1/* e_j0f.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 <math.h>
17#include <math-barriers.h>
18#include <math_private.h>
19
20static float pzerof(float), qzerof(float);
21
22static const float
23huge = 1e30,
24one = 1.0,
25invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
26tpi = 6.3661974669e-01, /* 0x3f22f983 */
27 /* R0/S0 on [0, 2.00] */
28R02 = 1.5625000000e-02, /* 0x3c800000 */
29R03 = -1.8997929874e-04, /* 0xb947352e */
30R04 = 1.8295404516e-06, /* 0x35f58e88 */
31R05 = -4.6183270541e-09, /* 0xb19eaf3c */
32S01 = 1.5619102865e-02, /* 0x3c7fe744 */
33S02 = 1.1692678527e-04, /* 0x38f53697 */
34S03 = 5.1354652442e-07, /* 0x3509daa6 */
35S04 = 1.1661400734e-09; /* 0x30a045e8 */
36
37static const float zero = 0.0;
38
39float
40__ieee754_j0f(float x)
41{
42 float z, s,c,ss,cc,r,u,v;
43 int32_t hx,ix;
44
45 GET_FLOAT_WORD(hx,x);
46 ix = hx&0x7fffffff;
47 if(ix>=0x7f800000) return one/(x*x);
48 x = fabsf(x);
49 if(ix >= 0x40000000) { /* |x| >= 2.0 */
50 __sincosf (x, &s, &c);
51 ss = s-c;
52 cc = s+c;
53 if(ix<0x7f000000) { /* make sure x+x not overflow */
54 z = -__cosf(x+x);
55 if ((s*c)<zero) cc = z/ss;
56 else ss = z/cc;
57 }
58 /*
59 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
60 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
61 */
62 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrtf(x);
63 else {
64 u = pzerof(x); v = qzerof(x);
65 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
66 }
67 return z;
68 }
69 if(ix<0x39000000) { /* |x| < 2**-13 */
70 math_force_eval(huge+x); /* raise inexact if x != 0 */
71 if(ix<0x32000000) return one; /* |x|<2**-27 */
72 else return one - (float)0.25*x*x;
73 }
74 z = x*x;
75 r = z*(R02+z*(R03+z*(R04+z*R05)));
76 s = one+z*(S01+z*(S02+z*(S03+z*S04)));
77 if(ix < 0x3F800000) { /* |x| < 1.00 */
78 return one + z*((float)-0.25+(r/s));
79 } else {
80 u = (float)0.5*x;
81 return((one+u)*(one-u)+z*(r/s));
82 }
83}
84strong_alias (__ieee754_j0f, __j0f_finite)
85
86static const float
87u00 = -7.3804296553e-02, /* 0xbd9726b5 */
88u01 = 1.7666645348e-01, /* 0x3e34e80d */
89u02 = -1.3818567619e-02, /* 0xbc626746 */
90u03 = 3.4745343146e-04, /* 0x39b62a69 */
91u04 = -3.8140706238e-06, /* 0xb67ff53c */
92u05 = 1.9559013964e-08, /* 0x32a802ba */
93u06 = -3.9820518410e-11, /* 0xae2f21eb */
94v01 = 1.2730483897e-02, /* 0x3c509385 */
95v02 = 7.6006865129e-05, /* 0x389f65e0 */
96v03 = 2.5915085189e-07, /* 0x348b216c */
97v04 = 4.4111031494e-10; /* 0x2ff280c2 */
98
99float
100__ieee754_y0f(float x)
101{
102 float z, s,c,ss,cc,u,v;
103 int32_t hx,ix;
104
105 GET_FLOAT_WORD(hx,x);
106 ix = 0x7fffffff&hx;
107 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf. */
108 if(ix>=0x7f800000) return one/(x+x*x);
109 if(ix==0) return -1/zero; /* -inf and divide by zero exception. */
110 if(hx<0) return zero/(zero*x);
111 if(ix >= 0x40000000) { /* |x| >= 2.0 */
112 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
113 * where x0 = x-pi/4
114 * Better formula:
115 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
116 * = 1/sqrt(2) * (sin(x) + cos(x))
117 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
118 * = 1/sqrt(2) * (sin(x) - cos(x))
119 * To avoid cancellation, use
120 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
121 * to compute the worse one.
