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0a099547019fbbd3f91bfdd30d1d965f6dbd48b9
libm/src/math/pow.rs
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Rahul Butani 0a09954701 pow!
2018-07-17 15:08:22 -05:00

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/* origin: FreeBSD /usr/src/lib/msun/src/e_pow.c */
/*
* ====================================================
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* pow(x,y) return x**y
*
* n
* Method: Let x = 2 * (1+f)
* 1. Compute and return log2(x) in two pieces:
* log2(x) = w1 + w2,
* where w1 has 53-24 = 29 bit trailing zeros.
* 2. Perform y*log2(x) = n+y' by simulating muti-precision
* arithmetic, where |y'|<=0.5.
* 3. Return x**y = 2**n*exp(y'*log2)
*
* Special cases:
* 1. (anything) ** 0 is 1
* 2. 1 ** (anything) is 1
* 3. (anything except 1) ** NAN is NAN
* 4. NAN ** (anything except 0) is NAN
* 5. +-(|x| > 1) ** +INF is +INF
* 6. +-(|x| > 1) ** -INF is +0
* 7. +-(|x| < 1) ** +INF is +0
* 8. +-(|x| < 1) ** -INF is +INF
* 9. -1 ** +-INF is 1
* 10. +0 ** (+anything except 0, NAN) is +0
* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
* 12. +0 ** (-anything except 0, NAN) is +INF, raise divbyzero
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF, raise divbyzero
* 14. -0 ** (+odd integer) is -0
* 15. -0 ** (-odd integer) is -INF, raise divbyzero
* 16. +INF ** (+anything except 0,NAN) is +INF
* 17. +INF ** (-anything except 0,NAN) is +0
* 18. -INF ** (+odd integer) is -INF
* 19. -INF ** (anything) = -0 ** (-anything), (anything except odd integer)
* 20. (anything) ** 1 is (anything)
* 21. (anything) ** -1 is 1/(anything)
* 22. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
* 23. (-anything except 0 and inf) ** (non-integer) is NAN
*
* Accuracy:
* pow(x,y) returns x**y nearly rounded. In particular
* pow(integer,integer)
* always returns the correct integer provided it is
* representable.
*
* Constants :
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
// #include "libm.h"
/* Concerns:
* - Some constants are shared; DRY?
* - FLT_EVAL_METHOD: the others sidestep this (epsilon or just always true in the case of hypot (#71))
*/
use super::{fabs, scalbn, sqrt, with_set_low_word, with_set_high_word, get_high_word};
const BP: [f64; 2] = [1.0, 1.5];
const DP_H: [f64; 2] = [0.0, 5.84962487220764160156e-01]; /* 0x3fe2b803_40000000 */
const DP_L: [f64; 2] = [0.0, 1.35003920212974897128e-08]; /* 0x3E4CFDEB, 0x43CFD006 */
const TWO53: f64 = 9007199254740992.0; /* 0x43400000_00000000 */
const HUGE: f64 = 1.0e300;
const TINY: f64 = 1.0e-300;
// poly coefs for (3/2)*(log(x)-2s-2/3*s**3:
const L1: f64 = 5.99999999999994648725e-01; /* 0x3fe33333_33333303 */
const L2: f64 = 4.28571428578550184252e-01; /* 0x3fdb6db6_db6fabff */
const L3: f64 = 3.33333329818377432918e-01; /* 0x3fd55555_518f264d */
const L4: f64 = 2.72728123808534006489e-01; /* 0x3fd17460_a91d4101 */
const L5: f64 = 2.30660745775561754067e-01; /* 0x3fcd864a_93c9db65 */
const L6: f64 = 2.06975017800338417784e-01; /* 0x3fca7e28_4a454eef */
const P1: f64 = 1.66666666666666019037e-01; /* 0x3fc55555_5555553e */
const P2: f64 = -2.77777777770155933842e-03; /* 0xbf66c16c_16bebd93 */
const P3: f64 = 6.61375632143793436117e-05; /* 0x3f11566a_af25de2c */
const P4: f64 = -1.65339022054652515390e-06; /* 0xbebbbd41_c5d26bf1 */
const P5: f64 = 4.13813679705723846039e-08; /* 0x3e663769_72bea4d0 */
const LG2: f64 = 6.93147180559945286227e-01; /* 0x3fe62e42_fefa39ef */
const LG2_H: f64 = 6.93147182464599609375e-01; /* 0x3fe62e43_00000000 */
const LG2_L: f64 = -1.90465429995776804525e-09; /* 0xbe205c61_0ca86c39 */
