cf99fb40c9
Squashed commit of the following:
libs/libxx: Fix some confusing in naming. If the directory is called libxx, then the library must be libxx.a (unless perhaps LIBCXX is selected).
libs/: Fix paths in moved library directories.
libs: Brute force move of libc, libnx, and libxx to libs. Cannot yet build it in that configuration.
457 lines
16 KiB
C
457 lines
16 KiB
C
/****************************************************************************
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* libs/libc/math/lib_gamma.c
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*
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* Ported to NuttX from FreeBSD by Alan Carvalho de Assis:
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*
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunSoft, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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*
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in
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* the documentation and/or other materials provided with the
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* distribution.
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* 3. Neither the name NuttX nor the names of its contributors may be
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* used to endorse or promote products derived from this software
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* without specific prior written permission.
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*
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* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
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* FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
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* COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
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* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
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* BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS
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* OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED
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* AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
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* ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
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* POSSIBILITY OF SUCH DAMAGE.
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*
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****************************************************************************/
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/* lgamma_r(x, signgamp)
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*
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* Reentrant version of the logarithm of the Gamma function
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* with user provide pointer for the sign of Gamma(x).
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*
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* Method:
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* 1. Argument Reduction for 0 < x <= 8
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* Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
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* reduce x to a number in [1.5,2.5] by
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* lgamma(1+s) = log(s) + lgamma(s)
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* for example,
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* lgamma(7.3) = log(6.3) + lgamma(6.3)
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* = log(6.3*5.3) + lgamma(5.3)
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* = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
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* 2. Polynomial approximation of lgamma around its
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* minimun ymin=1.461632144968362245 to maintain monotonicity.
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* On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
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* Let z = x-ymin;
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* lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
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* where
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* poly(z) is a 14 degree polynomial.
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* 2. Rational approximation in the primary interval [2,3]
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* We use the following approximation:
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* s = x-2.0;
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* lgamma(x) = 0.5*s + s*P(s)/Q(s)
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* with accuracy
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* |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
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* Our algorithms are based on the following observation
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*
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* zeta(2)-1 2 zeta(3)-1 3
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* lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
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* 2 3
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*
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* where Euler = 0.5771... is the Euler constant, which is very
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* close to 0.5.
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*
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* 3. For x>=8, we have
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* lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
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* (better formula:
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* lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
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* Let z = 1/x, then we approximation
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* f(z) = lgamma(x) - (x-0.5)(log(x)-1)
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* by
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* 3 5 11
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* w = w0 + w1*z + w2*z + w3*z + ... + w6*z
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* where
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* |w - f(z)| < 2**-58.74
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*
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* 4. For negative x, since (G is gamma function)
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* -x*G(-x)*G(x) = pi/sin(pi*x),
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* we have
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* G(x) = pi/(sin(pi*x)*(-x)*G(-x))
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* since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
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* Hence, for x<0, signgam = sign(sin(pi*x)) and
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* lgamma(x) = log(|Gamma(x)|)
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* = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
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* Note: one should avoid compute pi*(-x) directly in the
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* computation of sin(pi*(-x)).
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*
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* 5. Special Cases
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* lgamma(2+s) ~ s*(1-Euler) for tiny s
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* lgamma(1) = lgamma(2) = 0
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* lgamma(x) ~ -log(|x|) for tiny x
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* lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero
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* lgamma(inf) = inf
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* lgamma(-inf) = inf (bug for bug compatible with C99!?)
