mirror of git://sourceware.org/git/glibc.git
math: Use asinh from CORE-MATH
The current implementation precision shows the following accuracy, on
tthree different ranges ([-DBL_MAX, -10], [-10,10], and [10, DBL_MAX))
with 10e9 uniform randomly generated numbers for each range (first
column is the accuracy in ULP, with '0' being correctly rounded, second
is the number of samples with the corresponding precision):
* range [-DBL_MAX, -10]
* FE_TONEAREST
0: 5164019099 51.64%
1: 4835980901 48.36%
* FE_UPWARD
1: 4836053540 48.36%
2: 5163946460 51.64%
* FE_DOWNWARD
1: 5163926134 51.64%
2: 4836073866 48.36%
* FE_TOWARDZERO
0: 5163937001 51.64%
1: 4836062999 48.36%
* Range [-10, 10)
* FE_TONEAREST
0: 8679029381 86.79%
1: 1320934581 13.21%
2: 36038 0.00%
* FE_UPWARD
0: 3965704277 39.66%
1: 4993616710 49.94%
2: 1039680225 10.40%
3: 998788 0.01%
* FE_DOWNWARD
0: 3965806523 39.66%
1: 4993534438 49.94%
2: 1039601726 10.40%
3: 1057313 0.01%
* FE_TOWARDZEROA
0: 7734210130 77.34%
1: 2261868439 22.62%
2: 3921431 0.04%
* Range [10, DBL_MAX)
* FE_TONEAREST
0: 5163973212 51.64%
1: 4836026788 48.36%
* FE_UPWARD
0: 4835991071 48.36%
1: 5164008929 51.64%
* FE_DOWNWARD
0: 5163983594 51.64%
1: 4836016406 48.36%
* FE_TOWARDZERO
0: 5163993394 51.64%
1: 4836006606 48.36%
The CORE-MATH implementation is correctly rounded for any rounding mode.
The code was adapted to glibc style and to use the definition of
math_config.h (to handle errno, overflow, and underflow).
Benchtest on x64_64 (Ryzen 9 5900X, gcc 14.2.1), aarch64 (Neoverse-N1,
gcc 13.3.1), and powerpc (POWER10, gcc 13.2.1) shows:
reciprocal-throughput master patched improvement
x86_64 26.5178 45.3754 -71.11%
x86_64v2 26.3167 44.7870 -70.18%
x86_64v3 25.9109 25.4887 1.63%
aarch64 18.0555 17.3374 3.98%
power10 19.8535 20.5586 -3.55%
Latency master patched improvement
x86_64 82.6755 91.2026 -10.31%
x86_64v2 82.4581 90.7152 -10.01%
x86_64v3 80.7000 71.9454 10.85%
aarch64 32.8320 28.8565 12.11%
power10 44.5309 37.0096 16.89%
For x86_64/x86_64-v2, most performance hit came from the fma call
through the ifunc mechanism.
Checked on x86_64-linux-gnu, aarch64-linux-gnu, and
powerpc64le-linux-gnu.
Reviewed-by: DJ Delorie <dj@redhat.com>
This commit is contained in:
Adhemerval Zanella
parent
d1509f2ce3
commit
30e66b085c
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@ -241,6 +241,8 @@ tzdata:
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core-math:
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core-math:
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# src/binary64/acosh/acosh.c, revision 69062c4d
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# src/binary64/acosh/acosh.c, revision 69062c4d
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sysdeps/ieee754/dbl-64/e_acosh.c
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sysdeps/ieee754/dbl-64/e_acosh.c
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# src/binary64/asinh/asinh.c, revision fde815f8
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sysdeps/ieee754/dbl-64/s_asinh.c
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# src/binary32/acos/acosf.c, revision 56dd347
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# src/binary32/acos/acosf.c, revision 56dd347
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sysdeps/ieee754/flt-32/e_acosf.c
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sysdeps/ieee754/flt-32/e_acosf.c
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# src/binary32/acosh/acoshf.c, revision d0b9ddd
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# src/binary32/acosh/acoshf.c, revision d0b9ddd
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@ -23,3 +23,16 @@ float: 1
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Function: "cbrt_upward":
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Function: "cbrt_upward":
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float: 1
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float: 1
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# sysdeps/i386/fpu/s_asinh.S is not correctly rounded
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Function: "asinh":
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double: 1
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Function: "asinh_downward":
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double: 1
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Function: "asinh_towardzero":
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double: 1
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Function: "asinh_upward":
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double: 1
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@ -10,3 +10,15 @@ double: 0
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Function: "acosh_upward":
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Function: "acosh_upward":
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double: 0
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double: 0
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Function: "asinh":
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double: 0
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Function: "asinh_downward":
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double: 0
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Function: "asinh_towardzero":
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double: 0
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Function: "asinh_upward":
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double: 0
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@ -172,6 +172,8 @@ attribute_hidden double __math_edom (double x);
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attribute_hidden double __math_check_oflow (double);
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attribute_hidden double __math_check_oflow (double);
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/* Check if the result underflowed to 0. */
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/* Check if the result underflowed to 0. */
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attribute_hidden double __math_check_uflow (double);
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attribute_hidden double __math_check_uflow (double);
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/* Check if the |X| if less than Y. */
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attribute_hidden double __math_check_uflow_lt (double, double);
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/* Check if the result overflowed to infinity. */
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/* Check if the result overflowed to infinity. */
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static inline double
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static inline double
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@ -79,6 +79,12 @@ __math_may_uflow (uint32_t sign)
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}
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}
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#endif
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#endif
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attribute_hidden double
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__math_always_uflow (double x)
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{
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return with_errno (x, ERANGE);
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}
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attribute_hidden double
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attribute_hidden double
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__math_oflow (uint32_t sign)
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__math_oflow (uint32_t sign)
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{
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{
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@ -123,6 +129,13 @@ __math_check_uflow (double y)
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return y == 0.0 ? with_errno (y, ERANGE) : y;
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return y == 0.0 ? with_errno (y, ERANGE) : y;
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}
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}
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attribute_hidden double
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__math_check_uflow_lt (double x, double y)
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{
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return fabs (x) < y ? with_errno (x, ERANGE) : x;
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}
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attribute_hidden double
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attribute_hidden double
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__math_check_oflow (double y)
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__math_check_oflow (double y)
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{
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{
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@ -1,70 +1,583 @@
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/* @(#)s_asinh.c 5.1 93/09/24 */
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/* Correctly-rounded inverse hyperbolic sine function. Binary64 version.
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/* asinh(x)
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Copyright (c) 2023-2025 Alexei Sibidanov.
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* Method :
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* Based on
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* asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
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* we have
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* asinh(x) := x if 1+x*x=1,
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* := sign(x)*(log(x)+ln2)) for large |x|, else
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* := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
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* := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
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*/
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#include <float.h>
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The original version of this file was copied from the CORE-MATH
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project (file src/binary64/asinh/asinh.c, revision fde815f8).
