libm: Port gamma() and lgamma() from FreeBSD to NuttX.
This commit is contained in:
Alan Carvalho de Assiss
Gregory Nutt
parent
814fc2049d
commit
680368b656
@ -76,6 +76,8 @@ namespace std
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using ::sqrtf;
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using ::tanf;
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using ::tanhf;
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using ::gamma;
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using ::lgamma;
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#endif
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#ifdef CONFIG_HAVE_DOUBLE
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@ -230,6 +230,13 @@ long double expl (long double x);
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#define expm1l(x) (expl(x) - 1.0)
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#endif
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#ifdef CONFIG_HAVE_DOUBLE
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double __cos(double x, double y);
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double __sin(double x, double y, int iy);
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double gamma(double x);
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double lgamma(double x);
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#endif
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float logf (float x);
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#ifdef CONFIG_HAVE_DOUBLE
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double log (double x);
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@ -37,6 +37,7 @@
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"ldexp","math.h","defined(CONFIG_HAVE_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","double","double",int"
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"ldexpf","math.h","defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH)","float","float",int"
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"ldexpl","math.h","defined(CONFIG_HAVE_LONG_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","long double","long double","int"
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"lgamma","math.h","defined(CONFIG_HAVE_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","double","double"
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"log","math.h","defined(CONFIG_HAVE_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","double","double"
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"log10","math.h","defined(CONFIG_HAVE_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","double","double"
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"log10f","math.h","defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH)","float","float"
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@ -46,6 +47,7 @@
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"log2l","math.h","defined(CONFIG_HAVE_LONG_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","long double","long double"
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"logf","math.h","defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH)","float","float"
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"logl","math.h","defined(CONFIG_HAVE_LONG_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","long double","long double"
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"gamma","math.h","defined(CONFIG_HAVE_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","double","double"
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"modf","math.h","defined(CONFIG_HAVE_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","double","double","double *"
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"modff","math.h","defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH)","float","float","float *"
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"modfl","math.h","defined(CONFIG_HAVE_LONG_DOUBLE) && (defined(CONFIG_LIBM) || defined(CONFIG_ARCH_MATH))","long double","long double","long double *"
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Can't render this file because it contains an unexpected character in line 8 and column 127.
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@ -62,6 +62,8 @@ CSRCS += lib_truncl.c
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CSRCS += lib_libexpi.c lib_libsqrtapprox.c
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CSRCS += lib_libexpif.c
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CSRCS += __cos.c __sin.c lib_gamma.c lib_lgamma.c
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# Use the C versions of some functions only if architecture specific
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# optimized versions are not provided.
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124
libc/math/__cos.c
Normal file
124
libc/math/__cos.c
Normal file
@ -0,0 +1,124 @@
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/****************************************************************************
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* libc/math/__cos.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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/* __cos( x, y )
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*
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* kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
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* Input x is assumed to be bounded by ~pi/4 in magnitude.
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* Input y is the tail of x.
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*
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* Algorithm
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* 1. Since cos(-x) = cos(x), we need only to consider positive x.
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* 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
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* 3. cos(x) is approximated by a polynomial of degree 14 on
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* [0,pi/4]
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* 4 14
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* cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
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* where the remez error is
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*
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* | 2 4 6 8 10 12 14 | -58
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* |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
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* | |
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*
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* 4 6 8 10 12 14
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* 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
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* cos(x) ~ 1 - x*x/2 + r
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* since cos(x+y) ~ cos(x) - sin(x)*y
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* ~ cos(x) - x*y,
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* a correction term is necessary in cos(x) and hence
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* cos(x+y) = 1 - (x*x/2 - (r - x*y))
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* For better accuracy, rearrange to
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* cos(x+y) ~ w + (tmp + (r-x*y))
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* where w = 1 - x*x/2 and tmp is a tiny correction term
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* (1 - x*x/2 == w + tmp exactly in infinite precision).
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* The exactness of w + tmp in infinite precision depends on w
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* and tmp having the same precision as x. If they have extra
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* precision due to compiler bugs, then the extra precision is
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* only good provided it is retained in all terms of the final
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* expression for cos(). Retention happens in all cases tested
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* under FreeBSD, so don't pessimize things by forcibly clipping
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* any extra precision in w.
