1996-03-05 21:41:30 +00:00
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/* e_j1f.c -- float version of e_j1.c.
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* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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*/
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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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2000-10-25 22:17:16 +00:00
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* software is freely granted, provided that this notice
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1996-03-05 21:41:30 +00:00
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* is preserved.
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* ====================================================
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*/
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2014-06-23 20:17:13 +00:00
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#include <errno.h>
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2015-06-29 16:52:16 +00:00
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#include <float.h>
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2012-03-09 19:29:16 +00:00
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#include <math.h>
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2018-05-09 00:15:10 +00:00
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#include <math-narrow-eval.h>
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2012-03-09 19:29:16 +00:00
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#include <math_private.h>
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Do not include fenv_private.h in math_private.h.
Continuing the clean-up related to the catch-all math_private.h
header, this patch stops math_private.h from including fenv_private.h.
Instead, fenv_private.h is included directly from those users of
math_private.h that also used interfaces from fenv_private.h. No
attempt is made to remove unused includes of math_private.h, but that
is a natural followup.
(However, since math_private.h sometimes defines optimized versions of
math.h interfaces or __* variants thereof, as well as defining its own
interfaces, I think it might make sense to get all those optimized
versions included from include/math.h, not requiring a separate header
at all, before eliminating unused math_private.h includes - that
avoids a file quietly becoming less-optimized if someone adds a call
to one of those interfaces without restoring a math_private.h include
to that file.)
There is still a pitfall that if code uses plain fe* and __fe*
interfaces, but only includes fenv.h and not fenv_private.h or (before
this patch) math_private.h, it will compile on platforms with
exceptions and rounding modes but not get the optimized versions (and
possibly not compile) on platforms without exception and rounding mode
support, so making it easy to break the build for such platforms
accidentally.
I think it would be most natural to move the inlines / macros for fe*
and __fe* in the case of no exceptions and rounding modes into
include/fenv.h, so that all code including fenv.h with _ISOMAC not
defined automatically gets them. Then fenv_private.h would be purely
the header for the libc_fe*, SET_RESTORE_ROUND etc. internal
interfaces and the risk of breaking the build on other platforms than
the one you tested on because of a missing fenv_private.h include
would be much reduced (and there would be some unused fenv_private.h
includes to remove along with unused math_private.h includes).
Tested for x86_64 and x86, and tested with build-many-glibcs.py that
installed stripped shared libraries are unchanged by this patch.
* sysdeps/generic/math_private.h: Do not include <fenv_private.h>.
* math/fromfp.h: Include <fenv_private.h>.
* math/math-narrow.h: Likewise.
* math/s_cexp_template.c: Likewise.
* math/s_csin_template.c: Likewise.
* math/s_csinh_template.c: Likewise.
* math/s_ctan_template.c: Likewise.
* math/s_ctanh_template.c: Likewise.
* math/s_iseqsig_template.c: Likewise.
* math/w_acos_compat.c: Likewise.
* math/w_acosf_compat.c: Likewise.
* math/w_acosl_compat.c: Likewise.
* math/w_asin_compat.c: Likewise.
* math/w_asinf_compat.c: Likewise.
* math/w_asinl_compat.c: Likewise.
* math/w_ilogb_template.c: Likewise.
* math/w_j0_compat.c: Likewise.
* math/w_j0f_compat.c: Likewise.
* math/w_j0l_compat.c: Likewise.
* math/w_j1_compat.c: Likewise.
* math/w_j1f_compat.c: Likewise.
* math/w_j1l_compat.c: Likewise.
* math/w_jn_compat.c: Likewise.
* math/w_jnf_compat.c: Likewise.
* math/w_llogb_template.c: Likewise.
* math/w_log10_compat.c: Likewise.
* math/w_log10f_compat.c: Likewise.
* math/w_log10l_compat.c: Likewise.
* math/w_log2_compat.c: Likewise.
* math/w_log2f_compat.c: Likewise.
* math/w_log2l_compat.c: Likewise.
* math/w_log_compat.c: Likewise.
* math/w_logf_compat.c: Likewise.
* math/w_logl_compat.c: Likewise.
* sysdeps/aarch64/fpu/feholdexcpt.c: Likewise.
* sysdeps/aarch64/fpu/fesetround.c: Likewise.
* sysdeps/aarch64/fpu/fgetexcptflg.c: Likewise.
* sysdeps/aarch64/fpu/ftestexcept.c: Likewise.
* sysdeps/ieee754/dbl-64/e_atan2.c: Likewise.
* sysdeps/ieee754/dbl-64/e_exp.c: Likewise.
* sysdeps/ieee754/dbl-64/e_exp2.c: Likewise.
* sysdeps/ieee754/dbl-64/e_gamma_r.c: Likewise.
* sysdeps/ieee754/dbl-64/e_jn.c: Likewise.
* sysdeps/ieee754/dbl-64/e_pow.c: Likewise.
* sysdeps/ieee754/dbl-64/e_remainder.c: Likewise.
* sysdeps/ieee754/dbl-64/e_sqrt.c: Likewise.
* sysdeps/ieee754/dbl-64/gamma_product.c: Likewise.
* sysdeps/ieee754/dbl-64/lgamma_neg.c: Likewise.
* sysdeps/ieee754/dbl-64/s_atan.c: Likewise.
* sysdeps/ieee754/dbl-64/s_fma.c: Likewise.
* sysdeps/ieee754/dbl-64/s_fmaf.c: Likewise.
* sysdeps/ieee754/dbl-64/s_llrint.c: Likewise.
* sysdeps/ieee754/dbl-64/s_llround.c: Likewise.
* sysdeps/ieee754/dbl-64/s_lrint.c: Likewise.
* sysdeps/ieee754/dbl-64/s_lround.c: Likewise.
* sysdeps/ieee754/dbl-64/s_nearbyint.c: Likewise.
* sysdeps/ieee754/dbl-64/s_sin.c: Likewise.
* sysdeps/ieee754/dbl-64/s_sincos.c: Likewise.
* sysdeps/ieee754/dbl-64/s_tan.c: Likewise.
* sysdeps/ieee754/dbl-64/wordsize-64/s_lround.c: Likewise.
* sysdeps/ieee754/dbl-64/wordsize-64/s_nearbyint.c: Likewise.
* sysdeps/ieee754/dbl-64/x2y2m1.c: Likewise.
* sysdeps/ieee754/float128/float128_private.h: Likewise.
* sysdeps/ieee754/flt-32/e_gammaf_r.c: Likewise.
* sysdeps/ieee754/flt-32/e_j1f.c: Likewise.
* sysdeps/ieee754/flt-32/e_jnf.c: Likewise.
* sysdeps/ieee754/flt-32/lgamma_negf.c: Likewise.
* sysdeps/ieee754/flt-32/s_llrintf.c: Likewise.
* sysdeps/ieee754/flt-32/s_llroundf.c: Likewise.
* sysdeps/ieee754/flt-32/s_lrintf.c: Likewise.
* sysdeps/ieee754/flt-32/s_lroundf.c: Likewise.
* sysdeps/ieee754/flt-32/s_nearbyintf.c: Likewise.
* sysdeps/ieee754/k_standardl.c: Likewise.
* sysdeps/ieee754/ldbl-128/e_expl.c: Likewise.
* sysdeps/ieee754/ldbl-128/e_gammal_r.c: Likewise.
* sysdeps/ieee754/ldbl-128/e_j1l.c: Likewise.
* sysdeps/ieee754/ldbl-128/e_jnl.c: Likewise.
* sysdeps/ieee754/ldbl-128/gamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-128/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_fmal.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_llrintl.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_llroundl.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_lrintl.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_lroundl.c: Likewise.
* sysdeps/ieee754/ldbl-128/s_nearbyintl.c: Likewise.
* sysdeps/ieee754/ldbl-128/x2y2m1l.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/e_expl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/e_gammal_r.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/e_j1l.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/e_jnl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_fmal.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_llrintl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_llroundl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_lrintl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_lroundl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/s_rintl.c: Likewise.
* sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c: Likewise.
* sysdeps/ieee754/ldbl-96/e_gammal_r.c: Likewise.
* sysdeps/ieee754/ldbl-96/e_jnl.c: Likewise.
* sysdeps/ieee754/ldbl-96/gamma_productl.c: Likewise.
* sysdeps/ieee754/ldbl-96/lgamma_negl.c: Likewise.
* sysdeps/ieee754/ldbl-96/s_fma.c: Likewise.
* sysdeps/ieee754/ldbl-96/s_fmal.c: Likewise.
* sysdeps/ieee754/ldbl-96/s_llrintl.c: Likewise.
* sysdeps/ieee754/ldbl-96/s_llroundl.c: Likewise.
* sysdeps/ieee754/ldbl-96/s_lrintl.c: Likewise.
* sysdeps/ieee754/ldbl-96/s_lroundl.c: Likewise.
* sysdeps/ieee754/ldbl-96/x2y2m1l.c: Likewise.
* sysdeps/powerpc/fpu/e_sqrt.c: Likewise.
* sysdeps/powerpc/fpu/e_sqrtf.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_ceil.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_floor.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_nearbyint.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_round.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_roundeven.c: Likewise.
* sysdeps/riscv/rv64/rvd/s_trunc.c: Likewise.
* sysdeps/riscv/rvd/s_finite.c: Likewise.
* sysdeps/riscv/rvd/s_fmax.c: Likewise.
* sysdeps/riscv/rvd/s_fmin.c: Likewise.
* sysdeps/riscv/rvd/s_fpclassify.c: Likewise.
* sysdeps/riscv/rvd/s_isinf.c: Likewise.
* sysdeps/riscv/rvd/s_isnan.c: Likewise.
* sysdeps/riscv/rvd/s_issignaling.c: Likewise.
* sysdeps/riscv/rvf/fegetround.c: Likewise.
* sysdeps/riscv/rvf/feholdexcpt.c: Likewise.
* sysdeps/riscv/rvf/fesetenv.c: Likewise.
* sysdeps/riscv/rvf/fesetround.c: Likewise.
* sysdeps/riscv/rvf/feupdateenv.c: Likewise.
* sysdeps/riscv/rvf/fgetexcptflg.c: Likewise.
* sysdeps/riscv/rvf/ftestexcept.c: Likewise.
* sysdeps/riscv/rvf/s_ceilf.c: Likewise.
