lib/libm/src/s_fmal.c

   1 /*      $NetBSD: s_fmal.c,v 1.3 2013/02/12 21:40:19 martin Exp $        */
   2
   3 /*-
   4  * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
   5  * All rights reserved.
   6  *
   7  * Redistribution and use in source and binary forms, with or without
   8  * modification, are permitted provided that the following conditions
   9  * are met:
  10  * 1. Redistributions of source code must retain the above copyright
  11  *    notice, this list of conditions and the following disclaimer.
  12  * 2. Redistributions in binary form must reproduce the above copyright
  13  *    notice, this list of conditions and the following disclaimer in the
  14  *    documentation and/or other materials provided with the distribution.
  15  *
  16  * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
  17  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  18  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  19  * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
  20  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  21  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  22  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  23  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  24  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  25  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  26  * SUCH DAMAGE.
  27  */
  28
  29 #include <sys/cdefs.h>
  30 #if 0
  31 __FBSDID("$FreeBSD: src/lib/msun/src/s_fmal.c,v 1.7 2011/10/21 06:30:43 das Exp $");
  32 #else
  33 __RCSID("$NetBSD: s_fmal.c,v 1.3 2013/02/12 21:40:19 martin Exp $");
  34 #endif
  35
  36 #include <machine/ieee.h>
  37 #include <fenv.h>
  38 #include <float.h>
  39 #include <math.h>
  40
  41 #include "math_private.h"
  42
  43 #ifdef __HAVE_LONG_DOUBLE
  44 /*
  45  * A struct dd represents a floating-point number with twice the precision
  46  * of a long double.  We maintain the invariant that "hi" stores the high-order
  47  * bits of the result.
  48  */
  49 struct dd {
  50         long double hi;
  51         long double lo;
  52 };
  53
  54 /*
  55  * Compute a+b exactly, returning the exact result in a struct dd.  We assume
  56  * that both a and b are finite, but make no assumptions about their relative
  57  * magnitudes.
  58  */
  59 static inline struct dd
  60 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
  83 add_adjusted(long double a, long double b)
  84 {
  85         struct dd sum;
  86         union ieee_ext_u u;
  87
  88         sum = dd_add(a, b);
  89         if (sum.lo != 0) {
  90                 u.extu_ld = sum.hi;
  91                 if ((u.extu_ext.ext_fracl & 1) == 0)
  92                         sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
  93         }
  94         return (sum.hi);
  95 }
  96
  97 /*
  98  * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
  99  * that the result will be subnormal, and care is taken to ensure that
 100  * double rounding does not occur.
 101  */
 102 static inline long double
 103 add_and_denormalize(long double a, long double b, int scale)
 104 {
 105         struct dd sum;
 106         int bits_lost;
 107         union ieee_ext_u u;
 108
 109         sum = dd_add(a, b);
 110
 111         /*
 112          * If we are losing at least two bits of accuracy to denormalization,
 113          * then the first lost bit becomes a round bit, and we adjust the
 114          * lowest bit of sum.hi to make it a sticky bit summarizing all the
 115          * bits in sum.lo. With the sticky bit adjusted, the hardware will
 116          * break any ties in the correct direction.
 117          *
 118          * If we are losing only one bit to denormalization, however, we must
 119          * break the ties manually.
 120          */
 121         if (sum.lo != 0) {
 122                 u.extu_ld = sum.hi;
 123                 bits_lost = -u.extu_ext.ext_exp - scale + 1;
 124                 if ((bits_lost != 1) ^ (int)(u.extu_ext.ext_fracl & 1))
 125                         sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
 126         }
 127         return (ldexp((double)sum.hi, scale));
 128 }
 129
 130 /*
 131  * Compute a*b exactly, returning the exact result in a struct dd.  We assume
 132  * that both a and b are normalized, so no underflow or overflow will occur.
 133  * The current rounding mode must be round-to-nearest.
 134  */
 135 static inline struct dd
 136 dd_mul(long double a, long double b)
 137 {
 138 #if LDBL_MANT_DIG == 64
 139         static const long double split = 0x1p32L + 1.0;
 140 #elif LDBL_MANT_DIG == 113
 141         static const long double split = 0x1p57L + 1.0;
 142 #endif
 143         struct dd ret;
 144         long double ha, hb, la, lb, p, q;
 145
 146         p = a * split;
 147         ha = a - p;
 148         ha += p;
 149         la = a - ha;
 150
 151         p = b * split;
 152         hb = b - p;
 153         hb += p;
 154         lb = b - hb;
 155
 156         p = ha * hb;
 157         q = ha * lb + la * hb;
 158
 159         ret.hi = p + q;
 160         ret.lo = p - ret.hi + q + la * lb;
 161         return (ret);
 162 }
 163
 164 /*
 165  * Fused multiply-add: Compute x * y + z with a single rounding error.
