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libc/newlib/libm/machine/spu/headers/lgammad2.h

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/* -------------------------------------------------------------- */
/* (C)Copyright 2007,2008, */
/* International Business Machines Corporation */
/* All Rights Reserved. */
/* */
/* Redistribution and use in source and binary forms, with or */
/* without modification, are permitted provided that the */
/* following conditions are met: */
/* */
/* - Redistributions of source code must retain the above copyright*/
/* notice, this list of conditions and the following disclaimer. */
/* */
/* - 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. */
/* */
/* - Neither the name of IBM Corporation 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, */
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/* 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. */
/* -------------------------------------------------------------- */
/* PROLOG END TAG zYx */
#ifdef __SPU__
#ifndef _LGAMMAD2_H_
#define _LGAMMAD2_H_ 1
#include <spu_intrinsics.h>
#include "divd2.h"
#include "recipd2.h"
#include "logd2.h"
#include "sind2.h"
#include "truncd2.h"
/*
* FUNCTION
* vector double _lgammad2(vector double x) - Natural Log of Gamma Function
*
* DESCRIPTION
* _lgammad2 calculates the natural logarithm of the absolute value of the gamma
* function for the corresponding elements of the input vector.
*
* C99 Special Cases:
* lgamma(0) returns +infinite
* lgamma(1) returns +0
* lgamma(2) returns +0
* lgamma(negative integer) returns +infinite
* lgamma(+infinite) returns +infinite
* lgamma(-infinite) returns +infinite
*
* Other Cases:
* lgamma(Nan) returns Nan
* lgamma(Denorm) treated as lgamma(0) and returns +infinite
*
*/
#define PI 3.1415926535897932384626433832795028841971693993751058209749445923078164
#define HALFLOG2PI 9.1893853320467274178032973640561763986139747363778341281715154048276570E-1
#define EULER_MASCHERONI 0.5772156649015328606065
/*
* Zeta constants for Maclaurin approx. near zero
*/
#define ZETA_02_DIV_02 8.2246703342411321823620758332301E-1
#define ZETA_03_DIV_03 -4.0068563438653142846657938717048E-1
#define ZETA_04_DIV_04 2.7058080842778454787900092413529E-1
#define ZETA_05_DIV_05 -2.0738555102867398526627309729141E-1
#define ZETA_06_DIV_06 1.6955717699740818995241965496515E-1
/*
* More Maclaurin coefficients
*/
/*
#define ZETA_07_DIV_07 -1.4404989676884611811997107854997E-1
#define ZETA_08_DIV_08 1.2550966952474304242233565481358E-1
#define ZETA_09_DIV_09 -1.1133426586956469049087252991471E-1
#define ZETA_10_DIV_10 1.0009945751278180853371459589003E-1
#define ZETA_11_DIV_11 -9.0954017145829042232609298411497E-2
#define ZETA_12_DIV_12 8.3353840546109004024886499837312E-2
#define ZETA_13_DIV_13 -7.6932516411352191472827064348181E-2
#define ZETA_14_DIV_14 7.1432946295361336059232753221795E-2
#define ZETA_15_DIV_15 -6.6668705882420468032903448567376E-2
#define ZETA_16_DIV_16 6.2500955141213040741983285717977E-2
#define ZETA_17_DIV_17 -5.8823978658684582338957270605504E-2
#define ZETA_18_DIV_18 5.5555767627403611102214247869146E-2
#define ZETA_19_DIV_19 -5.2631679379616660733627666155673E-2
#define ZETA_20_DIV_20 5.0000047698101693639805657601934E-2
*/
/*
* Coefficients for Stirling's Series for Lgamma()
*/
#define STIRLING_01 8.3333333333333333333333333333333333333333333333333333333333333333333333E-2
#define STIRLING_02 -2.7777777777777777777777777777777777777777777777777777777777777777777778E-3
#define STIRLING_03 7.9365079365079365079365079365079365079365079365079365079365079365079365E-4
#define STIRLING_04 -5.9523809523809523809523809523809523809523809523809523809523809523809524E-4
#define STIRLING_05 8.4175084175084175084175084175084175084175084175084175084175084175084175E-4
#define STIRLING_06 -1.9175269175269175269175269175269175269175269175269175269175269175269175E-3
#define STIRLING_07 6.4102564102564102564102564102564102564102564102564102564102564102564103E-3
#define STIRLING_08 -2.9550653594771241830065359477124183006535947712418300653594771241830065E-2
#define STIRLING_09 1.7964437236883057316493849001588939669435025472177174963552672531000704E-1
#define STIRLING_10 -1.3924322169059011164274322169059011164274322169059011164274322169059011E0
#define STIRLING_11 1.3402864044168391994478951000690131124913733609385783298826777087646653E1
#define STIRLING_12 -1.5684828462600201730636513245208897382810426288687158252375643679991506E2
#define STIRLING_13 2.1931033333333333333333333333333333333333333333333333333333333333333333E3
#define STIRLING_14 -3.6108771253724989357173265219242230736483610046828437633035334184759472E4
#define STIRLING_15 6.9147226885131306710839525077567346755333407168779805042318946657100161E5
/*
* More Stirling's coefficients
*/
/*
#define STIRLING_16 -1.5238221539407416192283364958886780518659076533839342188488298545224541E7
#define STIRLING_17 3.8290075139141414141414141414141414141414141414141414141414141414141414E8
#define STIRLING_18 -1.0882266035784391089015149165525105374729434879810819660443720594096534E10
#define STIRLING_19 3.4732028376500225225225225225225225225225225225225225225225225225225225E11
#define STIRLING_20 -1.2369602142269274454251710349271324881080978641954251710349271324881081E13
#define STIRLING_21 4.8878806479307933507581516251802290210847053890567382180703629532735764E14
*/
static __inline vector double _lgammad2(vector double x)
{
vec_uchar16 dup_even = ((vec_uchar16) { 0,1,2,3, 0,1,2,3, 8, 9,10,11, 8, 9,10,11 });
vec_uchar16 dup_odd = ((vec_uchar16) { 4,5,6,7, 4,5,6,7, 12,13,14,15, 12,13,14,15 });
vec_uchar16 swap_word = ((vec_uchar16) { 4,5,6,7, 0,1,2,3, 12,13,14,15, 8, 9,10,11 });
vec_double2 infinited = (vec_double2)spu_splats(0x7FF0000000000000ull);
vec_double2 zerod = spu_splats(0.0);
vec_double2 oned = spu_splats(1.0);
vec_double2 twod = spu_splats(2.0);
vec_double2 pi = spu_splats(PI);
vec_double2 sign_maskd = spu_splats(-0.0);
/* This is where we switch from near zero approx. */
vec_float4 zero_switch = spu_splats(0.001f);
vec_float4 shift_switch = spu_splats(6.0f);
vec_float4 xf;
vec_double2 inv_x, inv_xsqu;
vec_double2 xtrunc, xstirling;
vec_double2 sum, xabs;
vec_uint4 xhigh, xlow, xthigh, xtlow;
vec_uint4 x1, isnaninf, isnposint, iszero, isint, isneg, isshifted, is1, is2;
vec_double2 result, stresult, shresult, mresult, nresult;
/* Force Denorms to 0 */
x = spu_add(x, zerod);
xabs = spu_andc(x, sign_maskd);
xf = spu_roundtf(xabs);
xf = spu_shuffle(xf, xf, dup_even);
/*
* For 0 < x <= 0.001.
* Approximation Near Zero
*
* Use Maclaurin Expansion of lgamma()
*
* lgamma(z) = -ln(z) - z * EulerMascheroni + Sum[(-1)^n * z^n * Zeta(n)/n]
*/
mresult = spu_madd(xabs, spu_splats(ZETA_06_DIV_06), spu_splats(ZETA_05_DIV_05));
mresult = spu_madd(xabs, mresult, spu_splats(ZETA_04_DIV_04));
mresult = spu_madd(xabs, mresult, spu_splats(ZETA_03_DIV_03));
mresult = spu_madd(xabs, mresult, spu_splats(ZETA_02_DIV_02));
mresult = spu_mul(xabs, spu_mul(xabs, mresult));
mresult = spu_sub(mresult, spu_add(_logd2(xabs), spu_mul(xabs, spu_splats(EULER_MASCHERONI))));
/*
* For 0.001 < x <= 6.0, we are going to push value
* out to an area where Stirling's approximation is
* accurate. Let's use a constant of 6.
*
* Use the recurrence relation:
* lgamma(x + 1) = ln(x) + lgamma(x)
*
* Note that we shift x here, before Stirling's calculation,
* then after Stirling's, we adjust the result.
*
*/
isshifted = spu_cmpgt(shift_switch, xf);
xstirling = spu_sel(xabs, spu_add(xabs, spu_splats(6.0)), (vec_ullong2)isshifted);
inv_x = _recipd2(xstirling);
inv_xsqu = spu_mul(inv_x, inv_x);
/*
* For 6.0 < x < infinite
*
* Use Stirling's Series.
*
* 1 1 1 1 1
* lgamma(x) = --- ln (2*pi) + (z - ---) ln(x) - x + --- - ----- + ------ ...
* 2 2 12x 360x^3 1260x^5
*
* Taking 10 terms of the sum gives good results for x > 6.0
*
*/
sum = spu_madd(inv_xsqu, spu_splats(STIRLING_15), spu_splats(STIRLING_14));
sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_13));
sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_12));
sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_11));
sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_10));
sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_09));
sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_08));
sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_07));
sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_06));
sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_05));
sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_04));
sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_03));
sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_02));
sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_01));
sum = spu_mul(sum, inv_x);
stresult = spu_madd(spu_sub(xstirling, spu_splats(0.5)), _logd2(xstirling), spu_splats(HALFLOG2PI));
stresult = spu_sub(stresult, xstirling);
stresult = spu_add(stresult, sum);
/*
* Adjust result if we shifted x into Stirling range.
*
* lgamma(x) = lgamma(x + n) - ln(x(x+1)(x+2)...(x+n-1)
*
*/
shresult = spu_mul(xabs, spu_add(xabs, spu_splats(1.0)));
shresult = spu_mul(shresult, spu_add(xabs, spu_splats(2.0)));
shresult = spu_mul(shresult, spu_add(xabs, spu_splats(3.0)));
shresult = spu_mul(shresult, spu_add(xabs, spu_splats(4.0)));
shresult = spu_mul(shresult, spu_add(xabs, spu_splats(5.0)));
shresult = _logd2(shresult);
shresult = spu_sub(stresult, shresult);
stresult = spu_sel(stresult, shresult, (vec_ullong2)isshifted);
/*
* Select either Maclaurin or Stirling result before Negative X calc.
*/
xf = spu_shuffle(xf, xf, dup_even);
vec_uint4 useStirlings = spu_cmpgt(xf, zero_switch);
result = spu_sel(mresult, stresult, (vec_ullong2)useStirlings);
/*
* Approximation for Negative X
*
* Use reflection relation
*
* gamma(x) * gamma(-x) = -pi/(x sin(pi x))
*
* lgamma(x) = log(pi/(-x sin(pi x))) - lgamma(-x)
*
*/
nresult = spu_mul(x, _sind2(spu_mul(x, pi)));
nresult = spu_andc(nresult, sign_maskd);
nresult = _logd2(_divd2(pi, nresult));
nresult = spu_sub(nresult, result);
/*
* Select between the negative or positive x approximations.
*/
isneg = (vec_uint4)spu_shuffle(x, x, dup_even);
isneg = spu_rlmaska(isneg, -32);
result = spu_sel(result, nresult, (vec_ullong2)isneg);
/*
* Finally, special cases/errors.
*/
xhigh = (vec_uint4)spu_shuffle(xabs, xabs, dup_even);
xlow = (vec_uint4)spu_shuffle(xabs, xabs, dup_odd);
/* x = zero, return infinite */
x1 = spu_or(xhigh, xlow);
iszero = spu_cmpeq(x1, 0);
/* x = negative integer, return infinite */
xtrunc = _truncd2(xabs);
xthigh = (vec_uint4)spu_shuffle(xtrunc, xtrunc, dup_even);
xtlow = (vec_uint4)spu_shuffle(xtrunc, xtrunc, dup_odd);
isint = spu_and(spu_cmpeq(xthigh, xhigh), spu_cmpeq(xtlow, xlow));
isnposint = spu_or(spu_and(isint, isneg), iszero);
result = spu_sel(result, infinited, (vec_ullong2)isnposint);
/* x = 1.0 or 2.0, return 0.0 */
is1 = spu_cmpeq((vec_uint4)x, (vec_uint4)oned);
is1 = spu_and(is1, spu_shuffle(is1, is1, swap_word));
is2 = spu_cmpeq((vec_uint4)x, (vec_uint4)twod);
is2 = spu_and(is2, spu_shuffle(is2, is2, swap_word));
result = spu_sel(result, zerod, (vec_ullong2)spu_or(is1,is2));
/* x = +/- infinite or nan, return |x| */
isnaninf = spu_cmpgt(xhigh, 0x7FEFFFFF);
result = spu_sel(result, xabs, (vec_ullong2)isnaninf);
return result;
}
#endif /* _LGAMMAD2_H_ */
#endif /* __SPU__ */
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