| 1 | /* origin: FreeBSD /usr/src/lib/msun/src/s_fmal.c */
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| 2 | /*-
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| 3 | * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
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| 4 | * All rights reserved.
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| 5 | *
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| 6 | * Redistribution and use in source and binary forms, with or without
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| 7 | * modification, are permitted provided that the following conditions
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| 8 | * are met:
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| 9 | * 1. Redistributions of source code must retain the above copyright
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| 10 | * notice, this list of conditions and the following disclaimer.
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| 11 | * 2. Redistributions in binary form must reproduce the above copyright
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| 12 | * notice, this list of conditions and the following disclaimer in the
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| 13 | * documentation and/or other materials provided with the distribution.
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| 14 | *
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| 15 | * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
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| 16 | * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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| 17 | * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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| 18 | * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
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| 19 | * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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| 20 | * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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| 21 | * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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| 22 | * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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| 23 | * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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| 24 | * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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| 25 | * SUCH DAMAGE.
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| 26 | */
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| 27 |
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| 28 |
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| 29 | #include "libm.h"
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| 30 | #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
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| 31 | long double fmal(long double x, long double y, long double z)
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| 32 | {
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| 33 | return fma(x, y, z);
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| 34 | }
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| 35 | #elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
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| 36 | #include <fenv.h>
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| 37 | #if LDBL_MANT_DIG == 64
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| 38 | #define LASTBIT(u) (u.i.m & 1)
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| 39 | #define SPLIT (0x1p32L + 1)
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| 40 | #elif LDBL_MANT_DIG == 113
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| 41 | #define LASTBIT(u) (u.i.lo & 1)
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| 42 | #define SPLIT (0x1p57L + 1)
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| 43 | #endif
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| 44 |
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| 45 | /*
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| 46 | * A struct dd represents a floating-point number with twice the precision
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| 47 | * of a long double. We maintain the invariant that "hi" stores the high-order
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| 48 | * bits of the result.
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| 49 | */
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| 50 | struct dd {
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| 51 | long double hi;
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| 52 | long double lo;
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| 53 | };
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| 54 |
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| 55 | /*
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| 56 | * Compute a+b exactly, returning the exact result in a struct dd. We assume
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| 57 | * that both a and b are finite, but make no assumptions about their relative
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| 58 | * magnitudes.
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| 59 | */
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| 60 | static inline struct dd dd_add(long double a, long double b)
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| 61 | {
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| 62 | struct dd ret;
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| 63 | long double s;
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| 64 |
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| 65 | ret.hi = a + b;
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| 66 | s = ret.hi - a;
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| 67 | ret.lo = (a - (ret.hi - s)) + (b - s);
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| 68 | return (ret);
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| 69 | }
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| 70 |
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| 71 | /*
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| 72 | * Compute a+b, with a small tweak: The least significant bit of the
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| 73 | * result is adjusted into a sticky bit summarizing all the bits that
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| 74 | * were lost to rounding. This adjustment negates the effects of double
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| 75 | * rounding when the result is added to another number with a higher
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| 76 | * exponent. For an explanation of round and sticky bits, see any reference
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| 77 | * on FPU design, e.g.,
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| 78 | *
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| 79 | * J. Coonen. An Implementation Guide to a Proposed Standard for
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| 80 | * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
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| 81 | */
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| 82 | static inline long double add_adjusted(long double a, long double b)
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| 83 | {
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| 84 | struct dd sum;
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| 85 | union ldshape u;
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| 86 |
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| 87 | sum = dd_add(a, b);
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| 88 | if (sum.lo != 0) {
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| 89 | u.f = sum.hi;
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| 90 | if (!LASTBIT(u))
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| 91 | sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
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| 92 | }
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| 93 | return (sum.hi);
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| 94 | }
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| 95 |
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| 96 | /*
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| 97 | * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
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| 98 | * that the result will be subnormal, and care is taken to ensure that
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| 99 | * double rounding does not occur.
