libm/math/
j1.rs

1/* origin: FreeBSD /usr/src/lib/msun/src/e_j1.c */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12/* j1(x), y1(x)
13 * Bessel function of the first and second kinds of order zero.
14 * Method -- j1(x):
15 *      1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
16 *      2. Reduce x to |x| since j1(x)=-j1(-x),  and
17 *         for x in (0,2)
18 *              j1(x) = x/2 + x*z*R0/S0,  where z = x*x;
19 *         (precision:  |j1/x - 1/2 - R0/S0 |<2**-61.51 )
20 *         for x in (2,inf)
21 *              j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
22 *              y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
23 *         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
24 *         as follow:
25 *              cos(x1) =  cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
26 *                      =  1/sqrt(2) * (sin(x) - cos(x))
27 *              sin(x1) =  sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
28 *                      = -1/sqrt(2) * (sin(x) + cos(x))
29 *         (To avoid cancellation, use
30 *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
31 *          to compute the worse one.)
32 *
33 *      3 Special cases
34 *              j1(nan)= nan
35 *              j1(0) = 0
36 *              j1(inf) = 0
37 *
38 * Method -- y1(x):
39 *      1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
40 *      2. For x<2.
41 *         Since
42 *              y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
43 *         therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
44 *         We use the following function to approximate y1,
45 *              y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
46 *         where for x in [0,2] (abs err less than 2**-65.89)
47 *              U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
48 *              V(z) = 1  + v0[0]*z + ... + v0[4]*z^5
49 *         Note: For tiny x, 1/x dominate y1 and hence
50 *              y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
51 *      3. For x>=2.
52 *              y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
53 *         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
54 *         by method mentioned above.
55 */
56
57use super::{cos, fabs, get_high_word, get_low_word, log, sin, sqrt};
58
59const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */
60const TPI: f64 = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */
61
62fn common(ix: u32, x: f64, y1: bool, sign: bool) -> f64 {
63    let z: f64;
64    let mut s: f64;
65    let c: f64;
66    let mut ss: f64;
67    let mut cc: f64;
68
69    /*
70     * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x-3pi/4)-q1(x)*sin(x-3pi/4))
71     * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x-3pi/4)+q1(x)*cos(x-3pi/4))
72     *
73     * sin(x-3pi/4) = -(sin(x) + cos(x))/sqrt(2)
74     * cos(x-3pi/4) = (sin(x) - cos(x))/sqrt(2)
75     * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
76     */
77    s = sin(x);
78    if y1 {
79        s = -s;
80    }
81    c = cos(x);
82    cc = s - c;
83    if ix < 0x7fe00000 {
84        /* avoid overflow in 2*x */
85        ss = -s - c;
86        z = cos(2.0 * x);
87        if s * c > 0.0 {
88            cc = z / ss;
89        } else {
90            ss = z / cc;
91        }
92        if ix < 0x48000000 {
93            if y1 {
94                ss = -ss;
95            }
96            cc = pone(x) * cc - qone(x) * ss;
97        }
98    }
99    if sign {
100        cc = -cc;
101    }
102    return INVSQRTPI * cc / sqrt(x);
103}
104
105/* R0/S0 on [0,2] */
106const R00: f64 = -6.25000000000000000000e-02; /* 0xBFB00000, 0x00000000 */
107const R01: f64 = 1.40705666955189706048e-03; /* 0x3F570D9F, 0x98472C61 */
108const R02: f64 = -1.59955631084035597520e-05; /* 0xBEF0C5C6, 0xBA169668 */
109const R03: f64 = 4.96727999609584448412e-08; /* 0x3E6AAAFA, 0x46CA0BD9 */
110const S01: f64 = 1.91537599538363460805e-02; /* 0x3F939D0B, 0x12637E53 */
111const S02: f64 = 1.85946785588630915560e-04; /* 0x3F285F56, 0xB9CDF664 */
112const S03: f64 = 1.17718464042623683263e-06; /* 0x3EB3BFF8, 0x333F8498 */
113const S04: f64 = 5.04636257076217042715e-09; /* 0x3E35AC88, 0xC97DFF2C */
114const S05: f64 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
115
116/// First order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the first kind (f64).