* sysdeps/riscv/rvf/s_finitef.c: Likewise.
* sysdeps/riscv/rvf/s_floorf.c: Likewise.
* sysdeps/riscv/rvf/s_fmaxf.c: Likewise.
* sysdeps/riscv/rvf/s_fminf.c: Likewise.
* sysdeps/riscv/rvf/s_fpclassifyf.c: Likewise.
* sysdeps/riscv/rvf/s_isinff.c: Likewise.
* sysdeps/riscv/rvf/s_isnanf.c: Likewise.
* sysdeps/riscv/rvf/s_issignalingf.c: Likewise.
* sysdeps/riscv/rvf/s_nearbyintf.c: Likewise.
* sysdeps/riscv/rvf/s_roundevenf.c: Likewise.
* sysdeps/riscv/rvf/s_roundf.c: Likewise.
* sysdeps/riscv/rvf/s_truncf.c: Likewise.
2018-09-03 21:09:04 +00:00
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#include <fenv_private.h>
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2018-05-10 00:53:04 +00:00
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#include <math-underflow.h>
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2019-07-16 15:17:22 +00:00
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#include <libm-alias-finite.h>
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Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
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#include <reduce_aux.h>
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1996-03-05 21:41:30 +00:00
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static float ponef(float), qonef(float);
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2000-10-25 22:17:16 +00:00
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static const float
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1996-03-05 21:41:30 +00:00
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huge = 1e30,
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one = 1.0,
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invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
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tpi = 6.3661974669e-01, /* 0x3f22f983 */
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/* R0/S0 on [0,2] */
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r00 = -6.2500000000e-02, /* 0xbd800000 */
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r01 = 1.4070566976e-03, /* 0x3ab86cfd */
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r02 = -1.5995563444e-05, /* 0xb7862e36 */
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r03 = 4.9672799207e-08, /* 0x335557d2 */
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s01 = 1.9153760746e-02, /* 0x3c9ce859 */
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s02 = 1.8594678841e-04, /* 0x3942fab6 */
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s03 = 1.1771846857e-06, /* 0x359dffc2 */
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s04 = 5.0463624390e-09, /* 0x31ad6446 */
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s05 = 1.2354227016e-11; /* 0x2d59567e */
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static const float zero = 0.0;
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Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
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/* This is the nearest approximation of the first positive zero of j1. */
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#define FIRST_ZERO_J1 0x3.d4eabp+0f
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#define SMALL_SIZE 64
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/* The following table contains successive zeros of j1 and degree-3
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polynomial approximations of j1 around these zeros: Pj[0] for the first
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positive zero (3.831705), Pj[1] for the second one (7.015586), and so on.
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Each line contains:
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{x0, xmid, x1, p0, p1, p2, p3}
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where [x0,x1] is the interval around the zero, xmid is the binary32 number
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closest to the zero, and p0+p1*x+p2*x^2+p3*x^3 is the approximation
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polynomial. Each polynomial was generated using Sollya on the interval
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[x0,x1] around the corresponding zero where the error exceeds 9 ulps
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for the alternate code. Degree 3 is enough to get an error at most
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9 ulps, except around the first zero.
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*/
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static const float Pj[SMALL_SIZE][7] = {
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/* For index 0, we use a degree-4 polynomial generated by Sollya, with the
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coefficient of degree 4 hard-coded in j1f_near_root(). */
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{ 0x1.e09e5ep+1, 0x1.ea7558p+1, 0x1.ef7352p+1, -0x8.4f069p-28,
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-0x6.71b3d8p-4, 0xd.744a2p-8, 0xd.acd9p-8/*, -0x1.3e51aap-8*/ }, /* 0 */
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{ 0x1.bdb4c2p+2, 0x1.c0ff6p+2, 0x1.c27a8cp+2, 0xe.c2858p-28,
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0x4.cd464p-4, -0x5.79b71p-8, -0xc.11124p-8 }, /* 1 */
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{ 0x1.43b214p+3, 0x1.458d0ep+3, 0x1.460ccep+3, -0x1.e7acecp-24,
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-0x3.feca9p-4, 0x3.2470f8p-8, 0xa.625b7p-8 }, /* 2 */
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{ 0x1.a9c98p+3, 0x1.aa5bbp+3, 0x1.aaa4d8p+3, 0x1.698158p-24,
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0x3.7e666cp-4, -0x2.1900ap-8, -0x9.2755p-8 }, /* 3 */
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{ 0x1.073be4p+4, 0x1.0787b4p+4, 0x1.07aed8p+4, -0x1.f5f658p-24,
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-0x3.24b8ep-4, 0x1.86e35cp-8, 0x8.4e4bbp-8 }, /* 4 */
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{ 0x1.39ae2ap+4, 0x1.39da8ep+4, 0x1.39f3dap+4, -0x1.4e744p-24,
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0x2.e18a24p-4, -0x1.2ccd16p-8, -0x7.a27ep-8 }, /* 5 */
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{ 0x1.6bfa46p+4, 0x1.6c294ep+4, 0x1.6c412p+4, 0xa.3fb7fp-28,
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-0x2.acc9c4p-4, 0xf.0b783p-12, 0x7.1c0d3p-8 }, /* 6 */
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{ 0x1.9e42bep+4, 0x1.9e757p+4, 0x1.9e876ep+4, -0x2.29f6f4p-24,
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0x2.81f21p-4, -0xc.641bp-12, -0x6.a7ea58p-8 }, /* 7 */
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{ 0x1.d08a3ep+4, 0x1.d0bfdp+4, 0x1.d0cd3cp+4, -0x1.b5d196p-24,
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-0x2.5e40e4p-4, 0xa.7059fp-12, 0x6.4d6bfp-8 }, /* 8 */
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{ 0x1.017794p+5, 0x1.018476p+5, 0x1.018b8cp+5, -0x4.0e001p-24,
