fusl/src/math/fmal.c - mojo-tools - Git at Google

 /* 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
	/* 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