compiler/stdc/math/s_fma.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_fma.c,v 1.8 2011/10/21 06:30:43 das Exp $");
  28
  29 #include <aros/system.h>
  30
  31 #include <fenv.h>
  32 #include <float.h>
  33
  34 #include "math.h"
  35 #include "math_private.h"
  36
  37 /*
  38  * A struct dd represents a floating-point number with twice the precision
  39  * of a double.  We maintain the invariant that "hi" stores the 53 high-order
  40  * bits of the result.
  41  */
  42 struct dd {
  43         double hi;
  44         double lo;
  45 };
  46
  47 /*
  48  * Compute a+b exactly, returning the exact result in a struct dd.  We assume
  49  * that both a and b are finite, but make no assumptions about their relative
  50  * magnitudes.
  51  */
  52 static inline struct dd
  53 dd_add(double a, double b)
  54 {
  55         struct dd ret;
  56         double s;
  57
  58         ret.hi = a + b;
  59         s = ret.hi - a;
  60         ret.lo = (a - (ret.hi - s)) + (b - s);
  61         return (ret);
  62 }
  63
  64 /*
  65  * Compute a+b, with a small tweak:  The least significant bit of the
  66  * result is adjusted into a sticky bit summarizing all the bits that
  67  * were lost to rounding.  This adjustment negates the effects of double
  68  * rounding when the result is added to another number with a higher
  69  * exponent.  For an explanation of round and sticky bits, see any reference
  70  * on FPU design, e.g.,
  71  *
  72  *     J. Coonen.  An Implementation Guide to a Proposed Standard for
  73  *     Floating-Point Arithmetic.  Computer, vol. 13, no. 1, Jan 1980.
  74  */
  75 static inline double
  76 add_adjusted(double a, double b)
  77 {
  78         struct dd sum;
  79         uint64_t hibits, lobits;
  80
  81         sum = dd_add(a, b);
  82         if (sum.lo != 0) {
  83                 EXTRACT_WORD64(hibits, sum.hi);
  84                 if ((hibits & 1) == 0) {
  85                         /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
  86                         EXTRACT_WORD64(lobits, sum.lo);
  87                         hibits += 1 - ((hibits ^ lobits) >> 62);
  88                         INSERT_WORD64(sum.hi, hibits);
  89                 }
  90         }
  91         return (sum.hi);
  92 }
  93
  94 /*
  95  * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
  96  * that the result will be subnormal, and care is taken to ensure that
  97  * double rounding does not occur.
  98  */
  99 static inline double
 100 add_and_denormalize(double a, double b, int scale)
 101 {
 102         struct dd sum;
 103         uint64_t hibits, lobits;
 104         int bits_lost;
 105
 106         sum = dd_add(a, b);
 107
 108         /*
 109          * If we are losing at least two bits of accuracy to denormalization,
 110          * then the first lost bit becomes a round bit, and we adjust the
 111          * lowest bit of sum.hi to make it a sticky bit summarizing all the
 112          * bits in sum.lo. With the sticky bit adjusted, the hardware will
 113          * break any ties in the correct direction.
 114          *
 115          * If we are losing only one bit to denormalization, however, we must
 116          * break the ties manually.
 117          */
 118         if (sum.lo != 0) {
 119                 EXTRACT_WORD64(hibits, sum.hi);
 120                 bits_lost = -((int)(hibits >> 52) & 0x7ff) - scale + 1;
 121                 if ((bits_lost != 1) ^ (int)(hibits & 1)) {
 122                         /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
 123                         EXTRACT_WORD64(lobits, sum.lo);
 124                         hibits += 1 - (((hibits ^ lobits) >> 62) & 2);
 125                         INSERT_WORD64(sum.hi, hibits);
 126                 }
 127         }
 128         return (ldexp(sum.hi, scale));
 129 }
 130
 131 /*
 132  * Compute a*b exactly, returning the exact result in a struct dd.  We assume
 133  * that both a and b are normalized, so no underflow or overflow will occur.
 134  * The current rounding mode must be round-to-nearest.
 135  */
 136 static inline struct dd
 137 dd_mul(double a, double b)
 138 {
 139         static const double split = 0x1p27 + 1.0;
 140         struct dd ret;
 141         double ha, hb, la, lb, p, q;
 142
 143         p = a * split;
 144         ha = a - p;
 145         ha += p;
 146         la = a - ha;
 147
 148         p = b * split;
 149         hb = b - p;
 150         hb += p;
 151         lb = b - hb;
 152
 153         p = ha * hb;
 154         q = ha * lb + la * hb;
 155
 156         ret.hi = p + q;
 157         ret.lo = p - ret.hi + q + la * lb;
 158         return (ret);
 159 }
 160
 161 /*
 162  * Fused multiply-add: Compute x * y + z with a single rounding error.
 163  *
 164  * We use scaling to avoid overflow/underflow, along with the
 165  * canonical precision-doubling technique adapted from:
 166  *
 167  *      Dekker, T.  A Floating-Point Technique for Extending the
 168  *      Available Precision.  Numer. Math. 18, 224-242 (1971).
