403 lines
15 KiB
Rust
403 lines
15 KiB
Rust
/* @(#)e_j0.c 1.3 95/01/18 */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunSoft, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/* j0(x), y0(x)
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* Bessel function of the first and second kinds of order zero.
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* Method -- j0(x):
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* 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
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* 2. Reduce x to |x| since j0(x)=j0(-x), and
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* for x in (0,2)
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* j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x;
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* (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
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* for x in (2,inf)
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* j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
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* where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
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* as follow:
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* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
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* = 1/sqrt(2) * (cos(x) + sin(x))
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* sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
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* = 1/sqrt(2) * (sin(x) - cos(x))
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* (To avoid cancellation, use
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* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
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* to compute the worse 1.)
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*
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* 3 Special cases
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* j0(nan)= nan
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* j0(0) = 1
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* j0(inf) = 0
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*
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* Method -- y0(x):
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* 1. For x<2.
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* Since
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* y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
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* therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
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* We use the following function to approximate y0,
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* y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
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* where
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* U(z) = u00 + u01*z + ... + u06*z^6
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* V(z) = 1 + v01*z + ... + v04*z^4
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* with absolute approximation error bounded by 2**-72.
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* Note: For tiny x, U/V = u0 and j0(x)~1, hence
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* y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
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* 2. For x>=2.
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* y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
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* where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
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* by the method menti1d above.
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* 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
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*/
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use std::f64::consts::FRAC_2_PI;
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use super::bessel_util::{high_word, split_words, FRAC_2_SQRT_PI, HUGE};
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// R0/S0 on [0, 2.00]
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const R02: f64 = 1.562_499_999_999_999_5e-2; // 0x3F8FFFFF, 0xFFFFFFFD
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const R03: f64 = -1.899_792_942_388_547_2e-4; // 0xBF28E6A5, 0xB61AC6E9
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const R04: f64 = 1.829_540_495_327_006_7e-6; // 0x3EBEB1D1, 0x0C503919
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const R05: f64 = -4.618_326_885_321_032e-9; // 0xBE33D5E7, 0x73D63FCE
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const S01: f64 = 1.561_910_294_648_900_1e-2; // 0x3F8FFCE8, 0x82C8C2A4
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const S02: f64 = 1.169_267_846_633_374_5e-4; // 0x3F1EA6D2, 0xDD57DBF4
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const S03: f64 = 5.135_465_502_073_181e-7; // 0x3EA13B54, 0xCE84D5A9
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const S04: f64 = 1.166_140_033_337_9e-9; // 0x3E1408BC, 0xF4745D8F
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/* The asymptotic expansions of pzero is
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* 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
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* For x >= 2, We approximate pzero by
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* pzero(x) = 1 + (R/S)
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* where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
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* S = 1 + pS0*s^2 + ... + pS4*s^10
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* and
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* | pzero(x)-1-R/S | <= 2 ** ( -60.26)
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*/
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const P_R8: [f64; 6] = [
