compiler/stdc/math/s_fmal.c

   1 /*-
   2  * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
   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, this list of conditions and the following disclaimer.
  10  * 2. Redistributions in binary form must reproduce the above copyright
  11  *    notice, this list of conditions and the following disclaimer in the
  12  *    documentation and/or other materials provided with the distribution.
  13  *
  14  * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
  15  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  16  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  17  * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
  18  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  19  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  20  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  21  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  22  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  23  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  24  * SUCH DAMAGE.
  25  */
  26
  27 __FBSDID("$FreeBSD: src/lib/msun/src/s_fmal.c,v 1.3 2007/01/07 07:54:21 das Exp $");
  28
  29 #include <fenv.h>
  30 #include <float.h>
  31 #include <math.h>
  32
  33 /*
  34  * Fused multiply-add: Compute x * y + z with a single rounding error.
  35  *
  36  * We use scaling to avoid overflow/underflow, along with the
  37  * canonical precision-doubling technique adapted from:
  38  *
  39  *      Dekker, T.  A Floating-Point Technique for Extending the
  40  *      Available Precision.  Numer. Math. 18, 224-242 (1971).
  41  */
  42 long double
  43 fmal(long double x, long double y, long double z)
  44 {
  45 #if LDBL_MANT_DIG == 64
  46         static const long double split = 0x1p32L + 1.0;
  47 #elif LDBL_MANT_DIG == 113
  48         static const long double split = 0x1p57L + 1.0;
  49 #endif
  50         long double xs, ys, zs;
  51         long double c, cc, hx, hy, p, q, tx, ty;
  52         long double r, rr, s;
  53         int oround;
  54         int ex, ey, ez;
  55         int spread;
  56
  57         if (z == 0.0)
  58                 return (x * y);
  59         if (x == 0.0 || y == 0.0)
  60                 return (x * y + z);
  61
  62         /* Results of frexp() are undefined for these cases. */
  63         if (!isfinite(x) || !isfinite(y) || !isfinite(z))
  64                 return (x * y + z);
  65
  66         xs = frexpl(x, &ex);
  67         ys = frexpl(y, &ey);
  68         zs = frexpl(z, &ez);
  69         oround = fegetround();
  70         spread = ex + ey - ez;
  71
  72         /*
  73          * If x * y and z are many orders of magnitude apart, the scaling
  74          * will overflow, so we handle these cases specially.  Rounding
  75          * modes other than FE_TONEAREST are painful.
  76          */
  77         if (spread > LDBL_MANT_DIG * 2) {
  78                 fenv_t env;
  79                 feraiseexcept(FE_INEXACT);
  80                 switch(oround) {
  81                 case FE_TONEAREST:
  82                         return (x * y);
  83                 case FE_TOWARDZERO:
  84                         if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
  85                                 return (x * y);
  86                         feholdexcept(&env);
  87                         r = x * y;
  88                         if (!fetestexcept(FE_INEXACT))
  89                                 r = nextafterl(r, 0);
  90                         feupdateenv(&env);
  91                         return (r);
  92                 case FE_DOWNWARD:
  93                         if (z > 0.0)
  94                                 return (x * y);
  95                         feholdexcept(&env);
  96                         r = x * y;
  97                         if (!fetestexcept(FE_INEXACT))
  98                                 r = nextafterl(r, -INFINITY);
  99                         feupdateenv(&env);
 100                         return (r);
 101                 default:        /* FE_UPWARD */
 102                         if (z < 0.0)
 103                                 return (x * y);
 104                         feholdexcept(&env);
 105                         r = x * y;
 106                         if (!fetestexcept(FE_INEXACT))
 107                                 r = nextafterl(r, INFINITY);
 108                         feupdateenv(&env);
 109                         return (r);
 110                 }
 111         }
 112         if (spread < -LDBL_MANT_DIG) {
 113                 feraiseexcept(FE_INEXACT);
 114                 if (!isnormal(z))
 115                         feraiseexcept(FE_UNDERFLOW);
 116                 switch (oround) {
 117                 case FE_TONEAREST:
 118                         return (z);
 119                 case FE_TOWARDZERO:
 120                         if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
 121                                 return (z);
 122                         else
 123                                 return (nextafterl(z, 0));
 124                 case FE_DOWNWARD:
 125                         if (x > 0.0 ^ y < 0.0)
 126                                 return (z);
 127                         else
 128                                 return (nextafterl(z, -INFINITY));
 129                 default:        /* FE_UPWARD */
 130                         if (x > 0.0 ^ y < 0.0)
 131                                 return (nextafterl(z, INFINITY));
 132                         else
 133                                 return (z);
 134                 }
 135         }
 136
 137         /*
 138          * Use Dekker's algorithm to perform the multiplication and
 139          * subsequent addition in twice the machine precision.
 140          * Arrange so that x * y = c + cc, and x * y + z = r + rr.
 141          */
 142         fesetround(FE_TONEAREST);
 143
 144         p = xs * split;
 145         hx = xs - p;
 146         hx += p;
 147         tx = xs - hx;
 148
 149         p = ys * split;
 150         hy = ys - p;
 151         hy += p;
 152         ty = ys - hy;
 153
 154         p = hx * hy;
 155         q = hx * ty + tx * hy;
 156         c = p + q;
 157         cc = p - c + q + tx * ty;
 158
 159         zs = ldexpl(zs, -spread);
 160         r = c + zs;
 161         s = r - c;
 162         rr = (c - (r - s)) + (zs - s) + cc;
 163
 164         spread = ex + ey;
 165         if (spread + ilogbl(r) > -16383) {
 166                 fesetround(oround);
 167                 r = r + rr;
 168         } else {
 169                 /*
 170                  * The result is subnormal, so we round before scaling to
 171                  * avoid double rounding.
 172                  */
 173                 p = ldexpl(copysignl(0x1p-16382L, r), -spread);
 174                 c = r + p;
 175                 s = c - r;
 176                 cc = (r - (c - s)) + (p - s) + rr;
 177                 fesetround(oround);
 178                 r = (c + cc) - p;
 179         }
 180         return (ldexpl(r, spread));
 181 }