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+/* origin: FreeBSD /usr/src/lib/msun/src/s_fma.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 <fenv.h>
+#include "libm.h"
+
+/*
+ * 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);
+
+	/*
+	 * 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 double vzs = zs; /* XXX gcc CSE bug workaround */
+		return (xy.hi + vzs + ldexp(xy.lo, spread));
+	}
+
+	if (oround != FE_TONEAREST) {
+		/*
+		 * There is no need to worry about double rounding in directed
+		 * rounding modes.
+		 */
+		fesetround(oround);
+		adj = r.lo + 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));
+}