b6b */ -const T13: f32 = -3.1275415677e-04; /* 0xb9a3f927 */ -const T14: f32 = 3.3552918467e-04; /* 0x39afe9f7 */ -const U0: f32 = -7.7215664089e-02; /* 0xbd9e233f */ -const U1: f32 = 6.3282704353e-01; /* 0x3f2200f4 */ -const U2: f32 = 1.4549225569e+00; /* 0x3fba3ae7 */ -const U3: f32 = 9.7771751881e-01; /* 0x3f7a4bb2 */ -const U4: f32 = 2.2896373272e-01; /* 0x3e6a7578 */ -const U5: f32 = 1.3381091878e-02; /* 0x3c5b3c5e */ -const V1: f32 = 2.4559779167e+00; /* 0x401d2ebe */ -const V2: f32 = 2.1284897327e+00; /* 0x4008392d */ -const V3: f32 = 7.6928514242e-01; /* 0x3f44efdf */ -const V4: f32 = 1.0422264785e-01; /* 0x3dd572af */ -const V5: f32 = 3.2170924824e-03; /* 0x3b52d5db */ -const S0: f32 = -7.7215664089e-02; /* 0xbd9e233f */ -const S1: f32 = 2.1498242021e-01; /* 0x3e5c245a */ -const S2: f32 = 3.2577878237e-01; /* 0x3ea6cc7a */ -const S3: f32 = 1.4635047317e-01; /* 0x3e15dce6 */ -const S4: f32 = 2.6642270386e-02; /* 0x3cda40e4 */ -const S5: f32 = 1.8402845599e-03; /* 0x3af135b4 */ -const S6: f32 = 3.1947532989e-05; /* 0x3805ff67 */ -const R1: f32 = 1.3920053244e+00; /* 0x3fb22d3b */ -const R2: f32 = 7.2193557024e-01; /* 0x3f38d0c5 */ -const R3: f32 = 1.7193385959e-01; /* 0x3e300f6e */ -const R4: f32 = 1.8645919859e-02; /* 0x3c98bf54 */ -const R5: f32 = 7.7794247773e-04; /* 0x3a4beed6 */ -const R6: f32 = 7.3266842264e-06; /* 0x36f5d7bd */ -const W0: f32 = 4.1893854737e-01; /* 0x3ed67f1d */ -const W1: f32 = 8.3333335817e-02; /* 0x3daaaaab */ -const W2: f32 = -2.7777778450e-03; /* 0xbb360b61 */ -const W3: f32 = 7.9365057172e-04; /* 0x3a500cfd */ -const W4: f32 = -5.9518753551e-04; /* 0xba1c065c */ -const W5: f32 = 8.3633989561e-04; /* 0x3a5b3dd2 */ -const W6: f32 = -1.6309292987e-03; /* 0xbad5c4e8 */ - -/* sin(PI*x) assuming x > 2^-100, if sin(PI*x)==0 the sign is arbitrary */ -fn sin_pi(mut x: f32) -> f32 -{ - let mut y: f64; - let mut n: isize; - - /* spurious inexact if odd int */ - x = 2.0*(x*0.5 - floorf(x*0.5)); /* x mod 2.0 */ - - n = (x*4.0) as isize; - n = (n+1)/2; - y = (x as f64) - (n as f64)*0.5; - y *= 3.14159265358979323846; - match n { - 1 => k_cosf(y), - 2 => k_sinf(-y), - 3 => -k_cosf(y), - 0|_ => k_sinf(y), - } -} - -pub fn lgammaf(x: f32) -> f32 { - lgammaf_r(x).0 -} - -pub fn lgammaf_r(mut x: f32) -> (f32, isize) -{ - let u = x.to_bits(); - let mut t: f32; - let y: f32; - let mut z: f32; - let nadj: f32; - let p: f32; - let p1: f32; - let p2: f32; - let p3: f32; - let q: f32; - let mut r: f32; - let w: f32; - let ix: u32; - let i: isize; - let sign: bool; - let mut signgam: isize; - - /* purge off +-inf, NaN, +-0, tiny and negative arguments */ - signgam = 1; - sign = (u>>31) != 0; - ix = u & 0x7fffffff; - if ix >= 0x7f800000 { - return (x*x, signgam); - } - if ix < 0x35000000 { /* |x| < 2**-21, return -log(|x|) */ - if sign { - signgam = -1; - x = -x; - } - return (-logf(x), signgam); - } - if sign { - x = -x; - t = sin_pi(x); - if t == 0.0 { /* -integer */ - return (1.0/(x-x), signgam); - } - if t > 0.0 { - signgam = -1; - } else { - t = -t; - } - nadj = logf(PI/(t*x)); - } else { - nadj = 0.0; - } - - /* purge off 1 and 2 */ - if ix == 0x3f800000 || ix == 0x40000000 { - r = 0.0; - } - /* for x < 2.0 */ - else if ix < 0x40000000 { - if ix <= 0x3f666666 { /* lgamma(x) = lgamma(x+1)-log(x) */ - r = -logf(x); - if ix >= 0x3f3b4a20 { - y = 1.0 - x; - i = 0; - } else if ix >= 0x3e6d3308 { - y = x - (TC-1.0); - i = 1; - } else { - y = x; - i = 2; - } - } else { - r = 0.0; - if ix >= 0x3fdda618 { /* [1.7316,2] */ - y = 2.0 - x; - i = 0; - } else if ix >= 0x3F9da620 { /* [1.23,1.73] */ - y = x - TC; - i = 1; - } else { - y = x - 1.0; - i = 2; - } - } - match i { - 0 => { - z = y*y; - p1 = A0+z*(A2+z*(A4+z*(A6+z*(A8+z*A10)))); - p2 = z*(A1+z*(A3+z*(A5+z*(A7+z*(A9+z*A11))))); - p = y*p1+p2; - r += p - 0.5*y; - } - 1 => { - z = y*y; - w = z*y; - p1 = T0+w*(T3+w*(T6+w*(T9 +w*T12))); /* parallel comp */ - p2 = T1+w*(T4+w*(T7+w*(T10+w*T13))); - p3 = T2+w*(T5+w*(T8+w*(T11+w*T14))); - p = z*p1-(TT-w*(p2+y*p3)); - r += TF + p; - } - 2 => { - p1 = y*(U0+y*(U1+y*(U2+y*(U3+y*(U4+y*U5))))); - p2 = 1.0+y*(V1+y*(V2+y*(V3+y*(V4+y*V5)))); - r += -0.5*y + p1/p2; - } - #[cfg(feature = "checked")] - _ => unreachable!(), - #[cfg(not(feature = "checked"))] - _ => {} - } - } else if ix < 0x41000000 { /* x < 8.0 */ - i = x as isize; - y = x - (i as f32); - p = y*(S0+y*(S1+y*(S2+y*(S3+y*(S4+y*(S5+y*S6)))))); - q = 1.0+y*(R1+y*(R2+y*(R3+y*(R4+y*(R5+y*R6))))); - r = 0.5*y+p/q; - z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */ - // TODO: In C, this was implemented using switch jumps with fallthrough. - // Does this implementation have performance problems? - if i >= 7 { z *= y + 6.0; } - if i >= 6 { z *= y + 5.0; } - if i >= 5 { z *= y + 4.0; } - if i >= 4 { z *= y + 3.0; } - if i >= 3 { - z *= y + 2.0; - r += logf(z); - } - } else if ix < 0x5c800000 { /* 8.0 <= x < 2**58 */ - t = logf(x); - z = 1.0/x; - y = z*z; - w = W0+z*(W1+y*(W2+y*(W3+y*(W4+y*(W5+y*W6))))); - r = (x-0.5)*(t-1.0)+w; - } else { /* 2**58 <= x <= inf */ - r = x*(logf(x)-1.0); - } - if sign { - r = nadj - r; - } - return (r, signgam); -} +/* origin: FreeBSD /usr/src/lib/msun/src/e_lgammaf_r.c */ +/* + * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. + */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +use super::{floorf, k_cosf, k_sinf, logf}; + +const PI: f32 = 3.1415927410e+00; /* 0x40490fdb */ +const A0: f32 = 7.7215664089e-02; /* 0x3d9e233f */ +const A1: f32 = 3.2246702909e-01; /* 0x3ea51a66 */ +const A2: f32 = 6.7352302372e-02; /* 0x3d89f001 */ +const A3: f32 = 2.0580807701e-02; /* 0x3ca89915 */ +const A4: f32 = 7.3855509982e-03; /* 0x3bf2027e */ +const A5: f32 = 2.8905137442e-03; /* 0x3b3d6ec6 */ +const A6: f32 = 1.1927076848e-03; /* 0x3a9c54a1 */ +const A7: f32 = 5.1006977446e-04; /* 0x3a05b634 */ +const A8: f32 = 2.2086278477e-04; /* 0x39679767 */ +const A9: f32 = 1.0801156895e-04; /* 0x38e28445 */ +const A10: f32 = 2.5214456400e-05; /* 0x37d383a2 */ +const A11: f32 = 4.4864096708e-05; /* 0x383c2c75 */ +const TC: f32 = 1.4616321325e+00; /* 0x3fbb16c3 */ +const TF: f32 = -1.2148628384e-01; /* 0xbdf8cdcd */ +/* TT = -(tail of TF) */ +const TT: f32 = 6.6971006518e-09; /* 0x31e61c52 */ +const T0: f32 = 4.8383611441e-01; /* 0x3ef7b95e */ +const T1: f32 = -1.4758771658e-01; /* 0xbe17213c */ +const T2: f32 = 6.4624942839e-02; /* 0x3d845a15 */ +const T3: f32 = -3.2788541168e-02; /* 0xbd064d47 */ +const T4: f32 = 1.7970675603e-02; /* 0x3c93373d */ +const T5: f32 = -1.0314224288e-02; /* 0xbc28fcfe */ +const T6: f32 = 6.1005386524e-03; /* 0x3bc7e707 */ +const T7: f32 = -3.6845202558e-03; /* 0xbb7177fe */ +const T8: f32 = 2.2596477065e-03; /* 0x3b141699 */ +const T9: f32 = -1.4034647029e-03; /* 0xbab7f476 */ +const T10: f32 = 8.8108185446e-04; /* 0x3a66f867 */ +const T11: f32 = -5.3859531181e-04; /* 0xba0d3085 */ +const T12: f32 = 3.1563205994e-04; /* 0x39a57b6b */ +const T13: f32 = -3.1275415677e-04; /* 0xb9a3f927 */ +const T14: f32 = 3.3552918467e-04; /* 0x39afe9f7 */ +const U0: f32 = -7.7215664089e-02; /* 0xbd9e233f */ +const U1: f32 = 6.3282704353e-01; /* 0x3f2200f4 */ +const U2: f32 = 1.4549225569e+00; /* 0x3fba3ae7 */ +const U3: f32 = 9.7771751881e-01; /* 0x3f7a4bb2 */ +const U4: f32 = 2.2896373272e-01; /* 0x3e6a7578 */ +const U5: f32 = 1.3381091878e-02; /* 0x3c5b3c5e */ +const V1: f32 = 2.4559779167e+00; /* 0x401d2ebe */ +const V2: f32 = 2.1284897327e+00; /* 0x4008392d */ +const V3: f32 = 7.6928514242e-01; /* 0x3f44efdf */ +const V4: f32 = 1.0422264785e-01; /* 0x3dd572af */ +const V5: f32 = 3.2170924824e-03; /* 0x3b52d5db */ +const S0: f32 = -7.7215664089e-02; /* 0xbd9e233f */ +const S1: f32 = 2.1498242021e-01; /* 0x3e5c245a */ +const S2: f32 = 3.2577878237e-01; /* 0x3ea6cc7a */ +const S3: f32 = 1.4635047317e-01; /* 0x3e15dce6 */ +const S4: f32 = 2.6642270386e-02; /* 0x3cda40e4 */ +const S5: f32 = 1.8402845599e-03; /* 0x3af135b4 */ +const S6: f32 = 3.1947532989e-05; /* 0x3805ff67 */ +const R1: f32 = 1.3920053244e+00; /* 0x3fb22d3b */ +const R2: f32 = 7.2193557024e-01; /* 0x3f38d0c5 */ +const R3: f32 = 1.7193385959e-01; /* 0x3e300f6e */ +const R4: f32 = 1.8645919859e-02; /* 0x3c98bf54 */ +const R5: f32 = 7.7794247773e-04; /* 0x3a4beed6 */ +const R6: f32 = 7.3266842264e-06; /* 0x36f5d7bd */ +const W0: f32 = 4.1893854737e-01; /* 0x3ed67f1d */ +const W1: f32 = 8.3333335817e-02; /* 0x3daaaaab */ +const W2: f32 = -2.7777778450e-03; /* 0xbb360b61 */ +const W3: f32 = 7.9365057172e-04; /* 0x3a500cfd */ +const W4: f32 = -5.9518753551e-04; /* 0xba1c065c */ +const W5: f32 = 8.3633989561e-04; /* 0x3a5b3dd2 */ +const W6: f32 = -1.6309292987e-03; /* 0xbad5c4e8 */ + +/* sin(PI*x) assuming x > 2^-100, if sin(PI*x)==0 the sign is arbitrary */ +fn sin_pi(mut x: f32) -> f32 { + let mut y: f64; + let mut n: isize; + + /* spurious inexact if odd int */ + x = 2.0 * (x * 0.5 - floorf(x * 0.5)); /* x mod 2.0 */ + + n = (x * 4.0) as isize; + n = (n + 1) / 2; + y = (x as f64) - (n as f64) * 0.5; + y *= 3.14159265358979323846; + match n { + 1 => k_cosf(y), + 2 => k_sinf(-y), + 3 => -k_cosf(y), + 0 | _ => k_sinf(y), + } +} + +pub fn lgammaf(x: f32) -> f32 { + lgammaf_r(x).0 +} + +pub fn lgammaf_r(mut x: f32) -> (f32, isize) { + let u = x.to_bits(); + let mut t: f32; + let y: f32; + let mut z: f32; + let nadj: f32; + let p: f32; + let p1: f32; + let p2: f32; + let p3: f32; + let q: f32; + let mut r: f32; + let w: f32; + let ix: u32; + let i: isize; + let sign: bool; + let mut signgam: isize; + + /* purge off +-inf, NaN, +-0, tiny and negative arguments */ + signgam = 1; + sign = (u >> 31) != 0; + ix = u & 0x7fffffff; + if ix >= 0x7f800000 { + return (x * x, signgam); + } + if ix < 0x35000000 { + /* |x| < 2**-21, return -log(|x|) */ + if sign { + signgam = -1; + x = -x; + } + return (-logf(x), signgam); + } + if sign { + x = -x; + t = sin_pi(x); + if t == 0.0 { + /* -integer */ + return (1.0 / (x - x), signgam); + } + if t > 0.0 { + signgam = -1; + } else { + t = -t; + } + nadj = logf(PI / (t * x)); + } else { + nadj = 0.0; + } + + /* purge off 1 and 2 */ + if ix == 0x3f800000 || ix == 0x40000000 { + r = 0.0; + } + /* for x < 2.0 */ + else if ix < 0x40000000 { + if ix <= 0x3f666666 { + /* lgamma(x) = lgamma(x+1)-log(x) */ + r = -logf(x); + if ix >= 0x3f3b4a20 { + y = 1.0 - x; + i = 0; + } else if ix >= 0x3e6d3308 { + y = x - (TC - 1.0); + i = 1; + } else { + y = x; + i = 2; + } + } else { + r = 0.0; + if ix >= 0x3fdda618 { + /* [1.7316,2] */ + y = 2.0 - x; + i = 0; + } else if ix >= 0x3F9da620 { + /* [1.23,1.73] */ + y = x - TC; + i = 1; + } else { + y = x - 1.0; + i = 2; + } + } + match i { + 0 => { + z = y * y; + p1 = A0 + z * (A2 + z * (A4 + z * (A6 + z * (A8 + z * A10)))); + p2 = z * (A1 + z * (A3 + z * (A5 + z * (A7 + z * (A9 + z * A11))))); + p = y * p1 + p2; + r += p - 0.5 * y; + } + 1 => { + z = y * y; + w = z * y; + p1 = T0 + w * (T3 + w * (T6 + w * (T9 + w * T12))); /* parallel comp */ + p2 = T1 + w * (T4 + w * (T7 + w * (T10 + w * T13))); + p3 = T2 + w * (T5 + w * (T8 + w * (T11 + w * T14))); + p = z * p1 - (TT - w * (p2 + y * p3)); + r += TF + p; + } + 2 => { + p1 = y * (U0 + y * (U1 + y * (U2 + y * (U3 + y * (U4 + y * U5))))); + p2 = 1.0 + y * (V1 + y * (V2 + y * (V3 + y * (V4 + y * V5)))); + r += -0.5 * y + p1 / p2; + } + #[cfg(feature = "checked")] + _ => unreachable!(), + #[cfg(not(feature = "checked"))] + _ => {} + } + } else if ix < 0x41000000 { + /* x < 8.0 */ + i = x as isize; + y = x - (i as f32); + p = y * (S0 + y * (S1 + y * (S2 + y * (S3 + y * (S4 + y * (S5 + y * S6)))))); + q = 1.0 + y * (R1 + y * (R2 + y * (R3 + y * (R4 + y * (R5 + y * R6))))); + r = 0.5 * y + p / q; + z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */ + // TODO: In C, this was implemented using switch jumps with fallthrough. + // Does this implementation have performance problems? + if i >= 7 { + z *= y + 6.0; + } + if i >= 6 { + z *= y + 5.0; + } + if i >= 5 { + z *= y + 4.0; + } + if i >= 4 { + z *= y + 3.0; + } + if i >= 3 { + z *= y + 2.0; + r += logf(z); + } + } else if ix < 0x5c800000 { + /* 8.0 <= x < 2**58 */ + t = logf(x); + z = 1.0 / x; + y = z * z; + w = W0 + z * (W1 + y * (W2 + y * (W3 + y * (W4 + y * (W5 + y * W6))))); + r = (x - 0.5) * (t - 1.0) + w; + } else { + /* 2**58 <= x <= inf */ + r = x * (logf(x) - 1.0); + } + if sign { + r = nadj - r; + } + return (r, signgam); +} diff --git a/src/math/mod.rs b/src/math/mod.rs index 6e53b02..b70b0cd 100644 --- a/src/math/mod.rs +++ b/src/math/mod.rs @@ -118,8 +118,6 @@ mod frexp; mod frexpf; mod hypot; mod hypotf; -mod ldexp; -mod ldexpf; mod ilogb; mod ilogbf; mod j0; @@ -128,6 +126,8 @@ mod j1; mod j1f; mod jn; mod jnf; +mod ldexp; +mod ldexpf; mod lgamma; mod lgammaf; mod log; @@ -192,8 +192,8 @@ pub use self::cosh::cosh; pub use self::coshf::coshf; pub use self::erf::erf; pub use self::erf::erfc; -pub use self::erff::erff; pub use self::erff::erfcf; +pub use self::erff::erff; pub use self::exp::exp; pub use self::exp10::exp10; pub use self::exp10f::exp10f; @@ -216,8 +216,6 @@ pub use self::frexp::frexp; pub use self::frexpf::frexpf; pub use self::hypot::hypot; pub use self::hypotf::hypotf; -pub use self::ldexp::ldexp; -pub use self::ldexpf::ldexpf; pub use self::ilogb::ilogb; pub use self::ilogbf::ilogbf; pub use self::j0::j0; @@ -232,6 +230,8 @@ pub use self::jn::jn; pub use self::jn::yn; pub use self::jnf::jnf; pub use self::jnf::ynf; +pub use self::ldexp::ldexp; +pub use self::ldexpf::ldexpf; pub use self::lgamma::lgamma; pub use self::lgamma::lgamma_r; pub use self::lgammaf::lgammaf; diff --git a/src/math/modf.rs b/src/math/modf.rs index 1ff8ee1..a37f8b9 100644 --- a/src/math/modf.rs +++ b/src/math/modf.rs @@ -1,33 +1,34 @@ -pub fn modf(x: f64) -> (f64, f64) { - let rv2: f64; - let mut u = x.to_bits(); - let mask: u64; - let e = ((u>>52 & 0x7ff) as isize) - 0x3ff; - - /* no fractional part */ - if e >= 52 { - rv2 = x; - if e == 0x400 && (u<<12) != 0 { /* nan */ - return (x, rv2); - } - u &= 1<<63; - return (f64::from_bits(u), rv2); - } - - /* no integral part*/ - if e < 0 { - u &= 1<<63; - rv2 = f64::from_bits(u); - return (x, rv2); - } - - mask = ((!0)>>12)>>e; - if (u & mask) == 0 { - rv2 = x; - u &= 1<<63; - return (f64::from_bits(u), rv2); - } - u &= !mask; - rv2 = f64::from_bits(u); - return (x - rv2, rv2); -} +pub fn modf(x: f64) -> (f64, f64) { + let rv2: f64; + let mut u = x.to_bits(); + let mask: u64; + let e = ((u >> 52 & 0x7ff) as isize) - 0x3ff; + + /* no fractional part */ + if e >= 52 { + rv2 = x; + if e == 0x400 && (u << 12) != 0 { + /* nan */ + return (x, rv2); + } + u &= 1 << 63; + return (f64::from_bits(u), rv2); + } + + /* no integral part*/ + if e < 0 { + u &= 1 << 63; + rv2 = f64::from_bits(u); + return (x, rv2); + } + + mask = ((!0) >> 12) >> e; + if (u & mask) == 0 { + rv2 = x; + u &= 1 << 63; + return (f64::from_bits(u), rv2); + } + u &= !mask; + rv2 = f64::from_bits(u); + return (x - rv2, rv2); +} diff --git a/src/math/modff.rs b/src/math/modff.rs index 5250e8d..4ce9052 100644 --- a/src/math/modff.rs +++ b/src/math/modff.rs @@ -1,32 +1,33 @@ -pub fn modff(x: f32) -> (f32, f32) { - let rv2: f32; - let mut u: u32 = x.to_bits(); - let mask: u32; - let e = ((u>>23 & 0xff) as isize) - 0x7f; - - /* no fractional part */ - if e >= 23 { - rv2 = x; - if e == 