| 1 | /*-
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| 2 | * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
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| 3 | * All rights reserved.
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| 4 | *
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| 5 | * Redistribution and use in source and binary forms, with or without
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| 6 | * modification, are permitted provided that the following conditions
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| 7 | * are met:
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| 8 | * 1. Redistributions of source code must retain the above copyright
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| 9 | * notice, this list of conditions and the following disclaimer.
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| 10 | * 2. Redistributions in binary form must reproduce the above copyright
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| 11 | * notice, this list of conditions and the following disclaimer in the
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| 12 | * documentation and/or other materials provided with the distribution.
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| 13 | *
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| 14 | * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
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| 15 | * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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| 16 | * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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| 17 | * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
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| 18 | * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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| 19 | * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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| 20 | * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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| 21 | * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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| 22 | * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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| 23 | * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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| 24 | * SUCH DAMAGE.
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| 25 | */
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| 26 |
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| 27 | #include <sys/cdefs.h>
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| 28 | __FBSDID("$FreeBSD: src/lib/msun/src/s_fma.c,v 1.4 2005/03/18 02:27:59 das Exp $");
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| 29 |
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| 30 | #include <fenv.h>
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| 31 | #include <float.h>
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| 32 | #include <math.h>
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| 33 |
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| 34 | /*
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| 35 | * Fused multiply-add: Compute x * y + z with a single rounding error.
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| 36 | *
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| 37 | * We use scaling to avoid overflow/underflow, along with the
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| 38 | * canonical precision-doubling technique adapted from:
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| 39 | *
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| 40 | * Dekker, T. A Floating-Point Technique for Extending the
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| 41 | * Available Precision. Numer. Math. 18, 224-242 (1971).
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| 42 | *
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| 43 | * This algorithm is sensitive to the rounding precision. FPUs such
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| 44 | * as the i387 must be set in double-precision mode if variables are
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| 45 | * to be stored in FP registers in order to avoid incorrect results.
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| 46 | * This is the default on FreeBSD, but not on many other systems.
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| 47 | *
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| 48 | * Hardware instructions should be used on architectures that support it,
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| 49 | * since this implementation will likely be several times slower.
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| 50 | */
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| 51 | #if LDBL_MANT_DIG != 113
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| 52 | double
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| 53 | fma(double x, double y, double z)
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| 54 | {
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| 55 | static const double split = 0x1p27 + 1.0;
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| 56 | double xs, ys, zs;
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| 57 | double c, cc, hx, hy, p, q, tx, ty;
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| 58 | double r, rr, s;
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| 59 | int oround;
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| 60 | int ex, ey, ez;
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| 61 | int spread;
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| 62 |
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| 63 | if (z == 0.0)
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| 64 | return (x * y);
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| 65 | if (x == 0.0 || y == 0.0)
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| 66 | return (x * y + z);
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| 67 |
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| 68 | /* Results of frexp() are undefined for these cases. */
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| 69 | if (!isfinite(x) || !isfinite(y) || !isfinite(z))
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| 70 | return (x * y + z);
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| 71 |
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| 72 | xs = frexp(x, &ex);
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| 73 | ys = frexp(y, &ey);
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| 74 | zs = frexp(z, &ez);
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| 75 | oround = fegetround();
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| 76 | spread = ex + ey - ez;
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| 77 |
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| 78 | /*
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| 79 | * If x * y and z are many orders of magnitude apart, the scaling
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| 80 | * will overflow, so we handle these cases specially. Rounding
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| 81 | * modes other than FE_TONEAREST are painful.
