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jeremybenn |
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/*
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* ====================================================
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* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
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*
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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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/* __ieee754_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 muti-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) ** 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 NAN
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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 "fdlibm.h"
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#ifndef _DOUBLE_IS_32BITS
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#ifdef __STDC__
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static const double
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#else
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static double
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#endif
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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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#ifdef __STDC__
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double __ieee754_pow(double x, double y)
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#else
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double __ieee754_pow(x,y)
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double x, y;
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#endif
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{
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double z,ax,z_h,z_l,p_h,p_l;
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double y1,t1,t2,r,s,t,u,v,w;
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int32_t i0,i1,i,j,k,yisint,n;
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int32_t hx,hy,ix,iy;
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uint32_t lx,ly;
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i0 = ((*(int*)&one)>>29)^1; i1=1-i0;
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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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if((iy|ly)==0) return one;
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/* +-NaN return x+y */
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if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
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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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if(hx<0) {
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if(iy>=0x43400000) yisint = 2; /* even integer y */
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else if(iy>=0x3ff00000) {
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k = (iy>>20)-0x3ff; /* exponent */
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if(k>20) {
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j = ly>>(52-k);
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if((uint32_t)(j<<(52-k))==ly) yisint = 2-(j&1);
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} else if(ly==0) {
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j = iy>>(20-k);
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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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if(ly==0) {
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if (iy==0x7ff00000) { /* y is +-inf */
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if(((ix-0x3ff00000)|lx)==0)
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return y - y; /* inf**+-1 is NaN */
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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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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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if(hy==0x40000000) return x*x; /* y is 2 */
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if(hy==0x3fe00000) { /* y is 0.5 */
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if(hx>=0) /* x >= +0 */
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return __ieee754_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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if(lx==0) {
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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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if(hy<0) z = one/z; /* z = (1/|x|) */
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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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n = (hx>>31)+1;
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/* (x<0)**(non-int) is NaN */
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if((n|yisint)==0) return (x-x)/(x-x);
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s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
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if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
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| 187 |
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/* |y| is huge */
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if(iy>0x41e00000) { /* if |y| > 2**31 */
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| 190 |
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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;
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if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
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}
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| 194 |
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/* over/underflow if x is not close to one */
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if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
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| 196 |
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if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
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| 197 |
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/* now |1-x| is tiny <= 2**-20, suffice to compute
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log(x) by x-x^2/2+x^3/3-x^4/4 */
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| 199 |
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t = ax-one; /* t has 20 trailing zeros */
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| 200 |
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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 */
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| 202 |
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v = t*ivln2_l-w*ivln2;
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| 203 |
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t1 = u+v;
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SET_LOW_WORD(t1,0);
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| 205 |
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t2 = v-(t1-u);
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| 206 |
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} else {
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| 207 |
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double ss,s2,s_h,s_l,t_h,t_l;
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| 208 |
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n = 0;
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| 209 |
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/* take care subnormal number */
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| 210 |
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if(ix<0x00100000)
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| 211 |
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{ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
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| 212 |
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n += ((ix)>>20)-0x3ff;
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| 213 |
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j = ix&0x000fffff;
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| 214 |
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/* determine interval */
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| 215 |
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ix = j|0x3ff00000; /* normalize ix */
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| 216 |
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if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
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| 217 |
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else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