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*/
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/****************************************************************************
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* Included Files
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****************************************************************************/
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#include <nuttx/config.h>
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#include <nuttx/compiler.h>
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#include <sys/types.h>
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#include <math.h>
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#ifdef CONFIG_HAVE_DOUBLE
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/****************************************************************************
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* Private Data
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****************************************************************************/
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static int g_signgam = 0;
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static const double g_pi = 3.14159265358979311600e+00; /* 0x400921FB, 0x54442D18 */
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static const double g_a0 = 7.72156649015328655494e-02; /* 0x3FB3C467, 0xE37DB0C8 */
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static const double g_a1 = 3.22467033424113591611e-01; /* 0x3FD4A34C, 0xC4A60FAD */
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static const double g_a2 = 6.73523010531292681824e-02; /* 0x3FB13E00, 0x1A5562A7 */
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static const double g_a3 = 2.05808084325167332806e-02; /* 0x3F951322, 0xAC92547B */
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static const double g_a4 = 7.38555086081402883957e-03; /* 0x3F7E404F, 0xB68FEFE8 */
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static const double g_a5 = 2.89051383673415629091e-03; /* 0x3F67ADD8, 0xCCB7926B */
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static const double g_a6 = 1.19270763183362067845e-03; /* 0x3F538A94, 0x116F3F5D */
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static const double g_a7 = 5.10069792153511336608e-04; /* 0x3F40B6C6, 0x89B99C00 */
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static const double g_a8 = 2.20862790713908385557e-04; /* 0x3F2CF2EC, 0xED10E54D */
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static const double g_a9 = 1.08011567247583939954e-04; /* 0x3F1C5088, 0x987DFB07 */
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static const double g_a10 = 2.52144565451257326939e-05; /* 0x3EFA7074, 0x428CFA52 */
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static const double g_a11 = 4.48640949618915160150e-05; /* 0x3F07858E, 0x90A45837 */
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static const double g_tc = 1.46163214496836224576e+00; /* 0x3FF762D8, 0x6356BE3F */
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static const double g_tf = -1.21486290535849611461e-01; /* 0xBFBF19B9, 0xBCC38A42 */
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/* tt = -(tail of tf) */
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static const double g_tt = -3.63867699703950536541e-18; /* 0xBC50C7CA, 0xA48A971F */
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static const double g_t0 = 4.83836122723810047042e-01; /* 0x3FDEF72B, 0xC8EE38A2 */
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static const double g_t1 = -1.47587722994593911752e-01; /* 0xBFC2E427, 0x8DC6C509 */
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static const double g_t2 = 6.46249402391333854778e-02; /* 0x3FB08B42, 0x94D5419B */
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static const double g_t3 = -3.27885410759859649565e-02; /* 0xBFA0C9A8, 0xDF35B713 */
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static const double g_t4 = 1.79706750811820387126e-02; /* 0x3F9266E7, 0x970AF9EC */
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static const double g_t5 = -1.03142241298341437450e-02; /* 0xBF851F9F, 0xBA91EC6A */
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static const double g_t6 = 6.10053870246291332635e-03; /* 0x3F78FCE0, 0xE370E344 */
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static const double g_t7 = -3.68452016781138256760e-03; /* 0xBF6E2EFF, 0xB3E914D7 */
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static const double g_t8 = 2.25964780900612472250e-03; /* 0x3F6282D3, 0x2E15C915 */
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static const double g_t9 = -1.40346469989232843813e-03; /* 0xBF56FE8E, 0xBF2D1AF1 */
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static const double g_t10 = 8.81081882437654011382e-04; /* 0x3F4CDF0C, 0xEF61A8E9 */
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static const double g_t11 = -5.38595305356740546715e-04; /* 0xBF41A610, 0x9C73E0EC */
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static const double g_t12 = 3.15632070903625950361e-04; /* 0x3F34AF6D, 0x6C0EBBF7 */
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static const double g_t13 = -3.12754168375120860518e-04; /* 0xBF347F24, 0xECC38C38 */
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static const double g_t14 = 3.35529192635519073543e-04; /* 0x3F35FD3E, 0xE8C2D3F4 */
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static const double g_u0 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */
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static const double g_u1 = 6.32827064025093366517e-01; /* 0x3FE4401E, 0x8B005DFF */
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static const double g_u2 = 1.45492250137234768737e+00; /* 0x3FF7475C, 0xD119BD6F */
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static const double g_u3 = 9.77717527963372745603e-01; /* 0x3FEF4976, 0x44EA8450 */
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static const double g_u4 = 2.28963728064692451092e-01; /* 0x3FCD4EAE, 0xF6010924 */
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static const double g_u5 = 1.33810918536787660377e-02; /* 0x3F8B678B, 0xBF2BAB09 */
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static const double g_v1 = 2.45597793713041134822e+00; /* 0x4003A5D7, 0xC2BD619C */
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static const double g_v2 = 2.12848976379893395361e+00; /* 0x40010725, 0xA42B18F5 */
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static const double g_v3 = 7.69285150456672783825e-01; /* 0x3FE89DFB, 0xE45050AF */
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static const double g_v4 = 1.04222645593369134254e-01; /* 0x3FBAAE55, 0xD6537C88 */
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static const double g_v5 = 3.21709242282423911810e-03; /* 0x3F6A5ABB, 0x57D0CF61 */
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static const double g_s0 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */
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static const double g_s1 = 2.14982415960608852501e-01; /* 0x3FCB848B, 0x36E20878 */
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static const double g_s2 = 3.25778796408930981787e-01; /* 0x3FD4D98F, 0x4F139F59 */
