1/* e_j0f.c -- float version of e_j0.c.
2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
3 * Bugs in __ieee754_j0f and __ieee754_y0f fixed by Scott Turner 01/16/2010
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
18static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_j0f.c,v 1.7 2002/05/28 18:15:03 alfred Exp $";
19#endif
20
21#include "math.h"
22#include "math_private.h"
23
24static float pzerof(float), qzerof(float);
25
26static const float
27huge 	= 1e30,
28one	= 1.0,
29invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
30tpi      =  6.3661974669e-01, /* 0x3f22f983 */
31 		/* R0/S0 on [0, 2.00] */
32R02  =  1.5625000000e-02, /* 0x3c800000 */
33R03  = -1.8997929874e-04, /* 0xb947352e */
34R04  =  1.8295404516e-06, /* 0x35f58e88 */
35R05  = -4.6183270541e-09, /* 0xb19eaf3c */
36S01  =  1.5619102865e-02, /* 0x3c7fe744 */
37S02  =  1.1692678527e-04, /* 0x38f53697 */
38S03  =  5.1354652442e-07, /* 0x3509daa6 */
39S04  =  1.1661400734e-09; /* 0x30a045e8 */
40
41static const float zero = 0.0;
42
43float
44__ieee754_j0f(float x)
45{
46	float z, s,c,ss,cc,r,u,v;
47	int32_t hx,ix;
48
49	GET_FLOAT_WORD(hx,x);
50	ix = hx&0x7fffffff;
51	if(ix>=0x7f800000) return one/(x*x);
52	x = fabsf(x);
53	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
54		s = sinf(x);
55		c = cosf(x);
56		ss = s-c;
57		cc = s+c;
58		if(ix<0x7f000000) {  /* make sure x+x not overflow */
59		    z = -cosf(x+x);
60		    if ((s*c)<zero) cc = z/ss;
61		    else 	    ss = z/cc;
62		}
63	/*
64	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
65	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
66	 */
67		if(((uint32_t)hx)>0x80000000) z = (invsqrtpi*cc)/sqrtf(x);
68		else {
69		    u = pzerof(x); v = qzerof(x);
70		    z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
71		}
72		return z;
73	}
74	if(ix<0x39000000) {	/* |x| < 2**-13 */
75	    if(huge+x>one) {	/* raise inexact if x != 0 */
76	        if(ix<0x32000000) return one;	/* |x|<2**-27 */
77	        else 	      return one - (float)0.25*x*x;
78	    }
79	}
80	z = x*x;
81	r =  z*(R02+z*(R03+z*(R04+z*R05)));
82	s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
83	if(ix < 0x3F800000) {	/* |x| < 1.00 */
84	    return one + z*((float)-0.25+(r/s));
85	} else {
86	    u = (float)0.5*x;
87	    return((one+u)*(one-u)+z*(r/s));
88	}
89}
90
91static const float
92u00  = -7.3804296553e-02, /* 0xbd9726b5 */
93u01  =  1.7666645348e-01, /* 0x3e34e80d */
94u02  = -1.3818567619e-02, /* 0xbc626746 */
95u03  =  3.4745343146e-04, /* 0x39b62a69 */
96u04  = -3.8140706238e-06, /* 0xb67ff53c */
97u05  =  1.9559013964e-08, /* 0x32a802ba */
98u06  = -3.9820518410e-11, /* 0xae2f21eb */
99v01  =  1.2730483897e-02, /* 0x3c509385 */
100v02  =  7.6006865129e-05, /* 0x389f65e0 */
101v03  =  2.5915085189e-07, /* 0x348b216c */
102v04  =  4.4111031494e-10; /* 0x2ff280c2 */
103
104float
105__ieee754_y0f(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 = hx&0x7fffffff;
112    /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
113	if(ix>=0x7f800000) return  one/(x+x*x);
114        if(ix==0) return -one/zero;
115        if(hx<0) return zero/zero;
116        if(ix >= 0x40000000) {  /* |x| >= 2.0 */
117        /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
118         * where x0 = x-pi/4
119         *      Better formula:
120         *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
121         *                      =  1/sqrt(2) * (sin(x) + cos(x))
122         *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
123         *                      =  1/sqrt(2) * (sin(x) - cos(x))
124         * To avoid cancellation, use
125         *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
126         * to compute the worse one.
