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347a5b592c
Converting double precision constants to float is now affected by the runtime dynamic rounding mode instead of being evaluated at compile time with default rounding mode (except static object initializers). This can change the computed result and cause performance regression. The known correctness issues (increased ulp errors) are already fixed, this patch fixes remaining cases of unnecessary runtime conversions. Add float M_* macros to math.h as new GNU extension API. To avoid conversions the new M_* macros are used and instead of casting double literals to float, use float literals (only required if the conversion is inexact). The patch was tested on aarch64 where the following symbols had new spurious conversion instructions that got fixed: __clog10f __gammaf_r_finite@GLIBC_2.17 __j0f_finite@GLIBC_2.17 __j1f_finite@GLIBC_2.17 __jnf_finite@GLIBC_2.17 __kernel_casinhf __lgamma_negf __log1pf __y0f_finite@GLIBC_2.17 __y1f_finite@GLIBC_2.17 cacosf cacoshf casinhf catanf catanhf clogf gammaf_positive Fixes bug 28713. Reviewed-by: Paul Zimmermann <Paul.Zimmermann@inria.fr>
789 lines
33 KiB
C
789 lines
33 KiB
C
/* e_j0f.c -- float version of e_j0.c.
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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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* 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 <math.h>
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#include <math-barriers.h>
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#include <math_private.h>
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#include <fenv_private.h>
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#include <libm-alias-finite.h>
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#include <reduce_aux.h>
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static float pzerof(float), qzerof(float);
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static const float
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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, 2.00] */
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R02 = 1.5625000000e-02, /* 0x3c800000 */
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R03 = -1.8997929874e-04, /* 0xb947352e */
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R04 = 1.8295404516e-06, /* 0x35f58e88 */
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R05 = -4.6183270541e-09, /* 0xb19eaf3c */
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S01 = 1.5619102865e-02, /* 0x3c7fe744 */
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S02 = 1.1692678527e-04, /* 0x38f53697 */
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S03 = 5.1354652442e-07, /* 0x3509daa6 */
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S04 = 1.1661400734e-09; /* 0x30a045e8 */
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static const float zero = 0.0;
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/* This is the nearest approximation of the first zero of j0. */
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#define FIRST_ZERO_J0 0x1.33d152p+1f
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#define SMALL_SIZE 64
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/* The following table contains successive zeros of j0 and degree-3
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polynomial approximations of j0 around these zeros: Pj[0] for the first
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zero (2.40482), Pj[1] for the second one (5.520078), 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 <= 9 ulps.
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*/
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static const float Pj[SMALL_SIZE][7] = {
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/* The following polynomial was optimized by hand with respect to the one
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generated by Sollya, to ensure the maximal error is at most 9 ulps,
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both if the polynomial is evaluated with fma or not. */
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{ 0x1.31e5c4p+1, 0x1.33d152p+1, 0x1.3b58dep+1, 0xf.2623fp-28,
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-0x8.4e6d7p-4, 0x1.ba2aaap-4, 0xe.4b9ap-8 }, /* 0 */
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{ 0x1.60eafap+2, 0x1.6148f6p+2, 0x1.62955cp+2, 0x6.9205fp-28,
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0x5.71b98p-4, -0x7.e3e798p-8, -0xd.87d1p-8 }, /* 1 */
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{ 0x1.14cde2p+3, 0x1.14eb56p+3, 0x1.1525c6p+3, 0x1.bcc1cap-24,
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-0x4.57de6p-4, 0x4.03e7cp-8, 0xb.39a37p-8 }, /* 2 */
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{ 0x1.7931d8p+3, 0x1.79544p+3, 0x1.7998d6p+3, -0xf.2976fp-32,
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0x3.b827ccp-4, -0x2.8603ep-8, -0x9.bf49bp-8 }, /* 3 */
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{ 0x1.ddb6d4p+3, 0x1.ddca14p+3, 0x1.ddf0c8p+3, -0x1.bd67d8p-28,
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-0x3.4e03ap-4, 0x1.c562a2p-8, 0x8.90ec2p-8 }, /* 4 */
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{ 0x1.2118e4p+4, 0x1.212314p+4, 0x1.21375p+4, 0x1.62209cp-28,
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0x3.00efecp-4, -0x1.5458dap-8, -0x8.10063p-8 }, /* 5 */
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{ 0x1.535d28p+4, 0x1.5362dep+4, 0x1.536e48p+4, -0x2.853f74p-24,
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-0x2.c5b274p-4, 0x1.0b9db4p-8, 0x7.8c3578p-8 }, /* 6 */
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{ 0x1.859ddp+4, 0x1.85a3bap+4, 0x1.85aff4p+4, 0x2.19ed1cp-24,
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0x2.96545cp-4, -0xd.997e6p-12, -0x6.d9af28p-8 }, /* 7 */
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{ 0x1.b7decap+4, 0x1.b7e54ap+4, 0x1.b7f038p+4, 0xe.959aep-28,
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-0x2.6f5594p-4, 0xb.538dp-12, 0x7.003ea8p-8 }, /* 8 */
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{ 0x1.ea21c6p+4, 0x1.ea275ap+4, 0x1.ea337ap+4, 0x2.0c3964p-24,
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0x2.4e80fcp-4, -0x9.a2216p-12, -0x6.61e0a8p-8 }, /* 9 */
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{ 0x1.0e3316p+5, 0x1.0e34e2p+5, 0x1.0e379ap+5, -0x3.642554p-24,
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-0x2.325e48p-4, 0x8.4f49cp-12, 0x7.d37c3p-8 }, /* 10 */
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{ 0x1.275456p+5, 0x1.275638p+5, 0x1.2759e2p+5, 0x1.6c015ap-24,