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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 const double g_c1 = 4.16666666666666019037e-02; /* 0x3fa55555, 0x5555554c */
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static const double g_c2 = -1.38888888888741095749e-03; /* 0xbf56C16c, 0x16c15177 */
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static const double g_c3 = 2.48015872894767294178e-05; /* 0x3efa01a0, 0x19cb1590 */
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static const double g_c4 = -2.75573143513906633035e-07; /* 0xbe927e4e, 0x809c52ad */
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static const double g_c5 = 2.08757232129817482790e-09; /* 0x3e21ee9E, 0xbdb4b1c4 */
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static const double g_c6 = -1.13596475577881948265e-11; /* 0xbda8fae9, 0xbe8838d4 */
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/****************************************************************************
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* Public Functions
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****************************************************************************/
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double __cos(double x, double y)
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{
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double hz;
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double z;
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double r;
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double w;
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z = x * x;
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w = z * z;
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r =
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z * (g_c1 + z * (g_c2 + z * g_c3)) + w * w * (g_c4 + z * (g_c5 + z * g_c6));
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hz = 0.5 * z;
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w = 1.0 - hz;
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return w + (((1.0 - w) - hz) + (z * r - x * y));
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}
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#endif
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121
libc/math/__sin.c
Normal file
121
libc/math/__sin.c
Normal file
@ -0,0 +1,121 @@
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/****************************************************************************
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* libc/math/__cos.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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/* __sin( x, y, iy)
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*
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* kernel sin function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
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* Input x is assumed to be bounded by ~pi/4 in magnitude.
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* Input y is the tail of x.
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* Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
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*
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* Algorithm
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* 1. Since sin(-x) = -sin(x), we need only to consider positive x.
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* 2. Callers must return sin(-0) = -0 without calling here since our
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* odd polynomial is not evaluated in a way that preserves -0.
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* Callers may do the optimization sin(x) ~ x for tiny x.
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* 3. sin(x) is approximated by a polynomial of degree 13 on
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* [0,pi/4]
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* 3 13
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* sin(x) ~ x + S1*x + ... + S6*x
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* where
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*
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* |sin(x) 2 4 6 8 10 12 | -58
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* |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2
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* | x |
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*
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* 4. sin(x+y) = sin(x) + sin'(x')*y
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* ~ sin(x) + (1-x*x/2)*y
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* For better accuracy, let
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* 3 2 2 2 2
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* r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
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* then 3 2
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* sin(x) = x + (S1*x + (x *(r-y/2)+y))
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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 const double g_s1 = -1.66666666666666324348e-01; /* 0xbfc55555, 0x55555549 */
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static const double g_s2 = 8.33333333332248946124e-03; /* 0x3f811111, 0x1110f8a6 */
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static const double g_s3 = -1.98412698298579493134e-04; /* 0xbf2a01a0, 0x19c161d5 */
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static const double g_s4 = 2.75573137070700676789e-06; /* 0x3ec71de3, 0x57b1fe7d */
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static const double g_s5 = -2.50507602534068634195e-08; /* 0xbe5ae5e6, 0x8a2b9ceb */
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static const double g_s6 = 1.58969099521155010221e-10; /* 0x3de5d93a, 0x5acfd57c */
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/****************************************************************************
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* Public Functions
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****************************************************************************/
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double __sin(double x, double y, int iy)
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{
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double z;
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double r;
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double v;
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double w;
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z = x * x;
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w = z * z;
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r = g_s2 + z * (g_s3 + z * g_s4) + z * w * (g_s5 + z * g_s6);
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v = z * x;
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if (iy == 0)
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{
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return x + v * (g_s1 + z * r);
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}
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else
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{
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return x - ((z * (0.5 * y - v * r) - y) - v * g_s1);
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}
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}
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#endif
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323
libc/math/lib_gamma.c
Normal file
323
libc/math/lib_gamma.c
Normal file
@ -0,0 +1,323 @@
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/****************************************************************************
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* 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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* 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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*
|
||||
* 1. Redistributions of source code must retain the above copyright
|
||||
* notice, this list of conditions and the following disclaimer.
|
||||
* 2. Redistributions in binary form must reproduce the above copyright
|
||||
* notice, this list of conditions and the following disclaimer in
|
||||
* 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
|
||||
* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
|
||||
* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
|
||||
* FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
|
||||
* COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
|
||||
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
|
||||
* BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS
|
||||
* OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED
|
||||
* AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
|
||||
* 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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/* "A Precision Approximation of the Gamma Function" - Cornelius Lanczos (1964)
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* "Lanczos Implementation of the Gamma Function" - Paul Godfrey (2001)
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* "An Analysis of the Lanczos Gamma Approximation" - Glendon Ralph Pugh (2004)
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*
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* approximation method:
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*
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* (x - 0.5) S(x)
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* Gamma(x) = (x + g - 0.5) * ----------------
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* exp(x + g - 0.5)
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*
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* with
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* a1 a2 a3 aN
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* S(x) ~= [ a0 + ----- + ----- + ----- + ... + ----- ]
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* x + 1 x + 2 x + 3 x + N
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*
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* with a0, a1, a2, a3,.. aN constants which depend on g.