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Permission is hereby granted, free of charge, to any person obtaining a copy
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of this software and associated documentation files (the "Software"), to deal
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in the Software without restriction, including without limitation the rights
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to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
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copies of the Software, and to permit persons to whom the Software is
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furnished to do so, subject to the following conditions:
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The above copyright notice and this permission notice shall be included in all
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copies or substantial portions of the Software.
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THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
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OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
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SOFTWARE.
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*/
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#include <array_length.h>
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#include <stdint.h>
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#include <math.h>
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#include <math.h>
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#include <math_private.h>
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#include <math-underflow.h>
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#include <libm-alias-double.h>
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#include <libm-alias-double.h>
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#include "math_config.h"
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static const double
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static inline double
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one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
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fasttwosum (double x, double y, double *e)
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ln2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
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{
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huge = 1.00000000000000000000e+300;
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double s = x + y, z = s - x;
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*e = y - z;
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return s;
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}
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static inline double
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adddd (double xh, double xl, double ch, double cl, double *l)
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{
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double s = xh + ch, d = s - xh;
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*l = ((ch - d) + (xh + (d - s))) + (xl + cl);
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return s;
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}
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static inline double
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muldd (double xh, double xl, double ch, double cl, double *l)
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{
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double ahlh = ch * xl, alhh = cl * xh, ahhh = ch * xh,
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ahhl = fma (ch, xh, -ahhh);
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ahhl += alhh + ahlh;
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ch = ahhh + ahhl;
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*l = (ahhh - ch) + ahhl;
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return ch;
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}
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static inline double
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mulddd (double xh, double xl, double ch, double *l)
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{
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double ahlh = ch * xl, ahhh = ch * xh, ahhl = fma (ch, xh, -ahhh);
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ahhl += ahlh;
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ch = ahhh + ahhl;
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*l = (ahhh - ch) + ahhl;
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return ch;
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}
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static inline double
