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

fusl/src/math/fma.c

+#include <fenv.h>
+#include "libm.h"
+
+#if LDBL_MANT_DIG==64 && LDBL_MAX_EXP==16384
+/* exact add, assumes exponent_x >= exponent_y */
+static void add(long double *hi, long double *lo, long double x, long double y)
+{
+        long double r;
+
+        r = x + y;
+        *hi = r;
+        r -= x;
+        *lo = y - r;
+}
+
+/* exact mul, assumes no over/underflow */
+static void mul(long double *hi, long double *lo, long double x, long double y)
+{
+        static const long double c = 1.0 + 0x1p32L;
+        long double cx, xh, xl, cy, yh, yl;
+
+        cx = c*x;
+        xh = (x - cx) + cx;
+        xl = x - xh;
+        cy = c*y;
+        yh = (y - cy) + cy;
+        yl = y - yh;
+        *hi = x*y;
+        *lo = (xh*yh - *hi) + xh*yl + xl*yh + xl*yl;
+}
+
+/*
+assume (long double)(hi+lo) == hi
+return an adjusted hi so that rounding it to double (or less) precision is correct
+*/
+static long double adjust(long double hi, long double lo)
+{
+        union ldshape uhi, ulo;
+
+        if (lo == 0)
+                return hi;
+        uhi.f = hi;
+        if (uhi.i.m & 0x3ff)
+                return hi;
+        ulo.f = lo;
+        if ((uhi.i.se & 0x8000) == (ulo.i.se & 0x8000))
+                uhi.i.m++;
+        else {
+                /* handle underflow and take care of ld80 implicit msb */
+                if (uhi.i.m << 1 == 0) {
+                        uhi.i.m = 0;
+                        uhi.i.se--;
+                }
+                uhi.i.m--;
+        }
+        return uhi.f;
+}
+
+/* adjusted add so the result is correct when rounded to double (or less) precision */
+static long double dadd(long double x, long double y)
+{
+        add(&x, &y, x, y);
+        return adjust(x, y);
+}
+
+/* adjusted mul so the result is correct when rounded to double (or less) precision */
+static long double dmul(long double x, long double y)
+{
+        mul(&x, &y, x, y);
+        return adjust(x, y);
+}
+
+static int getexp(long double x)
+{
+        union ldshape u;
+        u.f = x;
+        return u.i.se & 0x7fff;
+}
+
+double fma(double x, double y, double z)
+{
+        #pragma STDC FENV_ACCESS ON
+        long double hi, lo1, lo2, xy;
+        int round, ez, exy;
+
+        /* handle +-inf,nan */
+        if (!isfinite(x) || !isfinite(y))
+                return x*y + z;
+        if (!isfinite(z))
+                return z;
+        /* handle +-0 */
+        if (x == 0.0 || y == 0.0)
+                return x*y + z;
+        round = fegetround();
+        if (z == 0.0) {
+                if (round == FE_TONEAREST)
+                        return dmul(x, y);
+                return x*y;
+        }
+
+        /* exact mul and add require nearest rounding */
+        /* spurious inexact exceptions may be raised */
+        fesetround(FE_TONEAREST);
+        mul(&xy, &lo1, x, y);
+        exy = getexp(xy);
+        ez = getexp(z);
+        if (ez > exy) {
+                add(&hi, &lo2, z, xy);
+        } else if (ez > exy - 12) {
+                add(&hi, &lo2, xy, z);
+                if (hi == 0) {
+                        /*
+                        xy + z is 0, but it should be calculated with the
+                        original rounding mode so the sign is correct, if the
+                        compiler does not support FENV_ACCESS ON it does not
+                        know about the changed rounding mode and eliminates
+                        the xy + z below without the volatile memory access
+                        */
+                        volatile double z_;
+                        fesetround(round);
+                        z_ = z;
+                        return (xy + z_) + lo1;
+                }
+        } else {
+                /*
+                ez <= exy - 12
+                the 12 extra bits (1guard, 11round+sticky) are needed so with
+                        lo = dadd(lo1, lo2)
+                elo <= ehi - 11, and we use the last 10 bits in adjust so
+                        dadd(hi, lo)
+                gives correct result when rounded to double
+                */
+                hi = xy;
+                lo2 = z;