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0x2.19e7d8p-4, -0x7.4c1bf8p-12, -0x4.af7ef8p-8 }, /* 11 */
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{ 0x1.4075ecp+5, 0x1.4077a8p+5, 0x1.407b96p+5, -0x4.b18c9p-28,
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-0x2.046174p-4, 0x6.705618p-12, 0x5.f2d28p-8 }, /* 12 */
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{ 0x1.59973p+5, 0x1.59992cp+5, 0x1.599b2ap+5, -0x1.8b8792p-24,
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0x1.f13fbp-4, -0x5.c14938p-12, -0x5.73e0cp-8 }, /* 13 */
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{ 0x1.72b958p+5, 0x1.72bacp+5, 0x1.72bc5ap+5, 0x3.a26e0cp-24,
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-0x1.e018dap-4, 0x5.30e8dp-12, 0x2.81099p-8 }, /* 14 */
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{ 0x1.8bdb4ap+5, 0x1.8bdc62p+5, 0x1.8bde7ep+5, -0x2.18fabcp-24,
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0x1.d09b22p-4, -0x4.b0b688p-12, -0x5.5fd308p-8 }, /* 15 */
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{ 0x1.a4fcecp+5, 0x1.a4fe0ep+5, 0x1.a50042p+5, 0x3.2370e8p-24,
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-0x1.c28614p-4, 0x4.4647e8p-12, 0x5.68a28p-8 }, /* 16 */
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{ 0x1.be1ebcp+5, 0x1.be1fc4p+5, 0x1.be21fp+5, -0x5.9eae3p-28,
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0x1.b5a622p-4, -0x3.eb9054p-12, -0x5.12d8cp-8 }, /* 17 */
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{ 0x1.d7405p+5, 0x1.d7418p+5, 0x1.d74294p+5, 0x2.9fa1e8p-24,
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-0x1.a9d184p-4, 0x3.9d1e7p-12, 0x4.33d058p-8 }, /* 18 */
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{ 0x1.f0624p+5, 0x1.f06344p+5, 0x1.f0645ep+5, 0x9.9ac67p-28,
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0x1.9ee5eep-4, -0x3.5816e8p-12, -0x2.6e5004p-8 }, /* 19 */
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{ 0x1.04c22ep+6, 0x1.04c286p+6, 0x1.04c316p+6, 0xd.6ab94p-28,
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-0x1.94c6f6p-4, 0x3.174efcp-12, 0x7.9a092p-8 }, /* 20 */
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{ 0x1.1153p+6, 0x1.11536cp+6, 0x1.11541p+6, -0x4.4cb2d8p-24,
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0x1.8b5cccp-4, -0x2.e3c238p-12, -0x4.e5437p-8 }, /* 21 */
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{ 0x1.1de3d8p+6, 0x1.1de456p+6, 0x1.1de4dap+6, -0x4.4aa8c8p-24,
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-0x1.829356p-4, 0x2.b45124p-12, 0x5.baf638p-8 }, /* 22 */
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{ 0x1.2a74f8p+6, 0x1.2a754p+6, 0x1.2a75bp+6, 0x2.077c38p-24,
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0x1.7a597ep-4, -0x2.8a0414p-12, -0x2.838d3p-8 }, /* 23 */
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{ 0x1.3705d4p+6, 0x1.37062cp+6, 0x1.3706b2p+6, -0x2.6a6cd8p-24,
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-0x1.72a09ap-4, 0x2.623a3cp-12, 0x5.5256a8p-8 }, /* 24 */
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{ 0x1.4396dp+6, 0x1.439718p+6, 0x1.43976ep+6, -0x5.08287p-24,
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0x1.6b5c06p-4, -0x2.3da154p-12, -0x7.a2254p-8 }, /* 25 */
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{ 0x1.5027acp+6, 0x1.502808p+6, 0x1.50288cp+6, -0x3.4598dcp-24,
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-0x1.6480c4p-4, 0x2.1cb944p-12, 0x7.27c77p-8 }, /* 26 */
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{ 0x1.5cb89ap+6, 0x1.5cb8f8p+6, 0x1.5cb97ep+6, 0x5.4e74bp-24,
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0x1.5e0544p-4, -0x2.00b158p-12, -0x5.9bc4a8p-8 }, /* 27 */
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{ 0x1.69498cp+6, 0x1.6949e8p+6, 0x1.694a42p+6, -0x2.05751cp-24,
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-0x1.57e12p-4, 0x1.e78edcp-12, 0x9.9667dp-8 }, /* 28 */
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{ 0x1.75da7ep+6, 0x1.75dadap+6, 0x1.75db3p+6, 0x4.c5e278p-24,
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0x1.520ceep-4, -0x1.d0127cp-12, -0xd.62681p-8 }, /* 29 */
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{ 0x1.826b7ep+6, 0x1.826bccp+6, 0x1.826c2cp+6, -0x3.50e62cp-24,
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-0x1.4c822p-4, 0x1.ba5832p-12, -0x1.eb2ee2p-8 }, /* 30 */
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{ 0x1.8efc84p+6, 0x1.8efcbep+6, 0x1.8efd16p+6, -0x1.c39f38p-24,
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0x1.473ae6p-4, -0x1.a616c8p-12, 0xf.f352ap-12 }, /* 31 */
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{ 0x1.9b8d84p+6, 0x1.9b8db2p+6, 0x1.9b8e7p+6, -0x1.9245b6p-28,
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-0x1.42320ap-4, 0x1.932a04p-12, 0x2.dc113cp-8 }, /* 32 */
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{ 0x1.a81e72p+6, 0x1.a81ea6p+6, 0x1.a81f04p+6, -0x1.0acf8p-24,
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0x1.3d62e6p-4, -0x1.7c4b14p-12, -0x1.cfc5c2p-4 }, /* 33 */
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{ 0x1.b4af6ap+6, 0x1.b4af9ap+6, 0x1.b4afeep+6, 0x4.cd92d8p-24,
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-0x1.38c94ap-4, 0x1.643154p-12, 0x1.4c2a06p-4 }, /* 34 */
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{ 0x1.c1406p+6, 0x1.c1409p+6, 0x1.c140cp+6, -0x1.37bf8ap-24,
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0x1.34617p-4, -0x1.5f504ap-12, -0x1.e2d324p-4 }, /* 35 */
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{ 0x1.cdd154p+6, 0x1.cdd186p+6, 0x1.cdd1eap+6, -0x1.8f62dep-28,
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-0x1.3027fp-4, 0x1.534a02p-12, 0x2.c7f144p-12 }, /* 36 */
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{ 0x1.da6248p+6, 0x1.da627cp+6, 0x1.da62e6p+6, -0x9.81e79p-28,
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0x1.2c19b4p-4, -0x1.4b8288p-12, 0x7.2d8bap-8 }, /* 37 */
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{ 0x1.e6f33ep+6, 0x1.e6f372p+6, 0x1.e6f3a8p+6, 0x3.103b3p-24,
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-0x1.2833eep-4, 0x1.36f4d2p-12, 0x9.29f91p-8 }, /* 38 */
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{ 0x1.f38434p+6, 0x1.f3846ap+6, 0x1.f384d8p+6, 0x2.07b058p-24,
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0x1.24740ap-4, -0x1.2ee58ap-12, 0xd.f1393p-12 }, /* 39 */
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{ 0x1.000a98p+7, 0x1.000abp+7, 0x1.000ac8p+7, 0x3.87576cp-24,
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-0x1.20d7b6p-4, 0x1.2083e2p-12, 0x3.9a7aap-8 }, /* 40 */
