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+/* 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