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/* @(#)e_pow.c 5.1 93/09/24 */ | ||
/* | ||
* ==================================================== | ||
* 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. | ||
* ==================================================== | ||
*/ | ||
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||
#if defined(LIBM_SCCS) && !defined(lint) | ||
static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $"; | ||
#endif | ||
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||
/* __ieee754_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. (anything) ** 1 is itself | ||
* 3. (anything) ** 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 NAN | ||
* 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 | ||
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF | ||
* 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) | ||
* 15. +INF ** (+anything except 0,NAN) is +INF | ||
* 16. +INF ** (-anything except 0,NAN) is +0 | ||
* 17. -INF ** (anything) = -0 ** (-anything) | ||
* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) | ||
* 19. (-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. | ||
*/ | ||
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#include "math.h" | ||
#include "math_private.h" | ||
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libm_hidden_proto(scalbn) | ||
libm_hidden_proto(fabs) | ||
#ifdef __STDC__ | ||
static const double | ||
#else | ||
static double | ||
#endif | ||
bp[] = { 1.0, 1.5, }, dp_h[] = { | ||
0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ | ||
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dp_l[] = { | ||
0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ | ||
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zero = 0.0, one = 1.0, two = 2.0, two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ | ||
huge = 1.0e300, tiny = 1.0e-300, | ||
/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ | ||
L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ | ||
L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ | ||
L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ | ||
L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ | ||
L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ | ||
L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ | ||
P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ | ||
P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ | ||
P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ | ||
P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ | ||
P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ | ||
lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ | ||
lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ | ||
lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ | ||
ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ | ||
cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ | ||
cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ | ||
cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h */ | ||
ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ | ||
ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2 */ | ||
ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail */ | ||
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#ifdef __STDC__ | ||
double attribute_hidden __ieee754_pow(double x, double y) | ||
#else | ||
double attribute_hidden __ieee754_pow(x, y) | ||
double x, y; | ||
#endif | ||
{ | ||
double z, ax, z_h, z_l, p_h, p_l; | ||
double y1, t1, t2, r, s, t, u, v, w; | ||
int32_t i, j, k, yisint, n; | ||
int32_t hx, hy, ix, iy; | ||
u_int32_t lx, ly; | ||
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EXTRACT_WORDS(hx, lx, x); | ||
EXTRACT_WORDS(hy, ly, y); | ||
ix = hx & 0x7fffffff; | ||
iy = hy & 0x7fffffff; | ||
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/* y==zero: x**0 = 1 */ | ||
if ((iy | ly) == 0) | ||
return one; | ||
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/* +-NaN return x+y */ | ||
if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) || | ||
iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0))) | ||
return x + y; | ||
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/* 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 | ||
*/ | ||
yisint = 0; | ||
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); | ||
if ((j << (52 - k)) == ly) | ||
yisint = 2 - (j & 1); | ||
} else if (ly == 0) { | ||
j = iy >> (20 - k); | ||
if ((j << (20 - k)) == iy) | ||
yisint = 2 - (j & 1); | ||
} | ||
} | ||
} | ||
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/* special value of y */ | ||
if (ly == 0) { | ||
if (iy == 0x7ff00000) { /* y is +-inf */ | ||
if (((ix - 0x3ff00000) | lx) == 0) | ||
return y - y; /* inf**+-1 is NaN */ | ||
else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */ | ||
return (hy >= 0) ? y : zero; | ||
else /* (|x|<1)**-,+inf = inf,0 */ | ||
return (hy < 0) ? -y : zero; | ||
} | ||
if (iy == 0x3ff00000) { /* y is +-1 */ | ||
if (hy < 0) | ||
return one / x; | ||
else | ||
return x; | ||
} | ||
if (hy == 0x40000000) | ||
return x * x; /* y is 2 */ | ||