const OVT: f64 = 8.0085662595372944372e-017; /* -(1024-log2(ovfl+.5ulp)) */
const CP: f64 = 9.61796693925975554329e-01; /* 0x3feec709_dc3a03fd =2/(3ln2) */
const CP_H: f64 = 9.61796700954437255859e-01; /* 0x3feec709_e0000000 =(float)cp */
const CP_L: f64 = -7.02846165095275826516e-09; /* 0xbe3e2fe0_145b01f5 =tail of cp_h*/
const IVLN2: f64 = 1.44269504088896338700e+00; /* 0x3ff71547_652b82fe =1/ln2 */
const IVLN2_H: f64 = 1.44269502162933349609e+00; /* 0x3ff71547_60000000 =24b 1/ln2*/
const IVLN2_L: f64 = 1.92596299112661746887e-08; /* 0x3e54ae0b_f85ddf44 =1/ln2 tail*/
#[inline]
pub fn pow(x: f64, y: f64) -> f64 {
let t1: f64;
let t2: f64;
let (hx, lx): (i32, u32) = ((x.to_bits() >> 32) as i32, x.to_bits() as u32);
let (hy, ly): (i32, u32) = ((y.to_bits() >> 32) as i32, y.to_bits() as u32);
let mut ix: i32 = (hx & 0x7fffffff) as i32;
let iy: i32 = (hy & 0x7fffffff) as i32;
/* x**0 = 1, even if x is NaN */
if ((iy as u32) | ly) == 0 {
return 1.0;
}
/* 1**y = 1, even if y is NaN */
if hx == 0x3ff00000 && lx == 0 {
return 1.0;
}
/* NaN if either arg is NaN */
if ix > 0x7ff00000 || (ix == 0x7ff00000 && lx != 0) ||
iy > 0x7ff00000 || (iy == 0x7ff00000 && ly != 0) {
return x + y;
}
/* determine if y is an odd int when x < 0
* yisint = 0 ... y is not an integer
* yisint = 1 ... y is an odd int
* yisint = 2 ... y is an even int
*/
let mut yisint: i32 = 0;
let mut k: i32;
let mut j: i32;
if hx < 0 {
if iy >= 0x43400000 {
yisint = 2; /* even integer y */
} else if iy >= 0x3ff00000 {
k = (iy >> 20) - 0x3ff; /* exponent */
if k > 20 {
j = (ly >> (52 - k)) as i32;
if (j << (52 - k)) == (ly as i32) {
yisint = 2 - (j & 1);
}
} else if ly == 0 {
j = iy >> (20 - k);
if (j << (20 - k)) == iy {
yisint = 2 - (j & 1);
}
}
}
}
if ly == 0 {
/* special value of y */
if iy == 0x7ff00000 {
/* y is +-inf */
return if ((ix - 0x3ff00000) | (lx as i32)) == 0 {
/* (-1)**+-inf is 1 */
1.0
} else if ix >= 0x3ff00000 {
/* (|x|>1)**+-inf = inf,0 */
if hy >= 0 { y } else { 0.0 }
} else {
/* (|x|<1)**+-inf = 0,inf */
if hy >= 0 { 0.0 } else { -y }
};
}
if iy == 0x3ff00000 {
/* y is +-1 */
return if hy >= 0 { x } else { 1.0 / x };
}
if hy == 0x40000000 {
/* y is 2 */
return x * x;
}
if hy == 0x3fe00000 {
/* y is 0.5 */
if hx >= 0 {
/* x >= +0 */
return sqrt(x);
}
}
}
let mut ax: f64 = fabs(x);
if lx == 0 {
/* special value of x */
if ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000 {
/* x is +-0,+-inf,+-1 */
let mut z: f64 = ax;
if hy < 0 {
/* z = (1/|x|) */
z = 1.0 / z;
}
if hx < 0 {
if ((ix-0x3ff00000)|yisint) == 0 {
z = (z - z) / (z - z); /* (-1)**non-int is NaN */
} else if yisint == 1 {
z = -z; /* (x<0)**odd = -(|x|**odd) */
}
}
return z;
}
}
let mut s: f64 = 1.0; /* sign of result */
if hx < 0 {
if yisint == 0 {
/* (x<0)**(non-int) is NaN */
return (x - x) / (x - x);
}
if yisint == 1 {
/* (x<0)**(odd int) */
s = -1.0;
}
}
/* |y| is HUGE */
if iy > 0x41e00000 {
/* if |y| > 2**31 */
if iy > 0x43f00000 {
/* if |y| > 2**64, must o/uflow */
if ix <= 0x3fefffff {
return if hy < 0 { HUGE * HUGE } else { TINY * TINY };
}
if ix >= 0x3ff00000 {
return if hy > 0 { HUGE * HUGE } else { TINY * TINY };
}
}
/* over/underflow if x is not close to one */
if ix < 0x3fefffff {
return if hy < 0 { s * HUGE * HUGE } else { s * TINY * TINY };
}
if ix > 0x3ff00000 {
return if hy > 0 { s * HUGE * HUGE } else { s * TINY * TINY };
}
/* now |1-x| is TINY <= 2**-20, suffice to compute
log(x) by x-x^2/2+x^3/3-x^4/4 */
let t: f64 = ax - 1.0; /* t has 20 trailing zeros */
let w: f64 = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