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/* for x in [inf, 8]=1/[0,0.125] */
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0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
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-7.031_249_999_999_004e-2, /* 0xBFB1FFFF, 0xFFFFFD32 */
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-8.081_670_412_753_498, /* 0xC02029D0, 0xB44FA779 */
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-2.570_631_056_797_048_5e2, /* 0xC0701102, 0x7B19E863 */
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-2.485_216_410_094_288e3, /* 0xC0A36A6E, 0xCD4DCAFC */
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-5.253_043_804_907_295e3, /* 0xC0B4850B, 0x36CC643D */
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];
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const P_S8: [f64; 5] = [
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1.165_343_646_196_681_8e2, /* 0x405D2233, 0x07A96751 */
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3.833_744_753_641_218_3e3, /* 0x40ADF37D, 0x50596938 */
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4.059_785_726_484_725_5e4, /* 0x40E3D2BB, 0x6EB6B05F */
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1.167_529_725_643_759_2e5, /* 0x40FC810F, 0x8F9FA9BD */
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4.762_772_841_467_309_6e4, /* 0x40E74177, 0x4F2C49DC */
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];
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const P_R5: [f64; 6] = [
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/* for x in [8,4.5454]=1/[0.125,0.22001] */
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-1.141_254_646_918_945e-11, /* 0xBDA918B1, 0x47E495CC */
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-7.031_249_408_735_993e-2, /* 0xBFB1FFFF, 0xE69AFBC6 */
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-4.159_610_644_705_878, /* 0xC010A370, 0xF90C6BBF */
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-6.767_476_522_651_673e1, /* 0xC050EB2F, 0x5A7D1783 */
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-3.312_312_996_491_729_7e2, /* 0xC074B3B3, 0x6742CC63 */
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-3.464_333_883_656_049e2, /* 0xC075A6EF, 0x28A38BD7 */
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];
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const P_S5: [f64; 5] = [
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6.075_393_826_923_003_4e1, /* 0x404E6081, 0x0C98C5DE */
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1.051_252_305_957_045_8e3, /* 0x40906D02, 0x5C7E2864 */
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5.978_970_943_338_558e3, /* 0x40B75AF8, 0x8FBE1D60 */
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9.625_445_143_577_745e3, /* 0x40C2CCB8, 0xFA76FA38 */
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2.406_058_159_229_391e3, /* 0x40A2CC1D, 0xC70BE864 */
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];
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const P_R3: [f64; 6] = [
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/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
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-2.547_046_017_719_519e-9, /* 0xBE25E103, 0x6FE1AA86 */
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-7.031_196_163_814_817e-2, /* 0xBFB1FFF6, 0xF7C0E24B */
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-2.409_032_215_495_296, /* 0xC00345B2, 0xAEA48074 */
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-2.196_597_747_348_831e1, /* 0xC035F74A, 0x4CB94E14 */
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-5.807_917_047_017_376e1, /* 0xC04D0A22, 0x420A1A45 */
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-3.144_794_705_948_885e1, /* 0xC03F72AC, 0xA892D80F */
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];
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const P_S3: [f64; 5] = [
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3.585_603_380_552_097e1, /* 0x4041ED92, 0x84077DD3 */
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3.615_139_830_503_038_6e2, /* 0x40769839, 0x464A7C0E */
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1.193_607_837_921_115_3e3, /* 0x4092A66E, 0x6D1061D6 */
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1.127_996_798_569_074_1e3, /* 0x40919FFC, 0xB8C39B7E */
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1.735_809_308_133_357_5e2, /* 0x4065B296, 0xFC379081 */
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];
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const P_R2: [f64; 6] = [
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/* for x in [2.8570,2]=1/[0.3499,0.5] */
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-8.875_343_330_325_264e-8, /* 0xBE77D316, 0xE927026D */
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-7.030_309_954_836_247e-2, /* 0xBFB1FF62, 0x495E1E42 */
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-1.450_738_467_809_529_9, /* 0xBFF73639, 0x8A24A843 */
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-7.635_696_138_235_278, /* 0xC01E8AF3, 0xEDAFA7F3 */
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-1.119_316_688_603_567_5e1, /* 0xC02662E6, 0xC5246303 */
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-3.233_645_793_513_353_4, /* 0xC009DE81, 0xAF8FE70F */
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];
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const P_S2: [f64; 5] = [
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2.222_029_975_320_888e1, /* 0x40363865, 0x908B5959 */
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1.362_067_942_182_152e2, /* 0x4061069E, 0x0EE8878F */
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2.704_702_786_580_835e2, /* 0x4070E786, 0x42EA079B */
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1.538_753_942_083_203_3e2, /* 0x40633C03, 0x3AB6FAFF */
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1.465_761_769_482_562e1, /* 0x402D50B3, 0x44391809 */
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];
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// Note: This function is only called for ix>=0x40000000 (see above)