122 */
123 __sincosf (x, &s, &c);
124 ss = s-c;
125 cc = s+c;
126 /*
127 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
128 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
129 */
130 if(ix<0x7f000000) { /* make sure x+x not overflow */
131 z = -__cosf(x+x);
132 if ((s*c)<zero) cc = z/ss;
133 else ss = z/cc;
134 }
135 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x);
136 else {
137 u = pzerof(x); v = qzerof(x);
138 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
139 }
140 return z;
141 }
142 if(ix<=0x39800000) { /* x < 2**-13 */
143 return(u00 + tpi*__ieee754_logf(x));
144 }
145 z = x*x;
146 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
147 v = one+z*(v01+z*(v02+z*(v03+z*v04)));
148 return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
149}
150strong_alias (__ieee754_y0f, __y0f_finite)
151
152/* The asymptotic expansions of pzero is
153 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
154 * For x >= 2, We approximate pzero by
155 * pzero(x) = 1 + (R/S)
156 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
157 * S = 1 + pS0*s^2 + ... + pS4*s^10
158 * and
159 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
160 */
161static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
162 0.0000000000e+00, /* 0x00000000 */
163 -7.0312500000e-02, /* 0xbd900000 */
164 -8.0816707611e+00, /* 0xc1014e86 */
165 -2.5706311035e+02, /* 0xc3808814 */
166 -2.4852163086e+03, /* 0xc51b5376 */
167 -5.2530439453e+03, /* 0xc5a4285a */
168};
169static const float pS8[5] = {
170 1.1653436279e+02, /* 0x42e91198 */
171 3.8337448730e+03, /* 0x456f9beb */
172 4.0597855469e+04, /* 0x471e95db */
173 1.1675296875e+05, /* 0x47e4087c */
174 4.7627726562e+04, /* 0x473a0bba */
175};
176static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
177 -1.1412546255e-11, /* 0xad48c58a */
178 -7.0312492549e-02, /* 0xbd8fffff */
179 -4.1596107483e+00, /* 0xc0851b88 */
180 -6.7674766541e+01, /* 0xc287597b */
181 -3.3123129272e+02, /* 0xc3a59d9b */
182 -3.4643338013e+02, /* 0xc3ad3779 */
183};
184static const float pS5[5] = {
185 6.0753936768e+01, /* 0x42730408 */
186 1.0512523193e+03, /* 0x44836813 */
187 5.9789707031e+03, /* 0x45bad7c4 */
188 9.6254453125e+03, /* 0x461665c8 */
189 2.4060581055e+03, /* 0x451660ee */
190};
191
192static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
193 -2.5470459075e-09, /* 0xb12f081b */
194 -7.0311963558e-02, /* 0xbd8fffb8 */
195 -2.4090321064e+00, /* 0xc01a2d95 */
196 -2.1965976715e+01, /* 0xc1afba52 */
197 -5.8079170227e+01, /* 0xc2685112 */
198 -3.1447946548e+01, /* 0xc1fb9565 */
199};
200static const float pS3[5] = {
201 3.5856033325e+01, /* 0x420f6c94 */
202 3.6151397705e+02, /* 0x43b4c1ca */
203 1.1936077881e+03, /* 0x44953373 */
204 1.1279968262e+03, /* 0x448cffe6 */
205 1.7358093262e+02, /* 0x432d94b8 */
206};
207
208static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
209 -8.8753431271e-08, /* 0xb3be98b7 */
210 -7.0303097367e-02, /* 0xbd8ffb12 */
211 -1.4507384300e+00, /* 0xbfb9b1cc */
212 -7.6356959343e+00, /* 0xc0f4579f */
213 -1.1193166733e+01, /* 0xc1331736 */
214 -3.2336456776e+00, /* 0xc04ef40d */
215};
216static const float pS2[5] = {
217 2.2220300674e+01, /* 0x41b1c32d */
218 1.3620678711e+02, /* 0x430834f0 */
219 2.7047027588e+02, /* 0x43873c32 */
220 1.5387539673e+02, /* 0x4319e01a */
221 1.4657617569e+01, /* 0x416a859a */
222};
223
224static float
225pzerof(float x)
226{
227 const float *p,*q;
228 float z,r,s;
229 int32_t ix;
230 GET_FLOAT_WORD(ix,x);
231 ix &= 0x7fffffff;
232 /* ix >= 0x40000000 for all calls to this function. */