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*
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* for x < 0 the following reflection formula is used:
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*
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* Gamma(x)*Gamma(-x) = -pi/(x sin(pi x))
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*
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* most ideas and constants are from boost and python
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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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* Pre-processor Definitions
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****************************************************************************/
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#define FORCE_EVAL(x) \
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do \
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{ \
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if (sizeof(x) == sizeof(float)) \
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{ \
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volatile float __x; \
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UNUSED(__x); \
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__x = (x); \
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} \
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else if (sizeof(x) == sizeof(double)) \
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{ \
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volatile double __x; \
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UNUSED(__x); \
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__x = (x); \
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} \
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else \
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{ \
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volatile long double __x; \
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UNUSED(__x); \
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__x = (x); \
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} \
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} \
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while(0)
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#define N 12
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/****************************************************************************
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* Private Data
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****************************************************************************/
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static const double pi = 3.141592653589793238462643383279502884;
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static const double g_gmhalf = 5.524680040776729583740234375;
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static const double g_snum[N + 1] =
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{
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23531376880.410759688572007674451636754734846804940,
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42919803642.649098768957899047001988850926355848959,
|
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35711959237.355668049440185451547166705960488635843,
|
||||
17921034426.037209699919755754458931112671403265390,
|
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6039542586.3520280050642916443072979210699388420708,
|
||||
1439720407.3117216736632230727949123939715485786772,
|
||||
248874557.86205415651146038641322942321632125127801,
|
||||
31426415.585400194380614231628318205362874684987640,
|
||||
2876370.6289353724412254090516208496135991145378768,
|
||||
186056.26539522349504029498971604569928220784236328,
|
||||
8071.6720023658162106380029022722506138218516325024,
|
||||
210.82427775157934587250973392071336271166969580291,
|
||||
2.5066282746310002701649081771338373386264310793408,
|
||||
};
|
||||
|
||||
static const double g_sden[N + 1] =
|
||||
{
|
||||
0, 39916800, 120543840, 150917976, 105258076, 45995730, 13339535,
|
||||
2637558, 357423, 32670, 1925, 66, 1,
|
||||
};
|
||||
|
||||
/* n! for small integer n */
|
||||
|
||||
static const double g_fact[] =
|
||||
{
|
||||
1, 1, 2, 6, 24, 120, 720, 5040.0, 40320.0, 362880.0, 3628800.0, 39916800.0,
|
||||
479001600.0, 6227020800.0, 87178291200.0, 1307674368000.0, 20922789888000.0,
|
||||
355687428096000.0, 6402373705728000.0, 121645100408832000.0,
|
||||
2432902008176640000.0, 51090942171709440000.0, 1124000727777607680000.0,
|
||||
};
|
||||
|
||||
/* S(x) rational function for positive x */
