1 /* $OpenBSD: s_fma.c,v 1.6 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 * This algorithm is sensitive to the rounding precision. FPUs such 43 * as the i387 must be set in double-precision mode if variables are 44 * to be stored in FP registers in order to avoid incorrect results. 45 * This is the default on FreeBSD, but not on many other systems. 46 * 47 * Hardware instructions should be used on architectures that support it, 48 * since this implementation will likely be several times slower. 49 */ 50 #if LDBL_MANT_DIG != 113 51 double 52 fma(double x, double y, double z) 53 { 54 static const double split = 0x1p27 + 1.0; 55 double xs, ys, zs; 56 double c, cc, hx, hy, p, q, tx, ty; 57 double r, rr, s; 58 int oround; 59 int ex, ey, ez; 60 int spread; 61 62 /* 63 * Handle special cases. The order of operations and the particular 64 * return values here are crucial in handling special cases involving 65 * infinities, NaNs, overflows, and signed zeroes correctly. 66 */ 67 if (x == 0.0 || y == 0.0) 68 return (x * y + z); 69 if (z == 0.0) 70 return (x * y); 71 if (!isfinite(x) || !isfinite(y)) 72 return (x * y + z); 73 if (!isfinite(z)) 74 return (z); 75 76 xs = frexp(x, &ex); 77 ys = frexp(y, &ey); 78 zs = frexp(z, &ez); 79 oround = fegetround(); 80 spread = ex + ey - ez; 81 82 /* 83 * If x * y and z are many orders of magnitude apart, the scaling 84 * will overflow, so we handle these cases specially. Rounding 85 * modes other than FE_TONEAREST are painful. 86 */ 87 if (spread > DBL_MANT_DIG * 2) { 88 fenv_t env; 89 feraiseexcept(FE_INEXACT); 90 switch(oround) { 91 case FE_TONEAREST: 92 return (x * y); 93 case FE_TOWARDZERO: 94 if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0)) 95 return (x * y); 96 feholdexcept(&env); 97 r = x * y; 98 if (!fetestexcept(FE_INEXACT)) 99 r = nextafter(r, 0); 100 feupdateenv(&env); 101 return (r); 102 case FE_DOWNWARD: 103 if (z > 0.0) 104 return (x * y); 105 feholdexcept(&env); 106 r = x * y; 107 if (!fetestexcept(FE_INEXACT)) 108 r = nextafter(r, -INFINITY); 109 feupdateenv(&env); 110 return (r); 111 default: /* FE_UPWARD */ 112 if (z < 0.0) 113 return (x * y); 114 feholdexcept(&env); 115 r = x * y; 116 if (!fetestexcept(FE_INEXACT)) 117 r = nextafter(r, INFINITY); 118 feupdateenv(&env); 119 return (r); 120 } 121 } 122 if (spread < -DBL_MANT_DIG) { 123 feraiseexcept(FE_INEXACT); 124 if (!isnormal(z)) 125 feraiseexcept(FE_UNDERFLOW); 126 switch (oround) { 127 case FE_TONEAREST: 128 return (z); 129 case FE_TOWARDZERO: 130 if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0)) 131 return (z); 132 else 133 return (nextafter(z, 0)); 134 case FE_DOWNWARD: 135 if ((x > 0.0) ^ (y < 0.0)) 136 return (z); 137 else 138 return (nextafter(z, -INFINITY)); 139 default: /* FE_UPWARD */ 140 if ((x > 0.0) ^ (y < 0.0)) 141 return (nextafter(z, INFINITY)); 142 else 143 return (z); 144 } 145 } 146 147 /* 148 * Use Dekker's algorithm to perform the multiplication and 149 * subsequent addition in twice the machine precision. 150 * Arrange so that x * y = c + cc, and x * y + z = r + rr. 151 */ 152 fesetround(FE_TONEAREST); 153 154 p = xs * split; 155 hx = xs - p; 156 hx += p; 157 tx = xs - hx; 158 159 p = ys * split; 160 hy = ys - p; 161 hy += p; 162 ty = ys - hy; 163 164 p = hx * hy; 165 q = hx * ty + tx * hy; 166 c = p + q; 167 cc = p - c + q + tx * ty; 168 169 zs = ldexp(zs, -spread); 170 r = c + zs; 171 s = r - c; 172 rr = (c - (r - s)) + (zs - s) + cc; 173 174 spread = ex + ey; 175 if (spread + ilogb(r) > -1023) { 176 fesetround(oround); 177 r = r + rr; 178 } else { 179 /* 180 * The result is subnormal, so we round before scaling to 181 * avoid double rounding. 182 */ 183 p = ldexp(copysign(0x1p-1022, r), -spread); 184 c = r + p; 185 s = c - r; 186 cc = (r - (c - s)) + (p - s) + rr; 187 fesetround(oround); 188 r = (c + cc) - p; 189 } 190 return (ldexp(r, spread)); 191 } 192 #else /* LDBL_MANT_DIG == 113 */ 193 /* 194 * 113 bits of precision is more than twice the precision of a double, 195 * so it is enough to represent the intermediate product exactly. 196 */ 197 double 198 fma(double x, double y, double z) 199 { 200 return ((long double)x * y + z); 201 } 202 #endif /* LDBL_MANT_DIG != 113 */ 203 204 #if LDBL_MANT_DIG == DBL_MANT_DIG 205 __strong_alias(fmal, fma); 206 #endif /* LDBL_MANT_DIG == DBL_MANT_DIG */ 207