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{ 0x1.06531p+7, 0x1.06532cp+7, 0x1.065348p+7, -0x1.691ecp-24,
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0x1.1d5ccap-4, -0x1.166726p-12, -0x1.e4af48p-8 }, /* 41 */
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{ 0x1.0c9b9ap+7, 0x1.0c9ba8p+7, 0x1.0c9bbep+7, 0x9.b406dp-28,
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-0x1.1a015p-4, 0x1.038f9cp-12, -0x4.021058p-4 }, /* 42 */
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{ 0x1.12e412p+7, 0x1.12e424p+7, 0x1.12e436p+7, -0xf.bfd8fp-28,
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0x1.16c37ap-4, -0x1.039edep-12, 0x1.f0033p-4 }, /* 43 */
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{ 0x1.192c92p+7, 0x1.192cap+7, 0x1.192cb6p+7, 0x2.6d50c8p-24,
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-0x1.13a19ep-4, 0xf.9df8ap-16, 0x4.ecd978p-8 }, /* 44 */
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{ 0x1.1f7512p+7, 0x1.1f751cp+7, 0x1.1f753ap+7, -0x4.d475c8p-24,
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0x1.109a32p-4, -0x1.04fb3ap-12, -0xd.c271p-12 }, /* 45 */
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{ 0x1.25bd8ep+7, 0x1.25bd98p+7, 0x1.25bdap+7, 0x8.1982p-24,
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-0x1.0dabc8p-4, 0xe.88eabp-16, -0x4.ed75dp-4 }, /* 46 */
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{ 0x1.2c060cp+7, 0x1.2c0616p+7, 0x1.2c0644p+7, 0x4.864518p-24,
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0x1.0ad51p-4, -0xe.27196p-16, 0xb.97a3ep-8 }, /* 47 */
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{ 0x1.324e86p+7, 0x1.324e92p+7, 0x1.324e9ep+7, 0x6.8917a8p-28,
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-0x1.0814d4p-4, 0xd.4fe7ep-16, -0x6.8d8d6p-4 }, /* 48 */
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{ 0x1.389702p+7, 0x1.38970ep+7, 0x1.389728p+7, -0x5.fa18fp-24,
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0x1.0569fp-4, -0xd.5b0d4p-16, 0x1.50353ap-4 }, /* 49 */
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{ 0x1.3edf84p+7, 0x1.3edf8cp+7, 0x1.3edfaap+7, -0x4.0e5c98p-24,
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-0x1.02d354p-4, 0xb.7b255p-16, 0x7.8a916p-4 }, /* 50 */
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{ 0x1.4527fp+7, 0x1.452808p+7, 0x1.452812p+7, -0x2.c3ddbp-24,
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0x1.005004p-4, -0xd.7729cp-16, -0x3.bcc354p-8 }, /* 51 */
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{ 0x1.4b7076p+7, 0x1.4b7086p+7, 0x1.4b70a4p+7, -0x5.d052p-24,
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-0xf.ddf16p-8, 0xc.318c1p-16, 0x5.7947p-8 }, /* 52 */
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{ 0x1.51b8f4p+7, 0x1.51b902p+7, 0x1.51b90ep+7, -0x2.0b97dcp-24,
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0xf.b7fafp-8, -0xc.1429dp-16, -0x3.43c36p-4 }, /* 53 */
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{ 0x1.580168p+7, 0x1.58018p+7, 0x1.580188p+7, -0x5.4aab5p-24,
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-0xf.930fep-8, 0xa.ecc24p-16, 0x9.c62cdp-12 }, /* 54 */
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{ 0x1.5e49eap+7, 0x1.5e49fcp+7, 0x1.5e4a12p+7, -0x3.6dadd8p-24,
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0xf.6f245p-8, -0xb.6816cp-16, 0xa.d731ap-8 }, /* 55 */
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{ 0x1.649272p+7, 0x1.64927ap+7, 0x1.64929p+7, -0x2.d7e038p-24,
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-0xf.4c2cep-8, 0xb.118bep-16, 0xb.69a4ep-8 }, /* 56 */
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{ 0x1.6adae6p+7, 0x1.6adaf6p+7, 0x1.6adb04p+7, -0x6.977a1p-24,
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0xf.2a1fp-8, -0xa.a8911p-16, -0x4.bf6d2p-8 }, /* 57 */
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{ 0x1.712366p+7, 0x1.712374p+7, 0x1.71238ep+7, 0x1.3cc95ep-24,
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-0xf.08f0ap-8, 0x9.f0858p-16, 0x1.77f7f4p-4 }, /* 58 */
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{ 0x1.776beap+7, 0x1.776bf2p+7, 0x1.776bfap+7, 0x3.a4921p-24,
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0xe.e8986p-8, -0xa.39dfp-16, -0x6.7ba3dp-4 }, /* 59 */
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{ 0x1.7db464p+7, 0x1.7db46ep+7, 0x1.7db476p+7, 0x6.b45a7p-24,
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-0xe.c90d8p-8, 0xa.e586fp-16, -0x1.d66becp-4 }, /* 60 */
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{ 0x1.83fce2p+7, 0x1.83fcecp+7, 0x1.83fd0ep+7, -0x2.8f34a4p-24,
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0xe.aa478p-8, -0x9.810bp-16, -0x3.a5f3fcp-8 }, /* 61 */
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{ 0x1.8a455cp+7, 0x1.8a456ap+7, 0x1.8a4588p+7, -0x1.325968p-24,
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-0xe.8c3eap-8, 0x9.0a765p-16, 0x1.29a54ap-4 }, /* 62 */
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{ 0x1.908dd8p+7, 0x1.908de8p+7, 0x1.908df4p+7, 0x4.96b808p-24,
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0xe.6eeb5p-8, -0x9.0251bp-16, 0x1.41a488p-4 }, /* 63 */
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};
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/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
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j0f(x) ~ sqrt(2/(pi*x))*beta0(x)*cos(x-pi/4-alpha0(x))
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where beta0(x) = 1 - 1/(16*x^2) + 53/(512*x^4)
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and alpha0(x) = 1/(8*x) - 25/(384*x^3). */
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static float
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j0f_asympt (float x)
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{
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/* The following code fails to give an error <= 9 ulps in only two cases,
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for which we tabulate the result. */
|
|
if (x == 0x1.4665d2p+24f)
|
|
return 0xa.50206p-52f;
|
|
if (x == 0x1.a9afdep+7f)
|
|
return 0xf.47039p-28f;
|
|
double y = 1.0 / (double) x;
|
|
double y2 = y * y;
|
|
double beta0 = 1.0f + y2 * (-0x1p-4f + 0x1.a8p-4 * y2);
|
|
double alpha0 = y * (0x2p-4 - 0x1.0aaaaap-4 * y2);
|
|
double h;
|
|
int n;
|
|
h = reduce_aux (x, &n, alpha0);
|
|
/* Now x - pi/4 - alpha0 = h + n*pi/2 mod (2*pi). */
|
|
float xr = (float) h;
|
|
n = n & 3;
|
|
float cst = 0xc.c422ap-4f; /* sqrt(2/pi) rounded to nearest */
|
|
float t = cst / sqrtf (x) * (float) beta0;
|
|
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 j0.
|
|
z is the value computed by the generic code.
|
|
For small x, we use a polynomial approximating j0 around its root.