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| 100 | */
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| 101 | static inline long double add_and_denormalize(long double a, long double b, int scale)
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| 102 | {
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| 103 | struct dd sum;
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| 104 | int bits_lost;
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| 105 | union ldshape u;
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| 106 |
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| 107 | sum = dd_add(a, b);
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| 108 |
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| 109 | /*
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| 110 | * If we are losing at least two bits of accuracy to denormalization,
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| 111 | * then the first lost bit becomes a round bit, and we adjust the
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| 112 | * lowest bit of sum.hi to make it a sticky bit summarizing all the
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| 113 | * bits in sum.lo. With the sticky bit adjusted, the hardware will
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| 114 | * break any ties in the correct direction.
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| 115 | *
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| 116 | * If we are losing only one bit to denormalization, however, we must
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| 117 | * break the ties manually.
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| 118 | */
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| 119 | if (sum.lo != 0) {
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| 120 | u.f = sum.hi;
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| 121 | bits_lost = -u.i.se - scale + 1;
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| 122 | if ((bits_lost != 1) ^ LASTBIT(u))
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| 123 | sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
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| 124 | }
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| 125 | return scalbnl(sum.hi, scale);
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| 126 | }
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| 127 |
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| 128 | /*
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| 129 | * Compute a*b exactly, returning the exact result in a struct dd. We assume
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| 130 | * that both a and b are normalized, so no underflow or overflow will occur.
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| 131 | * The current rounding mode must be round-to-nearest.
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| 132 | */
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| 133 | static inline struct dd dd_mul(long double a, long double b)
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| 134 | {
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| 135 | struct dd ret;
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| 136 | long double ha, hb, la, lb, p, q;
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| 137 |
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| 138 | p = a * SPLIT;
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| 139 | ha = a - p;
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| 140 | ha += p;
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| 141 | la = a - ha;
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| 142 |
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| 143 | p = b * SPLIT;
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| 144 | hb = b - p;
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| 145 | hb += p;
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| 146 | lb = b - hb;
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| 147 |
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| 148 | p = ha * hb;
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| 149 | q = ha * lb + la * hb;
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| 150 |
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| 151 | ret.hi = p + q;
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| 152 | ret.lo = p - ret.hi + q + la * lb;
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| 153 | return (ret);
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| 154 | }
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| 155 |
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| 156 | /*
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| 157 | * Fused multiply-add: Compute x * y + z with a single rounding error.
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| 158 | *
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| 159 | * We use scaling to avoid overflow/underflow, along with the
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| 160 | * canonical precision-doubling technique adapted from:
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| 161 | *
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| 162 | * Dekker, T. A Floating-Point Technique for Extending the
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| 163 | * Available Precision. Numer. Math. 18, 224-242 (1971).
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| 164 | */
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| 165 | long double fmal(long double x, long double y, long double z)
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| 166 | {
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| 167 | #pragma STDC FENV_ACCESS ON
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| 168 | long double xs, ys, zs, adj;
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| 169 | struct dd xy, r;
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| 170 | int oround;
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| 171 | int ex, ey, ez;
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| 172 | int spread;
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| 173 |
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| 174 | /*
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| 175 | * Handle special cases. The order of operations and the particular
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| 176 | * return values here are crucial in handling special cases involving
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| 177 | * infinities, NaNs, overflows, and signed zeroes correctly.
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| 178 | */
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| 179 | if (!isfinite(x) || !isfinite(y))
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| 180 | return (x * y + z);
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| 181 | if (!isfinite(z))
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| 182 | return (z);
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| 183 | if (x == 0.0 || y == 0.0)
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| 184 | return (x * y + z);
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| 185 | if (z == 0.0)
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| 186 | return (x * y);
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| 187 |
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| 188 | xs = frexpl(x, &ex);
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| 189 | ys = frexpl(y, &ey);
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| 190 | zs = frexpl(z, &ez);
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| 191 | oround = fegetround();
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| 192 | spread = ex + ey - ez;
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| 193 |
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| 194 | /*
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| 195 | * If x * y and z are many orders of magnitude apart, the scaling
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| 196 | * will overflow, so we handle these cases specially. Rounding
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| 197 | * modes other than FE_TONEAREST are painful.
