src/video/e_pow.h
author Sam Lantinga <slouken@libsdl.org>
Sun, 28 May 2006 13:04:16 +0000
branchSDL-1.3
changeset 1662 782fd950bd46
parent 1424 7a610f25c12f
child 1668 4da1ee79c9af
permissions -rw-r--r--
Revamp of the video system in progress - adding support for multiple displays, multiple windows, and a full video mode selection API. WARNING: None of the video drivers have been updated for the new API yet! The API is still under design and very fluid. The code is now run through a consistent indent format: indent -i4 -nut -nsc -br -ce The headers are being converted to automatically generate doxygen documentation.
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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.h"*/
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#include "math_private.h"
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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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    /* 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
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__ieee754_pow (double x, double y)
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#else
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double
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__ieee754_pow (x, y)
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     double x, y;
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#endif
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   107
{
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   108
    double z, ax, z_h, z_l, p_h, p_l;
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   109
    double y1, t1, t2, r, s, t, u, v, w;
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   110
    int32_t i, j, k, yisint, n;
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   111
    int32_t hx, hy, ix, iy;
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   112
    u_int32_t lx, ly;
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   113
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   114
    EXTRACT_WORDS (hx, lx, x);
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   115
    EXTRACT_WORDS (hy, ly, y);
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   116
    ix = hx & 0x7fffffff;
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   117
    iy = hy & 0x7fffffff;
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   118
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   119
    /* y==zero: x**0 = 1 */
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   120
    if ((iy | ly) == 0)
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   121
        return one;
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   122
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   123
    /* +-NaN return x+y */
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   124
    if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) ||
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   125
        iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0)))
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   126
        return x + y;
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   127
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   128
    /* determine if y is an odd int when x < 0
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   129
     * yisint = 0       ... y is not an integer
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   130
     * yisint = 1       ... y is an odd int
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   131
     * yisint = 2       ... y is an even int
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   132
     */
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   133
    yisint = 0;
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   134
    if (hx < 0) {
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   135
        if (iy >= 0x43400000)
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   136
            yisint = 2;         /* even integer y */
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   137
        else if (iy >= 0x3ff00000) {
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   138
            k = (iy >> 20) - 0x3ff;     /* exponent */
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   139
            if (k > 20) {
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   140
                j = ly >> (52 - k);
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   141
                if ((u_int32_t) (j << (52 - k)) == ly)
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   142
                    yisint = 2 - (j & 1);
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   143
            } else if (ly == 0) {
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   144
                j = iy >> (20 - k);
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   145
                if ((j << (20 - k)) == iy)
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   146
                    yisint = 2 - (j & 1);
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   147
            }
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   148
        }
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   149
    }
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   150
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   151
    /* special value of y */
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   152
    if (ly == 0) {
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   153
        if (iy == 0x7ff00000) { /* y is +-inf */
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   154
            if (((ix - 0x3ff00000) | lx) == 0)
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   155
                return y - y;   /* inf**+-1 is NaN */
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   156
            else if (ix >= 0x3ff00000)  /* (|x|>1)**+-inf = inf,0 */
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   157
                return (hy >= 0) ? y : zero;
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   158
            else                /* (|x|<1)**-,+inf = inf,0 */
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   159
                return (hy < 0) ? -y : zero;
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   160
        }
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   161
        if (iy == 0x3ff00000) { /* y is  +-1 */
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   162
            if (hy < 0)
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   163
                return one / x;
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   164
            else
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   165
                return x;
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   166
        }
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   167
        if (hy == 0x40000000)
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   168
            return x * x;       /* y is  2 */
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   169
        if (hy == 0x3fe00000) { /* y is  0.5 */
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   170
            if (hx >= 0)        /* x >= +0 */
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   171
                return __ieee754_sqrt (x);
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   172
        }
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   173
    }
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   174
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   175
    ax = x < 0 ? -x : x;        /*fabs(x); */
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   176
    /* special value of x */
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   177
    if (lx == 0) {
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   178
        if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) {
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   179
            z = ax;             /*x is +-0,+-inf,+-1 */
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   180
            if (hy < 0)
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   181
                z = one / z;    /* z = (1/|x|) */
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   182
            if (hx < 0) {
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   183
                if (((ix - 0x3ff00000) | yisint) == 0) {
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   184
                    z = (z - z) / (z - z);      /* (-1)**non-int is NaN */
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   185
                } else if (yisint == 1)
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   186
                    z = -z;     /* (x<0)**odd = -(|x|**odd) */
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   187
            }
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   188
            return z;
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   189
        }
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   190
    }
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   191
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   192
    /* (x<0)**(non-int) is NaN */
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   193
    if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
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   194
        return (x - x) / (x - x);
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   195
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   196
    /* |y| is huge */
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   197
    if (iy > 0x41e00000) {      /* if |y| > 2**31 */
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   198
        if (iy > 0x43f00000) {  /* if |y| > 2**64, must o/uflow */
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   199
            if (ix <= 0x3fefffff)
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   200
                return (hy < 0) ? huge * huge : tiny * tiny;
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   201
            if (ix >= 0x3ff00000)
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   202
                return (hy > 0) ? huge * huge : tiny * tiny;
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diff changeset
   203
        }
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   204
        /* over/underflow if x is not close to one */
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   205
        if (ix < 0x3fefffff)
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   206
            return (hy < 0) ? huge * huge : tiny * tiny;
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diff changeset
   207
        if (ix > 0x3ff00000)
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   208
            return (hy > 0) ? huge * huge : tiny * tiny;
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   209
