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 #define _GNU_SOURCE
17 #include "libm.h"
18
19 static float pzerof(float), qzerof(float);
20
21 static const float
22 invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */
23 tpi = 6.3661974669e-01; /* 0x3f22f983 */
24
25 static float common(uint32_t ix, float x, int y0)
26 {
27 float z,s,c,ss,cc;
28 /*
29 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
30 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
31 */
32 s = sinf(x);
33 c = cosf(x);
34 if (y0)
35 c = -c;
36 cc = s+c;
37 if (ix < 0x7f000000) {
38 ss = s-c;
39 z = -cosf(2*x);
40 if (s*c < 0)
41 cc = z/ss;
42 else
43 ss = z/cc;
44 if (ix < 0x58800000) {
45 if (y0)
46 ss = -ss;
47 cc = pzerof(x)*cc-qzerof(x)*ss;
48 }
49 }
50 return invsqrtpi*cc/sqrtf(x);
51 }
52
53 /* R0/S0 on [0, 2.00] */
54 static const float
55 R02 = 1.5625000000e-02, /* 0x3c800000 */
56 R03 = -1.8997929874e-04, /* 0xb947352e */
57 R04 = 1.8295404516e-06, /* 0x35f58e88 */
58 R05 = -4.6183270541e-09, /* 0xb19eaf3c */
59 S01 = 1.5619102865e-02, /* 0x3c7fe744 */
60 S02 = 1.1692678527e-04, /* 0x38f53697 */
61 S03 = 5.1354652442e-07, /* 0x3509daa6 */
62 S04 = 1.1661400734e-09; /* 0x30a045e8 */
63
64 float j0f(float x)
65 {
66 float z,r,s;
67 uint32_t ix;
68
69 GET_FLOAT_WORD(ix, x);
70 ix &= 0x7fffffff;
71 if (ix >= 0x7f800000)
72 return 1/(x*x);
73 x = fabsf(x);
74
75 if (ix >= 0x40000000) { /* |x| >= 2 */
76 /* large ulp error near zeros */
77 return common(ix, x, 0);
78 }
79 if (ix >= 0x3a000000) { /* |x| >= 2**-11 */
80 /* up to 4ulp error near 2 */
81 z = x*x;
82 r = z*(R02+z*(R03+z*(R04+z*R05)));
83 s = 1+z*(S01+z*(S02+z*(S03+z*S04)));
84 return (1+x/2)*(1-x/2) + z*(r/s);
85 }
86 if (ix >= 0x21800000) /* |x| >= 2**-60 */
87 x = 0.25f*x*x;
88 return 1 - x;
89 }
90
91 static const float
92 u00 = -7.3804296553e-02, /* 0xbd9726b5 */
93 u01 = 1.7666645348e-01, /* 0x3e34e80d */
94 u02 = -1.3818567619e-02, /* 0xbc626746 */
95 u03 = 3.4745343146e-04, /* 0x39b62a69 */
96 u04 = -3.8140706238e-06, /* 0xb67ff53c */
97 u05 = 1.9559013964e-08, /* 0x32a802ba */
98 u06 = -3.9820518410e-11, /* 0xae2f21eb */
99 v01 = 1.2730483897e-02, /* 0x3c509385 */
100 v02 = 7.6006865129e-05, /* 0x389f65e0 */
101 v03 = 2.5915085189e-07, /* 0x348b216c */
102 v04 = 4.4111031494e-10; /* 0x2ff280c2 */
103
104 float y0f(float x)
105 {
106 float z,u,v;
107 uint32_t ix;
108
109 GET_FLOAT_WORD(ix, x);
110 if ((ix & 0x7fffffff) == 0)
111 return -1/0.0f;
112 if (ix>>31)
113 return 0/0.0f;
114 if (ix >= 0x7f800000)
115 return 1/x;
116 if (ix >= 0x40000000) { /* |x| >= 2.0 */
117 /* large ulp error near zeros */
118 return common(ix,x,1);
119 }
120 if (ix >= 0x39000000) { /* x >= 2**-13 */
121 /* large ulp error at x ~= 0.89 */
122 z = x*x;
123 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
124 v = 1+z*(v01+z*(v02+z*(v03+z*v04)));
125 return u/v + tpi*(j0f(x)*logf(x));
126 }
127 return u00 + tpi*logf(x);
128 }
129
130 /* The asymptotic expansions of pzero is
131 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
132 * For x >= 2, We approximate pzero by
