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
16 use super::{cosf, fabsf, logf, sinf, sqrtf};
17
18 const INVSQRTPI: f32 = 5.6418961287e-01; /* 0x3f106ebb */
19 const TPI: f32 = 6.3661974669e-01; /* 0x3f22f983 */
20
common(ix: u32, x: f32, y0: bool) -> f3221 fn 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] */
56 const R02: f32 = 1.5625000000e-02; /* 0x3c800000 */
57 const R03: f32 = -1.8997929874e-04; /* 0xb947352e */
58 const R04: f32 = 1.8295404516e-06; /* 0x35f58e88 */
59 const R05: f32 = -4.6183270541e-09; /* 0xb19eaf3c */
60 const S01: f32 = 1.5619102865e-02; /* 0x3c7fe744 */
61 const S02: f32 = 1.1692678527e-04; /* 0x38f53697 */
62 const S03: f32 = 5.1354652442e-07; /* 0x3509daa6 */
63 const 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).
j0f(mut x: f32) -> f3266 pub 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
99 const U00: f32 = -7.3804296553e-02; /* 0xbd9726b5 */
100 const U01: f32 = 1.7666645348e-01; /* 0x3e34e80d */
101 const U02: f32 = -1.3818567619e-02; /* 0xbc626746 */
102 const U03: f32 = 3.4745343146e-04; /* 0x39b62a69 */
103 const U04: f32 = -3.8140706238e-06; /* 0xb67ff53c */
104 const U05: f32 = 1.9559013964e-08; /* 0x32a802ba */
105 const U06: f32 = -3.9820518410e-11; /* 0xae2f21eb */
106 const V01: f32 = 1.2730483897e-02; /* 0x3c509385 */
107 const V02: f32 = 7.6006865129e-05; /* 0x389f65e0 */
108 const V03: f32 = 2.5915085189e-07; /* 0x348b216c */
109 const 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).
y0f(x: f32) -> f32112 pub 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 */
153 const 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 ];
162 const 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 ];
169 const 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 ];
178 const 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
186 const 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 ];
195 const 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
203 const 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 ];
212 const 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
pzerof(x: f32) -> f32220 fn 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 */
260 const 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 ];
269 const 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
278 const 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 ];
287 const 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
296 const 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 ];
305 const 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
314 const 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 ];
323 const 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
qzerof(x: f32) -> f32332 fn 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 }
362