src/video/e_pow.h
author Sam Lantinga <slouken@libsdl.org>
Mon, 10 Jul 2006 21:04:37 +0000
changeset 1895 c121d94672cb
parent 1424 7a610f25c12f
child 2752 edd2839b36f7
permissions -rw-r--r--
SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head.
     1 /* @(#)e_pow.c 5.1 93/09/24 */
     2 /*
     3  * ====================================================
     4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
     5  *
     6  * Developed at SunPro, a Sun Microsystems, Inc. business.
     7  * Permission to use, copy, modify, and distribute this
     8  * software is freely granted, provided that this notice
     9  * is preserved.
    10  * ====================================================
    11  */
    12 
    13 #if defined(LIBM_SCCS) && !defined(lint)
    14 static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $";
    15 #endif
    16 
    17 /* __ieee754_pow(x,y) return x**y
    18  *
    19  *		      n
    20  * Method:  Let x =  2   * (1+f)
    21  *	1. Compute and return log2(x) in two pieces:
    22  *		log2(x) = w1 + w2,
    23  *	   where w1 has 53-24 = 29 bit trailing zeros.
    24  *	2. Perform y*log2(x) = n+y' by simulating muti-precision
    25  *	   arithmetic, where |y'|<=0.5.
    26  *	3. Return x**y = 2**n*exp(y'*log2)
    27  *
    28  * Special cases:
    29  *	1.  (anything) ** 0  is 1
    30  *	2.  (anything) ** 1  is itself
    31  *	3.  (anything) ** NAN is NAN
    32  *	4.  NAN ** (anything except 0) is NAN
    33  *	5.  +-(|x| > 1) **  +INF is +INF
    34  *	6.  +-(|x| > 1) **  -INF is +0
    35  *	7.  +-(|x| < 1) **  +INF is +0
    36  *	8.  +-(|x| < 1) **  -INF is +INF
    37  *	9.  +-1         ** +-INF is NAN
    38  *	10. +0 ** (+anything except 0, NAN)               is +0
    39  *	11. -0 ** (+anything except 0, NAN, odd integer)  is +0
    40  *	12. +0 ** (-anything except 0, NAN)               is +INF
    41  *	13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
    42  *	14. -0 ** (odd integer) = -( +0 ** (odd integer) )
    43  *	15. +INF ** (+anything except 0,NAN) is +INF
    44  *	16. +INF ** (-anything except 0,NAN) is +0
    45  *	17. -INF ** (anything)  = -0 ** (-anything)
    46  *	18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
    47  *	19. (-anything except 0 and inf) ** (non-integer) is NAN
    48  *
    49  * Accuracy:
    50  *	pow(x,y) returns x**y nearly rounded. In particular
    51  *			pow(integer,integer)
    52  *	always returns the correct integer provided it is
    53  *	representable.
    54  *
    55  * Constants :
    56  * The hexadecimal values are the intended ones for the following
    57  * constants. The decimal values may be used, provided that the
    58  * compiler will convert from decimal to binary accurately enough
    59  * to produce the hexadecimal values shown.
