1 | /* e_j0f.c -- float version of e_j0.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 <math.h> |
17 | #include <math_private.h> |
18 | |
19 | static float pzerof(float), qzerof(float); |
20 | |
21 | static const float |
22 | huge = 1e30, |
23 | one = 1.0, |
24 | invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ |
25 | tpi = 6.3661974669e-01, /* 0x3f22f983 */ |
26 | /* R0/S0 on [0, 2.00] */ |
27 | R02 = 1.5625000000e-02, /* 0x3c800000 */ |
28 | R03 = -1.8997929874e-04, /* 0xb947352e */ |
29 | R04 = 1.8295404516e-06, /* 0x35f58e88 */ |
30 | R05 = -4.6183270541e-09, /* 0xb19eaf3c */ |
31 | S01 = 1.5619102865e-02, /* 0x3c7fe744 */ |
32 | S02 = 1.1692678527e-04, /* 0x38f53697 */ |
33 | S03 = 5.1354652442e-07, /* 0x3509daa6 */ |
34 | S04 = 1.1661400734e-09; /* 0x30a045e8 */ |
35 | |
36 | static const float zero = 0.0; |
37 | |
38 | float |
39 | __ieee754_j0f(float x) |
40 | { |
41 | float z, s,c,ss,cc,r,u,v; |
42 | int32_t hx,ix; |
43 | |
44 | GET_FLOAT_WORD(hx,x); |
45 | ix = hx&0x7fffffff; |
46 | if(ix>=0x7f800000) return one/(x*x); |
47 | x = fabsf(x); |
48 | if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
49 | __sincosf (x, &s, &c); |
50 | ss = s-c; |
51 | cc = s+c; |
52 | if(ix<0x7f000000) { /* make sure x+x not overflow */ |
53 | z = -__cosf(x+x); |
54 | if ((s*c)<zero) cc = z/ss; |
55 | else ss = z/cc; |
56 | } |
57 | /* |
58 | * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) |
59 | * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) |
60 | */ |
61 | if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(x); |
62 | else { |
63 | u = pzerof(x); v = qzerof(x); |
64 | z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrtf(x); |
65 | } |
66 | return z; |
67 | } |
68 | if(ix<0x39000000) { /* |x| < 2**-13 */ |
69 | math_force_eval(huge+x); /* raise inexact if x != 0 */ |
70 | if(ix<0x32000000) return one; /* |x|<2**-27 */ |
71 | else return one - (float)0.25*x*x; |
72 | } |
73 | z = x*x; |
74 | r = z*(R02+z*(R03+z*(R04+z*R05))); |
75 | s = one+z*(S01+z*(S02+z*(S03+z*S04))); |
76 | if(ix < 0x3F800000) { /* |x| < 1.00 */ |
77 | return one + z*((float)-0.25+(r/s)); |
78 | } else { |
79 | u = (float)0.5*x; |
80 | return((one+u)*(one-u)+z*(r/s)); |
81 | } |
82 | } |
83 | strong_alias (__ieee754_j0f, __j0f_finite) |
84 | |
85 | static const float |
86 | u00 = -7.3804296553e-02, /* 0xbd9726b5 */ |
87 | u01 = 1.7666645348e-01, /* 0x3e34e80d */ |
88 | u02 = -1.3818567619e-02, /* 0xbc626746 */ |
89 | u03 = 3.4745343146e-04, /* 0x39b62a69 */ |
90 | u04 = -3.8140706238e-06, /* 0xb67ff53c */ |
91 | u05 = 1.9559013964e-08, /* 0x32a802ba */ |
92 | u06 = -3.9820518410e-11, /* 0xae2f21eb */ |
93 | v01 = 1.2730483897e-02, /* 0x3c509385 */ |
94 | v02 = 7.6006865129e-05, /* 0x389f65e0 */ |
95 | v03 = 2.5915085189e-07, /* 0x348b216c */ |
96 | v04 = 4.4111031494e-10; /* 0x2ff280c2 */ |
97 | |
98 | float |
99 | __ieee754_y0f(float x) |
100 | { |
101 | float z, s,c,ss,cc,u,v; |
