libm/math/
j0f.rs

1/* origin: FreeBSD /usr/src/lib/msun/src/e_j0f.c */
2/*
3 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
4 */
5/*
6 * ====================================================
7 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
8 *
9 * Developed at SunPro, a Sun Microsystems, Inc. business.
10 * Permission to use, copy, modify, and distribute this
11 * software is freely granted, provided that this notice
12 * is preserved.
13 * ====================================================
14 */
15
16use super::{cosf, fabsf, logf, sinf, sqrtf};
17
18const INVSQRTPI: f32 = 5.6418961287e-01; /* 0x3f106ebb */
19const TPI: f32 = 6.3661974669e-01; /* 0x3f22f983 */
20
21fn common(ix: u32, x: f32, y0: bool) -> f32 {
22    let z: f32;
23    let s: f32;
24    let mut c: f32;
25    let mut ss: f32;
26    let mut cc: f32;
27    /*
28     * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
29     * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
30     */
31    s = sinf(x);
32    c = cosf(x);
33    if y0 {
34        c = -c;
35    }
36    cc = s + c;
37    if ix < 0x7f000000 {
38        ss = s - c;
39        z = -cosf(2.0 * x);
40        if s * c < 0.0 {
41            cc = z / ss;
42        } else {
43            ss = z / cc;
44        }
45        if ix < 0x58800000 {
46            if y0 {
47                ss = -ss;
48            }
49            cc = pzerof(x) * cc - qzerof(x) * ss;
50        }
51    }
52    return INVSQRTPI * cc / sqrtf(x);
53}
54
55/* R0/S0 on [0, 2.00] */
56const R02: f32 = 1.5625000000e-02; /* 0x3c800000 */
57const R03: f32 = -1.8997929874e-04; /* 0xb947352e */
58const R04: f32 = 1.8295404516e-06; /* 0x35f58e88 */
59const R05: f32 = -4.6183270541e-09; /* 0xb19eaf3c */
60const S01: f32 = 1.5619102865e-02; /* 0x3c7fe744 */
61const S02: f32 = 1.1692678527e-04; /* 0x38f53697 */
62const S03: f32 = 5.1354652442e-07; /* 0x3509daa6 */
63const S04: f32 = 1.1661400734e-09; /* 0x30a045e8 */
64
65/// Zeroth order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the first kind (f32).
66pub fn j0f(mut x: f32) -> f32 {
67    let z: f32;
68    let r: f32;
69    let s: f32;
70    let mut ix: u32;
71
72    ix = x.to_bits();
73    ix &= 0x7fffffff;
74    if ix >= 0x7f800000 {
75        return 1.0 / (x * x);
76    }
77    x = fabsf(x);
78
79    if ix >= 0x40000000 {
80        /* |x| >= 2 */
81        /* large ulp error near zeros */
82        return common(ix, x, false);
83    }
84    if ix >= 0x3a000000 {
85        /* |x| >= 2**-11 */
86        /* up to 4ulp error near 2 */
87        z = x * x;
88        r = z * (R02 + z * (R03 + z * (R04 + z * R05)));
89        s = 1.0 + z * (S01 + z * (S02 + z * (S03 + z * S04)));
90        return (1.0 + x / 2.0) * (1.0 - x / 2.0) + z * (r / s);
91    }
92    if ix >= 0x21800000 {
93        /* |x| >= 2**-60 */
94        x = 0.25 * x * x;
95    }
96    return 1.0 - x;
97}
98
99const U00: f32 = -7.3804296553e-02; /* 0xbd9726b5 */
100const U01: f32 = 1.7666645348e-01; /* 0x3e34e80d */
101const U02: f32 = -1.3818567619e-02; /* 0xbc626746 */
102const U03: f32 = 3.4745343146e-04; /* 0x39b62a69 */
103const U04: f32 = -3.8140706238e-06; /* 0xb67ff53c */
104const U05: f32 = 1.9559013964e-08; /* 0x32a802ba */
105const U06: f32 = -3.9820518410e-11; /* 0xae2f21eb */
106const V01: f32 = 1.2730483897e-02; /* 0x3c509385 */
107const V02: f32 = 7.6006865129e-05; /* 0x389f65e0 */
108const V03: f32 = 2.5915085189e-07; /* 0x348b216c */
109const V04: f32 = 4.4111031494e-10; /* 0x2ff280c2 */
110
111/// Zeroth order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the second kind (f32).
