1 /*- 2 * Copyright (c) 2005 David Schultz <das@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("$FreeBSD$"); 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 /* 64 * Handle special cases. The order of operations and the particular 65 * return values here are crucial in handling special cases involving 66 * infinities, NaNs, overflows, and signed zeroes correctly. 67 */ 68 if (x == 0.0 || y == 0.0) 69 return (x * y + z); 70 if (z == 0.0) 71 return (x * y); 72 if (!isfinite(x) || !isfinite(y)) 73 return (x * y + z); 74 if (!isfinite(z)) 75 return (z); 76 77 xs = frexp(x, &ex); 78 ys = frexp(y, &ey); 79 zs = frexp(z, &ez); 80 oround = fegetround(); 81 spread = ex + ey - ez; 82 83 /* 84 * If x * y and z are many orders of magnitude apart, the scaling 85 * will overflow, so we handle these cases specially. Rounding 86 * modes other than FE_TONEAREST are painful. 87 */ 88 if (spread > DBL_MANT_DIG * 2) { 89 fenv_t env; 90 feraiseexcept(FE_INEXACT); 91 switch(oround) { 92 case FE_TONEAREST: 93 return (x * y); 94 case FE_TOWARDZERO: 95 if (x > 0.0 ^ y < 0.0 ^ z < 0.0) 96 return (x * y); 97 feholdexcept(&env); 98 r = x * y; 99 if (!fetestexcept(FE_INEXACT)) 100 r = nextafter(r, 0); 101 feupdateenv(&env); 102 return (r); 103 case FE_DOWNWARD: 104 if (z > 0.0) 105 return (x * y); 106 feholdexcept(&env); 107 r = x * y; 108 if (!fetestexcept(FE_INEXACT)) 109 r = nextafter(r, -INFINITY); 110 feupdateenv(&env); 111 return (r); 112 default: /* FE_UPWARD */ 113 if (z < 0.0) 114 return (x * y); 115 feholdexcept(&env); 116 r = x * y; 117 if (!fetestexcept(FE_INEXACT)) 118 r = nextafter(r, INFINITY); 119 feupdateenv(&env); 120 return (r); 121 } 122 } 123 if (spread < -DBL_MANT_DIG) { 124 feraiseexcept(FE_INEXACT); 125 if (!isnormal(z)) 126 feraiseexcept(FE_UNDERFLOW); 127 switch (oround) { 128 case FE_TONEAREST: 129 return (z); 130 case FE_TOWARDZERO: 131 if (x > 0.0 ^ y < 0.0 ^ z < 0.0) 132 return (z); 133 else 134 return (nextafter(z, 0)); 135 case FE_DOWNWARD: 136 if (x > 0.0 ^ y < 0.0) 137 return (z); 138 else 139 return (nextafter(z, -INFINITY)); 140 default: /* FE_UPWARD */ 141 if (x > 0.0 ^ y < 0.0) 142 return (nextafter(z, INFINITY)); 143 else 144 return (z); 145 } 146 } 147 148 /* 149 * Use Dekker's algorithm to perform the multiplication and 150 * subsequent addition in twice the machine precision. 151 * Arrange so that x * y = c + cc, and x * y + z = r + rr. 152 */ 153 fesetround(FE_TONEAREST); 154 155 p = xs * split; 156 hx = xs - p; 157 hx += p; 158 tx = xs - hx; 159 160 p = ys * split; 161 hy = ys - p; 162 hy += p; 163 ty = ys - hy; 164 165 p = hx * hy; 166 q = hx * ty + tx * hy; 167 c = p + q; 168 cc = p - c + q + tx * ty; 169 170 zs = ldexp(zs, -spread); 171 r = c + zs; 172 s = r - c; 173 rr = (c - (r - s)) + (zs - s) + cc; 174 175 spread = ex + ey; 176 if (spread + ilogb(r) > -1023) { 177 fesetround(oround); 178 r = r + rr; 179 } else { 180 /* 181 * The result is subnormal, so we round before scaling to 182 * avoid double rounding. 183 */ 184 p = ldexp(copysign(0x1p-1022, r), -spread); 185 c = r + p; 186 s = c - r; 187 cc = (r - (c - s)) + (p - s) + rr; 188 fesetround(oround); 189 r = (c + cc) - p; 190 } 191 return (ldexp(r, spread)); 192 } 193 #else /* LDBL_MANT_DIG == 113 */ 194 /* 195 * 113 bits of precision is more than twice the precision of a double, 196 * so it is enough to represent the intermediate product exactly. 197 */ 198 double 199 fma(double x, double y, double z) 200 { 201 return ((long double)x * y + z); 202 } 203 #endif /* LDBL_MANT_DIG != 113 */ 204 205 #if (LDBL_MANT_DIG == 53) 206 __weak_reference(fma, fmal); 207 #endif 208