127         */
128                s = sinf(x);
129                c = cosf(x);
130                ss = s-c;
131                cc = s+c;
132	/*
133	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
134	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
135	 */
136                if(ix<0x7f000000) {  /* make sure x+x not overflow */
137                    z = -cosf(x+x);
138                    if ((s*c)<zero) cc = z/ss;
139                    else            ss = z/cc;
140                }
141                if(((uint32_t)hx)>0x80000000) z = (invsqrtpi*ss)/sqrtf(x);
142                else {
143                    u = pzerof(x); v = qzerof(x);
144                    z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
145                }
146                return z;
147	}
148	if(ix<=0x32000000) {	/* x < 2**-27 */
149	    return(u00 + tpi*__ieee754_logf(x));
150	}
151	z = x*x;
152	u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
153	v = one+z*(v01+z*(v02+z*(v03+z*v04)));
154	return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
155}
156
157/* The asymptotic expansions of pzero is
158 *	1 - 9/128 s^2 + 11025/98304 s^4 - ...,	where s = 1/x.
159 * For x >= 2, We approximate pzero by
160 * 	pzero(x) = 1 + (R/S)
161 * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
162 * 	  S = 1 + pS0*s^2 + ... + pS4*s^10
163 * and
164 *	| pzero(x)-1-R/S | <= 2  ** ( -60.26)
165 */
166static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
167  0.0000000000e+00, /* 0x00000000 */
168 -7.0312500000e-02, /* 0xbd900000 */
169 -8.0816707611e+00, /* 0xc1014e86 */
170 -2.5706311035e+02, /* 0xc3808814 */
171 -2.4852163086e+03, /* 0xc51b5376 */
172 -5.2530439453e+03, /* 0xc5a4285a */
173};
174static const float pS8[5] = {
175  1.1653436279e+02, /* 0x42e91198 */
176  3.8337448730e+03, /* 0x456f9beb */
177  4.0597855469e+04, /* 0x471e95db */
178  1.1675296875e+05, /* 0x47e4087c */
179  4.7627726562e+04, /* 0x473a0bba */
180};
181static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
182 -1.1412546255e-11, /* 0xad48c58a */
183 -7.0312492549e-02, /* 0xbd8fffff */
184 -4.1596107483e+00, /* 0xc0851b88 */
185 -6.7674766541e+01, /* 0xc287597b */
186 -3.3123129272e+02, /* 0xc3a59d9b */
187 -3.4643338013e+02, /* 0xc3ad3779 */
188};
189static const float pS5[5] = {
190  6.0753936768e+01, /* 0x42730408 */
191  1.0512523193e+03, /* 0x44836813 */
192  5.9789707031e+03, /* 0x45bad7c4 */
193  9.6254453125e+03, /* 0x461665c8 */
194  2.4060581055e+03, /* 0x451660ee */
195};
196
197static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
198 -2.5470459075e-09, /* 0xb12f081b */
199 -7.0311963558e-02, /* 0xbd8fffb8 */
200 -2.4090321064e+00, /* 0xc01a2d95 */
201 -2.1965976715e+01, /* 0xc1afba52 */
202 -5.8079170227e+01, /* 0xc2685112 */
203 -3.1447946548e+01, /* 0xc1fb9565 */
204};
205static const float pS3[5] = {
206  3.5856033325e+01, /* 0x420f6c94 */
207  3.6151397705e+02, /* 0x43b4c1ca */
208  1.1936077881e+03, /* 0x44953373 */
209  1.1279968262e+03, /* 0x448cffe6 */
210  1.7358093262e+02, /* 0x432d94b8 */
211};
212
213static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
214 -8.8753431271e-08, /* 0xb3be98b7 */
215 -7.0303097367e-02, /* 0xbd8ffb12 */
216 -1.4507384300e+00, /* 0xbfb9b1cc */
217 -7.6356959343e+00, /* 0xc0f4579f */
218 -1.1193166733e+01, /* 0xc1331736 */
219 -3.2336456776e+00, /* 0xc04ef40d */
220};
221static const float pS2[5] = {
222  2.2220300674e+01, /* 0x41b1c32d */
223  1.3620678711e+02, /* 0x430834f0 */
224  2.7047027588e+02, /* 0x43873c32 */
225  1.5387539673e+02, /* 0x4319e01a */
226  1.4657617569e+01, /* 0x416a859a */
227};
228
229	static float pzerof(float x)
230{
231	const float *p,*q;
232	float z,r,s;
233	int32_t ix;
234	GET_FLOAT_WORD(ix,x);
235	ix &= 0x7fffffff;
236	if(ix>=0x41000000)     {p = pR8; q= pS8;}
237	else if(ix>=0x40f71c58){p = pR5; q= pS5;}
238	else if(ix>=0x4036db68){p = pR3; q= pS3;}
239	else if(ix>=0x40000000){p = pR2; q= pS2;}
240	z = one/(x*x);
241	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
242	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
243	return one+ r/s;
244}
245
246
247/* For x >= 8, the asymptotic expansions of qzero is
248 *	-1/8 s + 75/1024 s^3 - ..., where s = 1/x.
