/* origin: FreeBSD /usr/src/lib/msun/src/s_fmal.c */ /*- * Copyright (c) 2005-2011 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 "libm.h" #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 long double fmal(long double x, long double y, long double z) { return fma(x, y, z); } #elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384 #include <fenv.h> #if LDBL_MANT_DIG == 64 #define LASTBIT(u) (u.i.m & 1) #define SPLIT (0x1p32L + 1) #elif LDBL_MANT_DIG == 113 #define LASTBIT(u) (u.i.lo & 1) #define SPLIT (0x1p57L + 1) #endif /* * A struct dd represents a floating-point number with twice the precision * of a long double. We maintain the invariant that "hi" stores the high-order * bits of the result. */ struct dd { long double hi; long double lo; }; /* * Compute a+b exactly, returning the exact result in a struct dd. We assume * that both a and b are finite, but make no assumptions about their relative * magnitudes. */ static inline struct dd dd_add(long double a, long double b) { struct dd ret; long double s; ret.hi = a + b; s = ret.hi - a; ret.lo = (a - (ret.hi - s)) + (b - s); return (ret); } /* * Compute a+b, with a small tweak: The least significant bit of the * result is adjusted into a sticky bit summarizing all the bits that * were lost to rounding. This adjustment negates the effects of double * rounding when the result is added to another number with a higher * exponent. For an explanation of round and sticky bits, see any reference * on FPU design, e.g., * * J. Coonen. An Implementation Guide to a Proposed Standard for * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980. */ static inline long double add_adjusted(long double a, long double b) { struct dd sum; union ldshape u; sum = dd_add(a, b); if (sum.lo != 0) { u.f = sum.hi; if (!LASTBIT(u)) sum.hi = nextafterl(sum.hi, INFINITY * sum.lo); } return (sum.hi); } /* * Compute ldexp(a+b, scale) with a single rounding error. It is assumed * that the result will be subnormal, and care is taken to ensure that * double rounding does not occur. */ static inline long double add_and_denormalize(long double a, long double b, int scale) { struct dd sum; int bits_lost; union ldshape u; sum = dd_add(a, b); /* * If we are losing at least two bits of accuracy to denormalization, * then the first lost bit becomes a round bit, and we adjust the * lowest bit of sum.hi to make it a sticky bit summarizing all the * bits in sum.lo. With the sticky bit adjusted, the hardware will * break any ties in the correct direction. * * If we are losing only one bit to denormalization, however, we must * break the ties manually. */ if (sum.lo != 0) { u.f = sum.hi; bits_lost = -u.i.se - scale + 1; if ((bits_lost != 1) ^ LASTBIT(u)) sum.hi = nextafterl(sum.hi, INFINITY * sum.lo); } return scalbnl(sum.hi, scale); } /* * Compute a*b exactly, returning the exact result in a struct dd. We assume * that both a and b are normalized, so no underflow or overflow will occur. * The current rounding mode must be round-to-nearest. */ static inline struct dd dd_mul(long double a, long double b) { struct dd ret; long double ha, hb, la, lb, p, q; p = a * SPLIT; ha = a - p; ha += p; la = a - ha; p = b * SPLIT; hb = b - p; hb += p; lb = b - hb; p = ha * hb; q = ha * lb + la * hb; ret.hi = p + q; ret.lo = p - ret.hi + q + la * lb; return (ret); } /* * 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) { #pragma STDC FENV_ACCESS ON long double xs, ys, zs, adj; struct dd xy, r; int oround; int ex, ey, ez; int spread; /* * Handle special cases. The order of operations and the particular * return values here are crucial in handling special cases involving * infinities, NaNs, overflows, and signed zeroes correctly. */ if (!isfinite(x) || !isfinite(y)) return (x * y + z); if (!isfinite(z)) return (z); if (x == 0.0 || y == 0.0) return (x * y + z); if (z == 0.0) return (x * y); 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) { #ifdef FE_INEXACT feraiseexcept(FE_INEXACT); #endif #ifdef FE_UNDERFLOW if (!isnormal(z)) feraiseexcept(FE_UNDERFLOW); #endif switch (oround) { default: /* FE_TONEAREST */ return (z); #ifdef FE_TOWARDZERO case FE_TOWARDZERO: if (x > 0.0 ^ y < 0.0 ^ z < 0.0) return (z); else return (nextafterl(z, 0)); #endif #ifdef FE_DOWNWARD case FE_DOWNWARD: if (x > 0.0 ^ y < 0.0) return (z); else return (nextafterl(z, -INFINITY)); #endif #ifdef FE_UPWARD case FE_UPWARD: if (x > 0.0 ^ y < 0.0) return (nextafterl(z, INFINITY)); else return (z); #endif } } if (spread <= LDBL_MANT_DIG * 2) zs = scalbnl(zs, -spread); else zs = copysignl(LDBL_MIN, zs); fesetround(FE_TONEAREST); /* * Basic approach for round-to-nearest: * * (xy.hi, xy.lo) = x * y (exact) * (r.hi, r.lo) = xy.hi + z (exact) * adj = xy.lo + r.lo (inexact; low bit is sticky) * result = r.hi + adj (correctly rounded) */ xy = dd_mul(xs, ys); r = dd_add(xy.hi, zs); spread = ex + ey; if (r.hi == 0.0) { /* * When the addends cancel to 0, ensure that the result has * the correct sign. */ fesetround(oround); volatile long double vzs = zs; /* XXX gcc CSE bug workaround */ return xy.hi + vzs + scalbnl(xy.lo, spread); } if (oround != FE_TONEAREST) { /* * There is no need to worry about double rounding in directed * rounding modes. * But underflow may not be raised correctly, example in downward rounding: * fmal(0x1.0000000001p-16000L, 0x1.0000000001p-400L, -0x1p-16440L) */ long double ret; #if defined(FE_INEXACT) && defined(FE_UNDERFLOW) int e = fetestexcept(FE_INEXACT); feclearexcept(FE_INEXACT); #endif fesetround(oround); adj = r.lo + xy.lo; ret = scalbnl(r.hi + adj, spread); #if defined(FE_INEXACT) && defined(FE_UNDERFLOW) if (ilogbl(ret) < -16382 && fetestexcept(FE_INEXACT)) feraiseexcept(FE_UNDERFLOW); else if (e) feraiseexcept(FE_INEXACT); #endif return ret; } adj = add_adjusted(r.lo, xy.lo); if (spread + ilogbl(r.hi) > -16383) return scalbnl(r.hi + adj, spread); else return add_and_denormalize(r.hi, adj, spread); } #endif
