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228 lines
7.0 KiB
C
228 lines
7.0 KiB
C
/* Compute x * y + z as ternary operation.
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Copyright (C) 2010, 2011 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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/* 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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double
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__fma (double x, double y, double z)
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{
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union ieee754_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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>= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG, 0)
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|| __builtin_expect (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
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|| __builtin_expect (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
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|| __builtin_expect (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
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|| __builtin_expect (u.ieee.exponent + v.ieee.exponent
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<= IEEE754_DOUBLE_BIAS + DBL_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 == 0x7ff
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&& u.ieee.exponent != 0x7ff
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&& v.ieee.exponent != 0x7ff)
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return (z + x) + y;
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/* If x or y or z is Inf/NaN, or if fma will certainly overflow,
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or if x * y is less than half of DBL_DENORM_MIN,
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compute as x * y + z. */
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if (u.ieee.exponent == 0x7ff
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|| v.ieee.exponent == 0x7ff
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|| w.ieee.exponent == 0x7ff
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|| u.ieee.exponent + v.ieee.exponent
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> 0x7ff + IEEE754_DOUBLE_BIAS
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|| u.ieee.exponent + v.ieee.exponent
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< IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2)
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return x * y + z;
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if (u.ieee.exponent + v.ieee.exponent
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>= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG)
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{
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/* Compute 1p-53 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 -= DBL_MANT_DIG;
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else
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v.ieee.exponent -= DBL_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 > DBL_MANT_DIG)
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w.ieee.exponent -= DBL_MANT_DIG;
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adjust = 1;
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}
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else if (w.ieee.exponent >= 0x7ff - DBL_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, it doesn't matter if we don't adjust it. */
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if (u.ieee.exponent > v.ieee.exponent)
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{
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if (u.ieee.exponent > DBL_MANT_DIG)
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u.ieee.exponent -= DBL_MANT_DIG;
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}
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else if (v.ieee.exponent > DBL_MANT_DIG)
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v.ieee.exponent -= DBL_MANT_DIG;
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w.ieee.exponent -= DBL_MANT_DIG;
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adjust = 1;
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}
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else if (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
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{
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u.ieee.exponent -= DBL_MANT_DIG;
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if (v.ieee.exponent)
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v.ieee.exponent += DBL_MANT_DIG;
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else
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v.d *= 0x1p53;
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}
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else if (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
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{
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v.ieee.exponent -= DBL_MANT_DIG;
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if (u.ieee.exponent)
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u.ieee.exponent += DBL_MANT_DIG;
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else
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u.d *= 0x1p53;
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}
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else /* if (u.ieee.exponent + v.ieee.exponent
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<= IEEE754_DOUBLE_BIAS + DBL_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 * DBL_MANT_DIG;
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else
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v.ieee.exponent += 2 * DBL_MANT_DIG;
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if (w.ieee.exponent <= 4 * DBL_MANT_DIG + 4)
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{
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if (w.ieee.exponent)
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w.ieee.exponent += 2 * DBL_MANT_DIG;
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else
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w.d *= 0x1p106;
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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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/* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
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#define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
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double x1 = x * C;
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double y1 = y * C;
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double m1 = x * y;
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x1 = (x - x1) + x1;
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y1 = (y - y1) + y1;
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double x2 = x - x1;
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double y2 = y - y1;
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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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double a1 = z + m1;
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double t1 = a1 - z;
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double t2 = a1 - t1;
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t1 = m1 - t1;
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t2 = z - t2;
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double a2 = t1 + t2;
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fenv_t env;
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libc_feholdexcept_setround (&env, 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 (__builtin_expect (adjust < 0, 0))
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{
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if ((u.ieee.mantissa1 & 1) == 0)
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u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0;
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v.d = a1 + u.d;
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}
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/* Reset rounding mode and test for inexact simultaneously. */
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int j = libc_feupdateenv_test (&env, FE_INEXACT) != 0;
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if (__builtin_expect (adjust == 0, 1))
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{
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if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
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u.ieee.mantissa1 |= j;
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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 (__builtin_expect (adjust > 0, 1))
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{
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if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
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u.ieee.mantissa1 |= j;
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/* Result is a1 + u.d, scaled up. */
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return (a1 + u.d) * 0x1p53;
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}
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else
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{
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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-106;
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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 > 106)
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return (a1 + u.d) * 0x1p-106;
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/* If v.d * 0x1p-106 with round to zero is a subnormal above
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or equal to DBL_MIN / 2, then v.d * 0x1p-106 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-106 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 == 106)
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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. In round-to-nearest 001 rounds down like 00,
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011 rounds up, even though 01 rounds down (thus we need
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to adjust), 101 rounds down like 10 and 111 rounds up
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like 11. */
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if ((v.ieee.mantissa1 & 3) == 1)
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{
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v.d *= 0x1p-106;
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if (v.ieee.negative)
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return v.d - 0x1p-1074 /* __DBL_DENORM_MIN__ */;
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else
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return v.d + 0x1p-1074 /* __DBL_DENORM_MIN__ */;
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}
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else
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return v.d * 0x1p-106;
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}
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v.ieee.mantissa1 |= j;
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return v.d * 0x1p-106;
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}
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}
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#ifndef __fma
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weak_alias (__fma, fma)
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#endif
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#ifdef NO_LONG_DOUBLE
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strong_alias (__fma, __fmal)
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weak_alias (__fmal, fmal)
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#endif
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