if (hy == 0x3fe00000) { /* y is 0.5 */ | ||
if (hx >= 0) /* x >= +0 */ | ||
return __ieee754_sqrt(x); | ||
} | ||
} | ||
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ax = fabs(x); | ||
/* special value of x */ | ||
if (lx == 0) { | ||
if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) { | ||
z = ax; /*x is +-0,+-inf,+-1 */ | ||
if (hy < 0) | ||
z = one / z; /* z = (1/|x|) */ | ||
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; | ||
} | ||
} | ||
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/* (x<0)**(non-int) is NaN */ | ||
if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0) | ||
return (x - x) / (x - x); | ||
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/* |y| is huge */ | ||
if (iy > 0x41e00000) { /* if |y| > 2**31 */ | ||
if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */ | ||
if (ix <= 0x3fefffff) | ||
return (hy < 0) ? huge * huge : tiny * tiny; | ||
if (ix >= 0x3ff00000) | ||
return (hy > 0) ? huge * huge : tiny * tiny; | ||
} | ||
/* over/underflow if x is not close to one */ | ||
if (ix < 0x3fefffff) | ||
return (hy < 0) ? huge * huge : tiny * tiny; | ||
if (ix > 0x3ff00000) | ||
return (hy > 0) ? huge * huge : 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 */ | ||
t = x - 1; /* t has 20 trailing zeros */ | ||
w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25)); | ||
u = ivln2_h * t; /* ivln2_h has 21 sig. bits */ | ||
v = t * ivln2_l - w * ivln2; | ||
t1 = u + v; | ||
SET_LOW_WORD(t1, 0); | ||
t2 = v - (t1 - u); | ||
} else { | ||
double s2, s_h, s_l, t_h, t_l; | ||
n = 0; | ||
/* take care subnormal number */ | ||
if (ix < 0x00100000) { | ||
ax *= two53; | ||
n -= 53; | ||
GET_HIGH_WORD(ix, ax); | ||
} | ||
n += ((ix) >> 20) - 0x3ff; | ||
j = ix & 0x000fffff; | ||
/* determine interval */ | ||
ix = j | 0x3ff00000; /* normalize ix */ | ||
if (j <= 0x3988E) | ||
k = 0; /* |x|<sqrt(3/2) */ | ||
else if (j < 0xBB67A) | ||
k = 1; /* |x|<sqrt(3) */ | ||
else { | ||
k = 0; | ||
n += 1; | ||
ix -= 0x00100000; | ||
} | ||
SET_HIGH_WORD(ax, ix); | ||
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/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ | ||
u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ | ||
v = one / (ax + bp[k]); | ||
s = u * v; | ||
s_h = s; | ||
SET_LOW_WORD(s_h, 0); | ||
/* t_h=ax+bp[k] High */ | ||
t_h = zero; | ||
SET_HIGH_WORD(t_h, | ||
((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18)); | ||
t_l = ax - (t_h - bp[k]); | ||
s_l = v * ((u - s_h * t_h) - s_h * t_l); | ||
/* compute log(ax) */ | ||
s2 = s * s; | ||
r = s2 * s2 * (L1 + | ||
s2 * (L2 + | ||
s2 * (L3 + | ||
s2 * (L4 + s2 * (L5 + s2 * L6))))); | ||
r += s_l * (s_h + s); | ||
s2 = s_h * s_h; | ||
t_h = 3.0 + s2 + r; | ||
SET_LOW_WORD(t_h, 0); | ||
t_l = r - ((t_h - 3.0) - s2); | ||
/* u+v = s*(1+...) */ | ||
u = s_h * t_h; | ||
v = s_l * t_h + t_l * s; | ||
/* 2/(3log2)*(s+...) */ | ||
p_h = u + v; | ||
SET_LOW_WORD(p_h, 0); | ||
p_l = v - (p_h - u); | ||
z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */ | ||
z_l = cp_l * p_h + p_l * cp + dp_l[k]; | ||
/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ | ||
t = (double) n; | ||
t1 = (((z_h + z_l) + dp_h[k]) + t); | ||
SET_LOW_WORD(t1, 0); | ||
t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); | ||
} | ||
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s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ | ||
if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0) | ||
s = -one; /* (-ve)**(odd int) */ | ||
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/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ | ||
y1 = y; | ||
SET_LOW_WORD(y1, 0); | ||
p_l = (y - y1) * t1 + y * t2; | ||
p_h = y1 * t1; | ||
z = p_l + p_h; | ||
EXTRACT_WORDS(j, i, z); | ||
if (j >= 0x40900000) { /* z >= 1024 */ | ||
if (((j - 0x40900000) | i) != 0) /* if z > 1024 */ | ||
return s * huge * huge; /* overflow */ | ||
else { | ||
if (p_l + ovt > z - p_h) | ||
return s * huge * huge; /* overflow */ | ||
} | ||
} else if ((j & 0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */ | ||
if (((j - 0xc090cc00) | i) != 0) /* z < -1075 */ | ||
return s * tiny * tiny; /* underflow */ | ||
else { | ||
if (p_l <= z - p_h) | ||
return s * tiny * tiny; /* underflow */ | ||
} | ||
} | ||
/* | ||
* compute 2**(p_h+p_l) | ||
*/ | ||
i = j & 0x7fffffff; | ||
k = (i >> 20) - 0x3ff; | ||
n = 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 */ | ||
t = zero; | ||
SET_HIGH_WORD(t, n & ~(0x000fffff >> k)); | ||
n = ((n & 0x000fffff) | 0x00100000) >> (20 - k); | ||
if (j < 0) | ||
n = -n; | ||
p_h -= t; | ||
} | ||
t = p_l + p_h; | ||
SET_LOW_WORD(t, 0); | ||
u = t * lg2_h; | ||
v = (p_l - (t - p_h)) * lg2 + t * lg2_l; | ||
z = u + v; | ||
w = v - (z - u); | ||
t = z * z; | ||
t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); | ||
r = (z * t1) / (t1 - two) - (w + z * w); | ||
z = one - (r - z); | ||
GET_HIGH_WORD(j, z); | ||
j += (n << 20); | ||
if ((j >> 20) <= 0) | ||
z = scalbn(z, n); /* subnormal output */ | ||
else | ||
SET_HIGH_WORD(z, j); | ||
return s * z; | ||
} |
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