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polydd (double xh, double xl, int n, const double c[][2], double *l)
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{
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int i = n - 1;
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double ch = c[i][0] + *l, cl = ((c[i][0] - ch) + *l) + c[i][1];
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while (--i >= 0)
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{
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ch = muldd (xh, xl, ch, cl, &cl);
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double th = ch + c[i][0], tl = (c[i][0] - th) + ch;
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ch = th;
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cl += tl + c[i][1];
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}
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*l = cl;
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return ch;
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}
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static double __attribute__ ((noinline)) as_asinh_refine (double, double,
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double, double);
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static double __attribute__ ((noinline))
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as_asinh_zero (double x, double x2h, double x2l)
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{
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static const double ch[][2]
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= { { -0x1.5555555555555p-3, -0x1.5555555555555p-57 },
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{ 0x1.3333333333333p-4, 0x1.99999999949dfp-59 },
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{ -0x1.6db6db6db6db7p-5, 0x1.2492496091b0cp-60 },
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{ 0x1.f1c71c71c71c7p-6, 0x1.c71a35cfa0671p-62 },
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{ -0x1.6e8ba2e8ba2e9p-6, 0x1.17f937248cf81p-60 },
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{ 0x1.1c4ec4ec4ec4fp-6, -0x1.74e3c1dfd4c3dp-60 },
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{ -0x1.c999999999977p-7, -0x1.38e7a467ecc55p-61 },
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{ 0x1.7a87878786c7ep-7, 0x1.a83c7bace55ebp-61 },
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{ -0x1.3fde50d764083p-7, -0x1.d024df7fa0542p-61 },
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{ 0x1.12ef3ceae4d12p-7, -0x1.ba9c13deb261fp-61 },
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{ -0x1.df3bd104aa267p-8, -0x1.546da9bc5b32ap-62 },
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{ 0x1.a685fc5de7a04p-8, 0x1.40d284a1d67f9p-62 } };
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static const double cl[]
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= { -0x1.7828d553ec8p-8, 0x1.51712f7bee368p-8, -0x1.2e6d98527bcc6p-8,
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0x1.0095da47b392cp-8, -0x1.3b92d6368192cp-9 };
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double y2
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= x2h
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* (cl[0]
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+ x2h * (cl[1] + x2h * (cl[2] + x2h * (cl[3] + x2h * (cl[4])))));
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double y1 = polydd (x2h, x2l, 12, ch, &y2);
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y1 = muldd (y1, y2, x2h, x2l, &y2);
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y1 = mulddd (y1, y2, x, &y2);
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double y0 = fasttwosum (x, y1, &y1);
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y1 = fasttwosum (y1, y2, &y2);
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uint64_t t = asuint64 (y1);
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if (__glibc_unlikely (!(t & (~UINT64_C (0) >> 12))))
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{
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uint64_t w = asuint64 (y2);
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if ((w ^ t) >> 63)
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t--;
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else
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t++;
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y1 = asdouble (t);
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}
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return y0 + y1;
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}
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static const struct
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{
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uint16_t c0;
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short c1;
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} B[] = {
