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: stable/11/lib/msun/src/e_j1f.c 347068 2019-05-03 22:56:50Z peterj $");
18
19/*
20 * See e_j1.c for complete comments.
21 */
22
23#include "math.h"
24#include "math_private.h"
25
26static __inline float ponef(float), qonef(float);
27
28static const volatile float vone = 1, vzero = 0;
29
30static const float
31huge    = 1e30,
32one	= 1.0,
33invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
34tpi      =  6.3661974669e-01, /* 0x3f22f983 */
35/* R0/S0 on [0,2] */
36r00  = -6.2500000000e-02, /* 0xbd800000 */
37r01  =  1.4070566976e-03, /* 0x3ab86cfd */
38r02  = -1.5995563444e-05, /* 0xb7862e36 */
39r03  =  4.9672799207e-08, /* 0x335557d2 */
40s01  =  1.9153760746e-02, /* 0x3c9ce859 */
41s02  =  1.8594678841e-04, /* 0x3942fab6 */
42s03  =  1.1771846857e-06, /* 0x359dffc2 */
43s04  =  5.0463624390e-09, /* 0x31ad6446 */
44s05  =  1.2354227016e-11; /* 0x2d59567e */
45
46static const float zero    = 0.0;
47
48float
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		sincosf(y, &s, &c);
60		ss = -s-c;
61		cc = s-c;
62		if(ix<0x7f000000) {  /* make sure y+y not overflow */
63		    z = cosf(y+y);
64		    if ((s*c)>zero) cc = z/ss;
65		    else 	    ss = z/cc;
66		}
67	/*
68	 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
69	 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
70	 */
71		if(ix>0x58000000) z = (invsqrtpi*cc)/sqrtf(y); /* |x|>2**49 */
72		else {
73		    u = ponef(y); v = qonef(y);
74		    z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
75		}
76		if(hx<0) return -z;
77		else  	 return  z;
78	}
79	if(ix<0x39000000) {	/* |x|<2**-13 */
80	    if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
81	}
82	z = x*x;
83	r =  z*(r00+z*(r01+z*(r02+z*r03)));
84	s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
85	r *= x;
86	return(x*(float)0.5+r/s);
87}
88
89static const float U0[5] = {
90 -1.9605709612e-01, /* 0xbe48c331 */
91  5.0443872809e-02, /* 0x3d4e9e3c */
92 -1.9125689287e-03, /* 0xbafaaf2a */
93  2.3525259166e-05, /* 0x37c5581c */
94 -9.1909917899e-08, /* 0xb3c56003 */
95};
96static const float V0[5] = {
97  1.9916731864e-02, /* 0x3ca3286a */
98  2.0255257550e-04, /* 0x3954644b */
99  1.3560879779e-06, /* 0x35b602d4 */
100  6.2274145840e-09, /* 0x31d5f8eb */
101  1.6655924903e-11, /* 0x2d9281cf */
102};
103
104float
105__ieee754_y1f(float x)
106{
107	float z, s,c,ss,cc,u,v;
108	int32_t hx,ix;
109
110	GET_FLOAT_WORD(hx,x);
111        ix = 0x7fffffff&hx;
112	if(ix>=0x7f800000) return  vone/(x+x*x);
113	if(ix==0) return -one/vzero;
114	if(hx<0) return vzero/vzero;
115        if(ix >= 0x40000000) {  /* |x| >= 2.0 */
116                sincosf(x, &s, &c);
117                ss = -s-c;
118                cc = s-c;
119                if(ix<0x7f000000) {  /* make sure x+x not overflow */
120                    z = cosf(x+x);
121                    if ((s*c)>zero) cc = z/ss;
122                    else            ss = z/cc;
123                }
124        /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
125         * where x0 = x-3pi/4
126         *      Better formula:
127         *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
128         *                      =  1/sqrt(2) * (sin(x) - cos(x))
129         *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
130         *                      = -1/sqrt(2) * (cos(x) + sin(x))
131         * To avoid cancellation, use
132         *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
133         * to compute the worse one.