112pub fn y0f(x: f32) -> f32 {
113    let z: f32;
114    let u: f32;
115    let v: f32;
116    let ix: u32;
117
118    ix = x.to_bits();
119    if (ix & 0x7fffffff) == 0 {
120        return -1.0 / 0.0;
121    }
122    if (ix >> 31) != 0 {
123        return 0.0 / 0.0;
124    }
125    if ix >= 0x7f800000 {
126        return 1.0 / x;
127    }
128    if ix >= 0x40000000 {
129        /* |x| >= 2.0 */
130        /* large ulp error near zeros */
131        return common(ix, x, true);
132    }
133    if ix >= 0x39000000 {
134        /* x >= 2**-13 */
135        /* large ulp error at x ~= 0.89 */
136        z = x * x;
137        u = U00 + z * (U01 + z * (U02 + z * (U03 + z * (U04 + z * (U05 + z * U06)))));
138        v = 1.0 + z * (V01 + z * (V02 + z * (V03 + z * V04)));
139        return u / v + TPI * (j0f(x) * logf(x));
140    }
141    return U00 + TPI * logf(x);
142}
143
144/* The asymptotic expansions of pzero is
145 *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
146 * For x >= 2, We approximate pzero by
147 *      pzero(x) = 1 + (R/S)
148 * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
149 *        S = 1 + pS0*s^2 + ... + pS4*s^10
150 * and
151 *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
152 */
153const PR8: [f32; 6] = [
154    /* for x in [inf, 8]=1/[0,0.125] */
155    0.0000000000e+00,  /* 0x00000000 */
156    -7.0312500000e-02, /* 0xbd900000 */
157    -8.0816707611e+00, /* 0xc1014e86 */
158    -2.5706311035e+02, /* 0xc3808814 */
159    -2.4852163086e+03, /* 0xc51b5376 */
160    -5.2530439453e+03, /* 0xc5a4285a */
161];
162const PS8: [f32; 5] = [
163    1.1653436279e+02, /* 0x42e91198 */
164    3.8337448730e+03, /* 0x456f9beb */
165    4.0597855469e+04, /* 0x471e95db */
166    1.1675296875e+05, /* 0x47e4087c */
167    4.7627726562e+04, /* 0x473a0bba */
168];
169const PR5: [f32; 6] = [
170    /* for x in [8,4.5454]=1/[0.125,0.22001] */
171    -1.1412546255e-11, /* 0xad48c58a */
172    -7.0312492549e-02, /* 0xbd8fffff */
173    -4.1596107483e+00, /* 0xc0851b88 */
174    -6.7674766541e+01, /* 0xc287597b */
175    -3.3123129272e+02, /* 0xc3a59d9b */
176    -3.4643338013e+02, /* 0xc3ad3779 */
177];
178const PS5: [f32; 5] = [
179    6.0753936768e+01, /* 0x42730408 */
180    1.0512523193e+03, /* 0x44836813 */
181    5.9789707031e+03, /* 0x45bad7c4 */
182    9.6254453125e+03, /* 0x461665c8 */
183    2.4060581055e+03, /* 0x451660ee */
184];
185
186const PR3: [f32; 6] = [
187    /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
188    -2.5470459075e-09, /* 0xb12f081b */
189    -7.0311963558e-02, /* 0xbd8fffb8 */
190    -2.4090321064e+00, /* 0xc01a2d95 */
191    -2.1965976715e+01, /* 0xc1afba52 */
192    -5.8079170227e+01, /* 0xc2685112 */
193    -3.1447946548e+01, /* 0xc1fb9565 */
194];
195const PS3: [f32; 5] = [
196    3.5856033325e+01, /* 0x420f6c94 */
197    3.6151397705e+02, /* 0x43b4c1ca */
198    1.1936077881e+03, /* 0x44953373 */
199    1.1279968262e+03, /* 0x448cffe6 */
200    1.7358093262e+02, /* 0x432d94b8 */
201];
202
203const PR2: [f32; 6] = [
204    /* for x in [2.8570,2]=1/[0.3499,0.5] */
205    -8.8753431271e-08, /* 0xb3be98b7 */
206    -7.0303097367e-02, /* 0xbd8ffb12 */
207    -1.4507384300e+00, /* 0xbfb9b1cc */
208    -7.6356959343e+00, /* 0xc0f4579f */
209    -1.1193166733e+01, /* 0xc1331736 */
210    -3.2336456776e+00, /* 0xc04ef40d */
211];
212const PS2: [f32; 5] = [
213    2.2220300674e+01, /* 0x41b1c32d */
214    1.3620678711e+02, /* 0x430834f0 */
215    2.7047027588e+02, /* 0x43873c32 */
216    1.5387539673e+02, /* 0x4319e01a */
217    1.4657617569e+01, /* 0x416a859a */
218];
219
220fn pzerof(x: f32) -> f32 {
221    let p: &[f32; 6];
222    let q: &[f32; 5];
223    let z: f32;
224    let r: f32;
225    let s: f32;
226    let mut ix: u32;
227
228    ix = x.to_bits();
229    ix &= 0x7fffffff;
230    if ix >= 0x41000000 {
231        p = &PR8;
232        q = &PS8;
233    } else if ix >= 0x409173eb {
234        p = &PR5;
235        q = &PS5;
236    } else if ix >= 0x4036d917 {