|
|
For large x, we use an asymptotic formula (j0f_asympt). */
|
|
static float
|
|
j0f_near_root (float x, float z)
|
|
{
|
|
float index_f;
|
|
int index;
|
|
|
|
index_f = roundf ((x - FIRST_ZERO_J0) / M_PIf);
|
|
/* j0f_asympt fails to give an error <= 9 ulps for x=0x1.324e92p+7
|
|
(index 48) thus we can't reduce SMALL_SIZE below 49. */
|
|
if (index_f >= SMALL_SIZE)
|
|
return j0f_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, we return the value z. */
|
|
if (! (x0 <= x && x <= x1))
|
|
return z;
|
|
float xmid = p[1];
|
|
float y = x - xmid;
|
|
return p[3] + y * (p[4] + y * (p[5] + y * p[6]));
|
|
}
|
|
|
|
float
|
|
__ieee754_j0f(float x)
|
|
{
|
|
float z, s,c,ss,cc,r,u,v;
|
|
int32_t hx,ix;
|
|
|
|
GET_FLOAT_WORD(hx,x);
|
|
ix = hx&0x7fffffff;
|
|
if(ix>=0x7f800000) return one/(x*x);
|
|
x = fabsf(x);
|
|
if(ix >= 0x40000000) { /* |x| >= 2.0 */
|
|
SET_RESTORE_ROUNDF (FE_TONEAREST);
|
|
__sincosf (x, &s, &c);
|
|
ss = s-c;
|
|
cc = s+c;
|
|
if (ix >= 0x7f000000)
|
|
/* x >= 2^127: use asymptotic expansion. */
|
|
return j0f_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;
|
|
/*
|
|
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
|
|
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
|
|
*/
|
|
if (ix <= 0x5c000000)
|
|
{
|
|
u = pzerof(x); v = qzerof(x);
|
|
cc = u*cc-v*ss;
|
|
}
|
|
z = (invsqrtpi * cc) / sqrtf(x);
|
|
/* The following threshold is optimal: for x=0x1.3b58dep+1
|
|
and rounding upwards, |cc|=0x1.579b26p-4 and z is 10 ulps
|
|
far from the correctly rounded value. */
|
|
float threshold = 0x1.579b26p-4f;
|
|
if (fabsf (cc) > threshold)
|
|
return z;
|
|
else
|
|
return j0f_near_root (x, z);
|
|
}
|
|
if(ix<0x39000000) { /* |x| < 2**-13 */
|
|
math_force_eval(huge+x); /* raise inexact if x != 0 */
|
|
if(ix<0x32000000) return one; /* |x|<2**-27 */
|
|
else return one - (float)0.25*x*x;
|
|
}
|
|
z = x*x;
|
|
r = z*(R02+z*(R03+z*(R04+z*R05)));
|
|
s = one+z*(S01+z*(S02+z*(S03+z*S04)));
|
|
if(ix < 0x3F800000) { /* |x| < 1.00 */
|
|
return one + z*((float)-0.25+(r/s));
|
|
} else {
|
|
u = (float)0.5*x;
|
|
return((one+u)*(one-u)+z*(r/s));
|
|
}
|
|
}
|
|
libm_alias_finite (__ieee754_j0f, __j0f)
|
|
|
|
static const float
|
|
u00 = -7.3804296553e-02, /* 0xbd9726b5 */
|
|
u01 = 1.7666645348e-01, /* 0x3e34e80d */
|
|
u02 = -1.3818567619e-02, /* 0xbc626746 */
|
|
u03 = 3.4745343146e-04, /* 0x39b62a69 */
|
|
u04 = -3.8140706238e-06, /* 0xb67ff53c */
|
|
u05 = 1.9559013964e-08, /* 0x32a802ba */
|
|
u06 = -3.9820518410e-11, /* 0xae2f21eb */
|
|
v01 = 1.2730483897e-02, /* 0x3c509385 */
|
|
v02 = 7.6006865129e-05, /* 0x389f65e0 */
|
|
v03 = 2.5915085189e-07, /* 0x348b216c */
|
|
v04 = 4.4111031494e-10; /* 0x2ff280c2 */
|
|
|
|
/* This is the nearest approximation of the first zero of y0. */
|
|
#define FIRST_ZERO_Y0 0xe.4c166p-4f
|
|
|
|
/* The following table contains successive zeros of y0 and degree-3
|
|
polynomial approximations of y0 around these zeros: Py[0] for the first
|
|
zero (0.89358), Py[1] for the second one (3.957678), 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 index 0 where we
|
|
use degree 5, and the coefficients of degree 4 and 5 are hard-coded in
|
|
y0f_near_root.
|
|
*/
|
|
static const float Py[SMALL_SIZE][7] = {
|
|
{ 0x1.a681dap-1, 0x1.c982ecp-1, 0x1.ef6bcap-1, 0x3.274468p-28,
|
|
0xe.121b8p-4, -0x7.df8b3p-4, 0x3.877be4p-4
|
|
/*, -0x3.a46c9p-4, 0x3.735478p-4*/ }, /* 0 */
|
|
{ 0x1.f957c6p+1, 0x1.fa9534p+1, 0x1.fd11b2p+1, 0xa.f1f83p-28,
|
|
-0x6.70d098p-4, 0xd.04d48p-8, 0xe.f61a9p-8 }, /* 1 */
|
|
{ 0x1.c51832p+2, 0x1.c581dcp+2, 0x1.c65164p+2, -0x5.e2a51p-28,
|
|
0x4.cd3328p-4, -0x5.6bbe08p-8, -0xc.46d8p-8 }, /* 2 */
|
|
{ 0x1.46fd84p+3, 0x1.471d74p+3, 0x1.475bfcp+3, -0x1.4b0aeep-24,
|
|
-0x3.fec6b8p-4, 0x3.2068a4p-8, 0xa.76e2dp-8 }, /* 3 */
|
|
{ 0x1.ab7afap+3, 0x1.ab8e1cp+3, 0x1.abb294p+3, -0x8.179d7p-28,
|
|
0x3.7e6544p-4, -0x2.1799fp-8, -0x9.0e1c4p-8 }, /* 4 */
|
|
{ 0x1.07f9aap+4, 0x1.0803c8p+4, 0x1.08170cp+4, -0x2.5b8078p-24,
|
|
-0x3.24b868p-4, 0x1.8631ecp-8, 0x8.3cb46p-8 }, /* 5 */
|
|
{ 0x1.3a38eap+4, 0x1.3a42cep+4, 0x1.3a4d8ap+4, 0x1.cd304ap-28,
|
|
0x2.e189ecp-4, -0x1.2c6954p-8, -0x7.8178ep-8 }, /* 6 */