133 * pzero(x) = 1 + (R/S)
134 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
135 * S = 1 + pS0*s^2 + ... + pS4*s^10
136 * and
137 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
138 */
139 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
140 0.0000000000e+00, /* 0x00000000 */
141 -7.0312500000e-02, /* 0xbd900000 */
142 -8.0816707611e+00, /* 0xc1014e86 */
143 -2.5706311035e+02, /* 0xc3808814 */
144 -2.4852163086e+03, /* 0xc51b5376 */
145 -5.2530439453e+03, /* 0xc5a4285a */
146 };
147 static const float pS8[5] = {
148 1.1653436279e+02, /* 0x42e91198 */
149 3.8337448730e+03, /* 0x456f9beb */
150 4.0597855469e+04, /* 0x471e95db */
151 1.1675296875e+05, /* 0x47e4087c */
152 4.7627726562e+04, /* 0x473a0bba */
153 };
154 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
155 -1.1412546255e-11, /* 0xad48c58a */
156 -7.0312492549e-02, /* 0xbd8fffff */
157 -4.1596107483e+00, /* 0xc0851b88 */
158 -6.7674766541e+01, /* 0xc287597b */
159 -3.3123129272e+02, /* 0xc3a59d9b */
160 -3.4643338013e+02, /* 0xc3ad3779 */
161 };
162 static const float pS5[5] = {
163 6.0753936768e+01, /* 0x42730408 */
164 1.0512523193e+03, /* 0x44836813 */
165 5.9789707031e+03, /* 0x45bad7c4 */
166 9.6254453125e+03, /* 0x461665c8 */
167 2.4060581055e+03, /* 0x451660ee */
168 };
169
170 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
171 -2.5470459075e-09, /* 0xb12f081b */
172 -7.0311963558e-02, /* 0xbd8fffb8 */
173 -2.4090321064e+00, /* 0xc01a2d95 */
174 -2.1965976715e+01, /* 0xc1afba52 */
175 -5.8079170227e+01, /* 0xc2685112 */
176 -3.1447946548e+01, /* 0xc1fb9565 */
177 };
178 static const float pS3[5] = {
179 3.5856033325e+01, /* 0x420f6c94 */
180 3.6151397705e+02, /* 0x43b4c1ca */
181 1.1936077881e+03, /* 0x44953373 */
182 1.1279968262e+03, /* 0x448cffe6 */
183 1.7358093262e+02, /* 0x432d94b8 */
184 };
185
186 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
187 -8.8753431271e-08, /* 0xb3be98b7 */
188 -7.0303097367e-02, /* 0xbd8ffb12 */
189 -1.4507384300e+00, /* 0xbfb9b1cc */
190 -7.6356959343e+00, /* 0xc0f4579f */
191 -1.1193166733e+01, /* 0xc1331736 */
192 -3.2336456776e+00, /* 0xc04ef40d */
193 };
194 static const float pS2[5] = {
195 2.2220300674e+01, /* 0x41b1c32d */
196 1.3620678711e+02, /* 0x430834f0 */
197 2.7047027588e+02, /* 0x43873c32 */
198 1.5387539673e+02, /* 0x4319e01a */
199 1.4657617569e+01, /* 0x416a859a */
200 };
201
202 static float pzerof(float x)
203 {
204 const float *p,*q;
205 float_t z,r,s;
206 uint32_t ix;
207
208 GET_FLOAT_WORD(ix, x);
209 ix &= 0x7fffffff;
210 if (ix >= 0x41000000){p = pR8; q = pS8;}
211 else if (ix >= 0x409173eb){p = pR5; q = pS5;}
212 else if (ix >= 0x4036d917){p = pR3; q = pS3;}
213 else /*ix >= 0x40000000*/ {p = pR2; q = pS2;}
214 z = 1.0f/(x*x);
215 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
216 s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
217 return 1.0f + r/s;
218 }
219
220
221 /* For x >= 8, the asymptotic expansions of qzero is
222 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
223 * We approximate pzero by