0x80 && (u<<9) != 0 { /* nan */ - return (x, rv2); - } - u &= 0x80000000; - return (f32::from_bits(u), rv2); - } - /* no integral part */ - if e < 0 { - u &= 0x80000000; - rv2 = f32::from_bits(u); - return (x, rv2); - } - - mask = 0x007fffff>>e; - if (u & mask) == 0 { - rv2 = x; - u &= 0x80000000; - return (f32::from_bits(u), rv2); - } - u &= !mask; - rv2 = f32::from_bits(u); - return (x - rv2, rv2); -} +pub fn modff(x: f32) -> (f32, f32) { + let rv2: f32; + let mut u: u32 = x.to_bits(); + let mask: u32; + let e = ((u >> 23 & 0xff) as isize) - 0x7f; + + /* no fractional part */ + if e >= 23 { + rv2 = x; + if e == 0x80 && (u << 9) != 0 { + /* nan */ + return (x, rv2); + } + u &= 0x80000000; + return (f32::from_bits(u), rv2); + } + /* no integral part */ + if e < 0 { + u &= 0x80000000; + rv2 = f32::from_bits(u); + return (x, rv2); + } + + mask = 0x007fffff >> e; + if (u & mask) == 0 { + rv2 = x; + u &= 0x80000000; + return (f32::from_bits(u), rv2); + } + u &= !mask; + rv2 = f32::from_bits(u); + return (x - rv2, rv2); +} diff --git a/src/math/remquo.rs b/src/math/remquo.rs index 98f4b38..3681b94 100644 --- a/src/math/remquo.rs +++ b/src/math/remquo.rs @@ -1,98 +1,97 @@ -pub fn remquo(mut x: f64, mut y: f64) -> (f64, isize) -{ - let ux: u64 = x.to_bits(); - let mut uy: u64 = y.to_bits(); - let mut ex = ((ux>>52) & 0x7ff) as isize; - let mut ey = ((uy>>52) & 0x7ff) as isize; - let sx = (ux>>63) != 0; - let sy = (uy>>63) != 0; - let mut q: u32; - let mut i: u64; - let mut uxi: u64 = ux; - - if (uy<<1) == 0 || y.is_nan() || ex == 0x7ff { - return ((x*y)/(x*y), 0); - } - if (ux<<1) == 0 { - return (x, 0); - } - - /* normalize x and y */ - if ex == 0 { - i = uxi << 12; - while (i>>63) == 0 { - ex -= 1; - i <<= 1; - } - uxi <<= -ex + 1; - } else { - uxi &= (!0) >> 12; - uxi |= 1 << 52; - } - if ey == 0 { - i = uy<<12; - while (i>>63) == 0 { - ey -= 1; - i <<= 1; - } - uy <<= -ey + 1; - } else { - uy &= (!0) >> 12; - uy |= 1 << 52; - } - - q = 0; - - if ex+1 != ey { - if ex < ey { - return (x, 0); - } - /* x mod y */ - while ex > ey { - i = uxi - uy; - if (i>>63) == 0 { - uxi = i; - q += 1; - } - uxi <<= 1; - q <<= 1; - ex -= 1; - } - i = uxi - uy; - if (i>>63) == 0 { - uxi = i; - q += 1; - } - if uxi == 0 { - ex = -60; - } else { - while (uxi>>52) == 0 { - uxi <<= 1; - ex -= 1; - } - } - } - - /* scale result and decide between |x| and |x|-|y| */ - if ex > 0 { - uxi -= 1 << 52; - uxi |= (ex as u64) << 52; - } else { - uxi >>= -ex + 1; - } - x = f64::from_bits(uxi); - if sy { - y = -y; - } - if ex == ey || (ex+1 == ey && (2.0*x > y || (2.0*x == y && (q%2) != 0))) { - x -= y; - q += 1; - } - q &= 0x7fffffff; - let quo = if sx ^ sy { -(q as isize) } else { q as isize }; - if sx { - (-x, quo) - } else { - (x, quo) - } -} +pub fn remquo(mut x: f64, mut y: f64) -> (f64, isize) { + let ux: u64 = x.to_bits(); + let mut uy: u64 = y.to_bits(); + let mut ex = ((ux >> 52) & 0x7ff) as isize; + let mut ey = ((uy >> 52) & 0x7ff) as isize; + let sx = (ux >> 63) != 0; + let sy = (uy >> 63) != 0; + let mut q: u32; + let mut i: u64; + let mut uxi: u64 = ux; + + if (uy << 1) == 0 || y.is_nan() || ex == 0x7ff { + return ((x * y) / (x * y), 0); + } + if (ux << 1) == 0 { + return (x, 0); + } + + /* normalize x and y */ + if ex == 0 { + i = uxi << 12; + while (i >> 63) == 0 { + ex -= 1; + i <<= 1; + } + uxi <<= -ex + 1; + } else { + uxi &= (!0) >> 12; + uxi |= 1 << 52; + } + if ey == 0 { + i = uy << 12; + while (i >> 63) == 0 { + ey -= 1; + i <<= 1; + } + uy <<= -ey + 1; + } else { + uy &= (!0) >> 12; + uy |= 1 << 52; + } + + q = 0; + + if ex + 1 != ey { + if ex < ey { + return (x, 0); + } + /* x mod y */ + while ex > ey { + i = uxi - uy; + if (i >> 63) == 0 { + uxi = i; + q += 1; + } + uxi <<= 1; + q <<= 1; + ex -= 1; + } + i = uxi - uy; + if (i >> 63) == 0 { + uxi = i; + q += 1; + } + if uxi == 0 { + ex = -60; + } else { + while (uxi >> 52) == 0 { + uxi <<= 1; + ex -= 1; + } + } + } + + /* scale result and decide between |x| and |x|-|y| */ + if ex > 0 { + uxi -= 1 << 52; + uxi |= (ex as u64) << 52; + } else { + uxi >>= -ex + 1; + } + x = f64::from_bits(uxi); + if sy { + y = -y; + } + if ex == ey || (ex + 1 == ey && (2.0 * x > y || (2.0 * x == y && (q % 2) != 0))) { + x -= y; + q += 1; + } + q &= 0x7fffffff; + let quo = if sx ^ sy { -(q as isize) } else { q as isize }; + if sx { + (-x, quo) + } else { + (x, quo) + } +} diff --git a/src/math/remquof.rs b/src/math/remquof.rs index 4307e19..40ded5d 100644 --- a/src/math/remquof.rs +++ b/src/math/remquof.rs @@ -1,97 +1,96 @@ -pub fn remquof(mut x: f32, mut y: f32) -> (f32, isize) -{ - let ux: u32 = x.to_bits(); - let mut uy: u32 = y.to_bits(); - let mut ex = ((ux>>23) & 0xff) as isize; - let mut ey = ((uy>>23) & 0xff) as isize; - let sx = (ux>>31) != 0; - let sy = (uy>>31) != 0; - let mut q: u32; - let mut i: u32; - let mut uxi: u32 = ux; - - if (uy<<1) == 0 || y.is_nan() || ex == 0xff { - return ((x*y)/(x*y), 0); - } - if (ux<<1) == 0 { - return (x, 0); - } - - /* normalize x and y */ - if ex == 0 { - i = uxi<<9; - while (i>>31) == 0 { - ex -= 1; - i <<= 1; - } - uxi <<= -ex + 1; - } else { - uxi &= (!0) >> 9; - uxi |= 1 << 23; - } - if ey == 0 { - i = uy<<9; - while (i>>31) == 0 { - ey -= 1; - i <<= 1; - } - uy <<= -ey + 1; - } else { - uy &= (!0) >> 9; - uy |= 1 << 23; - } - - q = 0; - if ex+1 != ey { - if ex < ey { - return (x, 0); - } - /* x mod y */ - while ex > ey { - i = uxi - uy; - if (i>>31) == 0 { - uxi = i; - q += 1; - } - uxi <<= 1; - q <<= 1; - ex -= 1; - } - i = uxi - uy; - if (i>>31) == 0 { - uxi = i; - q += 1; - } - if uxi == 0 { - ex = -30; - } else { - while (uxi>>23) == 0 { - uxi <<= 1; - ex -= 1; - } - } - } - - /* scale result and decide between |x| and |x|-|y| */ - if ex > 0 { - uxi -= 1 << 23; - uxi |= (ex as u32) << 23; - } else { - uxi >>= -ex + 1; - } - x = f32::from_bits(uxi); - if sy { - y = -y; - } - if ex == ey || (ex+1 == ey && (2.0*x > y || (2.0*x == y && (q%2) != 0))) { - x -= y; - q += 1; - } - q &= 0x7fffffff; - let quo = if sx^sy { -(q as isize) } else { q as isize }; - if sx { - (-x, quo) - } else { - (x, quo) - } -} +pub fn remquof(mut x: f32, mut y: f32) -> (f32, isize) { + let ux: u32 = x.to_bits(); + let mut uy: u32 = y.to_bits(); + let mut ex = ((ux >> 23) & 0xff) as isize; + let mut ey = ((uy >> 23) & 0xff) as isize; + let sx = (ux >> 31) != 0; + let sy = (uy >> 31) != 0; + let mut q: u32; + let mut i: u32; + let mut uxi: u32 = ux; + + if (uy << 1) == 0 || y.is_nan() || ex == 0xff { + return ((x * y) / (x * y), 0); + } + if (ux << 1) == 0 { + return (x, 0); + } + + /* normalize x and y */ + if ex == 0 { + i = uxi << 9; + while (i >> 31) == 0 { + ex -= 1; + i <<= 1; + } + uxi <<= -ex + 1; + } else { + uxi &= (!0) >> 9; + uxi |= 1 << 23; + } + if ey == 0 { + i = uy << 9; + while (i >> 31) == 0 { + ey -= 1; + i <<= 1; + } + uy <<= -ey + 1; + } else { + uy &= (!0) >> 9; + uy |= 1 << 23; + } + + q = 0; + if ex + 1 != ey { + if ex < ey { + return (x, 0); + } + /* x mod y */ + while ex > ey { + i = uxi - uy; + if (i >> 31) == 0 { + uxi = i; + q += 1; + } + uxi <<= 1; + q <<= 1; + ex -= 1; + } + i = uxi - uy; + if (i >> 31) == 0 { + uxi = i; + q += 1; + } + if uxi == 0 { + ex = -30; + } else { + while (uxi >> 23) == 0 { + uxi <<= 1; + ex -= 1; + } + } + } + + /* scale result and decide between |x| and |x|-|y| */ + if ex > 0 { + uxi -= 1 << 23; + uxi |= (ex as u32) << 23; + } else { + uxi >>= -ex + 1; + } + x = f32::from_bits(uxi); + if sy { + y = -y; + } + if ex == ey || (ex + 1 == ey && (2.0 * x > y || (2.0 * x == y && (q % 2) != 0))) { + x -= y; + q += 1; + } + q &= 0x7fffffff; + let quo = if sx ^ sy { -(q as isize) } else { q as isize }; + if sx { + (-x, quo) + } else { + (x, quo) + } +} diff --git a/src/math/sincos.rs b/src/math/sincos.rs index c15ee46..750908d 100644 --- a/src/math/sincos.rs +++ b/src/math/sincos.rs @@ -1,60 +1,59 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/s_sin.c */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -use super::{get_high_word, k_cos, k_sin, rem_pio2}; - -pub fn sincos(x: f64) -> (f64, f64) -{ - let s: f64; - let c: f64; - let mut ix: u32; - - ix = get_high_word(x); - ix &= 0x7fffffff; - - /* |x| ~< pi/4 */ - if ix <= 0x3fe921fb { - /* if |x| < 2**-27 * sqrt(2) */ - if ix < 0x3e46a09e { - /* raise inexact if x!=0 and underflow if subnormal */ - let x1p120 = f64::from_bits(0x4770000000000000); // 0x1p120 == 2^120 - if ix < 0x00100000 { - force_eval!(x/x1p120); - } else { - force_eval!(x+x1p120); - } - return (x, 1.0); - } - return (k_sin(x, 0.0, 0), k_cos(x, 0.0)); - } - - /* sincos(Inf or NaN) is NaN */ - if ix >= 0x7ff00000 { - let rv = x - x; - return (rv, rv); - } - - /* argument reduction needed */ - let (n, y0, y1) = rem_pio2(x); - s = k_sin(y0, y1, 1); - c = k_cos(y0, y1); - match n&3 { - 0 => (s, c), - 1 => (c, -s), - 2 => (-s, -c), - 3 => (-c, s), - #[cfg(feature = "checked")] - _ => unreachable!(), - #[cfg(not(feature = "checked"))] - _ => (0.0, 1.0), - } -} +/* origin: FreeBSD /usr/src/lib/msun/src/s_sin.c */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +use super::{get_high_word, k_cos, k_sin, rem_pio2}; + +pub fn sincos(x: f64) -> (f64, f64) { + let s: f64; + let c: f64; + let mut ix: u32; + + ix = get_high_word(x); + ix &= 0x7fffffff; + + /* |x| ~< pi/4 */ + if ix <= 0x3fe921fb { + /* if |x| < 2**-27 * sqrt(2) */ + if ix < 0x3e46a09e { + /* raise inexact if x!=0 and underflow if subnormal */ + let x1p120 = f64::from_bits(0x4770000000000000); // 0x1p120 == 2^120 + if ix < 0x00100000 { + force_eval!(x / x1p120); + } else { + force_eval!(x + x1p120); + } + return (x, 1.0); + } + return (k_sin(x, 0.0, 0), k_cos(x, 0.0)); + } + + /* sincos(Inf or NaN) is NaN */ + if ix >= 0x7ff00000 { + let rv = x - x; + return (rv, rv); + } + + /* argument reduction needed */ + let (n, y0, y1) = rem_pio2(x); + s = k_sin(y0, y1, 1); + c = k_cos(y0, y1); + match n & 3 { + 0 => (s, c), + 1 => (c, -s), + 2 => (-s, -c), + 3 => (-c, s), + #[cfg(feature = "checked")] + _ => unreachable!