 169  *
 170  * This algorithm is sensitive to the rounding precision.  FPUs such
 171  * as the i387 must be set in double-precision mode if variables are
 172  * to be stored in FP registers in order to avoid incorrect results.
 173  * This is the default on FreeBSD, but not on many other systems.
 174  *
 175  * Hardware instructions should be used on architectures that support it,
 176  * since this implementation will likely be several times slower.
 177  */
 178
 179 double
 180 fma(double x, double y, double z)
 181 {
 182         double xs, ys, zs, adj;
 183         struct dd xy, r;
 184         int oround;
 185         int ex, ey, ez;
 186         int spread;
 187
 188         /*
 189          * Handle special cases. The order of operations and the particular
 190          * return values here are crucial in handling special cases involving
 191          * infinities, NaNs, overflows, and signed zeroes correctly.
 192          */
 193         if (x == 0.0 || y == 0.0)
 194                 return (x * y + z);
 195         if (z == 0.0)
 196                 return (x * y);
 197         if (!isfinite(x) || !isfinite(y))
 198                 return (x * y + z);
 199         if (!isfinite(z))
 200                 return (z);
 201
 202         xs = frexp(x, &ex);
 203         ys = frexp(y, &ey);
 204         zs = frexp(z, &ez);
 205         oround = fegetround();
 206         spread = ex + ey - ez;
 207
 208         /*
 209          * If x * y and z are many orders of magnitude apart, the scaling
 210          * will overflow, so we handle these cases specially.  Rounding
 211          * modes other than FE_TONEAREST are painful.
 212          */
 213         if (spread < -DBL_MANT_DIG) {
 214                 feraiseexcept(FE_INEXACT);
 215                 if (!isnormal(z))
 216                         feraiseexcept(FE_UNDERFLOW);
 217                 switch (oround) {
 218                 case FE_TONEAREST:
 219                         return (z);
 220                 case FE_TOWARDZERO:
 221                         if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
 222                                 return (z);
 223                         else
 224                                 return (nextafter(z, 0));
 225                 case FE_DOWNWARD:
 226                         if (x > 0.0 ^ y < 0.0)
 227                                 return (z);
 228                         else
 229                                 return (nextafter(z, -INFINITY));
 230                 default:        /* FE_UPWARD */
 231                         if (x > 0.0 ^ y < 0.0)
 232                                 return (nextafter(z, INFINITY));
 233                         else
 234                                 return (z);
 235                 }
 236         }
 237         if (spread <= DBL_MANT_DIG * 2)
 238                 zs = ldexp(zs, -spread);
 239         else
 240                 zs = copysign(DBL_MIN, zs);
 241
 242         fesetround(FE_TONEAREST);
 243         /* work around clang bug 8100 */
 244         volatile double vxs = xs;
 245
 246         /*
 247          * Basic approach for round-to-nearest:
 248          *
 249          *     (xy.hi, xy.lo) = x * y           (exact)
 250          *     (r.hi, r.lo)   = xy.hi + z       (exact)
 251          *     adj = xy.lo + r.lo               (inexact; low bit is sticky)
 252          *     result = r.hi + adj              (correctly rounded)
 253          */
 254         xy = dd_mul(vxs, ys);
 255         r = dd_add(xy.hi, zs);
 256
 257         spread = ex + ey;
 258
 259         if (r.hi == 0.0) {
 260                 /*
 261                  * When the addends cancel to 0, ensure that the result has
 262                  * the correct sign.
 263                  */
 264                 fesetround(oround);
 265                 volatile double vzs = zs; /* XXX gcc CSE bug workaround */
 266                 return (xy.hi + vzs + ldexp(xy.lo, spread));
 267         }
 268
 269         if (oround != FE_TONEAREST) {
 270                 /*
 271                  * There is no need to worry about double rounding in directed
 272                  * rounding modes.
 273                  */
 274                 fesetround(oround);
 275                 /* work around clang bug 8100 */
 276                 volatile double vrlo = r.lo;
 277                 adj = vrlo + xy.lo;
 278                 return (ldexp(r.hi + adj, spread));
 279         }
 280
 281         adj = add_adjusted(r.lo, xy.lo);
 282         if (spread + ilogb(r.hi) > -1023)
 283                 return (ldexp(r.hi + adj, spread));
 284         else
 285                 return (add_and_denormalize(r.hi, adj, spread));
 286 }
 287
 288 #if (LDBL_MANT_DIG == 53)
 289 /* Alias fma -> fmal */
 290 AROS_MAKE_ASM_SYM(typeof(fmal), fmal, AROS_CSYM_FROM_ASM_NAME(fmal), AROS_CSYM_FROM_ASM_NAME(fma));
 291 AROS_EXPORT_ASM_SYM(AROS_CSYM_FROM_ASM_NAME(fmal));
 292 #endif