src/math/cbrtl.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/s_cbrtl.c */
   2 /*-
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  * Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz.
   6  *
   7  * Developed at SunPro, a Sun Microsystems, Inc. business.
   8  * Permission to use, copy, modify, and distribute this
   9  * software is freely granted, provided that this notice
  10  * is preserved.
  11  * ====================================================
  12  *
  13  * The argument reduction and testing for exceptional cases was
  14  * written by Steven G. Kargl with input from Bruce D. Evans
  15  * and David A. Schultz.
  16  */
  17
  18 #include "libm.h"
  19
  20 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
  21 long double cbrtl(long double x)
  22 {
  23         return cbrt(x);
  24 }
  25 #elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
  26 static const unsigned B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */
  27
  28 long double cbrtl(long double x)
  29 {
  30         union ldshape u = {x}, v;
  31         union {float f; uint32_t i;} uft;
  32         long double r, s, t, w;
  33         double_t dr, dt, dx;
  34         float_t ft;
  35         int e = u.i.se & 0x7fff;
  36         int sign = u.i.se & 0x8000;
  37
  38         /*
  39          * If x = +-Inf, then cbrt(x) = +-Inf.
  40          * If x = NaN, then cbrt(x) = NaN.
  41          */
  42         if (e == 0x7fff)
  43                 return x + x;
  44         if (e == 0) {
  45                 /* Adjust subnormal numbers. */
  46                 u.f *= 0x1p120;
  47                 e = u.i.se & 0x7fff;
  48                 /* If x = +-0, then cbrt(x) = +-0. */
  49                 if (e == 0)
  50                         return x;
  51                 e -= 120;
  52         }
  53         e -= 0x3fff;
  54         u.i.se = 0x3fff;
  55         x = u.f;
  56         switch (e % 3) {
  57         case 1:
  58         case -2:
  59                 x *= 2;
  60                 e--;
  61                 break;
  62         case 2:
  63         case -1:
  64                 x *= 4;
  65                 e -= 2;
  66                 break;
  67         }
  68         v.f = 1.0;
  69         v.i.se = sign | (0x3fff + e/3);
  70
  71         /*
  72          * The following is the guts of s_cbrtf, with the handling of
  73          * special values removed and extra care for accuracy not taken,
  74          * but with most of the extra accuracy not discarded.
  75          */
  76
  77         /* ~5-bit estimate: */
  78         uft.f = x;
  79         uft.i = (uft.i & 0x7fffffff)/3 + B1;
  80         ft = uft.f;
  81
  82         /* ~16-bit estimate: */
  83         dx = x;
  84         dt = ft;
  85         dr = dt * dt * dt;
  86         dt = dt * (dx + dx + dr) / (dx + dr + dr);
  87
  88         /* ~47-bit estimate: */
  89         dr = dt * dt * dt;
  90         dt = dt * (dx + dx + dr) / (dx + dr + dr);
  91
  92 #if LDBL_MANT_DIG == 64
  93         /*
  94          * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
  95          * Round it away from zero to 32 bits (32 so that t*t is exact, and
  96          * away from zero for technical reasons).
  97          */
  98         t = dt + (0x1.0p32L + 0x1.0p-31L) - 0x1.0p32;
  99 #elif LDBL_MANT_DIG == 113
 100         /*
 101          * Round dt away from zero to 47 bits.  Since we don't trust the 47,
 102          * add 2 47-bit ulps instead of 1 to round up.  Rounding is slow and
 103          * might be avoidable in this case, since on most machines dt will
 104          * have been evaluated in 53-bit precision and the technical reasons
 105          * for rounding up might not apply to either case in cbrtl() since
 106          * dt is much more accurate than needed.
 107          */
 108         t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
 109 #endif
 110
 111         /*
 112          * Final step Newton iteration to 64 or 113 bits with
 113          * error < 0.667 ulps
 114          */
 115         s = t*t;         /* t*t is exact */
 116         r = x/s;         /* error <= 0.5 ulps; |r| < |t| */
 117         w = t+t;         /* t+t is exact */
 118         r = (r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
 119         t = t+t*r;       /* error <= 0.5 + 0.5/3 + epsilon */
 120
 121         t *= v.f;
 122         return t;
 123 }
 124 #endif