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 * XXX May incur a small error for subnormal results due to double 44 * rounding induced by the final scaling operation. 45 */ 46 long double 47 fmal(long double x, long double y, long double z) 48 { 49 #if LDBL_MANT_DIG == 64 50 static const long double split = 0x1p32L + 1.0; 51 #elif LDBL_MANT_DIG == 113 52 static const long double split = 0x1p57L + 1.0; 53 #endif 54 long double xs, ys, zs; 55 long double c, cc, hx, hy, p, q, tx, ty; 56 long double r, rr, s; 57 int oround; 58 int ex, ey, ez; 59 int spread; 60 61 if (z == 0.0) 62 return (x * y); 63 if (x == 0.0 || y == 0.0) 64 return (x * y + z); 65 66 /* Results of frexp() are undefined for these cases. */ 67 if (!isfinite(x) || !isfinite(y) || !isfinite(z)) 68 return (x * y + z); 69 70 xs = frexpl(x, &ex); 71 ys = frexpl(y, &ey); 72 zs = frexpl(z, &ez); 73 oround = fegetround(); 74 spread = ex + ey - ez; 75 76 /* 77 * If x * y and z are many orders of magnitude apart, the scaling 78 * will overflow, so we handle these cases specially. Rounding 79 * modes other than FE_TONEAREST are painful. 80 */ 81 if (spread > LDBL_MANT_DIG * 2) { 82 fenv_t env; 83 feraiseexcept(FE_INEXACT); 84 switch(oround) { 85 case FE_TONEAREST: 86 return (x * y); 87 case FE_TOWARDZERO: 88 if (x > 0.0 ^ y < 0.0 ^ z < 0.0) 89 return (x * y); 90 feholdexcept(&env); 91 r = x * y; 92 if (!fetestexcept(FE_INEXACT)) 93 r = nextafterl(r, 0); 94 feupdateenv(&env); 95 return (r); 96 case FE_DOWNWARD: 97 if (z > 0.0) 98 return (x * y); 99 feholdexcept(&env); 100 r = x * y; 101 if (!fetestexcept(FE_INEXACT)) 102 r = nextafterl(r, -INFINITY); 103 feupdateenv(&env); 104 return (r); 105 default: /* FE_UPWARD */ 106 if (z < 0.0) 107 return (x * y); 108 feholdexcept(&env); 109 r = x * y; 110 if (!fetestexcept(FE_INEXACT)) 111 r = nextafterl(r, INFINITY); 112 feupdateenv(&env); 113 return (r); 114 } 115 } 116 if (spread < -LDBL_MANT_DIG) { 117 feraiseexcept(FE_INEXACT); 118 if (!isnormal(z)) 119 feraiseexcept(FE_UNDERFLOW); 120 switch (oround) { 121 case FE_TONEAREST: 122 return (z); 123 case FE_TOWARDZERO: 124 if (x > 0.0 ^ y < 0.0 ^ z < 0.0) 125 return (z); 126 else 127 return (nextafterl(z, 0)); 128 case FE_DOWNWARD: 129 if (x > 0.0 ^ y < 0.0) 130 return (z); 131 else 132 return (nextafterl(z, -INFINITY)); 133 default: /* FE_UPWARD */ 134 if (x > 0.0 ^ y < 0.0) 135 return (nextafterl(z, INFINITY)); 136 else 137 return (z); 138 } 139 } 140 141 /* 142 * Use Dekker's algorithm to perform the multiplication and 143 * subsequent addition in twice the machine precision. 144 * Arrange so that x * y = c + cc, and x * y + z = r + rr. 145 */ 146 fesetround(FE_TONEAREST); 147 148 p = xs * split; 149 hx = xs - p; 150 hx += p; 151 tx = xs - hx; 152 153 p = ys * split; 154 hy = ys - p; 155 hy += p; 156 ty = ys - hy; 157 158 p = hx * hy; 159 q = hx * ty + tx * hy; 160 c = p + q; 161 cc = p - c + q + tx * ty; 162 163 zs = ldexpl(zs, -spread); 164 r = c + zs; 165 s = r - c; 166 rr = (c - (r - s)) + (zs - s) + cc; 167 168 fesetround(oround); 169 return (ldexpl(r + rr, ex + ey)); 170 } 171