237        p = &PR3;
238        q = &PS3;
239    } else
240    /*ix >= 0x40000000*/
241    {
242        p = &PR2;
243        q = &PS2;
244    }
245    z = 1.0 / (x * x);
246    r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
247    s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))));
248    return 1.0 + r / s;
249}
250
251/* For x >= 8, the asymptotic expansions of qzero is
252 *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
253 * We approximate pzero by
254 *      qzero(x) = s*(-1.25 + (R/S))
255 * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
256 *        S = 1 + qS0*s^2 + ... + qS5*s^12
257 * and
258 *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
259 */
260const QR8: [f32; 6] = [
261    /* for x in [inf, 8]=1/[0,0.125] */
262    0.0000000000e+00, /* 0x00000000 */
263    7.3242187500e-02, /* 0x3d960000 */
264    1.1768206596e+01, /* 0x413c4a93 */
265    5.5767340088e+02, /* 0x440b6b19 */
266    8.8591972656e+03, /* 0x460a6cca */
267    3.7014625000e+04, /* 0x471096a0 */
268];
269const QS8: [f32; 6] = [
270    1.6377603149e+02,  /* 0x4323c6aa */
271    8.0983447266e+03,  /* 0x45fd12c2 */
272    1.4253829688e+05,  /* 0x480b3293 */
273    8.0330925000e+05,  /* 0x49441ed4 */
274    8.4050156250e+05,  /* 0x494d3359 */
275    -3.4389928125e+05, /* 0xc8a7eb69 */
276];
277
278const QR5: [f32; 6] = [
279    /* for x in [8,4.5454]=1/[0.125,0.22001] */
280    1.8408595828e-11, /* 0x2da1ec79 */
281    7.3242180049e-02, /* 0x3d95ffff */
282    5.8356351852e+00, /* 0x40babd86 */
283    1.3511157227e+02, /* 0x43071c90 */
284    1.0272437744e+03, /* 0x448067cd */
285    1.9899779053e+03, /* 0x44f8bf4b */
286];
287const QS5: [f32; 6] = [
288    8.2776611328e+01,  /* 0x42a58da0 */
289    2.0778142090e+03,  /* 0x4501dd07 */
290    1.8847289062e+04,  /* 0x46933e94 */
291    5.6751113281e+04,  /* 0x475daf1d */
292    3.5976753906e+04,  /* 0x470c88c1 */
293    -5.3543427734e+03, /* 0xc5a752be */
294];
295
296const QR3: [f32; 6] = [
297    /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
298    4.3774099900e-09, /* 0x3196681b */
299    7.3241114616e-02, /* 0x3d95ff70 */
300    3.3442313671e+00, /* 0x405607e3 */
301    4.2621845245e+01, /* 0x422a7cc5 */
302    1.7080809021e+02, /* 0x432acedf */
303    1.6673394775e+02, /* 0x4326bbe4 */
304];
305const QS3: [f32; 6] = [
306    4.8758872986e+01,  /* 0x42430916 */
307    7.0968920898e+02,  /* 0x44316c1c */
308    3.7041481934e+03,  /* 0x4567825f */
309    6.4604252930e+03,  /* 0x45c9e367 */
310    2.5163337402e+03,  /* 0x451d4557 */
311    -1.4924745178e+02, /* 0xc3153f59 */
312];
313
314const QR2: [f32; 6] = [
315    /* for x in [2.8570,2]=1/[0.3499,0.5] */
316    1.5044444979e-07, /* 0x342189db */
317    7.3223426938e-02, /* 0x3d95f62a */
318    1.9981917143e+00, /* 0x3fffc4bf */
319    1.4495602608e+01, /* 0x4167edfd */
320    3.1666231155e+01, /* 0x41fd5471 */
321    1.6252708435e+01, /* 0x4182058c */
322];
323const QS2: [f32; 6] = [
324    3.0365585327e+01,  /* 0x41f2ecb8 */
325    2.6934811401e+02,  /* 0x4386ac8f */
326    8.4478375244e+02,  /* 0x44533229 */
327    8.8293585205e+02,  /* 0x445cbbe5 */
328    2.1266638184e+02,  /* 0x4354aa98 */
329    -5.3109550476e+00, /* 0xc0a9f358 */
330];
331
332fn qzerof(x: f32) -> f32 {
333    let p: &[f32; 6];
334    let q: &[f32; 6];
335    let s: f32;
336    let r: f32;
337    let z: f32;
338    let mut ix: u32;
339
340    ix = x.to_bits();
341    ix &= 0x7fffffff;
342    if ix >= 0x41000000 {
343        p = &QR8;
344        q = &QS8;
345    } else if ix >= 0x409173eb {
346        p = &QR5;
347        q = &QS5;
348    } else if ix >= 0x4036d917 {
349        p = &QR3;
350        q = &QS3;
351    } else
352    /*ix >= 0x40000000*/
353    {
354        p = &QR2;
355        q = &QS2;
356    }
357    z = 1.0 / (x * x);
358    r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
359    s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))));
360    return (-0.125 + r / s) / x;
361}
libm/math/j0f.rs

libm/math/
j0f.rs
