Annotation of src/lib/libm/src/e_j0f.c, Revision 1.2
1.1 jtc 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: #ifndef lint
1.2 ! jtc 17: static char rcsid[] = "$Id: e_j0f.c,v 1.1 1994/08/10 20:30:59 jtc Exp $";
1.1 jtc 18: #endif
19:
20: #include "math.h"
21: #include "math_private.h"
22:
23: #ifdef __STDC__
24: static float pzerof(float), qzerof(float);
25: #else
26: static float pzerof(), qzerof();
27: #endif
28:
29: #ifdef __STDC__
30: static const float
31: #else
32: static float
33: #endif
34: huge = 1e30,
35: one = 1.0,
36: invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
37: tpi = 6.3661974669e-01, /* 0x3f22f983 */
38: /* R0/S0 on [0, 2.00] */
39: R02 = 1.5625000000e-02, /* 0x3c800000 */
40: R03 = -1.8997929874e-04, /* 0xb947352e */
41: R04 = 1.8295404516e-06, /* 0x35f58e88 */
42: R05 = -4.6183270541e-09, /* 0xb19eaf3c */
43: S01 = 1.5619102865e-02, /* 0x3c7fe744 */
44: S02 = 1.1692678527e-04, /* 0x38f53697 */
45: S03 = 5.1354652442e-07, /* 0x3509daa6 */
46: S04 = 1.1661400734e-09; /* 0x30a045e8 */
47:
48: #ifdef __STDC__
49: static const float zero = 0.0;
50: #else
51: static float zero = 0.0;
52: #endif
53:
54: #ifdef __STDC__
55: float __ieee754_j0f(float x)
56: #else
57: float __ieee754_j0f(x)
58: float x;
59: #endif
60: {
61: float z, s,c,ss,cc,r,u,v;
1.2 ! jtc 62: int32_t hx,ix;
1.1 jtc 63:
64: GET_FLOAT_WORD(hx,x);
65: ix = hx&0x7fffffff;
66: if(ix>=0x7f800000) return one/(x*x);
67: x = fabsf(x);
68: if(ix >= 0x40000000) { /* |x| >= 2.0 */
69: s = sinf(x);
70: c = cosf(x);
71: ss = s-c;
72: cc = s+c;
73: if(ix<0x7f000000) { /* make sure x+x not overflow */
74: z = -cosf(x+x);
75: if ((s*c)<zero) cc = z/ss;
76: else ss = z/cc;
77: }
78: /*
79: * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
80: * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
81: */
82: if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x);
83: else {
84: u = pzerof(x); v = qzerof(x);
85: z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
86: }
87: return z;
88: }
89: if(ix<0x39000000) { /* |x| < 2**-13 */
90: if(huge+x>one) { /* raise inexact if x != 0 */
91: if(ix<0x32000000) return one; /* |x|<2**-27 */
92: else return one - (float)0.25*x*x;
93: }
94: }
95: z = x*x;
96: r = z*(R02+z*(R03+z*(R04+z*R05)));
97: s = one+z*(S01+z*(S02+z*(S03+z*S04)));
98: if(ix < 0x3F800000) { /* |x| < 1.00 */
99: return one + z*((float)-0.25+(r/s));
100: } else {
101: u = (float)0.5*x;
102: return((one+u)*(one-u)+z*(r/s));
103: }
104: }
105:
106: #ifdef __STDC__
107: static const float
108: #else
109: static float
110: #endif
111: u00 = -7.3804296553e-02, /* 0xbd9726b5 */
112: u01 = 1.7666645348e-01, /* 0x3e34e80d */
113: u02 = -1.3818567619e-02, /* 0xbc626746 */
114: u03 = 3.4745343146e-04, /* 0x39b62a69 */
115: u04 = -3.8140706238e-06, /* 0xb67ff53c */
116: u05 = 1.9559013964e-08, /* 0x32a802ba */
117: u06 = -3.9820518410e-11, /* 0xae2f21eb */
118: v01 = 1.2730483897e-02, /* 0x3c509385 */
119: v02 = 7.6006865129e-05, /* 0x389f65e0 */
120: v03 = 2.5915085189e-07, /* 0x348b216c */
121: v04 = 4.4111031494e-10; /* 0x2ff280c2 */
122:
123: #ifdef __STDC__
124: float __ieee754_y0f(float x)
125: #else
126: float __ieee754_y0f(x)
127: float x;
128: #endif
129: {
130: float z, s,c,ss,cc,u,v;
1.2 ! jtc 131: int32_t hx,ix;
1.1 jtc 132:
133: GET_FLOAT_WORD(hx,x);
134: ix = 0x7fffffff&hx;
135: /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
136: if(ix>=0x7f800000) return one/(x+x*x);
137: if(ix==0) return -one/zero;
138: if(hx<0) return zero/zero;
139: if(ix >= 0x40000000) { /* |x| >= 2.0 */
140: /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
141: * where x0 = x-pi/4
142: * Better formula:
143: * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
144: * = 1/sqrt(2) * (sin(x) + cos(x))
145: * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
146: * = 1/sqrt(2) * (sin(x) - cos(x))
147: * To avoid cancellation, use
148: * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
149: * to compute the worse one.