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{ 301, 27565 }, { 7189, 24786 }, { 13383, 22167 }, { 18923, 19696 },
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{ 23845, 17361 }, { 28184, 15150 }, { 31969, 13054 }, { 35231, 11064 },
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{ 37996, 9173 }, { 40288, 7372 }, { 42129, 5657 }, { 43542, 4020 },
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{ 44546, 2457 }, { 45160, 962 }, { 45399, -468 }, { 45281, -1838 },
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{ 44821, -3151 }, { 44032, -4412 }, { 42929, -5622 }, { 41522, -6786 },
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{ 39825, -7905 }, { 37848, -8982 }, { 35602, -10020 }, { 33097, -11020 },
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{ 30341, -11985 }, { 27345, -12916 }, { 24115, -13816 }, { 20661, -14685 },
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{ 16989, -15526 }, { 13107, -16339 }, { 9022, -17126 }, { 4740, -17889 }
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};
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static const double r1[]
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= { 0x1p+0, 0x1.f5076p-1, 0x1.ea4bp-1, 0x1.dfc98p-1, 0x1.d5818p-1,
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0x1.cb72p-1, 0x1.c199cp-1, 0x1.b7f76p-1, 0x1.ae8ap-1, 0x1.a5504p-1,
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0x1.9c492p-1, 0x1.93738p-1, 0x1.8ace6p-1, 0x1.8258ap-1, 0x1.7a114p-1,
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0x1.71f76p-1, 0x1.6a09ep-1, 0x1.6247ep-1, 0x1.5ab08p-1, 0x1.5342cp-1,
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0x1.4bfdap-1, 0x1.44e08p-1, 0x1.3dea6p-1, 0x1.371a8p-1, 0x1.306fep-1,
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0x1.29e9ep-1, 0x1.2387ap-1, 0x1.1d488p-1, 0x1.172b8p-1, 0x1.11302p-1,
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0x1.0b558p-1, 0x1.059bp-1, 0x1p-1 };
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static const double r2[]
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= { 0x1p+0, 0x1.ffa74p-1, 0x1.ff4eap-1, 0x1.fef62p-1, 0x1.fe9dap-1,
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0x1.fe452p-1, 0x1.fdeccp-1, 0x1.fd946p-1, 0x1.fd3c2p-1, 0x1.fce3ep-1,
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0x1.fc8bcp-1, 0x1.fc33ap-1, 0x1.fbdbap-1, 0x1.fb83ap-1, 0x1.fb2bcp-1,
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0x1.fad3ep-1, 0x1.fa7c2p-1, 0x1.fa246p-1, 0x1.f9ccap-1, 0x1.f975p-1,
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0x1.f91d8p-1, 0x1.f8c6p-1, 0x1.f86e8p-1, 0x1.f8172p-1, 0x1.f7bfep-1,
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||||||
|
0x1.f768ap-1, 0x1.f7116p-1, 0x1.f6ba4p-1, 0x1.f6632p-1, 0x1.f60c2p-1,
|
||||||
|
0x1.f5b52p-1, 0x1.f55e4p-1, 0x1.f5076p-1 };
|
||||||
|
static const double l1[][2] = { { 0x0p+0, 0x0p+0 },
|
||||||
|
{ -0x1.269e2038315b3p-46, 0x1.62e4eacd4p-6 },
|
||||||
|
{ -0x1.3f2558bddfc47p-45, 0x1.62e3ce7218p-5 },
|
||||||
|
{ 0x1.07ea13c34efb5p-45, 0x1.0a2ab6d3ecp-4 },
|
||||||
|
{ 0x1.8f3e77084d3bap-44, 0x1.62e4a86d8cp-4 },
|
||||||
|
{ -0x1.8d92a005f1a7ep-46, 0x1.bb9db7062cp-4 },
|
||||||
|
{ 0x1.58239e799bfe5p-44, 0x1.0a2b1a22ccp-3 },
|
||||||
|
{ -0x1.a93fcf5f593b7p-44, 0x1.3687f0a298p-3 },
|
||||||
|
{ -0x1.db4cac32fd2b5p-46, 0x1.62e4116b64p-3 },
|
||||||
|
{ -0x1.0e65a92ee0f3bp-46, 0x1.8f409e4df6p-3 },
|
||||||
|
{ -0x1.8261383d475f1p-44, 0x1.bb9d15001cp-3 },
|
||||||
|
{ -0x1.359886207513bp-44, 0x1.e7f9a8c94p-3 },
|
||||||
|
{ 0x1.811f87496ceb7p-44, 0x1.0a2b052ddbp-2 },
|
||||||
|
{ 0x1.4991ec6cb435cp-44, 0x1.205955ef73p-2 },
|
||||||
|
{ -0x1.4581abfeb8927p-44, 0x1.3687bd9121p-2 },
|
||||||
|
{ 0x1.cab48f6942703p-44, 0x1.4cb5e8f2b5p-2 },
|
||||||
|
{ -0x1.df2c452fde132p-47, 0x1.62e4420e2p-2 },
|
||||||
|
{ 0x1.6109f4fdb74bdp-45, 0x1.791292c46ap-2 },
|
||||||
|
{ -0x1.6b95fbdac7696p-44, 0x1.8f40af84e7p-2 },
|
||||||
|
{ 0x1.7394fa880cbdap-46, 0x1.a56ed8f865p-2 },
|
||||||
|
{ -0x1.50b06a94eccabp-46, 0x1.bb9d6505b4p-2 },
|
||||||
|
{ -0x1.be2abf0b38989p-44, 0x1.d1cb91e728p-2 },
|
||||||
|
{ -0x1.7d6bf1e34da04p-44, 0x1.e7f9d139e2p-2 },
|
||||||
|
{ -0x1.423c1e14de6edp-44, 0x1.fe27db9b0ep-2 },
|
||||||
|
{ 0x1.c46f1a0efbbc2p-44, 0x1.0a2b25060a8p-1 },
|
||||||
|
{ 0x1.834fe4e3e6018p-45, 0x1.154244482ap-1 },
|
||||||
|
{ 0x1.6a03d0f02b65p-46, 0x1.20597312988p-1 },
|
||||||
|
{ 0x1.d437056526f3p-44, 0x1.2b707145dep-1 },
|
||||||
|
{ -0x1.a0233728405c5p-45, 0x1.3687b0e0b28p-1 },
|
||||||
|
{ -0x1.4dbdda10d2bf1p-45, 0x1.419ec5d3f68p-1 },
|
||||||
|
{ 0x1.f7d0a25d154f2p-44, 0x1.4cb5f9fc02p-1 },
|
||||||
|
{ 0x1.15ede4d803b18p-44, 0x1.57cd28421a8p-1 },
|
||||||
|
{ 0x1.ef35793c7673p-45, 0x1.62e42fefa38p-1 } };
|
||||||
|
static const double l2[][2] = { { 0x0p+0, 0x0p+0 },
|
||||||
|
{ 0x1.5abdac3638e99p-44, 0x1.631ec81ep-11 },
|
||||||
|
{ -0x1.16b8be9bbe239p-45, 0x1.62fd8127p-10 },
|
||||||
|
{ -0x1.364c6315542ebp-44, 0x1.0a2520508p-9 },
|
||||||
|
{ 0x1.734abe459c9p-45, 0x1.62dadc1dp-9 },
