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/* @(#)e_pow.c 5.1 93/09/24 */ |
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/* |
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* ==================================================== |
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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* |
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* Developed at SunPro, a Sun Microsystems, Inc. business. |
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* Permission to use, copy, modify, and distribute this |
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* software is freely granted, provided that this notice |
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* is preserved. |
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* ==================================================== |
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*/ |
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/* pow(x,y) return x**y |
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* |
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* n |
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* Method: Let x = 2 * (1+f) |
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* 1. Compute and return log2(x) in two pieces: |
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* log2(x) = w1 + w2, |
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* where w1 has 53-24 = 29 bit trailing zeros. |
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* 2. Perform y*log2(x) = n+y' by simulating multi-precision |
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* arithmetic, where |y'|<=0.5. |
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* 3. Return x**y = 2**n*exp(y'*log2) |
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* |
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* Special cases: |
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* 1. (anything) ** 0 is 1 |
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* 2. (anything) ** 1 is itself |
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* 3. (anything except 1) ** NAN is NAN |
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* 4. NAN ** (anything except 0) is NAN |
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* 5. +-(|x| > 1) ** +INF is +INF |
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* 6. +-(|x| > 1) ** -INF is +0 |
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* 7. +-(|x| < 1) ** +INF is +0 |
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* 8. +-(|x| < 1) ** -INF is +INF |
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* 9. +-1 ** +-INF is 1 |
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* 10. +0 ** (+anything except 0, NAN) is +0 |
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* 11. -0 ** (+anything except 0, NAN, odd integer) is +0 |
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* 12. +0 ** (-anything except 0, NAN) is +INF |
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* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF |
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* 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) |
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* 15. +INF ** (+anything except 0,NAN) is +INF |
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* 16. +INF ** (-anything except 0,NAN) is +0 |
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* 17. -INF ** (anything) = -0 ** (-anything) |
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* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) |
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* 19. (-anything except 0 and inf) ** (non-integer) is NAN |
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* |
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* Accuracy: |
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* pow(x,y) returns x**y nearly rounded. In particular |
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* pow(integer,integer) |
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* always returns the correct integer provided it is |
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* representable. |
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* |
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* Constants : |
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* The hexadecimal values are the intended ones for the following |
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* constants. The decimal values may be used, provided that the |
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* compiler will convert from decimal to binary accurately enough |
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* to produce the hexadecimal values shown. |
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*/ |
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#include <float.h> |
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#include <math.h> |
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#include "math_private.h" |
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static const double |
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bp[] = {1.0, 1.5,}, |
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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, |
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one = 1.0, |
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two = 2.0, |
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two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ |
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huge = 1.0e300, |
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tiny = 1.0e-300, |
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/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ |
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L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ |
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L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ |
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L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ |
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L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ |
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L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ |
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L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ |
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P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ |
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P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ |
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P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ |
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P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ |
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P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ |
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lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ |
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lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ |
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lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ |
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ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ |
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cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ |
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cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ |
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cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ |
