text: /*-
   * 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 <sys/cdefs.h>
  __FBSDID("$FreeBSD: releng/11.0/lib/msun/src/s_fmal.c 252170 2013-06-24 19:12:17Z eadler $");
  
  #include <fenv.h>
  #include <float.h>
  #include <math.h>
  
  #include "fpmath.h"
  
  /*
   * 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 IEEEl2bits u;
  
  	sum = dd_add(a, b);
  	if (sum.lo != 0) {
  		u.e = sum.hi;
  		if ((u.bits.manl & 1) == 0)
  			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 IEEEl2bits 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.e = sum.hi;
  		bits_lost = -u.bits.exp - scale + 1;
  		if ((bits_lost != 1) ^ (int)(u.bits.manl & 1))
  			sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
  	}
  	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(long double a, long double b)
  {
  #if LDBL_MANT_DIG == 64
  	static const long double split = 0x1p32L + 1.0;
  #elif LDBL_MANT_DIG == 113
  	static const long double split = 0x1p57L + 1.0;
  #endif
  	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)
  {
  	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 (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 = 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) {
  		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 (nextafterl(z, 0));
  		case FE_DOWNWARD:
  			if (x > 0.0 ^ y < 0.0)
  				return (z);
  			else
  				return (nextafterl(z, -INFINITY));
  		default:	/* FE_UPWARD */
  			if (x > 0.0 ^ y < 0.0)
  				return (nextafterl(z, INFINITY));
  			else
  				return (z);
  		}
  	}
  	if (spread <= LDBL_MANT_DIG * 2)
  		zs = ldexpl(zs, -spread);
  	else
  		zs = copysignl(LDBL_MIN, zs);
  
  	fesetround(FE_TONEAREST);
  	/* work around clang bug 8100 */
  	volatile long double vxs = xs;
  
  	/*
  	 * 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(vxs, 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 + ldexpl(xy.lo, spread));
  	}
  
  	if (oround != FE_TONEAREST) {
  		/*
  		 * There is no need to worry about double rounding in directed
  		 * rounding modes.
  		 */
  		fesetround(oround);
  		/* work around clang bug 8100 */
  		volatile long double vrlo = r.lo;
  		adj = vrlo + xy.lo;
  		return (ldexpl(r.hi + adj, spread));
  	}
  
  	adj = add_adjusted(r.lo, xy.lo);
  	if (spread + ilogbl(r.hi) > -16383)
  		return (ldexpl(r.hi + adj, spread));
  	else
  		return (add_and_denormalize(r.hi, adj, spread));
  }
  