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| 82 | */
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| 83 | if (spread > DBL_MANT_DIG * 2) {
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| 84 | fenv_t env;
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| 85 | feraiseexcept(FE_INEXACT);
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| 86 | switch(oround) {
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| 87 | case FE_TONEAREST:
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| 88 | return (x * y);
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| 89 | case FE_TOWARDZERO:
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| 90 | if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
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| 91 | return (x * y);
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| 92 | feholdexcept(&env);
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| 93 | r = x * y;
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| 94 | if (!fetestexcept(FE_INEXACT))
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| 95 | r = nextafter(r, 0);
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| 96 | feupdateenv(&env);
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| 97 | return (r);
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| 98 | case FE_DOWNWARD:
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| 99 | if (z > 0.0)
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| 100 | return (x * y);
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| 101 | feholdexcept(&env);
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| 102 | r = x * y;
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| 103 | if (!fetestexcept(FE_INEXACT))
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| 104 | r = nextafter(r, -INFINITY);
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| 105 | feupdateenv(&env);
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| 106 | return (r);
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| 107 | default: /* FE_UPWARD */
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| 108 | if (z < 0.0)
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| 109 | return (x * y);
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| 110 | feholdexcept(&env);
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| 111 | r = x * y;
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| 112 | if (!fetestexcept(FE_INEXACT))
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| 113 | r = nextafter(r, INFINITY);
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| 114 | feupdateenv(&env);
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| 115 | return (r);
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| 116 | }
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| 117 | }
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| 118 | if (spread < -DBL_MANT_DIG) {
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| 119 | feraiseexcept(FE_INEXACT);
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| 120 | if (!isnormal(z))
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| 121 | feraiseexcept(FE_UNDERFLOW);
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| 122 | switch (oround) {
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| 123 | case FE_TONEAREST:
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| 124 | return (z);
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| 125 | case FE_TOWARDZERO:
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| 126 | if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
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| 127 | return (z);
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| 128 | else
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| 129 | return (nextafter(z, 0));
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| 130 | case FE_DOWNWARD:
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| 131 | if (x > 0.0 ^ y < 0.0)
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| 132 | return (z);
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| 133 | else
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| 134 | return (nextafter(z, -INFINITY));
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| 135 | default: /* FE_UPWARD */
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| 136 | if (x > 0.0 ^ y < 0.0)
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| 137 | return (nextafter(z, INFINITY));
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| 138 | else
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| 139 | return (z);
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| 140 | }
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| 141 | }
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| 142 |
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| 143 | /*
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| 144 | * Use Dekker's algorithm to perform the multiplication and
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| 145 | * subsequent addition in twice the machine precision.
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| 146 | * Arrange so that x * y = c + cc, and x * y + z = r + rr.
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| 147 | */
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| 148 | fesetround(FE_TONEAREST);
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| 149 |
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| 150 | p = xs * split;
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| 151 | hx = xs - p;
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| 152 | hx += p;
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| 153 | tx = xs - hx;
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| 154 |
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| 155 | p = ys * split;
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| 156 | hy = ys - p;
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| 157 | hy += p;
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| 158 | ty = ys - hy;
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| 159 |
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| 160 | p = hx * hy;
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| 161 | q = hx * ty + tx * hy;
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| 162 | c = p + q;
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| 163 | cc = p - c + q + tx * ty;
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| 164 |
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| 165 | zs = ldexp(zs, -spread);
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| 166 | r = c + zs;
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| 167 | s = r - c;
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| 168 | rr = (c - (r - s)) + (zs - s) + cc;
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| 169 |
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| 170 | spread = ex + ey;
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| 171 | if (spread + ilogb(r) > -1023) {
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| 172 | fesetround(oround);
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| 173 | r = r + rr;
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| 174 | } else {
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| 175 | /*
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| 176 | * The result is subnormal, so we round before scaling to
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| 177 | * avoid double rounding.
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| 178 | */
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| 179 | p = ldexp(copysign(0x1p-1022, r), -spread);
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| 180 | c = r + p;
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| 181 | s = c - r;
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| 182 | cc = (r - (c - s)) + (p - s) + rr;
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| 183 | fesetround(oround);
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| 184 | r = (c + cc) - p;
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| 185 | }
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| 186 | return (ldexp(r, spread));
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| 187 | }
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| 188 | #else /* LDBL_MANT_DIG == 113 */
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| 189 | /*
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| 190 | * 113 bits of precision is more than twice the precision of a double,
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| 191 | * so it is enough to represent the intermediate product exactly.
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| 192 | */
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| 193 | double
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| 194 | fma(double x, double y, double z)
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| 195 | {
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| 196 | return ((long double)x * y + z);
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| 197 | }
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| 198 | #endif /* LDBL_MANT_DIG != 113 */
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| 199 |
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| 200 | #if (LDBL_MANT_DIG == 53)
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| 201 | __weak_reference(fma, fmal);
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| 202 | #endif
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