lib/libm/src/e_sqrtl.c

   1 /*-
   2  * Copyright (c) 2007 Steven G. Kargl
   3  * All rights reserved.
   4  *
   5  * Redistribution and use in source and binary forms, with or without
   6  * modification, are permitted provided that the following conditions
   7  * are met:
   8  * 1. Redistributions of source code must retain the above copyright
   9  *    notice unmodified, this list of conditions, and the following
  10  *    disclaimer.
  11  * 2. Redistributions in binary form must reproduce the above copyright
  12  *    notice, this list of conditions and the following disclaimer in the
  13  *    documentation and/or other materials provided with the distribution.
  14  *
  15  * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
  16  * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
  17  * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
  18  * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
  19  * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
  20  * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
  21  * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
  22  * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
  23  * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
  24  * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
  25  */
  26
  27 #include <sys/cdefs.h>
  28 #if 0
  29 __FBSDID("$FreeBSD: head/lib/msun/src/e_sqrtl.c 176720 2008-03-02 01:47:58Z das $");
  30 #endif
  31 __RCSID("$NetBSD: e_sqrtl.c,v 1.4 2013/11/22 20:15:06 martin Exp $");
  32
  33 #include <machine/ieee.h>
  34 #include <float.h>
  35
  36 #include "math.h"
  37 #include "math_private.h"
  38
  39 #ifdef __HAVE_LONG_DOUBLE
  40
  41 #ifdef HAVE_FENV_H
  42 #include <fenv.h>
  43 #endif
  44
  45 #ifdef LDBL_IMPLICIT_NBIT
  46 #define LDBL_NBIT       0
  47 #endif
  48
  49 #ifdef HAVE_FENV_H
  50
  51 /* Return (x + ulp) for normal positive x. Assumes no overflow. */
  52 static inline long double
  53 inc(long double x)
  54 {
  55         union ieee_ext_u ux = { .extu_ld = x, };
  56
  57         if (++ux.extu_fracl == 0) {
  58                 if (++ux.extu_frach == 0) {
  59                         ux.extu_exp++;
  60                         ux.extu_frach |= LDBL_NBIT;
  61                 }
  62         }
  63         return (ux.extu_ld);
  64 }
  65
  66 /* Return (x - ulp) for normal positive x. Assumes no underflow. */
  67 static inline long double
  68 dec(long double x)
  69 {
  70         union ieee_ext_u ux = { .extu_ld = x, };
  71
  72         if (ux.extu_fracl-- == 0) {
  73                 if (ux.extu_frach-- == LDBL_NBIT) {
  74                         ux.extu_exp--;
  75                         ux.extu_frach |= LDBL_NBIT;
  76                 }
  77         }
  78         return (ux.extu_ld);
  79 }
  80
  81 /*
  82  * This is slow, but simple and portable. You should use hardware sqrt
  83  * if possible.
  84  */
  85
  86 long double
  87 __ieee754_sqrtl(long double x)
  88 {
  89         union ieee_ext_u ux = { .extu_ld = x, };
  90         int k, r;
  91         long double lo, xn;
  92         fenv_t env;
  93
  94         /* If x = NaN, then sqrt(x) = NaN. */
  95         /* If x = Inf, then sqrt(x) = Inf. */
  96         /* If x = -Inf, then sqrt(x) = NaN. */
  97         if (ux.extu_exp == LDBL_MAX_EXP * 2 - 1)
  98                 return (x * x + x);
  99
 100         /* If x = +-0, then sqrt(x) = +-0. */
 101         if ((ux.extu_frach | ux.extu_fracl | ux.extu_exp) == 0)
 102                 return (x);
 103
 104         /* If x < 0, then raise invalid and return NaN */
 105         if (ux.extu_sign)
 106                 return ((x - x) / (x - x));
 107
 108         feholdexcept(&env);
 109
 110         if (ux.extu_exp == 0) {
 111                 /* Adjust subnormal numbers. */
 112                 ux.extu_ld *= 0x1.0p514;
 113                 k = -514;
 114         } else {
 115                 k = 0;
 116         }
 117         /*
 118          * ux.extu_ld is a normal number, so break it into ux.extu_ld = e*2^n where
 119          * ux.extu_ld = (2*e)*2^2k for odd n and ux.extu_ld = (4*e)*2^2k for even n.
 120          */
 121         if ((ux.extu_exp - EXT_EXP_BIAS) & 1) { /* n is even.     */
 122                 k += ux.extu_exp - EXT_EXP_BIAS - 1; /* 2k = n - 2.   */
 123                 ux.extu_exp = EXT_EXP_BIAS + 1; /* ux.extu_ld in [2,4). */
 124         } else {
 125                 k += ux.extu_exp - EXT_EXP_BIAS;        /* 2k = n - 1.   */
 126                 ux.extu_exp = EXT_EXP_BIAS;     /* ux.extu_ld in [1,2). */
 127         }
 128
 129         /*
 130          * Newton's iteration.
 131          * Split ux.extu_ld into a high and low part to achieve additional precision.
 132          */
 133         xn = sqrt(ux.extu_ld);                  /* 53-bit estimate of sqrtl(x). */
 134 #if LDBL_MANT_DIG > 100
 135         xn = (xn + (ux.extu_ld / xn)) * 0.5;    /* 106-bit estimate. */
 136 #endif
 137         lo = ux.extu_ld;
 138         ux.extu_fracl = 0;              /* Zero out lower bits. */
 139         lo = (lo - ux.extu_ld) / xn;    /* Low bits divided by xn. */
 140         xn = xn + (ux.extu_ld / xn);    /* High portion of estimate. */
 141         ux.extu_ld = xn + lo;           /* Combine everything. */
 142         ux.extu_exp += (k >> 1) - 1;
 143
 144         feclearexcept(FE_INEXACT);
 145         r = fegetround();
 146         fesetround(FE_TOWARDZERO);      /* Set to round-toward-zero. */
 147         xn = x / ux.extu_ld;            /* Chopped quotient (inexact?). */
 148
 149         if (!fetestexcept(FE_INEXACT)) { /* Quotient is exact. */
 150                 if (xn == ux.extu_ld) {
 151                         fesetenv(&env);
 152                         return (ux.extu_ld);
 153                 }
 154                 /* Round correctly for inputs like x = y**2 - ulp. */
 155                 xn = dec(xn);           /* xn = xn - ulp. */
 156         }
 157
 158         if (r == FE_TONEAREST) {
 159                 xn = inc(xn);           /* xn = xn + ulp. */
 160         } else if (r == FE_UPWARD) {
 161                 ux.extu_ld = inc(ux.extu_ld);   /* ux.extu_ld = ux.extu_ld + ulp. */
 162                 xn = inc(xn);           /* xn  = xn + ulp. */
 163         }
 164         ux.extu_ld = ux.extu_ld + xn;           /* Chopped sum. */
 165         feupdateenv(&env);      /* Restore env and raise inexact */
 166         ux.extu_exp--;
 167         return (ux.extu_ld);
 168 }
 169
 170 #else
 171
 172 /*
 173  * No fenv support:
 174  * poor man's version: just use double
 175  */
 176 long double
 177 __ieee754_sqrtl(long double x)
 178 {
 179         return __ieee754_sqrt((double)x);
 180 }
 181
 182 #endif
 183
 184 #endif