134         */
135                if(ix>0x58000000) z = (invsqrtpi*ss)/sqrtf(x); /* |x|>2**49 */
136                else {
137                    u = ponef(x); v = qonef(x);
138                    z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
139                }
140                return z;
141        }
142        if(ix<=0x33000000) {    /* x < 2**-25 */
143            return(-tpi/x);
144        }
145        z = x*x;
146        u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
147        v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
148        return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
149}
150
151/* For x >= 8, the asymptotic expansions of pone is
152 *	1 + 15/128 s^2 - 4725/2^15 s^4 - ...,	where s = 1/x.
153 * We approximate pone by
154 * 	pone(x) = 1 + (R/S)
155 * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
156 * 	  S = 1 + ps0*s^2 + ... + ps4*s^10
157 * and
158 *	| pone(x)-1-R/S | <= 2  ** ( -60.06)
159 */
160
161static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
162  0.0000000000e+00, /* 0x00000000 */
163  1.1718750000e-01, /* 0x3df00000 */
164  1.3239480972e+01, /* 0x4153d4ea */
165  4.1205184937e+02, /* 0x43ce06a3 */
166  3.8747453613e+03, /* 0x45722bed */
167  7.9144794922e+03, /* 0x45f753d6 */
168};
169static const float ps8[5] = {
170  1.1420736694e+02, /* 0x42e46a2c */
171  3.6509309082e+03, /* 0x45642ee5 */
172  3.6956207031e+04, /* 0x47105c35 */
173  9.7602796875e+04, /* 0x47bea166 */
174  3.0804271484e+04, /* 0x46f0a88b */
175};
176
177static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
178  1.3199052094e-11, /* 0x2d68333f */
179  1.1718749255e-01, /* 0x3defffff */
180  6.8027510643e+00, /* 0x40d9b023 */
181  1.0830818176e+02, /* 0x42d89dca */
182  5.1763616943e+02, /* 0x440168b7 */
183  5.2871520996e+02, /* 0x44042dc6 */
184};
185static const float ps5[5] = {
186  5.9280597687e+01, /* 0x426d1f55 */
187  9.9140142822e+02, /* 0x4477d9b1 */
188  5.3532670898e+03, /* 0x45a74a23 */
189  7.8446904297e+03, /* 0x45f52586 */
190  1.5040468750e+03, /* 0x44bc0180 */
191};
192
193static const float pr3[6] = {
194  3.0250391081e-09, /* 0x314fe10d */
195  1.1718686670e-01, /* 0x3defffab */
196  3.9329774380e+00, /* 0x407bb5e7 */
197  3.5119403839e+01, /* 0x420c7a45 */
198  9.1055007935e+01, /* 0x42b61c2a */
199  4.8559066772e+01, /* 0x42423c7c */
200};
201static const float ps3[5] = {
202  3.4791309357e+01, /* 0x420b2a4d */
203  3.3676245117e+02, /* 0x43a86198 */
204  1.0468714600e+03, /* 0x4482dbe3 */
205  8.9081134033e+02, /* 0x445eb3ed */
206  1.0378793335e+02, /* 0x42cf936c */
207};
208
209static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
210  1.0771083225e-07, /* 0x33e74ea8 */
211  1.1717621982e-01, /* 0x3deffa16 */
212  2.3685150146e+00, /* 0x401795c0 */
213  1.2242610931e+01, /* 0x4143e1bc */
214  1.7693971634e+01, /* 0x418d8d41 */
215  5.0735230446e+00, /* 0x40a25a4d */
216};
217static const float ps2[5] = {
218  2.1436485291e+01, /* 0x41ab7dec */
219  1.2529022980e+02, /* 0x42fa9499 */
220  2.3227647400e+02, /* 0x436846c7 */
221  1.1767937469e+02, /* 0x42eb5bd7 */
222  8.3646392822e+00, /* 0x4105d590 */
223};
224
225static __inline float
226ponef(float x)
227{
228	const float *p,*q;
229	float z,r,s;
230        int32_t ix;
231	GET_FLOAT_WORD(ix,x);
232	ix &= 0x7fffffff;
233        if(ix>=0x41000000)     {p = pr8; q= ps8;}
234        else if(ix>=0x409173eb){p = pr5; q= ps5;}
235        else if(ix>=0x4036d917){p = pr3; q= ps3;}
236	else                   {p = pr2; q= ps2;}	/* ix>=0x40000000 */
237        z = one/(x*x);
238        r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
239        s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
240        return one+ r/s;
241}
242
243
244/* For x >= 8, the asymptotic expansions of qone is
245 *	3/8 s - 105/1024 s^3 - ..., where s = 1/x.
