[musl] / src / math / fma.c

#include <fenv.h>
#include "libm.h"

#if LDBL_MANT_DIG==64 && LDBL_MAX_EXP==16384
union ld80 {
        long double x;
        struct {
                uint64_t m;
                uint16_t e : 15;
                uint16_t s : 1;
                uint16_t pad;
        } bits;
};

/* 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 ld80 uhi, ulo;

        if (lo == 0)
                return hi;
        uhi.x = hi;
        if (uhi.bits.m & 0x3ff)
                return hi;
        ulo.x = lo;
        if (uhi.bits.s == ulo.bits.s)
                uhi.bits.m++;
        else {
                uhi.bits.m--;
                /* handle underflow and take care of ld80 implicit msb */
                if (uhi.bits.m == (uint64_t)-1/2) {
                        uhi.bits.m *= 2;
                        uhi.bits.e--;
                }
        }
        return uhi.x;
}

/* 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 ld80 u;
        u.x = x;
        return u.bits.e;
}

double fma(double x, double y, double z)
{
        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) {
                        fesetround(round);
                        /* make sure that the sign of 0 is correct */
                        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;
        }
        fesetround(round);
        if (round == FE_TONEAREST)
                return dadd(hi, dadd(lo1, lo2));
        return hi + (lo1 + lo2);
}
#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;
        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 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)
{
        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.
                 */
                fesetround(oround);
                adj = r.lo + xy.lo;
                return scalbn(r.hi + adj, spread);
        }

        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