1 /* ef_j1.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 "fdlibm.h"
17 
18 static float ponef(float), qonef(float);
19 
20 static const float one = 1.0,
21                    invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */
22     tpi = 6.3661974669e-01, /* 0x3f22f983 */
23     /* R0/S0 on [0,2] */
24     r00 = -6.2500000000e-02, /* 0xbd800000 */
25     r01 = 1.4070566976e-03, /* 0x3ab86cfd */
26     r02 = -1.5995563444e-05, /* 0xb7862e36 */
27     r03 = 4.9672799207e-08, /* 0x335557d2 */
28     s01 = 1.9153760746e-02, /* 0x3c9ce859 */
29     s02 = 1.8594678841e-04, /* 0x3942fab6 */
30     s03 = 1.1771846857e-06, /* 0x359dffc2 */
31     s04 = 5.0463624390e-09, /* 0x31ad6446 */
32     s05 = 1.2354227016e-11; /* 0x2d59567e */
33 
34 static const float zero = 0.0;
35 
36 float
j1f(float x)37 j1f(float x)
38 {
39     float z, s, c, ss, cc, r, u, v, y;
40     __int32_t hx, ix;
41 
42     if (isnan(x))
43         return x + x;
44 
45     if (isinf(x))
46         return zero;
47 
48     GET_FLOAT_WORD(hx, x);
49     ix = hx & 0x7fffffff;
50 
51     y = fabsf(x);
52     if (ix >= 0x40000000) { /* |x| >= 2.0 */
53         s = sinf(y);
54         c = cosf(y);
55         ss = -s - c;
56         cc = s - c;
57         if (ix <= FLT_UWORD_HALF_MAX) { /* make sure y+y not overflow */
58             z = cosf(y + y);
59             if ((s * c) > zero)
60                 cc = z / ss;
61             else
62                 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 (ix > 0x5c000000)
69             z = (invsqrtpi * cc) / sqrtf(y);
70         else {
71             u = ponef(y);
72             v = qonef(y);
73             z = invsqrtpi * (u * cc - v * ss) / sqrtf(y);
74         }
75         if (hx < 0)
76             return -z;
77         else
78             return z;
79     }
80     if (ix < 0x32000000) { /* |x|<2**-27 */
81         if (ix == 0)
82             return x;
83         return check_uflowf(0.5f * x); /* inexact if x!=0 necessary */
84     }
85     z = x * x;
86     r = z * (r00 + z * (r01 + z * (r02 + z * r03)));
87     s = one + z * (s01 + z * (s02 + z * (s03 + z * (s04 + z * s05))));
88     r *= x;
89     return (x * (float)0.5 + r / s);
90 }
91 
92 static const float U0[5] = {
93     -1.9605709612e-01, /* 0xbe48c331 */
94     5.0443872809e-02, /* 0x3d4e9e3c */
95     -1.9125689287e-03, /* 0xbafaaf2a */
96     2.3525259166e-05, /* 0x37c5581c */
97     -9.1909917899e-08, /* 0xb3c56003 */
98 };
99 static const float V0[5] = {
100     1.9916731864e-02, /* 0x3ca3286a */
101     2.0255257550e-04, /* 0x3954644b */
102     1.3560879779e-06, /* 0x35b602d4 */
103     6.2274145840e-09, /* 0x31d5f8eb */
104     1.6655924903e-11, /* 0x2d9281cf */
105 };
106 
107 float
y1f(float x)108 y1f(float x)
109 {
110     float z, s, c, ss, cc, u, v;
111     __int32_t hx, ix;
112 
113     GET_FLOAT_WORD(hx, x);
114     ix = 0x7fffffff & hx;
115 
116     if (ix == 0)
117         return __math_divzerof(1);
118 
119     if (ix > 0x7f800000)
120         return x+x;
121 
122     if (hx < 0)
123         return __math_invalidf(x);
124 
125     if (ix == 0x7f800000)
126         return zero;
127 
128     if (ix >= 0x40000000) { /* |x| >= 2.0 */
129         s = sinf(x);
130         c = cosf(x);
131         ss = -s - c;
132         cc = s - c;
133         if (ix <= FLT_UWORD_HALF_MAX) { /* make sure x+x not overflow */
134             z = cosf(x + x);
135             if ((s * c) > zero)
136                 cc = z / ss;
137             else
138                 ss = z / cc;
139         }
140         /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
141          * where x0 = x-3pi/4
142          *      Better formula:
143          *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
144          *                      =  1/sqrt(2) * (sin(x) - cos(x))
145          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
146          *                      = -1/sqrt(2) * (cos(x) + sin(x))
147          * To avoid cancellation, use
148          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
149          * to compute the worse one.