117pub fn j1(x: f64) -> f64 {
118    let mut z: f64;
119    let r: f64;
120    let s: f64;
121    let mut ix: u32;
122    let sign: bool;
123
124    ix = get_high_word(x);
125    sign = (ix >> 31) != 0;
126    ix &= 0x7fffffff;
127    if ix >= 0x7ff00000 {
128        return 1.0 / (x * x);
129    }
130    if ix >= 0x40000000 {
131        /* |x| >= 2 */
132        return common(ix, fabs(x), false, sign);
133    }
134    if ix >= 0x38000000 {
135        /* |x| >= 2**-127 */
136        z = x * x;
137        r = z * (R00 + z * (R01 + z * (R02 + z * R03)));
138        s = 1.0 + z * (S01 + z * (S02 + z * (S03 + z * (S04 + z * S05))));
139        z = r / s;
140    } else {
141        /* avoid underflow, raise inexact if x!=0 */
142        z = x;
143    }
144    return (0.5 + z) * x;
145}
146
147const U0: [f64; 5] = [
148    -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
149    5.04438716639811282616e-02,  /* 0x3FA9D3C7, 0x76292CD1 */
150    -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
151    2.35252600561610495928e-05,  /* 0x3EF8AB03, 0x8FA6B88E */
152    -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
153];
154const V0: [f64; 5] = [
155    1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
156    2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
157    1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
158    6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
159    1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
160];
161
162/// First order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the second kind (f64).
163pub fn y1(x: f64) -> f64 {
164    let z: f64;
165    let u: f64;
166    let v: f64;
167    let ix: u32;
168    let lx: u32;
169
170    ix = get_high_word(x);
171    lx = get_low_word(x);
172
173    /* y1(nan)=nan, y1(<0)=nan, y1(0)=-inf, y1(inf)=0 */
174    if (ix << 1 | lx) == 0 {
175        return -1.0 / 0.0;
176    }
177    if (ix >> 31) != 0 {
178        return 0.0 / 0.0;
179    }
180    if ix >= 0x7ff00000 {
181        return 1.0 / x;
182    }
183
184    if ix >= 0x40000000 {
185        /* x >= 2 */
186        return common(ix, x, true, false);
187    }
188    if ix < 0x3c900000 {
189        /* x < 2**-54 */
190        return -TPI / x;
191    }
192    z = x * x;
193    u = U0[0] + z * (U0[1] + z * (U0[2] + z * (U0[3] + z * U0[4])));
194    v = 1.0 + z * (V0[0] + z * (V0[1] + z * (V0[2] + z * (V0[3] + z * V0[4]))));
195    return x * (u / v) + TPI * (j1(x) * log(x) - 1.0 / x);
196}
197
198/* For x >= 8, the asymptotic expansions of pone is
199 *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
200 * We approximate pone by
201 *      pone(x) = 1 + (R/S)
202 * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
203 *        S = 1 + ps0*s^2 + ... + ps4*s^10
204 * and
205 *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
206 */
207
208const PR8: [f64; 6] = [
209    /* for x in [inf, 8]=1/[0,0.125] */
210    0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
211    1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
212    1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
213    4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
214    3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
215    7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
216];
217const PS8: [f64; 5] = [
218    1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
219    3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
220    3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
221    9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
222    3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
223];
224
225const PR5: [f64; 6] = [
226    /* for x in [8,4.5454]=1/[0.125,0.22001] */
227    1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
228    1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
229    6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
230    1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
231    5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
232    5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
233];
234const PS5: [f64; 5] = [
235    5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
236    9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
237    5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
238    7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
239    1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
240];
241
242const PR3: [f64; 6] = [
243    3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
244    1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
245    3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
246    3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
247    9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
248    4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
249];
250const PS3: [f64; 5] = [
251    3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
252    3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
253    1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
254    8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
255    1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
256];
257
258const PR2: [f64; 6] = [
259    /* for x in [2.8570,2]=1/[0.3499,0.5] */
260    1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
261    1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
262    2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
263    1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
264    1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
265    5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
266];
267const PS2: [f64; 5] = [