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static const double g_s3 = 1.46350472652464452805e-01; /* 0x3FC2BB9C, 0xBEE5F2F7 */
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static const double g_s4 = 2.66422703033638609560e-02; /* 0x3F9B481C, 0x7E939961 */
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static const double g_s5 = 1.84028451407337715652e-03; /* 0x3F5E26B6, 0x7368F239 */
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static const double g_s6 = 3.19475326584100867617e-05; /* 0x3F00BFEC, 0xDD17E945 */
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static const double g_r1 = 1.39200533467621045958e+00; /* 0x3FF645A7, 0x62C4AB74 */
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static const double g_r2 = 7.21935547567138069525e-01; /* 0x3FE71A18, 0x93D3DCDC */
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static const double g_r3 = 1.71933865632803078993e-01; /* 0x3FC601ED, 0xCCFBDF27 */
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static const double g_r4 = 1.86459191715652901344e-02; /* 0x3F9317EA, 0x742ED475 */
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static const double g_r5 = 7.77942496381893596434e-04; /* 0x3F497DDA, 0xCA41A95B */
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static const double g_r6 = 7.32668430744625636189e-06; /* 0x3EDEBAF7, 0xA5B38140 */
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static const double g_w0 = 4.18938533204672725052e-01; /* 0x3FDACFE3, 0x90C97D69 */
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static const double g_w1 = 8.33333333333329678849e-02; /* 0x3FB55555, 0x5555553B */
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static const double g_w2 = -2.77777777728775536470e-03; /* 0xBF66C16C, 0x16B02E5C */
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static const double g_w3 = 7.93650558643019558500e-04; /* 0x3F4A019F, 0x98CF38B6 */
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static const double g_w4 = -5.95187557450339963135e-04; /* 0xBF4380CB, 0x8C0FE741 */
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static const double g_w5 = 8.36339918996282139126e-04; /* 0x3F4B67BA, 0x4CDAD5D1 */
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static const double g_w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
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/****************************************************************************
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* Private Functions
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****************************************************************************/
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/* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */
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static double sin_pi(double x)
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{
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int n;
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/* spurious inexact if odd int */
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x = 2.0 * (x * 0.5 - floor(x * 0.5)); /* x mod 2.0 */
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n = (int)(x * 4.0);
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n = (n + 1) / 2;
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x -= n * 0.5f;
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x *= g_pi;
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switch (n)
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{
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default: /* case 4: */
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case 0:
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return __sin(x, 0.0, 0);
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case 1:
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return __cos(x, 0.0);
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case 2:
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return __sin(-x, 0.0, 0);
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case 3:
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return -__cos(x, 0.0);
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}
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}
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/****************************************************************************
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* Public Functions
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****************************************************************************/
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double lgamma_r(double x, int *signgamp)
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{
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union
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{
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double f;
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uint64_t i;
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} u;
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u.f = x;
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double t;
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double y;
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double z;
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double nadj = 0.0;
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double p;
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double p1;
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double p2;
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double p3;
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double q;
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double r;
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double w;
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uint32_t ix;
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int sign;
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int i;
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/* purge off +-inf, NaN, +-0, tiny and negative arguments */
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*signgamp = 1;
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sign = u.i >> 63;
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ix = u.i >> 32 & 0x7fffffff;
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if (ix >= 0x7ff00000)
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{
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return x * x;
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}
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/* |x|<2**-70, return -log(|x|) */
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if (ix < (0x3ff - 70) << 20)
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{
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if (sign)
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{
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x = -x;
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*signgamp = -1;
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}
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return -log(x);
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}
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if (sign)
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{
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x = -x;
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t = sin_pi(x);
