src/libm/e_pow.c
author Sam Lantinga <slouken@libsdl.org>
Mon, 15 Sep 2008 06:33:23 +0000
changeset 2756 a98604b691c8
child 3337 9ac6f0782dd6
permissions -rw-r--r--
Expanded the libm support and put it into a separate directory.
     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 libm_hidden_proto(scalbn)
    66     libm_hidden_proto(fabs)
    67 #ifdef __STDC__
    68      static const double
    69 #else
    70      static double
    71 #endif
    72        bp[] = { 1.0, 1.5, }, dp_h[] = {
    73      0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
    74 
    75          dp_l[] = {
    76      0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
    77 
    78          zero = 0.0, one = 1.0, two = 2.0, two53 = 9007199254740992.0,  /* 0x43400000, 0x00000000 */
    79          huge = 1.0e300, tiny = 1.0e-300,
    80          /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
    81          L1 = 5.99999999999994648725e-01,       /* 0x3FE33333, 0x33333303 */
    82          L2 = 4.28571428578550184252e-01,       /* 0x3FDB6DB6, 0xDB6FABFF */
    83          L3 = 3.33333329818377432918e-01,       /* 0x3FD55555, 0x518F264D */
    84          L4 = 2.72728123808534006489e-01,       /* 0x3FD17460, 0xA91D4101 */
    85          L5 = 2.30660745775561754067e-01,       /* 0x3FCD864A, 0x93C9DB65 */
    86          L6 = 2.06975017800338417784e-01,       /* 0x3FCA7E28, 0x4A454EEF */
    87          P1 = 1.66666666666666019037e-01,       /* 0x3FC55555, 0x5555553E */
    88          P2 = -2.77777777770155933842e-03,      /* 0xBF66C16C, 0x16BEBD93 */
    89          P3 = 6.61375632143793436117e-05,       /* 0x3F11566A, 0xAF25DE2C */
    90          P4 = -1.65339022054652515390e-06,      /* 0xBEBBBD41, 0xC5D26BF1 */
    91          P5 = 4.13813679705723846039e-08,       /* 0x3E663769, 0x72BEA4D0 */
    92          lg2 = 6.93147180559945286227e-01,      /* 0x3FE62E42, 0xFEFA39EF */
    93          lg2_h = 6.93147182464599609375e-01,    /* 0x3FE62E43, 0x00000000 */
    94          lg2_l = -1.90465429995776804525e-09,   /* 0xBE205C61, 0x0CA86C39 */
    95          ovt = 8.0085662595372944372e-0017,     /* -(1024-log2(ovfl+.5ulp)) */
    96          cp = 9.61796693925975554329e-01,       /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
    97          cp_h = 9.61796700954437255859e-01,     /* 0x3FEEC709, 0xE0000000 =(float)cp */
    98          cp_l = -7.02846165095275826516e-09,    /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h */
    99          ivln2 = 1.44269504088896338700e+00,    /* 0x3FF71547, 0x652B82FE =1/ln2 */
   100          ivln2_h = 1.44269502162933349609e+00,  /* 0x3FF71547, 0x60000000 =24b 1/ln2 */
   101          ivln2_l = 1.92596299112661746887e-08;  /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail */
   102 
   103 #ifdef __STDC__
   104      double attribute_hidden __ieee754_pow(double x, double y)
   105 #else
   106      double attribute_hidden __ieee754_pow(x, y)
   107      double x, y;
   108 #endif
   109      {
   110          double z, ax, z_h, z_l, p_h, p_l;
   111          double y1, t1, t2, r, s, t, u, v, w;
   112          int32_t i, j, k, yisint, n;
   113          int32_t hx, hy, ix, iy;
   114          u_int32_t lx, ly;
   115 
   116          EXTRACT_WORDS(hx, lx, x);
   117          EXTRACT_WORDS(hy, ly, y);
   118          ix = hx & 0x7fffffff;
   119          iy = hy & 0x7fffffff;
   120 
   121          /* y==zero: x**0 = 1 */
   122          if ((iy | ly) == 0)
   123              return one;
   124 
   125          /* +-NaN return x+y */
   126          if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) ||
   127              iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0)))
   128              return x + y;
   129 
   130          /* determine if y is an odd int when x < 0
   131           * yisint = 0       ... y is not an integer
   132           * yisint = 1       ... y is an odd int
   133           * yisint = 2       ... y is an even int
   134           */
   135          yisint = 0;
   136          if (hx < 0) {
   137              if (iy >= 0x43400000)
   138                  yisint = 2;    /* even integer y */
   139              else if (iy >= 0x3ff00000) {
   140                  k = (iy >> 20) - 0x3ff;        /* exponent */
   141                  if (k > 20) {
   142                      j = ly >> (52 - k);
   143                      if ((j << (52 - k)) == ly)
   144                          yisint = 2 - (j & 1);