233 if(ix>=0x41000000) {p = pR8; q= pS8;}
234 else if(ix>=0x40f71c58){p = pR5; q= pS5;}
235 else if(ix>=0x4036db68){p = pR3; q= pS3;}
236 else {p = pR2; q= pS2;}
237 z = one/(x*x);
238 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
239 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
240 return one+ r/s;
241}
242
243
244/* For x >= 8, the asymptotic expansions of qzero is
245 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
246 * We approximate pzero by
247 * qzero(x) = s*(-1.25 + (R/S))
248 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
249 * S = 1 + qS0*s^2 + ... + qS5*s^12
250 * and
251 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
252 */
253static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
254 0.0000000000e+00, /* 0x00000000 */
255 7.3242187500e-02, /* 0x3d960000 */
256 1.1768206596e+01, /* 0x413c4a93 */
257 5.5767340088e+02, /* 0x440b6b19 */
258 8.8591972656e+03, /* 0x460a6cca */
259 3.7014625000e+04, /* 0x471096a0 */
260};
261static const float qS8[6] = {
262 1.6377603149e+02, /* 0x4323c6aa */
263 8.0983447266e+03, /* 0x45fd12c2 */
264 1.4253829688e+05, /* 0x480b3293 */
265 8.0330925000e+05, /* 0x49441ed4 */
266 8.4050156250e+05, /* 0x494d3359 */
267 -3.4389928125e+05, /* 0xc8a7eb69 */
268};
269
270static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
271 1.8408595828e-11, /* 0x2da1ec79 */
272 7.3242180049e-02, /* 0x3d95ffff */
273 5.8356351852e+00, /* 0x40babd86 */
274 1.3511157227e+02, /* 0x43071c90 */
275 1.0272437744e+03, /* 0x448067cd */
276 1.9899779053e+03, /* 0x44f8bf4b */
277};
278static const float qS5[6] = {
279 8.2776611328e+01, /* 0x42a58da0 */
280 2.0778142090e+03, /* 0x4501dd07 */
281 1.8847289062e+04, /* 0x46933e94 */
282 5.6751113281e+04, /* 0x475daf1d */
283 3.5976753906e+04, /* 0x470c88c1 */
284 -5.3543427734e+03, /* 0xc5a752be */
285};
286
287static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
288 4.3774099900e-09, /* 0x3196681b */
289 7.3241114616e-02, /* 0x3d95ff70 */
290 3.3442313671e+00, /* 0x405607e3 */
291 4.2621845245e+01, /* 0x422a7cc5 */
292 1.7080809021e+02, /* 0x432acedf */
293 1.6673394775e+02, /* 0x4326bbe4 */
294};
295static const float qS3[6] = {
296 4.8758872986e+01, /* 0x42430916 */
297 7.0968920898e+02, /* 0x44316c1c */
298 3.7041481934e+03, /* 0x4567825f */
299 6.4604252930e+03, /* 0x45c9e367 */
300 2.5163337402e+03, /* 0x451d4557 */
301 -1.4924745178e+02, /* 0xc3153f59 */
302};
303
304static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
305 1.5044444979e-07, /* 0x342189db */
306 7.3223426938e-02, /* 0x3d95f62a */
307 1.9981917143e+00, /* 0x3fffc4bf */
308 1.4495602608e+01, /* 0x4167edfd */
309 3.1666231155e+01, /* 0x41fd5471 */
310 1.6252708435e+01, /* 0x4182058c */
311};
312static const float qS2[6] = {
313 3.0365585327e+01, /* 0x41f2ecb8 */
314 2.6934811401e+02, /* 0x4386ac8f */
315 8.4478375244e+02, /* 0x44533229 */
316 8.8293585205e+02, /* 0x445cbbe5 */
317 2.1266638184e+02, /* 0x4354aa98 */
318 -5.3109550476e+00, /* 0xc0a9f358 */
319};
320
321static float
322qzerof(float x)
323{
324 const float *p,*q;
325 float s,r,z;
326 int32_t ix;
327 GET_FLOAT_WORD(ix,x);
328 ix &= 0x7fffffff;
329 /* ix >= 0x40000000 for all calls to this function. */
330 if(ix>=0x41000000) {p = qR8; q= qS8;}
331 else if(ix>=0x40f71c58){p = qR5; q= qS5;}
332 else if(ix>=0x4036db68){p = qR3; q= qS3;}
333 else {p = qR2; q= qS2;}
334 z = one/(x*x);
335 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
336 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
337 return (-(float).125 + r/s)/x;
338}
339