249 * We approximate pzero by
250 * 	qzero(x) = s*(-1.25 + (R/S))
251 * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
252 * 	  S = 1 + qS0*s^2 + ... + qS5*s^12
253 * and
254 *	| qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
255 */
256static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
257  0.0000000000e+00, /* 0x00000000 */
258  7.3242187500e-02, /* 0x3d960000 */
259  1.1768206596e+01, /* 0x413c4a93 */
260  5.5767340088e+02, /* 0x440b6b19 */
261  8.8591972656e+03, /* 0x460a6cca */
262  3.7014625000e+04, /* 0x471096a0 */
263};
264static const float qS8[6] = {
265  1.6377603149e+02, /* 0x4323c6aa */
266  8.0983447266e+03, /* 0x45fd12c2 */
267  1.4253829688e+05, /* 0x480b3293 */
268  8.0330925000e+05, /* 0x49441ed4 */
269  8.4050156250e+05, /* 0x494d3359 */
270 -3.4389928125e+05, /* 0xc8a7eb69 */
271};
272
273static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
274  1.8408595828e-11, /* 0x2da1ec79 */
275  7.3242180049e-02, /* 0x3d95ffff */
276  5.8356351852e+00, /* 0x40babd86 */
277  1.3511157227e+02, /* 0x43071c90 */
278  1.0272437744e+03, /* 0x448067cd */
279  1.9899779053e+03, /* 0x44f8bf4b */
280};
281static const float qS5[6] = {
282  8.2776611328e+01, /* 0x42a58da0 */
283  2.0778142090e+03, /* 0x4501dd07 */
284  1.8847289062e+04, /* 0x46933e94 */
285  5.6751113281e+04, /* 0x475daf1d */
286  3.5976753906e+04, /* 0x470c88c1 */
287 -5.3543427734e+03, /* 0xc5a752be */
288};
289
290static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
291  4.3774099900e-09, /* 0x3196681b */
292  7.3241114616e-02, /* 0x3d95ff70 */
293  3.3442313671e+00, /* 0x405607e3 */
294  4.2621845245e+01, /* 0x422a7cc5 */
295  1.7080809021e+02, /* 0x432acedf */
296  1.6673394775e+02, /* 0x4326bbe4 */
297};
298static const float qS3[6] = {
299  4.8758872986e+01, /* 0x42430916 */
300  7.0968920898e+02, /* 0x44316c1c */
301  3.7041481934e+03, /* 0x4567825f */
302  6.4604252930e+03, /* 0x45c9e367 */
303  2.5163337402e+03, /* 0x451d4557 */
304 -1.4924745178e+02, /* 0xc3153f59 */
305};
306
307static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
308  1.5044444979e-07, /* 0x342189db */
309  7.3223426938e-02, /* 0x3d95f62a */
310  1.9981917143e+00, /* 0x3fffc4bf */
311  1.4495602608e+01, /* 0x4167edfd */
312  3.1666231155e+01, /* 0x41fd5471 */
313  1.6252708435e+01, /* 0x4182058c */
314};
315static const float qS2[6] = {
316  3.0365585327e+01, /* 0x41f2ecb8 */
317  2.6934811401e+02, /* 0x4386ac8f */
318  8.4478375244e+02, /* 0x44533229 */
319  8.8293585205e+02, /* 0x445cbbe5 */
320  2.1266638184e+02, /* 0x4354aa98 */
321 -5.3109550476e+00, /* 0xc0a9f358 */
322};
323
324	static float qzerof(float x)
325{
326	const float *p,*q;
327	float s,r,z;
328	int32_t ix;
329	GET_FLOAT_WORD(ix,x);
330	ix &= 0x7fffffff;
331	if(ix>=0x41000000)     {p = qR8; q= qS8;}
332	else if(ix>=0x40f71c58){p = qR5; q= qS5;}
333	else if(ix>=0x4036db68){p = qR3; q= qS3;}
334	else if(ix>=0x40000000){p = qR2; q= qS2;}
335	z = one/(x*x);
336	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
337	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
338	return (-(float).125 + r/s)/x;
339}
340