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ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ |
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ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ |
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ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ |
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double |
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pow(double x, double y) |
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{ |
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double z,ax,z_h,z_l,p_h,p_l; |
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double yy1,t1,t2,r,s,t,u,v,w; |
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int32_t i,j,k,yisint,n; |
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int32_t hx,hy,ix,iy; |
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u_int32_t lx,ly; |
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12930 |
EXTRACT_WORDS(hx,lx,x); |
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EXTRACT_WORDS(hy,ly,y); |
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ix = hx&0x7fffffff; iy = hy&0x7fffffff; |
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/* y==zero: x**0 = 1 */ |
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✗✓ |
6465 |
if((iy|ly)==0) return one; |
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/* x==1: 1**y = 1, even if y is NaN */ |
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✗✓ |
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if (hx==0x3ff00000 && lx == 0) return one; |
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/* +-NaN return x+y */ |
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✓✗✓✗
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19395 |
if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || |
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✗✓ |
12930 |
iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) |
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return x+y; |
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/* determine if y is an odd int when x < 0 |
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* yisint = 0 ... y is not an integer |
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* yisint = 1 ... y is an odd int |
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* yisint = 2 ... y is an even int |
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*/ |
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yisint = 0; |
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✓✓ |
6465 |
if(hx<0) { |
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✗✓ |
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if(iy>=0x43400000) yisint = 2; /* even integer y */ |
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✓✗ |
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else if(iy>=0x3ff00000) { |
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k = (iy>>20)-0x3ff; /* exponent */ |
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✗✓ |
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if(k>20) { |
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j = ly>>(52-k); |
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if((j<<(52-k))==ly) yisint = 2-(j&1); |
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✓✗ |
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} else if(ly==0) { |
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j = iy>>(20-k); |
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✓✗ |
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if((j<<(20-k))==iy) yisint = 2-(j&1); |
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} |
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} |
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} |
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/* special value of y */ |
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✓✗ |
6465 |
if(ly==0) { |
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✗✓ |
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if (iy==0x7ff00000) { /* y is +-inf */ |
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if(((ix-0x3ff00000)|lx)==0) |
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return one; /* (-1)**+-inf is 1 */ |
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else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ |
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return (hy>=0)? y: zero; |
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else /* (|x|<1)**-,+inf = inf,0 */ |
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return (hy<0)?-y: zero; |
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} |
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✗✓ |
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if(iy==0x3ff00000) { /* y is +-1 */ |
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if(hy<0) return one/x; else return x; |
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} |
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✓✓ |
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if(hy==0x40000000) return x*x; /* y is 2 */ |
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✗✓ |
12926 |
if(hy==0x3fe00000) { /* y is 0.5 */ |
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if(hx>=0) /* x >= +0 */ |
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return sqrt(x); |
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} |
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} |
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ax = fabs(x); |
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/* special value of x */ |
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✓✓ |
6463 |
if(lx==0) { |
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✓✓ |
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if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ |
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z = ax; /*x is +-0,+-inf,+-1*/ |
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✗✓ |
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if(hy<0) z = one/z; /* z = (1/|x|) */ |
166 |
✗✓ |
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if(hx<0) { |
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if(((ix-0x3ff00000)|yisint)==0) { |
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z = (z-z)/(z-z); /* (-1)**non-int is NaN */ |
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} else if(yisint==1) |
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z = -z; /* (x<0)**odd = -(|x|**odd) */ |
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} |
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return z; |
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} |
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} |
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176 |
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n = (hx>>31)+1; |
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178 |
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/* (x<0)**(non-int) is NaN */ |
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✗✓ |
6457 |
if((n|yisint)==0) return (x-x)/(x-x); |
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181 |
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s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ |
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6457 |
if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ |
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184 |
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/* |y| is huge */ |
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✗✓ |
6457 |
if(iy>0x41e00000) { /* if |y| > 2**31 */ |
