116 lines
3.5 KiB
C
116 lines
3.5 KiB
C
/* Compute full X * Y for double type.
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Copyright (C) 2013-2022 Free Software Foundation, Inc.
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This file is part of the GNU C Library.
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The GNU C Library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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The GNU C Library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with the GNU C Library; if not, see
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<https://www.gnu.org/licenses/>. */
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#ifndef _MUL_SPLIT_H
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#define _MUL_SPLIT_H
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#include <float.h>
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/* Calculate X * Y exactly and store the result in *HI + *LO. It is
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given that the values are small enough that no overflow occurs and
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large enough (or zero) that no underflow occurs. */
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static void
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mul_split (double *hi, double *lo, double x, double y)
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{
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#ifdef __FP_FAST_FMA
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/* Fast built-in fused multiply-add. */
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*hi = x * y;
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*lo = __builtin_fma (x, y, -*hi);
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#else
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/* Apply Dekker's algorithm. */
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*hi = x * y;
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# define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
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double x1 = x * C;
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double y1 = y * C;
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# undef C
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x1 = (x - x1) + x1;
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y1 = (y - y1) + y1;
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double x2 = x - x1;
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double y2 = y - y1;
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*lo = (((x1 * y1 - *hi) + x1 * y2) + x2 * y1) + x2 * y2;
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#endif
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}
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/* Add a + b exactly, such that *hi + *lo = a + b.
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Assumes |a| >= |b| and rounding to nearest. */
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static inline void
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fast_two_sum (double *hi, double *lo, double a, double b)
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{
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double e;
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*hi = a + b;
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e = *hi - a; /* exact */
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*lo = b - e; /* exact */
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/* Now *hi + *lo = a + b exactly. */
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}
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/* Multiplication of two floating-point expansions: *hi + *lo is an
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approximation of (h1+l1)*(h2+l2), assuming |l1| <= 1/2*ulp(h1)
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and |l2| <= 1/2*ulp(h2) and rounding to nearest. */
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static inline void
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mul_expansion (double *hi, double *lo, double h1, double l1,
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double h2, double l2)
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{
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double r, e;
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mul_split (hi, lo, h1, h2);
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r = h1 * l2 + h2 * l1;
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/* Now add r to (hi,lo) using fast two-sum, where we know |r| < |hi|. */
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fast_two_sum (hi, &e, *hi, r);
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*lo -= e;
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}
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/* Calculate X / Y and store the approximate result in *HI + *LO. It is
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assumed that Y is not zero, that no overflow nor underflow occurs, and
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rounding is to nearest. */
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static inline void
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div_split (double *hi, double *lo, double x, double y)
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{
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double a, b;
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*hi = x / y;
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mul_split (&a, &b, *hi, y);
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/* a + b = hi*y, which should be near x. */
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a = x - a; /* huge cancellation */
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a = a - b;
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/* Now x ~ hi*y + a thus x/y ~ hi + a/y. */
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*lo = a / y;
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}
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/* Division of two floating-point expansions: *hi + *lo is an
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approximation of (h1+l1)/(h2+l2), assuming |l1| <= 1/2*ulp(h1)
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and |l2| <= 1/2*ulp(h2), h2+l2 is not zero, and rounding to nearest. */
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static inline void
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div_expansion (double *hi, double *lo, double h1, double l1,
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double h2, double l2)
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{
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double r, e;
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div_split (hi, lo, h1, h2);
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/* (h1+l1)/(h2+l2) ~ h1/h2 + (l1*h2 - l2*h1)/h2^2 */
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r = (l1 * h2 - l2 * h1) / (h2 * h2);
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/* Now add r to (hi,lo) using fast two-sum, where we know |r| < |hi|. */
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fast_two_sum (hi, &e, *hi, r);
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*lo += e;
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/* Renormalize since |lo| might be larger than 0.5 ulp(hi). */
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fast_two_sum (hi, lo, *hi, *lo);
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}
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#endif /* _MUL_SPLIT_H */
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