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#include <libm-alias-finite.h>
20
21static float pzerof(float), qzerof(float);
22
23static const float
24huge = 1e30,
25one = 1.0,
26invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
27tpi = 6.3661974669e-01, /* 0x3f22f983 */
28 /* R0/S0 on [0, 2.00] */
29R02 = 1.5625000000e-02, /* 0x3c800000 */
30R03 = -1.8997929874e-04, /* 0xb947352e */
31R04 = 1.8295404516e-06, /* 0x35f58e88 */
32R05 = -4.6183270541e-09, /* 0xb19eaf3c */
33S01 = 1.5619102865e-02, /* 0x3c7fe744 */
34S02 = 1.1692678527e-04, /* 0x38f53697 */
35S03 = 5.1354652442e-07, /* 0x3509daa6 */
36S04 = 1.1661400734e-09; /* 0x30a045e8 */
37
38static const float zero = 0.0;
39
40float
41__ieee754_j0f(float x)
42{
43 float z, s,c,ss,cc,r,u,v;
44 int32_t hx,ix;
45
46 GET_FLOAT_WORD(hx,x);
47 ix = hx&0x7fffffff;
48 if(ix>=0x7f800000) return one/(x*x);
49 x = fabsf(x);
50 if(ix >= 0x40000000) { /* |x| >= 2.0 */
51 __sincosf (x, &s, &c);
52 ss = s-c;
53 cc = s+c;
54 if(ix<0x7f000000) { /* make sure x+x not overflow */
55 z = -__cosf(x+x);
56 if ((s*c)<zero) cc = z/ss;
57 else ss = z/cc;
58 }
59 /*
60 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
61 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
62 */
63 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrtf(x);
64 else {
65 u = pzerof(x); v = qzerof(x);
66 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
67 }
68 return z;
69 }
70 if(ix<0x39000000) { /* |x| < 2**-13 */
71 math_force_eval(huge+x); /* raise inexact if x != 0 */
72 if(ix<0x32000000) return one; /* |x|<2**-27 */
73 else return one - (float)0.25*x*x;
74 }
75 z = x*x;
76 r = z*(R02+z*(R03+z*(R04+z*R05)));
77 s = one+z*(S01+z*(S02+z*(S03+z*S04)));
78 if(ix < 0x3F800000) { /* |x| < 1.00 */
79 return one + z*((float)-0.25+(r/s));
80 } else {
81 u = (float)0.5*x;
82 return((one+u)*(one-u)+z*(r/s));
83 }
84}
85libm_alias_finite (__ieee754_j0f, __j0f)
86
87static const float
88u00 = -7.3804296553e-02, /* 0xbd9726b5 */
89u01 = 1.7666645348e-01, /* 0x3e34e80d */
90u02 = -1.3818567619e-02, /* 0xbc626746 */
91u03 = 3.4745343146e-04, /* 0x39b62a69 */
92u04 = -3.8140706238e-06, /* 0xb67ff53c */
93u05 = 1.9559013964e-08, /* 0x32a802ba */
94u06 = -3.9820518410e-11, /* 0xae2f21eb */
95v01 = 1.2730483897e-02, /* 0x3c509385 */
96v02 = 7.6006865129e-05, /* 0x389f65e0 */
97v03 = 2.5915085189e-07, /* 0x348b216c */
98v04 = 4.4111031494e-10; /* 0x2ff280c2 */
99
100float
101__ieee754_y0f(float x)
102{
103 float z, s,c,ss,cc,u,v;
104 int32_t hx,ix;
105
106 GET_FLOAT_WORD(hx,x);
107 ix = 0x7fffffff&hx;
108 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf. */
109 if(ix>=0x7f800000) return one/(x+x*x);
110 if(ix==0) return -1/zero; /* -inf and divide by zero exception. */
111 if(hx<0) return zero/(zero*x);
112 if(ix >= 0x40000000) { /* |x| >= 2.0 */
113 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
114 * where x0 = x-pi/4
115 * Better formula:
116 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
117 * = 1/sqrt(2) * (sin(x) + cos(x))
118 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
119 * = 1/sqrt(2) * (sin(x) - cos(x))
120 * To avoid cancellation, use
121 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
122 * to compute the worse one.
