1 | /* @(#)e_j1.c 5.1 93/09/24 */ |
2 | /* |
3 | * ==================================================== |
4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
5 | * |
6 | * Developed at SunPro, a Sun Microsystems, Inc. business. |
7 | * Permission to use, copy, modify, and distribute this |
8 | * software is freely granted, provided that this notice |
9 | * is preserved. |
10 | * ==================================================== |
11 | */ |
12 | /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/26, |
13 | for performance improvement on pipelined processors. |
14 | */ |
15 | |
16 | /* __ieee754_j1(x), __ieee754_y1(x) |
17 | * Bessel function of the first and second kinds of order zero. |
18 | * Method -- j1(x): |
19 | * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ... |
20 | * 2. Reduce x to |x| since j1(x)=-j1(-x), and |
21 | * for x in (0,2) |
22 | * j1(x) = x/2 + x*z*R0/S0, where z = x*x; |
23 | * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 ) |
24 | * for x in (2,inf) |
25 | * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1)) |
26 | * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) |
27 | * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) |
28 | * as follow: |
29 | * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) |
30 | * = 1/sqrt(2) * (sin(x) - cos(x)) |
31 | * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) |
32 | * = -1/sqrt(2) * (sin(x) + cos(x)) |
33 | * (To avoid cancellation, use |
34 | * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) |
35 | * to compute the worse one.) |
36 | * |
37 | * 3 Special cases |
38 | * j1(nan)= nan |
39 | * j1(0) = 0 |
40 | * j1(inf) = 0 |
41 | * |
42 | * Method -- y1(x): |
43 | * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN |
44 | * 2. For x<2. |
45 | * Since |
46 | * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...) |
47 | * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function. |
48 | * We use the following function to approximate y1, |
49 | * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2 |
50 | * where for x in [0,2] (abs err less than 2**-65.89) |
51 | * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4 |
52 | * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5 |
53 | * Note: For tiny x, 1/x dominate y1 and hence |
54 | * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54) |
55 | * 3. For x>=2. |
56 | * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) |
57 | * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) |
58 | * by method mentioned above. |
59 | */ |
60 | |
61 | #include <errno.h> |
62 | #include <float.h> |
63 | #include <math.h> |
64 | #include <math-narrow-eval.h> |
65 | #include <math_private.h> |
66 | #include <math-underflow.h> |
67 | #include <libm-alias-finite.h> |
68 | |
69 | static double pone (double), qone (double); |
70 | |
71 | static const double |
72 | huge = 1e300, |
73 | one = 1.0, |
74 | invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ |
75 | tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ |
76 | /* R0/S0 on [0,2] */ |
77 | R[] = { -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */ |
78 | 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */ |
79 | -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */ |
80 | 4.96727999609584448412e-08 }, /* 0x3E6AAAFA, 0x46CA0BD9 */ |
81 | S[] = { 0.0, 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */ |
82 | 1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */ |
83 | 1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */ |
84 | 5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */ |
85 | 1.23542274426137913908e-11 }; /* 0x3DAB2ACF, 0xCFB97ED8 */ |
86 | |
87 | static const double zero = 0.0; |
88 | |
89 | double |
90 | __ieee754_j1 (double x) |
91 | { |
92 | double z, s, c, ss, cc, r, u, v, y, r1, r2, s1, s2, s3, z2, z4; |
93 | int32_t hx, ix; |
94 | |
95 | GET_HIGH_WORD (hx, x); |
96 | ix = hx & 0x7fffffff; |
97 | if (__glibc_unlikely (ix >= 0x7ff00000)) |
98 | return one / x; |
99 | y = fabs (x: x); |
100 | if (ix >= 0x40000000) /* |x| >= 2.0 */ |
101 | { |
102 | __sincos (x: y, sinx: &s, cosx: &c); |
103 | ss = -s - c; |
104 | cc = s - c; |
105 | if (ix < 0x7fe00000) /* make sure y+y not overflow */ |
106 | { |
107 | z = __cos (x: y + y); |
108 | if ((s * c) > zero) |
109 | cc = z / ss; |
110 | else |
111 | ss = z / cc; |
112 | } |
113 | /* |
114 | * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) |
115 | * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) |
116 | */ |
117 | if (ix > 0x48000000) |
118 | z = (invsqrtpi * cc) / sqrt (y); |
119 | else |
120 | { |
121 | u = pone (y); v = qone (y); |
122 | z = invsqrtpi * (u * cc - v * ss) / sqrt (y); |
123 | } |
124 | if (hx < 0) |
125 | return -z; |
126 | else |
127 | return z; |
128 | } |
129 | if (__glibc_unlikely (ix < 0x3e400000)) /* |x|<2**-27 */ |
130 | { |
131 | if (huge + x > one) /* inexact if x!=0 necessary */ |
132 | { |
133 | double ret = math_narrow_eval (0.5 * x); |
134 | math_check_force_underflow (ret); |
135 | if (ret == 0 && x != 0) |
136 | __set_errno (ERANGE); |
137 | return ret; |
138 | } |
139 | } |
140 | z = x * x; |
141 | r1 = z * R[0]; z2 = z * z; |
142 | r2 = R[1] + z * R[2]; z4 = z2 * z2; |
143 | r = r1 + z2 * r2 + z4 * R[3]; |
144 | r *= x; |
145 | s1 = one + z * S[1]; |
146 | s2 = S[2] + z * S[3]; |
147 | s3 = S[4] + z * S[5]; |
148 | s = s1 + z2 * s2 + z4 * s3; |
149 | return (x * 0.5 + r / s); |
150 | } |
151 | libm_alias_finite (__ieee754_j1, __j1) |
152 | |
153 | static const double U0[5] = { |
154 | -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */ |
155 | 5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */ |
156 | -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */ |
157 | 2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */ |
158 | -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */ |
159 | }; |
160 | static const double V0[5] = { |
161 | 1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */ |
162 | 2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */ |
163 | 1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */ |
164 | 6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */ |
165 | 1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */ |
166 | }; |
167 | |
168 | double |
169 | __ieee754_y1 (double x) |
170 | { |
171 | double z, s, c, ss, cc, u, v, u1, u2, v1, v2, v3, z2, z4; |
172 | int32_t hx, ix, lx; |
173 | |
174 | EXTRACT_WORDS (hx, lx, x); |
175 | ix = 0x7fffffff & hx; |
176 | /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ |
177 | if (__glibc_unlikely (ix >= 0x7ff00000)) |
178 | return one / (x + x * x); |
179 | if (__glibc_unlikely ((ix | lx) == 0)) |
180 | return -1 / zero; /* -inf and divide by zero exception. */ |
181 | /* -inf and overflow exception. */; |
182 | if (__glibc_unlikely (hx < 0)) |
183 | return zero / (zero * x); |
184 | if (ix >= 0x40000000) /* |x| >= 2.0 */ |
185 | { |
186 | __sincos (x: x, sinx: &s, cosx: &c); |
187 | ss = -s - c; |
188 | cc = s - c; |
189 | if (ix < 0x7fe00000) /* make sure x+x not overflow */ |
190 | { |
191 | z = __cos (x: x + x); |
192 | if ((s * c) > zero) |
193 | cc = z / ss; |
194 | else |
195 | ss = z / cc; |
196 | } |
197 | /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) |
198 | * where x0 = x-3pi/4 |
199 | * Better formula: |
200 | * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) |
201 | * = 1/sqrt(2) * (sin(x) - cos(x)) |
202 | * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) |
203 | * = -1/sqrt(2) * (cos(x) + sin(x)) |
204 | * To avoid cancellation, use |
205 | * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) |
206 | * to compute the worse one. |
207 | */ |
208 | if (ix > 0x48000000) |
209 | z = (invsqrtpi * ss) / sqrt (x); |
210 | else |
211 | { |
212 | u = pone (x); v = qone (x); |
213 | z = invsqrtpi * (u * ss + v * cc) / sqrt (x); |
214 | } |
215 | return z; |
216 | } |
217 | if (__glibc_unlikely (ix <= 0x3c900000)) /* x < 2**-54 */ |
218 | { |
219 | z = -tpi / x; |
220 | if (isinf (z)) |
221 | __set_errno (ERANGE); |
222 | return z; |
223 | } |
224 | z = x * x; |
225 | u1 = U0[0] + z * U0[1]; z2 = z * z; |
226 | u2 = U0[2] + z * U0[3]; z4 = z2 * z2; |
227 | u = u1 + z2 * u2 + z4 * U0[4]; |
228 | v1 = one + z * V0[0]; |
229 | v2 = V0[1] + z * V0[2]; |
230 | v3 = V0[3] + z * V0[4]; |
231 | v = v1 + z2 * v2 + z4 * v3; |
232 | return (x * (u / v) + tpi * (__ieee754_j1 (x) * __ieee754_log (x) - one / x)); |
233 | } |
234 | libm_alias_finite (__ieee754_y1, __y1) |
235 | |
236 | /* For x >= 8, the asymptotic expansions of pone is |
237 | * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. |
238 | * We approximate pone by |
239 | * pone(x) = 1 + (R/S) |
240 | * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 |
241 | * S = 1 + ps0*s^2 + ... + ps4*s^10 |
242 | * and |
243 | * | pone(x)-1-R/S | <= 2 ** ( -60.06) |
244 | */ |
245 | |
246 | static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
247 | 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ |
248 | 1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */ |
249 | 1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */ |
250 | 4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */ |
251 | 3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */ |
252 | 7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */ |
253 | }; |
254 | static const double ps8[5] = { |
255 | 1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */ |
256 | 3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */ |
257 | 3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */ |
258 | 9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */ |
259 | 3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */ |
260 | }; |
261 | |
262 | static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
263 | 1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */ |
264 | 1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */ |
265 | 6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */ |
266 | 1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */ |
267 | 5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */ |
268 | 5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */ |
269 | }; |
270 | static const double ps5[5] = { |
271 | 5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */ |
272 | 9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */ |
273 | 5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */ |
274 | 7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */ |
275 | 1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */ |
276 | }; |
277 | |
278 | static const double pr3[6] = { |
279 | 3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */ |
280 | 1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */ |
281 | 3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */ |
282 | 3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */ |
283 | 9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */ |
284 | 4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */ |
285 | }; |
286 | static const double ps3[5] = { |
287 | 3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */ |
288 | 3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */ |
289 | 1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */ |
290 | 8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */ |
291 | 1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */ |
292 | }; |
293 | |
294 | static const double pr2[6] = { /* for x in [2.8570,2]=1/[0.3499,0.5] */ |
295 | 1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */ |
296 | 1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */ |
297 | 2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */ |
298 | 1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */ |
299 | 1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */ |
300 | 5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */ |
301 | }; |
302 | static const double ps2[5] = { |
303 | 2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */ |
304 | 1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */ |
305 | 2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */ |
306 | 1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */ |
307 | 8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */ |
308 | }; |
309 | |
310 | static double |
311 | pone (double x) |
312 | { |
313 | const double *p, *q; |
314 | double z, r, s, r1, r2, r3, s1, s2, s3, z2, z4; |
315 | int32_t ix; |
316 | GET_HIGH_WORD (ix, x); |
317 | ix &= 0x7fffffff; |
318 | /* ix >= 0x40000000 for all calls to this function. */ |
319 | if (ix >= 0x41b00000) |
320 | { |
321 | return one; |
322 | } |
323 | else if (ix >= 0x40200000) |
324 | { |
325 | p = pr8; q = ps8; |
326 | } |
327 | else if (ix >= 0x40122E8B) |
328 | { |
329 | p = pr5; q = ps5; |
330 | } |
331 | else if (ix >= 0x4006DB6D) |
332 | { |
333 | p = pr3; q = ps3; |
334 | } |
335 | else |
336 | { |
337 | p = pr2; q = ps2; |
338 | } |
339 | z = one / (x * x); |
340 | r1 = p[0] + z * p[1]; z2 = z * z; |
341 | r2 = p[2] + z * p[3]; z4 = z2 * z2; |
342 | r3 = p[4] + z * p[5]; |
343 | r = r1 + z2 * r2 + z4 * r3; |
344 | s1 = one + z * q[0]; |
345 | s2 = q[1] + z * q[2]; |
346 | s3 = q[3] + z * q[4]; |
347 | s = s1 + z2 * s2 + z4 * s3; |
348 | return one + r / s; |
349 | } |
350 | |
351 | |
352 | /* For x >= 8, the asymptotic expansions of qone is |
353 | * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. |
354 | * We approximate pone by |
355 | * qone(x) = s*(0.375 + (R/S)) |
356 | * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 |
357 | * S = 1 + qs1*s^2 + ... + qs6*s^12 |
358 | * and |
359 | * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) |
360 | */ |
361 | |
362 | static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
363 | 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ |
364 | -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */ |
365 | -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */ |
366 | -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */ |
367 | -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */ |
368 | -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */ |
369 | }; |
370 | static const double qs8[6] = { |
371 | 1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */ |
372 | 7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */ |
373 | 1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */ |
374 | 7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */ |
375 | 6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */ |
376 | -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */ |
377 | }; |
378 | |
379 | static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
380 | -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */ |
381 | -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */ |
382 | -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */ |
383 | -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */ |
384 | -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */ |
385 | -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */ |
386 | }; |
387 | static const double qs5[6] = { |
388 | 8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */ |
389 | 1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */ |
390 | 1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */ |
391 | 4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */ |
392 | 2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */ |
393 | -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */ |
394 | }; |
395 | |
396 | static const double qr3[6] = { |
397 | -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */ |
398 | -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */ |
399 | -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */ |
400 | -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */ |
401 | -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */ |
402 | -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */ |
403 | }; |
404 | static const double qs3[6] = { |
405 | 4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */ |
406 | 6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */ |
407 | 3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */ |
408 | 5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */ |
409 | 1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */ |
410 | -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */ |
411 | }; |
412 | |
413 | static const double qr2[6] = { /* for x in [2.8570,2]=1/[0.3499,0.5] */ |
414 | -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */ |
415 | -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */ |
416 | -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */ |
417 | -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */ |
418 | -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */ |
419 | -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */ |
420 | }; |
421 | static const double qs2[6] = { |
422 | 2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */ |
423 | 2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */ |
424 | 7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */ |
425 | 7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */ |
426 | 1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */ |
427 | -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */ |
428 | }; |
429 | |
430 | static double |
431 | qone (double x) |
432 | { |
433 | const double *p, *q; |
434 | double s, r, z, r1, r2, r3, s1, s2, s3, z2, z4, z6; |
435 | int32_t ix; |
436 | GET_HIGH_WORD (ix, x); |
437 | ix &= 0x7fffffff; |
438 | /* ix >= 0x40000000 for all calls to this function. */ |
439 | if (ix >= 0x41b00000) |
440 | { |
441 | return .375 / x; |
442 | } |
443 | else if (ix >= 0x40200000) |
444 | { |
445 | p = qr8; q = qs8; |
446 | } |
447 | else if (ix >= 0x40122E8B) |
448 | { |
449 | p = qr5; q = qs5; |
450 | } |
451 | else if (ix >= 0x4006DB6D) |
452 | { |
453 | p = qr3; q = qs3; |
454 | } |
455 | else |
456 | { |
457 | p = qr2; q = qs2; |
458 | } |
459 | z = one / (x * x); |
460 | r1 = p[0] + z * p[1]; z2 = z * z; |
461 | r2 = p[2] + z * p[3]; z4 = z2 * z2; |
462 | r3 = p[4] + z * p[5]; z6 = z4 * z2; |
463 | r = r1 + z2 * r2 + z4 * r3; |
464 | s1 = one + z * q[0]; |
465 | s2 = q[1] + z * q[2]; |
466 | s3 = q[3] + z * q[4]; |
467 | s = s1 + z2 * s2 + z4 * s3 + z6 * q[5]; |
468 | return (.375 + r / s) / x; |
469 | } |
470 | |