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| 218 |
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else {k=0;n+=1;ix -= 0x00100000;}
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| 219 |
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SET_HIGH_WORD(ax,ix);
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| 220 |
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| 221 |
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/* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
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| 222 |
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u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
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| 223 |
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v = one/(ax+bp[k]);
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| 224 |
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ss = u*v;
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| 225 |
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s_h = ss;
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| 226 |
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SET_LOW_WORD(s_h,0);
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| 227 |
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/* t_h=ax+bp[k] High */
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| 228 |
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t_h = zero;
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| 229 |
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SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
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| 230 |
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t_l = ax - (t_h-bp[k]);
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| 231 |
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s_l = v*((u-s_h*t_h)-s_h*t_l);
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| 232 |
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/* compute log(ax) */
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| 233 |
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s2 = ss*ss;
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| 234 |
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r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
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| 235 |
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r += s_l*(s_h+ss);
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| 236 |
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s2 = s_h*s_h;
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| 237 |
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t_h = 3.0+s2+r;
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| 238 |
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SET_LOW_WORD(t_h,0);
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| 239 |
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t_l = r-((t_h-3.0)-s2);
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| 240 |
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/* u+v = ss*(1+...) */
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| 241 |
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u = s_h*t_h;
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| 242 |
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v = s_l*t_h+t_l*ss;
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| 243 |
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/* 2/(3log2)*(ss+...) */
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| 244 |
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p_h = u+v;
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| 245 |
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SET_LOW_WORD(p_h,0);
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| 246 |
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p_l = v-(p_h-u);
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| 247 |
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z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
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| 248 |
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z_l = cp_l*p_h+p_l*cp+dp_l[k];
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| 249 |
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/* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
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| 250 |
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t = (double)n;
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| 251 |
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t1 = (((z_h+z_l)+dp_h[k])+t);
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| 252 |
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SET_LOW_WORD(t1,0);
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| 253 |
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t2 = z_l-(((t1-t)-dp_h[k])-z_h);
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| 254 |
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}
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| 255 |
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| 256 |
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/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
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| 257 |
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y1 = y;
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| 258 |
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SET_LOW_WORD(y1,0);
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| 259 |
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p_l = (y-y1)*t1+y*t2;
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| 260 |
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p_h = y1*t1;
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| 261 |
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z = p_l+p_h;
|
| 262 |
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EXTRACT_WORDS(j,i,z);
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| 263 |
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if (j>=0x40900000) { /* z >= 1024 */
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| 264 |
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if(((j-0x40900000)|i)!=0) /* if z > 1024 */
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| 265 |
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return s*huge*huge; /* overflow */
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| 266 |
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else {
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| 267 |
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if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
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| 268 |
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}
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| 269 |
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} else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
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| 270 |
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if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
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| 271 |
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return s*tiny*tiny; /* underflow */
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| 272 |
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else {
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| 273 |
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if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
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| 274 |
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}
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| 275 |
|
|
}
|
| 276 |
|
|
/*
|
| 277 |
|
|
* compute 2**(p_h+p_l)
|
| 278 |
|
|
*/
|
| 279 |
|
|
i = j&0x7fffffff;
|
| 280 |
|
|
k = (i>>20)-0x3ff;
|
| 281 |
|
|
n = 0;
|
| 282 |
|
|
if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
|
| 283 |
|
|
n = j+(0x00100000>>(k+1));
|
| 284 |
|
|
k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
|
| 285 |
|
|
t = zero;
|
| 286 |
|
|
SET_HIGH_WORD(t,(n&~(0x000fffff>>k)));
|
| 287 |
|
|
n = ((n&0x000fffff)|0x00100000)>>(20-k);
|
| 288 |
|
|
if(j<0) n = -n;
|
| 289 |
|
|
p_h -= t;
|
| 290 |
|
|
}
|
| 291 |
|
|
t = p_l+p_h;
|
| 292 |
|
|
SET_LOW_WORD(t,0);
|
| 293 |
|
|
u = t*lg2_h;
|
| 294 |
|
|
v = (p_l-(t-p_h))*lg2+t*lg2_l;
|
| 295 |
|
|
z = u+v;
|
| 296 |
|
|
w = v-(z-u);
|
| 297 |
|
|
t = z*z;
|
| 298 |
|
|
t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
|
| 299 |
|
|
r = (z*t1)/(t1-two)-(w+z*w);
|
| 300 |
|
|
z = one-(r-z);
|
| 301 |
|
|
GET_HIGH_WORD(j,z);
|
| 302 |
|
|
j += (n<<20);
|
| 303 |
|
|
if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
|
| 304 |
|
|
else
|
| 305 |
|
|
{
|
| 306 |
|
|
uint32_t hz;
|
| 307 |
|
|
GET_HIGH_WORD(hz,z);
|
| 308 |
|
|
SET_HIGH_WORD(z,hz + (n<<20));
|
| 309 |
|
|
}
|
| 310 |
|
|
return s*z;
|
| 311 |
|
|
}
|
| 312 |
|
|
#endif /* defined(_DOUBLE_IS_32BITS) */
|