(), + #[cfg(not(feature = "checked"))] + _ => (0.0, 1.0), + } +} diff --git a/src/math/sincosf.rs b/src/math/sincosf.rs index 911421d..bb9a003 100644 --- a/src/math/sincosf.rs +++ b/src/math/sincosf.rs @@ -1,122 +1,123 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/s_sinf.c */ -/* - * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. - * Optimized by Bruce D. Evans. - */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ - -use super::{k_cosf, k_sinf, rem_pio2f}; - -/* Small multiples of pi/2 rounded to double precision. */ -const PI_2: f32 = 0.5 * 3.1415926535897931160E+00; -const S1PIO2: f32 = 1.0*PI_2; /* 0x3FF921FB, 0x54442D18 */ -const S2PIO2: f32 = 2.0*PI_2; /* 0x400921FB, 0x54442D18 */ -const S3PIO2: f32 = 3.0*PI_2; /* 0x4012D97C, 0x7F3321D2 */ -const S4PIO2: f32 = 4.0*PI_2; /* 0x401921FB, 0x54442D18 */ - -pub fn sincosf(x: f32) -> (f32, f32) -{ - let s: f32; - let c: f32; - let mut ix: u32; - let sign: bool; - - ix = x.to_bits(); - sign = (ix >> 31) != 0; - ix &= 0x7fffffff; - - /* |x| ~<= pi/4 */ - if ix <= 0x3f490fda { - /* |x| < 2**-12 */ - if ix < 0x39800000 { - /* raise inexact if x!=0 and underflow if subnormal */ - - let x1p120 = f32::from_bits(0x7b800000); // 0x1p120 == 2^120 - if ix < 0x00100000 { - force_eval!(x/x1p120); - } else { - force_eval!(x+x1p120); - } - return (x, 1.0); - } - return (k_sinf(x as f64), k_cosf(x as f64)); - } - - /* |x| ~<= 5*pi/4 */ - if ix <= 0x407b53d1 { - if ix <= 0x4016cbe3 { /* |x| ~<= 3pi/4 */ - if sign { - s = -k_cosf((x + S1PIO2) as f64); - c = k_sinf((x + S1PIO2) as f64); - } else { - s = k_cosf((S1PIO2 - x) as f64); - c = k_sinf((S1PIO2 - x) as f64); - } - } - /* -sin(x+c) is not correct if x+c could be 0: -0 vs +0 */ - else { - if sign { - s = k_sinf((x + S2PIO2) as f64); - c = k_cosf((x + S2PIO2) as f64); - } else { - s = k_sinf((x - S2PIO2) as f64); - c = k_cosf((x - S2PIO2) as f64); - } - } - - return (s, c); - } - - /* |x| ~<= 9*pi/4 */ - if ix <= 0x40e231d5 { - if ix <= 0x40afeddf { /* |x| ~<= 7*pi/4 */ - if sign { - s = k_cosf((x + S3PIO2) as f64); - c = -k_sinf((x + S3PIO2) as f64); - } else { - s = -k_cosf((x - S3PIO2) as f64); - c = k_sinf((x - S3PIO2) as f64); - } - } else { - if sign { - s = k_cosf((x + S4PIO2) as f64); - c = k_sinf((x + S4PIO2) as f64); - } else { - s = k_cosf((x - S4PIO2) as f64); - c = k_sinf((x - S4PIO2) as f64); - } - } - - return (s, c); - } - - /* sin(Inf or NaN) is NaN */ - if ix >= 0x7f800000 { - let rv = x - x; - return (rv, rv); - } - - /* general argument reduction needed */ - let (n, y) = rem_pio2f(x); - s = k_sinf(y); - c = k_cosf(y); - match n&3 { - 0 => (s, c), - 1 => (c, -s), - 2 => (-s, -c), - 3 => (-c, s), - #[cfg(feature = "checked")] - _ => unreachable!(), - #[cfg(not(feature = "checked"))] - _ => (0.0, 1.0), - } -} +/* origin: FreeBSD /usr/src/lib/msun/src/s_sinf.c */ +/* + * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. + * Optimized by Bruce D. Evans. + */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +use super::{k_cosf, k_sinf, rem_pio2f}; + +/* Small multiples of pi/2 rounded to double precision. */ +const PI_2: f32 = 0.5 * 3.1415926535897931160E+00; +const S1PIO2: f32 = 1.0 * PI_2; /* 0x3FF921FB, 0x54442D18 */ +const S2PIO2: f32 = 2.0 * PI_2; /* 0x400921FB, 0x54442D18 */ +const S3PIO2: f32 = 3.0 * PI_2; /* 0x4012D97C, 0x7F3321D2 */ +const S4PIO2: f32 = 4.0 * PI_2; /* 0x401921FB, 0x54442D18 */ + +pub fn sincosf(x: f32) -> (f32, f32) { + let s: f32; + let c: f32; + let mut ix: u32; + let sign: bool; + + ix = x.to_bits(); + sign = (ix >> 31) != 0; + ix &= 0x7fffffff; + + /* |x| ~<= pi/4 */ + if ix <= 0x3f490fda { + /* |x| < 2**-12 */ + if ix < 0x39800000 { + /* raise inexact if x!=0 and underflow if subnormal */ + + let x1p120 = f32::from_bits(0x7b800000); // 0x1p120 == 2^120 + if ix < 0x00100000 { + force_eval!(x / x1p120); + } else { + force_eval!(x + x1p120); + } + return (x, 1.0); + } + return (k_sinf(x as f64), k_cosf(x as f64)); + } + + /* |x| ~<= 5*pi/4 */ + if ix <= 0x407b53d1 { + if ix <= 0x4016cbe3 { + /* |x| ~<= 3pi/4 */ + if sign { + s = -k_cosf((x + S1PIO2) as f64); + c = k_sinf((x + S1PIO2) as f64); + } else { + s = k_cosf((S1PIO2 - x) as f64); + c = k_sinf((S1PIO2 - x) as f64); + } + } + /* -sin(x+c) is not correct if x+c could be 0: -0 vs +0 */ + else { + if sign { + s = k_sinf((x + S2PIO2) as f64); + c = k_cosf((x + S2PIO2) as f64); + } else { + s = k_sinf((x - S2PIO2) as f64); + c = k_cosf((x - S2PIO2) as f64); + } + } + + return (s, c); + } + + /* |x| ~<= 9*pi/4 */ + if ix <= 0x40e231d5 { + if ix <= 0x40afeddf { + /* |x| ~<= 7*pi/4 */ + if sign { + s = k_cosf((x + S3PIO2) as f64); + c = -k_sinf((x + S3PIO2) as f64); + } else { + s = -k_cosf((x - S3PIO2) as f64); + c = k_sinf((x - S3PIO2) as f64); + } + } else { + if sign { + s = k_cosf((x + S4PIO2) as f64); + c = k_sinf((x + S4PIO2) as f64); + } else { + s = k_cosf((x - S4PIO2) as f64); + c = k_sinf((x - S4PIO2) as f64); + } + } + + return (s, c); + } + + /* sin(Inf or NaN) is NaN */ + if ix >= 0x7f800000 { + let rv = x - x; + return (rv, rv); + } + + /* general argument reduction needed */ + let (n, y) = rem_pio2f(x); + s = k_sinf(y); + c = k_cosf(y); + match n & 3 { + 0 => (s, c), + 1 => (c, -s), + 2 => (-s, -c), + 3 => (-c, s), + #[cfg(feature = "checked")] + _ => unreachable!