268    2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
269    1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
270    2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
271    1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
272    8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
273];
274
275fn pone(x: f64) -> f64 {
276    let p: &[f64; 6];
277    let q: &[f64; 5];
278    let z: f64;
279    let r: f64;
280    let s: f64;
281    let mut ix: u32;
282
283    ix = get_high_word(x);
284    ix &= 0x7fffffff;
285    if ix >= 0x40200000 {
286        p = &PR8;
287        q = &PS8;
288    } else if ix >= 0x40122E8B {
289        p = &PR5;
290        q = &PS5;
291    } else if ix >= 0x4006DB6D {
292        p = &PR3;
293        q = &PS3;
294    } else
295    /*ix >= 0x40000000*/
296    {
297        p = &PR2;
298        q = &PS2;
299    }
300    z = 1.0 / (x * x);
301    r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
302    s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))));
303    return 1.0 + r / s;
304}
305
306/* For x >= 8, the asymptotic expansions of qone is
307 *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
308 * We approximate pone by
309 *      qone(x) = s*(0.375 + (R/S))
310 * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
311 *        S = 1 + qs1*s^2 + ... + qs6*s^12
312 * and
313 *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
314 */
315
316const QR8: [f64; 6] = [
317    /* for x in [inf, 8]=1/[0,0.125] */
318    0.00000000000000000000e+00,  /* 0x00000000, 0x00000000 */
319    -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
320    -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
321    -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
322    -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
323    -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
324];
325const QS8: [f64; 6] = [
326    1.61395369700722909556e+02,  /* 0x40642CA6, 0xDE5BCDE5 */
327    7.82538599923348465381e+03,  /* 0x40BE9162, 0xD0D88419 */
328    1.33875336287249578163e+05,  /* 0x4100579A, 0xB0B75E98 */
329    7.19657723683240939863e+05,  /* 0x4125F653, 0x72869C19 */
330    6.66601232617776375264e+05,  /* 0x412457D2, 0x7719AD5C */
331    -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
332];
333
334const QR5: [f64; 6] = [
335    /* for x in [8,4.5454]=1/[0.125,0.22001] */
336    -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
337    -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
338    -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
339    -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
340    -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
341    -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
342];
343const QS5: [f64; 6] = [
344    8.12765501384335777857e+01,  /* 0x405451B2, 0xFF5A11B2 */
345    1.99179873460485964642e+03,  /* 0x409F1F31, 0xE77BF839 */
346    1.74684851924908907677e+04,  /* 0x40D10F1F, 0x0D64CE29 */
347    4.98514270910352279316e+04,  /* 0x40E8576D, 0xAABAD197 */
348    2.79480751638918118260e+04,  /* 0x40DB4B04, 0xCF7C364B */
349    -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
350];
351
352const QR3: [f64; 6] = [
353    -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
354    -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
355    -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
356    -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
357    -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
358    -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
359];
360const QS3: [f64; 6] = [
361    4.76651550323729509273e+01,  /* 0x4047D523, 0xCCD367E4 */
362    6.73865112676699709482e+02,  /* 0x40850EEB, 0xC031EE3E */
363    3.38015286679526343505e+03,  /* 0x40AA684E, 0x448E7C9A */
364    5.54772909720722782367e+03,  /* 0x40B5ABBA, 0xA61D54A6 */
365    1.90311919338810798763e+03,  /* 0x409DBC7A, 0x0DD4DF4B */
366    -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
367];
368
369const QR2: [f64; 6] = [
370    /* for x in [2.8570,2]=1/[0.3499,0.5] */
371    -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
372    -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
373    -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
374    -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
375    -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
376    -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
377];
378const QS2: [f64; 6] = [
379    2.95333629060523854548e+01,  /* 0x403D888A, 0x78AE64FF */
380    2.52981549982190529136e+02,  /* 0x406F9F68, 0xDB821CBA */
381    7.57502834868645436472e+02,  /* 0x4087AC05, 0xCE49A0F7 */
382    7.39393205320467245656e+02,  /* 0x40871B25, 0x48D4C029 */
383    1.55949003336666123687e+02,  /* 0x40637E5E, 0x3C3ED8D4 */
384    -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
385];
386
387fn qone(x: f64) -> f64 {
388    let p: &[f64; 6];
389    let q: &[f64; 6];
390    let s: f64;
391    let r: f64;
392    let z: f64;
393    let mut ix: u32;
394
395    ix = get_high_word(x);
396    ix &= 0x7fffffff;
397    if ix >= 0x40200000 {
398        p = &QR8;
399        q = &QS8;
400    } else if ix >= 0x40122E8B {
401        p = &QR5;
402        q = &QS5;
403    } else if ix >= 0x4006DB6D {
404        p = &QR3;
405        q = &QS3;
406    } else
407    /*ix >= 0x40000000*/
408    {
409        p = &QR2;
410        q = &QS2;
411    }
412    z = 1.0 / (x * x);
413    r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
414    s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))));
415    return (0.375 + r / s) / x;
416}
libm/math/j1.rs

libm/math/
j1.rs
