src/libm/e_pow.c
author Gabriel Jacobo <gabomdq@gmail.com>
Wed, 21 Aug 2013 09:47:10 -0300
changeset 7678 286c42d7c5ed
parent 6044 35448a5ea044
child 11683 48bcba563d9c
permissions -rw-r--r--
OCD fixes: Adds a space after /* (glory to regular expressions!)
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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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#if defined(LIBM_SCCS) && !defined(lint)
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static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $";
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#endif
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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 "math_libm.h"
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#include "math_private.h"
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libm_hidden_proto(scalbn)
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    libm_hidden_proto(fabs)
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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, }, dp_h[] = {
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     0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
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         dp_l[] = {
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     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 */
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         huge_val = 1.0e300, 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 attribute_hidden __ieee754_pow(double x, double y)
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#else
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     double attribute_hidden __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 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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         EXTRACT_WORDS(hx, lx, x);
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         EXTRACT_WORDS(hy, ly, y);
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         ix = hx & 0x7fffffff;
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         iy = hy & 0x7fffffff;
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         /* y==zero: x**0 = 1 */
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         if ((iy | ly) == 0)
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             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)
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                 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 ((j << (52 - k)) == ly)
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                         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)
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                         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)
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                     return one / x;
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                 else
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                     return x;
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             }
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             if (hy == 0x40000000)
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                 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)
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                     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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         /* (x<0)**(non-int) is NaN */
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         if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
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             return (x - x) / (x - x);
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         /* |y| is huge */
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         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)
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                     return (hy < 0) ? huge_val * huge_val : tiny * tiny;
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                 if (ix >= 0x3ff00000)
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                     return (hy > 0) ? huge_val * huge_val : tiny * tiny;
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             }
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             /* over/underflow if x is not close to one */
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             if (ix < 0x3fefffff)
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                 return (hy < 0) ? huge_val * huge_val : tiny * tiny;
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             if (ix > 0x3ff00000)
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                 return (hy > 0) ? huge_val * huge_val : tiny * tiny;
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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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             t = x - 1;         /* 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 */
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             v = t * ivln2_l - w * ivln2;
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             t1 = u + v;
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             SET_LOW_WORD(t1, 0);
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             t2 = v - (t1 - u);
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         } else {
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             double s2, s_h, s_l, t_h, t_l;
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             n = 0;
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             /* take care subnormal number */
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             if (ix < 0x00100000) {
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                 ax *= two53;
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                 n -= 53;
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                 GET_HIGH_WORD(ix, ax);
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             }
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             n += ((ix) >> 20) - 0x3ff;
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             j = ix & 0x000fffff;
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             /* determine interval */
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             ix = j | 0x3ff00000;       /* normalize ix */
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             if (j <= 0x3988E)
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                 k = 0;         /* |x|<sqrt(3/2) */
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             else if (j < 0xBB67A)
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                 k = 1;         /* |x|<sqrt(3)   */
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             else {
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                 k = 0;
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                 n += 1;