|
|
{ 0x1.6c7d42p+4, 0x1.6c833p+4, 0x1.6c99fp+4, -0x6.c63f1p-28,
|
|
-0x2.acc9a8p-4, 0xf.09e31p-12, 0x7.0b5ab8p-8 }, /* 7 */
|
|
{ 0x1.9ebec4p+4, 0x1.9ec47p+4, 0x1.9ed016p+4, 0x1.e53838p-24,
|
|
0x2.81f2p-4, -0xc.5ff51p-12, -0x7.05ep-8 }, /* 8 */
|
|
{ 0x1.d1008ep+4, 0x1.d10644p+4, 0x1.d11262p+4, 0x1.636feep-24,
|
|
-0x2.5e40dcp-4, 0xa.6f81dp-12, 0x5.ff6p-8 }, /* 9 */
|
|
{ 0x1.01a27cp+5, 0x1.01a442p+5, 0x1.01a924p+5, -0x4.04e1bp-28,
|
|
0x2.3febd8p-4, -0x8.f11e2p-12, -0x6.0111ap-8 }, /* 10 */
|
|
{ 0x1.1ac3bcp+5, 0x1.1ac588p+5, 0x1.1ac912p+5, 0x3.6063d8p-24,
|
|
-0x2.25baacp-4, 0x7.c93cdp-12, 0x4.e7577p-8 }, /* 11 */
|
|
{ 0x1.33e508p+5, 0x1.33e6ecp+5, 0x1.33ea1ap+5, -0x3.f39ebcp-24,
|
|
0x2.0ed04cp-4, -0x6.d9434p-12, -0x4.fc0b7p-8 }, /* 12 */
|
|
{ 0x1.4d0666p+5, 0x1.4d0868p+5, 0x1.4d0c14p+5, -0x4.ea23p-28,
|
|
-0x1.fa8b4p-4, 0x6.1470e8p-12, 0x5.97f71p-8 }, /* 13 */
|
|
{ 0x1.6628b8p+5, 0x1.6629f4p+5, 0x1.662e0ep+5, -0x3.5df0c8p-24,
|
|
0x1.e8727ep-4, -0x5.76a038p-12, -0x4.ee37c8p-8 }, /* 14 */
|
|
{ 0x1.7f4a7cp+5, 0x1.7f4b9p+5, 0x1.7f4daap+5, 0x1.1ef09ep-24,
|
|
-0x1.d8293ap-4, 0x4.ed8a28p-12, 0x4.d43708p-8 }, /* 15 */
|
|
{ 0x1.986c5cp+5, 0x1.986d38p+5, 0x1.986f6p+5, 0x1.b70cecp-24,
|
|
0x1.c967p-4, -0x4.7a70b8p-12, -0x5.6840e8p-8 }, /* 16 */
|
|
{ 0x1.b18dcap+5, 0x1.b18ee8p+5, 0x1.b19122p+5, 0x1.abaadcp-24,
|
|
-0x1.bbf246p-4, 0x4.1a35bp-12, 0x3.c2d46p-8 }, /* 17 */
|
|
{ 0x1.caaf86p+5, 0x1.cab0a2p+5, 0x1.cab2fep+5, 0x1.63989ep-24,
|
|
0x1.af9cb4p-4, -0x3.c2f2dcp-12, -0x4.cf8108p-8 }, /* 18 */
|
|
{ 0x1.e3d146p+5, 0x1.e3d262p+5, 0x1.e3d492p+5, -0x1.68a8ecp-24,
|
|
-0x1.a4407ep-4, 0x3.7733ecp-12, 0x5.97916p-8 }, /* 19 */
|
|
{ 0x1.fcf316p+5, 0x1.fcf428p+5, 0x1.fcf59ap+5, 0x1.e1bb5p-24,
|
|
0x1.99be74p-4, -0x3.37210cp-12, -0x5.d19bf8p-8 }, /* 20 */
|
|
{ 0x1.0b0a7cp+6, 0x1.0b0afap+6, 0x1.0b0b9cp+6, -0x5.5bbcfp-24,
|
|
-0x1.8ffc9ap-4, 0x2.ffe638p-12, 0x2.ed28e8p-8 }, /* 21 */
|
|
{ 0x1.179b66p+6, 0x1.179bep+6, 0x1.179d0ap+6, -0x4.9e34a8p-24,
|
|
0x1.86e51cp-4, -0x2.cc7a68p-12, -0x3.3642c4p-8 }, /* 22 */
|
|
{ 0x1.242c5cp+6, 0x1.242ccap+6, 0x1.242d68p+6, 0x1.966706p-24,
|
|
-0x1.7e657p-4, 0x2.9aed4cp-12, 0x7.b87a58p-8 }, /* 23 */
|
|
{ 0x1.30bd62p+6, 0x1.30bdb6p+6, 0x1.30beb2p+6, 0x3.4b3b68p-24,
|
|
0x1.766dc2p-4, -0x2.72651cp-12, -0x3.e347f8p-8 }, /* 24 */
|
|
{ 0x1.3d4e56p+6, 0x1.3d4ea2p+6, 0x1.3d4f2ep+6, 0x6.a99008p-28,
|
|
-0x1.6ef07ep-4, 0x2.53aec4p-12, 0x2.9e3d88p-12 }, /* 25 */
|
|
{ 0x1.49df38p+6, 0x1.49df9p+6, 0x1.49e042p+6, -0x7.a9fa6p-32,
|
|
0x1.67e1dap-4, -0x2.324d7p-12, -0xc.0e669p-12 }, /* 26 */
|
|
{ 0x1.56702ep+6, 0x1.56708p+6, 0x1.567116p+6, -0x5.026808p-24,
|
|
-0x1.613798p-4, 0x2.114594p-12, 0x1.a22402p-8 }, /* 27 */
|
|
{ 0x1.630126p+6, 0x1.63017p+6, 0x1.630226p+6, 0x4.46aa2p-24,
|
|
0x1.5ae8c2p-4, -0x1.f4aaa4p-12, -0x3.58593cp-8 }, /* 28 */
|
|
{ 0x1.6f9234p+6, 0x1.6f926p+6, 0x1.6f92b2p+6, 0x1.5cfccp-24,
|
|
-0x1.54ed76p-4, 0x1.dd540ap-12, -0xb.e9429p-12 }, /* 29 */
|
|
{ 0x1.7c22fep+6, 0x1.7c2352p+6, 0x1.7c23c2p+6, -0xb.4dc4cp-28,
|
|
0x1.4f3ebcp-4, -0x1.c463fp-12, -0x1.e94c54p-8 }, /* 30 */
|
|
{ 0x1.88b412p+6, 0x1.88b444p+6, 0x1.88b50ap+6, 0x3.f5343p-24,
|
|
-0x1.49d668p-4, 0x1.a53f24p-12, 0x5.128198p-8 }, /* 31 */
|
|
{ 0x1.9544dcp+6, 0x1.954538p+6, 0x1.95459p+6, -0x6.e6f32p-28,
|
|
0x1.44aefap-4, -0x1.9a6ef8p-12, -0x6.c639cp-8 }, /* 32 */
|
|
{ 0x1.a1d5fap+6, 0x1.a1d62cp+6, 0x1.a1d674p+6, 0x1.f359c2p-28,
|
|
-0x1.3fc386p-4, 0x1.887ebep-12, 0x1.6c813cp-8 }, /* 33 */
|
|
{ 0x1.ae66cp+6, 0x1.ae672p+6, 0x1.ae6788p+6, -0x2.9de748p-24,
|
|
0x1.3b0fa4p-4, -0x1.777f26p-12, 0x1.c128ccp-8 }, /* 34 */
|
|
{ 0x1.baf7c2p+6, 0x1.baf816p+6, 0x1.baf86cp+6, -0x2.24ccc8p-24,
|
|
-0x1.368f5cp-4, 0x1.62bd9ep-12, 0xa.df002p-8 }, /* 35 */
|
|
{ 0x1.c788dap+6, 0x1.c7890cp+6, 0x1.c7896cp+6, 0x4.7dcea8p-24,
|
|
0x1.323f16p-4, -0x1.61abf4p-12, 0xa.ad73ep-8 }, /* 36 */
|
|
{ 0x1.d419ccp+6, 0x1.d41a02p+6, 0x1.d41a68p+6, -0x4.b538p-24,
|
|
-0x1.2e1b98p-4, 0x1.4a4d64p-12, 0x3.a47674p-8 }, /* 37 */
|
|
{ 0x1.e0aacep+6, 0x1.e0aaf8p+6, 0x1.e0ab5ep+6, 0x3.09dc4cp-24,
|
|
0x1.2a21ecp-4, -0x1.3fa20cp-12, 0x2.216e8cp-8 }, /* 38 */
|
|
{ 0x1.ed3bb8p+6, 0x1.ed3beep+6, 0x1.ed3c56p+6, 0x4.d5a58p-28,