150: */
151: s = sinf(x);
152: c = cosf(x);
153: ss = s-c;
154: cc = s+c;
155: /*
156: * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
157: * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
158: */
159: if(ix<0x7f000000) { /* make sure x+x not overflow */
160: z = -cosf(x+x);
161: if ((s*c)<zero) cc = z/ss;
162: else ss = z/cc;
163: }
164: if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x);
165: else {
166: u = pzerof(x); v = qzerof(x);
167: z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
168: }
169: return z;
170: }
171: if(ix<=0x32000000) { /* x < 2**-27 */
172: return(u00 + tpi*__ieee754_logf(x));
173: }
174: z = x*x;
175: u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
176: v = one+z*(v01+z*(v02+z*(v03+z*v04)));
177: return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
178: }
179:
180: /* The asymptotic expansions of pzero is
181: * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
182: * For x >= 2, We approximate pzero by
183: * pzero(x) = 1 + (R/S)
184: * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
185: * S = 1 + pS0*s^2 + ... + pS4*s^10
186: * and
187: * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
188: */
189: #ifdef __STDC__
190: static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
191: #else
192: static float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
193: #endif
194: 0.0000000000e+00, /* 0x00000000 */
195: -7.0312500000e-02, /* 0xbd900000 */
196: -8.0816707611e+00, /* 0xc1014e86 */
197: -2.5706311035e+02, /* 0xc3808814 */
198: -2.4852163086e+03, /* 0xc51b5376 */
199: -5.2530439453e+03, /* 0xc5a4285a */
200: };
201: #ifdef __STDC__
202: static const float pS8[5] = {
203: #else
204: static float pS8[5] = {
205: #endif
206: 1.1653436279e+02, /* 0x42e91198 */
207: 3.8337448730e+03, /* 0x456f9beb */
208: 4.0597855469e+04, /* 0x471e95db */
209: 1.1675296875e+05, /* 0x47e4087c */
210: 4.7627726562e+04, /* 0x473a0bba */
211: };
212: #ifdef __STDC__
213: static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
214: #else
215: static float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
216: #endif
217: -1.1412546255e-11, /* 0xad48c58a */
218: -7.0312492549e-02, /* 0xbd8fffff */
219: -4.1596107483e+00, /* 0xc0851b88 */
220: -6.7674766541e+01, /* 0xc287597b */
221: -3.3123129272e+02, /* 0xc3a59d9b */
222: -3.4643338013e+02, /* 0xc3ad3779 */
223: };
224: #ifdef __STDC__
225: static const float pS5[5] = {
226: #else
227: static float pS5[5] = {
228: #endif
229: 6.0753936768e+01, /* 0x42730408 */
230: 1.0512523193e+03, /* 0x44836813 */
231: 5.9789707031e+03, /* 0x45bad7c4 */
232: 9.6254453125e+03, /* 0x461665c8 */
233: 2.4060581055e+03, /* 0x451660ee */
234: };
235:
236: #ifdef __STDC__
237: static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
238: #else
239: static float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
240: #endif
241: -2.5470459075e-09, /* 0xb12f081b */
242: -7.0311963558e-02, /* 0xbd8fffb8 */
243: -2.4090321064e+00, /* 0xc01a2d95 */
244: -2.1965976715e+01, /* 0xc1afba52 */
