/* Compute x * y + z as ternary operation.

   Copyright (C) 2010-2018 Free Software Foundation, Inc.

   This file is part of the GNU C Library.

   Contributed by Jakub Jelinek <jakub@redhat.com>, 2010.

   The GNU C Library is free software; you can redistribute it and/or

   modify it under the terms of the GNU Lesser General Public

   License as published by the Free Software Foundation; either

   version 2.1 of the License, or (at your option) any later version.

   The GNU C Library is distributed in the hope that it will be useful,

   but WITHOUT ANY WARRANTY; without even the implied warranty of

   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU

   Lesser General Public License for more details.

   You should have received a copy of the GNU Lesser General Public

   License along with the GNU C Library; if not, see

   <http://www.gnu.org/licenses/>.  */

#include <float.h>

#include <math.h>

#include <fenv.h>

#include <ieee754.h>

#include <math-barriers.h>

#include <math_private.h>

#include <libm-alias-ldouble.h>

#include <tininess.h>

/* This implementation uses rounding to odd to avoid problems with

   double rounding.  See a paper by Boldo and Melquiond:

   http://www.lri.fr/~melquion/doc/08-tc.pdf  */

long double

__fmal (long double x, long double y, long double z)

{

  union ieee854_long_double u, v, w;

  int adjust = 0;

  u.d = x;

  v.d = y;

  w.d = z;

  if (__builtin_expect (u.ieee.exponent + v.ieee.exponent

			>= 0x7fff + IEEE854_LONG_DOUBLE_BIAS

			   - LDBL_MANT_DIG, 0)

      || __builtin_expect (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)

      || __builtin_expect (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)

      || __builtin_expect (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)

      || __builtin_expect (u.ieee.exponent + v.ieee.exponent

			   <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG, 0))

    {

      /* If z is Inf, but x and y are finite, the result should be

	 z rather than NaN.  */

      if (w.ieee.exponent == 0x7fff

	  && u.ieee.exponent != 0x7fff

          && v.ieee.exponent != 0x7fff)

	return (z + x) + y;

      /* If z is zero and x are y are nonzero, compute the result

	 as x * y to avoid the wrong sign of a zero result if x * y

	 underflows to 0.  */

      if (z == 0 && x != 0 && y != 0)

	return x * y;

      /* If x or y or z is Inf/NaN, or if x * y is zero, compute as

	 x * y + z.  */

      if (u.ieee.exponent == 0x7fff

	  || v.ieee.exponent == 0x7fff

	  || w.ieee.exponent == 0x7fff

	  || x == 0

	  || y == 0)

	return x * y + z;

      /* If fma will certainly overflow, compute as x * y.  */

      if (u.ieee.exponent + v.ieee.exponent

	  > 0x7fff + IEEE854_LONG_DOUBLE_BIAS)

	return x * y;

      /* If x * y is less than 1/4 of LDBL_TRUE_MIN, neither the

	 result nor whether there is underflow depends on its exact

	 value, only on its sign.  */

      if (u.ieee.exponent + v.ieee.exponent

	  < IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG - 2)

	{

	  int neg = u.ieee.negative ^ v.ieee.negative;

	  long double tiny = neg ? -0x1p-16445L : 0x1p-16445L;

	  if (w.ieee.exponent >= 3)

	    return tiny + z;

	  /* Scaling up, adding TINY and scaling down produces the

	     correct result, because in round-to-nearest mode adding

	     TINY has no effect and in other modes double rounding is

	     harmless.  But it may not produce required underflow

	     exceptions.  */

	  v.d = z * 0x1p65L + tiny;

	  if (TININESS_AFTER_ROUNDING

	      ? v.ieee.exponent < 66

	      : (w.ieee.exponent == 0

		 || (w.ieee.exponent == 1

		     && w.ieee.negative != neg

		     && w.ieee.mantissa1 == 0

		     && w.ieee.mantissa0 == 0x80000000)))

	    {

	      long double force_underflow = x * y;

	      math_force_eval (force_underflow);

	    }

	  return v.d * 0x1p-65L;

	}

      if (u.ieee.exponent + v.ieee.exponent

	  >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG)

	{

	  /* Compute 1p-64 times smaller result and multiply

	     at the end.  */

	  if (u.ieee.exponent > v.ieee.exponent)

	    u.ieee.exponent -= LDBL_MANT_DIG;

	  else

	    v.ieee.exponent -= LDBL_MANT_DIG;

	  /* If x + y exponent is very large and z exponent is very small,

	     it doesn't matter if we don't adjust it.  */

	  if (w.ieee.exponent > LDBL_MANT_DIG)

	    w.ieee.exponent -= LDBL_MANT_DIG;

	  adjust = 1;

	}

      else if (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)

	{

	  /* Similarly.