102 | int32_t hx,ix; |
103 | |
104 | GET_FLOAT_WORD(hx,x); |
105 | ix = 0x7fffffff&hx; |
106 | /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf. */ |
107 | if(ix>=0x7f800000) return one/(x+x*x); |
108 | if(ix==0) return -1/zero; /* -inf and divide by zero exception. */ |
109 | if(hx<0) return zero/(zero*x); |
110 | if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
111 | /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) |
112 | * where x0 = x-pi/4 |
113 | * Better formula: |
114 | * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) |
115 | * = 1/sqrt(2) * (sin(x) + cos(x)) |
116 | * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) |
117 | * = 1/sqrt(2) * (sin(x) - cos(x)) |
118 | * To avoid cancellation, use |
119 | * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) |
120 | * to compute the worse one. |
121 | */ |
122 | __sincosf (x, &s, &c); |
123 | ss = s-c; |
124 | cc = s+c; |
125 | /* |
126 | * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) |
127 | * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) |
128 | */ |
129 | if(ix<0x7f000000) { /* make sure x+x not overflow */ |
130 | z = -__cosf(x+x); |
131 | if ((s*c)<zero) cc = z/ss; |
132 | else ss = z/cc; |
133 | } |
134 | if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrtf(x); |
135 | else { |
136 | u = pzerof(x); v = qzerof(x); |
137 | z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrtf(x); |
138 | } |
139 | return z; |
140 | } |
141 | if(ix<=0x39800000) { /* x < 2**-13 */ |
142 | return(u00 + tpi*__ieee754_logf(x)); |
143 | } |
144 | z = x*x; |
145 | u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); |
146 | v = one+z*(v01+z*(v02+z*(v03+z*v04))); |
147 | return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x))); |
148 | } |
149 | strong_alias (__ieee754_y0f, __y0f_finite) |
150 | |
151 | /* The asymptotic expansions of pzero is |
152 | * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. |
153 | * For x >= 2, We approximate pzero by |
154 | * pzero(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 | * | pzero(x)-1-R/S | <= 2 ** ( -60.26) |
159 | */ |
160 | static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
161 | 0.0000000000e+00, /* 0x00000000 */ |
162 | -7.0312500000e-02, /* 0xbd900000 */ |
163 | -8.0816707611e+00, /* 0xc1014e86 */ |
164 | -2.5706311035e+02, /* 0xc3808814 */ |
165 | -2.4852163086e+03, /* 0xc51b5376 */ |
166 | -5.2530439453e+03, /* 0xc5a4285a */ |
167 | }; |
168 | static const float pS8[5] = { |
169 | 1.1653436279e+02, /* 0x42e91198 */ |
170 | 3.8337448730e+03, /* 0x456f9beb */ |
171 | 4.0597855469e+04, /* 0x471e95db */ |
172 | 1.1675296875e+05, /* 0x47e4087c */ |
173 | 4.7627726562e+04, /* 0x473a0bba */ |
174 | }; |
175 | static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
176 | -1.1412546255e-11, /* 0xad48c58a */ |
177 | -7.0312492549e-02, /* 0xbd8fffff */ |
178 | -4.1596107483e+00, /* 0xc0851b88 */ |
