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_fma.c,v 1.4 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 * This algorithm is sensitive to the rounding precision. FPUs such 44 * as the i387 must be set in double-precision mode if variables are 45 * to be stored in FP registers in order to avoid incorrect results. 46 * This is the default on FreeBSD, but not on many other systems. 47 * 48 * Hardware instructions should be used on architectures that support it, 49 * since this implementation will likely be several times slower. 50 */ 51 #if LDBL_MANT_DIG != 113 52 double 53 fma(double x, double y, double z) 54 { 55 static const double split = 0x1p27 + 1.0; 56 double xs, ys, zs; 57 double c, cc, hx, hy, p, q, tx, ty; 58 double r, rr, s; 59 int oround; 60 int ex, ey, ez; 61 int spread; 62 63 if (z == 0.0) 64 return (x * y); 65 if (x == 0.0 || y == 0.0) 66 return (x * y + z); 67 68 /* Results of frexp() are undefined for these cases. */ 69 if (!isfinite(x) || !isfinite(y) || !isfinite(z)) 70 return (x * y + z); 71 72 xs = frexp(x, &ex); 73 ys = frexp(y, &ey); 74 zs = frexp(z, &ez); 75 oround = fegetround(); 76 spread = ex + ey - ez; 77 78 /* 79 * If x * y and z are many orders of magnitude apart, the scaling 80 * will overflow, so we handle these cases specially. Rounding 81 * modes other than FE_TONEAREST are painful. 82 */ 83 if (spread > DBL_MANT_DIG * 2) { 84 fenv_t env; 85 feraiseexcept(FE_INEXACT); 86 switch(oround) { 87 case FE_TONEAREST: 88 return (x * y); 89 case FE_TOWARDZERO: 90 if (x > 0.0 ^ y < 0.0 ^ z < 0.0) 91 return (x * y); 92 feholdexcept(&env); 93 r = x * y; 94 if (!fetestexcept(FE_INEXACT)) 95 r = nextafter(r, 0); 96 feupdateenv(&env); 97 return (r); 98 case FE_DOWNWARD: 99 if (z > 0.0) 100 return (x * y); 101 feholdexcept(&env); 102 r = x * y; 103 if (!fetestexcept(FE_INEXACT)) 104 r = nextafter(r, -INFINITY); 105 feupdateenv(&env); 106 return (r); 107 default: /* FE_UPWARD */ 108 if (z < 0.0) 109 return (x * y); 110 feholdexcept(&env); 111 r = x * y; 112 if (!fetestexcept(FE_INEXACT)) 113 r = nextafter(r, INFINITY); 114 feupdateenv(&env); 115 return (r); 116 } 117 } 118 if (spread < -DBL_MANT_DIG) { 119 feraiseexcept(FE_INEXACT); 120 if (!isnormal(z)) 121 feraiseexcept(FE_UNDERFLOW); 122 switch (oround) { 123 case FE_TONEAREST: 124 return (z); 125 case FE_TOWARDZERO: 126 if (x > 0.0 ^ y < 0.0 ^ z < 0.0) 127 return (z); 128 else 129 return (nextafter(z, 0)); 130 case FE_DOWNWARD: 131 if (x > 0.0 ^ y < 0.0) 132 return (z); 133 else 134 return (nextafter(z, -INFINITY)); 135 default: /* FE_UPWARD */ 136 if (x > 0.0 ^ y < 0.0) 137 return (nextafter(z, INFINITY)); 138 else 139 return (z); 140 } 141 } 142 143 /* 144 * Use Dekker's algorithm to perform the multiplication and 145 * subsequent addition in twice the machine precision. 146 * Arrange so that x * y = c + cc, and x * y + z = r + rr. 147 */ 148 fesetround(FE_TONEAREST); 149 150 p = xs * split; 151 hx = xs - p; 152 hx += p; 153 tx = xs - hx; 154 155 p = ys * split; 156 hy = ys - p; 157 hy += p; 158 ty = ys - hy; 159 160 p = hx * hy; 161 q = hx * ty + tx * hy; 162 c = p + q; 163 cc = p - c + q + tx * ty; 164 165 zs = ldexp(zs, -spread); 166 r = c + zs; 167 s = r - c; 168 rr = (c - (r - s)) + (zs - s) + cc; 169 170 spread = ex + ey; 171 if (spread + ilogb(r) > -1023) { 172 fesetround(oround); 173 r = r + rr; 174 } else { 175 /* 176 * The result is subnormal, so we round before scaling to 177 * avoid double rounding. 178 */ 179 p = ldexp(copysign(0x1p-1022, r), -spread); 180 c = r + p; 181 s = c - r; 182 cc = (r - (c - s)) + (p - s) + rr; 183 fesetround(oround); 184 r = (c + cc) - p; 185 } 186 return (ldexp(r, spread)); 187 } 188 #else /* LDBL_MANT_DIG == 113 */ 189 /* 190 * 113 bits of precision is more than twice the precision of a double, 191 * so it is enough to represent the intermediate product exactly. 192 */ 193 double 194 fma(double x, double y, double z) 195 { 196 return ((long double)x * y + z); 197 } 198 #endif /* LDBL_MANT_DIG != 113 */ 199 200 #if (LDBL_MANT_DIG == 53) 201 __weak_reference(fma, fmal); 202 #endif 203