        /* now |1-x| is tiny <= 2**-20, suffice to compute
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   210
           log(x) by x-x^2/2+x^3/3-x^4/4 */
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   211
        t = x - 1;              /* t has 20 trailing zeros */
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   212
        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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   225
            GET_HIGH_WORD (ix, ax);
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   226
        }
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   227
        n += ((ix) >> 20) - 0x3ff;
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        j = ix & 0x000fffff;
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   229
        /* determine interval */
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   230
        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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   233
        else if (j < 0xBB67A)
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   234
            k = 1;              /* |x|<sqrt(3)   */
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   235
        else {
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   236
            k = 0;
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   237
            n += 1;
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   238
            ix -= 0x00100000;
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   239
        }
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   240
        SET_HIGH_WORD (ax, ix);
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1662
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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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   243
        u = ax - bp[k];         /* bp[0]=1.0, bp[1]=1.5 */
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   244
        v = one / (ax + bp[k]);
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   245
        s = u * v;
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   246
        s_h = s;
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   247
        SET_LOW_WORD (s_h, 0);
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   248
        /* t_h=ax+bp[k] High */
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   249
        t_h = zero;
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   250
        SET_HIGH_WORD (t_h,
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   251
                       ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
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   252
        t_l = ax - (t_h - bp[k]);
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   253
        s_l = v * ((u - s_h * t_h) - s_h * t_l);
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   254
        /* compute log(ax) */
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   255
        s2 = s * s;
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   256
        r = s2 * s2 * (L1 +
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   257
                       s2 * (L2 +
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   258
                             s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
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   259
        r += s_l * (s_h + s);
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   260
        s2 = s_h * s_h;
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   261
        t_h = 3.0 + s2 + r;
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   262
        SET_LOW_WORD (t_h, 0);
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   263
        t_l = r - ((t_h - 3.0) - s2);
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   264
        /* u+v = s*(1+...) */
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   265
        u = s_h * t_h;
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   266
        v = s_l * t_h + t_l * s;
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   267
        /* 2/(3log2)*(s+...) */
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   268
        p_h = u + v;
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   269
        SET_LOW_WORD (p_h, 0);
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   270
        p_l = v - (p_h - u);
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   271
        z_h = cp_h * p_h;       /* cp_h+cp_l = 2/(3*log2) */
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   272
        z_l = cp_l * p_h + p_l * cp + dp_l[k];
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   273
        /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
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   274
        t = (double) n;
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   275
        t1 = (((z_h + z_l) + dp_h[k]) + t);
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   276
        SET_LOW_WORD (t1, 0);
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   277
        t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
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   278
    }
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   279
1662
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   280
    s = one;                    /* s (sign of result -ve**odd) = -1 else = 1 */
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   281
    if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
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   282
        s = -one;               /* (-ve)**(odd int) */
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   283
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   284
    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
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   285
    y1 = y;
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   286
    SET_LOW_WORD (y1, 0);
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   287
    p_l = (y - y1) * t1 + y * t2;
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   288
    p_h = y1 * t1;
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   289
    z = p_l + p_h;
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   290
    EXTRACT_WORDS (j, i, z);
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   291
    if (j >= 0x40900000) {      /* z >= 1024 */
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   292
        if (((j - 0x40900000) | i) != 0)        /* if z > 1024 */
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   293
            return s * huge * huge;     /* overflow */
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   294
        else {
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   295
            if (p_l + ovt > z - p_h)
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   296
                return s * huge * huge; /* overflow */
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   297
        }
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   298
    } else if ((j & 0x7fffffff) >= 0x4090cc00) {        /* z <= -1075 */
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   299
        if (((j - 0xc090cc00) | i) != 0)        /* z < -1075 */
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   300
            return s * tiny * tiny;     /* underflow */
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   301
        else {
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   302
            if (p_l <= z - p_h)
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   303
                return s * tiny * tiny; /* underflow */
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   304
        }
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   305
    }
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   306
    /*
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   307
     * compute 2**(p_h+p_l)
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   308
     */
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   309
    i = j & 0x7fffffff;
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   310
    k = (i >> 20) - 0x3ff;
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   311
    n = 0;
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   312
    if (i > 0x3fe00000) {       /* if |z| > 0.5, set n = [z+0.5] */
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   313
        n = j + (0x00100000 >> (k + 1));
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   314
        k = ((n & 0x7fffffff) >> 20) - 0x3ff;   /* new k for n */
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   315
        t = zero;
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   316
        SET_HIGH_WORD (t, n & ~(0x000fffff >> k));
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        n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
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   318
        if (j < 0)
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   319
            n = -n;
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   320
        p_h -= t;
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   321
    }
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   322
    t = p_l + p_h;
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   323
    SET_LOW_WORD (t, 0);
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   324
    u = t * lg2_h;
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   325
    v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
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   326
    z = u + v;
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   327
    w = v - (z - u);
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   328
    t = z * z;
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   329
    t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
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   330
    r = (z * t1) / (t1 - two) - (w + z * w);
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   331
    z = one - (r - z);
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   332
    GET_HIGH_WORD (j, z);
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   333
    j += (n << 20);
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   334
    if ((j >> 20) <= 0)
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   335
        z = SDL_NAME (scalbn) (z, n);   /* subnormal output */
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   336
    else
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   337
        SET_HIGH_WORD (z, j);
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   338
    return s * z;
1330
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diff changeset
   339
}
1662
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   340
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   341
/* vi: set ts=4 sw=4 expandtab: */