1 /* $OpenBSD: s_fmal.c,v 1.3 2013/11/12 19:00:38 martynas Exp $ */ 2 3 /*- 4 * Copyright (c) 2005 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 <fenv.h> 30 #include <float.h> 31 #include <math.h> 32 33 /* 34 * Fused multiply-add: Compute x * y + z with a single rounding error. 35 * 36 * We use scaling to avoid overflow/underflow, along with the 37 * canonical precision-doubling technique adapted from: 38 * 39 * Dekker, T. A Floating-Point Technique for Extending the 40 * Available Precision. Numer. Math. 18, 224-242 (1971). 41 */ 42 long double 43 fmal(long double x, long double y, long double z) 44 { 45 #if LDBL_MANT_DIG == 64 46 static const long double split = 0x1p32L + 1.0; 47 #elif LDBL_MANT_DIG == 113 48 static const long double split = 0x1p57L + 1.0; 49 #endif 50 long double xs, ys, zs; 51 long double c, cc, hx, hy, p, q, tx, ty; 52 long double r, rr, s; 53 int oround; 54 int ex, ey, ez; 55 int spread; 56 57 /* 58 * Handle special cases. The order of operations and the particular 59 * return values here are crucial in handling special cases involving 60 * infinities, NaNs, overflows, and signed zeroes correctly. 61 */ 62 if (x == 0.0 || y == 0.0) 63 return (x * y + z); 64 if (z == 0.0) 65 return (x * y); 66 if (!isfinite(x) || !isfinite(y)) 67 return (x * y + z); 68 if (!isfinite(z)) 69 return (z); 70 71 xs = frexpl(x, &ex); 72 ys = frexpl(y, &ey); 73 zs = frexpl(z, &ez); 74 oround = fegetround(); 75 spread = ex + ey - ez; 76 77 /* 78 * If x * y and z are many orders of magnitude apart, the scaling 79 * will overflow, so we handle these cases specially. Rounding 80 * modes other than FE_TONEAREST are painful. 81 */ 82 if (spread > LDBL_MANT_DIG * 2) { 83 fenv_t env; 84 feraiseexcept(FE_INEXACT); 85 switch(oround) { 86 case FE_TONEAREST: 87 return (x * y); 88 case FE_TOWARDZERO: 89 if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0)) 90 return (x * y); 91 feholdexcept(&env); 92 r = x * y; 93 if (!fetestexcept(FE_INEXACT)) 94 r = nextafterl(r, 0); 95 feupdateenv(&env); 96 return (r); 97 case FE_DOWNWARD: 98 if (z > 0.0) 99 return (x * y); 100 feholdexcept(&env); 101 r = x * y; 102 if (!fetestexcept(FE_INEXACT)) 103 r = nextafterl(r, -INFINITY); 104 feupdateenv(&env); 105 return (r); 106 default: /* FE_UPWARD */ 107 if (z < 0.0) 108 return (x * y); 109 feholdexcept(&env); 110 r = x * y; 111 if (!fetestexcept(FE_INEXACT)) 112 r = nextafterl(r, INFINITY); 113 feupdateenv(&env); 114 return (r); 115 } 116 } 117 if (spread < -LDBL_MANT_DIG) { 118 feraiseexcept(FE_INEXACT); 119 if (!isnormal(z)) 120 feraiseexcept(FE_UNDERFLOW); 121 switch (oround) { 122 case FE_TONEAREST: 123 return (z); 124 case FE_TOWARDZERO: 125 if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0)) 126 return (z); 127 else 128 return (nextafterl(z, 0)); 129 case FE_DOWNWARD: 130 if ((x > 0.0) ^ (y < 0.0)) 131 return (z); 132 else 133 return (nextafterl(z, -INFINITY)); 134 default: /* FE_UPWARD */ 135 if ((x > 0.0) ^ (y < 0.0)) 136 return (nextafterl(z, INFINITY)); 137 else 138 return (z); 139 } 140 } 141 142 /* 143 * Use Dekker's algorithm to perform the multiplication and 144 * subsequent addition in twice the machine precision. 145 * Arrange so that x * y = c + cc, and x * y + z = r + rr. 146 */ 147 fesetround(FE_TONEAREST); 148 149 p = xs * split; 150 hx = xs - p; 151 hx += p; 152 tx = xs - hx; 153 154 p = ys * split; 155 hy = ys - p; 156 hy += p; 157 ty = ys - hy; 158 159 p = hx * hy; 160 q = hx * ty + tx * hy; 161 c = p + q; 162 cc = p - c + q + tx * ty; 163 164 zs = ldexpl(zs, -spread); 165 r = c + zs; 166 s = r - c; 167 rr = (c - (r - s)) + (zs - s) + cc; 168 169 spread = ex + ey; 170 if (spread + ilogbl(r) > -16383) { 171 fesetround(oround); 172 r = r + rr; 173 } else { 174 /* 175 * The result is subnormal, so we round before scaling to 176 * avoid double rounding. 177 */ 178 p = ldexpl(copysignl(0x1p-16382L, r), -spread); 179 c = r + p; 180 s = c - r; 181 cc = (r - (c - s)) + (p - s) + rr; 182 fesetround(oround); 183 r = (c + cc) - p; 184 } 185 return (ldexpl(r, spread)); 186 } 187