123 */
124 __sincosf (x, &s, &c);
125 ss = s-c;
126 cc = s+c;
127 /*
128 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
129 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
130 */
131 if(ix<0x7f000000) { /* make sure x+x not overflow */
132 z = -__cosf(x+x);
133 if ((s*c)<zero) cc = z/ss;
134 else ss = z/cc;
135 }
136 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x);
137 else {
138 u = pzerof(x); v = qzerof(x);
139 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
140 }
141 return z;
142 }
143 if(ix<=0x39800000) { /* x < 2**-13 */
144 return(u00 + tpi*__ieee754_logf(x));
145 }
146 z = x*x;
147 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
148 v = one+z*(v01+z*(v02+z*(v03+z*v04)));
149 return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
150}
151libm_alias_finite (__ieee754_y0f, __y0f)
152
153/* The asymptotic expansions of pzero is
154 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
155 * For x >= 2, We approximate pzero by
156 * pzero(x) = 1 + (R/S)
157 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
158 * S = 1 + pS0*s^2 + ... + pS4*s^10
159 * and
160 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
161 */
162static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
163 0.0000000000e+00, /* 0x00000000 */
164 -7.0312500000e-02, /* 0xbd900000 */
165 -8.0816707611e+00, /* 0xc1014e86 */
166 -2.5706311035e+02, /* 0xc3808814 */
167 -2.4852163086e+03, /* 0xc51b5376 */
168 -5.2530439453e+03, /* 0xc5a4285a */
169};
170static const float pS8[5] = {
171 1.1653436279e+02, /* 0x42e91198 */
172 3.8337448730e+03, /* 0x456f9beb */
173 4.0597855469e+04, /* 0x471e95db */
174 1.1675296875e+05, /* 0x47e4087c */
175 4.7627726562e+04, /* 0x473a0bba */
176};
177static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
178 -1.1412546255e-11, /* 0xad48c58a */
179 -7.0312492549e-02, /* 0xbd8fffff */
180 -4.1596107483e+00, /* 0xc0851b88 */
181 -6.7674766541e+01, /* 0xc287597b */
182 -3.3123129272e+02, /* 0xc3a59d9b */
183 -3.4643338013e+02, /* 0xc3ad3779 */
184};
185static const float pS5[5] = {
186 6.0753936768e+01, /* 0x42730408 */
187 1.0512523193e+03, /* 0x44836813 */
188 5.9789707031e+03, /* 0x45bad7c4 */
189 9.6254453125e+03, /* 0x461665c8 */
190 2.4060581055e+03, /* 0x451660ee */
191};
192
193static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
194 -2.5470459075e-09, /* 0xb12f081b */
195 -7.0311963558e-02, /* 0xbd8fffb8 */
196 -2.4090321064e+00, /* 0xc01a2d95 */
197 -2.1965976715e+01, /* 0xc1afba52 */
198 -5.8079170227e+01, /* 0xc2685112 */
199 -3.1447946548e+01, /* 0xc1fb9565 */
200};
201static const float pS3[5] = {
202 3.5856033325e+01, /* 0x420f6c94 */
203 3.6151397705e+02, /* 0x43b4c1ca */
204 1.1936077881e+03, /* 0x44953373 */
205 1.1279968262e+03, /* 0x448cffe6 */
206 1.7358093262e+02, /* 0x432d94b8 */
207};
208
209static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
210 -8.8753431271e-08, /* 0xb3be98b7 */
211 -7.0303097367e-02, /* 0xbd8ffb12 */
212 -1.4507384300e+00, /* 0xbfb9b1cc */
213 -7.6356959343e+00, /* 0xc0f4579f */
214 -1.1193166733e+01, /* 0xc1331736 */
215 -3.2336456776e+00, /* 0xc04ef40d */
216};
217static const float pS2[5] = {
218 2.2220300674e+01, /* 0x41b1c32d */
219 1.3620678711e+02, /* 0x430834f0 */
220 2.7047027588e+02, /* 0x43873c32 */
221 1.5387539673e+02, /* 0x4319e01a */
222 1.4657617569e+01, /* 0x416a859a */
223};
224
225static float
226pzerof(float x)
227{
228 const float *p,*q;
229 float z,r,s;
230 int32_t ix;
231 GET_FLOAT_WORD(ix,x);
232 ix &= 0x7fffffff;
233 /* ix >= 0x40000000 for all calls to this function. */