179 | -6.7674766541e+01, /* 0xc287597b */ |
180 | -3.3123129272e+02, /* 0xc3a59d9b */ |
181 | -3.4643338013e+02, /* 0xc3ad3779 */ |
182 | }; |
183 | static const float pS5[5] = { |
184 | 6.0753936768e+01, /* 0x42730408 */ |
185 | 1.0512523193e+03, /* 0x44836813 */ |
186 | 5.9789707031e+03, /* 0x45bad7c4 */ |
187 | 9.6254453125e+03, /* 0x461665c8 */ |
188 | 2.4060581055e+03, /* 0x451660ee */ |
189 | }; |
190 | |
191 | static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
192 | -2.5470459075e-09, /* 0xb12f081b */ |
193 | -7.0311963558e-02, /* 0xbd8fffb8 */ |
194 | -2.4090321064e+00, /* 0xc01a2d95 */ |
195 | -2.1965976715e+01, /* 0xc1afba52 */ |
196 | -5.8079170227e+01, /* 0xc2685112 */ |
197 | -3.1447946548e+01, /* 0xc1fb9565 */ |
198 | }; |
199 | static const float pS3[5] = { |
200 | 3.5856033325e+01, /* 0x420f6c94 */ |
201 | 3.6151397705e+02, /* 0x43b4c1ca */ |
202 | 1.1936077881e+03, /* 0x44953373 */ |
203 | 1.1279968262e+03, /* 0x448cffe6 */ |
204 | 1.7358093262e+02, /* 0x432d94b8 */ |
205 | }; |
206 | |
207 | static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
208 | -8.8753431271e-08, /* 0xb3be98b7 */ |
209 | -7.0303097367e-02, /* 0xbd8ffb12 */ |
210 | -1.4507384300e+00, /* 0xbfb9b1cc */ |
211 | -7.6356959343e+00, /* 0xc0f4579f */ |
212 | -1.1193166733e+01, /* 0xc1331736 */ |
213 | -3.2336456776e+00, /* 0xc04ef40d */ |
214 | }; |
215 | static const float pS2[5] = { |
216 | 2.2220300674e+01, /* 0x41b1c32d */ |
217 | 1.3620678711e+02, /* 0x430834f0 */ |
218 | 2.7047027588e+02, /* 0x43873c32 */ |
219 | 1.5387539673e+02, /* 0x4319e01a */ |
220 | 1.4657617569e+01, /* 0x416a859a */ |
221 | }; |
222 | |
223 | static float |
224 | pzerof(float x) |
225 | { |
226 | const float *p,*q; |
227 | float z,r,s; |
228 | int32_t ix; |
229 | GET_FLOAT_WORD(ix,x); |
230 | ix &= 0x7fffffff; |
231 | /* ix >= 0x40000000 for all calls to this function. */ |
232 | if(ix>=0x41000000) {p = pR8; q= pS8;} |
233 | else if(ix>=0x40f71c58){p = pR5; q= pS5;} |
234 | else if(ix>=0x4036db68){p = pR3; q= pS3;} |
235 | else {p = pR2; q= pS2;} |
236 | z = one/(x*x); |
237 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
238 | s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); |
239 | return one+ r/s; |
240 | } |
241 | |
242 | |
243 | /* For x >= 8, the asymptotic expansions of qzero is |
244 | * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. |
245 | * We approximate pzero by |
246 | * qzero(x) = s*(-1.25 + (R/S)) |
247 | * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 |
248 | * S = 1 + qS0*s^2 + ... + qS5*s^12 |
249 | * and |
250 | * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) |
251 | */ |
252 | static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
253 | 0.0000000000e+00, /* 0x00000000 */ |
254 | 7.3242187500e-02, /* 0x3d960000 */ |
255 | 1.1768206596e+01, /* 0x413c4a93 */ |
256 | 5.5767340088e+02, /* 0x440b6b19 */ |
257 | 8.8591972656e+03, /* 0x460a6cca */ |