|
|
-0x1.264f66p-4, 0x1.32c4cep-12, 0x1.53cbb4p-8 }, /* 39 */
|
|
{ 0x1.f9ccaep+6, 0x1.f9cce6p+6, 0x1.f9cd52p+6, 0x3.f4c44cp-24,
|
|
0x1.22a192p-4, -0x1.1f8514p-12, -0xc.0de32p-8 }, /* 40 */
|
|
{ 0x1.032ed6p+7, 0x1.032eeep+7, 0x1.032f0cp+7, 0x2.4beae8p-24,
|
|
-0x1.1f1634p-4, 0x1.171664p-12, 0x1.72a654p-4 }, /* 41 */
|
|
{ 0x1.097756p+7, 0x1.09776ap+7, 0x1.09779cp+7, -0xd.a581ep-28,
|
|
0x1.1bab3cp-4, -0xf.9f507p-16, -0xc.ba2d4p-8 }, /* 42 */
|
|
{ 0x1.0fbfdp+7, 0x1.0fbfe6p+7, 0x1.0fbff6p+7, 0xa.7c0bdp-28,
|
|
-0x1.185eccp-4, 0x1.01d7dep-12, -0x1.a2186ep-4 }, /* 43 */
|
|
{ 0x1.160856p+7, 0x1.160862p+7, 0x1.16087ap+7, -0x1.9452ecp-24,
|
|
0x1.152f26p-4, -0x1.07b4aap-12, 0x1.6bbd7ep-4 }, /* 44 */
|
|
{ 0x1.1c50dp+7, 0x1.1c50dep+7, 0x1.1c5118p+7, 0x3.83b7b8p-24,
|
|
-0x1.121ab2p-4, 0x1.0e938cp-12, -0x5.1a6dfp-8 }, /* 45 */
|
|
{ 0x1.22995p+7, 0x1.22995ap+7, 0x1.229976p+7, -0x6.5ca3a8p-24,
|
|
0x1.0f1ff2p-4, -0xe.f198p-16, -0x3.8e98b8p-8 }, /* 46 */
|
|
{ 0x1.28e1ccp+7, 0x1.28e1d8p+7, 0x1.28e1f4p+7, -0x6.bb61ap-24,
|
|
-0x1.0c3d8ap-4, 0xf.a14b9p-16, 0x9.81e82p-4 }, /* 47 */
|
|
{ 0x1.2f2a48p+7, 0x1.2f2a54p+7, 0x1.2f2a74p+7, 0x2.2438p-24,
|
|
0x1.097236p-4, -0xd.fed5ep-16, -0x3.19eb5cp-8 }, /* 48 */
|
|
{ 0x1.3572b8p+7, 0x1.3572dp+7, 0x1.3572ecp+7, 0x3.1e0054p-24,
|
|
-0x1.06bcc8p-4, 0xd.d2596p-16, -0x1.67e00ap-4 }, /* 49 */
|
|
{ 0x1.3bbb3ep+7, 0x1.3bbb4ep+7, 0x1.3bbb6ap+7, 0x7.46c908p-24,
|
|
0x1.041c28p-4, -0xd.04045p-16, -0x8.fb297p-8 }, /* 50 */
|
|
{ 0x1.4203aep+7, 0x1.4203cap+7, 0x1.4203e6p+7, -0xb.4f158p-28,
|
|
-0x1.018f52p-4, 0xc.ccf6fp-16, 0x1.4d5dp-4 }, /* 51 */
|
|
{ 0x1.484c38p+7, 0x1.484c46p+7, 0x1.484c56p+7, -0x6.5a89c8p-24,
|
|
0xf.f155p-8, -0xc.5d21dp-16, -0xd.aca34p-8 }, /* 52 */
|
|
{ 0x1.4e94b8p+7, 0x1.4e94c4p+7, 0x1.4e94d4p+7, -0x1.ef16c8p-24,
|
|
-0xf.cad3fp-8, 0xb.d75f8p-16, 0x1.f36732p-4 }, /* 53 */
|
|
{ 0x1.54dd36p+7, 0x1.54dd4p+7, 0x1.54dd52p+7, -0x6.1e7b68p-24,
|
|
0xf.a564cp-8, -0xb.ec1cfp-16, 0xe.e7421p-8 }, /* 54 */
|
|
{ 0x1.5b25aep+7, 0x1.5b25bep+7, 0x1.5b25d4p+7, -0xf.8c858p-28,
|
|
-0xf.80faep-8, 0xb.8b6c5p-16, -0x5.835ed8p-8 }, /* 55 */
|
|
{ 0x1.616e34p+7, 0x1.616e3cp+7, 0x1.616e4ep+7, 0x7.75d858p-24,
|
|
0xf.5d8abp-8, -0xb.b3779p-16, 0x2.40b948p-4 }, /* 56 */
|
|
{ 0x1.67b6bp+7, 0x1.67b6b8p+7, 0x1.67b6dp+7, 0x1.d78632p-24,
|
|
-0xf.3b096p-8, 0xa.daf89p-16, 0x1.aa8548p-8 }, /* 57 */
|
|
{ 0x1.6dff28p+7, 0x1.6dff36p+7, 0x1.6dff54p+7, 0x3.b24794p-24,
|
|
0xf.196c7p-8, -0xb.1afe1p-16, -0x1.77538cp-8 }, /* 58 */
|
|
{ 0x1.7447a2p+7, 0x1.7447b2p+7, 0x1.7447cap+7, 0x6.39cbc8p-24,
|
|
-0xe.f8aa5p-8, 0xa.50daap-16, 0x1.9592c2p-8 }, /* 59 */
|
|
{ 0x1.7a902p+7, 0x1.7a903p+7, 0x1.7a903ep+7, -0x1.820e3ap-24,
|
|
0xe.d8b9dp-8, -0xa.998cp-16, -0x2.c35d78p-4 }, /* 60 */
|
|
{ 0x1.80d89ep+7, 0x1.80d8aep+7, 0x1.80d8bep+7, -0x2.c7e038p-24,
|
|
-0xe.b9925p-8, 0x9.ce06p-16, -0x2.2b3054p-4 }, /* 61 */
|
|
{ 0x1.87211cp+7, 0x1.87212cp+7, 0x1.872144p+7, 0x6.ab31c8p-24,
|
|
0xe.9b2bep-8, -0x9.4de7p-16, -0x1.32cb5ep-4 }, /* 62 */
|
|
{ 0x1.8d699ap+7, 0x1.8d69a8p+7, 0x1.8d69bp+7, 0x4.4ef25p-24,
|
|
-0xe.7d7ecp-8, 0x9.a0f1ep-16, 0x1.6ac076p-4 }, /* 63 */
|
|
};
|
|
|
|
/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
|
|
y0(x) ~ sqrt(2/(pi*x))*beta0(x)*sin(x-pi/4-alpha0(x))
|
|
where beta0(x) = 1 - 1/(16*x^2) + 53/(512*x^4)
|
|
and alpha0(x) = 1/(8*x) - 25/(384*x^3). */
|
|
static float
|
|
y0f_asympt (float x)
|
|
{
|
|
/* The following code fails to give an error <= 9 ulps in only two cases,
|
|
for which we tabulate the correctly-rounded result. */
|
|
if (x == 0x1.bfad96p+7f)
|
|
return -0x7.f32bdp-32f;
|
|
if (x == 0x1.2e2a42p+17f)
|
|
return 0x1.a48974p-40f;
|
|
double y = 1.0 / (double) x;
|
|
double y2 = y * y;
|
|
double beta0 = 1.0f + y2 * (-0x1p-4f + 0x1.a8p-4 * y2);
|
|
double alpha0 = y * (0x2p-4 - 0x1.0aaaaap-4 * y2);
|
|
double h;
|
|
int n;
|
|
h = reduce_aux (x, &n, alpha0);
|
|
/* Now x - pi/4 - alpha0 = h + n*pi/2 mod (2*pi). */
|
|
float xr = (float) h;
|
|
n = n & 3;
|
|
float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */
|
|
float t = cst / sqrtf (x) * (float) beta0;
|
|
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 y0.