245: -5.8079170227e+01, /* 0xc2685112 */
246: -3.1447946548e+01, /* 0xc1fb9565 */
247: };
248: #ifdef __STDC__
249: static const float pS3[5] = {
250: #else
251: static float pS3[5] = {
252: #endif
253: 3.5856033325e+01, /* 0x420f6c94 */
254: 3.6151397705e+02, /* 0x43b4c1ca */
255: 1.1936077881e+03, /* 0x44953373 */
256: 1.1279968262e+03, /* 0x448cffe6 */
257: 1.7358093262e+02, /* 0x432d94b8 */
258: };
259:
260: #ifdef __STDC__
261: static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
262: #else
263: static float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
264: #endif
265: -8.8753431271e-08, /* 0xb3be98b7 */
266: -7.0303097367e-02, /* 0xbd8ffb12 */
267: -1.4507384300e+00, /* 0xbfb9b1cc */
268: -7.6356959343e+00, /* 0xc0f4579f */
269: -1.1193166733e+01, /* 0xc1331736 */
270: -3.2336456776e+00, /* 0xc04ef40d */
271: };
272: #ifdef __STDC__
273: static const float pS2[5] = {
274: #else
275: static float pS2[5] = {
276: #endif
277: 2.2220300674e+01, /* 0x41b1c32d */
278: 1.3620678711e+02, /* 0x430834f0 */
279: 2.7047027588e+02, /* 0x43873c32 */
280: 1.5387539673e+02, /* 0x4319e01a */
281: 1.4657617569e+01, /* 0x416a859a */
282: };
283:
284: #ifdef __STDC__
285: static float pzerof(float x)
286: #else
287: static float pzerof(x)
288: float x;
289: #endif
290: {
291: #ifdef __STDC__
292: const float *p,*q;
293: #else
294: float *p,*q;
295: #endif
296: float z,r,s;
1.2 ! jtc 297: int32_t ix;
1.1 jtc 298: GET_FLOAT_WORD(ix,x);
299: ix &= 0x7fffffff;
300: if(ix>=0x41000000) {p = pR8; q= pS8;}
301: else if(ix>=0x40f71c58){p = pR5; q= pS5;}
302: else if(ix>=0x4036db68){p = pR3; q= pS3;}
303: else if(ix>=0x40000000){p = pR2; q= pS2;}
304: z = one/(x*x);
305: r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
306: s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
307: return one+ r/s;
308: }
309:
310:
311: /* For x >= 8, the asymptotic expansions of qzero is
312: * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
313: * We approximate pzero by
314: * qzero(x) = s*(-1.25 + (R/S))
315: * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
316: * S = 1 + qS0*s^2 + ... + qS5*s^12
317: * and
318: * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
319: */
320: #ifdef __STDC__
321: static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
322: #else
323: static float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
324: #endif
325: 0.0000000000e+00, /* 0x00000000 */
326: 7.3242187500e-02, /* 0x3d960000 */
327: 1.1768206596e+01, /* 0x413c4a93 */
328: 5.5767340088e+02, /* 0x440b6b19 */
329: 8.8591972656e+03, /* 0x460a6cca */
330: 3.7014625000e+04, /* 0x471096a0 */
331: };
332: #ifdef __STDC__
333: static const float qS8[6] = {
334: #else
335: static float qS8[6] = {
336: #endif
337: 1.6377603149e+02, /* 0x4323c6aa */
338: 8.0983447266e+03, /* 0x45fd12c2 */
339: 1.4253829688e+05, /* 0x480b3293 */
340: 8.0330925000e+05, /* 0x49441ed4 */
341: 8.4050156250e+05, /* 0x494d3359 */
342: -3.4389928125e+05, /* 0xc8a7eb69 */
343: };
344:
345: #ifdef __STDC__
346: static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