    60  */
    61 
    62 /*#include "math.h"*/
    63 #include "math_private.h"
    64 
    65 #ifdef __STDC__
    66 static const double
    67 #else
    68 static double
    69 #endif
    70   bp[] = { 1.0, 1.5, }, dp_h[] = {
    71 0.0, 5.84962487220764160156e-01,},      /* 0x3FE2B803, 0x40000000 */
    72 
    73     dp_l[] = {
    74 0.0, 1.35003920212974897128e-08,},      /* 0x3E4CFDEB, 0x43CFD006 */
    75 
    76     /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
    77     L1 = 5.99999999999994648725e-01,    /* 0x3FE33333, 0x33333303 */
    78     L2 = 4.28571428578550184252e-01,    /* 0x3FDB6DB6, 0xDB6FABFF */
    79     L3 = 3.33333329818377432918e-01,    /* 0x3FD55555, 0x518F264D */
    80     L4 = 2.72728123808534006489e-01,    /* 0x3FD17460, 0xA91D4101 */
    81     L5 = 2.30660745775561754067e-01,    /* 0x3FCD864A, 0x93C9DB65 */
    82     L6 = 2.06975017800338417784e-01,    /* 0x3FCA7E28, 0x4A454EEF */
    83     P1 = 1.66666666666666019037e-01,    /* 0x3FC55555, 0x5555553E */
    84     P2 = -2.77777777770155933842e-03,   /* 0xBF66C16C, 0x16BEBD93 */
    85     P3 = 6.61375632143793436117e-05,    /* 0x3F11566A, 0xAF25DE2C */
    86     P4 = -1.65339022054652515390e-06,   /* 0xBEBBBD41, 0xC5D26BF1 */
    87     P5 = 4.13813679705723846039e-08,    /* 0x3E663769, 0x72BEA4D0 */
    88     lg2 = 6.93147180559945286227e-01,   /* 0x3FE62E42, 0xFEFA39EF */
    89     lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
    90     lg2_l = -1.90465429995776804525e-09,        /* 0xBE205C61, 0x0CA86C39 */
    91     ovt = 8.0085662595372944372e-0017,  /* -(1024-log2(ovfl+.5ulp)) */
    92     cp = 9.61796693925975554329e-01,    /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
    93     cp_h = 9.61796700954437255859e-01,  /* 0x3FEEC709, 0xE0000000 =(float)cp */
    94     cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h */
    95     ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
    96     ivln2_h = 1.44269502162933349609e+00,       /* 0x3FF71547, 0x60000000 =24b 1/ln2 */
    97     ivln2_l = 1.92596299112661746887e-08;       /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail */
    98 
    99 #ifdef __STDC__
   100 double
   101 __ieee754_pow(double x, double y)
   102 #else
   103 double
   104 __ieee754_pow(x, y)
   105      double x, y;
   106 #endif
   107 {
   108     double z, ax, z_h, z_l, p_h, p_l;
   109     double y1, t1, t2, r, s, t, u, v, w;
   110     int32_t i, j, k, yisint, n;
   111     int32_t hx, hy, ix, iy;
   112     u_int32_t lx, ly;
   113 
   114     EXTRACT_WORDS(hx, lx, x);
   115     EXTRACT_WORDS(hy, ly, y);
   116     ix = hx & 0x7fffffff;
   117     iy = hy & 0x7fffffff;
   118 
   119     /* y==zero: x**0 = 1 */
   120     if ((iy | ly) == 0)
   121         return one;
   122 
   123     /* +-NaN return x+y */
   124     if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) ||
   125         iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0)))
   126         return x + y;
   127 
   128     /* determine if y is an odd int when x < 0
   129      * yisint = 0       ... y is not an integer
   130      * yisint = 1       ... y is an odd int
   131      * yisint = 2       ... y is an even int
   132      */
   133     yisint = 0;
   134     if (hx < 0) {
   135         if (iy >= 0x43400000)
   136             yisint = 2;         /* even integer y */
   137         else if (iy >= 0x3ff00000) {
   138             k = (iy >> 20) - 0x3ff;     /* exponent */
   139             if (k > 20) {
   140                 j = ly >> (52 - k);
   141                 if ((u_int32_t) (j << (52 - k)) == ly)
   142                     yisint = 2 - (j & 1);
   143             } else if (ly == 0) {
   144                 j = iy >> (20 - k);
   145                 if ((j << (20 - k)) == iy)
   146                     yisint = 2 - (j & 1);
   147             }
   148         }
   149     }