|
||||
|
||||
/****************************************************************************
|
||||
* Private Functions
|
||||
****************************************************************************/
|
||||
|
||||
/* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */
|
||||
|
||||
static double sinpi(double x)
|
||||
{
|
||||
int n;
|
||||
|
||||
/* argument reduction: x = |x| mod 2 */
|
||||
/* spurious inexact when x is odd int */
|
||||
|
||||
x = x * 0.5;
|
||||
x = 2 * (x - floor(x));
|
||||
|
||||
/* reduce x into [-.25,.25] */
|
||||
|
||||
n = 4 * x;
|
||||
n = (n + 1) / 2;
|
||||
x -= n * 0.5;
|
||||
|
||||
x *= pi;
|
||||
switch (n)
|
||||
{
|
||||
default: /* case 4 */
|
||||
case 0:
|
||||
return __sin(x, 0, 0);
|
||||
|
||||
case 1:
|
||||
return __cos(x, 0);
|
||||
|
||||
case 2:
|
||||
return __sin(-x, 0, 0);
|
||||
|
||||
case 3:
|
||||
return -__cos(x, 0);
|
||||
}
|
||||
}
|
||||
|
||||
static double s(double x)
|
||||
{
|
||||
double num = 0;
|
||||
double den = 0;
|
||||
int i;
|
||||
|
||||
/* to avoid overflow handle large x differently */
|
||||
|
||||
if (x < 8)
|
||||
{
|
||||
for (i = N; i >= 0; i--)
|
||||
{
|
||||
num = num * x + g_snum[i];
|
||||
den = den * x + g_sden[i];
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
for (i = 0; i <= N; i++)
|
||||
{
|
||||
num = num / x + g_snum[i];
|
||||
den = den / x + g_sden[i];
|
||||
}
|
||||
}
|
||||
|
||||
return num/den;
|
||||
}
|
||||
|
||||
/****************************************************************************
|
||||
* Public Functions
|
||||
****************************************************************************/
|
||||
|
||||
double tgamma(double x)
|
||||
{
|
||||
union
|
||||
{
|
||||
double f;
|
||||
uint64_t i;
|
||||
} u;
|
||||
u.f = x;
|
||||
|
||||
double absx;
|
||||
double y;
|
||||
double dy;
|
||||
double z;
|
||||
double r;
|
||||
uint32_t ix = u.i >> 32 & 0x7fffffff;
|
||||
int sign = u.i >> 63;
|
||||
|
||||
/* special cases */
|
||||
|
||||
if (ix >= 0x7ff00000)
|
||||
{
|
||||
/* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */
|
||||
|
||||
return x + INFINITY;
|
||||
}
|
||||
|
||||
if (ix < (0x3ff - 54) << 20)
|
||||
{
|
||||
/* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */
|
||||
return 1 / x;
|
||||
}
|
||||
|
||||
/* integer arguments */
|
||||
/* raise inexact when non-integer */
|
||||
|
||||
if (x == floor(x))
|
||||
{
|
||||
if (sign)
|
||||
{
|
||||
return 0 / 0.0;
|
||||
}
|
||||
|
||||
if (x <= sizeof g_fact / sizeof *g_fact)
|
||||
{
|
||||
return g_fact[(int)x - 1];
|
||||
}
|
||||
}
|
||||
|
||||
/* x >= 172: tgamma(x)=inf with overflow */
|
||||
/* x =< -184: tgamma(x)=+-0 with underflow */
|
||||
|
||||
if (ix >= 0x40670000)
|
||||
{
|
||||
/* |x| >= 184 */
|
||||
|
||||
if (sign)
|
||||
{
|
||||
FORCE_EVAL((float)(0x1p-126 / x));
|
||||
if (floor(x) * 0.5 == floor(x * 0.5))
|
||||
{
|
||||
return 0;
|
||||
}
|
||||
return -0.0;
|
||||
}
|
||||
|
||||
x *= 0x1p1023;
|
||||
return x;
|
||||
}
|
||||
|
||||
absx = sign ? -x : x;
|
||||
|
||||
/* handle the error of x + g - 0.5 */
|
||||
|
||||
y = absx + g_gmhalf;
|
||||
if (absx > g_gmhalf)
|
||||
{
|
||||
dy = y - absx;
|
||||
dy -= g_gmhalf;
|
||||
}
|
||||
else
|
||||
{
|
||||
dy = y - g_gmhalf;
|
||||
dy -= absx;
|
||||
}
|
||||
|
||||
z = absx - 0.5;
|
||||
r = s(absx) * exp(-y);
|
||||
if (x < 0)
|
||||
{
|
||||
/* reflection formula for negative x */
|
||||
/* sinpi(absx) is not 0, integers are already handled */
|
||||
|
||||
r = -pi / (sinpi(absx) * absx * r);
|
||||
dy = -dy;
|
||||
z = -z;
|
||||
}
|
||||
|
||||
r += dy * (g_gmhalf + 0.5) * r / y;
|
||||
z = pow(y, 0.5 * z);
|
||||
y = r * z * z;
|
||||
|
||||
return y;
|
||||
}
|
||||
|
||||
double gamma(double x)
|
||||
{
|
||||
return tgamma(x);
|
||||
}
|
||||
#endif
|
||||
456
libc/math/lib_lgamma.c
Normal file
456
libc/math/lib_lgamma.c
Normal file
@ -0,0 +1,456 @@
|
||||
/****************************************************************************
|
||||
* libc/math/lib_gamma.c
|
||||
*
|
||||
* Ported to NuttX from FreeBSD by Alan Carvalho de Assis:
|
||||
*
|
||||
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
*
|
||||
* Redistribution and use in source and binary forms, with or without
|
||||
* modification, are permitted provided that the following conditions
|
||||
* are met:
|
||||
*
|
||||
* 1. Redistributions of source code must retain the above copyright
|
||||
* notice, this list of conditions and the following disclaimer.
|
||||
* 2. Redistributions in binary form must reproduce the above copyright
|
||||
* notice, this list of conditions and the following disclaimer in
|
||||
* the documentation and/or other materials provided with the
|
||||
* distribution.