|
||||||
|
{ 0x1.0cf8a761431bfp-44, 0x1.bb9ff94dp-9 },
|
||||||
|
{ 0x1.da2718eb78708p-45, 0x1.0a2a2def8p-8 },
|
||||||
|
{ 0x1.34ada62c59b93p-44, 0x1.368c0fae4p-8 },
|
||||||
|
{ 0x1.d09ab376682d4p-44, 0x1.62e58e4f8p-8 },
|
||||||
|
{ -0x1.3cb7b94329211p-45, 0x1.8f46bd28cp-8 },
|
||||||
|
{ -0x1.eec5c297c41dp-45, 0x1.bb9f8312p-8 },
|
||||||
|
{ -0x1.6411b9395d15p-44, 0x1.e7fff8f3p-8 },
|
||||||
|
{ -0x1.1c0e59a43053cp-44, 0x1.0a2c0006ep-7 },
|
||||||
|
{ 0x1.6506596e077b6p-46, 0x1.205bdb6fp-7 },
|
||||||
|
{ 0x1.e256bce6faa27p-44, 0x1.36877c86ep-7 },
|
||||||
|
{ 0x1.bd42467b0c8d1p-51, 0x1.4cb6f5578p-7 },
|
||||||
|
{ -0x1.c4f92132ff0fp-44, 0x1.62e230e8cp-7 },
|
||||||
|
{ -0x1.80be08bfab39p-44, 0x1.7911440f6p-7 },
|
||||||
|
{ -0x1.f0b1319ceb1f7p-44, 0x1.8f443020ap-7 },
|
||||||
|
{ 0x1.a65fcfb8de99bp-45, 0x1.a572dbef4p-7 },
|
||||||
|
{ 0x1.4233885d3779cp-46, 0x1.bb9d449a6p-7 },
|
||||||
|
{ 0x1.f46a59e646edbp-44, 0x1.d1cb8491cp-7 },
|
||||||
|
{ -0x1.c3d2f11c11446p-44, 0x1.e7fd9d2aap-7 },
|
||||||
|
{ 0x1.7763f78a1e0ccp-45, 0x1.fe2b6f978p-7 },
|
||||||
|
{ 0x1.b4c37fc60c043p-44, 0x1.0a2a7c7a5p-6 },
|
||||||
|
{ -0x1.5b8a822859be3p-46, 0x1.15412ca86p-6 },
|
||||||
|
{ -0x1.f2d8c9fc064p-44, 0x1.2059c9005p-6 },
|
||||||
|
{ -0x1.e80e79c20378dp-44, 0x1.2b703f49bp-6 },
|
||||||
|
{ 0x1.68256e4329bdbp-44, 0x1.3688a1a8dp-6 },
|
||||||
|
{ 0x1.7e9741da248c3p-44, 0x1.419edc7bap-6 },
|
||||||
|
{ 0x1.e330dccce602bp-45, 0x1.4cb7034fap-6 },
|
||||||
|
{ 0x1.2f32b5d18eefbp-49, 0x1.57cd01187p-6 },
|
||||||
|
{ -0x1.269e2038315b3p-46, 0x1.62e4eacd4p-6 } };
|
||||||
|
static const double c[] = { -0x1p-1, 0x1.555555555553p-2, -0x1.fffffffffffap-3,
|
||||||
|
0x1.99999e33a6366p-3, -0x1.555559ef9525fp-3 };
|
||||||
|
|
||||||
double
|
double
|
||||||
__asinh (double x)
|
__asinh (double x)
|
||||||
{
|
{
|
||||||
double w;
|
double ax = fabs (x);
|
||||||
int32_t hx, ix;
|
uint64_t u = asuint64 (ax);;
|
||||||
GET_HIGH_WORD (hx, x);
|
if (__glibc_unlikely (u < UINT64_C (0x3fbb000000000000)))
|
||||||
ix = hx & 0x7fffffff;
|
{ // |x| < 0x1.bp-4
|
||||||
if (__glibc_unlikely (ix < 0x3e300000)) /* |x|<2**-28 */
|
// for |x| < 0x1.7137449123ef7p-26, asinh(x) rounds to x to nearest
|
||||||
{
|
// for |x| < 0x1p-1022 we have underflow but not for 0x1p-1022 (to
|
||||||
math_check_force_underflow (x);
|
// nearest)
|
||||||
if (huge + x > one)
|
if (__glibc_unlikely (u < UINT64_C (0x3e57137449123ef7)))
|
||||||
return x; /* return x inexact except 0 */
|
{ // |x| < 0x1.7137449123ef7p-26
|
||||||
|
if (__glibc_unlikely (!u))
|
||||||
|
return x;
|
||||||
|
double res = fma (-0x1p-60, x, x);
|
||||||
|
return __math_check_uflow_lt (res, 0x1p-1022);
|
||||||
|
}
|
||||||
|
double x2h = x * x, x2l = fma (x, x, -x2h);
|
||||||
|
double x3h = x2h * x, sl;
|
||||||
|
if (__glibc_unlikely (u < UINT64_C (0x3f93000000000000)))
|
||||||
|
{ // |x| < 0x1.3p-6
|
||||||
|
if (__glibc_unlikely (u < UINT64_C (0x3f30000000000000)))
|
||||||
|
{ // |x| < 0x1p-12
|
||||||
|
if (__glibc_unlikely (u < UINT64_C (0x3e5a000000000000)))
|
||||||
|
{ // |x| < 0x1.ap-26
|
||||||
|
static const double cl[] = { -0x1.5555555555555p-3 };
|
||||||
|
sl = x3h * cl[0];
|
||||||
|
}
|
||||||
|
else
|
||||||
|
{
|
||||||
|
static const double cl[]
|
||||||
|
= { -0x1.5555555555555p-3, 0x1.3333327c57c6p-4 };
|
||||||
|
sl = x3h * (cl[0] + x2h * cl[1]);
|
||||||
|
}
|
||||||
|
}
|
||||||
|
else
|
||||||
|
{
|
||||||
|
static const double cl[]
|
||||||
|
= { -0x1.5555555555555p-3, 0x1.333333332f2ffp-4,
|
||||||
|
-0x1.6db6d9a665159p-5, 0x1.f186866d775fp-6 };
|
||||||
|
sl = x3h * (cl[0] + x2h * (cl[1] + x2h * (cl[2] + x2h * cl[3])));
|
||||||
|
}
|
||||||
|
}
|
||||||
|
else
|
||||||
|
{
|
||||||
|
static const double cl[]
|
||||||
|
= { -0x1.5555555555555p-3, 0x1.333333333331p-4,
|
||||||
|
-0x1.6db6db6da466cp-5, 0x1.f1c71c2ea7be4p-6,
|
||||||
|
-0x1.6e8b651b09d72p-6, 0x1.1c309fc0e69c2p-6,
|
||||||
|
-0x1.bab7833c1ep-7 };
|
||||||
|
double c1 = cl[1] + x2h * cl[2];
|
||||||
|
double c3 = cl[3] + x2h * cl[4];
|
||||||
|
double c5 = cl[5] + x2h * cl[6];
|
||||||
|
double x4 = x2h * x2h;
|
||||||
|
sl = x3h * (cl[0] + x2h * (c1 + x4 * (c3 + x4 * c5)));
|
||||||
|
}
|
||||||
|
double eps = 0x1.6p-53 * x3h;
|
||||||
|
double lb = x + (sl - eps), ub = x + (sl + eps);
|
||||||
|
if (lb == ub)
|
||||||
|
return lb;
|
||||||
|
return as_asinh_zero (x, x2h, x2l);
|
||||||
}
|
}
|
||||||
if (__glibc_unlikely (ix > 0x41b00000)) /* |x| > 2**28 */
|
// |x| >= 0x1.bp-4
|
||||||
|
double x2h = 0, x2l = 0;
|
||||||
|
double ah, al;
|
||||||
|
int off = 0x3ff;
|
||||||
|
if (__glibc_likely (u < UINT64_C (0x4190000000000000)))
|
||||||
|
{ // x < 0x1p+26
|
||||||
|
double th, tl;
|
||||||
|
x2h = x * x;
|
||||||
|
x2l = fma (x, x, -x2h);
|
||||||
|
if (__glibc_unlikely (u < UINT64_C (0x3ff0000000000000)))
|
||||||
|
th = fasttwosum (1, x2h, &tl);
|
||||||
|
else
|
||||||
|
th = fasttwosum (x2h, 1, &tl);
|
||||||
|
tl += x2l;
|
||||||
|
ah = sqrt (th);
|
||||||
|
double rs = 0.5 / th;