   150 
   151     /* special value of y */
   152     if (ly == 0) {
   153         if (iy == 0x7ff00000) { /* y is +-inf */
   154             if (((ix - 0x3ff00000) | lx) == 0)
   155                 return y - y;   /* inf**+-1 is NaN */
   156             else if (ix >= 0x3ff00000)  /* (|x|>1)**+-inf = inf,0 */
   157                 return (hy >= 0) ? y : zero;
   158             else                /* (|x|<1)**-,+inf = inf,0 */
   159                 return (hy < 0) ? -y : zero;
   160         }
   161         if (iy == 0x3ff00000) { /* y is  +-1 */
   162             if (hy < 0)
   163                 return one / x;
   164             else
   165                 return x;
   166         }
   167         if (hy == 0x40000000)
   168             return x * x;       /* y is  2 */
   169         if (hy == 0x3fe00000) { /* y is  0.5 */
   170             if (hx >= 0)        /* x >= +0 */
   171                 return __ieee754_sqrt(x);
   172         }
   173     }
   174 
   175     ax = x < 0 ? -x : x;        /*fabs(x); */
   176     /* special value of x */
   177     if (lx == 0) {
   178         if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) {
   179             z = ax;             /*x is +-0,+-inf,+-1 */
   180             if (hy < 0)
   181                 z = one / z;    /* z = (1/|x|) */
   182             if (hx < 0) {
   183                 if (((ix - 0x3ff00000) | yisint) == 0) {
   184                     z = (z - z) / (z - z);      /* (-1)**non-int is NaN */
   185                 } else if (yisint == 1)
   186                     z = -z;     /* (x<0)**odd = -(|x|**odd) */
   187             }
   188             return z;
   189         }
   190     }
   191 
   192     /* (x<0)**(non-int) is NaN */
   193     if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
   194         return (x - x) / (x - x);
   195 
   196     /* |y| is huge */
   197     if (iy > 0x41e00000) {      /* if |y| > 2**31 */
   198         if (iy > 0x43f00000) {  /* if |y| > 2**64, must o/uflow */
   199             if (ix <= 0x3fefffff)
   200                 return (hy < 0) ? huge * huge : tiny * tiny;
   201             if (ix >= 0x3ff00000)
   202                 return (hy > 0) ? huge * huge : tiny * tiny;
   203         }
   204         /* over/underflow if x is not close to one */
   205         if (ix < 0x3fefffff)
   206             return (hy < 0) ? huge * huge : tiny * tiny;
   207         if (ix > 0x3ff00000)
   208             return (hy > 0) ? huge * huge : tiny * tiny;
   209         /* now |1-x| is tiny <= 2**-20, suffice to compute
   210            log(x) by x-x^2/2+x^3/3-x^4/4 */
   211         t = x - 1;              /* t has 20 trailing zeros */
   212         w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
   213         u = ivln2_h * t;        /* ivln2_h has 21 sig. bits */
   214         v = t * ivln2_l - w * ivln2;
   215         t1 = u + v;
   216         SET_LOW_WORD(t1, 0);
   217         t2 = v - (t1 - u);
   218     } else {
   219         double s2, s_h, s_l, t_h, t_l;
   220         n = 0;
   221         /* take care subnormal number */
   222         if (ix < 0x00100000) {
   223             ax *= two53;
   224             n -= 53;
   225             GET_HIGH_WORD(ix, ax);
   226         }
   227         n += ((ix) >> 20) - 0x3ff;
   228         j = ix & 0x000fffff;
   229         /* determine interval */
   230         ix = j | 0x3ff00000;    /* normalize ix */
   231         if (j <= 0x3988E)
   232             k = 0;              /* |x|<sqrt(3/2) */
   233         else if (j < 0xBB67A)
   234             k = 1;              /* |x|<sqrt(3)   */
   235         else {
   236             k = 0;
   237             n += 1;
   238             ix -= 0x00100000;
   239         }
   240         SET_HIGH_WORD(ax, ix);
   241 
   242         /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
