1 /* e_j1f.c -- float version of e_j1.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 <sys/cdefs.h>
17 __FBSDID("$FreeBSD: head/lib/msun/src/e_j1f.c 283032 2015-05-17 16:27:06Z kargl $");
18
19 /*
20 * See e_j1.c for complete comments.
21 */
22
23 #include "math.h"
24 #include "math_private.h"
25
26 static __inline float ponef(float), qonef(float);
27
28 static const volatile float vone = 1, vzero = 0;
29
30 static const float
31 huge = 1e30,
32 one = 1.0,
33 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
34 tpi = 6.3661974669e-01, /* 0x3f22f983 */
35 /* R0/S0 on [0,2] */
36 r00 = -6.2500000000e-02, /* 0xbd800000 */
37 r01 = 1.4070566976e-03, /* 0x3ab86cfd */
38 r02 = -1.5995563444e-05, /* 0xb7862e36 */
39 r03 = 4.9672799207e-08, /* 0x335557d2 */
40 s01 = 1.9153760746e-02, /* 0x3c9ce859 */
41 s02 = 1.8594678841e-04, /* 0x3942fab6 */
42 s03 = 1.1771846857e-06, /* 0x359dffc2 */
43 s04 = 5.0463624390e-09, /* 0x31ad6446 */
44 s05 = 1.2354227016e-11; /* 0x2d59567e */
45
46 static const float zero = 0.0;
47
48 float
__ieee754_j1f(float x)49 __ieee754_j1f(float x)
50 {
51 float z, s,c,ss,cc,r,u,v,y;
52 int32_t hx,ix;
53
54 GET_FLOAT_WORD(hx,x);
55 ix = hx&0x7fffffff;
56 if(ix>=0x7f800000) return one/x;
57 y = fabsf(x);
58 if(ix >= 0x40000000) { /* |x| >= 2.0 */
59 s = sinf(y);
60 c = cosf(y);
61 ss = -s-c;
62 cc = s-c;
63 if(ix<0x7f000000) { /* make sure y+y not overflow */
64 z = cosf(y+y);
65 if ((s*c)>zero) cc = z/ss;
66 else ss = z/cc;
67 }
68 /*
69 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
70 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
71 */
72 if(ix>0x58000000) z = (invsqrtpi*cc)/sqrtf(y); /* |x|>2**49 */
73 else {
74 u = ponef(y); v = qonef(y);
75 z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
76 }
77 if(hx<0) return -z;
78 else return z;
79 }
80 if(ix<0x39000000) { /* |x|<2**-13 */
81 if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
82 }
83 z = x*x;
84 r = z*(r00+z*(r01+z*(r02+z*r03)));
85 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
86 r *= x;
87 return(x*(float)0.5+r/s);
88 }
89
90 static const float U0[5] = {
91 -1.9605709612e-01, /* 0xbe48c331 */
92 5.0443872809e-02, /* 0x3d4e9e3c */
93 -1.9125689287e-03, /* 0xbafaaf2a */
94 2.3525259166e-05, /* 0x37c5581c */
95 -9.1909917899e-08, /* 0xb3c56003 */
96 };
97 static const float V0[5] = {
98 1.9916731864e-02, /* 0x3ca3286a */
99 2.0255257550e-04, /* 0x3954644b */
100 1.3560879779e-06, /* 0x35b602d4 */
101 6.2274145840e-09, /* 0x31d5f8eb */
102 1.6655924903e-11, /* 0x2d9281cf */
103 };
104
105 float
__ieee754_y1f(float x)106 __ieee754_y1f(float x)
107 {
108 float z, s,c,ss,cc,u,v;
109 int32_t hx,ix;
110
111 GET_FLOAT_WORD(hx,x);
112 ix = 0x7fffffff&hx;
113 if(ix>=0x7f800000) return vone/(x+x*x);
114 if(ix==0) return -one/vzero;
115 if(hx<0) return vzero/vzero;
116 if(ix >= 0x40000000) { /* |x| >= 2.0 */
117 s = sinf(x);
118 c = cosf(x);
119 ss = -s-c;
120 cc = s-c;
121 if(ix<0x7f000000) { /* make sure x+x not overflow */
122 z = cosf(x+x);
123 if ((s*c)>zero) cc = z/ss;
124 else ss = z/cc;
125 }
126 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
127 * where x0 = x-3pi/4
128 * Better formula:
129 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
130 * = 1/sqrt(2) * (sin(x) - cos(x))
131 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
132 * = -1/sqrt(2) * (cos(x) + sin(x))
133 * To avoid cancellation, use
134 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
135 * to compute the worse one.
136 */
137 if(ix>0x58000000) z = (invsqrtpi*ss)/sqrtf(x); /* |x|>2**49 */
138 else {
139 u = ponef(x); v = qonef(x);
140 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
141 }
142 return z;
143 }
144 if(ix<=0x33000000) { /* x < 2**-25 */
145 return(-tpi/x);
146 }
147 z = x*x;
148 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
149 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
150 return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
151 }
152
153 /* For x >= 8, the asymptotic expansions of pone is
154 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
155 * We approximate pone by
156 * pone(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 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
161 */
162
163 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
164 0.0000000000e+00, /* 0x00000000 */
165 1.1718750000e-01, /* 0x3df00000 */
166 1.3239480972e+01, /* 0x4153d4ea */
167 4.1205184937e+02, /* 0x43ce06a3 */
168 3.8747453613e+03, /* 0x45722bed */
169 7.9144794922e+03, /* 0x45f753d6 */
170 };
171 static const float ps8[5] = {
172 1.1420736694e+02, /* 0x42e46a2c */
173 3.6509309082e+03, /* 0x45642ee5 */
174 3.6956207031e+04, /* 0x47105c35 */
175 9.7602796875e+04, /* 0x47bea166 */
176 3.0804271484e+04, /* 0x46f0a88b */
177 };
178
179 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
180 1.3199052094e-11, /* 0x2d68333f */
181 1.1718749255e-01, /* 0x3defffff */
182 6.8027510643e+00, /* 0x40d9b023 */
183 1.0830818176e+02, /* 0x42d89dca */
184 5.1763616943e+02, /* 0x440168b7 */
185 5.2871520996e+02, /* 0x44042dc6 */
186 };
187 static const float ps5[5] = {
188 5.9280597687e+01, /* 0x426d1f55 */
189 9.9140142822e+02, /* 0x4477d9b1 */
190 5.3532670898e+03, /* 0x45a74a23 */
191 7.8446904297e+03, /* 0x45f52586 */
192 1.5040468750e+03, /* 0x44bc0180 */
193 };
194
195 static const float pr3[6] = {
196 3.0250391081e-09, /* 0x314fe10d */
197 1.1718686670e-01, /* 0x3defffab */
198 3.9329774380e+00, /* 0x407bb5e7 */
199 3.5119403839e+01, /* 0x420c7a45 */
200 9.1055007935e+01, /* 0x42b61c2a */
201 4.8559066772e+01, /* 0x42423c7c */
202 };
203 static const float ps3[5] = {
204 3.4791309357e+01, /* 0x420b2a4d */
205 3.3676245117e+02, /* 0x43a86198 */
206 1.0468714600e+03, /* 0x4482dbe3 */
207 8.9081134033e+02, /* 0x445eb3ed */
208 1.0378793335e+02, /* 0x42cf936c */
209 };
210
211 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
212 1.0771083225e-07, /* 0x33e74ea8 */
213 1.1717621982e-01, /* 0x3deffa16 */
214 2.3685150146e+00, /* 0x401795c0 */
215 1.2242610931e+01, /* 0x4143e1bc */
216 1.7693971634e+01, /* 0x418d8d41 */
217 5.0735230446e+00, /* 0x40a25a4d */
218 };
219 static const float ps2[5] = {
220 2.1436485291e+01, /* 0x41ab7dec */
221 1.2529022980e+02, /* 0x42fa9499 */
222 2.3227647400e+02, /* 0x436846c7 */
223 1.1767937469e+02, /* 0x42eb5bd7 */
224 8.3646392822e+00, /* 0x4105d590 */
225 };
226
227 static __inline float
ponef(float x)228 ponef(float x)
229 {
230 const float *p,*q;
231 float z,r,s;
232 int32_t ix;
233 GET_FLOAT_WORD(ix,x);
234 ix &= 0x7fffffff;
235 if(ix>=0x41000000) {p = pr8; q= ps8;}
236 else if(ix>=0x409173eb){p = pr5; q= ps5;}
237 else if(ix>=0x4036d917){p = pr3; q= ps3;}
238 else {p = pr2; q= ps2;} /* ix>=0x40000000 */
239 z = one/(x*x);
240 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
241 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
242 return one+ r/s;
243 }
244
245
246 /* For x >= 8, the asymptotic expansions of qone is
247 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
248 * We approximate pone by
249 * qone(x) = s*(0.375 + (R/S))
250 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
251 * S = 1 + qs1*s^2 + ... + qs6*s^12
252 * and
253 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
254 */
255
256 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
257 0.0000000000e+00, /* 0x00000000 */
258 -1.0253906250e-01, /* 0xbdd20000 */
259 -1.6271753311e+01, /* 0xc1822c8d */
260 -7.5960174561e+02, /* 0xc43de683 */
261 -1.1849806641e+04, /* 0xc639273a */
262 -4.8438511719e+04, /* 0xc73d3683 */
263 };
264 static const float qs8[6] = {
265 1.6139537048e+02, /* 0x43216537 */
266 7.8253862305e+03, /* 0x45f48b17 */
267 1.3387534375e+05, /* 0x4802bcd6 */
268 7.1965775000e+05, /* 0x492fb29c */
269 6.6660125000e+05, /* 0x4922be94 */
270 -2.9449025000e+05, /* 0xc88fcb48 */
271 };
272
273 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
274 -2.0897993405e-11, /* 0xadb7d219 */
275 -1.0253904760e-01, /* 0xbdd1fffe */
276 -8.0564479828e+00, /* 0xc100e736 */
277 -1.8366960144e+02, /* 0xc337ab6b */
278 -1.3731937256e+03, /* 0xc4aba633 */
279 -2.6124443359e+03, /* 0xc523471c */
280 };
281 static const float qs5[6] = {
282 8.1276550293e+01, /* 0x42a28d98 */
283 1.9917987061e+03, /* 0x44f8f98f */
284 1.7468484375e+04, /* 0x468878f8 */
285 4.9851425781e+04, /* 0x4742bb6d */
286 2.7948074219e+04, /* 0x46da5826 */
287 -4.7191835938e+03, /* 0xc5937978 */
288 };
289
290 static const float qr3[6] = {
291 -5.0783124372e-09, /* 0xb1ae7d4f */
292 -1.0253783315e-01, /* 0xbdd1ff5b */
293 -4.6101160049e+00, /* 0xc0938612 */
294 -5.7847221375e+01, /* 0xc267638e */
295 -2.2824453735e+02, /* 0xc3643e9a */
296 -2.1921012878e+02, /* 0xc35b35cb */
297 };
298 static const float qs3[6] = {
299 4.7665153503e+01, /* 0x423ea91e */
300 6.7386511230e+02, /* 0x4428775e */
301 3.3801528320e+03, /* 0x45534272 */
302 5.5477290039e+03, /* 0x45ad5dd5 */
303 1.9031191406e+03, /* 0x44ede3d0 */
304 -1.3520118713e+02, /* 0xc3073381 */
305 };
306
307 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
308 -1.7838172539e-07, /* 0xb43f8932 */
309 -1.0251704603e-01, /* 0xbdd1f475 */
310 -2.7522056103e+00, /* 0xc0302423 */
311 -1.9663616180e+01, /* 0xc19d4f16 */
312 -4.2325313568e+01, /* 0xc2294d1f */
313 -2.1371921539e+01, /* 0xc1aaf9b2 */
314 };
315 static const float qs2[6] = {
316 2.9533363342e+01, /* 0x41ec4454 */
317 2.5298155212e+02, /* 0x437cfb47 */
318 7.5750280762e+02, /* 0x443d602e */
319 7.3939318848e+02, /* 0x4438d92a */
320 1.5594900513e+02, /* 0x431bf2f2 */
321 -4.9594988823e+00, /* 0xc09eb437 */
322 };
323
324 static __inline float
qonef(float x)325 qonef(float x)
326 {
327 const float *p,*q;
328 float s,r,z;
329 int32_t ix;
330 GET_FLOAT_WORD(ix,x);
331 ix &= 0x7fffffff;
332 if(ix>=0x41000000) {p = qr8; q= qs8;}
333 else if(ix>=0x409173eb){p = qr5; q= qs5;}
334 else if(ix>=0x4036d917){p = qr3; q= qs3;}
335 else {p = qr2; q= qs2;} /* ix>=0x40000000 */
336 z = one/(x*x);
337 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
338 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
339 return ((float).375 + r/s)/x;
340 }
341