|
||||||
|
al = (tl - fma (ah, ah, -th)) * (rs * ah);
|
||||||
|
ah = fasttwosum (ah, ax, &tl);
|
||||||
|
al += tl;
|
||||||
|
}
|
||||||
|
else if (u < UINT64_C (0x4330000000000000))
|
||||||
{
|
{
|
||||||
if (ix >= 0x7ff00000)
|
ah = 2 * ax;
|
||||||
return x + x; /* x is inf or NaN */
|
al = 0.5 / ax;
|
||||||
w = __ieee754_log (fabs (x)) + ln2;
|
|
||||||
}
|
}
|
||||||
else
|
else
|
||||||
{
|
{
|
||||||
double xa = fabs (x);
|
if (__glibc_unlikely (u >= UINT64_C (0x7ff0000000000000)))
|
||||||
if (ix > 0x40000000) /* 2**28 > |x| > 2.0 */
|
return x + x; // +-inf or nan
|
||||||
{
|
off = 0x3fe;
|
||||||
w = __ieee754_log (2.0 * xa + one / (sqrt (xa * xa + one) +
|
ah = ax;
|
||||||
xa));
|
al = 0;
|
||||||
}
|
|
||||||
else /* 2.0 > |x| > 2**-28 */
|
|
||||||
{
|
|
||||||
double t = xa * xa;
|
|
||||||
w = __log1p (xa + t / (one + sqrt (one + t)));
|
|
||||||
}
|
|
||||||
}
|
}
|
||||||
return copysign (w, x);
|
|
||||||
|
uint64_t t = asuint64 (ah);
|
||||||
|
int ex = t >> 52, e = ex - off;
|
||||||
|
t &= ~UINT64_C (0) >> 12;
|
||||||
|
double ed = e;
|
||||||
|
uint64_t i = t >> (52 - 5);
|
||||||
|
int64_t d = t & (~UINT64_C (0) >> 17);
|
||||||
|
uint64_t j
|
||||||
|
= (t + ((uint64_t) B[i].c0 << 33) + ((int64_t) B[i].c1 * (d >> 16)))
|
||||||
|
>> (52 - 10);
|
||||||
|
t |= UINT64_C (0x3ff) << 52;
|
||||||
|
int i1 = j >> 5, i2 = j & 0x1f;
|
||||||
|
double r = r1[i1] * r2[i2], dx = fma (r, asdouble (t), -1), dx2 = dx * dx;
|
||||||
|
double f
|
||||||
|
= dx2 * ((c[0] + dx * c[1]) + dx2 * ((c[2] + dx * c[3]) + dx2 * c[4]));
|
||||||
|
const double l2h = 0x1.62e42fefa38p-1, l2l = 0x1.ef35793c7673p-45;
|
||||||
|
double lh = l2h * ed + (l1[i1][1] + l2[i2][1]);
|
||||||
|
double ll = l2l * ed + l1[i1][0] + l2[i2][0] + al / ah + f;
|
||||||
|
ll += dx;
|
||||||
|
lh *= copysign (1, x);
|
||||||
|
ll *= copysign (1, x);
|
||||||
|
double eps = 1.63e-19;
|
||||||
|
double lb = lh + (ll - eps), ub = lh + (ll + eps);
|
||||||
|
if (lb == ub)
|
||||||
|
return lb;
|
||||||
|
if (ax < 0x1p-2)
|
||||||
|
return as_asinh_zero (x, x2h, x2l);
|
||||||
|
return as_asinh_refine (x, ah, al,
|
||||||
|
0x1.71547652b82fep+0 * fabs (lb));
|
||||||
}
|
}
|
||||||
libm_alias_double (__asinh, asinh)
|
libm_alias_double (__asinh, asinh)
|
||||||
|
|
||||||
|
static __attribute__ ((noinline)) double
|
||||||
|
as_asinh_database (double x, double f)
|
||||||
|
{
|
||||||
|
static const double db[][3] = {
|
||||||
|
{ 0x1.00f9476450863p-2, 0x1.fcb35067f343cp-3, 0x1p-57 },
|
||||||
|
{ 0x1.1f0a79315b287p-2, 0x1.1b68aae88febap-2, 0x1p-56 },
|
||||||
|
{ 0x1.2b9618ff7acb7p-2, 0x1.27781d9aa4e25p-2, -0x1p-56 },
|
||||||
|
{ 0x1.389ef683f3aa7p-2, 0x1.33f52db6df1afp-2, 0x1p-56 },
|
||||||
|
{ 0x1.3b07e0c779ddap-2, 0x1.364303e1ad8f6p-2, 0x1p-56 },
|
||||||
|
{ 0x1.48441df33b6d3p-2, 0x1.42e385800f0a4p-2, 0x1p-56 },
|
||||||
|
{ 0x1.687bd068c1c1ep-2, 0x1.616cc75d49226p-2, -0x1p-56 },
|
||||||
|
{ 0x1.8740c4453a056p-2, 0x1.7e4f2ad132a1dp-2, 0x1p-56 },
|
||||||
|
{ 0x1.891acda11167ep-2, 0x1.8009d924a3ffdp-2, 0x1p-56 },
|
||||||
|
{ 0x1.bafc3479fc9ccp-2, 0x1.ae3773250e7d2p-2, 0x1p-56 },
|
||||||
|
{ 0x1.c59869f17b483p-2, 0x1.b7efa91915c95p-2, 0x1p-56 },
|
||||||
|
{ 0x1.c8be879787986p-2, 0x1.bad0485e0fe0ap-2, -0x1p-56 },
|
||||||
|
{ 0x1.e73b46abb01e1p-2, 0x1.d68039861ab53p-2, 0x1p-56 },
|
||||||
|
{ 0x1.ed6236da268bp-2, 0x1.dc0cb8f638126p-2, 0x1p-56 },
|
||||||
|
{ 0x1.f399ebafc1951p-2, 0x1.e1a4f519fab77p-2, -0x1p-56 },
|
||||||
|
{ 0x1.f70975ab0d471p-2, 0x1.e4bae8bcd6ea6p-2, 0x1p-56 },
|
||||||
|
{ 0x1.fbdd4a37760b7p-2, 0x1.e90f16eb88c09p-2, 0x1p-56 },
|
||||||
|
{ 0x1.fee72efb4bfddp-2, 0x1.ebc791a88bed8p-2, 0x1p-56 },
|
||||||
|
{ 0x1.02339d6bdb741p-1, 0x1.f0b2264e34555p-2, 0x1p-56 },
|
||||||
|
{ 0x1.09e7c831b1a23p-1, 0x1.fe694c3c89138p-2, 0x1p-56 },
|
||||||
|
{ 0x1.16d32c862fc3bp-1, 0x1.0a9c9334066dbp-1, -0x1p-55 },
|
||||||
|
{ 0x1.857954132083dp-1, 0x1.67425fe575c88p-1, -0x1p-55 },
|
||||||
|
{ 0x1.8a5c3b60f7e11p-1, 0x1.6b23ad4415a17p-1, -0x1p-55 },
|
||||||
|
{ 0x1.9740eb419dd04p-1, 0x1.754ab7535d47dp-1, 0x1p-55 },
|
||||||
|
{ 0x1.a16d9cc06011ap-1, 0x1.7d3755d851062p-1, -0x1p-55 },
|
||||||
|
{ 0x1.bb635be2213d1p-1, 0x1.91167cae3cfa9p-1, 0x1p-55 },
|
||||||
|
{ 0x1.d4b21ebf542fp-1, 0x1.a3fc7e4dd47d1p-1, -0x1p-55 },
|
||||||
|
{ 0x1.7b8516ffd2406p+0, 0x1.2f5d3b178914ap+0, 0x1p-54 },
|
||||||
|
{ 0x1.9295b9116e2e2p+0, 0x1.3bffa8863976p+0, 0x1p-54 },
|
||||||
|
{ 0x1.fedc65e32714p+0, 0x1.710f91e844f9bp+0, 0x1p-54 },
|
||||||
|
{ 0x1.57e377b3f0b4bp+1, 0x1.b6e2c73f41415p+0, 0x1p-54 },
|
||||||
|
{ 0x1.6056b06a21918p+3, 0x1.8c0a26d055288p+1, 0x1p-53 },
|
||||||
|
{ 0x1.843e1b5e5979cp+4, 0x1.f0f978201eb84p+1, 0x1p-53 },
|
||||||
|
{ 0x1.fee8f69c4cd25p+10, 0x1.0a19aebb51e9p+3, -0x1p-51 },
|
||||||
|
{ 0x1.0fbc6c02b1c9p+24, 0x1.16369cd53bb69p+4, 0x1p-50 },
|
||||||
|
};
|
||||||
|
double ax = fabs (x);
|
||||||
|
int a = 0, b = array_length (db) - 1, m = (a + b) / 2;
|
||||||
|
while (a <= b)
|
||||||
|
{ // binary search
