1 /*- 2 * Copyright (c) 2005 David Schultz <das (at) 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 #include <sys/cdefs.h> 28 /* __FBSDID("$FreeBSD: src/lib/msun/src/s_fmal.c,v 1.2 2005/03/18 02:27:59 das Exp $"); */ 29 30 #include <fenv.h> 31 #include <float.h> 32 #include <math.h> 33 34 /* 35 * Fused multiply-add: Compute x * y + z with a single rounding error. 36 * 37 * We use scaling to avoid overflow/underflow, along with the 38 * canonical precision-doubling technique adapted from: 39 * 40 * Dekker, T. A Floating-Point Technique for Extending the 41 * Available Precision. Numer. Math. 18, 224-242 (1971). 42 */ 43 long double 44 fmal(long double x, long double y, long double z) 45 { 46 #if LDBL_MANT_DIG == 64 47 static const long double split = 0x1p32L + 1.0; 48 #elif LDBL_MANT_DIG == 113 49 static const long double split = 0x1p57L + 1.0; 50 #endif 51 long double xs, ys, zs; 52 long double c, cc, hx, hy, p, q, tx, ty; 53 long double r, rr, s; 54 int oround; 55 int ex, ey, ez; 56 int spread; 57 58 if (z == 0.0) 59 return (x * y); 60 if (x == 0.0 || y == 0.0) 61 return (x * y + z); 62 63 /* Results of frexp() are undefined for these cases. */ 64 if (!isfinite(x) || !isfinite(y) || !isfinite(z)) 65 return (x * y + z); 66 67 xs = frexpl(x, &ex); 68 ys = frexpl(y, &ey); 69 zs = frexpl(z, &ez); 70 oround = fegetround(); 71 spread = ex + ey - ez; 72 73 /* 74 * If x * y and z are many orders of magnitude apart, the scaling 75 * will overflow, so we handle these cases specially. Rounding 76 * modes other than FE_TONEAREST are painful. 77 */ 78 if (spread > LDBL_MANT_DIG * 2) { 79 fenv_t env; 80 feraiseexcept(FE_INEXACT); 81 switch(oround) { 82 case FE_TONEAREST: 83 return (x * y); 84 case FE_TOWARDZERO: 85 if (x > 0.0 ^ y < 0.0 ^ z < 0.0) 86 return (x * y); 87 feholdexcept(&env); 88 r = x * y; 89 if (!fetestexcept(FE_INEXACT)) 90 r = nextafterl(r, 0); 91 feupdateenv(&env); 92 return (r); 93 case FE_DOWNWARD: 94 if (z > 0.0) 95 return (x * y); 96 feholdexcept(&env); 97 r = x * y; 98 if (!fetestexcept(FE_INEXACT)) 99 r = nextafterl(r, -INFINITY); 100 feupdateenv(&env); 101 return (r); 102 default: /* FE_UPWARD */ 103 if (z < 0.0) 104 return (x * y); 105 feholdexcept(&env); 106 r = x * y; 107 if (!fetestexcept(FE_INEXACT)) 108 r = nextafterl(r, INFINITY); 109 feupdateenv(&env); 110 return (r); 111 } 112 } 113 if (spread < -LDBL_MANT_DIG) { 114 feraiseexcept(FE_INEXACT); 115 if (!isnormal(z)) 116 feraiseexcept(FE_UNDERFLOW); 117 switch (oround) { 118 case FE_TONEAREST: 119 return (z); 120 case FE_TOWARDZERO: 121 if (x > 0.0 ^ y < 0.0 ^ z < 0.0) 122 return (z); 123 else 124 return (nextafterl(z, 0)); 125 case FE_DOWNWARD: 126 if (x > 0.0 ^ y < 0.0) 127 return (z); 128 else 129 return (nextafterl(z, -INFINITY)); 130 default: /* FE_UPWARD */ 131 if (x > 0.0 ^ y < 0.0) 132 return (nextafterl(z, INFINITY)); 133 else 134 return (z); 135 } 136 } 137 138 /* 139 * Use Dekker's algorithm to perform the multiplication and 140 * subsequent addition in twice the machine precision. 141 * Arrange so that x * y = c + cc, and x * y + z = r + rr. 142 */ 143 fesetround(FE_TONEAREST); 144 145 p = xs * split; 146 hx = xs - p; 147 hx += p; 148 tx = xs - hx; 149 150 p = ys * split; 151 hy = ys - p; 152 hy += p; 153 ty = ys - hy; 154 155 p = hx * hy; 156 q = hx * ty + tx * hy; 157 c = p + q; 158 cc = p - c + q + tx * ty; 159 160 zs = ldexpl(zs, -spread); 161 r = c + zs; 162 s = r - c; 163 rr = (c - (r - s)) + (zs - s) + cc; 164 165 spread = ex + ey; 166 if (spread + ilogbl(r) > -16383) { 167 fesetround(oround); 168 r = r + rr; 169 } else { 170 /* 171 * The result is subnormal, so we round before scaling to 172 * avoid double rounding. 173 */ 174 p = ldexpl(copysignl(0x1p-16382L, r), -spread); 175 c = r + p; 176 s = c - r; 177 cc = (r - (c - s)) + (p - s) + rr; 178 fesetround(oround); 179 r = (c + cc) - p; 180 } 181 return (ldexpl(r, spread)); 182 } 183
