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294 lines
9.5 KiB
C
294 lines
9.5 KiB
C
/* Compute x * y + z as ternary operation.
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Copyright (C) 2010-2014 Free Software Foundation, Inc.
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This file is part of the GNU C Library.
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Contributed by Jakub Jelinek <jakub@redhat.com>, 2010.
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The GNU C Library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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The GNU C Library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with the GNU C Library; if not, see
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<http://www.gnu.org/licenses/>. */
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#include <float.h>
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#include <math.h>
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#include <fenv.h>
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#include <ieee754.h>
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#include <math_private.h>
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#include <tininess.h>
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/* This implementation uses rounding to odd to avoid problems with
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double rounding. See a paper by Boldo and Melquiond:
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http://www.lri.fr/~melquion/doc/08-tc.pdf */
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long double
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__fmal (long double x, long double y, long double z)
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{
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union ieee854_long_double u, v, w;
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int adjust = 0;
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u.d = x;
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v.d = y;
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w.d = z;
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if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
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>= 0x7fff + IEEE854_LONG_DOUBLE_BIAS
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- LDBL_MANT_DIG, 0)
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|| __builtin_expect (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
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|| __builtin_expect (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
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|| __builtin_expect (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
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|| __builtin_expect (u.ieee.exponent + v.ieee.exponent
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<= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG, 0))
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{
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/* If z is Inf, but x and y are finite, the result should be
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z rather than NaN. */
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if (w.ieee.exponent == 0x7fff
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&& u.ieee.exponent != 0x7fff
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&& v.ieee.exponent != 0x7fff)
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return (z + x) + y;
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/* If z is zero and x are y are nonzero, compute the result
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as x * y to avoid the wrong sign of a zero result if x * y
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underflows to 0. */
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if (z == 0 && x != 0 && y != 0)
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return x * y;
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/* If x or y or z is Inf/NaN, or if x * y is zero, compute as
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x * y + z. */
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if (u.ieee.exponent == 0x7fff
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|| v.ieee.exponent == 0x7fff
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|| w.ieee.exponent == 0x7fff
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|| x == 0
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|| y == 0)
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return x * y + z;
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/* If fma will certainly overflow, compute as x * y. */
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if (u.ieee.exponent + v.ieee.exponent
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> 0x7fff + IEEE854_LONG_DOUBLE_BIAS)
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return x * y;
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/* If x * y is less than 1/4 of LDBL_DENORM_MIN, neither the
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result nor whether there is underflow depends on its exact
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value, only on its sign. */
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if (u.ieee.exponent + v.ieee.exponent
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< IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG - 2)
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{
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int neg = u.ieee.negative ^ v.ieee.negative;
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long double tiny = neg ? -0x1p-16445L : 0x1p-16445L;
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if (w.ieee.exponent >= 3)
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return tiny + z;
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/* Scaling up, adding TINY and scaling down produces the
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correct result, because in round-to-nearest mode adding
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TINY has no effect and in other modes double rounding is
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harmless. But it may not produce required underflow
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exceptions. */
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v.d = z * 0x1p65L + tiny;
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if (TININESS_AFTER_ROUNDING
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? v.ieee.exponent < 66
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: (w.ieee.exponent == 0
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|| (w.ieee.exponent == 1
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&& w.ieee.negative != neg
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&& w.ieee.mantissa1 == 0
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&& w.ieee.mantissa0 == 0x80000000)))
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{
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volatile long double force_underflow = x * y;
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(void) force_underflow;
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}
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return v.d * 0x1p-65L;
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}
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if (u.ieee.exponent + v.ieee.exponent
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>= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG)
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{
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/* Compute 1p-64 times smaller result and multiply
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at the end. */
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if (u.ieee.exponent > v.ieee.exponent)
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u.ieee.exponent -= LDBL_MANT_DIG;
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else
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v.ieee.exponent -= LDBL_MANT_DIG;
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/* If x + y exponent is very large and z exponent is very small,
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it doesn't matter if we don't adjust it. */
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if (w.ieee.exponent > LDBL_MANT_DIG)
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w.ieee.exponent -= LDBL_MANT_DIG;
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adjust = 1;
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}
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else if (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
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{
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/* Similarly.
