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 } else {
59 /* We subtract (exactly) a value x0 such that
60 cos(x0)+sin(x0) is very near to 0, and use the identity
61 sin(x-x0) = sin(x)*cos(x0)-cos(x)*sin(x0) to get
62 sin(x) + cos(x) with extra accuracy. */
63 float x0 = 0xe.d4108p+124f;
64 float y = x - x0; /* exact */
65 /* sin(y) = sin(x)*cos(x0)-cos(x)*sin(x0) */
66 z = __sinf (y);
67 float eps = 0x1.5f263ep-24f;
68 /* cos(x0) ~ -sin(x0) + eps */
69 z += eps * __cosf (x);
70 /* now z ~ (sin(x)-cos(x))*cos(x0) */
71 float cosx0 = -0xb.504f3p-4f;
72 cc = z / cosx0;
73 }
74 /*
75 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
76 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
77 */
78 if(ix>0x5c000000) z = (invsqrtpi*cc)/sqrtf(x);
79 else {
80 u = pzerof(x); v = qzerof(x);
81 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
82 }
83 return z;
84 }
85 if(ix<0x39000000) { /* |x| < 2**-13 */
86 math_force_eval(huge+x); /* raise inexact if x != 0 */
87 if(ix<0x32000000) return one; /* |x|<2**-27 */
88 else return one - (float)0.25*x*x;
89 }
90 z = x*x;
91 r = z*(R02+z*(R03+z*(R04+z*R05)));
92 s = one+z*(S01+z*(S02+z*(S03+z*S04)));
93 if(ix < 0x3F800000) { /* |x| < 1.00 */
94 return one + z*((float)-0.25+(r/s));
95 } else {
96 u = (float)0.5*x;
97 return((one+u)*(one-u)+z*(r/s));
98 }
99}
100libm_alias_finite (__ieee754_j0f, __j0f)
101
102static const float
103u00 = -7.3804296553e-02, /* 0xbd9726b5 */
104u01 = 1.7666645348e-01, /* 0x3e34e80d */
105u02 = -1.3818567619e-02, /* 0xbc626746 */
106u03 = 3.4745343146e-04, /* 0x39b62a69 */
107u04 = -3.8140706238e-06, /* 0xb67ff53c */
108u05 = 1.9559013964e-08, /* 0x32a802ba */
109u06 = -3.9820518410e-11, /* 0xae2f21eb */
110v01 = 1.2730483897e-02, /* 0x3c509385 */
111v02 = 7.6006865129e-05, /* 0x389f65e0 */
112v03 = 2.5915085189e-07, /* 0x348b216c */
113v04 = 4.4111031494e-10; /* 0x2ff280c2 */
114
115float
116__ieee754_y0f(float x)
117{
118 float z, s,c,ss,cc,u,v;
119 int32_t hx,ix;
120
121 GET_FLOAT_WORD(hx,x);
122 ix = 0x7fffffff&hx;
123 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf. */
124 if(ix>=0x7f800000) return one/(x+x*x);
125 if(ix==0) return -1/zero; /* -inf and divide by zero exception. */
126 if(hx<0) return zero/(zero*x);
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 __sincosf (x, &s, &c);
140 ss = s-c;
141 cc = s+c;
142 /*
143 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
144 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
145 */
146 if(ix<0x7f000000) { /* make sure x+x not overflow */
147 z = -__cosf(x+x);
148 if ((s*c)<zero) cc = z/ss;
149 else ss = z/cc;
150 }
151 if(ix>0x5c000000) z = (invsqrtpi*ss)/sqrtf(x);
152 else {
153 u = pzerof(x); v = qzerof(x);
154 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
155 }
156 return z;
157 }
158 if(ix<=0x39800000) { /* x < 2**-13 */
159 return(u00 + tpi*__ieee754_logf(x));
160 }
161 z = x*x;
162 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
163 v = one+z*(v01+z*(v02+z*(v03+z*v04)));
164 return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
165}
166libm_alias_finite (__ieee754_y0f, __y0f)
167