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if (t == 0.0)
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{
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/* -integer */
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return 1.0 / (x - x);
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}
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if (t > 0.0)
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{
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*signgamp = -1;
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}
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else
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{
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t = -t;
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}
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nadj = log(g_pi / (t * x));
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}
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/* purge off 1 and 2 */
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if ((ix == 0x3ff00000 || ix == 0x40000000) && (uint32_t) u.i == 0)
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{
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r = 0;
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}
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else /* for x < 2.0 */
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{
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if (ix < 0x40000000)
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{
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if (ix <= 0x3feccccc)
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{
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/* lgamma(x) = lgamma(x+1)-log(x) */
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r = -log(x);
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if (ix >= 0x3FE76944)
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{
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y = 1.0 - x;
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i = 0;
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}
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else
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{
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if (ix >= 0x3FCDA661)
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{
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y = x - (g_tc - 1.0);
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i = 1;
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}
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else
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{
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y = x;
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i = 2;
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}
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}
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}
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else
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{
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r = 0.0;
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if (ix >= 0x3FFBB4C3)
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{
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/* [1.7316,2] */
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y = 2.0 - x;
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i = 0;
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}
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else
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{
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if (ix >= 0x3FF3B4C4)
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{
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/* [1.23,1.73] */
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y = x - g_tc;
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i = 1;
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}
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else
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{
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y = x - 1.0;
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i = 2;
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}
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}
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}
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switch (i)
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{
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case 0:
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z = y*y;
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p1 = g_a0+z*(g_a2+z*(g_a4+z*(g_a6+z*(g_a8+z*g_a10))));
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p2 = z*(g_a1+z*(g_a3+z*(g_a5+z*(g_a7+z*(g_a9+z*g_a11)))));
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p = y*p1+p2;
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r += (p-0.5*y);
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break;
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case 1:
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z = y*y;
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w = z*y;
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p1 = g_t0+w*(g_t3+w*(g_t6+w*(g_t9+w*g_t12))); /* parallel comp */
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p2 = g_t1+w*(g_t4+w*(g_t7+w*(g_t10+w*g_t13)));
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p3 = g_t2+w*(g_t5+w*(g_t8+w*(g_t11+w*g_t14)));
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p = z*p1-(g_tt-w*(p2+y*p3));
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r += g_tf + p;
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break;
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case 2:
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p1 = y*(g_u0+y*(g_u1+y*(g_u2+y*(g_u3+y*(g_u4+y*g_u5)))));
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p2 = 1.0+y*(g_v1+y*(g_v2+y*(g_v3+y*(g_v4+y*g_v5))));
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r += -0.5*y + p1/p2;
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}
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}
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else
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{
|
|
if (ix < 0x40200000)
|
|
{
|
|
/* x < 8.0 */
|
|
|
|
i = (int)x;
|
|
y = x - (double)i;
|
|
p = y*(g_s0+y*(g_s1+y*(g_s2+y*(g_s3+y*(g_s4+y*(g_s5+y*g_s6))))));
|
|
q = 1.0+y*(g_r1+y*(g_r2+y*(g_r3+y*(g_r4+y*(g_r5+y*g_r6)))));
|
|
r = 0.5*y+p/q;
|
|
z = 1.0;
|
|
|
|
/* lgamma(1+s) = log(s) + lgamma(s) */
|
|
|
|
switch (i)
|
|
{
|
|
case 7:
|
|
z *= y + 6.0; /* FALLTHRU */
|
|
case 6:
|
|
z *= y + 5.0; /* FALLTHRU */
|
|
case 5:
|
|
z *= y + 4.0; /* FALLTHRU */
|
|
case 4:
|
|
z *= y + 3.0; /* FALLTHRU */
|
|
case 3:
|
|
z *= y + 2.0; /* FALLTHRU */
|
|
r += log(z);
|
|
break;
|
|
}
|
|
}
|
|
else
|
|
{
|
|
if (ix < 0x43900000)
|
|
{
|
|
/* 8.0 <= x < 2**58 */
|
|
|
|
t = log(x);
|
|
z = 1.0 / x;
|
|
y = z * z;
|
|
w = g_w0+z*(g_w1+y*(g_w2+y*(g_w3+y*(g_w4+y*(g_w5+y*g_w6)))));
|
|
r = (x-0.5)*(t-1.0)+w;
|
|
}
|
|
else
|
|
{
|
|
/* 2**58 <= x <= inf */
|
|
|
|
r = x * (log(x) - 1.0);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
if (sign)
|
|
{
|
|
r = nadj - r;
|
|
}
|
|
|
|
return r;
|
|
}
|
|
|
|
double lgamma(double x)
|
|
{
|
|
return lgamma_r(x, &g_signgam);
|
|
}
|
|
#endif
|