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0x2.3febep-4, -0x8.f23aap-12, -0x6.0102cp-8 }, /* 9 */
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{ 0x1.1a9e78p+5, 0x1.1aa89p+5, 0x1.1aaf88p+5, 0x3.b26f2p-24,
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-0x2.25babp-4, 0x7.c6d948p-12, 0x5.a1d988p-8 }, /* 10 */
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{ 0x1.33bddep+5, 0x1.33cc52p+5, 0x1.33d2e4p+5, -0xf.c8cdap-28,
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0x2.0ed05p-4, -0x6.d97dbp-12, -0x5.8da498p-8 }, /* 11 */
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{ 0x1.4ce7cp+5, 0x1.4cefdp+5, 0x1.4cf7d4p+5, -0x3.9940e4p-24,
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|
-0x1.fa8b4p-4, 0x6.16108p-12, 0x5.4355e8p-8 }, /* 12 */
|
|
|
|
|
{ 0x1.6603e8p+5, 0x1.661316p+5, 0x1.66173ap+5, 0x9.da15dp-28,
|
|
|
|
|
0x1.e8727ep-4, -0x5.742468p-12, -0x5.117c28p-8 }, /* 13 */
|
|
|
|
|
{ 0x1.7f2ebcp+5, 0x1.7f3632p+5, 0x1.7f3a7ep+5, -0x3.39b218p-24,
|
|
|
|
|
-0x1.d8293ap-4, 0x4.ee3348p-12, 0x4.f9bep-8 }, /* 14 */
|
|
|
|
|
{ 0x1.9850e6p+5, 0x1.985928p+5, 0x1.985d9ep+5, -0x3.7b5108p-24,
|
|
|
|
|
0x1.c96702p-4, -0x4.7b0d08p-12, -0x4.c784a8p-8 }, /* 15 */
|
|
|
|
|
{ 0x1.b172e8p+5, 0x1.b17c04p+5, 0x1.b1805cp+5, -0x1.91e43ep-24,
|
|
|
|
|
-0x1.bbf246p-4, 0x4.18ad78p-12, 0x4.9bfae8p-8 }, /* 16 */
|
|
|
|
|
{ 0x1.ca955ap+5, 0x1.ca9ec6p+5, 0x1.caa2a4p+5, 0x1.28453cp-24,
|
|
|
|
|
0x1.af9cb4p-4, -0x3.c3a494p-12, -0x4.78b69p-8 }, /* 17 */
|
|
|
|
|
{ 0x1.e3bc94p+5, 0x1.e3c174p+5, 0x1.e3c64p+5, -0x2.e7fef4p-24,
|
|
|
|
|
-0x1.a4407ep-4, 0x3.79b228p-12, 0x4.874f7p-8 }, /* 18 */
|
|
|
|
|
{ 0x1.fcdf16p+5, 0x1.fce40ep+5, 0x1.fce71p+5, -0x3.23b2fcp-24,
|
|
|
|
|
0x1.99be76p-4, -0x3.39ad7cp-12, -0x4.92a55p-8 }, /* 19 */
|
|
|
|
|
{ 0x1.0afe34p+6, 0x1.0b034ep+6, 0x1.0b054ap+6, -0xd.19e93p-28,
|
|
|
|
|
-0x1.8ffc9cp-4, 0x2.fee7f8p-12, 0x4.2d33b8p-8 }, /* 20 */
|
|
|
|
|
{ 0x1.179344p+6, 0x1.17948ep+6, 0x1.1795bp+6, 0x1.c2ac48p-24,
|
|
|
|
|
0x1.86e51cp-4, -0x2.cc5abp-12, -0x4.866d08p-8 }, /* 21 */
|
|
|
|
|
{ 0x1.24231ep+6, 0x1.2425c8p+6, 0x1.2426e2p+6, -0xd.31027p-28,
|
|
|
|
|
-0x1.7e656ep-4, 0x2.9db23cp-12, 0x3.cc63c8p-8 }, /* 22 */
|
|
|
|
|
{ 0x1.30b5a8p+6, 0x1.30b6fep+6, 0x1.30b84ep+6, 0x5.b5e53p-24,
|
|
|
|
|
0x1.766dc2p-4, -0x2.754cfcp-12, -0x3.c39bb4p-8 }, /* 23 */
|
|
|
|
|
{ 0x1.3d46ccp+6, 0x1.3d482ep+6, 0x1.3d495ep+6, -0x1.340a8ap-24,
|
|
|
|
|
-0x1.6ef07ep-4, 0x2.4ff9d4p-12, 0x3.9b36e4p-8 }, /* 24 */
|
|
|
|
|
{ 0x1.49d688p+6, 0x1.49d95ap+6, 0x1.49dabep+6, -0x3.ba66p-24,
|
|
|
|
|
0x1.67e1dcp-4, -0x2.2f32b8p-12, -0x3.e2aaf4p-8 }, /* 25 */
|
|
|
|
|
{ 0x1.566916p+6, 0x1.566a84p+6, 0x1.566bcp+6, 0x6.47ca5p-28,
|
|
|
|
|
-0x1.61379ap-4, 0x2.1096acp-12, 0x4.2d0968p-8 }, /* 26 */
|
|
|
|
|
{ 0x1.62f8dap+6, 0x1.62fbaap+6, 0x1.62fc9cp+6, -0x2.12affp-24,
|
|
|
|
|
0x1.5ae8c4p-4, -0x1.f32444p-12, -0x3.9e592p-8 }, /* 27 */
|
|
|
|
|
{ 0x1.6f89e6p+6, 0x1.6f8ccep+6, 0x1.6f8e34p+6, -0x7.4853ap-28,
|
|
|
|
|
-0x1.54ed76p-4, 0x1.db004ap-12, 0x3.907034p-8 }, /* 28 */
|
|
|
|
|
{ 0x1.7c1c6ap+6, 0x1.7c1deep+6, 0x1.7c1f4cp+6, -0x4.f0a998p-24,
|
|
|
|
|
0x1.4f3ebcp-4, -0x1.c26808p-12, -0x2.da8df8p-8 }, /* 29 */
|
|
|
|
|
{ 0x1.88adaep+6, 0x1.88af0ep+6, 0x1.88afc4p+6, -0x1.80c246p-24,
|
|
|
|
|
-0x1.49d668p-4, 0x1.aebc26p-12, 0x3.af7b5cp-8 }, /* 30 */
|
|
|
|
|
{ 0x1.953d7p+6, 0x1.95402ap+6, 0x1.9540ep+6, -0x2.22aff8p-24,
|
|
|
|
|
0x1.44aefap-4, -0x1.99f25p-12, -0x3.5e9198p-8 }, /* 31 */
|
|
|
|
|
{ 0x1.a1d01ep+6, 0x1.a1d146p+6, 0x1.a1d20ap+6, -0x3.aad6d4p-24,
|
|
|
|
|
-0x1.3fc386p-4, 0x1.892858p-12, 0x3.fe0184p-8 }, /* 32 */
|
|
|
|
|
{ 0x1.ae60ecp+6, 0x1.ae625ep+6, 0x1.ae6326p+6, -0x2.010be4p-24,
|
|
|
|
|
0x1.3b0fa4p-4, -0x1.7539ap-12, -0x2.b2c9bp-8 }, /* 33 */
|
|
|
|
|
{ 0x1.baf234p+6, 0x1.baf376p+6, 0x1.baf442p+6, -0xd.4fd17p-32,
|
|
|
|
|
-0x1.368f5cp-4, 0x1.6734e4p-12, 0x3.59f514p-8 }, /* 34 */
|
|
|
|
|
{ 0x1.c782e6p+6, 0x1.c7848cp+6, 0x1.c78516p+6, -0xa.d662dp-28,
|
|
|
|
|
0x1.323f18p-4, -0x1.571c02p-12, -0x3.2be5bp-8 }, /* 35 */
|
|
|
|
|
{ 0x1.d4144ep+6, 0x1.d415ap+6, 0x1.d41622p+6, 0x4.9f217p-24,
|
|
|
|
|
-0x1.2e1b9ap-4, 0x1.4a2edap-12, 0x3.a4e96p-8 }, /* 36 */
|
|
|
|
|
{ 0x1.e0a5ep+6, 0x1.e0a6b4p+6, 0x1.e0a788p+6, -0x2.d167p-24,
|
|
|
|
|
0x1.2a21eep-4, -0x1.3c4b46p-12, -0x4.9e0978p-8 }, /* 37 */
|
|
|
|
|
{ 0x1.ed36eep+6, 0x1.ed37c8p+6, 0x1.ed3892p+6, -0x4.15a83p-24,
|
|
|
|
|
-0x1.264f66p-4, 0x1.31dea4p-12, 0x3.d125ecp-8 }, /* 38 */
|
|
|
|
|
{ 0x1.f9c77p+6, 0x1.f9c8d8p+6, 0x1.f9c9acp+6, -0x2.a5bbbp-24,
|
|
|
|
|
0x1.22a192p-4, -0x1.25e59ep-12, -0x2.ef6934p-8 }, /* 39 */
|
|
|
|
|
{ 0x1.032c54p+7, 0x1.032cf4p+7, 0x1.032d66p+7, 0x4.e828bp-24,
|
|
|
|
|
-0x1.1f1634p-4, 0x1.1c2394p-12, 0x3.6d744cp-8 }, /* 40 */
|
|
|
|
|
{ 0x1.09750cp+7, 0x1.09757cp+7, 0x1.0975b6p+7, -0x3.28a3bcp-24,
|
|
|
|
|
0x1.1bab3ep-4, -0x1.1569cep-12, -0x5.84da7p-8 }, /* 41 */
|
|
|
|
|
{ 0x1.0fbd9ap+7, 0x1.0fbe04p+7, 0x1.0fbe5ep+7, -0x2.2f667p-24,
|
|
|
|
|
-0x1.185eccp-4, 0x1.07f42cp-12, 0x2.87896cp-8 }, /* 42 */
|
|
|
|
|
{ 0x1.160628p+7, 0x1.16068ap+7, 0x1.1606cep+7, -0x6.9097dp-24,
|
|
|
|
|
0x1.152f28p-4, -0x1.0227fep-12, -0x5.da6e6p-8 }, /* 43 */
|
|
|
|
|
{ 0x1.1c4e9ap+7, 0x1.1c4f12p+7, 0x1.1c4f7cp+7, -0x5.1b408p-24,
|
|
|
|
|
-0x1.121abp-4, 0xf.6efcp-16, 0x2.c5e954p-8 }, /* 44 */
|
|
|
|
|
{ 0x1.2296aap+7, 0x1.229798p+7, 0x1.2297d4p+7, 0x2.70d0dp-24,
|
|
|
|
|
0x1.0f1ffp-4, -0xf.523f5p-16, -0x3.5c0568p-8 }, /* 45 */
|
|
|
|
|
{ 0x1.28dfa4p+7, 0x1.28e01ep+7, 0x1.28e054p+7, -0x2.7c176p-24,
|
|
|
|
|
-0x1.0c3d8ap-4, 0xe.8329ap-16, 0x3.5eb34p-8 }, /* 46 */
|
|
|
|
|
{ 0x1.2f282ap+7, 0x1.2f28a4p+7, 0x1.2f28dep+7, 0x4.fd6368p-24,
|
|
|
|
|
0x1.097236p-4, -0xe.17299p-16, -0x3.73a2e4p-8 }, /* 47 */
|
|
|
|
|
{ 0x1.3570bp+7, 0x1.357128p+7, 0x1.35716p+7, 0x6.b05f68p-24,
|
|
|
|
|
-0x1.06bccap-4, 0xd.527b8p-16, 0x2.b8bf9cp-8 }, /* 48 */
|
|
|
|
|
{ 0x1.3bb932p+7, 0x1.3bb9aep+7, 0x1.3bb9eap+7, 0x4.0d622p-28,
|
|
|
|
|
0x1.041c28p-4, -0xd.0ac11p-16, -0x1.65f2ccp-8 }, /* 49 */
|
|
|
|
|
{ 0x1.4201b6p+7, 0x1.420232p+7, 0x1.42027p+7, 0x7.0d98cp-24,
|
|
|
|
|
-0x1.018f52p-4, 0xc.c4d8ep-16, 0x2.8f250cp-8 }, /* 50 */
|
|
|
|
|
{ 0x1.484a78p+7, 0x1.484ab8p+7, 0x1.484af4p+7, 0x3.93d10cp-24,
|
|
|
|
|
0xf.f154fp-8, -0xc.7b7fep-16, -0x3.6b6e4cp-8 }, /* 51 */
|
|
|
|
|
{ 0x1.4e92c8p+7, 0x1.4e933cp+7, 0x1.4e9368p+7, 0xd.88185p-32,
|
|
|
|
|
-0xf.cad3fp-8, 0xc.1462p-16, 0x2.bd66p-8 }, /* 52 */
|
|
|
|
|
{ 0x1.54db84p+7, 0x1.54dbcp+7, 0x1.54dbf4p+7, -0x1.fe6b92p-24,
|
|
|
|
|
0xf.a564cp-8, -0xb.c4e6cp-16, -0x3.d51decp-8 }, /* 53 */
|
|
|
|
|
{ 0x1.5b23c4p+7, 0x1.5b2444p+7, 0x1.5b2486p+7, 0x2.6137f4p-24,
|
|
|
|
|
-0xf.80faep-8, 0xb.5199ep-16, 0x1.9ca85ap-8 }, /* 54 */
|
|
|
|
|
{ 0x1.616c62p+7, 0x1.616cc8p+7, 0x1.616d0ap+7, -0x1.55468p-24,
|
|
|
|
|
0xf.5d8acp-8, -0xb.21d16p-16, -0x1.b8809ap-8 }, /* 55 */
|
|
|
|
|
{ 0x1.67b4fp+7, 0x1.67b54cp+7, 0x1.67b588p+7, -0x1.08c6bep-24,
|
|
|
|
|
-0xf.3b096p-8, 0xa.e65efp-16, 0x3.642304p-8 }, /* 56 */
|
|
|
|
|
{ 0x1.6dfd8ep+7, 0x1.6dfddp+7, 0x1.6dfe0ap+7, 0x4.9ebfa8p-24,
|
|
|
|
|