258 | 3.7014625000e+04, /* 0x471096a0 */ |
259 | }; |
260 | static const float qS8[6] = { |
261 | 1.6377603149e+02, /* 0x4323c6aa */ |
262 | 8.0983447266e+03, /* 0x45fd12c2 */ |
263 | 1.4253829688e+05, /* 0x480b3293 */ |
264 | 8.0330925000e+05, /* 0x49441ed4 */ |
265 | 8.4050156250e+05, /* 0x494d3359 */ |
266 | -3.4389928125e+05, /* 0xc8a7eb69 */ |
267 | }; |
268 | |
269 | static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
270 | 1.8408595828e-11, /* 0x2da1ec79 */ |
271 | 7.3242180049e-02, /* 0x3d95ffff */ |
272 | 5.8356351852e+00, /* 0x40babd86 */ |
273 | 1.3511157227e+02, /* 0x43071c90 */ |
274 | 1.0272437744e+03, /* 0x448067cd */ |
275 | 1.9899779053e+03, /* 0x44f8bf4b */ |
276 | }; |
277 | static const float qS5[6] = { |
278 | 8.2776611328e+01, /* 0x42a58da0 */ |
279 | 2.0778142090e+03, /* 0x4501dd07 */ |
280 | 1.8847289062e+04, /* 0x46933e94 */ |
281 | 5.6751113281e+04, /* 0x475daf1d */ |
282 | 3.5976753906e+04, /* 0x470c88c1 */ |
283 | -5.3543427734e+03, /* 0xc5a752be */ |
284 | }; |
285 | |
286 | static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
287 | 4.3774099900e-09, /* 0x3196681b */ |
288 | 7.3241114616e-02, /* 0x3d95ff70 */ |
289 | 3.3442313671e+00, /* 0x405607e3 */ |
290 | 4.2621845245e+01, /* 0x422a7cc5 */ |
291 | 1.7080809021e+02, /* 0x432acedf */ |
292 | 1.6673394775e+02, /* 0x4326bbe4 */ |
293 | }; |
294 | static const float qS3[6] = { |
295 | 4.8758872986e+01, /* 0x42430916 */ |
296 | 7.0968920898e+02, /* 0x44316c1c */ |
297 | 3.7041481934e+03, /* 0x4567825f */ |
298 | 6.4604252930e+03, /* 0x45c9e367 */ |
299 | 2.5163337402e+03, /* 0x451d4557 */ |
300 | -1.4924745178e+02, /* 0xc3153f59 */ |
301 | }; |
302 | |
303 | static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
304 | 1.5044444979e-07, /* 0x342189db */ |
305 | 7.3223426938e-02, /* 0x3d95f62a */ |
306 | 1.9981917143e+00, /* 0x3fffc4bf */ |
307 | 1.4495602608e+01, /* 0x4167edfd */ |
308 | 3.1666231155e+01, /* 0x41fd5471 */ |
309 | 1.6252708435e+01, /* 0x4182058c */ |
310 | }; |
311 | static const float qS2[6] = { |
312 | 3.0365585327e+01, /* 0x41f2ecb8 */ |
313 | 2.6934811401e+02, /* 0x4386ac8f */ |
314 | 8.4478375244e+02, /* 0x44533229 */ |
315 | 8.8293585205e+02, /* 0x445cbbe5 */ |
316 | 2.1266638184e+02, /* 0x4354aa98 */ |
317 | -5.3109550476e+00, /* 0xc0a9f358 */ |
318 | }; |
319 | |
320 | static float |
321 | qzerof(float x) |
322 | { |
323 | const float *p,*q; |
324 | float s,r,z; |
325 | int32_t ix; |
326 | GET_FLOAT_WORD(ix,x); |
327 | ix &= 0x7fffffff; |
328 | /* ix >= 0x40000000 for all calls to this function. */ |
329 | if(ix>=0x41000000) {p = qR8; q= qS8;} |
330 | else if(ix>=0x40f71c58){p = qR5; q= qS5;} |
331 | else if(ix>=0x4036db68){p = qR3; q= qS3;} |
332 | else {p = qR2; q= qS2;} |
333 | z = one/(x*x); |
334 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
335 | s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); |
336 | return (-(float).125 + r/s)/x; |
337 | } |
338 | |