(), + #[cfg(not(feature = "checked"))] + _ => (0.0, 1.0), + } +} diff --git a/src/math/tgamma.rs b/src/math/tgamma.rs index 598f46f..f8ccf66 100644 --- a/src/math/tgamma.rs +++ b/src/math/tgamma.rs @@ -1,179 +1,207 @@ -/* -"A Precision Approximation of the Gamma Function" - Cornelius Lanczos (1964) -"Lanczos Implementation of the Gamma Function" - Paul Godfrey (2001) -"An Analysis of the Lanczos Gamma Approximation" - Glendon Ralph Pugh (2004) - -approximation method: - - (x - 0.5) S(x) -Gamma(x) = (x + g - 0.5) * ---------------- - exp(x + g - 0.5) - -with - a1 a2 a3 aN -S(x) ~= [ a0 + ----- + ----- + ----- + ... + ----- ] - x + 1 x + 2 x + 3 x + N - -with a0, a1, a2, a3,.. aN constants which depend on g. - -for x < 0 the following reflection formula is used: - -Gamma(x)*Gamma(-x) = -pi/(x sin(pi x)) - -most ideas and constants are from boost and python -*/ -extern crate core; -use super::{exp, floor, k_cos, k_sin, pow}; - -const PI: f64 = 3.141592653589793238462643383279502884; - -/* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */ -fn sinpi(mut x: f64) -> f64 -{ - let mut n: isize; - - /* argument reduction: x = |x| mod 2 */ - /* spurious inexact when x is odd int */ - x = x * 0.5; - x = 2.0 * (x - floor(x)); - - /* reduce x into [-.25,.25] */ - n = (4.0 * x) as isize; - n = (n+1)/2; - x -= (n as f64) * 0.5; - - x *= PI; - match n { - 1 => k_cos(x, 0.0), - 2 => k_sin(-x, 0.0, 0), - 3 => -k_cos(x, 0.0), - 0|_ => k_sin(x, 0.0, 0), - } -} - -const N: usize = 12; -//static const double g = 6.024680040776729583740234375; -const GMHALF: f64 = 5.524680040776729583740234375; -const SNUM: [f64; N+1] = [ - 23531376880.410759688572007674451636754734846804940, - 42919803642.649098768957899047001988850926355848959, - 35711959237.355668049440185451547166705960488635843, - 17921034426.037209699919755754458931112671403265390, - 6039542586.3520280050642916443072979210699388420708, - 1439720407.3117216736632230727949123939715485786772, - 248874557.86205415651146038641322942321632125127801, - 31426415.585400194380614231628318205362874684987640, - 2876370.6289353724412254090516208496135991145378768, - 186056.26539522349504029498971604569928220784236328, - 8071.6720023658162106380029022722506138218516325024, - 210.82427775157934587250973392071336271166969580291, - 2.5066282746310002701649081771338373386264310793408, -]; -const SDEN: [f64; N+1] = [ - 0.0, 39916800.0, 120543840.0, 150917976.0, 105258076.0, - 45995730.0, 13339535.0, 2637558.0, 357423.0, 32670.0, 1925.0, 66.0, 1.0, -]; -/* n! for small integer n */ -const FACT: [f64; 23] = [ - 1.0, 1.0, 2.0, 6.0, 24.0, 120.0, 720.0, 5040.0, 40320.0, 362880.0, 3628800.0, - 39916800.0, 479001600.0, 6227020800.0, 87178291200.0, 1307674368000.0, - 20922789888000.0, 355687428096000.0, 6402373705728000.0, 121645100408832000.0, - 2432902008176640000.0, 51090942171709440000.0, 1124000727777607680000.0, -]; - -/* S(x) rational function for positive x */ -fn s(x: f64) -> f64 -{ - let mut num: f64 = 0.0; - let mut den: f64 = 0.0; - - /* to avoid overflow handle large x differently */ - if x < 8.0 { - for i in (0..=N).rev() { - num = num * x + SNUM[i]; - den = den * x + SDEN[i]; - } - } else { - for i in 0..=N { - num = num / x + SNUM[i]; - den = den / x + SDEN[i]; - } - } - return num/den; -} - -pub fn tgamma(mut x: f64) -> f64 -{ - let u: u64 = x.to_bits(); - let absx: f64; - let mut y: f64; - let mut dy: f64; - let mut z: f64; - let mut r: f64; - let ix: u32 = ((u >> 32) as u32) & 0x7fffffff; - let sign: bool = (u>>64) != 0; - - /* special cases */ - if ix >= 0x7ff00000 { - /* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */ - return x + core::f64::INFINITY; - } - if ix < ((0x3ff-54)<<20) { - /* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */ - return 1.0/x; - } - - /* integer arguments */ - /* raise inexact when non-integer */ - if x == floor(x) { - if sign { - return 0.0/0.0; - } - if x <= FACT.len() as f64 { - return FACT[(x as usize) - 1]; - } - } - - /* x >= 172: tgamma(x)=inf with overflow */ - /* x =< -184: tgamma(x)=+-0 with underflow */ - if ix >= 0x40670000 { /* |x| >= 184 */ - if sign { - let x1p_126 = f64::from_bits(0x3810000000000000); // 0x1p-126 == 2^-126 - force_eval!((x1p_126/x) as f32); - if floor(x) * 0.5 == floor(x * 0.5) { - return 0.0; - } else { - return -0.0; - } - } - let x1p1023 = f64::from_bits(0x7fe0000000000000); // 0x1p1023 == 2^1023 - x *= x1p1023; - return x; - } - - absx = if sign { -x } else { x }; - - /* handle the error of x + g - 0.5 */ - y = absx + GMHALF; - if absx > GMHALF { - dy = y - absx; - dy -= GMHALF; - } else { - dy = y - GMHALF; - dy -= absx; - } - - z = absx - 0.5; - r = s(absx) * exp(-y); - if x < 0.0 { - /* reflection formula for negative x */ - /* sinpi(absx) is not 0, integers are already handled */ - r = -PI / (sinpi(absx) * absx * r); - dy = -dy; - z = -z; - } - r += dy * (GMHALF+0.5) * r / y; - z = pow(y, 0.5*z); - y = r * z * z; - return y; -} +/* +"A Precision Approximation of the Gamma Function" - Cornelius