Lines Matching +full:a +full:- +full:z
1 /*-
2 * SPDX-License-Identifier: BSD-3-Clause
21 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
64 long double a; member
72 static const volatile double tiny = 1e-300;
77 * equal-ripples:
79 * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x))
81 * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid
82 * premature round-off.
84 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
88 * The following is a decomposition of 0.5 * (log(2*pi) - 1) into the
93 ln2pi_hiu = LD80C(0xd680000000000000, -2, 4.18945312500000000000e-01L),
94 ln2pi_lou = LD80C(0xe379b414b596d687, -18, -6.77929532725821967032e-06L);
99 Pa0u = LD80C(0xaaaaaaaaaaaaaaaa, -4, 8.33333333333333333288e-02L),
100 Pa1u = LD80C(0xb60b60b60b5fcd59, -9, -2.77777777777776516326e-03L),
101 Pa2u = LD80C(0xd00d00cffbb47014, -11, 7.93650793635429639018e-04L),
102 Pa3u = LD80C(0x9c09c07c0805343e, -11, -5.95238087960599252215e-04L),
103 Pa4u = LD80C(0xdca8d31f8e6e5e8f, -11, 8.41749082509607342883e-04L),
104 Pa5u = LD80C(0xfb4d4289632f1638, -10, -1.91728055205541624556e-03L),
105 Pa6u = LD80C(0xd15a4ba04078d3f8, -8, 6.38893788027752396194e-03L),
106 Pa7u = LD80C(0xe877283110bcad95, -6, -2.83771309846297590312e-02L),
107 Pa8u = LD80C(0x8da97eed13717af8, -3, 1.38341887683837576925e-01L),
108 Pa9u = LD80C(0xf093b1c1584e30ce, -2, -4.69876818515470146031e-01L);
123 long double p, z, thi, tlo, xhi, xlo; in large_gam() local
127 z = 1 / (x * x); in large_gam()
128 p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 + in large_gam()
129 z * (Pa6 + z * (Pa7 + z * (Pa8 + z * Pa9)))))))); in large_gam()
133 u.a -= 1; in large_gam()
135 /* Split (x - 0.5) in high and low parts. */ in large_gam()
136 x -= 0.5L; in large_gam()
138 xlo = x - xhi; in large_gam()
140 /* Compute t = (x-.5)*(log(x)-1) in extra precision. */ in large_gam()
141 thi = xhi * u.a; in large_gam()
142 tlo = xlo * u.a + x * u.b; in large_gam()
147 u.a = ln2pi_hi + tlo; in large_gam()
148 u.a += thi; in large_gam()
149 u.b = thi - u.a; in large_gam()
156 * [1.066.., 2.066..] accurate to 4.25e-19.
158 * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated.
161 a0_hiu = LD80C(0xe2b6e4153a57746c, -1, 8.85603194410888700265e-01L),
162 a0_lou = LD80C(0x851566d40f32c76d, -66, 1.40907742727049706207e-20L);
167 P0u = LD80C(0xdb629fb9bbdc1c1d, -2, 4.28486815855585429733e-01L),
168 P1u = LD80C(0xe6f4f9f5641aa6be, -3, 2.25543885805587730552e-01L),
169 P2u = LD80C(0xead1bd99fdaf7cc1, -6, 2.86644652514293482381e-02L),
170 P3u = LD80C(0x9ccc8b25838ab1e0, -8, 4.78512567772456362048e-03L),
171 P4u = LD80C(0x8f0c4383ef9ce72a, -9, 2.18273781132301146458e-03L),
172 P5u = LD80C(0xe732ab2c0a2778da, -13, 2.20487522485636008928e-04L),
173 P6u = LD80C(0xce70b27ca822b297, -16, 2.46095923774929264284e-05L),
174 P7u = LD80C(0xa309e2e16fb63663, -19, 2.42946473022376182921e-06L),
175 P8u = LD80C(0xaf9c110efb2c633d, -23, 1.63549217667765869987e-07L),
176 Q1u = LD80C(0xd4d7422719f48f15, -1, 8.31409582658993993626e-01L),
177 Q2u = LD80C(0xe13138ea404f1268, -5, -5.49785826915643198508e-02L),
178 Q3u = LD80C(0xd1c6cc91989352c0, -4, -1.02429960435139887683e-01L),
179 Q4u = LD80C(0xa7e9435a84445579, -7, 1.02484853505908820524e-02L),
180 Q5u = LD80C(0x83c7c34db89b7bda, -8, 4.02161632832052872697e-03L),
181 Q6u = LD80C(0xbed06bf6e1c14e5b, -11, -7.27898206351223022157e-04L),