0xf.196c7p-8, -0xa.ba8c8p-16, -0x5.ad6a2p-8 }, /* 57 */
|
|
|
|
|
{ 0x1.74461p+7, 0x1.744652p+7, 0x1.744692p+7, 0x5.a4017p-24,
|
|
|
|
|
-0xe.f8aa5p-8, 0xa.49748p-16, 0x2.a86498p-8 }, /* 58 */
|
|
|
|
|
{ 0x1.7a8e5ep+7, 0x1.7a8ed6p+7, 0x1.7a8ef8p+7, 0x3.bcb2a8p-28,
|
|
|
|
|
0xe.d8b9dp-8, -0x9.c43bep-16, -0x1.e7124ap-8 }, /* 59 */
|
|
|
|
|
{ 0x1.80d6cep+7, 0x1.80d75ap+7, 0x1.80d78ap+7, -0x7.1091fp-24,
|
|
|
|
|
-0xe.b9925p-8, 0x9.c43dap-16, 0x1.aba86p-8 }, /* 60 */
|
|
|
|
|
{ 0x1.871f58p+7, 0x1.871fdcp+7, 0x1.87201ep+7, 0x2.ca1cf4p-28,
|
|
|
|
|
0xe.9b2bep-8, -0x9.843b3p-16, -0x2.093e68p-8 }, /* 61 */
|
|
|
|
|
{ 0x1.8d67e8p+7, 0x1.8d685ep+7, 0x1.8d688ep+7, 0x5.aa8908p-24,
|
|
|
|
|
-0xe.7d7ecp-8, 0x9.501a8p-16, 0x2.54a754p-8 }, /* 62 */
|
|
|
|
|
{ 0x1.93b09cp+7, 0x1.93b0e2p+7, 0x1.93b10ep+7, 0x3.d9cd9cp-24,
|
|
|
|
|
0xe.6083ap-8, -0x9.45dadp-16, -0x5.112908p-8 }, /* 63 */
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
|
|
|
|
|
j1f(x) ~ sqrt(2/(pi*x))*beta1(x)*cos(x-3pi/4-alpha1(x))
|
|
|
|
|
where beta1(x) = 1 + 3/(16*x^2) - 99/(512*x^4)
|
|
|
|
|
and alpha1(x) = -3/(8*x) + 21/(128*x^3) - 1899/(5120*x^5). */
|
|
|
|
|
static float
|
|
|
|
|
j1f_asympt (float x)
|
|
|
|
|
{
|
|
|
|
|
float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */
|
|
|
|
|
if (x < 0)
|
|
|
|
|
{
|
|
|
|
|
x = -x;
|
|
|
|
|
cst = -cst;
|
|
|
|
|
}
|
|
|
|
|
double y = 1.0 / (double) x;
|
|
|
|
|
double y2 = y * y;
|
|
|
|
|
double beta1 = 1.0f + y2 * (0x3p-4 - 0x3.18p-4 * y2);
|
|
|
|
|
double alpha1;
|
|
|
|
|
alpha1 = y * (-0x6p-4 + y2 * (0x2.ap-4 - 0x5.ef33333333334p-4 * y2));
|
|
|
|
|
double h;
|
|
|
|
|
int n;
|
|
|
|
|
h = reduce_aux (x, &n, alpha1);
|
|
|
|
|
n--; /* Subtract pi/2. */
|
|
|
|
|
/* Now x - 3pi/4 - alpha1 = h + n*pi/2 mod (2*pi). */
|
|
|
|
|
float xr = (float) h;
|
|
|
|
|
n = n & 3;
|
|
|
|
|
float t = cst / sqrtf (x) * (float) beta1;
|
|
|
|
|
if (n == 0)
|
|
|
|
|
return t * __cosf (xr);
|
|
|
|
|
else if (n == 2) /* cos(x+pi) = -cos(x) */
|
|
|
|
|
return -t * __cosf (xr);
|
|
|
|
|
else if (n == 1) /* cos(x+pi/2) = -sin(x) */
|
|
|
|
|
return -t * __sinf (xr);
|
|
|
|
|
else /* cos(x+3pi/2) = sin(x) */
|
|
|
|
|
return t * __sinf (xr);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* Special code for x near a root of j1.
|
|
|
|
|
z is the value computed by the generic code.
|
|
|
|
|
For small x, we use a polynomial approximating j1 around its root.
|
|
|
|
|
For large x, we use an asymptotic formula (j1f_asympt). */
|
|
|
|
|
static float
|
|
|
|
|
j1f_near_root (float x, float z)
|
|
|
|
|
{
|
|
|
|
|
float index_f, sign = 1.0f;
|
|
|
|
|
int index;
|
|
|
|
|
|
|
|
|
|
if (x < 0)
|
|
|
|
|
{
|
|
|
|
|
x = -x;
|
|
|
|
|
sign = -1.0f;
|
|
|
|
|
}
|
|
|
|
|
index_f = roundf ((x - FIRST_ZERO_J1) / (float) M_PI);
|
|
|
|
|
if (index_f >= SMALL_SIZE)
|
|
|
|
|
return sign * j1f_asympt (x);
|
|
|
|
|
index = (int) index_f;
|
|
|
|
|
const float *p = Pj[index];
|
|
|
|
|
float x0 = p[0];
|
|
|
|
|
float x1 = p[2];
|
|
|
|
|
/* If not in the interval [x0,x1] around xmid, return the value z. */
|
|
|
|
|
if (! (x0 <= x && x <= x1))
|
|
|
|
|
return z;
|
|
|
|
|
float xmid = p[1];
|
|
|
|
|
float y = x - xmid;
|
|
|
|
|
float p6 = (index > 0) ? p[6] : p[6] + y * -0x1.3e51aap-8f;
|
|
|
|
|
return sign * (p[3] + y * (p[4] + y * (p[5] + y * p6)));
|
|
|
|
|
}
|
|
|
|
|
|
2011-10-12 15:27:51 +00:00
|
|
|
float
|
|
|
|
|
__ieee754_j1f(float x)
|
1996-03-05 21:41:30 +00:00
|
|
|
{
|
|
|
|
|
float z, s,c,ss,cc,r,u,v,y;
|
|
|
|
|
int32_t hx,ix;
|
|
|
|
|
|
|
|
|
|
GET_FLOAT_WORD(hx,x);
|
|
|
|
|
ix = hx&0x7fffffff;
|
2011-10-12 15:27:51 +00:00
|
|
|
if(__builtin_expect(ix>=0x7f800000, 0)) return one/x;
|
1996-03-05 21:41:30 +00:00
|
|
|
y = fabsf(x);
|
|
|
|
|
if(ix >= 0x40000000) { /* |x| >= 2.0 */
|
Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
|
|
|
SET_RESTORE_ROUNDF (FE_TONEAREST);
|
2001-02-13 01:23:48 +00:00
|
|
|
__sincosf (y, &s, &c);
|
1996-03-05 21:41:30 +00:00
|
|
|
ss = -s-c;
|
|
|
|
|
cc = s-c;
|
Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
|
|
|
if (ix >= 0x7f000000)
|
|
|
|
|
/* x >= 2^127: use asymptotic expansion. */
|
|
|
|
|
return j1f_asympt (x);
|
|
|
|
|
/* Now we are sure that x+x cannot overflow. */
|
|
|
|
|
z = __cosf(y+y);
|
|
|
|
|
if ((s*c)>zero) cc = z/ss;
|
|
|
|
|
else ss = z/cc;
|
1996-03-05 21:41:30 +00:00
|
|
|
/*
|
|
|
|
|
* j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
|
|
|
|
|
* y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
|
|
|
|
|
*/
|
Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
|
|
|
if (ix <= 0x5c000000)
|
|
|
|
|
{
|
1996-03-05 21:41:30 +00:00
|
|
|
u = ponef(y); v = qonef(y);
|
Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
|
|
|
cc = u*cc-v*ss;
|
|
|
|
|
}
|
|
|
|
|
z = (invsqrtpi * cc) / sqrtf(y);
|
|
|
|
|
/* Adjust sign of z. */
|
|
|
|
|
z = (hx < 0) ? -z : z;
|
|
|
|
|
/* The following threshold is optimal: for x=0x1.e09e5ep+1
|
|
|
|
|
and rounding upwards, cc=0x1.b79638p-4 and z is 10 ulps
|
|
|
|
|
far from the correctly rounded value. */
|
|
|
|
|
float threshold = 0x1.b79638p-4;
|
|
|
|
|
if (fabsf (cc) > threshold)
|
|
|
|
|
return z;
|
|
|
|
|
else
|
|
|
|
|
return j1f_near_root (x, z);
|
1996-03-05 21:41:30 +00:00
|
|
|
}
|
2011-10-12 15:27:51 +00:00
|
|
|
if(__builtin_expect(ix<0x32000000, 0)) { /* |x|<2**-27 */
|
2015-06-29 16:52:16 +00:00
|
|
|
if(huge+x>one) { /* inexact if x!=0 necessary */
|
2015-10-23 21:37:33 +00:00
|
|
|
float ret = math_narrow_eval ((float) 0.5 * x);
|
2015-09-23 22:42:30 +00:00
|
|
|
math_check_force_underflow (ret);
|
2015-10-23 21:37:33 +00:00
|
|
|
if (ret == 0 && x != 0)
|
|
|
|
|
__set_errno (ERANGE);
|
2015-06-29 16:52:16 +00:00
|
|
|
return ret;
|
|
|
|
|
}
|
1996-03-05 21:41:30 +00:00
|
|
|
}
|
|
|
|
|
z = x*x;
|
|
|
|
|
r = z*(r00+z*(r01+z*(r02+z*r03)));
|
|
|
|
|
s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
|
|
|
|
|
r *= x;
|
|
|
|
|
return(x*(float)0.5+r/s);
|
|
|
|
|
}
|
2019-07-16 15:17:22 +00:00
|
|
|
libm_alias_finite (__ieee754_j1f, __j1f)
|
1996-03-05 21:41:30 +00:00
|
|
|
|
|
|
|
|
static const float U0[5] = {
|
|
|
|
|
-1.9605709612e-01, /* 0xbe48c331 */
|
|
|
|
|
5.0443872809e-02, /* 0x3d4e9e3c */
|
|
|
|
|
-1.9125689287e-03, /* 0xbafaaf2a */
|
|
|
|
|
2.3525259166e-05, /* 0x37c5581c */
|
|
|
|
|
-9.1909917899e-08, /* 0xb3c56003 */
|
|
|
|
|
};
|
|
|
|
|
static const float V0[5] = {
|
|
|
|
|
1.9916731864e-02, /* 0x3ca3286a */
|
|
|
|
|
2.0255257550e-04, /* 0x3954644b */
|
|
|
|
|
1.3560879779e-06, /* 0x35b602d4 */
|
|
|
|
|
6.2274145840e-09, /* 0x31d5f8eb */
|
|
|
|
|
1.6655924903e-11, /* 0x2d9281cf */
|
|
|
|
|
};
|
|
|
|
|
|
Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
|
|
|
/* This is the nearest approximation of the first zero of y1. */
|
|
|
|
|
#define FIRST_ZERO_Y1 0x2.3277dcp+0f
|
|
|
|
|
|
|
|
|
|
/* The following table contains successive zeros of y1 and degree-3
|
|
|
|
|
polynomial approximations of y1 around these zeros: Py[0] for the first
|
|
|
|
|
positive zero (2.197141), Py[1] for the second one (5.429681), and so on.
