diff options
| -rw-r--r-- | math/libm-test.inc | 25 | ||||
| -rw-r--r-- | sysdeps/i386/fpu/s_fma.S | 31 | ||||
| -rw-r--r-- | sysdeps/i386/fpu/s_fmaf.S | 31 | ||||
| -rw-r--r-- | sysdeps/i386/fpu/s_fmal.S | 32 | ||||
| -rw-r--r-- | sysdeps/ieee754/dbl-64/s_fma.c | 8 | ||||
| -rw-r--r-- | sysdeps/ieee754/ldbl-128/s_fmal.c | 221 | ||||
| -rw-r--r-- | sysdeps/ieee754/ldbl-64-128/s_fmal.c | 5 | ||||
| -rw-r--r-- | sysdeps/ieee754/ldbl-96/s_fma.c | 15 | ||||
| -rw-r--r-- | sysdeps/ieee754/ldbl-96/s_fmal.c | 221 |
9 files changed, 492 insertions, 97 deletions
diff --git a/math/libm-test.inc b/math/libm-test.inc index 1bec476fbf..ceed18d7fd 100644 --- a/math/libm-test.inc +++ b/math/libm-test.inc @@ -2787,8 +2787,24 @@ fma_test (void) TEST_fff_f (fma, minus_infty, plus_infty, plus_infty, nan_value, INVALID_EXCEPTION); TEST_fff_f (fma, plus_infty, minus_infty, plus_infty, nan_value, INVALID_EXCEPTION); TEST_fff_f (fma, minus_infty, minus_infty, minus_infty, nan_value, INVALID_EXCEPTION); + TEST_fff_f (fma, plus_infty, 3.5L, minus_infty, nan_value, INVALID_EXCEPTION); + TEST_fff_f (fma, minus_infty, -7.5L, minus_infty, nan_value, INVALID_EXCEPTION); + TEST_fff_f (fma, -13.5L, plus_infty, plus_infty, nan_value, INVALID_EXCEPTION); + TEST_fff_f (fma, minus_infty, 7.5L, plus_infty, nan_value, INVALID_EXCEPTION); TEST_fff_f (fma, 1.25L, 0.75L, 0.0625L, 1.0L); + + FLOAT fltmax = CHOOSE (LDBL_MAX, DBL_MAX, FLT_MAX, + LDBL_MAX, DBL_MAX, FLT_MAX); + TEST_fff_f (fma, -fltmax, -fltmax, minus_infty, minus_infty); + TEST_fff_f (fma, fltmax / 2, fltmax / 2, minus_infty, minus_infty); + TEST_fff_f (fma, -fltmax, fltmax, plus_infty, plus_infty); + TEST_fff_f (fma, fltmax / 2, -fltmax / 4, plus_infty, plus_infty); + TEST_fff_f (fma, plus_infty, 4, plus_infty, plus_infty); + TEST_fff_f (fma, 2, minus_infty, minus_infty, minus_infty); + TEST_fff_f (fma, minus_infty, minus_infty, plus_infty, plus_infty); + TEST_fff_f (fma, plus_infty, minus_infty, minus_infty, minus_infty); + #if defined (TEST_FLOAT) && FLT_MANT_DIG == 24 TEST_fff_f (fma, 0x1.7ff8p+13, 0x1.000002p+0, 0x1.ffffp-24, 0x1.7ff802p+13); TEST_fff_f (fma, 0x1.fffp+0, 0x1.00001p+0, -0x1.fffp+0, 0x1.fffp-20); @@ -2818,6 +2834,15 @@ fma_test (void) TEST_fff_f (fma, -0x1.19cab66d73e17p-959, 0x1.c7108a8c5ff51p-107, -0x0.80b0ad65d9b64p-1022, -0x0.80b0ad65d9d59p-1022); TEST_fff_f (fma, -0x1.d2eaed6e8e9d3p-979, -0x1.4e066c62ac9ddp-63, -0x0.9245e6b003454p-1022, -0x0.9245c09c5fb5dp-1022); TEST_fff_f (fma, 0x1.153d650bb9f06p-907, 0x1.2d01230d48407p-125, -0x0.b278d5acfc3cp-1022, -0x0.b22757123bbe9p-1022); + TEST_fff_f (fma, -0x1.fffffffffffffp-711, 0x1.fffffffffffffp-275, 0x1.fffffe00007ffp-983, 0x1.7ffffe00007ffp-983); +#endif +#if defined (TEST_LDOUBLE) && LDBL_MANT_DIG == 64 + TEST_fff_f (fma, -0x8.03fcp+3696L, 0xf.fffffffffffffffp-6140L, 0x8.3ffffffffffffffp-2450L, -0x8.01ecp-2440L); + TEST_fff_f (fma, 0x9.fcp+2033L, -0x8.000e1f000ff800fp-3613L, -0xf.fffffffffffc0ffp-1579L, -0xd.fc119fb093ed092p-1577L); + TEST_fff_f (fma, 0xc.7fc000003ffffffp-1194L, 0x8.1e0003fffffffffp+15327L, -0x8.fffep+14072L, 0xc.ae9f164020effffp+14136L); + TEST_fff_f (fma, -0x8.0001fc000000003p+1798L, 0xcp-2230L, 0x8.f7e000000000007p-468L, -0xc.0002f9ffee10404p-429L); + TEST_fff_f (fma, 0xc.0000000000007ffp+10130L, -0x8.000000000000001p+4430L, 0xc.07000000001ffffp+14513L, -0xb.fffffffffffd7e4p+14563L); + TEST_fff_f (fma, 0xb.ffffp-4777L, 0x8.000000fffffffffp-11612L, -0x0.3800fff8p-16385L, 0x5.c7fe80c7ffeffffp-16385L); #endif END (fma); diff --git a/sysdeps/i386/fpu/s_fma.S b/sysdeps/i386/fpu/s_fma.S deleted file mode 100644 index db4959ccfc..0000000000 --- a/sysdeps/i386/fpu/s_fma.S +++ /dev/null @@ -1,31 +0,0 @@ -/* Compute (X * Y) + Z as ternary operation. - Copyright (C) 1997, 1998 Free Software Foundation, Inc. - This file is part of the GNU C Library. - Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. - - 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, write to the Free - Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA - 02111-1307 USA. */ - -#include <sysdep.h> - - .text -ENTRY(__fma) - fldl 4(%esp) // x - fmull 12(%esp) // x * y - fldl 20(%esp) // z : x * y - faddp // (x * y) + z - ret -END(__fma) -weak_alias (__fma, fma) diff --git a/sysdeps/i386/fpu/s_fmaf.S b/sysdeps/i386/fpu/s_fmaf.S deleted file mode 100644 index 5f875532a8..0000000000 --- a/sysdeps/i386/fpu/s_fmaf.S +++ /dev/null @@ -1,31 +0,0 @@ -/* Compute (X * Y) + Z as ternary operation. - Copyright (C) 1997 Free Software Foundation, Inc. - This file is part of the GNU C Library. - Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. - - 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, write to the Free - Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA - 02111-1307 USA. */ - -#include <sysdep.h> - - .text -ENTRY(__fmaf) - flds 4(%esp) // x - fmuls 8(%esp) // x * y - flds 12(%esp) // z : x * y - faddp // (x * y) + z - ret -END(__fmaf) -weak_alias (__fmaf, fmaf) diff --git a/sysdeps/i386/fpu/s_fmal.S b/sysdeps/i386/fpu/s_fmal.S deleted file mode 100644 index 1837f848c0..0000000000 --- a/sysdeps/i386/fpu/s_fmal.S +++ /dev/null @@ -1,32 +0,0 @@ -/* Compute (X * Y) + Z as ternary operation. - Copyright (C) 1997 Free Software Foundation, Inc. - This file is part of the GNU C Library. - Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997. - - 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, write to the Free - Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA - 02111-1307 USA. */ - -#include <sysdep.h> - - .text -ENTRY(__fmal) - fldt 4(%esp) // x - fldt 16(%esp) // x : y - fmulp // x * y - fldt 28(%esp) // z : x * y - faddp // (x * y) + z - ret -END(__fmal) -weak_alias (__fmal, fmal) diff --git a/sysdeps/ieee754/dbl-64/s_fma.c b/sysdeps/ieee754/dbl-64/s_fma.c index 911682eceb..3b0bfd5ce6 100644 --- a/sysdeps/ieee754/dbl-64/s_fma.c +++ b/sysdeps/ieee754/dbl-64/s_fma.c @@ -43,6 +43,12 @@ __fma (double x, double y, double z) || __builtin_expect (u.ieee.exponent + v.ieee.exponent <= IEEE754_DOUBLE_BIAS + DBL_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 == 0x7ff + && u.ieee.exponent != 0x7ff + && v.ieee.exponent != 0x7ff) + return (z + x) + y; /* If x or y or z is Inf/NaN, or if fma will certainly overflow, or if x * y is less than half of DBL_DENORM_MIN, compute as x * y + z. */ @@ -165,6 +171,8 @@ __fma (double x, double y, double z) } else { + if ((u.ieee.mantissa1 & 1) == 0) + u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0; v.d = a1 + u.d; int j = fetestexcept (FE_INEXACT) != 0; feupdateenv (&env); diff --git a/sysdeps/ieee754/ldbl-128/s_fmal.c b/sysdeps/ieee754/ldbl-128/s_fmal.c new file mode 100644 index 0000000000..9ec5ba9ee9 --- /dev/null +++ b/sysdeps/ieee754/ldbl-128/s_fmal.c @@ -0,0 +1,221 @@ +/* Compute x * y + z as ternary operation. + Copyright (C) 2010 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, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include <float.h> +#include <math.h> +#include <fenv.h> +#include <ieee754.