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If z exponent is very large and x and y exponents are
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very small, adjust them up to avoid spurious underflows,
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rather than down. */
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if (u.ieee.exponent + v.ieee.exponent
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<= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG)
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{
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if (u.ieee.exponent > v.ieee.exponent)
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u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
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else
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v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
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}
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else if (u.ieee.exponent > v.ieee.exponent)
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{
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if (u.ieee.exponent > LDBL_MANT_DIG)
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u.ieee.exponent -= LDBL_MANT_DIG;
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}
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else if (v.ieee.exponent > LDBL_MANT_DIG)
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v.ieee.exponent -= LDBL_MANT_DIG;
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w.ieee.exponent -= LDBL_MANT_DIG;
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adjust = 1;
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}
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else if (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
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{
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u.ieee.exponent -= LDBL_MANT_DIG;
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if (v.ieee.exponent)
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v.ieee.exponent += LDBL_MANT_DIG;
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else
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v.d *= 0x1p64L;
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}
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else if (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
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{
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v.ieee.exponent -= LDBL_MANT_DIG;
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if (u.ieee.exponent)
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u.ieee.exponent += LDBL_MANT_DIG;
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else
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u.d *= 0x1p64L;
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}
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else /* if (u.ieee.exponent + v.ieee.exponent
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<= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG) */
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{
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if (u.ieee.exponent > v.ieee.exponent)
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u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
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else
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v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
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if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 6)
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{
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if (w.ieee.exponent)
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w.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
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else
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w.d *= 0x1p130L;
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adjust = -1;
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}
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/* Otherwise x * y should just affect inexact
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and nothing else. */
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}
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x = u.d;
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y = v.d;
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z = w.d;
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}
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/* Ensure correct sign of exact 0 + 0. */
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if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
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return x * y + z;
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fenv_t env;
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feholdexcept (&env);
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fesetround (FE_TONEAREST);
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/* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
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#define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
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long double x1 = x * C;
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long double y1 = y * C;
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long double m1 = x * y;
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x1 = (x - x1) + x1;
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y1 = (y - y1) + y1;
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long double x2 = x - x1;
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long double y2 = y - y1;
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long double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
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/* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
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long double a1 = z + m1;
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long double t1 = a1 - z;
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long double t2 = a1 - t1;
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t1 = m1 - t1;
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t2 = z - t2;
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long double a2 = t1 + t2;
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/* Ensure the arithmetic is not scheduled after feclearexcept call. */
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math_force_eval (m2);
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math_force_eval (a2);
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feclearexcept (FE_INEXACT);
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/* If the result is an exact zero, ensure it has the correct sign. */
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if (a1 == 0 && m2 == 0)
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{
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feupdateenv (&env);
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/* Ensure that round-to-nearest value of z + m1 is not reused. */
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z = math_opt_barrier (z);
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return z + m1;
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}
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fesetround (FE_TOWARDZERO);
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/* Perform m2 + a2 addition with round to odd. */
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u.d = a2 + m2;
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if (__glibc_likely (adjust == 0))
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{
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if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7fff)
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u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;
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feupdateenv (&env);
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/* Result is a1 + u.d. */
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return a1 + u.d;
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}
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else if (__glibc_likely (adjust > 0))
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{
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if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7fff)
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u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;
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feupdateenv (&env);
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/* Result is a1 + u.d, scaled up. */
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return (a1 + u.d) * 0x1p64L;
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}
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else
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{
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if ((u.ieee.mantissa1 & 1) == 0)
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u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;
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v.d = a1 + u.d;
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/* Ensure the addition is not scheduled after fetestexcept call. */
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math_force_eval (v.d);
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int j = fetestexcept (FE_INEXACT) != 0;
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feupdateenv (&env);
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/* Ensure the following computations are performed in default rounding
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mode instead of just reusing the round to zero computation. */
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asm volatile ("" : "=m" (u) : "m" (u));
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/* If a1 + u.d is exact, the only rounding happens during
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scaling down. */
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if (j == 0)
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return v.d * 0x1p-130L;
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/* If result rounded to zero is not subnormal, no double
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rounding will occur. */
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if (v.ieee.exponent > 130)
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return (a1 + u.d) * 0x1p-130L;
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/* If v.d * 0x1p-130L with round to zero is a subnormal above
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or equal to LDBL_MIN / 2, then v.d * 0x1p-130L shifts mantissa
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down just by 1 bit, which means v.ieee.mantissa1 |= j would
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change the round bit, not sticky or guard bit.
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v.d * 0x1p-130L never normalizes by shifting up,
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so round bit plus sticky bit should be already enough
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for proper rounding. */
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if (v.ieee.exponent == 130)
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{
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/* If the exponent would be in the normal range when
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rounding to normal precision with unbounded exponent
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range, the exact result is known and spurious underflows
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must be avoided on systems detecting tininess after
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rounding. */
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if (TININESS_AFTER_ROUNDING)
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{
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w.d = a1 + u.d;
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if (w.ieee.exponent == 131)
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return w.d * 0x1p-130L;
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}
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/* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,
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v.ieee.mantissa1 & 1 is the round bit and j is our sticky
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bit. */
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w.d = 0.0L;
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w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) | j;
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w.ieee.negative = v.ieee.negative;
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v.ieee.mantissa1 &= ~3U;
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v.d *= 0x1p-130L;
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w.d *= 0x1p-2L;
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return v.d + w.d;
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}
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v.ieee.mantissa1 |= j;
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return v.d * 0x1p-130L;
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}
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}
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weak_alias (__fmal, fmal)
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