+        }
+        /*
+        the result is stored before return for correct precision and exceptions
+
+        one corner case is when the underflow flag should be raised because
+        the precise result is an inexact subnormal double, but the calculated
+        long double result is an exact subnormal double
+        (so rounding to double does not raise exceptions)
+
+        in nearest rounding mode dadd takes care of this: the last bit of the
+        result is adjusted so rounding sees an inexact value when it should
+
+        in non-nearest rounding mode fenv is used for the workaround
+        */
+        fesetround(round);
+        if (round == FE_TONEAREST)
+                z = dadd(hi, dadd(lo1, lo2));
+        else {
+#if defined(FE_INEXACT) && defined(FE_UNDERFLOW)
+                int e = fetestexcept(FE_INEXACT);
+                feclearexcept(FE_INEXACT);
+#endif
+                z = hi + (lo1 + lo2);
+#if defined(FE_INEXACT) && defined(FE_UNDERFLOW)
+                if (getexp(z) < 0x3fff-1022 && fetestexcept(FE_INEXACT))
+                        feraiseexcept(FE_UNDERFLOW);
+                else if (e)
+                        feraiseexcept(FE_INEXACT);
+#endif
+        }
+        return z;
+}
+#else
+/* 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.
+ */
+
+/*
+ * 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;
+        union {double f; uint64_t i;} uhi, ulo;
+
+        sum = dd_add(a, b);
+        if (sum.lo != 0) {
+                uhi.f = sum.hi;
+                if ((uhi.i & 1) == 0) {
+                        /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
+                        ulo.f = sum.lo;
+                        uhi.i += 1 - ((uhi.i ^ ulo.i) >> 62);
+                        sum.hi = uhi.f;
+                }
+        }
+        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;
+        union {double f; uint64_t i;} uhi, ulo;
+        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) {
+                uhi.f = sum.hi;
+                bits_lost = -((int)(uhi.i >> 52) & 0x7ff) - scale + 1;
+                if ((bits_lost != 1) ^ (int)(uhi.i & 1)) {
+                        /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
+                        ulo.f = sum.lo;
+                        uhi.i += 1 - (((uhi.i ^ ulo.i) >> 62) & 2);
+                        sum.hi = uhi.f;
+                }
+        }
+        return scalbn(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)
+{
+        #pragma STDC FENV_ACCESS ON
+        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 = 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) {
+#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 (nextafter(z, 0));
+#endif
+#ifdef FE_DOWNWARD
+                case FE_DOWNWARD:
+                        if (x > 0.0 ^ y < 0.0)
+                                return (z);
+                        else
+                                return (nextafter(z, -INFINITY));
+#endif
+#ifdef FE_UPWARD
+                case FE_UPWARD:
+                        if (x > 0.0 ^ y < 0.0)
+                                return (nextafter(z, INFINITY));
+                        else
+                                return (z);
+#endif
+                }
+        }
+        if (spread <= DBL_MANT_DIG * 2)
+                zs = scalbn(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 + scalbn(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 properly, example in downward rounding:
+                 * fma(0x1.000000001p-1000, 0x1.000000001p-30, -0x1p-1066)
+                 */
+                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 = scalbn(r.hi + adj, spread);
+#if defined(FE_INEXACT) && defined(FE_UNDERFLOW)
+                if (ilogb(ret) < -1022 && fetestexcept(FE_INEXACT))
+                        feraiseexcept(FE_UNDERFLOW);
+                else if (e)
+                        feraiseexcept(FE_INEXACT);
+#endif
+                return ret;
+        }
+
+        adj = add_adjusted(r.lo, xy.lo);
+        if (spread + ilogb(r.hi) > -1023)
+                return scalbn(r.hi + adj, spread);
+        else
+                return add_and_denormalize(r.hi, adj, spread);
+}
+#endif