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Add kernel functions for 80-bit long doubles. Many thanks to Steve and
Bruce for putting lots of effort into these; getting them right isn't easy, and they went through many iterations. Submitted by: Steve Kargl <sgk@apl.washington.edu> with revisions from bde
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David Schultz
parent
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3 changed files with 264 additions and 0 deletions
78
lib/msun/ld80/k_cosl.c
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78
lib/msun/ld80/k_cosl.c
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/* From: @(#)k_cos.c 1.3 95/01/18 */
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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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* Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
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*
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* Developed at SunSoft, 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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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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/*
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* ld80 version of k_cos.c. See ../src/k_cos.c for most comments.
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*/
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#include "math_private.h"
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/*
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* Domain [-0.7854, 0.7854], range ~[-2.43e-23, 2.425e-23]:
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* |cos(x) - c(x)| < 2**-75.1
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*
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* The coefficients of c(x) were generated by a pari-gp script using
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* a Remez algorithm that searches for the best higher coefficients
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* after rounding leading coefficients to a specified precision.
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*
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* Simpler methods like Chebyshev or basic Remez barely suffice for
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* cos() in 64-bit precision, because we want the coefficient of x^2
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* to be precisely -0.5 so that multiplying by it is exact, and plain
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* rounding of the coefficients of a good polynomial approximation only
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* gives this up to about 64-bit precision. Plain rounding also gives
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* a mediocre approximation for the coefficient of x^4, but a rounding
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* error of 0.5 ulps for this coefficient would only contribute ~0.01
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* ulps to the final error, so this is unimportant. Rounding errors in
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* higher coefficients are even less important.
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*
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* In fact, coefficients above the x^4 one only need to have 53-bit
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* precision, and this is more efficient. We get this optimization
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* almost for free from the complications needed to search for the best
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* higher coefficients.
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*/
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static const double
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one = 1.0;
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#if defined(__amd64__) || defined(__i386__)
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/* Long double constants are slow on these arches, and broken on i386. */
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static const volatile double
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C1hi = 0.041666666666666664, /* 0x15555555555555.0p-57 */
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C1lo = 2.2598839032744733e-18; /* 0x14d80000000000.0p-111 */
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#define C1 ((long double)C1hi + C1lo)
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#else
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static const long double
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C1 = 0.0416666666666666666136L; /* 0xaaaaaaaaaaaaaa9b.0p-68 */
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#endif
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static const double
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C2 = -0.0013888888888888874, /* -0x16c16c16c16c10.0p-62 */
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C3 = 0.000024801587301571716, /* 0x1a01a01a018e22.0p-68 */
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C4 = -0.00000027557319215507120, /* -0x127e4fb7602f22.0p-74 */
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C5 = 0.0000000020876754400407278, /* 0x11eed8caaeccf1.0p-81 */
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C6 = -1.1470297442401303e-11, /* -0x19393412bd1529.0p-89 */
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C7 = 4.7383039476436467e-14; /* 0x1aac9d9af5c43e.0p-97 */
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long double
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__kernel_cosl(long double x, long double y)
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{
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long double hz,z,r,w;
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z = x*x;
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r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7))))));
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hz = 0.5*z;
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w = one-hz;
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return w + (((one-w)-hz) + (z*r-x*y));
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}
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62
lib/msun/ld80/k_sinl.c
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62
lib/msun/ld80/k_sinl.c
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/* From: @(#)k_sin.c 1.3 95/01/18 */
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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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* Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
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*
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* Developed at SunSoft, 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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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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/*
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* ld80 version of k_sin.c. See ../src/k_sin.c for most comments.
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*/
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#include "math_private.h"
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static const double
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half = 0.5;
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/*
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* Domain [-0.7854, 0.7854], range ~[-1.89e-22, 1.915e-22]
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* |sin(x)/x - s(x)| < 2**-72.1
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*
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* See ../ld80/k_cosl.c for more details about the polynomial.
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*/
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#if defined(__amd64__) || defined(__i386__)
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/* Long double constants are slow on these arches, and broken on i386. */
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static const volatile double
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S1hi = -0.16666666666666666, /* -0x15555555555555.0p-55 */
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S1lo = -9.2563760475949941e-18; /* -0x15580000000000.0p-109 */
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#define S1 ((long double)S1hi + S1lo)
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#else
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static const long double
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S1 = -0.166666666666666666671L; /* -0xaaaaaaaaaaaaaaab.0p-66 */
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#endif
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static const double
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S2 = 0.0083333333333333332, /* 0x11111111111111.0p-59 */
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S3 = -0.00019841269841269427, /* -0x1a01a01a019f81.0p-65 */
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S4 = 0.0000027557319223597490, /* 0x171de3a55560f7.0p-71 */
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S5 = -0.000000025052108218074604, /* -0x1ae64564f16cad.0p-78 */
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S6 = 1.6059006598854211e-10, /* 0x161242b90243b5.0p-85 */
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S7 = -7.6429779983024564e-13, /* -0x1ae42ebd1b2e00.0p-93 */
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S8 = 2.6174587166648325e-15; /* 0x179372ea0b3f64.0p-101 */
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long double
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__kernel_sinl(long double x, long double y, int iy)
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{
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long double z,r,v;
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z = x*x;
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v = z*x;
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r = S2+z*(S3+z*(S4+z*(S5+z*(S6+z*(S7+z*S8)))));
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if(iy==0) return x+v*(S1+z*r);
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else return x-((z*(half*y-v*r)-y)-v*S1);
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}
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124
lib/msun/ld80/k_tanl.c
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124
lib/msun/ld80/k_tanl.c
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/* From: @(#)k_tan.c 1.5 04/04/22 SMI */
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/*
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* ====================================================
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* Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
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* Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
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*
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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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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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/*
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* ld80 version of k_tan.c. See ../src/k_tan.c for most comments.
