/*- * SPDX-License-Identifier: BSD-2-Clause * * 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 <fenv.h> #include <float.h> #include <math.h> #include "math_private.h" #ifdef USE_BUILTIN_FMA double fma(double x, double y, double z) { return (__builtin_fma(x, y, z)); } #else /* * A struct dd represents a floating-point number with twice the precision * of a double. We maintain the invariant that "hi" stores the 53 high-order * bits of the result. */ struct dd { double hi; 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(double a, double b) { struct dd ret; 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 double add_adjusted(double a, double b) { struct dd sum; uint64_t hibits, lobits; sum = dd_add(a, b); if (sum.lo != 0) { EXTRACT_WORD64(hibits, sum.hi); if ((hibits & 1) == 0) { /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */ EXTRACT_WORD64(lobits, sum.lo); hibits += 1 - ((hibits ^ lobits) >> 62); INSERT_WORD64(sum.hi, hibits); } } 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 double add_and_denormalize(double a, double b, int scale) { struct dd sum; uint64_t hibits, lobits; int bits_lost; 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) { EXTRACT_WORD64(hibits, sum.hi); bits_lost = -((int)(hibits >> 52) & 0x7ff) - scale + 1; if ((bits_lost != 1) ^ (int)(hibits & 1)) { /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */ EXTRACT_WORD64(lobits, sum.lo); hibits += 1 - (((hibits ^ lobits) >> 62) & 2); INSERT_WORD64(sum.hi, hibits); } } return (ldexp(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(double a, double b) { static const double split = 0x1p27 + 1.0; struct dd ret; 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). * * This algorithm is sensitive to the rounding precision. FPUs such * as the i387 must be set in double-precision mode if variables are * to be stored in FP registers in order to avoid incorrect results. * This is the default on FreeBSD, but not on many other systems. * * Hardware instructions should be used on architectures that support it, * since this implementation will likely be several times slower. */ double fma(double x, double y, double z) { 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 (x == 0.0 || y == 0.0) return (x * y + z); if (z == 0.0) return (x * y); if (!isfinite(x) || !isfinite(y)) return (x * y + z); if (!isfinite(z)) return (z); xs = frexp(x, &ex); ys = frexp(y, &ey); zs = frexp(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 < -DBL_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 (nextafter(z, 0)); case FE_DOWNWARD: if ((x > 0.0) ^ (y < 0.0)) return (z); else return (nextafter(z, -INFINITY)); default: /* FE_UPWARD */ if ((x > 0.0) ^ (y < 0.0)) return (nextafter(z, INFINITY)); else return (z); } } if (spread <= DBL_MANT_DIG * 2) zs = ldexp(zs, -spread); else zs = copysign(DBL_MIN, zs); fesetround(FE_TONEAREST); /* work around clang issue #8472 */ volatile double vxs = xs; /* * 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(vxs, ys); r = dd_add(xy.hi, zs); spread = ex + ey; if (r.hi == 0.0 && xy.lo == 0) { /* * When the addends cancel to 0, ensure that the result has * the correct sign. */ fesetround(oround); volatile double vzs = zs; /* XXX gcc CSE bug workaround */ return (xy.hi + vzs); } if (oround != FE_TONEAREST) { /* * There is no need to worry about double rounding in directed * rounding modes. */ fesetround(oround); /* work around clang issue #8472 */ volatile double vrlo = r.lo; adj = vrlo + xy.lo; return (ldexp(r.hi + adj, spread)); } adj = add_adjusted(r.lo, xy.lo); if (spread + ilogb(r.hi) > -1023) return (ldexp(r.hi + adj, spread)); else return (add_and_denormalize(r.hi, adj, spread)); } #endif /* !USE_BUILTIN_FMA */ #if (LDBL_MANT_DIG == 53) __weak_reference(fma, fmal); #endif