224 * qzero(x) = s*(-1.25 + (R/S))
225 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
226 * S = 1 + qS0*s^2 + ... + qS5*s^12
227 * and
228 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
229 */
230 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
231 0.0000000000e+00, /* 0x00000000 */
232 7.3242187500e-02, /* 0x3d960000 */
233 1.1768206596e+01, /* 0x413c4a93 */
234 5.5767340088e+02, /* 0x440b6b19 */
235 8.8591972656e+03, /* 0x460a6cca */
236 3.7014625000e+04, /* 0x471096a0 */
237 };
238 static const float qS8[6] = {
239 1.6377603149e+02, /* 0x4323c6aa */
240 8.0983447266e+03, /* 0x45fd12c2 */
241 1.4253829688e+05, /* 0x480b3293 */
242 8.0330925000e+05, /* 0x49441ed4 */
243 8.4050156250e+05, /* 0x494d3359 */
244 -3.4389928125e+05, /* 0xc8a7eb69 */
245 };
246
247 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
248 1.8408595828e-11, /* 0x2da1ec79 */
249 7.3242180049e-02, /* 0x3d95ffff */
250 5.8356351852e+00, /* 0x40babd86 */
251 1.3511157227e+02, /* 0x43071c90 */
252 1.0272437744e+03, /* 0x448067cd */
253 1.9899779053e+03, /* 0x44f8bf4b */
254 };
255 static const float qS5[6] = {
256 8.2776611328e+01, /* 0x42a58da0 */
257 2.0778142090e+03, /* 0x4501dd07 */
258 1.8847289062e+04, /* 0x46933e94 */
259 5.6751113281e+04, /* 0x475daf1d */
260 3.5976753906e+04, /* 0x470c88c1 */
261 -5.3543427734e+03, /* 0xc5a752be */
262 };
263
264 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
265 4.3774099900e-09, /* 0x3196681b */
266 7.3241114616e-02, /* 0x3d95ff70 */
267 3.3442313671e+00, /* 0x405607e3 */
268 4.2621845245e+01, /* 0x422a7cc5 */
269 1.7080809021e+02, /* 0x432acedf */
270 1.6673394775e+02, /* 0x4326bbe4 */
271 };
272 static const float qS3[6] = {
273 4.8758872986e+01, /* 0x42430916 */
274 7.0968920898e+02, /* 0x44316c1c */
275 3.7041481934e+03, /* 0x4567825f */
276 6.4604252930e+03, /* 0x45c9e367 */
277 2.5163337402e+03, /* 0x451d4557 */
278 -1.4924745178e+02, /* 0xc3153f59 */
279 };
280
281 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
282 1.5044444979e-07, /* 0x342189db */
283 7.3223426938e-02, /* 0x3d95f62a */
284 1.9981917143e+00, /* 0x3fffc4bf */
285 1.4495602608e+01, /* 0x4167edfd */
286 3.1666231155e+01, /* 0x41fd5471 */
287 1.6252708435e+01, /* 0x4182058c */
288 };
289 static const float qS2[6] = {
290 3.0365585327e+01, /* 0x41f2ecb8 */
291 2.6934811401e+02, /* 0x4386ac8f */
292 8.4478375244e+02, /* 0x44533229 */
293 8.8293585205e+02, /* 0x445cbbe5 */
294 2.1266638184e+02, /* 0x4354aa98 */
295 -5.3109550476e+00, /* 0xc0a9f358 */
296 };
297
298 static float qzerof(float x)
299 {
300 const float *p,*q;
301 float_t s,r,z;
302 uint32_t ix;
303
304 GET_FLOAT_WORD(ix, x);
305 ix &= 0x7fffffff;
306 if (ix >= 0x41000000){p = qR8; q = qS8;}
307 else if (ix >= 0x409173eb){p = qR5; q = qS5;}
308 else if (ix >= 0x4036d917){p = qR3; q = qS3;}
309 else /*ix >= 0x40000000*/ {p = qR2; q = qS2;}
310 z = 1.0f/(x*x);
311 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
312 s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
313 return (-.125f + r/s)/x;
314 }
315