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fn pzero(x: f64) -> f64 {
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let ix = high_word(x) & 0x7fffffff;
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// ix>=0x40000000 && ix<=0x48000000);
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let (p, q) = if ix >= 0x40200000 {
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(P_R8, P_S8)
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} else if ix >= 0x40122E8B {
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(P_R5, P_S5)
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} else if ix >= 0x4006DB6D {
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(P_R3, P_S3)
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} else {
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(P_R2, P_S2)
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};
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let z = 1.0 / (x * x);
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let r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
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let s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))));
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1.0 + r / s
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}
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/* For x >= 8, the asymptotic expansions of qzero is
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* -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
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* We approximate pzero by
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* qzero(x) = s*(-1.25 + (R/S))
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* where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
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* S = 1 + qS0*s^2 + ... + qS5*s^12
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* and
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* | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
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*/
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const Q_R8: [f64; 6] = [
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/* for x in [inf, 8]=1/[0,0.125] */
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0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
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7.324_218_749_999_35e-2, /* 0x3FB2BFFF, 0xFFFFFE2C */
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1.176_820_646_822_527e1, /* 0x40278952, 0x5BB334D6 */
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5.576_733_802_564_019e2, /* 0x40816D63, 0x15301825 */
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8.859_197_207_564_686e3, /* 0x40C14D99, 0x3E18F46D */
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3.701_462_677_768_878e4, /* 0x40E212D4, 0x0E901566 */
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];
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const Q_S8: [f64; 6] = [
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1.637_760_268_956_898_2e2, /* 0x406478D5, 0x365B39BC */
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8.098_344_946_564_498e3, /* 0x40BFA258, 0x4E6B0563 */
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1.425_382_914_191_204_8e5, /* 0x41016652, 0x54D38C3F */
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8.033_092_571_195_144e5, /* 0x412883DA, 0x83A52B43 */
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8.405_015_798_190_605e5, /* 0x4129A66B, 0x28DE0B3D */
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-3.438_992_935_378_666e5, /* 0xC114FD6D, 0x2C9530C5 */
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];
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const Q_R5: [f64; 6] = [
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/* for x in [8,4.5454]=1/[0.125,0.22001] */
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1.840_859_635_945_155_3e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
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7.324_217_666_126_848e-2, /* 0x3FB2BFFF, 0xD172B04C */
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5.835_635_089_620_569_5, /* 0x401757B0, 0xB9953DD3 */
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1.351_115_772_864_498_3e2, /* 0x4060E392, 0x0A8788E9 */
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1.027_243_765_961_641e3, /* 0x40900CF9, 0x9DC8C481 */
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1.989_977_858_646_053_8e3, /* 0x409F17E9, 0x53C6E3A6 */
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];
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const Q_S5: [f64; 6] = [
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8.277_661_022_365_378e1, /* 0x4054B1B3, 0xFB5E1543 */
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2.077_814_164_213_93e3, /* 0x40A03BA0, 0xDA21C0CE */
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1.884_728_877_857_181e4, /* 0x40D267D2, 0x7B591E6D */
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5.675_111_228_949_473e4, /* 0x40EBB5E3, 0x97E02372 */
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3.597_675_384_251_145e4, /* 0x40E19118, 0x1F7A54A0 */
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-5.354_342_756_019_448e3, /* 0xC0B4EA57, 0xBEDBC609 */
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];
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const Q_R3: [f64; 6] = [
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/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
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4.377_410_140_897_386e-9, /* 0x3E32CD03, 0x6ADECB82 */
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7.324_111_800_429_114e-2, /* 0x3FB2BFEE, 0x0E8D0842 */
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3.344_231_375_161_707, /* 0x400AC0FC, 0x61149CF5 */
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4.262_184_407_454_126_5e1, /* 0x40454F98, 0x962DAEDD */
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1.708_080_913_405_656e2, /* 0x406559DB, 0xE25EFD1F */