168/* The asymptotic expansions of pzero is
169 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
170 * For x >= 2, We approximate pzero by
171 * pzero(x) = 1 + (R/S)
172 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
173 * S = 1 + pS0*s^2 + ... + pS4*s^10
174 * and
175 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
176 */
177static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
178 0.0000000000e+00, /* 0x00000000 */
179 -7.0312500000e-02, /* 0xbd900000 */
180 -8.0816707611e+00, /* 0xc1014e86 */
181 -2.5706311035e+02, /* 0xc3808814 */
182 -2.4852163086e+03, /* 0xc51b5376 */
183 -5.2530439453e+03, /* 0xc5a4285a */
184};
185static const float pS8[5] = {
186 1.1653436279e+02, /* 0x42e91198 */
187 3.8337448730e+03, /* 0x456f9beb */
188 4.0597855469e+04, /* 0x471e95db */
189 1.1675296875e+05, /* 0x47e4087c */
190 4.7627726562e+04, /* 0x473a0bba */
191};
192static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
193 -1.1412546255e-11, /* 0xad48c58a */
194 -7.0312492549e-02, /* 0xbd8fffff */
195 -4.1596107483e+00, /* 0xc0851b88 */
196 -6.7674766541e+01, /* 0xc287597b */
197 -3.3123129272e+02, /* 0xc3a59d9b */
198 -3.4643338013e+02, /* 0xc3ad3779 */
199};
200static const float pS5[5] = {
201 6.0753936768e+01, /* 0x42730408 */
202 1.0512523193e+03, /* 0x44836813 */
203 5.9789707031e+03, /* 0x45bad7c4 */
204 9.6254453125e+03, /* 0x461665c8 */
205 2.4060581055e+03, /* 0x451660ee */
206};
207
208static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
209 -2.5470459075e-09, /* 0xb12f081b */
210 -7.0311963558e-02, /* 0xbd8fffb8 */
211 -2.4090321064e+00, /* 0xc01a2d95 */
212 -2.1965976715e+01, /* 0xc1afba52 */
213 -5.8079170227e+01, /* 0xc2685112 */
214 -3.1447946548e+01, /* 0xc1fb9565 */
215};
216static const float pS3[5] = {
217 3.5856033325e+01, /* 0x420f6c94 */
218 3.6151397705e+02, /* 0x43b4c1ca */
219 1.1936077881e+03, /* 0x44953373 */
220 1.1279968262e+03, /* 0x448cffe6 */
221 1.7358093262e+02, /* 0x432d94b8 */
222};
223
224static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
225 -8.8753431271e-08, /* 0xb3be98b7 */
226 -7.0303097367e-02, /* 0xbd8ffb12 */
227 -1.4507384300e+00, /* 0xbfb9b1cc */
228 -7.6356959343e+00, /* 0xc0f4579f */
229 -1.1193166733e+01, /* 0xc1331736 */
230 -3.2336456776e+00, /* 0xc04ef40d */
231};
232static const float pS2[5] = {
233 2.2220300674e+01, /* 0x41b1c32d */
234 1.3620678711e+02, /* 0x430834f0 */
235 2.7047027588e+02, /* 0x43873c32 */
236 1.5387539673e+02, /* 0x4319e01a */
237 1.4657617569e+01, /* 0x416a859a */
238};
239
240static float
241pzerof(float x)
242{
243 const float *p,*q;
244 float z,r,s;
245 int32_t ix;
246 GET_FLOAT_WORD(ix,x);
247 ix &= 0x7fffffff;
248 /* ix >= 0x40000000 for all calls to this function. */
249 if(ix>=0x41000000) {p = pR8; q= pS8;}
250 else if(ix>=0x40f71c58){p = pR5; q= pS5;}
251 else if(ix>=0x4036db68){p = pR3; q= pS3;}
252 else {p = pR2; q= pS2;}
253 z = one/(x*x);
254 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
255 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
256 return one+ r/s;
257}
258
259
260/* For x >= 8, the asymptotic expansions of qzero is
261 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