246 * We approximate pone by
247 * 	qone(x) = s*(0.375 + (R/S))
248 * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
249 * 	  S = 1 + qs1*s^2 + ... + qs6*s^12
250 * and
251 *	| qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
252 */
253
254static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
255  0.0000000000e+00, /* 0x00000000 */
256 -1.0253906250e-01, /* 0xbdd20000 */
257 -1.6271753311e+01, /* 0xc1822c8d */
258 -7.5960174561e+02, /* 0xc43de683 */
259 -1.1849806641e+04, /* 0xc639273a */
260 -4.8438511719e+04, /* 0xc73d3683 */
261};
262static const float qs8[6] = {
263  1.6139537048e+02, /* 0x43216537 */
264  7.8253862305e+03, /* 0x45f48b17 */
265  1.3387534375e+05, /* 0x4802bcd6 */
266  7.1965775000e+05, /* 0x492fb29c */
267  6.6660125000e+05, /* 0x4922be94 */
268 -2.9449025000e+05, /* 0xc88fcb48 */
269};
270
271static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
272 -2.0897993405e-11, /* 0xadb7d219 */
273 -1.0253904760e-01, /* 0xbdd1fffe */
274 -8.0564479828e+00, /* 0xc100e736 */
275 -1.8366960144e+02, /* 0xc337ab6b */
276 -1.3731937256e+03, /* 0xc4aba633 */
277 -2.6124443359e+03, /* 0xc523471c */
278};
279static const float qs5[6] = {
280  8.1276550293e+01, /* 0x42a28d98 */
281  1.9917987061e+03, /* 0x44f8f98f */
282  1.7468484375e+04, /* 0x468878f8 */
283  4.9851425781e+04, /* 0x4742bb6d */
284  2.7948074219e+04, /* 0x46da5826 */
285 -4.7191835938e+03, /* 0xc5937978 */
286};
287
288static const float qr3[6] = {
289 -5.0783124372e-09, /* 0xb1ae7d4f */
290 -1.0253783315e-01, /* 0xbdd1ff5b */
291 -4.6101160049e+00, /* 0xc0938612 */
292 -5.7847221375e+01, /* 0xc267638e */
293 -2.2824453735e+02, /* 0xc3643e9a */
294 -2.1921012878e+02, /* 0xc35b35cb */
295};
296static const float qs3[6] = {
297  4.7665153503e+01, /* 0x423ea91e */
298  6.7386511230e+02, /* 0x4428775e */
299  3.3801528320e+03, /* 0x45534272 */
300  5.5477290039e+03, /* 0x45ad5dd5 */
301  1.9031191406e+03, /* 0x44ede3d0 */
302 -1.3520118713e+02, /* 0xc3073381 */
303};
304
305static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
306 -1.7838172539e-07, /* 0xb43f8932 */
307 -1.0251704603e-01, /* 0xbdd1f475 */
308 -2.7522056103e+00, /* 0xc0302423 */
309 -1.9663616180e+01, /* 0xc19d4f16 */
310 -4.2325313568e+01, /* 0xc2294d1f */
311 -2.1371921539e+01, /* 0xc1aaf9b2 */
312};
313static const float qs2[6] = {
314  2.9533363342e+01, /* 0x41ec4454 */
315  2.5298155212e+02, /* 0x437cfb47 */
316  7.5750280762e+02, /* 0x443d602e */
317  7.3939318848e+02, /* 0x4438d92a */
318  1.5594900513e+02, /* 0x431bf2f2 */
319 -4.9594988823e+00, /* 0xc09eb437 */
320};
321
322static __inline float
323qonef(float x)
324{
325	const float *p,*q;
326	float  s,r,z;
327	int32_t ix;
328	GET_FLOAT_WORD(ix,x);
329	ix &= 0x7fffffff;
330	if(ix>=0x41000000)     {p = qr8; q= qs8;}
331	else if(ix>=0x409173eb){p = qr5; q= qs5;}
332	else if(ix>=0x4036d917){p = qr3; q= qs3;}
333	else                   {p = qr2; q= qs2;}	/* ix>=0x40000000 */
334	z = one/(x*x);
335	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
336	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
337	return ((float).375 + r/s)/x;
338}
339