|
|
|
|
|
Each line contains:
|
|
|
|
|
{x0, xmid, x1, p0, p1, p2, p3}
|
|
|
|
|
where [x0,x1] is the interval around the zero, xmid is the binary32 number
|
|
|
|
|
closest to the zero, and p0+p1*x+p2*x^2+p3*x^3 is the approximation
|
|
|
|
|
polynomial. Each polynomial was generated using Sollya on the interval
|
|
|
|
|
[x0,x1] around the corresponding zero where the error exceeds 9 ulps
|
|
|
|
|
for the alternate code. Degree 3 is enough, except for the first roots.
|
|
|
|
|
*/
|
|
|
|
|
static const float Py[SMALL_SIZE][7] = {
|
|
|
|
|
/* For index 0, we use a degree-5 polynomial generated by Sollya, with the
|
|
|
|
|
coefficients of degree 4 and 5 hard-coded in y1f_near_root(). */
|
|
|
|
|
{ 0x1.f7f16ap+0, 0x1.193beep+1, 0x1.2105dcp+1, 0xb.96749p-28,
|
|
|
|
|
0x8.55241p-4, -0x1.e570bp-4, -0x8.68b61p-8
|
|
|
|
|
/*, -0x1.28043p-8, 0x2.50e83p-8*/ }, /* 0 */
|
|
|
|
|
/* For index 1, we use a degree-4 polynomial generated by Sollya, with the
|
|
|
|
|
coefficient of degree 4 hard-coded in y1f_near_root(). */
|
|
|
|
|
{ 0x1.55c6d2p+2, 0x1.5b7fe4p+2, 0x1.5cf8cap+2, 0x1.3c7822p-24,
|
|
|
|
|
-0x5.71f158p-4, 0x8.05cb4p-8, 0xd.0b15p-8/*, -0xf.ff6b8p-12*/ }, /* 1 */
|
|
|
|
|
{ 0x1.113c6p+3, 0x1.13127ap+3, 0x1.1387dcp+3, -0x1.f3ad8ep-24,
|
|
|
|
|
0x4.57e66p-4, -0x4.0afb58p-8, -0xb.29207p-8 }, /* 2 */
|
|
|
|
|
{ 0x1.76e7dep+3, 0x1.77f914p+3, 0x1.786a6ap+3, -0xd.5608fp-28,
|
|
|
|
|
-0x3.b829d4p-4, 0x2.8852cp-8, 0x9.b70e3p-8 }, /* 3 */
|
|
|
|
|
{ 0x1.dc2794p+3, 0x1.dcb7d8p+3, 0x1.dd032p+3, -0xe.a7c04p-28,
|
|
|
|
|
0x3.4e0458p-4, -0x1.c64b18p-8, -0x8.b0e7fp-8 }, /* 4 */
|
|
|
|
|
{ 0x1.20874p+4, 0x1.20b1c6p+4, 0x1.20c71p+4, 0x1.c2626p-24,
|
|
|
|
|
-0x3.00f03cp-4, 0x1.54f806p-8, 0x7.f9cf9p-8 }, /* 5 */
|
|
|
|
|
{ 0x1.52d848p+4, 0x1.530254p+4, 0x1.531962p+4, -0x1.9503ecp-24,
|
|
|
|
|
0x2.c5b29cp-4, -0x1.0bf28p-8, -0x7.562e58p-8 }, /* 6 */
|
|
|
|
|
{ 0x1.851e64p+4, 0x1.854fa4p+4, 0x1.85679p+4, -0x2.8d40fcp-24,
|
|
|
|
|
-0x2.96547p-4, 0xd.9c38bp-12, 0x6.dcbf8p-8 }, /* 7 */
|
|
|
|
|
{ 0x1.b7808ep+4, 0x1.b79acep+4, 0x1.b7b2a8p+4, -0x2.36df5cp-24,
|
|
|
|
|
0x2.6f55ap-4, -0xb.57f9fp-12, -0x6.82569p-8 }, /* 8 */
|
|
|
|
|
{ 0x1.e9c8fp+4, 0x1.e9e48p+4, 0x1.e9f24p+4, 0xd.e2eb7p-28,
|
|
|
|
|
-0x2.4e8104p-4, 0x9.a4be2p-12, 0x6.2541fp-8 }, /* 9 */
|
|
|
|
|
{ 0x1.0e0808p+5, 0x1.0e169p+5, 0x1.0e1d92p+5, -0x2.3070f4p-24,
|
|
|
|
|
0x2.325e4cp-4, -0x8.53604p-12, -0x5.ca03a8p-8 }, /* 10 */
|
|
|
|
|
{ 0x1.272e08p+5, 0x1.273a7cp+5, 0x1.2741fcp+5, -0x3.525508p-24,
|
|
|
|
|
-0x2.19e7dcp-4, 0x7.49d1dp-12, 0x5.9cb02p-8 }, /* 11 */
|
|
|
|
|
{ 0x1.404ec6p+5, 0x1.405e18p+5, 0x1.4065cep+5, -0xe.6e158p-28,
|
|
|
|
|
0x2.046174p-4, -0x6.71b3dp-12, -0x5.4c3c8p-8 }, /* 12 */
|
|
|
|
|
{ 0x1.5971dcp+5, 0x1.598178p+5, 0x1.598592p+5, 0x1.e72698p-24,
|
|
|
|
|
-0x1.f13fb2p-4, 0x5.c0f938p-12, 0x5.28ca78p-8 }, /* 13 */
|
|
|
|
|
{ 0x1.729c4ep+5, 0x1.72a4a8p+5, 0x1.72a8eap+5, -0x1.5bed9cp-24,
|
|
|
|
|
0x1.e018dcp-4, -0x5.2f11e8p-12, -0x5.16ce48p-8 }, /* 14 */
|
|
|
|
|
{ 0x1.8bbf4ep+5, 0x1.8bc7b2p+5, 0x1.8bcc1p+5, -0x3.6b654cp-24,
|
|
|
|
|
-0x1.d09b2p-4, 0x4.b1747p-12, 0x4.bd22fp-8 }, /* 15 */
|
|
|
|
|
{ 0x1.a4e272p+5, 0x1.a4ea9ap+5, 0x1.a4eef4p+5, 0x1.6f11bp-24,
|
|
|
|
|
0x1.c28612p-4, -0x4.47462p-12, -0x4.947c5p-8 }, /* 16 */
|
|
|
|
|
{ 0x1.be08bep+5, 0x1.be0d68p+5, 0x1.be1088p+5, -0x2.0bc074p-24,
|
|
|
|
|
-0x1.b5a622p-4, 0x3.ed52d4p-12, 0x4.b76fc8p-8 }, /* 17 */
|
|
|
|
|
{ 0x1.d7272ap+5, 0x1.d7301ep+5, 0x1.d734aep+5, -0x2.87dd4p-24,
|
|
|
|
|
0x1.a9d184p-4, -0x3.9cf494p-12, -0x4.6303ep-8 }, /* 18 */
|
|
|
|
|
{ 0x1.f0499ap+5, 0x1.f052c4p+5, 0x1.f05758p+5, -0x2.fb964p-24,
|
|
|
|
|
-0x1.9ee5eep-4, 0x3.5800dp-12, 0x4.4e9f9p-8 }, /* 19 */
|
|
|
|
|
{ 0x1.04b63ap+6, 0x1.04baacp+6, 0x1.04bc92p+6, 0x2.cf5adp-24,
|
|
|
|
|
0x1.94c6f4p-4, -0x3.1a83e4p-12, -0x4.2311fp-8 }, /* 20 */
|
|
|
|
|
{ 0x1.1146dp+6, 0x1.114beep+6, 0x1.114e12p+6, 0x3.6766fp-24,
|
|
|
|
|
-0x1.8b5cccp-4, 0x2.e4a4e4p-12, 0x4.20bf9p-8 }, /* 21 */
|
|
|
|
|
{ 0x1.1dda8cp+6, 0x1.1ddd2cp+6, 0x1.1dde7ap+6, 0x3.501424p-24,
|
|
|
|
|
0x1.829356p-4, -0x2.b47524p-12, -0x4.04bf18p-8 }, /* 22 */
|
|
|
|
|
{ 0x1.2a6bcp+6, 0x1.2a6e64p+6, 0x1.2a6faap+6, -0x5.c05808p-24,
|
|
|
|
|
-0x1.7a597ep-4, 0x2.8a0498p-12, 0x4.187258p-8 }, /* 23 */
|
|
|
|
|
{ 0x1.36fcd6p+6, 0x1.36ff96p+6, 0x1.3700f6p+6, 0x7.1e1478p-28,
|
|
|
|
|
0x1.72a09ap-4, -0x2.61a7fp-12, -0x3.c0b54p-8 }, /* 24 */
|
|
|
|
|
{ 0x1.438f46p+6, 0x1.4390c4p+6, 0x1.4392p+6, 0x3.e36e6cp-24,
|
|
|
|
|
-0x1.6b5c06p-4, 0x2.3f612p-12, 0x4.18f868p-8 }, /* 25 */
|
|
|
|
|
{ 0x1.501f4cp+6, 0x1.5021fp+6, 0x1.50235p+6, 0x1.3f9e5ap-24,
|
|
|
|
|
0x1.6480c4p-4, -0x2.1f28fcp-12, -0x3.bb4e3cp-8 }, /* 26 */
|
|
|
|
|
{ 0x1.5cb07cp+6, 0x1.5cb318p+6, 0x1.5cb464p+6, -0x2.39e41cp-24,
|
|
|
|
|
-0x1.5e0544p-4, 0x2.0189f4p-12, 0x3.8b55acp-8 }, /* 27 */
|
|
|
|
|
{ 0x1.694166p+6, 0x1.69443cp+6, 0x1.694594p+6, -0x2.912f84p-24,
|
|
|
|
|
0x1.57e12p-4, -0x1.e6fabep-12, -0x3.850174p-8 }, /* 28 */
|
|
|
|
|
{ 0x1.75d27cp+6, 0x1.75d55ep+6, 0x1.75d67ep+6, 0x3.d5b00cp-24,
|
|
|
|
|
-0x1.520ceep-4, 0x1.d0286ep-12, 0x3.8e7d1p-8 }, /* 29 */
|
|
|
|
|
{ 0x1.82653ep+6, 0x1.82667ep+6, 0x1.82674p+6, -0x3.1726ecp-24,
|
|
|
|
|
0x1.4c8222p-4, -0x1.b98206p-12, -0x3.f34978p-8 }, /* 30 */
|
|
|
|
|
{ 0x1.8ef4b4p+6, 0x1.8ef79cp+6, 0x1.8ef888p+6, 0x1.949e22p-24,
|
|
|
|
|
-0x1.473ae6p-4, 0x1.a47388p-12, 0x3.69eefcp-8 }, /* 31 */
|
|
|
|
|
{ 0x1.9b8728p+6, 0x1.9b88b8p+6, 0x1.9b896cp+6, -0x5.5553bp-28,
|
|
|
|
|
0x1.42320ap-4, -0x1.90f0b8p-12, -0x3.6565p-8 }, /* 32 */
|
|
|
|
|
{ 0x1.a8183cp+6, 0x1.a819d2p+6, 0x1.a81aecp+6, 0x3.2df7ecp-28,
|
|
|
|
|
-0x1.3d62e4p-4, 0x1.7dae28p-12, 0x2.9eb128p-8 }, /* 33 */
|
|
|
|
|
{ 0x1.b4aa1cp+6, 0x1.b4aaeap+6, 0x1.b4abb8p+6, -0x1.e13fcep-24,
|
|
|
|
|
0x1.38c948p-4, -0x1.6eb0ecp-12, -0x1.f9ddf8p-8 }, /* 34 */
|
|
|
|
|
{ 0x1.c13a7ap+6, 0x1.c13c02p+6, 0x1.c13cbp+6, -0x3.ad9974p-24,
|
|
|
|
|
-0x1.34616ep-4, 0x1.5e36ecp-12, 0x2.a9fc5p-8 }, /* 35 */
|
|
|
|
|
{ 0x1.cdcb76p+6, 0x1.cdcd16p+6, 0x1.cdcde4p+6, -0x3.6977e8p-24,
|
|
|
|
|
0x1.3027fp-4, -0x1.4f703p-12, -0x2.9817d4p-8 }, /* 36 */
|
|
|
|
|
{ 0x1.da5cdep+6, 0x1.da5e2ap+6, 0x1.da5efp+6, 0x4.654cbp-24,
|
|
|
|
|
-0x1.2c19b6p-4, 0x1.455982p-12, 0x3.f1c564p-8 }, /* 37 */
|
|
|
|
|
{ 0x1.e6edccp+6, 0x1.e6ef3ep+6, 0x1.e6f00ap+6, 0x8.825c8p-32,
|
|
|
|
|
0x1.2833eep-4, -0x1.39097p-12, -0x3.b2646p-8 }, /* 38 */
|
|
|
|
|
{ 0x1.f37f72p+6, 0x1.f3805p+6, 0x1.f3812ap+6, -0x2.0d11d8p-28,
|
|
|
|
|
-0x1.24740ap-4, 0x1.2c16p-12, 0x1.fc3804p-8 }, /* 39 */
|
|
|
|
|
{ 0x1.000842p+7, 0x1.0008bp+7, 0x1.000908p+7, -0x4.4e495p-24,
|
|
|
|
|
0x1.20d7b6p-4, -0x1.20816p-12, -0x2.d1ebe8p-8 }, /* 40 */
|
|
|
|
|
{ 0x1.06505cp+7, 0x1.065138p+7, 0x1.06518p+7, 0x4.81c1c8p-24,
|
|
|
|
|
-0x1.1d5ccap-4, 0x1.17ad5ap-12, 0x2.fda33p-8 }, /* 41 */
|
|
|
|
|
{ 0x1.0c98dap+7, 0x1.0c99cp+7, 0x1.0c9a28p+7, -0xe.99386p-28,
|
|
|
|
|
0x1.1a015p-4, -0x1.0bd50ap-12, -0x2.9dfb68p-8 }, /* 42 */
|
|
|
|
|
{ 0x1.12e212p+7, 0x1.12e248p+7, 0x1.12e29p+7, -0x6.16f1c8p-24,
|
|
|
|
|
-0x1.16c37ap-4, 0x1.0303dcp-12, 0x4.34316p-8 }, /* 43 */
|
|
|
|
|
{ 0x1.192a68p+7, 0x1.192acep+7, 0x1.192b02p+7, -0x1.129336p-24,
|
|
|
|
|
0x1.13a19ep-4, -0xf.bd247p-16, -0x3.851d18p-8 }, /* 44 */
|
|
|
|
|
{ 0x1.1f727p+7, 0x1.1f7354p+7, 0x1.1f73ap+7, 0x5.19c09p-24,
|
|
|
|
|
-0x1.109a32p-4, 0xf.09644p-16, 0x2.d78194p-8 }, /* 45 */
|
|
|
|
|
{ 0x1.25bb8p+7, 0x1.25bbdap+7, 0x1.25bc12p+7, -0x6.497dp-24,
|
|
|
|
|
0x1.0dabc8p-4, -0xe.a1d25p-16, -0x2.3378bp-8 }, /* 46 */
|
|
|
|
|
{ 0x1.2c04p+7, 0x1.2c046p+7, 0x1.2c04ap+7, 0x4.e4f338p-24,
|
|
|
|
|
-0x1.0ad512p-4, 0xe.52d84p-16, 0x4.3bfa08p-8 }, /* 47 */
|
|
|
|
|
{ 0x1.324cbp+7, 0x1.324ce6p+7, 0x1.324d4p+7, -0x1.287c58p-24,
|
|
|
|
|
0x1.0814d4p-4, -0xe.03a95p-16, 0x3.9930ap-12 }, /* 48 */
|
|
|
|
|
{ 0x1.3894f6p+7, 0x1.38956cp+7, 0x1.3895ap+7, -0x4.b594ep-24,
|
|
|
|
|
-0x1.0569fp-4, 0xd.6787ep-16, 0x4.0a5148p-8 }, /* 49 */
|
|
|
|
|
{ 0x1.3edd98p+7, 0x1.3eddfp+7, 0x1.3ede2ap+7, -0x3.a8f164p-24,
|
|
|
|
|
0x1.02d354p-4, -0xd.0309dp-16, -0x3.2ebfb4p-8 }, /* 50 */
|
|
|
|
|
{ 0x1.452638p+7, 0x1.452676p+7, 0x1.4526b4p+7, -0x6.12505p-24,
|
|
|
|
|
-0x1.005004p-4, 0xc.a0045p-16, 0x4.87c67p-8 }, /* 51 */
|
|
|
|
|
{ 0x1.4b6e8p+7, 0x1.4b6efap+7, 0x1.4b6f34p+7, 0x1.8acf4ep-24,
|
|
|
|
|
0xf.ddf16p-8, -0xc.2d207p-16, -0x1.da6c36p-8 }, /* 52 */
|
|
|
|
|
{ 0x1.51b742p+7, 0x1.51b77ep+7, 0x1.51b7b2p+7, 0x1.39cf86p-24,
|
|
|
|
|
-0xf.b7faep-8, 0xb.db598p-16, -0x8.945b1p-12 }, /* 53 */
|
|
|
|
|
{ 0x1.57ffc4p+7, 0x1.580002p+7, 0x1.58003cp+7, -0x2.5f8de8p-24,
|
|
|
|
|
0xf.930fep-8, -0xb.91889p-16, -0xa.30df9p-12 }, /* 54 */
|
|
|
|
|
{ 0x1.5e483p+7, 0x1.5e4886p+7, 0x1.5e48c8p+7, 0x2.073d64p-24,
|
|
|
|
|
-0xf.6f245p-8, 0xb.4085fp-16, 0x2.128188p-8 }, /* 55 */
|
|
|
|
|
{ 0x1.64908cp+7, 0x1.64910ap+7, 0x1.64912ap+7, -0x4.ed26ep-28,
|
|
|
|
|
0xf.4c2cep-8, -0xa.fe719p-16, -0x2.9374b8p-8 }, /* 56 */
|
|
|
|
|
{ 0x1.6ad91ep+7, 0x1.6ad98ep+7, 0x1.6ad9cep+7, -0x2.ae5204p-24,
|
|
|
|
|
-0xf.2a1efp-8, 0xa.aa585p-16, 0x2.1c0834p-8 }, /* 57 */
|
|
|
|
|
{ 0x1.7121cep+7, 0x1.712212p+7, 0x1.712238p+7, 0x6.d72168p-24,
|
|
|
|
|
0xf.08f09p-8, -0xa.7da49p-16, -0x3.4f5f1cp-8 }, /* 58 */
|
|
|
|
|
{ 0x1.776a0cp+7, 0x1.776a94p+7, 0x1.776accp+7, 0x2.d3f294p-24,
|
|
|
|
|
-0xe.e8986p-8, 0xa.23ccdp-16, 0x2.2a6678p-8 }, /* 59 */
|
|
|
|
|
{ 0x1.7db2e8p+7, 0x1.7db318p+7, 0x1.7db35ap+7, 0x3.88c0fp-24,
|
|
|
|
|
0xe.c90d7p-8, -0x9.eaeap-16, -0x2.86438cp-8 }, /* 60 */
|
|
|
|
|
{ 0x1.83fb56p+7, 0x1.83fb9ap+7, 0x1.83fbep+7, 0x3.d94d34p-24,
|
|
|
|
|
-0xe.aa478p-8, 0x9.abac7p-16, 0x1.ac2d84p-8 }, /* 61 */
|
|
|
|
|
{ 0x1.8a43e8p+7, 0x1.8a441ep+7, 0x1.8a446p+7, 0x4.66b7ep-24,
|
|
|
|
|
0xe.8c3e9p-8, -0x9.87682p-16, -0x7.9ab4a8p-12 }, /* 62 */
|
|
|
|
|
{ 0x1.908c6p+7, 0x1.908cap+7, 0x1.908ce6p+7, 0xf.f7ac9p-28,
|
|
|
|
|
-0xe.6eeb6p-8, 0x9.4423p-16, 0x4.54c4d8p-8 }, /* 63 */
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
|
|
|
|
|
y1f(x) ~ sqrt(2/(pi*x))*beta1(x)*sin(x-3pi/4-alpha1(x))
|
|
|
|
|
where beta1(x) = 1 + 3/(16*x^2) - 99/(512*x^4)
|
|
|
|
|
and alpha1(x) = -3/(8*x) + 21/(128*x^3) - 1899/(5120*x^5). */
|
|
|
|
|
static float
|
|
|
|
|
y1f_asympt (float x)
|
|
|
|
|
{
|
|
|
|
|
float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */
|
|
|
|
|
double y = 1.0 / (double) x;
|
|
|
|
|
double y2 = y * y;
|
|
|
|
|
double beta1 = 1.0f + y2 * (0x3p-4 - 0x3.18p-4 * y2);
|
|
|
|
|
double alpha1;
|
|
|
|
|
alpha1 = y * (-0x6p-4 + y2 * (0x2.ap-4 - 0x5.ef33333333334p-4 * y2));
|
|
|
|
|
double h;
|
|
|
|
|
int n;
|
|
|
|
|
h = reduce_aux (x, &n, alpha1);
|
|
|
|
|
n--; /* Subtract pi/2. */
|
|
|
|
|
/* Now x - 3pi/4 - alpha1 = h + n*pi/2 mod (2*pi). */
|
|
|
|
|
float xr = (float) h;
|
|
|
|
|
n = n & 3;
|
|
|
|
|
float t = cst / sqrtf (x) * (float) beta1;
|
|
|
|
|
if (n == 0)
|
|
|
|
|
return t * __sinf (xr);
|
|
|
|
|
else if (n == 2) /* sin(x+pi) = -sin(x) */
|
|
|
|
|
return -t * __sinf (xr);
|
|
|
|
|
else if (n == 1) /* sin(x+pi/2) = cos(x) */
|
|
|
|
|
return t * __cosf (xr);
|
|
|
|
|
else /* sin(x+3pi/2) = -cos(x) */
|
|
|
|
|
return -t * __cosf (xr);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* Special code for x near a root of y1.
|
|
|
|
|
z is the value computed by the generic code.
|
|
|
|
|
For small x, we use a polynomial approximating y1 around its root.
|
|
|
|
|
For large x, we use an asymptotic formula (y1f_asympt). */
|
|
|
|
|
static float
|
|
|
|
|
y1f_near_root (float x, float z)
|
|
|
|
|
{
|
|
|
|
|
float index_f;
|
|
|
|
|
int index;
|
|
|
|
|
|
|
|
|
|
index_f = roundf ((x - FIRST_ZERO_Y1) / (float) M_PI);
|
|
|
|
|
if (index_f >= SMALL_SIZE)
|
|
|
|
|
return y1f_asympt (x);
|
|
|
|
|
index = (int) index_f;
|
|
|
|
|
const float *p = Py[index];
|
|
|
|
|
float x0 = p[0];
|
|
|
|
|
float x1 = p[2];
|
|
|
|
|
/* If not in the interval [x0,x1] around xmid, return the value z. */
|
|
|
|
|
if (! (x0 <= x && x <= x1))
|
|
|
|
|
return z;
|
|
|
|
|
float xmid = p[1];
|
|
|
|
|
float y = x - xmid, p6;
|
|
|
|
|
if (index == 0)
|
|
|
|
|
p6 = p[6] + y * (-0x1.28043p-8 + y * 0x2.50e83p-8);
|
|
|
|
|
else if (index == 1)
|
|
|
|
|
p6 = p[6] + y * -0xf.ff6b8p-12;
|
|
|
|
|
else
|
|
|
|
|
p6 = p[6];
|
|
|
|
|
return p[3] + y * (p[4] + y * (p[5] + y * p6));
|
|
|
|
|
}
|
|
|
|
|
|
2011-10-12 15:27:51 +00:00
|
|
|
float
|
|
|
|
|
__ieee754_y1f(float x)
|
1996-03-05 21:41:30 +00:00
|
|
|
{
|
|
|
|
|
float z, s,c,ss,cc,u,v;
|
|
|
|
|
int32_t hx,ix;
|
|
|
|
|
|
|
|
|
|
GET_FLOAT_WORD(hx,x);
|
2011-10-12 15:27:51 +00:00
|
|
|
ix = 0x7fffffff&hx;
|
1996-03-05 21:41:30 +00:00
|
|
|
/* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
|
2011-10-12 15:27:51 +00:00
|
|
|
if(__builtin_expect(ix>=0x7f800000, 0)) return one/(x+x*x);
|
|
|
|
|
if(__builtin_expect(ix==0, 0))
|
2017-02-13 00:36:27 +00:00
|
|
|
return -1/zero; /* -inf and divide by zero exception. */
|
2011-10-12 15:27:51 +00:00
|
|
|
if(__builtin_expect(hx<0, 0)) return zero/(zero*x);
|
Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
|
|
|
if (ix >= 0x3fe0dfbc) { /* |x| >= 0x1.c1bf78p+0 */
|
2014-05-25 01:58:01 +00:00
|
|
|
SET_RESTORE_ROUNDF (FE_TONEAREST);
|
2001-02-13 01:23:48 +00:00
|
|
|
__sincosf (x, &s, &c);
|
2011-10-12 15:27:51 +00:00
|
|
|
ss = -s-c;
|
|
|
|
|
cc = s-c;
|
Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
|
|
|
if (ix >= 0x7f000000)
|
|
|
|
|
/* x >= 2^127: use asymptotic expansion. */
|
|
|
|
|
return y1f_asympt (x);
|
|
|
|
|
/* Now we are sure that x+x cannot overflow. */
|
|
|
|
|
z = __cosf(x+x);
|
|
|
|
|
if ((s*c)>zero) cc = z/ss;
|
|
|
|
|
else ss = z/cc;
|
2011-10-12 15:27:51 +00:00
|
|
|
/* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
|
|
|
|
|
* where x0 = x-3pi/4
|
|
|
|
|
* Better formula:
|
|
|
|
|
* cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
|
|
|
|
|
* = 1/sqrt(2) * (sin(x) - cos(x))
|
|
|
|
|
* sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
|
|
|
|
|
* = -1/sqrt(2) * (cos(x) + sin(x))
|
|
|
|
|
* To avoid cancellation, use
|
|
|
|
|
* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
|
|
|
|
|
* to compute the worse one.
|
|
|
|
|
*/
|
Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
|
|
|
if (ix <= 0x5c000000)
|
|
|
|
|
{
|
|
|
|
|
u = ponef(x); v = qonef(x);
|
|
|
|
|
ss = u*ss+v*cc;
|
|
|
|
|
}
|
|
|
|
|
z = (invsqrtpi * ss) / sqrtf(x);
|
|
|
|
|
float threshold = 0x1.3e014cp-2;
|
|
|
|
|
/* The following threshold is optimal: for x=0x1.f7f16ap+0
|
|
|
|
|
and rounding upwards, |ss|=-0x1.3e014cp-2 and z is 11 ulps
|
|
|
|
|
far from the correctly rounded value. */
|
|
|
|
|
if (fabsf (ss) > threshold)
|
|
|
|
|
return z;
|
|
|
|
|
else
|
|
|
|
|
return y1f_near_root (x, z);
|
2011-10-12 15:27:51 +00:00
|
|
|
}
|
2012-11-18 20:33:53 +00:00
|
|
|
if(__builtin_expect(ix<=0x33000000, 0)) { /* x < 2**-25 */
|
2014-06-23 20:17:13 +00:00
|
|
|
z = -tpi / x;
|
2015-06-03 14:36:34 +00:00
|
|
|
if (isinf (z))
|
2014-06-23 20:17:13 +00:00
|
|
|
__set_errno (ERANGE);
|
|
|
|
|
return z;
|
2011-10-12 15:27:51 +00:00
|
|
|
}
|
Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
|
|
|
/* Now 2**-25 <= x < 0x1.c1bf78p+0. */
|
2011-10-12 15:27:51 +00:00
|
|
|
z = x*x;
|
|
|
|
|
u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
|
|
|
|
|
v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
|
|
|
|
|
return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
|
1996-03-05 21:41:30 +00:00
|
|
|
}
|
2019-07-16 15:17:22 +00:00
|
|
|
libm_alias_finite (__ieee754_y1f, __y1f)
|
1996-03-05 21:41:30 +00:00
|
|
|
|
Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
|
|
|
/* For x >= 8, the asymptotic expansion of pone is
|
1996-03-05 21:41:30 +00:00
|
|
|
* 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
|
|
|
|
|
* We approximate pone by
|
2011-10-12 15:27:51 +00:00
|
|
|
* pone(x) = 1 + (R/S)
|
1996-03-05 21:41:30 +00:00
|
|
|
* where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
|
2011-10-12 15:27:51 +00:00
|
|
|
* S = 1 + ps0*s^2 + ... + ps4*s^10
|
1996-03-05 21:41:30 +00:00
|
|
|
* and
|
|
|
|
|
* | pone(x)-1-R/S | <= 2 ** ( -60.06)
|
|
|
|
|
*/
|
|
|
|
|
|
|
|
|
|
static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
|
|
|
|
|
0.0000000000e+00, /* 0x00000000 */
|
|
|
|
|
1.1718750000e-01, /* 0x3df00000 */
|
|
|
|
|
1.3239480972e+01, /* 0x4153d4ea */
|
|
|
|
|
4.1205184937e+02, /* 0x43ce06a3 */
|
|
|
|
|
3.8747453613e+03, /* 0x45722bed */
|
|
|
|
|
7.9144794922e+03, /* 0x45f753d6 */
|
|
|
|
|
};
|
|
|
|
|
static const float ps8[5] = {
|
|
|
|
|
1.1420736694e+02, /* 0x42e46a2c */
|
|
|
|
|
3.6509309082e+03, /* 0x45642ee5 */
|
|
|
|
|
3.6956207031e+04, /* 0x47105c35 */
|
|
|
|
|
9.7602796875e+04, /* 0x47bea166 */
|
|
|
|
|
3.0804271484e+04, /* 0x46f0a88b */
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
|
|
|
|
|
1.3199052094e-11, /* 0x2d68333f */
|
|
|
|
|
1.1718749255e-01, /* 0x3defffff */
|
|
|
|
|
6.8027510643e+00, /* 0x40d9b023 */
|
|
|
|
|
1.0830818176e+02, /* 0x42d89dca */
|
|
|
|
|
5.1763616943e+02, /* 0x440168b7 */
|
|
|
|
|
5.2871520996e+02, /* 0x44042dc6 */
|
|
|
|
|
};
|
|
|
|
|
static const float ps5[5] = {
|
|
|
|
|
5.9280597687e+01, /* 0x426d1f55 */
|
|
|
|
|
9.9140142822e+02, /* 0x4477d9b1 */
|
|
|
|
|
5.3532670898e+03, /* 0x45a74a23 */
|
|
|
|
|
7.8446904297e+03, /* 0x45f52586 */
|
|
|
|
|
1.5040468750e+03, /* 0x44bc0180 */
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
static const float pr3[6] = {
|
|
|
|
|
3.0250391081e-09, /* 0x314fe10d */
|
|
|
|
|
1.1718686670e-01, /* 0x3defffab */
|
|
|
|
|
3.9329774380e+00, /* 0x407bb5e7 */
|
|
|
|
|
3.5119403839e+01, /* 0x420c7a45 */
|
|
|
|
|
9.1055007935e+01, /* 0x42b61c2a */
|
|
|
|
|
4.8559066772e+01, /* 0x42423c7c */
|
|
|
|
|
};