262 * We approximate pzero by
263 * qzero(x) = s*(-1.25 + (R/S))
264 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
265 * S = 1 + qS0*s^2 + ... + qS5*s^12
266 * and
267 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
268 */
269static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
270 0.0000000000e+00, /* 0x00000000 */
271 7.3242187500e-02, /* 0x3d960000 */
272 1.1768206596e+01, /* 0x413c4a93 */
273 5.5767340088e+02, /* 0x440b6b19 */
274 8.8591972656e+03, /* 0x460a6cca */
275 3.7014625000e+04, /* 0x471096a0 */
276};
277static const float qS8[6] = {
278 1.6377603149e+02, /* 0x4323c6aa */
279 8.0983447266e+03, /* 0x45fd12c2 */
280 1.4253829688e+05, /* 0x480b3293 */
281 8.0330925000e+05, /* 0x49441ed4 */
282 8.4050156250e+05, /* 0x494d3359 */
283 -3.4389928125e+05, /* 0xc8a7eb69 */
284};
285
286static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
287 1.8408595828e-11, /* 0x2da1ec79 */
288 7.3242180049e-02, /* 0x3d95ffff */
289 5.8356351852e+00, /* 0x40babd86 */
290 1.3511157227e+02, /* 0x43071c90 */
291 1.0272437744e+03, /* 0x448067cd */
292 1.9899779053e+03, /* 0x44f8bf4b */
293};
294static const float qS5[6] = {
295 8.2776611328e+01, /* 0x42a58da0 */
296 2.0778142090e+03, /* 0x4501dd07 */
297 1.8847289062e+04, /* 0x46933e94 */
298 5.6751113281e+04, /* 0x475daf1d */
299 3.5976753906e+04, /* 0x470c88c1 */
300 -5.3543427734e+03, /* 0xc5a752be */
301};
302
303static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
304 4.3774099900e-09, /* 0x3196681b */
305 7.3241114616e-02, /* 0x3d95ff70 */
306 3.3442313671e+00, /* 0x405607e3 */
307 4.2621845245e+01, /* 0x422a7cc5 */
308 1.7080809021e+02, /* 0x432acedf */
309 1.6673394775e+02, /* 0x4326bbe4 */
310};
311static const float qS3[6] = {
312 4.8758872986e+01, /* 0x42430916 */
313 7.0968920898e+02, /* 0x44316c1c */
314 3.7041481934e+03, /* 0x4567825f */
315 6.4604252930e+03, /* 0x45c9e367 */
316 2.5163337402e+03, /* 0x451d4557 */
317 -1.4924745178e+02, /* 0xc3153f59 */
318};
319
320static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
321 1.5044444979e-07, /* 0x342189db */
322 7.3223426938e-02, /* 0x3d95f62a */
323 1.9981917143e+00, /* 0x3fffc4bf */
324 1.4495602608e+01, /* 0x4167edfd */
325 3.1666231155e+01, /* 0x41fd5471 */
326 1.6252708435e+01, /* 0x4182058c */
327};
328static const float qS2[6] = {
329 3.0365585327e+01, /* 0x41f2ecb8 */
330 2.6934811401e+02, /* 0x4386ac8f */
331 8.4478375244e+02, /* 0x44533229 */
332 8.8293585205e+02, /* 0x445cbbe5 */
333 2.1266638184e+02, /* 0x4354aa98 */
334 -5.3109550476e+00, /* 0xc0a9f358 */
335};
336
337static float
338qzerof(float x)
339{
340 const float *p,*q;
341 float s,r,z;
342 int32_t ix;
343 GET_FLOAT_WORD(ix,x);
344 ix &= 0x7fffffff;
345 /* ix >= 0x40000000 for all calls to this function. */
346 if(ix>=0x41000000) {p = qR8; q= qS8;}
347 else if(ix>=0x40f71c58){p = qR5; q= qS5;}
348 else if(ix>=0x4036db68){p = qR3; q= qS3;}
349 else {p = qR2; q= qS2;}
350 z = one/(x*x);
351 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
352 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
353 return (-(float).125 + r/s)/x;
354}
355