	     If z exponent is very large and x and y exponents are

	     very small, adjust them up to avoid spurious underflows,

	     rather than down.  */

	  if (u.ieee.exponent + v.ieee.exponent

	      <= IEEE854_LONG_DOUBLE_BIAS + 2 * LDBL_MANT_DIG)

	    {

	      if (u.ieee.exponent > v.ieee.exponent)

		u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;

	      else

		v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;

	    }

	  else if (u.ieee.exponent > v.ieee.exponent)

	    {

	      if (u.ieee.exponent > LDBL_MANT_DIG)

		u.ieee.exponent -= LDBL_MANT_DIG;

	    }

	  else if (v.ieee.exponent > LDBL_MANT_DIG)

	    v.ieee.exponent -= LDBL_MANT_DIG;

	  w.ieee.exponent -= LDBL_MANT_DIG;

	  adjust = 1;

	}

      else if (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)

	{

	  u.ieee.exponent -= LDBL_MANT_DIG;

	  if (v.ieee.exponent)

	    v.ieee.exponent += LDBL_MANT_DIG;

	  else

	    v.d *= 0x1p64L;

	}

      else if (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)

	{

	  v.ieee.exponent -= LDBL_MANT_DIG;

	  if (u.ieee.exponent)

	    u.ieee.exponent += LDBL_MANT_DIG;

	  else

	    u.d *= 0x1p64L;

	}

      else /* if (u.ieee.exponent + v.ieee.exponent

		  <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG) */

	{

	  if (u.ieee.exponent > v.ieee.exponent)

	    u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;

	  else

	    v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;

	  if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 6)

	    {

	      if (w.ieee.exponent)

		w.ieee.exponent += 2 * LDBL_MANT_DIG + 2;

	      else

		w.d *= 0x1p130L;

	      adjust = -1;

	    }

	  /* Otherwise x * y should just affect inexact

	     and nothing else.  */

	}

      x = u.d;

      y = v.d;

      z = w.d;

    }

  /* Ensure correct sign of exact 0 + 0.  */

  if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))

    {

      x = math_opt_barrier (x);

      return x * y + z;

    }

  fenv_t env;

  feholdexcept (&env;;

  fesetround (FE_TONEAREST);

  /* Multiplication m1 + m2 = x * y using Dekker's algorithm.  */

#define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)

  long double x1 = x * C;

  long double y1 = y * C;

  long double m1 = x * y;

  x1 = (x - x1) + x1;

  y1 = (y - y1) + y1;

  long double x2 = x - x1;

  long double y2 = y - y1;

  long double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;

  /* Addition a1 + a2 = z + m1 using Knuth's algorithm.  */

  long double a1 = z + m1;

  long double t1 = a1 - z;

  long double t2 = a1 - t1;

  t1 = m1 - t1;

  t2 = z - t2;

  long double a2 = t1 + t2;

  /* Ensure the arithmetic is not scheduled after feclearexcept call.  */

  math_force_eval (m2);

  math_force_eval (a2);

  feclearexcept (FE_INEXACT);

  /* If the result is an exact zero, ensure it has the correct sign.  */

  if (a1 == 0 && m2 == 0)

    {

      feupdateenv (&env;;

      /* Ensure that round-to-nearest value of z + m1 is not reused.  */

      z = math_opt_barrier (z);

      return z + m1;

    }

  fesetround (FE_TOWARDZERO);

  /* Perform m2 + a2 addition with round to odd.  */

  u.d = a2 + m2;

  if (__glibc_likely (adjust == 0))

    {

      if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7fff)

	u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;

      feupdateenv (&env;;

      /* Result is a1 + u.d.  */

      return a1 + u.d;

    }

  else if (__glibc_likely (adjust > 0))

    {

      if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7fff)

	u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;

      feupdateenv (&env;;

      /* Result is a1 + u.d, scaled up.  */

      return (a1 + u.d) * 0x1p64L;

    }

  else

    {

      if ((u.ieee.mantissa1 & 1) == 0)

	u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;

      v.d = a1 + u.d;

      /* Ensure the addition is not scheduled after fetestexcept call.  */

      math_force_eval (v.d);

      int j = fetestexcept (FE_INEXACT) != 0;

      feupdateenv (&env;;

      /* Ensure the following computations are performed in default rounding

	 mode instead of just reusing the round to zero computation.  */

      asm volatile ("" : "=m" (u) : "m" (u));

      /* If a1 + u.d is exact, the only rounding happens during

	 scaling down.  */

      if (j == 0)

	return v.d * 0x1p-130L;

      /* If result rounded to zero is not subnormal, no double

	 rounding will occur.  */

      if (v.ieee.exponent > 130)

	return (a1 + u.d) * 0x1p-130L;

      /* If v.d * 0x1p-130L with round to zero is a subnormal above

	 or equal to LDBL_MIN / 2, then v.d * 0x1p-130L shifts mantissa

	 down just by 1 bit, which means v.ieee.mantissa1 |= j would

	 change the round bit, not sticky or guard bit.

	 v.d * 0x1p-130L never normalizes by shifting up,

	 so round bit plus sticky bit should be already enough

	 for proper rounding.  */

      if (v.ieee.exponent == 130)

	{

	  /* If the exponent would be in the normal range when

	     rounding to normal precision with unbounded exponent

	     range, the exact result is known and spurious underflows

	     must be avoided on systems detecting tininess after

	     rounding.  */

	  if (TININESS_AFTER_ROUNDING)

	    {

	      w.d = a1 + u.d;

	      if (w.ieee.exponent == 131)

		return w.d * 0x1p-130L;

	    }

	  /* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,

	     v.ieee.mantissa1 & 1 is the round bit and j is our sticky

	     bit.  */

	  w.d = 0.0L;

	  w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) | j;

	  w.ieee.negative = v.ieee.negative;

	  v.ieee.mantissa1 &= ~3U;

	  v.d *= 0x1p-130L;

	  w.d *= 0x1p-2L;

	  return v.d + w.d;

	}

      v.ieee.mantissa1 |= j;

      return v.d * 0x1p-130L;

    }

}

libm_alias_ldouble (__fma, fma)