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*/
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#include "math.h"
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#include "math_private.h"
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/*
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* Domain [-0.67434, 0.67434], range ~[-2.25e-22, 1.921e-22]
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* |tan(x)/x - t(x)| < 2**-71.9
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*
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* See k_cosl.c for more details about the polynomial.
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*/
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#if defined(__amd64__) || defined(__i386__)
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/* Long double constants are slow on these arches, and broken on i386. */
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static const volatile double
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T3hi = 0.33333333333333331, /* 0x15555555555555.0p-54 */
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T3lo = 1.8350121769317163e-17, /* 0x15280000000000.0p-108 */
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T5hi = 0.13333333333333336, /* 0x11111111111112.0p-55 */
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T5lo = 1.3051083651294260e-17, /* 0x1e180000000000.0p-109 */
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T7hi = 0.053968253968250494, /* 0x1ba1ba1ba1b827.0p-57 */
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T7lo = 3.1509625637859973e-18, /* 0x1d100000000000.0p-111 */
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pio4_hi = 0.78539816339744828, /* 0x1921fb54442d18.0p-53 */
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pio4_lo = 3.0628711372715500e-17, /* 0x11a80000000000.0p-107 */
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pio4lo_hi = -1.2541394031670831e-20, /* -0x1d9cceba3f91f2.0p-119 */
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pio4lo_lo = 6.1493048227390915e-37; /* 0x1a280000000000.0p-173 */
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#define T3 ((long double)T3hi + T3lo)
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#define T5 ((long double)T5hi + T5lo)
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#define T7 ((long double)T7hi + T7lo)
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#define pio4 ((long double)pio4_hi + pio4_lo)
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#define pio4lo ((long double)pio4lo_hi + pio4lo_lo)
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#else
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static const long double
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T3 = 0.333333333333333333180L, /* 0xaaaaaaaaaaaaaaa5.0p-65 */
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T5 = 0.133333333333333372290L, /* 0x88888888888893c3.0p-66 */
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T7 = 0.0539682539682504975744L; /* 0xdd0dd0dd0dc13ba2.0p-68 */
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pio4 = 0.785398163397448309628, /* 0xc90fdaa22168c235.0p-64 */
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pio4lo = -1.25413940316708300586e-20; /* -0xece675d1fc8f8cbb.0p-130 */
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#endif
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static const double
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T9 = 0.021869488536312216, /* 0x1664f4882cc1c2.0p-58 */
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T11 = 0.0088632355256619590, /* 0x1226e355c17612.0p-59 */
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T13 = 0.0035921281113786528, /* 0x1d6d3d185d7ff8.0p-61 */
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T15 = 0.0014558334756312418, /* 0x17da354aa3f96b.0p-62 */
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T17 = 0.00059003538700862256, /* 0x13559358685b83.0p-63 */
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T19 = 0.00023907843576635544, /* 0x1f56242026b5be.0p-65 */
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T21 = 0.000097154625656538905, /* 0x1977efc26806f4.0p-66 */
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T23 = 0.000038440165747303162, /* 0x14275a09b3ceac.0p-67 */
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T25 = 0.000018082171885432524, /* 0x12f5e563e5487e.0p-68 */
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T27 = 0.0000024196006108814377, /* 0x144c0d80cc6896.0p-71 */
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T29 = 0.0000078293456938132840, /* 0x106b59141a6cb3.0p-69 */
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T31 = -0.0000032609076735050182, /* -0x1b5abef3ba4b59.0p-71 */
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T33 = 0.0000023261313142559411; /* 0x13835436c0c87f.0p-71 */
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long double
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__kernel_tanl(long double x, long double y, int iy) {
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long double z, r, v, w, s;
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long double osign;
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int i;
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iy = (iy == 1 ? -1 : 1); /* XXX recover original interface */
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osign = (x >= 0 ? 1.0 : -1.0); /* XXX slow, probably wrong for -0 */
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if (fabsl(x) >= 0.67434) {
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if (x < 0) {
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x = -x;
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y = -y;
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}
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z = pio4 - x;
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w = pio4lo - y;
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x = z + w;
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y = 0.0;
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i = 1;
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} else
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i = 0;
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z = x * x;
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w = z * z;
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r = T5 + w * (T9 + w * (T13 + w * (T17 + w * (T21 +
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w * (T25 + w * (T29 + w * T33))))));
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v = z * (T7 + w * (T11 + w * (T15 + w * (T19 + w * (T23 +
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w * (T27 + w * T31))))));
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s = z * x;
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r = y + z * (s * (r + v) + y);
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r += T3 * s;
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w = x + r;
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if (i == 1) {
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v = (long double) iy;
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return osign *
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(v - 2.0 * (x - (w * w / (w + v) - r)));
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}
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if (iy == 1)
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return w;
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else {
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/*
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* if allow error up to 2 ulp, simply return
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* -1.0 / (x+r) here
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*/
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/* compute -1.0 / (x+r) accurately */
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long double a, t;
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z = w;
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z = z + 0x1p32 - 0x1p32;
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v = r - (z - x); /* z+v = r+x */
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t = a = -1.0 / w; /* a = -1.0/w */
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t = t + 0x1p32 - 0x1p32;
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s = 1.0 + t * z;
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return t + a * (s + t * v);
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}
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}
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