   145                  } else if (ly == 0) {
   146                      j = iy >> (20 - k);
   147                      if ((j << (20 - k)) == iy)
   148                          yisint = 2 - (j & 1);
   149                  }
   150              }
   151          }
   152 
   153          /* special value of y */
   154          if (ly == 0) {
   155              if (iy == 0x7ff00000) {    /* y is +-inf */
   156                  if (((ix - 0x3ff00000) | lx) == 0)
   157                      return y - y;      /* inf**+-1 is NaN */
   158                  else if (ix >= 0x3ff00000)     /* (|x|>1)**+-inf = inf,0 */
   159                      return (hy >= 0) ? y : zero;
   160                  else           /* (|x|<1)**-,+inf = inf,0 */
   161                      return (hy < 0) ? -y : zero;
   162              }
   163              if (iy == 0x3ff00000) {    /* y is  +-1 */
   164                  if (hy < 0)
   165                      return one / x;
   166                  else
   167                      return x;
   168              }
   169              if (hy == 0x40000000)
   170                  return x * x;  /* y is  2 */
   171              if (hy == 0x3fe00000) {    /* y is  0.5 */
   172                  if (hx >= 0)   /* x >= +0 */
   173                      return __ieee754_sqrt(x);
   174              }
   175          }
   176 
   177          ax = fabs(x);
   178          /* special value of x */
   179          if (lx == 0) {
   180              if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) {
   181                  z = ax;        /*x is +-0,+-inf,+-1 */
   182                  if (hy < 0)
   183                      z = one / z;       /* z = (1/|x|) */
   184                  if (hx < 0) {
   185                      if (((ix - 0x3ff00000) | yisint) == 0) {
   186                          z = (z - z) / (z - z); /* (-1)**non-int is NaN */
   187                      } else if (yisint == 1)
   188                          z = -z;        /* (x<0)**odd = -(|x|**odd) */
   189                  }
   190                  return z;
   191              }
   192          }
   193 
   194          /* (x<0)**(non-int) is NaN */
   195          if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
   196              return (x - x) / (x - x);
   197 
   198          /* |y| is huge */
   199          if (iy > 0x41e00000) { /* if |y| > 2**31 */
   200              if (iy > 0x43f00000) {     /* if |y| > 2**64, must o/uflow */
   201                  if (ix <= 0x3fefffff)
   202                      return (hy < 0) ? huge * huge : tiny * tiny;
   203                  if (ix >= 0x3ff00000)
   204                      return (hy > 0) ? huge * huge : tiny * tiny;
   205              }
   206              /* over/underflow if x is not close to one */
   207              if (ix < 0x3fefffff)
   208                  return (hy < 0) ? huge * huge : tiny * tiny;
   209              if (ix > 0x3ff00000)
   210                  return (hy > 0) ? huge * huge : tiny * tiny;
   211              /* now |1-x| is tiny <= 2**-20, suffice to compute
   212                 log(x) by x-x^2/2+x^3/3-x^4/4 */
   213              t = x - 1;         /* t has 20 trailing zeros */
   214              w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
   215              u = ivln2_h * t;   /* ivln2_h has 21 sig. bits */
   216              v = t * ivln2_l - w * ivln2;
   217              t1 = u + v;
   218              SET_LOW_WORD(t1, 0);
   219              t2 = v - (t1 - u);
   220          } else {
   221              double s2, s_h, s_l, t_h, t_l;
   222              n = 0;
   223              /* take care subnormal number */
   224              if (ix < 0x00100000) {
   225                  ax *= two53;
   226                  n -= 53;
   227                  GET_HIGH_WORD(ix, ax);
   228              }
   229              n += ((ix) >> 20) - 0x3ff;
   230              j = ix & 0x000fffff;
   231              /* determine interval */
   232              ix = j | 0x3ff00000;       /* normalize ix */
   233              if (j <= 0x3988E)
   234                  k = 0;         /* |x|<sqrt(3/2) */
   235              else if (j < 0xBB67A)
   236                  k = 1;         /* |x|<sqrt(3)   */
   237              else {
   238                  k = 0;
   239                  n += 1;
   240                  ix -= 0x00100000;
   241              }
   242              SET_HIGH_WORD(ax, ix);