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 x or y or z is Inf/NaN, or if fma will certainly overflow, + or if x * y is less than half of LDBL_DENORM_MIN, + compute as x * y + z. */ + if (u.ieee.exponent == 0x7fff + || v.ieee.exponent == 0x7fff + || w.ieee.exponent == 0x7fff + || u.ieee.exponent + v.ieee.exponent + > 0x7fff + IEEE854_LONG_DOUBLE_BIAS + || u.ieee.exponent + v.ieee.exponent + < IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG - 2) + return x * y + z; + if (u.ieee.exponent + v.ieee.exponent + >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG) + { + /* Compute 1p-113 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, it doesn't matter if we don't adjust it. */ + 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 *= 0x1p113L; + } + 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 *= 0x1p113L; + } + 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; + else + v.ieee.exponent += 2 * LDBL_MANT_DIG; + if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 4) + { + if (w.ieee.exponent) + w.ieee.exponent += 2 * LDBL_MANT_DIG; + else + w.d *= 0x1p226L; + adjust = -1; + } + /* Otherwise x * y should just affect inexact + and nothing else. */ + } + x = u.d; + y = v.d; + z = w.d; + } + /* 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; + + fenv_t env; + feholdexcept (&env); + fesetround (FE_TOWARDZERO); + /* Perform m2 + a2 addition with round to odd. */ + u.d = a2 + m2; + + if (__builtin_expect (adjust == 0, 1)) + { + if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff) + u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0; + feupdateenv (&env); + /* Result is a1 + u.d. */ + return a1 + u.d; + } + else if (__builtin_expect (adjust > 0, 1)) + { + if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff) + u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0; + feupdateenv (&env); + /* Result is a1 + u.d, scaled up. */ + return (a1 + u.d) * 0x1p113L; + } + else + { + if ((u.ieee.mantissa3 & 1) == 0) + u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0; + v.d = a1 + u.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-226L; + /* If result rounded to zero is not subnormal, no double + rounding will occur. */ + if (v.ieee.exponent > 226) + return (a1 + u.d) * 0x1p-226L; + /* If v.d * 0x1p-226L with round to zero is a subnormal above + or equal to LDBL_MIN / 2, then v.d * 0x1p-226L shifts mantissa + down just by 1 bit, which means v.ieee.mantissa3 |= j would + change the round bit, not sticky or guard bit. + v.d * 0x1p-226L never normalizes by shifting up, + so round bit plus sticky bit should be already enough + for proper rounding. */ + if (v.ieee.exponent == 226) + { + /* v.ieee.mantissa3 & 2 is LSB bit of the result before rounding, + v.ieee.mantissa3 & 1 is the round bit and j is our sticky + bit. In round-to-nearest 001 rounds down like 00, + 011 rounds up, even though 01 rounds down (thus we need + to adjust), 101 rounds down like 10 and 111 rounds up + like 11. */ + if ((v.ieee.mantissa3 & 3) == 1) + { + v.d *= 0x1p-226L; + if (v.ieee.negative) + return v.d - 0x1p-16493L /* __LDBL_DENORM_MIN__ */; + else + return v.d + 0x1p-16493L /* __LDBL_DENORM_MIN__ */; + } + else + return v.d * 0x1p-226L; + } + v.ieee.mantissa3 |= j; + return v.d * 0x1p-226L; + } +} +weak_alias (__fmal, fmal) diff --git a/sysdeps/ieee754/ldbl-64-128/s_fmal.c b/sysdeps/ieee754/ldbl-64-128/s_fmal.c new file mode 100644 index 0000000000..218aa52b35 --- /dev/null +++ b/sysdeps/ieee754/ldbl-64-128/s_fmal.c @@ -0,0 +1,5 @@ +#include <math_ldbl_opt.h> +#undef weak_alias +#define weak_alias(n,a) +#include <sysdeps/ieee754/ldbl-128/s_fmal.c> +long_double_symbol (libm, __fmal, fmal); diff --git a/sysdeps/ieee754/ldbl-96/s_fma.c b/sysdeps/ieee754/ldbl-96/s_fma.c index 90c6d1f29d..6c7e9d0d36 100644 --- a/sysdeps/ieee754/ldbl-96/s_fma.c +++ b/sysdeps/ieee754/ldbl-96/s_fma.c @@ -30,11 +30,20 @@ double __fma (double x, double y, double z) { + if (__builtin_expect (isinf (z), 0)) + { + /* If z is Inf, but x and y are finite, the result should be + z rather than NaN. */ + if (finite (x) && finite (y)) + return (z + x) + y; + return (x * y) + z; + } + /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */ #define C ((1ULL << (LDBL_MANT_DIG + 1) / 2) + 1) - long double x1 = x * C; - long double y1 = y * C; - long double m1 = x * y; + long double x1 = (long double) x * C; + long double y1 = (long double) y * C; + long double m1 = (long double) x * y; x1 = (x - x1) + x1; y1 = (y - y1) + y1; long double x2 = x - x1; diff --git a/sysdeps/ieee754/ldbl-96/s_fmal.c b/sysdeps/ieee754/ldbl-96/s_fmal.c new file mode 100644 index 0000000000..340f02536c --- /dev/null +++ b/sysdeps/ieee754/ldbl-96/s_fmal.c @@ -0,0 +1,221 @@ +/* Compute x * y + z as ternary operation. + Copyright (C) 2010 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, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +#include <float.h> +#include <math.h> +#include <fenv.h> +#include <ieee754.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 x or y or z is Inf/NaN, or if fma will certainly overflow, + or if x * y is less than half of LDBL_DENORM_MIN, + compute as x * y + z. */ + if (u.ieee.exponent == 0x7fff + || v.ieee.exponent == 0x7fff + || w.ieee.exponent == 0x7fff + || u.ieee.exponent + v.ieee.exponent + > 0x7fff + IEEE854_LONG_DOUBLE_BIAS + || u.ieee.exponent + v.ieee.exponent + < IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG - 2) + return x * y + z; + 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, it doesn't matter if we don't adjust it. */ + 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; + else + v.ieee.exponent += 2 * LDBL_MANT_DIG; + if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 4) + { + if (w.ieee.exponent) + w.ieee.exponent += 2 * LDBL_MANT_DIG; + else + w.d *= 0x1p128L; + adjust = -1; + } + /* Otherwise x * y should just affect inexact + and nothing else. */ + } + x = u.d; + y = v.d; + z = w.d; + } + /* 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; + + fenv_t env; + feholdexcept (&env); + fesetround (FE_TOWARDZERO); + /* Perform m2 + a2 addition with round to odd. */ + u.d = a2 + m2; + + if (__builtin_expect (adjust == 0, 1)) + { + 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 (__builtin_expect (adjust > 0, 1)) + { + 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; + 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-128L; + /* If result rounded to zero is not subnormal, no double + rounding will occur. */ + if (v.ieee.exponent > 128) + return (a1 + u.d) * 0x1p-128L; + /* If v.d * 0x1p-128L with round to zero is a subnormal above + or equal to LDBL_MIN / 2, then v.d * 0x1p-128L 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-128L never normalizes by shifting up, + so round bit plus sticky bit should be already enough + for proper rounding. */ + if (v.ieee.exponent == 128) + { + /* 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. In round-to-nearest 001 rounds down like 00, + 011 rounds up, even though 01 rounds down (thus we need + to adjust), 101 rounds down like 10 and 111 rounds up + like 11. */ + if ((v.ieee.mantissa1 & 3) == 1) + { + v.d *= 0x1p-128L; + if (v.ieee.negative) + return v.d - 0x1p-16445L /* __LDBL_DENORM_MIN__ */; + else + return v.d + 0x1p-16445L /* __LDBL_DENORM_MIN__ */; + } + else + return v.d * 0x1p-128L; + } + v.ieee.mantissa1 |= j; + return v.d * 0x1p-128L; + } +} +weak_alias (__fmal, fmal) |