|
||||||
|
if (db[m][0] < ax)
|
||||||
|
a = m + 1;
|
||||||
|
else if (db[m][0] == ax)
|
||||||
|
{
|
||||||
|
double sgn = copysign (1, x);
|
||||||
|
f = sgn * db[m][1] + sgn * db[m][2];
|
||||||
|
break;
|
||||||
|
}
|
||||||
|
else
|
||||||
|
b = m - 1;
|
||||||
|
m = (a + b) / 2;
|
||||||
|
}
|
||||||
|
return f;
|
||||||
|
}
|
||||||
|
|
||||||
|
static double
|
||||||
|
as_asinh_refine (double x, double zh, double zl, double a)
|
||||||
|
{
|
||||||
|
static const double t1[]
|
||||||
|
= { 0x1p+0, 0x1.ea4afap-1, 0x1.d5818ep-1, 0x1.c199bep-1,
|
||||||
|
0x1.ae89f98p-1, 0x1.9c4918p-1, 0x1.8ace54p-1, 0x1.7a1147p-1,
|
||||||
|
0x1.6a09e68p-1, 0x1.5ab07ep-1, 0x1.4bfdad8p-1, 0x1.3dea65p-1,
|
||||||
|
0x1.306fe08p-1, 0x1.2387a7p-1, 0x1.172b84p-1, 0x1.0b5587p-1,
|
||||||
|
0x1p-1 };
|
||||||
|
static const double t2[]
|
||||||
|
= { 0x1p+0, 0x1.fe9d968p-1, 0x1.fd3c228p-1, 0x1.fbdba38p-1,
|
||||||
|
0x1.fa7c18p-1, 0x1.f91d8p-1, 0x1.f7bfdbp-1, 0x1.f663278p-1,
|
||||||
|
0x1.f507658p-1, 0x1.f3ac948p-1, 0x1.f252b38p-1, 0x1.f0f9c2p-1,
|
||||||
|
0x1.efa1bfp-1, 0x1.ee4aaap-1, 0x1.ecf483p-1, 0x1.eb9f488p-1 };
|
||||||
|
static const double t3[]
|
||||||
|
= { 0x1p+0, 0x1.ffe9d2p-1, 0x1.ffd3a58p-1, 0x1.ffbd798p-1,
|
||||||
|
0x1.ffa74e8p-1, 0x1.ff91248p-1, 0x1.ff7afb8p-1, 0x1.ff64d38p-1,
|
||||||
|
0x1.ff4eac8p-1, 0x1.ff38868p-1, 0x1.ff22618p-1, 0x1.ff0c3dp-1,
|
||||||
|
0x1.fef61ap-1, 0x1.fedff78p-1, 0x1.fec9d68p-1, 0x1.feb3b6p-1 };
|
||||||
|
static const double t4[]
|
||||||
|
= { 0x1p+0, 0x1.fffe9dp-1, 0x1.fffd3ap-1, 0x1.fffbd78p-1,
|
||||||
|
0x1.fffa748p-1, 0x1.fff9118p-1, 0x1.fff7ae8p-1, 0x1.fff64cp-1,
|
||||||
|
0x1.fff4e9p-1, 0x1.fff386p-1, 0x1.fff2238p-1, 0x1.fff0c08p-1,
|
||||||
|
0x1.ffef5d8p-1, 0x1.ffedfa8p-1, 0x1.ffec98p-1, 0x1.ffeb35p-1 };
|
||||||
|
static const double LL[4][17][3] = {
|
||||||
|
{
|
||||||
|
{ 0x0p+0, 0x0p+0, 0x0p+0 },
|
||||||
|
{ 0x1.62e432b24p-6, -0x1.745af34bb54b8p-42, -0x1.17e3ec05cde7p-97 },
|
||||||
|
{ 0x1.62e42e4a8p-5, 0x1.111a4eadf312p-44, 0x1.cff3027abb119p-93 },
|
||||||
|
{ 0x1.0a2b233f1p-4, -0x1.88ac4ec78af8p-42, 0x1.4fa087ca75dfdp-93 },
|
||||||
|
{ 0x1.62e43056cp-4, 0x1.6bd65e8b0b7p-46, -0x1.b18e160362c24p-95 },
|
||||||
|
{ 0x1.bb9d3cbd6p-4, 0x1.de14aa55ec2bp-42, -0x1.c6ac3f1862a6bp-94 },
|
||||||
|
{ 0x1.0a2b244dap-3, 0x1.94def487fea7p-42, -0x1.dead1a4581acfp-94 },
|
||||||
|
{ 0x1.3687aa9b78p-3, 0x1.9cec9a50db22p-43, 0x1.34a70684f8e0ep-93 },
|
||||||
|
{ 0x1.62e42fabap-3, -0x1.d69047a3aebp-44, -0x1.4e061f79144e2p-95 },
|
||||||
|
{ 0x1.8f40b56d28p-3, 0x1.de7d755fd2e2p-42, 0x1.bdc7ecf001489p-94 },
|
||||||
|
{ 0x1.bb9d3b61fp-3, 0x1.c14f1445b12p-46, 0x1.a1d78cbdc5b58p-93 },
|
||||||
|
{ 0x1.e7f9c11f08p-3, -0x1.6e3e0000dae7p-43, 0x1.6a4559fadde98p-94 },
|
||||||
|
{ 0x1.0a2b242ec4p-2, 0x1.bb7cf852a5fe8p-42, 0x1.a6aef11ee43bdp-93 },
|
||||||
|
{ 0x1.205966c764p-2, 0x1.ad3a5f214294p-45, 0x1.5cc344fa10652p-93 },
|
||||||
|
{ 0x1.3687a98aacp-2, 0x1.1623671842fp-45, -0x1.0b428fe1f9e43p-94 },
|
||||||
|
{ 0x1.4cb5ec93f4p-2, 0x1.3d50980ea513p-42, 0x1.67f0ea083b1c4p-93 },
|
||||||
|
{ 0x1.62e42fefa4p-2, -0x1.8432a1b0e264p-44, 0x1.803f2f6af40f3p-93 },
|
||||||
|
},
|
||||||
|
{
|
||||||
|
{ 0x0p+0, 0x0p+0, 0x0p+0 },
|
||||||
|
{ 0x1.62e462b4p-10, 0x1.061d003b97318p-42, 0x1.d7faee66a2e1ep-93 },
|
||||||
|
{ 0x1.62e44c92p-9, 0x1.95a7bff5e239p-42, -0x1.f7e788a87135p-95 },
|
||||||
|
{ 0x1.0a2b1e33p-8, 0x1.2a3a1a65aa3ap-43, -0x1.54599c9605442p-93 },
|
||||||
|
{ 0x1.62e4367cp-8, -0x1.4a995b6d9ddcp-45, -0x1.56bb79b254f33p-100 },
|
||||||
|
{ 0x1.bb9d449ap-8, 0x1.8a119c42e9bcp-42, -0x1.8ecf7d8d661f1p-93 },
|
||||||
|
{ 0x1.0a2b1f19p-7, 0x1.8863771bd10a8p-42, 0x1.e9731de7f0155p-94 },
|
||||||
|
{ 0x1.3687ad11p-7, 0x1.e026a347ca1c8p-42, 0x1.fadc62522444dp-97 },
|
||||||
|
{ 0x1.62e436f28p-7, 0x1.25b84f71b70b8p-42, -0x1.fcb3f98612d27p-96 },
|
||||||
|
{ 0x1.8f40b7b38p-7, -0x1.62a0a4fd4758p-43, 0x1.3cb3c35d9f6a1p-93 },
|
||||||
|
{ 0x1.bb9d3abbp-7, -0x1.0ec48f94d786p-42, -0x1.6b47d410e4cc7p-93 },
|
||||||
|
{ 0x1.e7f9bb23p-7, 0x1.e4415cbc97ap-43, -0x1.3729fdb677231p-93 },
|
||||||
|
{ 0x1.0a2b22478p-6, -0x1.cb73f4505b03p-42, -0x1.1b3b3a3bc370ap-93 },
|
||||||
|
{ 0x1.2059691e8p-6, -0x1.abcc3412f264p-43, -0x1.fe6e998e48673p-95 },
|
||||||
|
{ 0x1.3687a768p-6, -0x1.43901e5c97a9p-42, 0x1.b54cdd52a5d88p-96 },
|
||||||
|
{ 0x1.4cb5eb5d8p-6, -0x1.8f106f00f13b8p-42, -0x1.8f793f5fce148p-93 },
|
||||||
|
{ 0x1.62e432b24p-6, -0x1.745af34bb54b8p-42, -0x1.17e3ec05cde7p-97 },
|
||||||
|
},
|
||||||
|
{
|
||||||
|
{ 0x0p+0, 0x0p+0, 0x0p+0 },
|
||||||
|
{ 0x1.62e7bp-14, -0x1.868625640a68p-44, -0x1.34bf0db910f65p-93 },
|
||||||
|
{ 0x1.62e35f6p-13, -0x1.2ee3d96b696ap-43, 0x1.a2948cd558655p-94 },
|
||||||
|
{ 0x1.0a2b4b2p-12, 0x1.53edbcf1165p-47, -0x1.cfc26ccf6d0e4p-97 },
|
||||||
|
{ 0x1.62e4be1p-12, 0x1.783e334614p-52, -0x1.04b96da30e63ap-93 },
|
||||||
|
{ 0x1.bb9e085p-12, -0x1.60785f20acb2p-43, -0x1.f33369bf7dff1p-96 },
|
||||||
|
{ 0x1.0a2b94dp-11, 0x1.fd4b3a273353p-42, -0x1.685a35575eff1p-96 },