182 Q7u = LD80C(0xef05bf841d4504c0, -18, 7.12342421869453515194e-06L),
183 Q8u = LD80C(0xf348d08a1ff53cb1, -19, 3.62522053809474067060e-06L);
203 ratfun_gam(long double z, long double c) in ratfun_gam() argument
208 q = 1 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 + in ratfun_gam()
209 z * (Q6 + z * (Q7 + z * Q8))))))); in ratfun_gam()
210 p = P0 + z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 + in ratfun_gam()
211 z * (P6 + z * (P7 + z * P8))))))); in ratfun_gam()
214 /* Split z into high and low parts. */ in ratfun_gam()
215 thi = (float)z; in ratfun_gam()
216 tlo = (z - thi) + c; in ratfun_gam()
217 tlo *= (thi + z); in ratfun_gam()
219 /* Split (z+c)^2 into high and low parts. */ in ratfun_gam()
223 tlo += (q - thi); in ratfun_gam()
226 r.a = (float)p; in ratfun_gam()
227 r.b = p - r.a; in ratfun_gam()
230 thi *= r.a; /* t = (z+c)^2*(P/Q) */ in ratfun_gam()
231 r.a = (float)(thi + a0_hi); in ratfun_gam()
232 r.b = ((a0_hi - r.a) + thi) + tlo; in ratfun_gam()
239 * 2.066124]. Use a rational approximation centered at the minimum
246 xm1u = LD80C(0xec5b0c6ad7c7edc3, -2, 4.61632144968362341254e-01L);
250 left = -0.3955078125; /* left boundary for rat. approx */
258 y = x - 1; in small_gam()
261 yy = ratfun_gam(y - x0, 0); in small_gam()
262 return (yy.a + yy.b); in small_gam()
265 r.a = (float)y; in small_gam()
266 yy.a = r.a - 1; in small_gam()
267 y = y - 1 ; in small_gam()
268 r.b = yy.b = y - yy.a; in small_gam()
271 for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) { in small_gam()
272 t = r.a * yy.a; in small_gam()
273 r.b = r.a * yy.b + y * r.b; in small_gam()
274 r.a = (float)t; in small_gam()
275 r.b += (t - r.a); in small_gam()
279 yy = ratfun_gam(y - x0, 0); in small_gam()
280 y = r.b * (yy.a + yy.b) + r.a * yy.b; in small_gam()
281 y += yy.a * r.a; in small_gam()
295 d = (t + x) * (x - t); in smaller_gam()
298 xlo = x - xhi; in smaller_gam()
301 t = 1 - x0; in smaller_gam()
303 d = 1 - x0; in smaller_gam()
304 d -= t; in smaller_gam()
309 xlo = x - xhi; in smaller_gam()
310 t = x - x0; in smaller_gam()
311 d = - x0 - t; in smaller_gam()
316 d = (float)(r.a / x); in smaller_gam()
317 r.a -= d * xhi; in smaller_gam()
318 r.a -= d * xlo; in smaller_gam()
319 r.a += r.b; in smaller_gam()
321 return (d + r.a / x); in smaller_gam()
338 long double y, z; in neg_gam() local
342 return ((x - x) / zero); in neg_gam()
344 z = y - x; in neg_gam()
345 if (z > 0.5) in neg_gam()
346 z = 1 - z; in neg_gam()
350 sgn = -1; in neg_gam()
352 if (z < 0.25) in neg_gam()
353 z = sinpil(z); in neg_gam()
355 z = cospil(0.5 - z); in neg_gam()
357 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ in neg_gam()
358 if (x < -1753) { in neg_gam()
360 if (x < -1760) in neg_gam()
364 return (sgn < 0 ? -y : y); in neg_gam()
368 y = 1 - x; in neg_gam()
369 if (1 - y == x) in neg_gam()
371 else /* 1-x is inexact */ in neg_gam()
372 y = - x * tgammal(-x); in neg_gam()
374 if (sgn < 0) y = -y; in neg_gam()
375 return (pi / (y * z)); in neg_gam()
378 * xmax comes from lgamma(xmax) - emax * log(2) = 0.
384 * iota is a sloppy threshold to isolate x = 0.
387 static const double iota = 0x1p-116;
400 RETURNI(__exp__D(u.a, u.b)); in tgammal()
409 if (x > -iota) { in tgammal()
411 u.a = 1 - tiny; /* raise inexact */ in tgammal()
416 RETURNI(x - x); /* x is NaN or -Inf */ in tgammal()