Lanczos (1964) +"Lanczos Implementation of the Gamma Function" - Paul Godfrey (2001) +"An Analysis of the Lanczos Gamma Approximation" - Glendon Ralph Pugh (2004) + +approximation method: + + (x - 0.5) S(x) +Gamma(x) = (x + g - 0.5) * ---------------- + exp(x + g - 0.5) + +with + a1 a2 a3 aN +S(x) ~= [ a0 + ----- + ----- + ----- + ... + ----- ] + x + 1 x + 2 x + 3 x + N + +with a0, a1, a2, a3,.. aN constants which depend on g. + +for x < 0 the following reflection formula is used: + +Gamma(x)*Gamma(-x) = -pi/(x sin(pi x)) + +most ideas and constants are from boost and python +*/ +extern crate core; +use super::{exp, floor, k_cos, k_sin, pow}; + +const PI: f64 = 3.141592653589793238462643383279502884; + +/* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */ +fn sinpi(mut x: f64) -> f64 { + let mut n: isize; + + /* argument reduction: x = |x| mod 2 */ + /* spurious inexact when x is odd int */ + x = x * 0.5; + x = 2.0 * (x - floor(x)); + + /* reduce x into [-.25,.25] */ + n = (4.0 * x) as isize; + n = (n + 1) / 2; + x -= (n as f64) * 0.5; + + x *= PI; + match n { + 1 => k_cos(x, 0.0), + 2 => k_sin(-x, 0.0, 0), + 3 => -k_cos(x, 0.0), + 0 | _ => k_sin(x, 0.0, 0), + } +} + +const N: usize = 12; +//static const double g = 6.024680040776729583740234375; +const GMHALF: f64 = 5.524680040776729583740234375; +const SNUM: [f64; N + 1] = [ + 23531376880.410759688572007674451636754734846804940, + 42919803642.649098768957899047001988850926355848959, + 35711959237.355668049440185451547166705960488635843, + 17921034426.037209699919755754458931112671403265390, + 6039542586.3520280050642916443072979210699388420708, + 1439720407.3117216736632230727949123939715485786772, + 248874557.86205415651146038641322942321632125127801, + 31426415.585400194380614231628318205362874684987640, + 2876370.6289353724412254090516208496135991145378768, + 186056.26539522349504029498971604569928220784236328, + 8071.6720023658162106380029022722506138218516325024, + 210.82427775157934587250973392071336271166969580291, + 2.5066282746310002701649081771338373386264310793408, +]; +const SDEN: [f64; N + 1] = [ + 0.0, + 39916800.0, + 120543840.0, + 150917976.0, + 105258076.0, + 45995730.0, + 13339535.0, + 2637558.0, + 357423.0, + 32670.0, + 1925.0, + 66.0, + 1.0, +]; +/* n! for small integer n */ +const FACT: [f64; 23] = [ + 1.0, + 1.0, + 2.0, + 6.0, + 24.0, + 120.0, + 720.0, + 5040.0, + 40320.0, + 362880.0, + 3628800.0, + 39916800.0, + 479001600.0, + 6227020800.0, + 87178291200.0, + 1307674368000.0, + 20922789888000.0, + 355687428096000.0, + 6402373705728000.0, + 121645100408832000.0, + 2432902008176640000.0, + 51090942171709440000.0, + 1124000727777607680000.0, +]; + +/* S(x) rational function for positive x */ +fn s(x: f64) -> f64 { + let mut num: f64 = 0.0; + let mut den: f64 = 0.0; + + /* to avoid overflow handle large x differently */ + if x < 8.0 { + for i in (0..=N).rev() { + num = num * x + SNUM[i]; + den = den * x + SDEN[i]; + } + } else { + for i in 0..=N { + num = num / x + SNUM[i]; + den = den / x + SDEN[i]; + } + } + return num / den; +} + +pub fn tgamma(mut x: f64) -> f64 { + let u: u64 = x.to_bits(); + let absx: f64; + let mut y: f64; + let mut dy: f64; + let mut z: f64; + let mut r: f64; + let ix: u32 = ((u >> 32) as u32) & 0x7fffffff; + let sign: bool = (u >> 63) != 0; + + /* special cases */ + if ix >= 0x7ff00000 { + /* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */ + return x + core::f64::INFINITY; + } + if ix < ((0x3ff - 54) << 20) { + /* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */ + return 1.0 / x; + } + + /* integer arguments */ + /* raise inexact when non-integer */ + if x == floor(x) { + if sign { + return 0.0 / 0.0; + } + if x <= FACT.len() as f64 { + return FACT[(x as usize) - 1]; + } + } + + /* x >= 172: tgamma(x)=inf with overflow */ + /* x =< -184: tgamma(x)=+-0 with underflow */ + if ix >= 0x40670000 { + /* |x| >= 184 */ + if sign { + let x1p_126 = f64::from_bits(0x3810000000000000); // 0x1p-126 == 2^-126 + force_eval!((x1p_126 / x) as f32); + if floor(x) * 0.5 == floor(x * 0.5) { + return 0.0; + } else { + return -0.0; + } + } + let x1p1023 = f64::from_bits(0x7fe0000000000000); // 0x1p1023 == 2^1023 + x *= x1p1023; + return x; + } + + absx = if sign { -x } else { x }; + + /* handle the error of x + g - 0.5 */ + y = absx + GMHALF; + if absx > GMHALF { + dy = y - absx; + dy -= GMHALF; + } else { + dy = y - GMHALF; + dy -= absx; + } + + z = absx - 0.5; + r = s(absx) * exp(-y); + if x < 0.0 { + /* reflection formula for negative x */ + /* sinpi(absx) is not 0, integers are already handled */ + r = -PI / (sinpi(absx) * absx * r); + dy = -dy; + z = -z; + } + r += dy * (GMHALF + 0.5) * r / y; + z = pow(y, 0.5 * z); + y = r * z * z; + return y; +} diff --git a/src/math/tgammaf.rs b/src/math/tgammaf.rs index b9c799c..a8f161f 100644 --- a/src/math/tgammaf.rs +++ b/src/math/tgammaf.rs @@ -1,5 +1,5 @@ -use super::{tgamma}; - -pub fn tgammaf(x: f32) -> f32 { - tgamma(x as f64) as f32 -} +use super::tgamma; + +pub fn tgammaf(x: f32) -> f32 { + tgamma(x as f64) as f32 +}