|
||||||
|
{ 0x1.368810f8p-11, 0x1.7ded26dc813p-47, -0x1.4c4d1abca79bfp-96 },
|
||||||
|
{ 0x1.62e47878p-11, 0x1.7d2bee9a1f63p-42, 0x1.860233b7ad13p-93 },
|
||||||
|
{ 0x1.8f40cb48p-11, -0x1.af034eaf471cp-42, 0x1.ae748822d57b7p-94 },
|
||||||
|
{ 0x1.bb9d094p-11, -0x1.7a223013a20fp-42, -0x1.1e499087075b6p-93 },
|
||||||
|
{ 0x1.e7fa32c8p-11, -0x1.b2e67b1b59bdp-43, -0x1.54a41eda30fa6p-93 },
|
||||||
|
{ 0x1.0a2b237p-10, -0x1.7ad97ff4ac7ap-44, 0x1.f932da91371ddp-93 },
|
||||||
|
{ 0x1.2059a338p-10, -0x1.96422d90df4p-44, -0x1.90800fbbf2ed3p-94 },
|
||||||
|
{ 0x1.36879824p-10, 0x1.0f9054001812p-44, 0x1.9567e01e48f9ap-93 },
|
||||||
|
{ 0x1.4cb602cp-10, -0x1.0d709a5ec0b5p-43, 0x1.253dfd44635d2p-94 },
|
||||||
|
{ 0x1.62e462b4p-10, 0x1.061d003b97318p-42, 0x1.d7faee66a2e1ep-93 },
|
||||||
|
},
|
||||||
|
{
|
||||||
|
{ 0x0p+0, 0x0p+0, 0x0p+0 },
|
||||||
|
{ 0x1.63007cp-18, -0x1.db0e38e5aaaap-43, 0x1.259a7b94815b9p-93 },
|
||||||
|
{ 0x1.6300f6p-17, 0x1.2b1c75580438p-44, 0x1.78cabba01e3e4p-93 },
|
||||||
|
{ 0x1.0a2115p-16, -0x1.5ff223730759p-42, 0x1.8074feacfe49dp-95 },
|
||||||
|
{ 0x1.62e1ecp-16, -0x1.85d6f6487ce4p-45, 0x1.05485074b9276p-93 },
|
||||||
|
{ 0x1.bba301p-16, -0x1.af5d58a7c921p-43, -0x1.30a8c0fd2ff5fp-93 },
|
||||||
|
{ 0x1.0a32298p-15, 0x1.590faa0883bdp-43, 0x1.95e9bda999947p-93 },
|
||||||
|
{ 0x1.3682f1p-15, 0x1.f0224376efaf8p-42, -0x1.5843c0db50d1p-93 },
|
||||||
|
{ 0x1.62e3d8p-15, -0x1.142c13daed4ap-43, 0x1.c68a61183ce87p-93 },
|
||||||
|
{ 0x1.8f44dd8p-15, -0x1.aa489f399931p-43, 0x1.11c5c376854eap-94 },
|
||||||
|
{ 0x1.bb9601p-15, 0x1.9904d8b6a3638p-42, 0x1.8c89554493c8fp-93 },
|
||||||
|
{ 0x1.e7f744p-15, 0x1.5785ddbe7cba8p-42, 0x1.e7ff3cde7d70cp-94 },
|
||||||
|
{ 0x1.0a2c53p-14, -0x1.6d9e8780d0d5p-43, 0x1.ad9c178106693p-94 },
|
||||||
|
{ 0x1.205d134p-14, -0x1.214a2e893fccp-43, 0x1.548a9500c9822p-93 },
|
||||||
|
{ 0x1.3685e28p-14, 0x1.e23588646103p-43, 0x1.2a97b26da2d88p-94 },
|
||||||
|
{ 0x1.4cb6c18p-14, 0x1.2b7cfcea9e0d8p-42, -0x1.5095048a6b824p-93 },
|
||||||
|
{ 0x1.62e7bp-14, -0x1.868625640a68p-44, -0x1.34bf0db910f65p-93 },
|
||||||
|
},
|
||||||
|
};
|
||||||
|
static const double ch[][2] = {
|
||||||
|
{ 0x1p-1, 0x1.24b67ee516e3bp-111 },
|
||||||
|
{ -0x1p-2, -0x1.932ce43199a8dp-110 },
|
||||||
|
{ 0x1.5555555555555p-3, 0x1.55540c15cf91fp-57 },
|
||||||
|
};
|
||||||
|
static const double cl[3]
|
||||||
|
= { -0x1p-3, 0x1.9999999a0754fp-4, -0x1.55555555c3157p-4 };
|
||||||
|
uint64_t t = asuint64 (zh);
|
||||||
|
|
||||||
|
int ex = t >> 52, e = ex - 0x3ff + (zl == 0.0);
|
||||||
|
t &= ~UINT64_C (0) >> 12;
|
||||||
|
t |= UINT64_C (0x3ff) << 52;
|
||||||
|
double ed = e;
|
||||||
|
uint64_t v = asuint64 (a - ed + 0x1.00008p+0);
|
||||||
|
uint64_t i = (v - (UINT64_C (0x3ff) << 52)) >> (52 - 16);
|
||||||
|
int i1 = (i >> 12) & 0x1f, i2 = (i >> 8) & 0xf, i3 = (i >> 4) & 0xf,
|
||||||
|
i4 = i & 0xf;
|
||||||
|
const double l20 = 0x1.62e42fefa38p-2, l21 = 0x1.ef35793c768p-46,
|
||||||
|
l22 = -0x1.9ff0342542fc3p-91;
|
||||||
|
double el2 = l22 * ed, el1 = l21 * ed, el0 = l20 * ed;
|
||||||
|
double L[3];
|
||||||
|
L[0] = LL[0][i1][0] + LL[1][i2][0] + (LL[2][i3][0] + LL[3][i4][0]);
|
||||||
|
L[1] = LL[0][i1][1] + LL[1][i2][1] + (LL[2][i3][1] + LL[3][i4][1]);
|
||||||
|
L[2] = LL[0][i1][2] + LL[1][i2][2] + (LL[2][i3][2] + LL[3][i4][2]);
|
||||||
|
L[0] += el0;
|
||||||
|
double t12 = t1[i1] * t2[i2], t34 = t3[i3] * t4[i4];
|
||||||
|
double th = t12 * t34, tl = fma (t12, t34, -th);
|
||||||
|
double tf = asdouble (t);
|
||||||
|
double dh = th * tf, dl = fma (th, tf, -dh);
|
||||||
|
double sh = tl * tf, sl = fma (tl, tf, -sh);
|
||||||
|
double xl, xh = fasttwosum (dh - 1, dl, &xl);
|
||||||
|
if (zl != 0.0)
|
||||||
|
{
|
||||||
|
t = asuint64 (zl);
|
||||||
|
t -= (int64_t) e << 52;
|
||||||
|
xl += th * asdouble (t);
|
||||||
|
}
|
||||||
|
xh = adddd (xh, xl, sh, sl, &xl);
|
||||||
|
sl = xh * (cl[0] + xh * (cl[1] + xh * cl[2]));
|
||||||
|
sh = polydd (xh, xl, 3, ch, &sl);
|
||||||
|
sh = muldd (xh, xl, sh, sl, &sl);
|
||||||
|
sh = adddd (sh, sl, el1, el2, &sl);
|
||||||
|
sh = adddd (sh, sl, L[1], L[2], &sl);
|
||||||
|
double v2, v0 = fasttwosum (L[0], sh, &v2);
|
||||||
|
double v1 = fasttwosum (v2, sl, &v2);
|
||||||
|
v0 *= copysign (2, x);
|
||||||
|
v1 *= copysign (2, x);
|
||||||
|
v2 *= copysign (2, x);
|
||||||
|
t = asuint64 (v1);
|
||||||
|
if (__glibc_unlikely (!(t & (~UINT64_C (0) >> 12))))
|
||||||
|
{
|
||||||
|
uint64_t w = asuint64 (v2);
|
||||||
|
if ((w ^ t) >> 63)
|
||||||
|
t--;
|
||||||
|
else
|
||||||
|
t++;
|
||||||
|
v1 = asdouble (t);
|
||||||
|
}
|
||||||
|
uint64_t t0 = asuint64 (v0);
|
||||||
|
uint64_t er = ((t + 33) & (~UINT64_C (0) >> 12)),
|
||||||
|
de = ((t0 >> 52) & 0x7ff) - ((t >> 52) & 0x7ff);
|
||||||
|
double res = v0 + v1;
|
||||||
|
if (__glibc_unlikely (de > 99 || er < 66))
|
||||||
|
return as_asinh_database (x, res);
|
||||||
|
return res;
|
||||||
|
}
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue