1 /* e_j1f.c -- float version of e_j1.c.
2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
3 * Bug in __ieee754_j1f fixed by Scott Turner 1/16/2010
4 */
5
6 /*
7 * ====================================================
8 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
9 *
10 * Developed at SunPro, a Sun Microsystems, Inc. business.
11 * Permission to use, copy, modify, and distribute this
12 * software is freely granted, provided that this notice
13 * is preserved.
14 * ====================================================
15 */
16
17 #ifndef lint
18 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_j1f.c,v 1.7 2002/05/28 18:15:04 alfred Exp $";
19 #endif
20
21 #include "math.h"
22 #include "math_private.h"
23
24 static float ponef(float), qonef(float);
25
26 static const float
27 huge = 1e30,
28 one = 1.0,
29 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
30 tpi = 6.3661974669e-01, /* 0x3f22f983 */
31 /* R0/S0 on [0,2] */
32 r00 = -6.2500000000e-02, /* 0xbd800000 */
33 r01 = 1.4070566976e-03, /* 0x3ab86cfd */
34 r02 = -1.5995563444e-05, /* 0xb7862e36 */
35 r03 = 4.9672799207e-08, /* 0x335557d2 */
36 s01 = 1.9153760746e-02, /* 0x3c9ce859 */
37 s02 = 1.8594678841e-04, /* 0x3942fab6 */
38 s03 = 1.1771846857e-06, /* 0x359dffc2 */
39 s04 = 5.0463624390e-09, /* 0x31ad6446 */
40 s05 = 1.2354227016e-11; /* 0x2d59567e */
41
42 static const float zero = 0.0;
43
44 float
__ieee754_j1f(float x)45 __ieee754_j1f(float x)
46 {
47 float z, s,c,ss,cc,r,u,v,y;
48 int32_t hx,ix;
49
50 GET_FLOAT_WORD(hx,x);
51 ix = hx&0x7fffffff;
52 if(ix>=0x7f800000) return one/x;
53 y = fabsf(x);
54 if(ix >= 0x40000000) { /* |x| >= 2.0 */
55 s = sinf(y);
56 c = cosf(y);
57 ss = -s-c;
58 cc = s-c;
59 if(ix<0x7f000000) { /* make sure y+y not overflow */
60 z = cosf(y+y);
61 if ((s*c)>zero) cc = z/ss;
62 else ss = z/cc;
63 }
64 /*
65 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
66 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
67 */
68 if(((uint32_t)hx)>0x80000000) z = (invsqrtpi*cc)/sqrtf(y);
69 else {
70 u = ponef(y); v = qonef(y);
71 z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
72 }
73 if(hx<0) return -z;
74 else return z;
75 }
76 if(ix<0x32000000) { /* |x|<2**-27 */
77 if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
78 }
79 z = x*x;
80 r = z*(r00+z*(r01+z*(r02+z*r03)));
81 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
82 r *= x;
83 return(x*(float)0.5+r/s);
84 }
85
86 static const float U0[5] = {
87 -1.9605709612e-01, /* 0xbe48c331 */
88 5.0443872809e-02, /* 0x3d4e9e3c */
89 -1.9125689287e-03, /* 0xbafaaf2a */
90 2.3525259166e-05, /* 0x37c5581c */
91 -9.1909917899e-08, /* 0xb3c56003 */
92 };
93 static const float V0[5] = {
94 1.9916731864e-02, /* 0x3ca3286a */
95 2.0255257550e-04, /* 0x3954644b */
96 1.3560879779e-06, /* 0x35b602d4 */
97 6.2274145840e-09, /* 0x31d5f8eb */
98 1.6655924903e-11, /* 0x2d9281cf */
99 };
100
101 float
__ieee754_y1f(float x)102 __ieee754_y1f(float x)
103 {
104 float z, s,c,ss,cc,u,v;
105 int32_t hx,ix;
106
107 GET_FLOAT_WORD(hx,x);
108 ix = 0x7fffffff&hx;
109 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
110 if(ix>=0x7f800000) return one/(x+x*x);
111 if(ix==0) return -one/zero;
112 if(hx<0) return zero/zero;
113 if(ix >= 0x40000000) { /* |x| >= 2.0 */
114 s = sinf(x);
115 c = cosf(x);
116 ss = -s-c;
117 cc = s-c;
118 if(ix<0x7f000000) { /* make sure x+x not overflow */
119 z = cosf(x+x);
120 if ((s*c)>zero) cc = z/ss;
121 else ss = z/cc;
122 }
123 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
124 * where x0 = x-3pi/4
125 * Better formula:
126 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
127 * = 1/sqrt(2) * (sin(x) - cos(x))
128 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
129 * = -1/sqrt(2) * (cos(x) + sin(x))
130 * To avoid cancellation, use
131 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
132 * to compute the worse one.
133 */
134 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x);
135 else {
136 u = ponef(x); v = qonef(x);
137 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
138 }
139 return z;
140 }
141 if(ix<=0x24800000) { /* x < 2**-54 */
142 return(-tpi/x);
143 }
144 z = x*x;
145 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
146 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
147 return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
148 }
149
150 /* For x >= 8, the asymptotic expansions of pone is
151 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
152 * We approximate pone by
153 * pone(x) = 1 + (R/S)
154 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
155 * S = 1 + ps0*s^2 + ... + ps4*s^10
156 * and
157 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
158 */
159
160 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
161 0.0000000000e+00, /* 0x00000000 */
162 1.1718750000e-01, /* 0x3df00000 */
163 1.3239480972e+01, /* 0x4153d4ea */
164 4.1205184937e+02, /* 0x43ce06a3 */
165 3.8747453613e+03, /* 0x45722bed */
166 7.9144794922e+03, /* 0x45f753d6 */
167 };
168 static const float ps8[5] = {
169 1.1420736694e+02, /* 0x42e46a2c */
170 3.6509309082e+03, /* 0x45642ee5 */
171 3.6956207031e+04, /* 0x47105c35 */
172 9.7602796875e+04, /* 0x47bea166 */
173 3.0804271484e+04, /* 0x46f0a88b */
174 };
175
176 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
177 1.3199052094e-11, /* 0x2d68333f */
178 1.1718749255e-01, /* 0x3defffff */
179 6.8027510643e+00, /* 0x40d9b023 */
180 1.0830818176e+02, /* 0x42d89dca */
181 5.1763616943e+02, /* 0x440168b7 */
182 5.2871520996e+02, /* 0x44042dc6 */
183 };
184 static const float ps5[5] = {
185 5.9280597687e+01, /* 0x426d1f55 */
186 9.9140142822e+02, /* 0x4477d9b1 */
187 5.3532670898e+03, /* 0x45a74a23 */
188 7.8446904297e+03, /* 0x45f52586 */
189 1.5040468750e+03, /* 0x44bc0180 */
190 };
191
192 static const float pr3[6] = {
193 3.0250391081e-09, /* 0x314fe10d */
194 1.1718686670e-01, /* 0x3defffab */
195 3.9329774380e+00, /* 0x407bb5e7 */
196 3.5119403839e+01, /* 0x420c7a45 */
197 9.1055007935e+01, /* 0x42b61c2a */
198 4.8559066772e+01, /* 0x42423c7c */
199 };
200 static const float ps3[5] = {
201 3.4791309357e+01, /* 0x420b2a4d */
202 3.3676245117e+02, /* 0x43a86198 */
203 1.0468714600e+03, /* 0x4482dbe3 */
204 8.9081134033e+02, /* 0x445eb3ed */
205 1.0378793335e+02, /* 0x42cf936c */
206 };
207
208 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
209 1.0771083225e-07, /* 0x33e74ea8 */
210 1.1717621982e-01, /* 0x3deffa16 */
211 2.3685150146e+00, /* 0x401795c0 */
212 1.2242610931e+01, /* 0x4143e1bc */
213 1.7693971634e+01, /* 0x418d8d41 */
214 5.0735230446e+00, /* 0x40a25a4d */
215 };
216 static const float ps2[5] = {
217 2.1436485291e+01, /* 0x41ab7dec */
218 1.2529022980e+02, /* 0x42fa9499 */
219 2.3227647400e+02, /* 0x436846c7 */
220 1.1767937469e+02, /* 0x42eb5bd7 */
221 8.3646392822e+00, /* 0x4105d590 */
222 };
223
ponef(float x)224 static float ponef(float x)
225 {
226 const float *p,*q;
227 float z,r,s;
228 int32_t ix;
229 GET_FLOAT_WORD(ix,x);
230 ix &= 0x7fffffff;
231 if(ix>=0x41000000) {p = pr8; q= ps8;}
232 else if(ix>=0x40f71c58){p = pr5; q= ps5;}
233 else if(ix>=0x4036db68){p = pr3; q= ps3;}
234 else if(ix>=0x40000000){p = pr2; q= ps2;}
235 z = one/(x*x);
236 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
237 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
238 return one+ r/s;
239 }
240
241
242 /* For x >= 8, the asymptotic expansions of qone is
243 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
244 * We approximate pone by
245 * qone(x) = s*(0.375 + (R/S))
246 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
247 * S = 1 + qs1*s^2 + ... + qs6*s^12
248 * and
249 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
250 */
251
252 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
253 0.0000000000e+00, /* 0x00000000 */
254 -1.0253906250e-01, /* 0xbdd20000 */
255 -1.6271753311e+01, /* 0xc1822c8d */
256 -7.5960174561e+02, /* 0xc43de683 */
257 -1.1849806641e+04, /* 0xc639273a */
258 -4.8438511719e+04, /* 0xc73d3683 */
259 };
260 static const float qs8[6] = {
261 1.6139537048e+02, /* 0x43216537 */
262 7.8253862305e+03, /* 0x45f48b17 */
263 1.3387534375e+05, /* 0x4802bcd6 */
264 7.1965775000e+05, /* 0x492fb29c */
265 6.6660125000e+05, /* 0x4922be94 */
266 -2.9449025000e+05, /* 0xc88fcb48 */
267 };
268
269 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
270 -2.0897993405e-11, /* 0xadb7d219 */
271 -1.0253904760e-01, /* 0xbdd1fffe */
272 -8.0564479828e+00, /* 0xc100e736 */
273 -1.8366960144e+02, /* 0xc337ab6b */
274 -1.3731937256e+03, /* 0xc4aba633 */
275 -2.6124443359e+03, /* 0xc523471c */
276 };
277 static const float qs5[6] = {
278 8.1276550293e+01, /* 0x42a28d98 */
279 1.9917987061e+03, /* 0x44f8f98f */
280 1.7468484375e+04, /* 0x468878f8 */
281 4.9851425781e+04, /* 0x4742bb6d */
282 2.7948074219e+04, /* 0x46da5826 */
283 -4.7191835938e+03, /* 0xc5937978 */
284 };
285
286 static const float qr3[6] = {
287 -5.0783124372e-09, /* 0xb1ae7d4f */
288 -1.0253783315e-01, /* 0xbdd1ff5b */
289 -4.6101160049e+00, /* 0xc0938612 */
290 -5.7847221375e+01, /* 0xc267638e */
291 -2.2824453735e+02, /* 0xc3643e9a */
292 -2.1921012878e+02, /* 0xc35b35cb */
293 };
294 static const float qs3[6] = {
295 4.7665153503e+01, /* 0x423ea91e */
296 6.7386511230e+02, /* 0x4428775e */
297 3.3801528320e+03, /* 0x45534272 */
298 5.5477290039e+03, /* 0x45ad5dd5 */
299 1.9031191406e+03, /* 0x44ede3d0 */
300 -1.3520118713e+02, /* 0xc3073381 */
301 };
302
303 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
304 -1.7838172539e-07, /* 0xb43f8932 */
305 -1.0251704603e-01, /* 0xbdd1f475 */
306 -2.7522056103e+00, /* 0xc0302423 */
307 -1.9663616180e+01, /* 0xc19d4f16 */
308 -4.2325313568e+01, /* 0xc2294d1f */
309 -2.1371921539e+01, /* 0xc1aaf9b2 */
310 };
311 static const float qs2[6] = {
312 2.9533363342e+01, /* 0x41ec4454 */
313 2.5298155212e+02, /* 0x437cfb47 */
314 7.5750280762e+02, /* 0x443d602e */
315 7.3939318848e+02, /* 0x4438d92a */
316 1.5594900513e+02, /* 0x431bf2f2 */
317 -4.9594988823e+00, /* 0xc09eb437 */
318 };
319
qonef(float x)320 static float qonef(float x)
321 {
322 const float *p,*q;
323 float s,r,z;
324 int32_t ix;
325 GET_FLOAT_WORD(ix,x);
326 ix &= 0x7fffffff;
327 if(ix>=0x40200000) {p = qr8; q= qs8;}
328 else if(ix>=0x40f71c58){p = qr5; q= qs5;}
329 else if(ix>=0x4036db68){p = qr3; q= qs3;}
330 else if(ix>=0x40000000){p = qr2; q= qs2;}
331 z = one/(x*x);
332 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
333 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
334 return ((float).375 + r/s)/x;
335 }
336