Add a "fork" of musl as //fusl.

fusl/src/math/fmal.c

+/* 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