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| 198 | */
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| 199 | if (spread < -LDBL_MANT_DIG) {
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| 200 | #ifdef FE_INEXACT
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| 201 | feraiseexcept(FE_INEXACT);
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| 202 | #endif
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| 203 | #ifdef FE_UNDERFLOW
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| 204 | if (!isnormal(z))
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| 205 | feraiseexcept(FE_UNDERFLOW);
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| 206 | #endif
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| 207 | switch (oround) {
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| 208 | default: /* FE_TONEAREST */
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| 209 | return (z);
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| 210 | #ifdef FE_TOWARDZERO
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| 211 | case FE_TOWARDZERO:
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| 212 | if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
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| 213 | return (z);
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| 214 | else
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| 215 | return (nextafterl(z, 0));
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| 216 | #endif
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| 217 | #ifdef FE_DOWNWARD
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| 218 | case FE_DOWNWARD:
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| 219 | if (x > 0.0 ^ y < 0.0)
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| 220 | return (z);
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| 221 | else
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| 222 | return (nextafterl(z, -INFINITY));
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| 223 | #endif
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| 224 | #ifdef FE_UPWARD
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| 225 | case FE_UPWARD:
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| 226 | if (x > 0.0 ^ y < 0.0)
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| 227 | return (nextafterl(z, INFINITY));
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| 228 | else
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| 229 | return (z);
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| 230 | #endif
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| 231 | }
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| 232 | }
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| 233 | if (spread <= LDBL_MANT_DIG * 2)
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| 234 | zs = scalbnl(zs, -spread);
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| 235 | else
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| 236 | zs = copysignl(LDBL_MIN, zs);
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| 237 |
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| 238 | fesetround(FE_TONEAREST);
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| 239 |
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| 240 | /*
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| 241 | * Basic approach for round-to-nearest:
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| 242 | *
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| 243 | * (xy.hi, xy.lo) = x * y (exact)
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| 244 | * (r.hi, r.lo) = xy.hi + z (exact)
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| 245 | * adj = xy.lo + r.lo (inexact; low bit is sticky)
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| 246 | * result = r.hi + adj (correctly rounded)
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| 247 | */
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| 248 | xy = dd_mul(xs, ys);
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| 249 | r = dd_add(xy.hi, zs);
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| 250 |
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| 251 | spread = ex + ey;
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| 252 |
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| 253 | if (r.hi == 0.0) {
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| 254 | /*
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| 255 | * When the addends cancel to 0, ensure that the result has
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| 256 | * the correct sign.
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| 257 | */
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| 258 | fesetround(oround);
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| 259 | volatile long double vzs = zs; /* XXX gcc CSE bug workaround */
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| 260 | return xy.hi + vzs + scalbnl(xy.lo, spread);
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| 261 | }
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| 262 |
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| 263 | if (oround != FE_TONEAREST) {
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| 264 | /*
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| 265 | * There is no need to worry about double rounding in directed
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| 266 | * rounding modes.
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| 267 | * But underflow may not be raised correctly, example in downward rounding:
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| 268 | * fmal(0x1.0000000001p-16000L, 0x1.0000000001p-400L, -0x1p-16440L)
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| 269 | */
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| 270 | long double ret;
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| 271 | #if defined(FE_INEXACT) && defined(FE_UNDERFLOW)
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| 272 | int e = fetestexcept(FE_INEXACT);
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| 273 | feclearexcept(FE_INEXACT);
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| 274 | #endif
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| 275 | fesetround(oround);
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| 276 | adj = r.lo + xy.lo;
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| 277 | ret = scalbnl(r.hi + adj, spread);
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| 278 | #if defined(FE_INEXACT) && defined(FE_UNDERFLOW)
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| 279 | if (ilogbl(ret) < -16382 && fetestexcept(FE_INEXACT))
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| 280 | feraiseexcept(FE_UNDERFLOW);
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| 281 | else if (e)
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| 282 | feraiseexcept(FE_INEXACT);
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| 283 | #endif
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| 284 | return ret;
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| 285 | }
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| 286 |
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| 287 | adj = add_adjusted(r.lo, xy.lo);
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| 288 | if (spread + ilogbl(r.hi) > -16383)
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| 289 | return scalbnl(r.hi + adj, spread);
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| 290 | else
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| 291 | return add_and_denormalize(r.hi, adj, spread);
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| 292 | }
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| 293 | #endif
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