|
|
z is the value computed by the generic code.
|
|
For small x, use a polynomial approximating y0 around its root.
|
|
For large x, use an asymptotic formula (y0f_asympt). */
|
|
static float
|
|
y0f_near_root (float x, float z)
|
|
{
|
|
float index_f;
|
|
int index;
|
|
|
|
index_f = roundf ((x - FIRST_ZERO_Y0) / M_PIf);
|
|
if (index_f >= SMALL_SIZE)
|
|
return y0f_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;
|
|
/* For degree 0 use a degree-5 polynomial, where the coefficients of
|
|
degree 4 and 5 are hard-coded. */
|
|
float p6 = (index > 0) ? p[6]
|
|
: p[6] + y * (-0x3.a46c9p-4 + y * 0x3.735478p-4);
|
|
float res = p[3] + y * (p[4] + y * (p[5] + y * p6));
|
|
return res;
|
|
}
|
|
|
|
float
|
|
__ieee754_y0f(float x)
|
|
{
|
|
float z, s,c,ss,cc,u,v;
|
|
int32_t hx,ix;
|
|
|
|
GET_FLOAT_WORD(hx,x);
|
|
ix = 0x7fffffff&hx;
|
|
/* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf. */
|
|
if(ix>=0x7f800000) return one/(x+x*x);
|
|
if(ix==0) return -1/zero; /* -inf and divide by zero exception. */
|
|
if(hx<0) return zero/(zero*x);
|
|
if(ix >= 0x40000000 || (0x3f5340ed <= ix && ix <= 0x3f77b5e5)) {
|
|
/* |x| >= 2.0 or
|
|
0x1.a681dap-1 <= |x| <= 0x1.ef6bcap-1 (around 1st zero) */
|
|
/* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
|
|
* where x0 = x-pi/4
|
|
* Better formula:
|
|
* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
|
|
* = 1/sqrt(2) * (sin(x) + cos(x))
|
|
* sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
|
|
* = 1/sqrt(2) * (sin(x) - cos(x))
|
|
* To avoid cancellation, use
|
|
* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
|
|
* to compute the worse one.
|
|
*/
|
|
SET_RESTORE_ROUNDF (FE_TONEAREST);
|
|
__sincosf (x, &s, &c);
|
|
ss = s-c;
|
|
cc = s+c;
|
|
/*
|
|
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
|
|
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
|
|
*/
|
|
if (ix >= 0x7f000000)
|
|
/* x >= 2^127: use asymptotic expansion. */
|
|
return y0f_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;
|
|
if (ix <= 0x5c000000)
|
|
{
|
|
u = pzerof(x); v = qzerof(x);
|
|
ss = u*ss+v*cc;
|
|
}
|
|
z = (invsqrtpi*ss)/sqrtf(x);
|
|
/* The following threshold is optimal (determined on
|
|
aarch64-linux-gnu). */
|
|
float threshold = 0x1.be585ap-4;
|
|
if (fabsf (ss) > threshold)
|
|
return z;
|
|
else
|
|
return y0f_near_root (x, z);
|
|
}
|
|
if(ix<=0x39800000) { /* x < 2**-13 */
|
|
return(u00 + tpi*__ieee754_logf(x));
|
|
}
|
|
z = x*x;
|
|
u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
|
|
v = one+z*(v01+z*(v02+z*(v03+z*v04)));
|
|
return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
|
|
}
|
|
libm_alias_finite (__ieee754_y0f, __y0f)
|
|
|
|
/* The asymptotic expansion of pzero is
|
|
* 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
|
|
* For x >= 2, We approximate pzero by
|
|
* pzero(x) = 1 + (R/S)
|
|
* where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
|
|
* S = 1 + pS0*s^2 + ... + pS4*s^10
|
|
* and
|
|
* | pzero(x)-1-R/S | <= 2 ** ( -60.26)
|
|
*/
|
|
static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
|
|
0.0000000000e+00, /* 0x00000000 */
|
|
-7.0312500000e-02, /* 0xbd900000 */
|
|
-8.0816707611e+00, /* 0xc1014e86 */
|
|
-2.5706311035e+02, /* 0xc3808814 */
|
|
-2.4852163086e+03, /* 0xc51b5376 */
|
|
-5.2530439453e+03, /* 0xc5a4285a */
|
|
};
|
|
static const float pS8[5] = {
|
|
1.1653436279e+02, /* 0x42e91198 */
|
|
3.8337448730e+03, /* 0x456f9beb */
|
|
4.0597855469e+04, /* 0x471e95db */
|
|
1.1675296875e+05, /* 0x47e4087c */
|
|
4.7627726562e+04, /* 0x473a0bba */
|
|
};
|
|
static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
|
|
-1.1412546255e-11, /* 0xad48c58a */
|
|
-7.0312492549e-02, /* 0xbd8fffff */
|
|
-4.1596107483e+00, /* 0xc0851b88 */
|
|
-6.7674766541e+01, /* 0xc287597b */
|
|
-3.3123129272e+02, /* 0xc3a59d9b */
|
|
-3.4643338013e+02, /* 0xc3ad3779 */
|
|
};
|
|
static const float pS5[5] = {
|
|
6.0753936768e+01, /* 0x42730408 */
|
|
1.0512523193e+03, /* 0x44836813 */
|
|
5.9789707031e+03, /* 0x45bad7c4 */
|
|
9.6254453125e+03, /* 0x461665c8 */
|
|
2.4060581055e+03, /* 0x451660ee */
|
|
};
|
|
|
|
static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
|
|
-2.5470459075e-09, /* 0xb12f081b */
|
|
-7.0311963558e-02, /* 0xbd8fffb8 */
|
|
-2.4090321064e+00, /* 0xc01a2d95 */
|
|
-2.1965976715e+01, /* 0xc1afba52 */
|
|
-5.8079170227e+01, /* 0xc2685112 */
|
|
-3.1447946548e+01, /* 0xc1fb9565 */
|
|
};
|
|
static const float pS3[5] = {
|
|
3.5856033325e+01, /* 0x420f6c94 */