|
||||
* 3. Neither the name NuttX nor the names of its contributors may be
|
||||
* used to endorse or promote products derived from this software
|
||||
* without specific prior written permission.
|
||||
*
|
||||
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
|
||||
* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
|
||||
* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
|
||||
* FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
|
||||
* COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
|
||||
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
|
||||
* BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS
|
||||
* OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED
|
||||
* AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
|
||||
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
|
||||
* ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
|
||||
* POSSIBILITY OF SUCH DAMAGE.
|
||||
*
|
||||
****************************************************************************/
|
||||
|
||||
/* lgamma_r(x, signgamp)
|
||||
*
|
||||
* Reentrant version of the logarithm of the Gamma function
|
||||
* with user provide pointer for the sign of Gamma(x).
|
||||
*
|
||||
* Method:
|
||||
* 1. Argument Reduction for 0 < x <= 8
|
||||
* Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
|
||||
* reduce x to a number in [1.5,2.5] by
|
||||
* lgamma(1+s) = log(s) + lgamma(s)
|
||||
* for example,
|
||||
* lgamma(7.3) = log(6.3) + lgamma(6.3)
|
||||
* = log(6.3*5.3) + lgamma(5.3)
|
||||
* = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
|
||||
* 2. Polynomial approximation of lgamma around its
|
||||
* minimun ymin=1.461632144968362245 to maintain monotonicity.
|
||||
* On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
|
||||
* Let z = x-ymin;
|
||||
* lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
|
||||
* where
|
||||
* poly(z) is a 14 degree polynomial.
|
||||
* 2. Rational approximation in the primary interval [2,3]
|
||||
* We use the following approximation:
|
||||
* s = x-2.0;
|
||||
* lgamma(x) = 0.5*s + s*P(s)/Q(s)
|
||||
* with accuracy
|
||||
* |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
|
||||
* Our algorithms are based on the following observation
|
||||
*
|
||||
* zeta(2)-1 2 zeta(3)-1 3
|
||||
* lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
|
||||
* 2 3
|
||||
*
|
||||
* where Euler = 0.5771... is the Euler constant, which is very
|
||||
* close to 0.5.
|
||||
*
|
||||
* 3. For x>=8, we have
|
||||
* lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
|
||||
* (better formula:
|
||||
* lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
|
||||
* Let z = 1/x, then we approximation
|
||||
* f(z) = lgamma(x) - (x-0.5)(log(x)-1)
|
||||
* by
|
||||
* 3 5 11
|
||||
* w = w0 + w1*z + w2*z + w3*z + ... + w6*z
|
||||
* where
|
||||
* |w - f(z)| < 2**-58.74
|
||||
*
|
||||
* 4. For negative x, since (G is gamma function)
|
||||
* -x*G(-x)*G(x) = pi/sin(pi*x),
|
||||
* we have
|
||||
* G(x) = pi/(sin(pi*x)*(-x)*G(-x))
|
||||
* since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
|
||||
* Hence, for x<0, signgam = sign(sin(pi*x)) and
|
||||
* lgamma(x) = log(|Gamma(x)|)
|
||||
* = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
|
||||
* Note: one should avoid compute pi*(-x) directly in the
|
||||
* computation of sin(pi*(-x)).
|
||||
*
|
||||
* 5. Special Cases
|
||||
* lgamma(2+s) ~ s*(1-Euler) for tiny s
|
||||
* lgamma(1) = lgamma(2) = 0
|
||||
* lgamma(x) ~ -log(|x|) for tiny x
|
||||
* lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero
|
||||
* lgamma(inf) = inf
|
||||
* lgamma(-inf) = inf (bug for bug compatible with C99!?)
