1 /* ef_j0.c -- float version of e_j0.c.
2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
3 */
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 #include "fdlibm.h"
17
18 static float pzerof(float), qzerof(float);
19
20 static const float huge = 1e30, one = 1.0,
21 invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */
22 tpi = 6.3661974669e-01, /* 0x3f22f983 */
23 /* R0/S0 on [0, 2.00] */
24 R02 = 1.5625000000e-02, /* 0x3c800000 */
25 R03 = -1.8997929874e-04, /* 0xb947352e */
26 R04 = 1.8295404516e-06, /* 0x35f58e88 */
27 R05 = -4.6183270541e-09, /* 0xb19eaf3c */
28 S01 = 1.5619102865e-02, /* 0x3c7fe744 */
29 S02 = 1.1692678527e-04, /* 0x38f53697 */
30 S03 = 5.1354652442e-07, /* 0x3509daa6 */
31 S04 = 1.1661400734e-09; /* 0x30a045e8 */
32
33 static const float zero = 0.0;
34
35 float
j0f(float x)36 j0f(float x)
37 {
38 float z, s, c, ss, cc, r, u, v;
39 __int32_t hx, ix;
40
41 if (isnan(x))
42 return x + x;
43
44 if (isinf(x))
45 return zero;
46
47 GET_FLOAT_WORD(hx, x);
48 ix = hx & 0x7fffffff;
49 x = fabsf(x);
50 if (ix >= 0x40000000) { /* |x| >= 2.0 */
51 s = sinf(x);
52 c = cosf(x);
53 ss = s - c;
54 cc = s + c;
55 if (ix <= FLT_UWORD_HALF_MAX) { /* make sure x+x not overflow */
56 z = -cosf(x + x);
57 if ((s * c) < zero)
58 cc = z / ss;
59 else
60 ss = z / cc;
61 }
62 /*
63 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
64 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
65 */
66 if (ix > 0x5c000000)
67 z = (invsqrtpi * cc) / sqrtf(x);
68 else {
69 u = pzerof(x);
70 v = qzerof(x);
71 z = invsqrtpi * (u * cc - v * ss) / sqrtf(x);
72 }
73 return z;
74 }
75 if (ix < 0x39000000) { /* |x| < 2**-13 */
76 if (huge + x > one) { /* raise inexact if x != 0 */
77 if (ix < 0x32000000)
78 return one; /* |x|<2**-27 */
79 else
80 return one - (float)0.25 * x * x;
81 }
82 }
83 z = x * x;
84 r = z * (R02 + z * (R03 + z * (R04 + z * R05)));
85 s = one + z * (S01 + z * (S02 + z * (S03 + z * S04)));
86 if (ix < 0x3F800000) { /* |x| < 1.00 */
87 return one + z * ((float)-0.25 + (r / s));
88 } else {
89 u = (float)0.5 * x;
90 return ((one + u) * (one - u) + z * (r / s));
91 }
92 }
93
94 static const float u00 = -7.3804296553e-02, /* 0xbd9726b5 */
95 u01 = 1.7666645348e-01, /* 0x3e34e80d */
96 u02 = -1.3818567619e-02, /* 0xbc626746 */
97 u03 = 3.4745343146e-04, /* 0x39b62a69 */
98 u04 = -3.8140706238e-06, /* 0xb67ff53c */
99 u05 = 1.9559013964e-08, /* 0x32a802ba */
100 u06 = -3.9820518410e-11, /* 0xae2f21eb */
101 v01 = 1.2730483897e-02, /* 0x3c509385 */
102 v02 = 7.6006865129e-05, /* 0x389f65e0 */
103 v03 = 2.5915085189e-07, /* 0x348b216c */
104 v04 = 4.4111031494e-10; /* 0x2ff280c2 */
105
106 float
y0f(float x)107 y0f(float x)
108 {
109 float z, s, c, ss, cc, u, v;
110 __int32_t hx, ix;
111
112 GET_FLOAT_WORD(hx, x);
113 ix = 0x7fffffff & hx;
114
115 if (ix == 0)
116 return __math_divzerof(1);
117
118 if (ix > 0x7f800000)
119 return x + x;
120
121 if (hx < 0)
122 return __math_invalidf(x);
123
124 if (ix == 0x7f800000)
125 return zero;
126
127 if (ix >= 0x40000000) { /* |x| >= 2.0 */
128 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
129 * where x0 = x-pi/4
130 * Better formula:
131 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
132 * = 1/sqrt(2) * (sin(x) + cos(x))
133 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
134 * = 1/sqrt(2) * (sin(x) - cos(x))
135 * To avoid cancellation, use
136 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
137 * to compute the worse one.