|
|
3.6151397705e+02, /* 0x43b4c1ca */
|
|
1.1936077881e+03, /* 0x44953373 */
|
|
1.1279968262e+03, /* 0x448cffe6 */
|
|
1.7358093262e+02, /* 0x432d94b8 */
|
|
};
|
|
|
|
static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
|
|
-8.8753431271e-08, /* 0xb3be98b7 */
|
|
-7.0303097367e-02, /* 0xbd8ffb12 */
|
|
-1.4507384300e+00, /* 0xbfb9b1cc */
|
|
-7.6356959343e+00, /* 0xc0f4579f */
|
|
-1.1193166733e+01, /* 0xc1331736 */
|
|
-3.2336456776e+00, /* 0xc04ef40d */
|
|
};
|
|
static const float pS2[5] = {
|
|
2.2220300674e+01, /* 0x41b1c32d */
|
|
1.3620678711e+02, /* 0x430834f0 */
|
|
2.7047027588e+02, /* 0x43873c32 */
|
|
1.5387539673e+02, /* 0x4319e01a */
|
|
1.4657617569e+01, /* 0x416a859a */
|
|
};
|
|
|
|
static float
|
|
pzerof(float x)
|
|
{
|
|
const float *p,*q;
|
|
float z,r,s;
|
|
int32_t ix;
|
|
GET_FLOAT_WORD(ix,x);
|
|
ix &= 0x7fffffff;
|
|
/* ix >= 0x40000000 for all calls to this function. */
|
|
if(ix>=0x41000000) {p = pR8; q= pS8;}
|
|
else if(ix>=0x40f71c58){p = pR5; q= pS5;}
|
|
else if(ix>=0x4036db68){p = pR3; q= pS3;}
|
|
else {p = pR2; q= pS2;}
|
|
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;
|
|
}
|
|
|
|
|
|
/* For x >= 8, the asymptotic expansion of qzero is
|
|
* -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
|
|
* We approximate pzero by
|
|
* qzero(x) = s*(-1.25 + (R/S))
|
|
* where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
|
|
* S = 1 + qS0*s^2 + ... + qS5*s^12
|
|
* and
|
|
* | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
|
|
*/
|
|
static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
|
|
0.0000000000e+00, /* 0x00000000 */
|
|
7.3242187500e-02, /* 0x3d960000 */
|
|
1.1768206596e+01, /* 0x413c4a93 */
|
|
5.5767340088e+02, /* 0x440b6b19 */
|
|
8.8591972656e+03, /* 0x460a6cca */
|
|
3.7014625000e+04, /* 0x471096a0 */
|
|
};
|
|
static const float qS8[6] = {
|
|
1.6377603149e+02, /* 0x4323c6aa */
|
|
8.0983447266e+03, /* 0x45fd12c2 */
|
|
1.4253829688e+05, /* 0x480b3293 */
|
|
8.0330925000e+05, /* 0x49441ed4 */
|
|
8.4050156250e+05, /* 0x494d3359 */
|
|
-3.4389928125e+05, /* 0xc8a7eb69 */
|
|
};
|
|
|
|
static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
|
|
1.8408595828e-11, /* 0x2da1ec79 */
|
|
7.3242180049e-02, /* 0x3d95ffff */
|
|
5.8356351852e+00, /* 0x40babd86 */
|
|
1.3511157227e+02, /* 0x43071c90 */
|
|
1.0272437744e+03, /* 0x448067cd */
|
|
1.9899779053e+03, /* 0x44f8bf4b */
|
|
};
|
|
static const float qS5[6] = {
|
|
8.2776611328e+01, /* 0x42a58da0 */
|
|
2.0778142090e+03, /* 0x4501dd07 */
|
|
1.8847289062e+04, /* 0x46933e94 */
|
|
5.6751113281e+04, /* 0x475daf1d */
|
|
3.5976753906e+04, /* 0x470c88c1 */
|
|
-5.3543427734e+03, /* 0xc5a752be */
|
|
};
|
|
|
|
static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
|
|
4.3774099900e-09, /* 0x3196681b */
|
|
7.3241114616e-02, /* 0x3d95ff70 */
|
|
3.3442313671e+00, /* 0x405607e3 */
|
|
4.2621845245e+01, /* 0x422a7cc5 */
|
|
1.7080809021e+02, /* 0x432acedf */
|
|
1.6673394775e+02, /* 0x4326bbe4 */
|
|
};
|
|
static const float qS3[6] = {
|
|
4.8758872986e+01, /* 0x42430916 */
|
|
7.0968920898e+02, /* 0x44316c1c */
|
|
3.7041481934e+03, /* 0x4567825f */
|
|
6.4604252930e+03, /* 0x45c9e367 */
|
|
2.5163337402e+03, /* 0x451d4557 */
|
|
-1.4924745178e+02, /* 0xc3153f59 */
|
|
};
|
|
|
|
static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
|
|
1.5044444979e-07, /* 0x342189db */
|
|
7.3223426938e-02, /* 0x3d95f62a */
|
|
1.9981917143e+00, /* 0x3fffc4bf */
|
|
1.4495602608e+01, /* 0x4167edfd */
|
|
3.1666231155e+01, /* 0x41fd5471 */
|
|
1.6252708435e+01, /* 0x4182058c */
|
|
};
|
|
static const float qS2[6] = {
|
|
3.0365585327e+01, /* 0x41f2ecb8 */
|
|
2.6934811401e+02, /* 0x4386ac8f */
|
|
8.4478375244e+02, /* 0x44533229 */
|
|
8.8293585205e+02, /* 0x445cbbe5 */
|
|
2.1266638184e+02, /* 0x4354aa98 */
|
|
-5.3109550476e+00, /* 0xc0a9f358 */
|
|
};
|
|
|
|
static float
|
|
qzerof(float x)
|
|
{
|
|
const float *p,*q;
|
|
float s,r,z;
|
|
int32_t ix;
|
|
GET_FLOAT_WORD(ix,x);
|
|
ix &= 0x7fffffff;
|
|
/* ix >= 0x40000000 for all calls to this function. */
|
|
if(ix>=0x41000000) {p = qR8; q= qS8;}
|
|
else if(ix>=0x40f71c58){p = qR5; q= qS5;}
|
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else if(ix>=0x4036db68){p = qR3; q= qS3;}
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else {p = qR2; q= qS2;}
|
|
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]+z*q[5])))));
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|
return (-(float).125 + r/s)/x;
|
|
}
|