|
||||
*/
|
||||
|
||||
/****************************************************************************
|
||||
* Included Files
|
||||
****************************************************************************/
|
||||
|
||||
#include <nuttx/config.h>
|
||||
#include <nuttx/compiler.h>
|
||||
|
||||
#include <sys/types.h>
|
||||
#include <math.h>
|
||||
|
||||
#ifdef CONFIG_HAVE_DOUBLE
|
||||
|
||||
/****************************************************************************
|
||||
* Private Data
|
||||
****************************************************************************/
|
||||
|
||||
static int g_signgam = 0;
|
||||
|
||||
static const double g_pi = 3.14159265358979311600e+00; /* 0x400921FB, 0x54442D18 */
|
||||
static const double g_a0 = 7.72156649015328655494e-02; /* 0x3FB3C467, 0xE37DB0C8 */
|
||||
static const double g_a1 = 3.22467033424113591611e-01; /* 0x3FD4A34C, 0xC4A60FAD */
|
||||
static const double g_a2 = 6.73523010531292681824e-02; /* 0x3FB13E00, 0x1A5562A7 */
|
||||
static const double g_a3 = 2.05808084325167332806e-02; /* 0x3F951322, 0xAC92547B */
|
||||
static const double g_a4 = 7.38555086081402883957e-03; /* 0x3F7E404F, 0xB68FEFE8 */
|
||||
static const double g_a5 = 2.89051383673415629091e-03; /* 0x3F67ADD8, 0xCCB7926B */
|
||||
static const double g_a6 = 1.19270763183362067845e-03; /* 0x3F538A94, 0x116F3F5D */
|
||||
static const double g_a7 = 5.10069792153511336608e-04; /* 0x3F40B6C6, 0x89B99C00 */
|
||||
static const double g_a8 = 2.20862790713908385557e-04; /* 0x3F2CF2EC, 0xED10E54D */
|
||||
static const double g_a9 = 1.08011567247583939954e-04; /* 0x3F1C5088, 0x987DFB07 */
|
||||
static const double g_a10 = 2.52144565451257326939e-05; /* 0x3EFA7074, 0x428CFA52 */
|
||||
static const double g_a11 = 4.48640949618915160150e-05; /* 0x3F07858E, 0x90A45837 */
|
||||
static const double g_tc = 1.46163214496836224576e+00; /* 0x3FF762D8, 0x6356BE3F */
|
||||
static const double g_tf = -1.21486290535849611461e-01; /* 0xBFBF19B9, 0xBCC38A42 */
|
||||
|
||||
/* tt = -(tail of tf) */
|
||||
|
||||
static const double g_tt = -3.63867699703950536541e-18; /* 0xBC50C7CA, 0xA48A971F */
|
||||
static const double g_t0 = 4.83836122723810047042e-01; /* 0x3FDEF72B, 0xC8EE38A2 */
|
||||
static const double g_t1 = -1.47587722994593911752e-01; /* 0xBFC2E427, 0x8DC6C509 */
|
||||
static const double g_t2 = 6.46249402391333854778e-02; /* 0x3FB08B42, 0x94D5419B */
|
||||
static const double g_t3 = -3.27885410759859649565e-02; /* 0xBFA0C9A8, 0xDF35B713 */
|
||||
static const double g_t4 = 1.79706750811820387126e-02; /* 0x3F9266E7, 0x970AF9EC */
|
||||
static const double g_t5 = -1.03142241298341437450e-02; /* 0xBF851F9F, 0xBA91EC6A */
|
||||
static const double g_t6 = 6.10053870246291332635e-03; /* 0x3F78FCE0, 0xE370E344 */
|
||||
static const double g_t7 = -3.68452016781138256760e-03; /* 0xBF6E2EFF, 0xB3E914D7 */
|
||||
static const double g_t8 = 2.25964780900612472250e-03; /* 0x3F6282D3, 0x2E15C915 */
|
||||
static const double g_t9 = -1.40346469989232843813e-03; /* 0xBF56FE8E, 0xBF2D1AF1 */
|
||||
static const double g_t10 = 8.81081882437654011382e-04; /* 0x3F4CDF0C, 0xEF61A8E9 */
|
||||
static const double g_t11 = -5.38595305356740546715e-04; /* 0xBF41A610, 0x9C73E0EC */
|
||||
static const double g_t12 = 3.15632070903625950361e-04; /* 0x3F34AF6D, 0x6C0EBBF7 */
|
||||
static const double g_t13 = -3.12754168375120860518e-04; /* 0xBF347F24, 0xECC38C38 */
|
||||
static const double g_t14 = 3.35529192635519073543e-04; /* 0x3F35FD3E, 0xE8C2D3F4 */
