src/math/fmal.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/s_fmal.c */
   2 /*-
   3  * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
   4  * All rights reserved.
   5  *
   6  * Redistribution and use in source and binary forms, with or without
   7  * modification, are permitted provided that the following conditions
   8  * are met:
   9  * 1. Redistributions of source code must retain the above copyright
  10  *    notice, this list of conditions and the following 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 AND CONTRIBUTORS ``AS IS'' AND
  16  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  17  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  18  * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
  19  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  20  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  21  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  22  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  23  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  24  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  25  * SUCH DAMAGE.
  26  */
  27
  28
  29 #include "libm.h"
  30 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
  31 long double fmal(long double x, long double y, long double z)
  32 {
  33         return fma(x, y, z);
  34 }
  35 #elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
  36 #include <fenv.h>
  37 #if LDBL_MANT_DIG == 64
  38 #define LASTBIT(u) (u.i.m & 1)
  39 #define SPLIT (0x1p32L + 1)
  40 #elif LDBL_MANT_DIG == 113
  41 #define LASTBIT(u) (u.i.lo & 1)
  42 #define SPLIT (0x1p57L + 1)
  43 #endif
  44
  45 /*
  46  * A struct dd represents a floating-point number with twice the precision
  47  * of a long double.  We maintain the invariant that "hi" stores the high-order
  48  * bits of the result.
  49  */
  50 struct dd {
  51         long double hi;
  52         long double lo;
  53 };
  54
  55 /*
  56  * Compute a+b exactly, returning the exact result in a struct dd.  We assume
  57  * that both a and b are finite, but make no assumptions about their relative
  58  * magnitudes.
  59  */
  60 static inline struct dd dd_add(long double a, long double b)
  61 {
  62         struct dd ret;
  63         long double s;
  64
  65         ret.hi = a + b;
  66         s = ret.hi - a;
  67         ret.lo = (a - (ret.hi - s)) + (b - s);
  68         return (ret);
  69 }
  70
  71 /*
  72  * Compute a+b, with a small tweak:  The least significant bit of the
  73  * result is adjusted into a sticky bit summarizing all the bits that
  74  * were lost to rounding.  This adjustment negates the effects of double
  75  * rounding when the result is added to another number with a higher
  76  * exponent.  For an explanation of round and sticky bits, see any reference
  77  * on FPU design, e.g.,
  78  *
  79  *     J. Coonen.  An Implementation Guide to a Proposed Standard for
  80  *     Floating-Point Arithmetic.  Computer, vol. 13, no. 1, Jan 1980.
  81  */
  82 static inline long double add_adjusted(long double a, long double b)
  83 {
  84         struct dd sum;
  85         union ldshape u;
  86
  87         sum = dd_add(a, b);
  88         if (sum.lo != 0) {
  89                 u.f = sum.hi;
  90                 if (!LASTBIT(u))
  91                         sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
  92         }
  93         return (sum.hi);
  94 }
  95
  96 /*
  97  * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
  98  * that the result will be subnormal, and care is taken to ensure that
  99  * double rounding does not occur.
 100  */
 101 static inline long double add_and_denormalize(long double a, long double b, int scale)
 102 {
 103         struct dd sum;
 104         int bits_lost;
 105         union ldshape u;
 106
 107         sum = dd_add(a, b);
 108
 109         /*
 110          * If we are losing at least two bits of accuracy to denormalization,
 111          * then the first lost bit becomes a round bit, and we adjust the
 112          * lowest bit of sum.hi to make it a sticky bit summarizing all the
 113          * bits in sum.lo. With the sticky bit adjusted, the hardware will
 114          * break any ties in the correct direction.
 115          *
 116          * If we are losing only one bit to denormalization, however, we must
 117          * break the ties manually.
 118          */
 119         if (sum.lo != 0) {
 120                 u.f = sum.hi;
 121                 bits_lost = -u.i.se - scale + 1;
 122                 if ((bits_lost != 1) ^ LASTBIT(u))
 123                         sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
 124         }
 125         return scalbnl(sum.hi, scale);
 126 }
 127
 128 /*
 129  * Compute a*b exactly, returning the exact result in a struct dd.  We assume
 130  * that both a and b are normalized, so no underflow or overflow will occur.
 131  * The current rounding mode must be round-to-nearest.
 132  */
 133 static inline struct dd dd_mul(long double a, long double b)
 134 {
 135         struct dd ret;
 136         long double ha, hb, la, lb, p, q;
 137
 138         p = a * SPLIT;
 139         ha = a - p;
 140         ha += p;
 141         la = a - ha;
 142
 143         p = b * SPLIT;
 144         hb = b - p;
 145         hb += p;
 146         lb = b - hb;
 147
 148         p = ha * hb;
 149         q = ha * lb + la * hb;
 150
 151         ret.hi = p + q;
 152         ret.lo = p - ret.hi + q + la * lb;
 153         return (ret);
 154 }
 155
 156 /*
 157  * Fused multiply-add: Compute x * y + z with a single rounding error.
 158  *
 159  * We use scaling to avoid overflow/underflow, along with the
 160  * canonical precision-doubling technique adapted from:
 161  *
 162  *      Dekker, T.  A Floating-Point Technique for Extending the
 163  *      Available Precision.  Numer. Math. 18, 224-242 (1971).