|
|
|
|
|
static const float ps3[5] = {
|
|
|
|
|
3.4791309357e+01, /* 0x420b2a4d */
|
|
|
|
|
3.3676245117e+02, /* 0x43a86198 */
|
|
|
|
|
1.0468714600e+03, /* 0x4482dbe3 */
|
|
|
|
|
8.9081134033e+02, /* 0x445eb3ed */
|
|
|
|
|
1.0378793335e+02, /* 0x42cf936c */
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
|
|
|
|
|
1.0771083225e-07, /* 0x33e74ea8 */
|
|
|
|
|
1.1717621982e-01, /* 0x3deffa16 */
|
|
|
|
|
2.3685150146e+00, /* 0x401795c0 */
|
|
|
|
|
1.2242610931e+01, /* 0x4143e1bc */
|
|
|
|
|
1.7693971634e+01, /* 0x418d8d41 */
|
|
|
|
|
5.0735230446e+00, /* 0x40a25a4d */
|
|
|
|
|
};
|
|
|
|
|
static const float ps2[5] = {
|
|
|
|
|
2.1436485291e+01, /* 0x41ab7dec */
|
|
|
|
|
1.2529022980e+02, /* 0x42fa9499 */
|
|
|
|
|
2.3227647400e+02, /* 0x436846c7 */
|
|
|
|
|
1.1767937469e+02, /* 0x42eb5bd7 */
|
|
|
|
|
8.3646392822e+00, /* 0x4105d590 */
|
|
|
|
|
};
|
|
|
|
|
|
2011-10-12 15:27:51 +00:00
|
|
|
static float
|
|
|
|
|
ponef(float x)
|
1996-03-05 21:41:30 +00:00
|
|
|
{
|
|
|
|
|
const float *p,*q;
|
|
|
|
|
float z,r,s;
|
2011-10-12 15:27:51 +00:00
|
|
|
int32_t ix;
|
1996-03-05 21:41:30 +00:00
|
|
|
GET_FLOAT_WORD(ix,x);
|
|
|
|
|
ix &= 0x7fffffff;
|
2015-02-26 21:49:19 +00:00
|
|
|
/* ix >= 0x40000000 for all calls to this function. */
|
2011-10-12 15:27:51 +00:00
|
|
|
if(ix>=0x41000000) {p = pr8; q= ps8;}
|
|
|
|
|
else if(ix>=0x40f71c58){p = pr5; q= ps5;}
|
|
|
|
|
else if(ix>=0x4036db68){p = pr3; q= ps3;}
|
2015-02-26 21:49:19 +00:00
|
|
|
else {p = pr2; q= ps2;}
|
2011-10-12 15:27:51 +00:00
|
|
|
z = one/(x*x);
|
|
|
|
|
r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
|
|
|
|
|
s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
|
|
|
|
|
return one+ r/s;
|
1996-03-05 21:41:30 +00:00
|
|
|
}
|
2000-10-25 22:17:16 +00:00
|
|
|
|
Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
|
|
|
/* For x >= 8, the asymptotic expansion of qone is
|
1996-03-05 21:41:30 +00:00
|
|
|
* 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
|
|
|
|
|
* We approximate pone by
|
2011-10-12 15:27:51 +00:00
|
|
|
* qone(x) = s*(0.375 + (R/S))
|
1996-03-05 21:41:30 +00:00
|
|
|
* where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
|
2011-10-12 15:27:51 +00:00
|
|
|
* S = 1 + qs1*s^2 + ... + qs6*s^12
|
1996-03-05 21:41:30 +00:00
|
|
|
* and
|
|
|
|
|
* | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
|
|
|
|
|
*/
|
|
|
|
|
|
|
|
|
|
static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
|
|
|
|
|
0.0000000000e+00, /* 0x00000000 */
|
|
|
|
|
-1.0253906250e-01, /* 0xbdd20000 */
|
|
|
|
|
-1.6271753311e+01, /* 0xc1822c8d */
|
|
|
|
|
-7.5960174561e+02, /* 0xc43de683 */
|
|
|
|
|
-1.1849806641e+04, /* 0xc639273a */
|
|
|
|
|
-4.8438511719e+04, /* 0xc73d3683 */
|
|
|
|
|
};
|
|
|
|
|
static const float qs8[6] = {
|
|
|
|
|
1.6139537048e+02, /* 0x43216537 */
|
|
|
|
|
7.8253862305e+03, /* 0x45f48b17 */
|
|
|
|
|
1.3387534375e+05, /* 0x4802bcd6 */
|
|
|
|
|
7.1965775000e+05, /* 0x492fb29c */
|
|
|
|
|
6.6660125000e+05, /* 0x4922be94 */
|
|
|
|
|
-2.9449025000e+05, /* 0xc88fcb48 */
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
|
|
|
|
|
-2.0897993405e-11, /* 0xadb7d219 */
|
|
|
|
|
-1.0253904760e-01, /* 0xbdd1fffe */
|
|
|
|
|
-8.0564479828e+00, /* 0xc100e736 */
|
|
|
|
|
-1.8366960144e+02, /* 0xc337ab6b */
|
|
|
|
|
-1.3731937256e+03, /* 0xc4aba633 */
|
|
|
|
|
-2.6124443359e+03, /* 0xc523471c */
|
|
|
|
|
};
|
|
|
|
|
static const float qs5[6] = {
|
|
|
|
|
8.1276550293e+01, /* 0x42a28d98 */
|
|
|
|
|
1.9917987061e+03, /* 0x44f8f98f */
|
|
|
|
|
1.7468484375e+04, /* 0x468878f8 */
|
|
|
|
|
4.9851425781e+04, /* 0x4742bb6d */
|
|
|
|
|
2.7948074219e+04, /* 0x46da5826 */
|
|
|
|
|
-4.7191835938e+03, /* 0xc5937978 */
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
static const float qr3[6] = {
|
|
|
|
|
-5.0783124372e-09, /* 0xb1ae7d4f */
|
|
|
|
|
-1.0253783315e-01, /* 0xbdd1ff5b */
|
|
|
|
|
-4.6101160049e+00, /* 0xc0938612 */
|
|
|
|
|
-5.7847221375e+01, /* 0xc267638e */
|
|
|
|
|
-2.2824453735e+02, /* 0xc3643e9a */
|
|
|
|
|
-2.1921012878e+02, /* 0xc35b35cb */
|
|
|
|
|
};
|
|
|
|
|
static const float qs3[6] = {
|
|
|
|
|
4.7665153503e+01, /* 0x423ea91e */
|
|
|
|
|
6.7386511230e+02, /* 0x4428775e */
|
|
|
|
|
3.3801528320e+03, /* 0x45534272 */
|
|
|
|
|
5.5477290039e+03, /* 0x45ad5dd5 */
|
|
|
|
|
1.9031191406e+03, /* 0x44ede3d0 */
|
|
|
|
|
-1.3520118713e+02, /* 0xc3073381 */
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
|
|
|
|
|
-1.7838172539e-07, /* 0xb43f8932 */
|
|
|
|
|
-1.0251704603e-01, /* 0xbdd1f475 */
|
|
|
|
|
-2.7522056103e+00, /* 0xc0302423 */
|
|
|
|
|
-1.9663616180e+01, /* 0xc19d4f16 */
|
|
|
|
|
-4.2325313568e+01, /* 0xc2294d1f */
|
|
|
|
|
-2.1371921539e+01, /* 0xc1aaf9b2 */
|
|
|
|
|
};
|
|
|
|
|
static const float qs2[6] = {
|
|
|
|
|
2.9533363342e+01, /* 0x41ec4454 */
|
|
|
|
|
2.5298155212e+02, /* 0x437cfb47 */
|
|
|
|
|
7.5750280762e+02, /* 0x443d602e */
|
|
|
|
|
7.3939318848e+02, /* 0x4438d92a */
|
|
|
|
|
1.5594900513e+02, /* 0x431bf2f2 */
|
|
|
|
|
-4.9594988823e+00, /* 0xc09eb437 */
|
|
|
|
|
};
|
|
|
|
|
|
2011-10-12 15:27:51 +00:00
|
|
|
static float
|
|
|
|
|
qonef(float x)
|
1996-03-05 21:41:30 +00:00
|
|
|
{
|
|
|
|
|
const float *p,*q;
|
|
|
|
|
float s,r,z;
|
|
|
|
|
int32_t ix;
|
|
|
|
|
GET_FLOAT_WORD(ix,x);
|
|
|
|
|
ix &= 0x7fffffff;
|
2015-02-26 21:49:19 +00:00
|
|
|
/* ix >= 0x40000000 for all calls to this function. */
|
Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
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if(ix>=0x41000000) {p = qr8; q= qs8;} /* x >= 8 */
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else if(ix>=0x40f71c58){p = qr5; q= qs5;} /* x >= 7.722209930e+00 */
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else if(ix>=0x4036db68){p = qr3; q= qs3;} /* x >= 2.857141495e+00 */
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else {p = qr2; q= qs2;} /* x >= 2 */
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1996-03-05 21:41:30 +00:00
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z = one/(x*x);
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r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
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s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
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return ((float).375 + r/s)/x;
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}
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