   243 
   244              /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
   245              u = ax - bp[k];    /* bp[0]=1.0, bp[1]=1.5 */
   246              v = one / (ax + bp[k]);
   247              s = u * v;
   248              s_h = s;
   249              SET_LOW_WORD(s_h, 0);
   250              /* t_h=ax+bp[k] High */
   251              t_h = zero;
   252              SET_HIGH_WORD(t_h,
   253                            ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
   254              t_l = ax - (t_h - bp[k]);
   255              s_l = v * ((u - s_h * t_h) - s_h * t_l);
   256              /* compute log(ax) */
   257              s2 = s * s;
   258              r = s2 * s2 * (L1 +
   259                             s2 * (L2 +
   260                                   s2 * (L3 +
   261                                         s2 * (L4 + s2 * (L5 + s2 * L6)))));
   262              r += s_l * (s_h + s);
   263              s2 = s_h * s_h;
   264              t_h = 3.0 + s2 + r;
   265              SET_LOW_WORD(t_h, 0);
   266              t_l = r - ((t_h - 3.0) - s2);
   267              /* u+v = s*(1+...) */
   268              u = s_h * t_h;
   269              v = s_l * t_h + t_l * s;
   270              /* 2/(3log2)*(s+...) */
   271              p_h = u + v;
   272              SET_LOW_WORD(p_h, 0);
   273              p_l = v - (p_h - u);
   274              z_h = cp_h * p_h;  /* cp_h+cp_l = 2/(3*log2) */
   275              z_l = cp_l * p_h + p_l * cp + dp_l[k];
   276              /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
   277              t = (double) n;
   278              t1 = (((z_h + z_l) + dp_h[k]) + t);
   279              SET_LOW_WORD(t1, 0);
   280              t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
   281          }
   282 
   283          s = one;               /* s (sign of result -ve**odd) = -1 else = 1 */
   284          if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
   285              s = -one;          /* (-ve)**(odd int) */
   286 
   287          /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
   288          y1 = y;
   289          SET_LOW_WORD(y1, 0);
   290          p_l = (y - y1) * t1 + y * t2;
   291          p_h = y1 * t1;
   292          z = p_l + p_h;
   293          EXTRACT_WORDS(j, i, z);
   294          if (j >= 0x40900000) { /* z >= 1024 */
   295              if (((j - 0x40900000) | i) != 0)   /* if z > 1024 */
   296                  return s * huge * huge;        /* overflow */
   297              else {
   298                  if (p_l + ovt > z - p_h)
   299                      return s * huge * huge;    /* overflow */
   300              }
   301          } else if ((j & 0x7fffffff) >= 0x4090cc00) {   /* z <= -1075 */
   302              if (((j - 0xc090cc00) | i) != 0)   /* z < -1075 */
   303                  return s * tiny * tiny;        /* underflow */
   304              else {
   305                  if (p_l <= z - p_h)
   306                      return s * tiny * tiny;    /* underflow */
   307              }
   308          }
   309          /*
   310           * compute 2**(p_h+p_l)
   311           */
   312          i = j & 0x7fffffff;
   313          k = (i >> 20) - 0x3ff;
   314          n = 0;
   315          if (i > 0x3fe00000) {  /* if |z| > 0.5, set n = [z+0.5] */
   316              n = j + (0x00100000 >> (k + 1));
   317              k = ((n & 0x7fffffff) >> 20) - 0x3ff;      /* new k for n */
   318              t = zero;
   319              SET_HIGH_WORD(t, n & ~(0x000fffff >> k));
   320              n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
   321              if (j < 0)
   322                  n = -n;
   323              p_h -= t;
   324          }
   325          t = p_l + p_h;
   326          SET_LOW_WORD(t, 0);
   327          u = t * lg2_h;
   328          v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
   329          z = u + v;
   330          w = v - (z - u);
   331          t = z * z;
   332          t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
   333          r = (z * t1) / (t1 - two) - (w + z * w);
   334          z = one - (r - z);
   335          GET_HIGH_WORD(j, z);
   336          j += (n << 20);
   337          if ((j >> 20) <= 0)
   338              z = scalbn(z, n);  /* subnormal output */
   339          else
   340              SET_HIGH_WORD(z, j);
   341          return s * z;
   342      }