 164  */
 165 long double fmal(long double x, long double y, long double z)
 166 {
 167         #pragma STDC FENV_ACCESS ON
 168         long double xs, ys, zs, adj;
 169         struct dd xy, r;
 170         int oround;
 171         int ex, ey, ez;
 172         int spread;
 173
 174         /*
 175          * Handle special cases. The order of operations and the particular
 176          * return values here are crucial in handling special cases involving
 177          * infinities, NaNs, overflows, and signed zeroes correctly.
 178          */
 179         if (!isfinite(x) || !isfinite(y))
 180                 return (x * y + z);
 181         if (!isfinite(z))
 182                 return (z);
 183         if (x == 0.0 || y == 0.0)
 184                 return (x * y + z);
 185         if (z == 0.0)
 186                 return (x * y);
 187
 188         xs = frexpl(x, &ex);
 189         ys = frexpl(y, &ey);
 190         zs = frexpl(z, &ez);
 191         oround = fegetround();
 192         spread = ex + ey - ez;
 193
 194         /*
 195          * If x * y and z are many orders of magnitude apart, the scaling
 196          * will overflow, so we handle these cases specially.  Rounding
 197          * modes other than FE_TONEAREST are painful.
 198          */
 199         if (spread < -LDBL_MANT_DIG) {
 200 #ifdef FE_INEXACT
 201                 feraiseexcept(FE_INEXACT);
 202 #endif
 203 #ifdef FE_UNDERFLOW
 204                 if (!isnormal(z))
 205                         feraiseexcept(FE_UNDERFLOW);
 206 #endif
 207                 switch (oround) {
 208                 default: /* FE_TONEAREST */
 209                         return (z);
 210 #ifdef FE_TOWARDZERO
 211                 case FE_TOWARDZERO:
 212                         if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
 213                                 return (z);
 214                         else
 215                                 return (nextafterl(z, 0));
 216 #endif
 217 #ifdef FE_DOWNWARD
 218                 case FE_DOWNWARD:
 219                         if (x > 0.0 ^ y < 0.0)
 220                                 return (z);
 221                         else
 222                                 return (nextafterl(z, -INFINITY));
 223 #endif
 224 #ifdef FE_UPWARD
 225                 case FE_UPWARD:
 226                         if (x > 0.0 ^ y < 0.0)
 227                                 return (nextafterl(z, INFINITY));
 228                         else
 229                                 return (z);
 230 #endif
 231                 }
 232         }
 233         if (spread <= LDBL_MANT_DIG * 2)
 234                 zs = scalbnl(zs, -spread);
 235         else
 236                 zs = copysignl(LDBL_MIN, zs);
 237
 238         fesetround(FE_TONEAREST);
 239
 240         /*
 241          * Basic approach for round-to-nearest:
 242          *
 243          *     (xy.hi, xy.lo) = x * y           (exact)
 244          *     (r.hi, r.lo)   = xy.hi + z       (exact)
 245          *     adj = xy.lo + r.lo               (inexact; low bit is sticky)
 246          *     result = r.hi + adj              (correctly rounded)
 247          */
 248         xy = dd_mul(xs, ys);
 249         r = dd_add(xy.hi, zs);
 250
 251         spread = ex + ey;
 252
 253         if (r.hi == 0.0) {
 254                 /*
 255                  * When the addends cancel to 0, ensure that the result has
 256                  * the correct sign.
 257                  */
 258                 fesetround(oround);
 259                 volatile long double vzs = zs; /* XXX gcc CSE bug workaround */
 260                 return xy.hi + vzs + scalbnl(xy.lo, spread);
 261         }
 262
 263         if (oround != FE_TONEAREST) {
 264                 /*
 265                  * There is no need to worry about double rounding in directed
 266                  * rounding modes.
 267                  * But underflow may not be raised correctly, example in downward rounding:
 268                  * fmal(0x1.0000000001p-16000L, 0x1.0000000001p-400L, -0x1p-16440L)
 269                  */
 270                 long double ret;
 271 #if defined(FE_INEXACT) && defined(FE_UNDERFLOW)
 272                 int e = fetestexcept(FE_INEXACT);
 273                 feclearexcept(FE_INEXACT);
 274 #endif
 275                 fesetround(oround);
 276                 adj = r.lo + xy.lo;
 277                 ret = scalbnl(r.hi + adj, spread);
 278 #if defined(FE_INEXACT) && defined(FE_UNDERFLOW)
 279                 if (ilogbl(ret) < -16382 && fetestexcept(FE_INEXACT))
 280                         feraiseexcept(FE_UNDERFLOW);
 281                 else if (e)
 282                         feraiseexcept(FE_INEXACT);
 283 #endif
 284                 return ret;
 285         }
 286
 287         adj = add_adjusted(r.lo, xy.lo);
 288         if (spread + ilogbl(r.hi) > -16383)
 289                 return scalbnl(r.hi + adj, spread);
 290         else
 291                 return add_and_denormalize(r.hi, adj, spread);
 292 }
 293 #endif