138 */
139 s = sinf(x);
140 c = cosf(x);
141 ss = s - c;
142 cc = s + c;
143 /*
144 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
145 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
146 */
147 if (ix <= FLT_UWORD_HALF_MAX) { /* make sure x+x not overflow */
148 z = -cosf(x + x);
149 if ((s * c) < zero)
150 cc = z / ss;
151 else
152 ss = z / cc;
153 }
154 if (ix > 0x5c000000)
155 z = (invsqrtpi * ss) / sqrtf(x);
156 else {
157 u = pzerof(x);
158 v = qzerof(x);
159 z = invsqrtpi * (u * ss + v * cc) / sqrtf(x);
160 }
161 return z;
162 }
163 if (ix <= 0x39800000) { /* x < 2**-27 */
164 return (u00 + tpi * logf(x));
165 }
166 z = x * x;
167 u = u00 +
168 z * (u01 + z * (u02 + z * (u03 + z * (u04 + z * (u05 + z * u06)))));
169 v = one + z * (v01 + z * (v02 + z * (v03 + z * v04)));
170 return (u / v + tpi * (j0f(x) * logf(x)));
171 }
172
173 /* The asymptotic expansions of pzero is
174 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
175 * For x >= 2, We approximate pzero by
176 * pzero(x) = 1 + (R/S)
177 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
178 * S = 1 + pS0*s^2 + ... + pS4*s^10
179 * and
180 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
181 */
182 static const float pR8[6] = {
183 /* for x in [inf, 8]=1/[0,0.125] */
184 0.0000000000e+00, /* 0x00000000 */
185 -7.0312500000e-02, /* 0xbd900000 */
186 -8.0816707611e+00, /* 0xc1014e86 */
187 -2.5706311035e+02, /* 0xc3808814 */
188 -2.4852163086e+03, /* 0xc51b5376 */
189 -5.2530439453e+03, /* 0xc5a4285a */
190 };
191 static const float pS8[5] = {
192 1.1653436279e+02, /* 0x42e91198 */
193 3.8337448730e+03, /* 0x456f9beb */
194 4.0597855469e+04, /* 0x471e95db */
195 1.1675296875e+05, /* 0x47e4087c */
196 4.7627726562e+04, /* 0x473a0bba */
197 };
198 static const float pR5[6] = {
199 /* for x in [8,4.5454]=1/[0.125,0.22001] */
200 -1.1412546255e-11, /* 0xad48c58a */
201 -7.0312492549e-02, /* 0xbd8fffff */
202 -4.1596107483e+00, /* 0xc0851b88 */
203 -6.7674766541e+01, /* 0xc287597b */
204 -3.3123129272e+02, /* 0xc3a59d9b */
205 -3.4643338013e+02, /* 0xc3ad3779 */
206 };
207 static const float pS5[5] = {
208 6.0753936768e+01, /* 0x42730408 */
209 1.0512523193e+03, /* 0x44836813 */
210 5.9789707031e+03, /* 0x45bad7c4 */
211 9.6254453125e+03, /* 0x461665c8 */
212 2.4060581055e+03, /* 0x451660ee */
213 };
214
215 static const float pR3[6] = {
216 /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
217 -2.5470459075e-09, /* 0xb12f081b */
218 -7.0311963558e-02, /* 0xbd8fffb8 */
219 -2.4090321064e+00, /* 0xc01a2d95 */
220 -2.1965976715e+01, /* 0xc1afba52 */
221 -5.8079170227e+01, /* 0xc2685112 */
222 -3.1447946548e+01, /* 0xc1fb9565 */
223 };
224 static const float pS3[5] = {
225 3.5856033325e+01, /* 0x420f6c94 */
226 3.6151397705e+02, /* 0x43b4c1ca */
227 1.1936077881e+03, /* 0x44953373 */
228 1.1279968262e+03, /* 0x448cffe6 */
229 1.7358093262e+02, /* 0x432d94b8 */
230 };
231
232 static const float pR2[6] = {
233 /* for x in [2.8570,2]=1/[0.3499,0.5] */
234 -8.8753431271e-08, /* 0xb3be98b7 */
235 -7.0303097367e-02, /* 0xbd8ffb12 */
236 -1.4507384300e+00, /* 0xbfb9b1cc */
237 -7.6356959343e+00, /* 0xc0f4579f */
238 -1.1193166733e+01, /* 0xc1331736 */
239 -3.2336456776e+00, /* 0xc04ef40d */
240 };
241 static const float pS2[5] = {
242 2.2220300674e+01, /* 0x41b1c32d */
243 1.3620678711e+02, /* 0x430834f0 */
244 2.7047027588e+02, /* 0x43873c32 */
245 1.5387539673e+02, /* 0x4319e01a */
246 1.4657617569e+01, /* 0x416a859a */
247 };
248
249 static float
pzerof(float x)250 pzerof(float x)
251 {
252 const float *p, *q;
253 float z, r, s;
254 __int32_t ix;
255 GET_FLOAT_WORD(ix, x);
256 ix &= 0x7fffffff;
257 if (ix >= 0x41000000) {
258 p = pR8;
259 q = pS8;
260 } else if (ix >= 0x40f71c58) {
261 p = pR5;
