| /*- |
| * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG> |
| * All rights reserved. |
| * |
| * Redistribution and use in source and binary forms, with or without |
| * modification, are permitted provided that the following conditions |
| * are met: |
| * 1. Redistributions of source code must retain the above copyright |
| * notice, this list of conditions and the following disclaimer. |
| * 2. Redistributions in binary form must reproduce the above copyright |
| * notice, this list of conditions and the following disclaimer in the |
| * documentation and/or other materials provided with the distribution. |
| * |
| * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND |
| * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE |
| * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE |
| * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE |
| * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL |
| * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS |
| * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) |
| * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT |
| * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY |
| * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF |
| * SUCH DAMAGE. |
| */ |
| |
| #include <sys/cdefs.h> |
| /* __FBSDID("$FreeBSD: src/lib/msun/src/s_fmal.c,v 1.2 2005/03/18 02:27:59 das Exp $"); */ |
| |
| #include <fenv.h> |
| #include <float.h> |
| #include <math.h> |
| |
| /* |
| * Fused multiply-add: Compute x * y + z with a single rounding error. |
| * |
| * We use scaling to avoid overflow/underflow, along with the |
| * canonical precision-doubling technique adapted from: |
| * |
| * Dekker, T. A Floating-Point Technique for Extending the |
| * Available Precision. Numer. Math. 18, 224-242 (1971). |
| */ |
| long double |
| fmal(long double x, long double y, long double z) |
| { |
| #if LDBL_MANT_DIG == 64 |
| static const long double split = 0x1p32L + 1.0; |
| #elif LDBL_MANT_DIG == 113 |
| static const long double split = 0x1p57L + 1.0; |
| #endif |
| long double xs, ys, zs; |
| long double c, cc, hx, hy, p, q, tx, ty; |
| long double r, rr, s; |
| int oround; |
| int ex, ey, ez; |
| int spread; |
| |
| if (z == 0.0) |
| return (x * y); |
| if (x == 0.0 || y == 0.0) |
| return (x * y + z); |
| |
| /* Results of frexp() are undefined for these cases. */ |
| if (!isfinite(x) || !isfinite(y) || !isfinite(z)) |
| return (x * y + z); |
| |
| xs = frexpl(x, &ex); |
| ys = frexpl(y, &ey); |
| zs = frexpl(z, &ez); |
| oround = fegetround(); |
| spread = ex + ey - ez; |
| |
| /* |
| * If x * y and z are many orders of magnitude apart, the scaling |
| * will overflow, so we handle these cases specially. Rounding |
| * modes other than FE_TONEAREST are painful. |
| */ |
| if (spread > LDBL_MANT_DIG * 2) { |
| fenv_t env; |
| feraiseexcept(FE_INEXACT); |
| switch(oround) { |
| case FE_TONEAREST: |
| return (x * y); |
| case FE_TOWARDZERO: |
| if (x > 0.0 ^ y < 0.0 ^ z < 0.0) |
| return (x * y); |
| feholdexcept(&env); |
| r = x * y; |
| if (!fetestexcept(FE_INEXACT)) |
| r = nextafterl(r, 0); |
| feupdateenv(&env); |
| return (r); |
| case FE_DOWNWARD: |
| if (z > 0.0) |
| return (x * y); |
| feholdexcept(&env); |
| r = x * y; |
| if (!fetestexcept(FE_INEXACT)) |
| r = nextafterl(r, -INFINITY); |
| feupdateenv(&env); |
| return (r); |
| default: /* FE_UPWARD */ |
| if (z < 0.0) |
| return (x * y); |
| feholdexcept(&env); |
| r = x * y; |
| if (!fetestexcept(FE_INEXACT)) |
| r = nextafterl(r, INFINITY); |
| feupdateenv(&env); |
| return (r); |
| } |
| } |
| if (spread < -LDBL_MANT_DIG) { |
| feraiseexcept(FE_INEXACT); |
| if (!isnormal(z)) |
| feraiseexcept(FE_UNDERFLOW); |
| switch (oround) { |
| case FE_TONEAREST: |
| return (z); |
| case FE_TOWARDZERO: |
| if (x > 0.0 ^ y < 0.0 ^ z < 0.0) |
| return (z); |
| else |
| return (nextafterl(z, 0)); |
| case FE_DOWNWARD: |
| if (x > 0.0 ^ y < 0.0) |
| return (z); |
| else |
| return (nextafterl(z, -INFINITY)); |
| default: /* FE_UPWARD */ |
| if (x > 0.0 ^ y < 0.0) |
| return (nextafterl(z, INFINITY)); |
| else |
| return (z); |
| } |
| } |
| |
| /* |
| * Use Dekker's algorithm to perform the multiplication and |
| * subsequent addition in twice the machine precision. |
| * Arrange so that x * y = c + cc, and x * y + z = r + rr. |
| */ |
| fesetround(FE_TONEAREST); |
| |
| p = xs * split; |
| hx = xs - p; |
| hx += p; |
| tx = xs - hx; |
| |
| p = ys * split; |
| hy = ys - p; |
| hy += p; |
| ty = ys - hy; |
| |
| p = hx * hy; |
| q = hx * ty + tx * hy; |
| c = p + q; |
| cc = p - c + q + tx * ty; |
| |
| zs = ldexpl(zs, -spread); |
| r = c + zs; |
| s = r - c; |
| rr = (c - (r - s)) + (zs - s) + cc; |
| |
| spread = ex + ey; |
| if (spread + ilogbl(r) > -16383) { |
| fesetround(oround); |
| r = r + rr; |
| } else { |
| /* |
| * The result is subnormal, so we round before scaling to |
| * avoid double rounding. |
| */ |
| p = ldexpl(copysignl(0x1p-16382L, r), -spread); |
| c = r + p; |
| s = c - r; |
| cc = (r - (c - s)) + (p - s) + rr; |
| fesetround(oround); |
| r = (c + cc) - p; |
| } |
| return (ldexpl(r, spread)); |
| } |