   243         u = ax - bp[k];         /* bp[0]=1.0, bp[1]=1.5 */
   244         v = one / (ax + bp[k]);
   245         s = u * v;
   246         s_h = s;
   247         SET_LOW_WORD(s_h, 0);
   248         /* t_h=ax+bp[k] High */
   249         t_h = zero;
   250         SET_HIGH_WORD(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
   251         t_l = ax - (t_h - bp[k]);
   252         s_l = v * ((u - s_h * t_h) - s_h * t_l);
   253         /* compute log(ax) */
   254         s2 = s * s;
   255         r = s2 * s2 * (L1 +
   256                        s2 * (L2 +
   257                              s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
   258         r += s_l * (s_h + s);
   259         s2 = s_h * s_h;
   260         t_h = 3.0 + s2 + r;
   261         SET_LOW_WORD(t_h, 0);
   262         t_l = r - ((t_h - 3.0) - s2);
   263         /* u+v = s*(1+...) */
   264         u = s_h * t_h;
   265         v = s_l * t_h + t_l * s;
   266         /* 2/(3log2)*(s+...) */
   267         p_h = u + v;
   268         SET_LOW_WORD(p_h, 0);
   269         p_l = v - (p_h - u);
   270         z_h = cp_h * p_h;       /* cp_h+cp_l = 2/(3*log2) */
   271         z_l = cp_l * p_h + p_l * cp + dp_l[k];
   272         /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
   273         t = (double) n;
   274         t1 = (((z_h + z_l) + dp_h[k]) + t);
   275         SET_LOW_WORD(t1, 0);
   276         t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
   277     }
   278 
   279     s = one;                    /* s (sign of result -ve**odd) = -1 else = 1 */
   280     if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
   281         s = -one;               /* (-ve)**(odd int) */
   282 
   283     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
   284     y1 = y;
   285     SET_LOW_WORD(y1, 0);
   286     p_l = (y - y1) * t1 + y * t2;
   287     p_h = y1 * t1;
   288     z = p_l + p_h;
   289     EXTRACT_WORDS(j, i, z);
   290     if (j >= 0x40900000) {      /* z >= 1024 */
   291         if (((j - 0x40900000) | i) != 0)        /* if z > 1024 */
   292             return s * huge * huge;     /* overflow */
   293         else {
   294             if (p_l + ovt > z - p_h)
   295                 return s * huge * huge; /* overflow */
   296         }
   297     } else if ((j & 0x7fffffff) >= 0x4090cc00) {        /* z <= -1075 */
   298         if (((j - 0xc090cc00) | i) != 0)        /* z < -1075 */
   299             return s * tiny * tiny;     /* underflow */
   300         else {
   301             if (p_l <= z - p_h)
   302                 return s * tiny * tiny; /* underflow */
   303         }
   304     }
   305     /*
   306      * compute 2**(p_h+p_l)
   307      */
   308     i = j & 0x7fffffff;
   309     k = (i >> 20) - 0x3ff;
   310     n = 0;
   311     if (i > 0x3fe00000) {       /* if |z| > 0.5, set n = [z+0.5] */
   312         n = j + (0x00100000 >> (k + 1));
   313         k = ((n & 0x7fffffff) >> 20) - 0x3ff;   /* new k for n */
   314         t = zero;
   315         SET_HIGH_WORD(t, n & ~(0x000fffff >> k));
   316         n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
   317         if (j < 0)
   318             n = -n;
   319         p_h -= t;
   320     }
   321     t = p_l + p_h;
   322     SET_LOW_WORD(t, 0);
   323     u = t * lg2_h;
   324     v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
   325     z = u + v;
   326     w = v - (z - u);
   327     t = z * z;
   328     t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
   329     r = (z * t1) / (t1 - two) - (w + z * w);
   330     z = one - (r - z);
   331     GET_HIGH_WORD(j, z);
   332     j += (n << 20);
   333     if ((j >> 20) <= 0)
   334         z = SDL_NAME(scalbn) (z, n);    /* subnormal output */
   335     else
   336         SET_HIGH_WORD(z, j);
   337     return s * z;
   338 }
   339 
   340 /* vi: set ts=4 sw=4 expandtab: */