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                 ix -= 0x00100000;
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             }
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             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) */
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             u = ax - bp[k];    /* bp[0]=1.0, bp[1]=1.5 */
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             v = one / (ax + bp[k]);
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             s = u * v;
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             s_h = s;
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             SET_LOW_WORD(s_h, 0);
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             /* t_h=ax+bp[k] High */
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             t_h = zero;
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             SET_HIGH_WORD(t_h,
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                           ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
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             t_l = ax - (t_h - bp[k]);
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             s_l = v * ((u - s_h * t_h) - s_h * t_l);
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             /* compute log(ax) */
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             s2 = s * s;
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             r = s2 * s2 * (L1 +
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                            s2 * (L2 +
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                                  s2 * (L3 +
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                                        s2 * (L4 + s2 * (L5 + s2 * L6)))));
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             r += s_l * (s_h + s);
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             s2 = s_h * s_h;
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             t_h = 3.0 + s2 + r;
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             SET_LOW_WORD(t_h, 0);
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             t_l = r - ((t_h - 3.0) - s2);
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             /* u+v = s*(1+...) */
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             u = s_h * t_h;
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             v = s_l * t_h + t_l * s;
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             /* 2/(3log2)*(s+...) */
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             p_h = u + v;
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             SET_LOW_WORD(p_h, 0);
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             p_l = v - (p_h - u);
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             z_h = cp_h * p_h;  /* cp_h+cp_l = 2/(3*log2) */
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             z_l = cp_l * p_h + p_l * cp + dp_l[k];
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             /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
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             t = (double) n;
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             t1 = (((z_h + z_l) + dp_h[k]) + t);
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             SET_LOW_WORD(t1, 0);
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             t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
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         }
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         s = one;               /* s (sign of result -ve**odd) = -1 else = 1 */
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         if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
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             s = -one;          /* (-ve)**(odd int) */
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         /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
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         y1 = y;
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         SET_LOW_WORD(y1, 0);
slouken@2756
   290
         p_l = (y - y1) * t1 + y * t2;
slouken@2756
   291
         p_h = y1 * t1;
slouken@2756
   292
         z = p_l + p_h;
slouken@2756
   293
         EXTRACT_WORDS(j, i, z);
slouken@2756
   294
         if (j >= 0x40900000) { /* z >= 1024 */
slouken@2756
   295
             if (((j - 0x40900000) | i) != 0)   /* if z > 1024 */
slouken@3337
   296
                 return s * huge_val * huge_val;        /* overflow */
slouken@2756
   297
             else {
slouken@2756
   298
                 if (p_l + ovt > z - p_h)
slouken@3337
   299
                     return s * huge_val * huge_val;    /* overflow */
slouken@2756
   300
             }
slouken@2756
   301
         } else if ((j & 0x7fffffff) >= 0x4090cc00) {   /* z <= -1075 */
slouken@2756
   302
             if (((j - 0xc090cc00) | i) != 0)   /* z < -1075 */
slouken@2756
   303
                 return s * tiny * tiny;        /* underflow */
slouken@2756
   304
             else {
slouken@2756
   305
                 if (p_l <= z - p_h)
slouken@2756
   306
                     return s * tiny * tiny;    /* underflow */
slouken@2756
   307
             }
slouken@2756
   308
         }
slouken@2756
   309
         /*
slouken@2756
   310
          * compute 2**(p_h+p_l)
slouken@2756
   311
          */
slouken@2756
   312
         i = j & 0x7fffffff;
slouken@2756
   313
         k = (i >> 20) - 0x3ff;
slouken@2756
   314
         n = 0;
slouken@2756
   315
         if (i > 0x3fe00000) {  /* if |z| > 0.5, set n = [z+0.5] */
slouken@2756
   316
             n = j + (0x00100000 >> (k + 1));
slouken@2756
   317
             k = ((n & 0x7fffffff) >> 20) - 0x3ff;      /* new k for n */
slouken@2756
   318
             t = zero;
slouken@2756
   319
             SET_HIGH_WORD(t, n & ~(0x000fffff >> k));
slouken@2756
   320
             n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
slouken@2756
   321
             if (j < 0)
slouken@2756
   322
                 n = -n;
slouken@2756
   323
             p_h -= t;
slouken@2756
   324
         }
slouken@2756
   325
         t = p_l + p_h;
slouken@2756
   326
         SET_LOW_WORD(t, 0);
slouken@2756
   327
         u = t * lg2_h;
slouken@2756
   328
         v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
slouken@2756
   329
         z = u + v;
slouken@2756
   330
         w = v - (z - u);
slouken@2756
   331
         t = z * z;
slouken@2756
   332
         t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
slouken@2756
   333
         r = (z * t1) / (t1 - two) - (w + z * w);
slouken@2756
   334
         z = one - (r - z);
slouken@2756
   335
         GET_HIGH_WORD(j, z);
slouken@2756
   336
         j += (n << 20);
slouken@2756
   337
         if ((j >> 20) <= 0)
slouken@2756
   338
             z = scalbn(z, n);  /* subnormal output */
slouken@2756
   339
         else
slouken@2756
   340
             SET_HIGH_WORD(z, j);
slouken@2756
   341
         return s * z;
slouken@2756
   342
     }