234 if(ix>=0x41000000) {p = pR8; q= pS8;}
235 else if(ix>=0x40f71c58){p = pR5; q= pS5;}
236 else if(ix>=0x4036db68){p = pR3; q= pS3;}
237 else {p = pR2; q= pS2;}
238 z = one/(x*x);
239 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
240 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
241 return one+ r/s;
242}
243
244
245/* For x >= 8, the asymptotic expansions of qzero is
246 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
247 * We approximate pzero by
248 * qzero(x) = s*(-1.25 + (R/S))
249 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
250 * S = 1 + qS0*s^2 + ... + qS5*s^12
251 * and
252 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
253 */
254static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
255 0.0000000000e+00, /* 0x00000000 */
256 7.3242187500e-02, /* 0x3d960000 */
257 1.1768206596e+01, /* 0x413c4a93 */
258 5.5767340088e+02, /* 0x440b6b19 */
259 8.8591972656e+03, /* 0x460a6cca */
260 3.7014625000e+04, /* 0x471096a0 */
261};
262static const float qS8[6] = {
263 1.6377603149e+02, /* 0x4323c6aa */
264 8.0983447266e+03, /* 0x45fd12c2 */
265 1.4253829688e+05, /* 0x480b3293 */
266 8.0330925000e+05, /* 0x49441ed4 */
267 8.4050156250e+05, /* 0x494d3359 */
268 -3.4389928125e+05, /* 0xc8a7eb69 */
269};
270
271static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
272 1.8408595828e-11, /* 0x2da1ec79 */
273 7.3242180049e-02, /* 0x3d95ffff */
274 5.8356351852e+00, /* 0x40babd86 */
275 1.3511157227e+02, /* 0x43071c90 */
276 1.0272437744e+03, /* 0x448067cd */
277 1.9899779053e+03, /* 0x44f8bf4b */
278};
279static const float qS5[6] = {
280 8.2776611328e+01, /* 0x42a58da0 */
281 2.0778142090e+03, /* 0x4501dd07 */
282 1.8847289062e+04, /* 0x46933e94 */
283 5.6751113281e+04, /* 0x475daf1d */
284 3.5976753906e+04, /* 0x470c88c1 */
285 -5.3543427734e+03, /* 0xc5a752be */
286};
287
288static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
289 4.3774099900e-09, /* 0x3196681b */
290 7.3241114616e-02, /* 0x3d95ff70 */
291 3.3442313671e+00, /* 0x405607e3 */
292 4.2621845245e+01, /* 0x422a7cc5 */
293 1.7080809021e+02, /* 0x432acedf */
294 1.6673394775e+02, /* 0x4326bbe4 */
295};
296static const float qS3[6] = {
297 4.8758872986e+01, /* 0x42430916 */
298 7.0968920898e+02, /* 0x44316c1c */
299 3.7041481934e+03, /* 0x4567825f */
300 6.4604252930e+03, /* 0x45c9e367 */
301 2.5163337402e+03, /* 0x451d4557 */
302 -1.4924745178e+02, /* 0xc3153f59 */
303};
304
305static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
306 1.5044444979e-07, /* 0x342189db */
307 7.3223426938e-02, /* 0x3d95f62a */
308 1.9981917143e+00, /* 0x3fffc4bf */
309 1.4495602608e+01, /* 0x4167edfd */
310 3.1666231155e+01, /* 0x41fd5471 */
311 1.6252708435e+01, /* 0x4182058c */
312};
313static const float qS2[6] = {
314 3.0365585327e+01, /* 0x41f2ecb8 */
315 2.6934811401e+02, /* 0x4386ac8f */
316 8.4478375244e+02, /* 0x44533229 */
317 8.8293585205e+02, /* 0x445cbbe5 */
318 2.1266638184e+02, /* 0x4354aa98 */
319 -5.3109550476e+00, /* 0xc0a9f358 */
320};
321
322static float
323qzerof(float x)
324{
325 const float *p,*q;
326 float s,r,z;
327 int32_t ix;
328 GET_FLOAT_WORD(ix,x);
329 ix &= 0x7fffffff;
330 /* ix >= 0x40000000 for all calls to this function. */
331 if(ix>=0x41000000) {p = qR8; q= qS8;}
332 else if(ix>=0x40f71c58){p = qR5; q= qS5;}
333 else if(ix>=0x4036db68){p = qR3; q= qS3;}
334 else {p = qR2; q= qS2;}
335 z = one/(x*x);
336 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
337 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
338 return (-(float).125 + r/s)/x;
339}
340