262 q = pS5;
263 } else if (ix >= 0x4036db68) {
264 p = pR3;
265 q = pS3;
266 } else {
267 p = pR2;
268 q = pS2;
269 }
270 z = one / (x * x);
271 r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
272 s = one + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))));
273 return one + r / s;
274 }
275
276 /* For x >= 8, the asymptotic expansions of qzero is
277 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
278 * We approximate qzero by
279 * qzero(x) = s*(-1.25 + (R/S))
280 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
281 * S = 1 + qS0*s^2 + ... + qS5*s^12
282 * and
283 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
284 */
285 static const float qR8[6] = {
286 /* for x in [inf, 8]=1/[0,0.125] */
287 0.0000000000e+00, /* 0x00000000 */
288 7.3242187500e-02, /* 0x3d960000 */
289 1.1768206596e+01, /* 0x413c4a93 */
290 5.5767340088e+02, /* 0x440b6b19 */
291 8.8591972656e+03, /* 0x460a6cca */
292 3.7014625000e+04, /* 0x471096a0 */
293 };
294 static const float qS8[6] = {
295 1.6377603149e+02, /* 0x4323c6aa */
296 8.0983447266e+03, /* 0x45fd12c2 */
297 1.4253829688e+05, /* 0x480b3293 */
298 8.0330925000e+05, /* 0x49441ed4 */
299 8.4050156250e+05, /* 0x494d3359 */
300 -3.4389928125e+05, /* 0xc8a7eb69 */
301 };
302
303 static const float qR5[6] = {
304 /* for x in [8,4.5454]=1/[0.125,0.22001] */
305 1.8408595828e-11, /* 0x2da1ec79 */
306 7.3242180049e-02, /* 0x3d95ffff */
307 5.8356351852e+00, /* 0x40babd86 */
308 1.3511157227e+02, /* 0x43071c90 */
309 1.0272437744e+03, /* 0x448067cd */
310 1.9899779053e+03, /* 0x44f8bf4b */
311 };
312 static const float qS5[6] = {
313 8.2776611328e+01, /* 0x42a58da0 */
314 2.0778142090e+03, /* 0x4501dd07 */
315 1.8847289062e+04, /* 0x46933e94 */
316 5.6751113281e+04, /* 0x475daf1d */
317 3.5976753906e+04, /* 0x470c88c1 */
318 -5.3543427734e+03, /* 0xc5a752be */
319 };
320
321 static const float qR3[6] = {
322 /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
323 4.3774099900e-09, /* 0x3196681b */
324 7.3241114616e-02, /* 0x3d95ff70 */
325 3.3442313671e+00, /* 0x405607e3 */
326 4.2621845245e+01, /* 0x422a7cc5 */
327 1.7080809021e+02, /* 0x432acedf */
328 1.6673394775e+02, /* 0x4326bbe4 */
329 };
330 static const float qS3[6] = {
331 4.8758872986e+01, /* 0x42430916 */
332 7.0968920898e+02, /* 0x44316c1c */
333 3.7041481934e+03, /* 0x4567825f */
334 6.4604252930e+03, /* 0x45c9e367 */
335 2.5163337402e+03, /* 0x451d4557 */
336 -1.4924745178e+02, /* 0xc3153f59 */
337 };
338
339 static const float qR2[6] = {
340 /* for x in [2.8570,2]=1/[0.3499,0.5] */
341 1.5044444979e-07, /* 0x342189db */
342 7.3223426938e-02, /* 0x3d95f62a */
343 1.9981917143e+00, /* 0x3fffc4bf */
344 1.4495602608e+01, /* 0x4167edfd */
345 3.1666231155e+01, /* 0x41fd5471 */
346 1.6252708435e+01, /* 0x4182058c */
347 };
348 static const float qS2[6] = {
349 3.0365585327e+01, /* 0x41f2ecb8 */
350 2.6934811401e+02, /* 0x4386ac8f */
351 8.4478375244e+02, /* 0x44533229 */
352 8.8293585205e+02, /* 0x445cbbe5 */
353 2.1266638184e+02, /* 0x4354aa98 */
354 -5.3109550476e+00, /* 0xc0a9f358 */
355 };
356
357 static float
qzerof(float x)358 qzerof(float x)
359 {
360 const float *p, *q;
361 float s, r, z;
362 __int32_t ix;
363 GET_FLOAT_WORD(ix, x);
364 ix &= 0x7fffffff;
365 if (ix >= 0x41000000) {
366 p = qR8;
367 q = qS8;
368 } else if (ix >= 0x40f71c58) {
369 p = qR5;
370 q = qS5;
371 } else if (ix >= 0x4036db68) {
372 p = qR3;
373 q = qS3;
374 } else {
375 p = qR2;
376 q = qS2;
377 }
378 z = one / (x * x);
379 r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
380 s = one +
381 z * (q[0] +
382 z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))));
383 return (-(float).125 + r / s) / x;
384 }
385
386 _MATH_ALIAS_f_f(j0)
387
388 _MATH_ALIAS_f_f(y0)
389
390