150          */
151         if (ix > 0x5c000000)
152             z = (invsqrtpi * ss) / sqrtf(x);
153         else {
154             u = ponef(x);
155             v = qonef(x);
156             z = invsqrtpi * (u * ss + v * cc) / sqrtf(x);
157         }
158         return z;
159     }
160     if (ix <= 0x24800000) { /* x < 2**-54 */
161         return check_oflowf(-tpi / x);
162     }
163     z = x * x;
164     u = U0[0] + z * (U0[1] + z * (U0[2] + z * (U0[3] + z * U0[4])));
165     v = one + z * (V0[0] + z * (V0[1] + z * (V0[2] + z * (V0[3] + z * V0[4]))));
166     return (x * (u / v) + tpi * (j1f(x) * logf(x) - one / x));
167 }
168 
169 /* For x >= 8, the asymptotic expansions of pone is
170  *	1 + 15/128 s^2 - 4725/2^15 s^4 - ...,	where s = 1/x.
171  * We approximate pone by
172  * 	pone(x) = 1 + (R/S)
173  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
174  * 	  S = 1 + ps0*s^2 + ... + ps4*s^10
175  * and
176  *	| pone(x)-1-R/S | <= 2  ** ( -60.06)
177  */
178 
179 static const float pr8[6] = {
180     /* for x in [inf, 8]=1/[0,0.125] */
181     0.0000000000e+00, /* 0x00000000 */
182     1.1718750000e-01, /* 0x3df00000 */
183     1.3239480972e+01, /* 0x4153d4ea */
184     4.1205184937e+02, /* 0x43ce06a3 */
185     3.8747453613e+03, /* 0x45722bed */
186     7.9144794922e+03, /* 0x45f753d6 */
187 };
188 static const float ps8[5] = {
189     1.1420736694e+02, /* 0x42e46a2c */
190     3.6509309082e+03, /* 0x45642ee5 */
191     3.6956207031e+04, /* 0x47105c35 */
192     9.7602796875e+04, /* 0x47bea166 */
193     3.0804271484e+04, /* 0x46f0a88b */
194 };
195 
196 static const float pr5[6] = {
197     /* for x in [8,4.5454]=1/[0.125,0.22001] */
198     1.3199052094e-11, /* 0x2d68333f */
199     1.1718749255e-01, /* 0x3defffff */
200     6.8027510643e+00, /* 0x40d9b023 */
201     1.0830818176e+02, /* 0x42d89dca */
202     5.1763616943e+02, /* 0x440168b7 */
203     5.2871520996e+02, /* 0x44042dc6 */
204 };
205 static const float ps5[5] = {
206     5.9280597687e+01, /* 0x426d1f55 */
207     9.9140142822e+02, /* 0x4477d9b1 */
208     5.3532670898e+03, /* 0x45a74a23 */
209     7.8446904297e+03, /* 0x45f52586 */
210     1.5040468750e+03, /* 0x44bc0180 */
211 };
212 
213 static const float pr3[6] = {
214     3.0250391081e-09, /* 0x314fe10d */
215     1.1718686670e-01, /* 0x3defffab */
216     3.9329774380e+00, /* 0x407bb5e7 */
217     3.5119403839e+01, /* 0x420c7a45 */
218     9.1055007935e+01, /* 0x42b61c2a */
219     4.8559066772e+01, /* 0x42423c7c */
220 };
221 static const float ps3[5] = {
222     3.4791309357e+01, /* 0x420b2a4d */
223     3.3676245117e+02, /* 0x43a86198 */
224     1.0468714600e+03, /* 0x4482dbe3 */
225     8.9081134033e+02, /* 0x445eb3ed */
226     1.0378793335e+02, /* 0x42cf936c */
227 };
228 
229 static const float pr2[6] = {
230     /* for x in [2.8570,2]=1/[0.3499,0.5] */
231     1.0771083225e-07, /* 0x33e74ea8 */
232     1.1717621982e-01, /* 0x3deffa16 */
233     2.3685150146e+00, /* 0x401795c0 */
234     1.2242610931e+01, /* 0x4143e1bc */
235     1.7693971634e+01, /* 0x418d8d41 */
236     5.0735230446e+00, /* 0x40a25a4d */
237 };
238 static const float ps2[5] = {
239     2.1436485291e+01, /* 0x41ab7dec */
240     1.2529022980e+02, /* 0x42fa9499 */
241     2.3227647400e+02, /* 0x436846c7 */
242     1.1767937469e+02, /* 0x42eb5bd7 */
243     8.3646392822e+00, /* 0x4105d590 */
244 };
245 
246 static float
ponef(float x)247 ponef(float x)
248 {
249     const float *p, *q;
250     float z, r, s;
251     __int32_t ix;
252     GET_FLOAT_WORD(ix, x);
253     ix &= 0x7fffffff;
254     if (ix >= 0x41000000) {
255         p = pr8;
256         q = ps8;
257     } else if (ix >= 0x40f71c58) {
258         p = pr5;
259         q = ps5;
260     } else if (ix >= 0x4036db68) {
261         p = pr3;
262         q = ps3;
263     } else {
264         p = pr2;
265         q = ps2;
266     }
267     z = one / (x * x);
268     r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
269     s = one + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))));
270     return one + r / s;
271 }
272 
273 /* For x >= 8, the asymptotic expansions of qone is
274  *	3/8 s - 105/1024 s^3 - ..., where s = 1/x.
275  * We approximate qone by
276  * 	qone(x) = s*(0.375 + (R/S))
277  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
278  * 	  S = 1 + qs1*s^2 + ... + qs6*s^12
279  * and
280  *	| qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
281  */
282 
283 static const float qr8[6] = {
284     /* for x in [inf, 8]=1/[0,0.125] */
285     0.0000000000e+00, /* 0x00000000 */
286     -1.0253906250e-01, /* 0xbdd20000 */
287     -1.6271753311e+01, /* 0xc1822c8d */
288     -7.5960174561e+02, /* 0xc43de683 */
289     -1.1849806641e+04, /* 0xc639273a */
290     -4.8438511719e+04, /* 0xc73d3683 */
291 };
292 static const float qs8[6] = {
293     1.6139537048e+02, /* 0x43216537 */
294     7.8253862305e+03, /* 0x45f48b17 */
295     1.3387534375e+05, /* 0x4802bcd6 */
296     7.1965775000e+05, /* 0x492fb29c */
297     6.6660125000e+05, /* 0x4922be94 */
298     -2.9449025000e+05, /* 0xc88fcb48 */
299 };
300 
301 static const float qr5[6] = {
302     /* for x in [8,4.5454]=1/[0.125,0.22001] */
303     -2.0897993405e-11, /* 0xadb7d219 */
304     -1.0253904760e-01, /* 0xbdd1fffe */
305     -8.0564479828e+00, /* 0xc100e736 */
306     -1.8366960144e+02, /* 0xc337ab6b */
307     -1.3731937256e+03, /* 0xc4aba633 */
308     -2.6124443359e+03, /* 0xc523471c */
309 };
310 static const float qs5[6] = {
311     8.1276550293e+01, /* 0x42a28d98 */
312     1.9917987061e+03, /* 0x44f8f98f */
313     1.7468484375e+04, /* 0x468878f8 */
314     4.9851425781e+04, /* 0x4742bb6d */
315     2.7948074219e+04, /* 0x46da5826 */
316     -4.7191835938e+03, /* 0xc5937978 */
317 };
318 
319 static const float qr3[6] = {
320     -5.0783124372e-09, /* 0xb1ae7d4f */
321     -1.0253783315e-01, /* 0xbdd1ff5b */
322     -4.6101160049e+00, /* 0xc0938612 */
323     -5.7847221375e+01, /* 0xc267638e */
324     -2.2824453735e+02, /* 0xc3643e9a */
325     -2.1921012878e+02, /* 0xc35b35cb */
326 };
327 static const float qs3[6] = {
328     4.7665153503e+01, /* 0x423ea91e */
329     6.7386511230e+02, /* 0x4428775e */
330     3.3801528320e+03, /* 0x45534272 */
331     5.5477290039e+03, /* 0x45ad5dd5 */
332     1.9031191406e+03, /* 0x44ede3d0 */
333     -1.3520118713e+02, /* 0xc3073381 */
334 };
335 
336 static const float qr2[6] = {
337     /* for x in [2.8570,2]=1/[0.3499,0.5] */
338     -1.7838172539e-07, /* 0xb43f8932 */
339     -1.0251704603e-01, /* 0xbdd1f475 */
340     -2.7522056103e+00, /* 0xc0302423 */
341     -1.9663616180e+01, /* 0xc19d4f16 */
342     -4.2325313568e+01, /* 0xc2294d1f */
343     -2.1371921539e+01, /* 0xc1aaf9b2 */
344 };
345 static const float qs2[6] = {
346     2.9533363342e+01, /* 0x41ec4454 */
347     2.5298155212e+02, /* 0x437cfb47 */
348     7.5750280762e+02, /* 0x443d602e */
349     7.3939318848e+02, /* 0x4438d92a */
350     1.5594900513e+02, /* 0x431bf2f2 */
351     -4.9594988823e+00, /* 0xc09eb437 */
352 };
353 
354 static float
qonef(float x)355 qonef(float x)
356 {
357     const float *p, *q;
358     float s, r, z;
359     __int32_t ix;
360     GET_FLOAT_WORD(ix, x);
361     ix &= 0x7fffffff;
362     if (ix >= 0x40200000) {
363         p = qr8;
364         q = qs8;
365     } else if (ix >= 0x40f71c58) {
366         p = qr5;
367         q = qs5;
368     } else if (ix >= 0x4036db68) {
369         p = qr3;
370         q = qs3;
371     } else {
372         p = qr2;
373         q = qs2;
374     }
375     z = one / (x * x);
376     r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
377     s = one +
378         z * (q[0] +
379              z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))));
380     return ((float).375 + r / s) / x;
381 }
382 
383 _MATH_ALIAS_f_f(j1)
384 
385 _MATH_ALIAS_f_f(y1)
386