let u: f64 = IVLN2_H * t; /* ivln2_h has 21 sig. bits */
let v: f64 = t * IVLN2_L - w * IVLN2;
t1 = with_set_low_word(u + v, 0);
t2 = v - (t1 - u);
} else {
// double ss,s2,s_h,s_l,t_h,t_l;
let mut n: i32 = 0;
if ix < 0x00100000 {
/* take care subnormal number */
ax *= TWO53;
n -= 53;
ix = get_high_word(ax) as i32;
}
n += (ix >> 20) - 0x3ff;
j = ix & 0x000fffff;
/* determine interval */
let k: i32;
ix = j | 0x3ff00000; /* normalize ix */
if j <= 0x3988E {
/* |x|<sqrt(3/2) */
k = 0;
}
else if j < 0xBB67A {
/* |x|<sqrt(3) */
k = 1;
} else {
k = 0;
n += 1;
ix -= 0x00100000;
}
ax = with_set_high_word(ax, ix as u32);
/* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
let u: f64 = ax - BP[k as usize]; /* bp[0]=1.0, bp[1]=1.5 */
let v: f64 = 1.0 / (ax + BP[k as usize]);
let ss: f64 = u * v;
let s_h = with_set_low_word(ss, 0);
/* t_h=ax+bp[k] High */
let t_h: f64 = with_set_high_word(0.0,
((ix as u32 >> 1) | 0x20000000) + 0x00080000 + ((k as u32) << 18));
let t_l: f64 = ax - (t_h - BP[k as usize]);
let s_l: f64 = v * ((u - s_h * t_h) - s_h * t_l);
/* compute log(ax) */
let s2: f64 = ss * ss;
let mut r: f64 = s2 * s2 * (L1 + s2 * (L2 + s2 *(L3 + s2 *(L4 + s2 *(L5 + s2 * L6)))));
r += s_l * (s_h + ss);
let s2: f64 = s_h * s_h;
let t_h: f64 = with_set_low_word(3.0 + s2 + r, 0);
let t_l: f64 = r - ((t_h - 3.0) - s2);
/* u+v = ss*(1+...) */
let u: f64 = s_h * t_h;
let v: f64 = s_l * t_h + t_l * ss;
/* 2/(3log2)*(ss+...) */
let p_h: f64 = with_set_low_word(u + v, 0);
let p_l = v - (p_h-u);
let z_h: f64 = CP_H * p_h; /* cp_h+cp_l = 2/(3*log2) */
let z_l: f64 = CP_L * p_h + p_l * CP + DP_L[k as usize];
/* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
let t: f64 = n as f64;
t1 = with_set_low_word(((z_h + z_l) + DP_H[k as usize]) + t, 0);
t2 = z_l - (((t1 - t) - DP_H[k as usize]) - z_h);
}
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
let y1: f64 = with_set_low_word(y, 0);
let p_l: f64 = (y - y1) * t1 + y * t2;
let mut p_h: f64 = y1 * t1;
let z: f64 = p_l + p_h;
let mut j: i32 = (z.to_bits() >> 32) as i32;
let i: i32 = z.to_bits() as i32;
// let (j, i): (i32, i32) = ((z.to_bits() >> 32) as i32, z.to_bits() as i32);
if j >= 0x40900000 {
/* z >= 1024 */
if (j - 0x40900000) | i != 0 {
/* if z > 1024 */
return s * HUGE * HUGE; /* overflow */
}
if p_l + OVT > z - p_h {
return s * HUGE * HUGE; /* overflow */
}
} else if (j & 0x7fffffff) >= 0x4090cc00 {
/* z <= -1075 */
// FIXME: instead of abs(j) use unsigned j
if (((j as u32) - 0xc090cc00) | (i as u32)) != 0 {
/* z < -1075 */
return s * TINY * TINY; /* underflow */
}
if p_l <= z - p_h {
return s * TINY * TINY; /* underflow */
}
}
/* compute 2**(p_h+p_l) */
let i: i32 = j & (0x7fffffff as i32);
k = (i >> 20) - 0x3ff;
let mut n: i32 = 0;
if i > 0x3fe00000 {
/* if |z| > 0.5, set n = [z+0.5] */
n = j + (0x00100000 >> (k + 1));
k = ((n&0x7fffffff) >> 20) - 0x3ff; /* new k for n */
let t: f64 = with_set_high_word(0.0, (n & !(0x000fffff >> k)) as u32);
n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
if j < 0 {
n = -n;
}
p_h -= t;
}
let t: f64 = with_set_low_word(p_l + p_h, 0);
let u: f64 = t * LG2_H;
let v: f64 = (p_l - (t - p_h)) * LG2 + t * LG2_L;
let mut z: f64 = u + v;
let w: f64 = v - (z - u);
let t: f64 = z * z;
let t1: f64 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
let r: f64 = (z * t1) / (t1 - 2.0) - (w + z*w);
z = 1.0 - (r - z);
j = get_high_word(z) as i32;
j += n << 20;
if (j >> 20) <= 0 {
/* subnormal output */
z = scalbn(z,n);
} else {
z = with_set_high_word(z, j as u32);
}
return s*z;
}
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