|
||||
static const double g_u0 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */
|
||||
static const double g_u1 = 6.32827064025093366517e-01; /* 0x3FE4401E, 0x8B005DFF */
|
||||
static const double g_u2 = 1.45492250137234768737e+00; /* 0x3FF7475C, 0xD119BD6F */
|
||||
static const double g_u3 = 9.77717527963372745603e-01; /* 0x3FEF4976, 0x44EA8450 */
|
||||
static const double g_u4 = 2.28963728064692451092e-01; /* 0x3FCD4EAE, 0xF6010924 */
|
||||
static const double g_u5 = 1.33810918536787660377e-02; /* 0x3F8B678B, 0xBF2BAB09 */
|
||||
static const double g_v1 = 2.45597793713041134822e+00; /* 0x4003A5D7, 0xC2BD619C */
|
||||
static const double g_v2 = 2.12848976379893395361e+00; /* 0x40010725, 0xA42B18F5 */
|
||||
static const double g_v3 = 7.69285150456672783825e-01; /* 0x3FE89DFB, 0xE45050AF */
|
||||
static const double g_v4 = 1.04222645593369134254e-01; /* 0x3FBAAE55, 0xD6537C88 */
|
||||
static const double g_v5 = 3.21709242282423911810e-03; /* 0x3F6A5ABB, 0x57D0CF61 */
|
||||
static const double g_s0 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */
|
||||
static const double g_s1 = 2.14982415960608852501e-01; /* 0x3FCB848B, 0x36E20878 */
|
||||
static const double g_s2 = 3.25778796408930981787e-01; /* 0x3FD4D98F, 0x4F139F59 */
|
||||
static const double g_s3 = 1.46350472652464452805e-01; /* 0x3FC2BB9C, 0xBEE5F2F7 */
|
||||
static const double g_s4 = 2.66422703033638609560e-02; /* 0x3F9B481C, 0x7E939961 */
|
||||
static const double g_s5 = 1.84028451407337715652e-03; /* 0x3F5E26B6, 0x7368F239 */
|
||||
static const double g_s6 = 3.19475326584100867617e-05; /* 0x3F00BFEC, 0xDD17E945 */
|
||||
static const double g_r1 = 1.39200533467621045958e+00; /* 0x3FF645A7, 0x62C4AB74 */
|
||||
static const double g_r2 = 7.21935547567138069525e-01; /* 0x3FE71A18, 0x93D3DCDC */
|
||||
static const double g_r3 = 1.71933865632803078993e-01; /* 0x3FC601ED, 0xCCFBDF27 */
|
||||
static const double g_r4 = 1.86459191715652901344e-02; /* 0x3F9317EA, 0x742ED475 */
|
||||
static const double g_r5 = 7.77942496381893596434e-04; /* 0x3F497DDA, 0xCA41A95B */
|
||||
static const double g_r6 = 7.32668430744625636189e-06; /* 0x3EDEBAF7, 0xA5B38140 */
|
||||
static const double g_w0 = 4.18938533204672725052e-01; /* 0x3FDACFE3, 0x90C97D69 */
|
||||
static const double g_w1 = 8.33333333333329678849e-02; /* 0x3FB55555, 0x5555553B */
|
||||
static const double g_w2 = -2.77777777728775536470e-03; /* 0xBF66C16C, 0x16B02E5C */
|
||||
static const double g_w3 = 7.93650558643019558500e-04; /* 0x3F4A019F, 0x98CF38B6 */
|
||||
static const double g_w4 = -5.95187557450339963135e-04; /* 0xBF4380CB, 0x8C0FE741 */
|
||||
static const double g_w5 = 8.36339918996282139126e-04; /* 0x3F4B67BA, 0x4CDAD5D1 */
|
||||
static const double g_w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
|
||||
|
||||
/****************************************************************************
|
||||
* Private Functions
|
||||
****************************************************************************/
|
||||
|
||||
/* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */
|
||||
|
||||
static double sin_pi(double x)
|
||||
{
|
||||
int n;
|
||||
|
||||
/* spurious inexact if odd int */
|
||||
|
||||
x = 2.0 * (x * 0.5 - floor(x * 0.5)); /* x mod 2.0 */
|
||||
|
||||
n = (int)(x * 4.0);
|
||||
n = (n + 1) / 2;
|
||||
x -= n * 0.5f;
|
||||
x *= g_pi;
|
||||
|
||||
switch (n)
|
||||
{
|
||||
default: /* case 4: */
|
||||