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if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ |
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if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; |
188 |
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if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; |
189 |
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} |
190 |
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/* over/underflow if x is not close to one */ |
191 |
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if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny; |
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if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny; |
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/* now |1-x| is tiny <= 2**-20, suffice to compute |
194 |
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log(x) by x-x^2/2+x^3/3-x^4/4 */ |
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t = ax-one; /* t has 20 trailing zeros */ |
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w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); |
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u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ |
198 |
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v = t*ivln2_l-w*ivln2; |
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t1 = u+v; |
200 |
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SET_LOW_WORD(t1,0); |
201 |
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t2 = v-(t1-u); |
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} else { |
203 |
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double ss,s2,s_h,s_l,t_h,t_l; |
204 |
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n = 0; |
205 |
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/* take care subnormal number */ |
206 |
✗✓ |
6457 |
if(ix<0x00100000) |
207 |
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{ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } |
208 |
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6457 |
n += ((ix)>>20)-0x3ff; |
209 |
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6457 |
j = ix&0x000fffff; |
210 |
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/* determine interval */ |
211 |
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6457 |
ix = j|0x3ff00000; /* normalize ix */ |
212 |
✓✓ |
7520 |
if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ |
213 |
✓✓ |
9666 |
else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ |
214 |
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1122 |
else {k=0;n+=1;ix -= 0x00100000;} |
215 |
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6457 |
SET_HIGH_WORD(ax,ix); |
216 |
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217 |
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/* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ |
218 |
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6457 |
u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ |
219 |
|
6457 |
v = one/(ax+bp[k]); |
220 |
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6457 |
ss = u*v; |
221 |
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s_h = ss; |
222 |
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6457 |
SET_LOW_WORD(s_h,0); |
223 |
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/* t_h=ax+bp[k] High */ |
224 |
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t_h = zero; |
225 |
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6457 |
SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); |
226 |
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6457 |
t_l = ax - (t_h-bp[k]); |
227 |
|
6457 |
s_l = v*((u-s_h*t_h)-s_h*t_l); |
228 |
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/* compute log(ax) */ |
229 |
|
6457 |
s2 = ss*ss; |
230 |
|
6457 |
r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); |
231 |
|
6457 |
r += s_l*(s_h+ss); |
232 |
|
6457 |
s2 = s_h*s_h; |
233 |
|
6457 |
t_h = 3.0+s2+r; |
234 |
|
6457 |
SET_LOW_WORD(t_h,0); |
235 |
|
6457 |
t_l = r-((t_h-3.0)-s2); |
236 |
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/* u+v = ss*(1+...) */ |
237 |
|
6457 |
u = s_h*t_h; |
238 |
|
6457 |
v = s_l*t_h+t_l*ss; |
239 |
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/* 2/(3log2)*(ss+...) */ |
240 |
|
6457 |
p_h = u+v; |
241 |
|
6457 |
SET_LOW_WORD(p_h,0); |
242 |
|
6457 |
p_l = v-(p_h-u); |
243 |
|
6457 |
z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ |
244 |
|
6457 |
z_l = cp_l*p_h+p_l*cp+dp_l[k]; |
245 |
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/* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ |
246 |
|
6457 |
t = (double)n; |
247 |
|
6457 |
t1 = (((z_h+z_l)+dp_h[k])+t); |
248 |
|
6457 |
SET_LOW_WORD(t1,0); |
249 |
|
6457 |
t2 = z_l-(((t1-t)-dp_h[k])-z_h); |
250 |
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} |
251 |
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252 |
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/* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */ |
253 |
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yy1 = y; |
254 |
|
6457 |
SET_LOW_WORD(yy1,0); |
255 |
|
6457 |
p_l = (y-yy1)*t1+y*t2; |
256 |
|
6457 |
p_h = yy1*t1; |
257 |
|
6457 |
z = p_l+p_h; |
258 |
|
6457 |
EXTRACT_WORDS(j,i,z); |
259 |
✓✓ |
6457 |
if (j>=0x40900000) { /* z >= 1024 */ |
260 |
✗✓ |
6 |
if(((j-0x40900000)|i)!=0) /* if z > 1024 */ |
261 |
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return s*huge*huge; /* overflow */ |
262 |
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else { |
263 |
✗✓ |
6 |
if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ |
264 |
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} |
265 |
✗✓ |
6451 |
} else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ |
266 |
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if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ |
267 |
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return s*tiny*tiny; /* underflow */ |
268 |
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else { |
269 |
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if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ |
270 |
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} |
271 |
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} |
272 |
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/* |
273 |
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* compute 2**(p_h+p_l) |
274 |
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*/ |
275 |
|
6457 |
i = j&0x7fffffff; |
276 |
|
6457 |
k = (i>>20)-0x3ff; |
277 |
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n = 0; |
278 |
✓✓ |
6457 |
if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ |
279 |
|
6271 |
n = j+(0x00100000>>(k+1)); |
280 |
|
6271 |
k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ |
281 |
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t = zero; |
282 |
|
6271 |
SET_HIGH_WORD(t,n&~(0x000fffff>>k)); |
283 |
|
6271 |
n = ((n&0x000fffff)|0x00100000)>>(20-k); |
284 |
✓✓ |
9380 |
if(j<0) n = -n; |
285 |
|
6271 |
p_h -= t; |
286 |
|
6271 |
} |
287 |
|
6457 |
t = p_l+p_h; |
288 |
|
6457 |
SET_LOW_WORD(t,0); |
289 |
|
6457 |
u = t*lg2_h; |
290 |
|
6457 |
v = (p_l-(t-p_h))*lg2+t*lg2_l; |
291 |
|
6457 |
z = u+v; |
292 |
|
6457 |
w = v-(z-u); |
293 |
|
6457 |
t = z*z; |
294 |
|
6457 |
t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); |
295 |
|
6457 |
r = (z*t1)/(t1-two)-(w+z*w); |
296 |
|
6457 |
z = one-(r-z); |
297 |
|
6457 |
GET_HIGH_WORD(j,z); |
298 |
|
6457 |
j += (n<<20); |
299 |
✗✓ |
6457 |
if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ |
300 |
|
6457 |
else SET_HIGH_WORD(z,j); |
301 |
|
6457 |
return s*z; |
302 |
|
6465 |
} |
303 |
|
|
DEF_STD(pow); |
304 |
|
|
LDBL_MAYBE_CLONE(pow); |