lib/libm/src/s_cbrtl.c

   1 /*-
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  * Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz.
   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  * The argument reduction and testing for exceptional cases was
  13  * written by Steven G. Kargl with input from Bruce D. Evans
  14  * and David A. Schultz.
  15  */
  16
  17 #include <sys/cdefs.h>
  18 __RCSID("$NetBSD: s_cbrtl.c,v 1.1 2013/11/19 19:24:34 joerg Exp $");
  19 #if 0
  20 __FBSDID("$FreeBSD: head/lib/msun/src/s_cbrtl.c 238924 2012-07-30 21:58:28Z kargl $");
  21 #endif
  22
  23 #include "namespace.h"
  24 #include <machine/ieee.h>
  25 #include <float.h>
  26
  27 #include "math.h"
  28 #include "math_private.h"
  29
  30 #ifdef __HAVE_LONG_DOUBLE
  31 __weak_alias(cbrtl, _cbrtl)
  32
  33 static const unsigned
  34     B1 = 709958130;     /* B1 = (127-127.0/3-0.03306235651)*2**23 */
  35
  36 long double
  37 cbrtl(long double x)
  38 {
  39         union ieee_ext_u ux, vx;
  40         long double r, s, t, w;
  41         double dr, dt, dx;
  42         float ft, fx;
  43         uint32_t hx;
  44         int k;
  45
  46         ux.extu_ld = x;
  47
  48
  49         /*
  50          * If x = +-Inf, then cbrt(x) = +-Inf.
  51          * If x = NaN, then cbrt(x) = NaN.
  52          */
  53         if (ux.extu_exp == EXT_EXP_INFNAN)
  54                 return (x + x);
  55         if ((ux.extu_frach | ux.extu_fracl | ux.extu_exp) == 0)
  56                 return (x);
  57
  58         vx.extu_ld = 1;
  59         vx.extu_ext.ext_sign = ux.extu_ext.ext_sign;
  60         ux.extu_ext.ext_sign = 0;
  61         if (ux.extu_exp == 0) {
  62                 /* Adjust subnormal numbers. */
  63                 ux.extu_ld *= 0x1.0p514;
  64                 k = ux.extu_exp - EXT_EXP_BIAS - 514;
  65         } else {
  66                 k = ux.extu_exp - EXT_EXP_BIAS;
  67         }
  68
  69         ux.extu_exp = EXT_EXP_BIAS;
  70         x = ux.extu_ld;
  71         switch (k % 3) {
  72         case 1:
  73         case -2:
  74                 x = 2*x;
  75                 k--;
  76                 break;
  77         case 2:
  78         case -1:
  79                 x = 4*x;
  80                 k -= 2;
  81                 break;
  82         }
  83         vx.extu_exp = EXT_EXP_BIAS + k / 3;
  84
  85         /*
  86          * The following is the guts of s_cbrtf, with the handling of
  87          * special values removed and extra care for accuracy not taken,
  88          * but with most of the extra accuracy not discarded.
  89          */
  90
  91         /* ~5-bit estimate: */
  92         fx = x;
  93         GET_FLOAT_WORD(hx, fx);
  94         SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
  95
  96         /* ~16-bit estimate: */
  97         dx = x;
  98         dt = ft;
  99         dr = dt * dt * dt;
 100         dt = dt * (dx + dx + dr) / (dx + dr + dr);
 101
 102         /* ~47-bit estimate: */
 103         dr = dt * dt * dt;
 104         dt = dt * (dx + dx + dr) / (dx + dr + dr);
 105
 106 #if LDBL_MANT_DIG == 64
 107         /*
 108          * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
 109          * Round it away from zero to 32 bits (32 so that t*t is exact, and
 110          * away from zero for technical reasons).
 111          */
 112         volatile double vd2 = 0x1.0p32;
 113         volatile double vd1 = 0x1.0p-31;
 114         #define vd ((long double)vd2 + vd1)
 115
 116         t = dt + vd - 0x1.0p32;
 117 #elif LDBL_MANT_DIG == 113
 118         /*
 119          * Round dt away from zero to 47 bits.  Since we don't trust the 47,
 120          * add 2 47-bit ulps instead of 1 to round up.  Rounding is slow and
 121          * might be avoidable in this case, since on most machines dt will
 122          * have been evaluated in 53-bit precision and the technical reasons
 123          * for rounding up might not apply to either case in cbrtl() since
 124          * dt is much more accurate than needed.
 125          */
 126         t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
 127 #else
 128 #error "Unsupported long double format"
 129 #endif
 130
 131         /*
 132          * Final step Newton iteration to 64 or 113 bits with
 133          * error < 0.667 ulps
 134          */
 135         s=t*t;                          /* t*t is exact */
 136         r=x/s;                          /* error <= 0.5 ulps; |r| < |t| */
 137         w=t+t;                          /* t+t is exact */
 138         r=(r-t)/(w+r);                  /* r-t is exact; w+r ~= 3*t */
 139         t=t+t*r;                        /* error <= 0.5 + 0.5/3 + epsilon */
 140
 141         t *= vx.extu_ld;
 142         return t;
 143 }
 144
 145 #endif /* __HAVE_LONG_DOUBLE */