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1.667_339_486_966_511_7e2, /* 0x4064D77C, 0x81FA21E0 */
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];
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const Q_S3: [f64; 6] = [
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4.875_887_297_245_872e1, /* 0x40486122, 0xBFE343A6 */
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7.096_892_210_566_06e2, /* 0x40862D83, 0x86544EB3 */
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3.704_148_226_201_113_6e3, /* 0x40ACF04B, 0xE44DFC63 */
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6.460_425_167_525_689e3, /* 0x40B93C6C, 0xD7C76A28 */
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2.516_333_689_203_689_6e3, /* 0x40A3A8AA, 0xD94FB1C0 */
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-1.492_474_518_361_564e2, /* 0xC062A7EB, 0x201CF40F */
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];
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const Q_R2: [f64; 6] = [
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/* for x in [2.8570,2]=1/[0.3499,0.5] */
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1.504_444_448_869_832_7e-7, /* 0x3E84313B, 0x54F76BDB */
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7.322_342_659_630_793e-2, /* 0x3FB2BEC5, 0x3E883E34 */
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1.998_191_740_938_16, /* 0x3FFFF897, 0xE727779C */
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1.449_560_293_478_857_4e1, /* 0x402CFDBF, 0xAAF96FE5 */
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3.166_623_175_047_815_4e1, /* 0x403FAA8E, 0x29FBDC4A */
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1.625_270_757_109_292_7e1, /* 0x403040B1, 0x71814BB4 */
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];
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const Q_S2: [f64; 6] = [
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3.036_558_483_552_192e1, /* 0x403E5D96, 0xF7C07AED */
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2.693_481_186_080_498_4e2, /* 0x4070D591, 0xE4D14B40 */
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8.447_837_575_953_201e2, /* 0x408A6645, 0x22B3BF22 */
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8.829_358_451_124_886e2, /* 0x408B977C, 0x9C5CC214 */
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2.126_663_885_117_988_3e2, /* 0x406A9553, 0x0E001365 */
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-5.310_954_938_826_669_5, /* 0xC0153E6A, 0xF8B32931 */
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];
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fn qzero(x: f64) -> f64 {
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let ix = high_word(x) & 0x7fffffff;
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let (p, q) = if ix >= 0x40200000 {
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(Q_R8, Q_S8)
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} else if ix >= 0x40122E8B {
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(Q_R5, Q_S5)
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} else if ix >= 0x4006DB6D {
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(Q_R3, Q_S3)
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} else {
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(Q_R2, Q_S2)
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};
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let z = 1.0 / (x * x);
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let r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
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let s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))));
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(-0.125 + r / s) / x
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}
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const U00: f64 = -7.380_429_510_868_723e-2; /* 0xBFB2E4D6, 0x99CBD01F */
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const U01: f64 = 1.766_664_525_091_811_2e-1; /* 0x3FC69D01, 0x9DE9E3FC */
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const U02: f64 = -1.381_856_719_455_969e-2; /* 0xBF8C4CE8, 0xB16CFA97 */
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const U03: f64 = 3.474_534_320_936_836_5e-4; /* 0x3F36C54D, 0x20B29B6B */
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const U04: f64 = -3.814_070_537_243_641_6e-6; /* 0xBECFFEA7, 0x73D25CAD */
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const U05: f64 = 1.955_901_370_350_229_2e-8; /* 0x3E550057, 0x3B4EABD4 */
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const U06: f64 = -3.982_051_941_321_034e-11; /* 0xBDC5E43D, 0x693FB3C8 */
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const V01: f64 = 1.273_048_348_341_237e-2; /* 0x3F8A1270, 0x91C9C71A */
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const V02: f64 = 7.600_686_273_503_533e-5; /* 0x3F13ECBB, 0xF578C6C1 */
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const V03: f64 = 2.591_508_518_404_578e-7; /* 0x3E91642D, 0x7FF202FD */
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const V04: f64 = 4.411_103_113_326_754_7e-10; /* 0x3DFE5018, 0x3BD6D9EF */
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pub(crate) fn y0(x: f64) -> f64 {
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let (lx, hx) = split_words(x);
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let ix = 0x7fffffff & hx;
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// Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0
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if ix >= 0x7ff00000 {
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return 1.0 / (x + x * x);
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}
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if (ix | lx) == 0 {
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return f64::NEG_INFINITY;
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}
|
|
if hx < 0 {
|
|
return f64::NAN;
|
|
}
|
|
|
|
if ix >= 0x40000000 {
|
|
// |x| >= 2.0
|
|
// 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 1.
