compiler/stdc/math/s_fmal.c

   1 /*-
   2  * Copyright (c) 2005-2011 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 1.7 2011/10/21 06:30:43 das Exp $");
  28
  29 #include <fenv.h>
  30 #include <float.h>
  31 #include <math.h>
  32
  33 #include "fpmath.h"
  34 #include "math_private.h"
  35
  36 /*
  37  * A struct dd represents a floating-point number with twice the precision
  38  * of a long double.  We maintain the invariant that "hi" stores the high-order
  39  * bits of the result.
  40  */
  41 struct dd {
  42         long double hi;
  43         long double lo;
  44 };
  45
  46 /*
  47  * Compute a+b exactly, returning the exact result in a struct dd.  We assume
  48  * that both a and b are finite, but make no assumptions about their relative
  49  * magnitudes.
  50  */
  51 static inline struct dd
  52 dd_add(long double a, long double b)
  53 {
  54         struct dd ret;
  55         long double s;
  56
  57         ret.hi = a + b;
  58         s = ret.hi - a;
  59         ret.lo = (a - (ret.hi - s)) + (b - s);
  60         return (ret);
  61 }
  62
  63 /*
  64  * Compute a+b, with a small tweak:  The least significant bit of the
  65  * result is adjusted into a sticky bit summarizing all the bits that
  66  * were lost to rounding.  This adjustment negates the effects of double
  67  * rounding when the result is added to another number with a higher
  68  * exponent.  For an explanation of round and sticky bits, see any reference
  69  * on FPU design, e.g.,
  70  *
  71  *     J. Coonen.  An Implementation Guide to a Proposed Standard for
  72  *     Floating-Point Arithmetic.  Computer, vol. 13, no. 1, Jan 1980.
  73  */
  74 static inline long double
  75 add_adjusted(long double a, long double b)
  76 {
  77         struct dd sum;
  78         union IEEEl2bits u;
  79
  80         sum = dd_add(a, b);
  81         if (sum.lo != 0) {
  82                 u.e = sum.hi;
  83                 if ((u.bits.manl & 1) == 0)
  84                         sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
  85         }
  86         return (sum.hi);
  87 }
  88
  89 /*
  90  * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
  91  * that the result will be subnormal, and care is taken to ensure that
  92  * double rounding does not occur.
  93  */
  94 static inline long double
  95 add_and_denormalize(long double a, long double b, int scale)
  96 {
  97         struct dd sum;
  98         int bits_lost;
  99         union IEEEl2bits u;
 100
 101         sum = dd_add(a, b);
 102
 103         /*
 104          * If we are losing at least two bits of accuracy to denormalization,
 105          * then the first lost bit becomes a round bit, and we adjust the
 106          * lowest bit of sum.hi to make it a sticky bit summarizing all the
 107          * bits in sum.lo. With the sticky bit adjusted, the hardware will
 108          * break any ties in the correct direction.
 109          *
 110          * If we are losing only one bit to denormalization, however, we must
 111          * break the ties manually.
 112          */
 113         if (sum.lo != 0) {
 114                 u.e = sum.hi;
 115                 bits_lost = -u.bits.exp - scale + 1;
 116                 if ((bits_lost != 1) ^ (int)(u.bits.manl & 1))
 117                         sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
 118         }
 119         return (ldexp(sum.hi, scale));
 120 }
 121
 122 /*
 123  * Compute a*b exactly, returning the exact result in a struct dd.  We assume
 124  * that both a and b are normalized, so no underflow or overflow will occur.
 125  * The current rounding mode must be round-to-nearest.
 126  */
 127 static inline struct dd
 128 dd_mul(long double a, long double b)
 129 {
 130 #if LDBL_MANT_DIG == 64
 131         static const long double split = 0x1p32L + 1.0;
 132 #elif LDBL_MANT_DIG == 113
 133         static const long double split = 0x1p57L + 1.0;
 134 #endif
 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
 166 fmal(long double x, long double y, long double z)
 167 {
 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 (x == 0.0 || y == 0.0)
 180                 return (x * y + z);
 181         if (z == 0.0)
 182                 return (x * y);
 183         if (!isfinite(x) || !isfinite(y))
 184                 return (x * y + z);
 185         if (!isfinite(z))
 186                 return (z);
 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                 feraiseexcept(FE_INEXACT);
 201                 if (!isnormal(z))
 202                         feraiseexcept(FE_UNDERFLOW);
 203                 switch (oround) {
 204                 case FE_TONEAREST:
 205                         return (z);
 206                 case FE_TOWARDZERO:
 207                         if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
 208                                 return (z);
 209                         else
 210                                 return (nextafterl(z, 0));
 211                 case FE_DOWNWARD:
 212                         if (x > 0.0 ^ y < 0.0)
 213                                 return (z);
 214                         else
 215                                 return (nextafterl(z, -INFINITY));
 216                 default:        /* FE_UPWARD */
 217                         if (x > 0.0 ^ y < 0.0)
 218                                 return (nextafterl(z, INFINITY));
 219                         else
 220                                 return (z);
 221                 }
 222         }
 223         if (spread <= LDBL_MANT_DIG * 2)
 224                 zs = ldexpl(zs, -spread);
 225         else
 226                 zs = copysignl(LDBL_MIN, zs);
 227
 228         fesetround(FE_TONEAREST);
 229         /* work around clang bug 8100 */
 230         volatile long double vxs = xs;
 231
 232         /*
 233          * Basic approach for round-to-nearest:
 234          *
 235          *     (xy.hi, xy.lo) = x * y           (exact)
 236          *     (r.hi, r.lo)   = xy.hi + z       (exact)
 237          *     adj = xy.lo + r.lo               (inexact; low bit is sticky)
 238          *     result = r.hi + adj              (correctly rounded)
 239          */
 240         xy = dd_mul(vxs, ys);
 241         r = dd_add(xy.hi, zs);
 242
 243         spread = ex + ey;
 244
 245         if (r.hi == 0.0) {
 246                 /*
 247                  * When the addends cancel to 0, ensure that the result has
 248                  * the correct sign.
 249                  */
 250                 fesetround(oround);
 251                 volatile long double vzs = zs; /* XXX gcc CSE bug workaround */
 252                 return (xy.hi + vzs + ldexpl(xy.lo, spread));
 253         }
 254
 255         if (oround != FE_TONEAREST) {
 256                 /*
 257                  * There is no need to worry about double rounding in directed
 258                  * rounding modes.
 259                  */
 260                 fesetround(oround);
 261                 /* work around clang bug 8100 */
 262                 volatile long double vrlo = r.lo;
 263                 adj = vrlo + xy.lo;
 264                 return (ldexpl(r.hi + adj, spread));
 265         }
 266
 267         adj = add_adjusted(r.lo, xy.lo);
 268         if (spread + ilogbl(r.hi) > -16383)
 269                 return (ldexpl(r.hi + adj, spread));
 270         else
 271                 return (add_and_denormalize(r.hi, adj, spread));
 272 }