 166  *
 167  * We use scaling to avoid overflow/underflow, along with the
 168  * canonical precision-doubling technique adapted from:
 169  *
 170  *      Dekker, T.  A Floating-Point Technique for Extending the
 171  *      Available Precision.  Numer. Math. 18, 224-242 (1971).
 172  */
 173 long double
 174 fmal(long double x, long double y, long double z)
 175 {
 176         long double xs, ys, zs, adj;
 177         struct dd xy, r;
 178         int oround;
 179         int ex, ey, ez;
 180         int spread;
 181
 182         /*
 183          * Handle special cases. The order of operations and the particular
 184          * return values here are crucial in handling special cases involving
 185          * infinities, NaNs, overflows, and signed zeroes correctly.
 186          */
 187         if (x == 0.0 || y == 0.0)
 188                 return (x * y + z);
 189         if (z == 0.0)
 190                 return (x * y);
 191         if (!isfinite(x) || !isfinite(y))
 192                 return (x * y + z);
 193         if (!isfinite(z))
 194                 return (z);
 195
 196         xs = frexpl(x, &ex);
 197         ys = frexpl(y, &ey);
 198         zs = frexpl(z, &ez);
 199         oround = fegetround();
 200         spread = ex + ey - ez;
 201
 202         /*
 203          * If x * y and z are many orders of magnitude apart, the scaling
 204          * will overflow, so we handle these cases specially.  Rounding
 205          * modes other than FE_TONEAREST are painful.
 206          */
 207         if (spread < -LDBL_MANT_DIG) {
 208                 feraiseexcept(FE_INEXACT);
 209                 if (!isnormal(z))
 210                         feraiseexcept(FE_UNDERFLOW);
 211                 switch (oround) {
 212                 case FE_TONEAREST:
 213                         return (z);
 214                 case FE_TOWARDZERO:
 215                         if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0))
 216                                 return (z);
 217                         else
 218                                 return (nextafterl(z, 0));
 219                 case FE_DOWNWARD:
 220                         if ((x > 0.0) ^ (y < 0.0))
 221                                 return (z);
 222                         else
 223                                 return (nextafterl(z, (long double)-INFINITY));
 224                 default:        /* FE_UPWARD */
 225                         if ((x > 0.0) ^ (y < 0.0))
 226                                 return (nextafterl(z, (long double)INFINITY));
 227                         else
 228                                 return (z);
 229                 }
 230         }
 231         if (spread <= LDBL_MANT_DIG * 2)
 232                 zs = ldexpl(zs, -spread);
 233         else
 234                 zs = copysignl(LDBL_MIN, zs);
 235
 236         fesetround(FE_TONEAREST);
 237
 238         /*
 239          * Basic approach for round-to-nearest:
 240          *
 241          *     (xy.hi, xy.lo) = x * y           (exact)
 242          *     (r.hi, r.lo)   = xy.hi + z       (exact)
 243          *     adj = xy.lo + r.lo               (inexact; low bit is sticky)
 244          *     result = r.hi + adj              (correctly rounded)
 245          */
 246         xy = dd_mul(xs, ys);
 247         r = dd_add(xy.hi, zs);
 248
 249         spread = ex + ey;
 250
 251         if (r.hi == 0.0) {
 252                 /*
 253                  * When the addends cancel to 0, ensure that the result has
 254                  * the correct sign.
 255                  */
 256                 fesetround(oround);
 257                 {
 258                 volatile long double vzs = zs; /* XXX gcc CSE bug workaround */
 259                 return (xy.hi + vzs + ldexpl(xy.lo, spread));
 260                 }
 261         }
 262
 263         if (oround != FE_TONEAREST) {
 264                 /*
 265                  * There is no need to worry about double rounding in directed
 266                  * rounding modes.
 267                  */
 268                 fesetround(oround);
 269                 adj = r.lo + xy.lo;
 270                 return (ldexpl(r.hi + adj, spread));
 271         }
 272
 273         adj = add_adjusted(r.lo, xy.lo);
 274         if (spread + ilogbl(r.hi) > -16383)
 275                 return (ldexpl(r.hi + adj, spread));
 276         else
 277                 return (add_and_denormalize(r.hi, adj, spread));
 278 }
 279 #endif