|
|
|
|
let s = x.sin();
|
|
let c = x.cos();
|
|
let mut ss = s - c;
|
|
let mut 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 < 0x7fe00000 {
|
|
// make sure x+x not overflow
|
|
let z = -(x + x).cos();
|
|
if (s * c) < 0.0 {
|
|
cc = z / ss;
|
|
} else {
|
|
ss = z / cc;
|
|
}
|
|
}
|
|
return if ix > 0x48000000 {
|
|
FRAC_2_SQRT_PI * ss / x.sqrt()
|
|
} else {
|
|
let u = pzero(x);
|
|
let v = qzero(x);
|
|
FRAC_2_SQRT_PI * (u * ss + v * cc) / x.sqrt()
|
|
};
|
|
}
|
|
|
|
if ix <= 0x3e400000 {
|
|
// x < 2^(-27)
|
|
return U00 + FRAC_2_PI * x.ln();
|
|
}
|
|
let z = x * x;
|
|
let u = U00 + z * (U01 + z * (U02 + z * (U03 + z * (U04 + z * (U05 + z * U06)))));
|
|
let v = 1.0 + z * (V01 + z * (V02 + z * (V03 + z * V04)));
|
|
u / v + FRAC_2_PI * (j0(x) * x.ln())
|
|
}
|
|
|
|
pub(crate) fn j0(x: f64) -> f64 {
|
|
let hx = high_word(x);
|
|
let ix = hx & 0x7fffffff;
|
|
if x.is_nan() {
|
|
return x;
|
|
} else if x.is_infinite() {
|
|
return 0.0;
|
|
}
|
|
// the function is even
|
|
let x = x.abs();
|
|
if ix >= 0x40000000 {
|
|
// |x| >= 2.0
|
|
// let (s, c) = x.sin_cos()
|
|
let s = x.sin();
|
|
let c = x.cos();
|
|
let mut ss = s - c;
|
|
let mut cc = s + c;
|
|
// makes sure that x+x does not overflow
|
|
if ix < 0x7fe00000 {
|
|
// |x| < f64::MAX / 2.0
|
|
let z = -(x + x).cos();
|
|
if s * c < 0.0 {
|
|
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)
|
|
return if ix > 0x48000000 {
|
|
// x < 5.253807105661922e-287 (2^(-951))
|
|
FRAC_2_SQRT_PI * cc / (x.sqrt())
|
|
} else {
|
|
let u = pzero(x);
|
|
let v = qzero(x);
|
|
FRAC_2_SQRT_PI * (u * cc - v * ss) / x.sqrt()
|
|
};
|
|
};
|
|
if ix < 0x3f200000 {
|
|
// |x| < 2^(-13)
|
|
if HUGE + x > 1.0 {
|
|
// raise inexact if x != 0
|
|
if ix < 0x3e400000 {
|
|
return 1.0; // |x|<2^(-27)
|
|
} else {
|
|
return 1.0 - 0.25 * x * x;
|
|
}
|
|
}
|
|
}
|
|
let z = x * x;
|
|
let r = z * (R02 + z * (R03 + z * (R04 + z * R05)));
|
|
let s = 1.0 + z * (S01 + z * (S02 + z * (S03 + z * S04)));
|
|
if ix < 0x3FF00000 {
|
|
// |x| < 1.00
|
|
1.0 + z * (-0.25 + (r / s))
|
|
} else {
|
|
let u = 0.5 * x;
|
|
(1.0 + u) * (1.0 - u) + z * (r / s)
|
|
}
|
|
}
|