347: #else
348: static float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
349: #endif
350: 1.8408595828e-11, /* 0x2da1ec79 */
351: 7.3242180049e-02, /* 0x3d95ffff */
352: 5.8356351852e+00, /* 0x40babd86 */
353: 1.3511157227e+02, /* 0x43071c90 */
354: 1.0272437744e+03, /* 0x448067cd */
355: 1.9899779053e+03, /* 0x44f8bf4b */
356: };
357: #ifdef __STDC__
358: static const float qS5[6] = {
359: #else
360: static float qS5[6] = {
361: #endif
362: 8.2776611328e+01, /* 0x42a58da0 */
363: 2.0778142090e+03, /* 0x4501dd07 */
364: 1.8847289062e+04, /* 0x46933e94 */
365: 5.6751113281e+04, /* 0x475daf1d */
366: 3.5976753906e+04, /* 0x470c88c1 */
367: -5.3543427734e+03, /* 0xc5a752be */
368: };
369:
370: #ifdef __STDC__
371: static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
372: #else
373: static float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
374: #endif
375: 4.3774099900e-09, /* 0x3196681b */
376: 7.3241114616e-02, /* 0x3d95ff70 */
377: 3.3442313671e+00, /* 0x405607e3 */
378: 4.2621845245e+01, /* 0x422a7cc5 */
379: 1.7080809021e+02, /* 0x432acedf */
380: 1.6673394775e+02, /* 0x4326bbe4 */
381: };
382: #ifdef __STDC__
383: static const float qS3[6] = {
384: #else
385: static float qS3[6] = {
386: #endif
387: 4.8758872986e+01, /* 0x42430916 */
388: 7.0968920898e+02, /* 0x44316c1c */
389: 3.7041481934e+03, /* 0x4567825f */
390: 6.4604252930e+03, /* 0x45c9e367 */
391: 2.5163337402e+03, /* 0x451d4557 */
392: -1.4924745178e+02, /* 0xc3153f59 */
393: };
394:
395: #ifdef __STDC__
396: static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
397: #else
398: static float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
399: #endif
400: 1.5044444979e-07, /* 0x342189db */
401: 7.3223426938e-02, /* 0x3d95f62a */
402: 1.9981917143e+00, /* 0x3fffc4bf */
403: 1.4495602608e+01, /* 0x4167edfd */
404: 3.1666231155e+01, /* 0x41fd5471 */
405: 1.6252708435e+01, /* 0x4182058c */
406: };
407: #ifdef __STDC__
408: static const float qS2[6] = {
409: #else
410: static float qS2[6] = {
411: #endif
412: 3.0365585327e+01, /* 0x41f2ecb8 */
413: 2.6934811401e+02, /* 0x4386ac8f */
414: 8.4478375244e+02, /* 0x44533229 */
415: 8.8293585205e+02, /* 0x445cbbe5 */
416: 2.1266638184e+02, /* 0x4354aa98 */
417: -5.3109550476e+00, /* 0xc0a9f358 */
418: };
419:
420: #ifdef __STDC__
421: static float qzerof(float x)
422: #else
423: static float qzerof(x)
424: float x;
425: #endif
426: {
427: #ifdef __STDC__
428: const float *p,*q;
429: #else
430: float *p,*q;
431: #endif
432: float s,r,z;
1.2 ! jtc 433: int32_t ix;
1.1 jtc 434: GET_FLOAT_WORD(ix,x);
435: ix &= 0x7fffffff;
436: if(ix>=0x41000000) {p = qR8; q= qS8;}
437: else if(ix>=0x40f71c58){p = qR5; q= qS5;}
438: else if(ix>=0x4036db68){p = qR3; q= qS3;}
439: else if(ix>=0x40000000){p = qR2; q= qS2;}
440: z = one/(x*x);
441: r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
442: s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
443: return (-(float).125 + r/s)/x;
444: }
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