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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
18 static 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 
24 static float pzerof(float), qzerof(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, 2.00] */
32 R02  =  1.5625000000e-02, /* 0x3c800000 */
33 R03  = -1.8997929874e-04, /* 0xb947352e */
34 R04  =  1.8295404516e-06, /* 0x35f58e88 */
35 R05  = -4.6183270541e-09, /* 0xb19eaf3c */
36 S01  =  1.5619102865e-02, /* 0x3c7fe744 */
37 S02  =  1.1692678527e-04, /* 0x38f53697 */
38 S03  =  5.1354652442e-07, /* 0x3509daa6 */
39 S04  =  1.1661400734e-09; /* 0x30a045e8 */
40 
41 static const float zero = 0.0;
42 
43 float
__ieee754_j0f(float x)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 
91 static const float
92 u00  = -7.3804296553e-02, /* 0xbd9726b5 */
93 u01  =  1.7666645348e-01, /* 0x3e34e80d */
94 u02  = -1.3818567619e-02, /* 0xbc626746 */
95 u03  =  3.4745343146e-04, /* 0x39b62a69 */
96 u04  = -3.8140706238e-06, /* 0xb67ff53c */
97 u05  =  1.9559013964e-08, /* 0x32a802ba */
98 u06  = -3.9820518410e-11, /* 0xae2f21eb */
99 v01  =  1.2730483897e-02, /* 0x3c509385 */
100 v02  =  7.6006865129e-05, /* 0x389f65e0 */
101 v03  =  2.5915085189e-07, /* 0x348b216c */
102 v04  =  4.4111031494e-10; /* 0x2ff280c2 */
103 
104 float
__ieee754_y0f(float x)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  */
166 static 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 };
174 static 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 };
181 static 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 };
189 static 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 
197 static 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 };
205 static 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 
213 static 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 };
221 static 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 
pzerof(float x)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  */
256 static 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 };
264 static 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 
273 static 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 };
281 static 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 
290 static 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 };
298 static 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 
307 static 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 };
315 static 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 
qzerof(float x)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