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