case 0:
|
||||
return __sin(x, 0.0, 0);
|
||||
|
||||
case 1:
|
||||
return __cos(x, 0.0);
|
||||
|
||||
case 2:
|
||||
return __sin(-x, 0.0, 0);
|
||||
|
||||
case 3:
|
||||
return -__cos(x, 0.0);
|
||||
}
|
||||
}
|
||||
|
||||
/****************************************************************************
|
||||
* Public Functions
|
||||
****************************************************************************/
|
||||
|
||||
double lgamma_r(double x, int *signgamp)
|
||||
{
|
||||
union
|
||||
{
|
||||
double f;
|
||||
uint64_t i;
|
||||
} u;
|
||||
u.f = x;
|
||||
|
||||
double t;
|
||||
double y;
|
||||
double z;
|
||||
double nadj;
|
||||
double p;
|
||||
double p1;
|
||||
double p2;
|
||||
double p3;
|
||||
double q;
|
||||
double r;
|
||||
double w;
|
||||
uint32_t ix;
|
||||
int sign;
|
||||
int i;
|
||||
|
||||
/* purge off +-inf, NaN, +-0, tiny and negative arguments */
|
||||
|
||||
*signgamp = 1;
|
||||
sign = u.i >> 63;
|
||||
|
||||
ix = u.i >> 32 & 0x7fffffff;
|
||||
if (ix >= 0x7ff00000)
|
||||
{
|
||||
return x * x;
|
||||
}
|
||||
|
||||
/* |x|<2**-70, return -log(|x|) */
|
||||
|
||||
if (ix < (0x3ff - 70) << 20)
|
||||
{
|
||||
if (sign)
|
||||
{
|
||||
x = -x;
|
||||
*signgamp = -1;
|
||||
}
|
||||
return -log(x);
|
||||
}
|
||||
|
||||
if (sign)
|
||||
{
|
||||
x = -x;
|
||||
t = sin_pi(x);
|
||||
|
||||
if (t == 0.0)
|
||||
{
|
||||
/* -integer */
|
||||
return 1.0 / (x - x);
|
||||
}
|
||||
|
||||
if (t > 0.0)
|
||||
{
|
||||
*signgamp = -1;
|
||||
}
|
||||
else
|
||||
{
|
||||
t = -t;
|
||||
}
|
||||
|
||||
nadj = log(g_pi / (t * x));
|
||||
}
|
||||
|
||||
/* purge off 1 and 2 */
|
||||
|
||||
if ((ix == 0x3ff00000 || ix == 0x40000000) && (uint32_t) u.i == 0)
|
||||
{
|
||||
r = 0;
|
||||
}
|
||||
else /* for x < 2.0 */
|
||||
{
|
||||
if (ix < 0x40000000)
|
||||
{
|
||||
if (ix <= 0x3feccccc)
|
||||
{
|
||||
/* lgamma(x) = lgamma(x+1)-log(x) */
|
||||
|
||||
r = -log(x);
|
||||
|
||||
if (ix >= 0x3FE76944)
|
||||
{
|
||||
y = 1.0 - x;
|
||||
i = 0;
|
||||
}
|
||||
else
|
||||
{
|
||||
if (ix >= 0x3FCDA661)
|
||||
{
|
||||
y = x - (g_tc - 1.0);
|
||||
i = 1;
|
||||
}
|
||||
else
|
||||
{
|
||||
y = x;
|
||||
i = 2;
|
||||
}
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
r = 0.0;
|
||||
|
||||
if (ix >= 0x3FFBB4C3)
|
||||
{
|
||||
/* [1.7316,2] */
|
||||
y = 2.0 - x;
|
||||
i = 0;
|
||||
}
|
||||
else
|
||||
{
|
||||
if (ix >= 0x3FF3B4C4)
|
||||
{
|
||||
/* [1.23,1.73] */
|
||||
y = x - g_tc;
|
||||
i = 1;
|
||||
}
|
||||
else
|
||||
{
|
||||
y = x - 1.0;
|
||||
i = 2;
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
switch (i)
|
||||
{
|
||||
case 0:
|
||||
z = y*y;
|
||||
p1 = g_a0+z*(g_a2+z*(g_a4+z*(g_a6+z*(g_a8+z*g_a10))));
|
||||
p2 = z*(g_a1+z*(g_a3+z*(g_a5+z*(g_a7+z*(g_a9+z*g_a11)))));
|
||||
p = y*p1+p2;
|
||||
r += (p-0.5*y);
|
||||
break;
|
||||
|
||||
case 1:
|
||||
z = y*y;
|
||||
w = z*y;
|
||||
p1 = g_t0+w*(g_t3+w*(g_t6+w*(g_t9+w*g_t12))); /* parallel comp */
|
||||
p2 = g_t1+w*(g_t4+w*(g_t7+w*(g_t10+w*g_t13)));
|
||||
p3 = g_t2+w*(g_t5+w*(g_t8+w*(g_t11+w*g_t14)));
|
||||
p = z*p1-(g_tt-w*(p2+y*p3));
|
||||
r += g_tf + p;
|
||||
break;
|
||||
|
||||
case 2:
|
||||
p1 = y*(g_u0+y*(g_u1+y*(g_u2+y*(g_u3+y*(g_u4+y*g_u5)))));
|
||||
p2 = 1.0+y*(g_v1+y*(g_v2+y*(g_v3+y*(g_v4+y*g_v5))));
|
||||
r += -0.5*y + p1/p2;
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
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
|
||||
Loading…
Reference in New Issue
Block a user