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/* unity.c
*
* Relative error approximations for function arguments near
* unity.
*
* log1p(x) = log(1+x)
* expm1(x) = exp(x) - 1
* cosm1(x) = cos(x) - 1
* lgam1p(x) = lgam(1+x)
*
*/
/* Scipy changes:
* - 06-10-2016: added lgam1p
*/
#include "mconf.h"
extern double MACHEP;
/* log1p(x) = log(1 + x) */
/* Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
* 1/sqrt(2) <= x < sqrt(2)
* Theoretical peak relative error = 2.32e-20
*/
static const double LP[] = {
4.5270000862445199635215E-5,
4.9854102823193375972212E-1,
6.5787325942061044846969E0,
2.9911919328553073277375E1,
6.0949667980987787057556E1,
5.7112963590585538103336E1,
2.0039553499201281259648E1,
};
static const double LQ[] = {
/* 1.0000000000000000000000E0, */
1.5062909083469192043167E1,
8.3047565967967209469434E1,
2.2176239823732856465394E2,
3.0909872225312059774938E2,
2.1642788614495947685003E2,
6.0118660497603843919306E1,
};
double log1p(double x)
{
double z;
z = 1.0 + x;
if ((z < NPY_SQRT1_2) || (z > NPY_SQRT2))
return (log(z));
z = x * x;
z = -0.5 * z + x * (z * polevl(x, LP, 6) / p1evl(x, LQ, 6));
return (x + z);
}
/* log(1 + x) - x */
double log1pmx(double x)
{
if (fabs(x) < 0.5) {
int n;
double xfac = x;
double term;
double res = 0;
for(n = 2; n < MAXITER; n++) {
xfac *= -x;
term = xfac / n;
res += term;
if (fabs(term) < MACHEP * fabs(res)) {
break;
}
}
return res;
}
else {
return log1p(x) - x;
}
}
/* expm1(x) = exp(x) - 1 */
/* e^x = 1 + 2x P(x^2)/( Q(x^2) - P(x^2) )
* -0.5 <= x <= 0.5
*/
static double EP[3] = {
1.2617719307481059087798E-4,
3.0299440770744196129956E-2,
9.9999999999999999991025E-1,
};
static double EQ[4] = {
3.0019850513866445504159E-6,
2.5244834034968410419224E-3,
2.2726554820815502876593E-1,
2.0000000000000000000897E0,
};
double expm1(double x)
{
double r, xx;
if (!cephes_isfinite(x)) {
if (cephes_isnan(x)) {
return x;
}
else if (x > 0) {
return x;
}
else {
return -1.0;
}
}
if ((x < -0.5) || (x > 0.5))
return (exp(x) - 1.0);
xx = x * x;
r = x * polevl(xx, EP, 2);
r = r / (polevl(xx, EQ, 3) - r);
return (r + r);
}
/* cosm1(x) = cos(x) - 1 */
static double coscof[7] = {
4.7377507964246204691685E-14,
-1.1470284843425359765671E-11,
2.0876754287081521758361E-9,
-2.7557319214999787979814E-7,
2.4801587301570552304991E-5,
-1.3888888888888872993737E-3,
4.1666666666666666609054E-2,
};
double cosm1(double x)
{
double xx;
if ((x < -NPY_PI_4) || (x > NPY_PI_4))
return (cos(x) - 1.0);
xx = x * x;
xx = -0.5 * xx + xx * xx * polevl(xx, coscof, 6);
return xx;
}
/* Compute lgam(x + 1) around x = 0 using its Taylor series. */
static double lgam1p_taylor(double x)
{
int n;
double xfac, coeff, res;
if (x == 0) {
return 0;
}
res = -NPY_EULER * x;
xfac = -x;
for (n = 2; n < 42; n++) {
xfac *= -x;
coeff = zeta(n, 1) * xfac / n;
res += coeff;
if (fabs(coeff) < MACHEP * fabs(res)) {
break;
}
}
return res;
}
/* Compute lgam(x + 1). */
double lgam1p(double x)
{
if (fabs(x) <= 0.5) {
return lgam1p_taylor(x);
} else if (fabs(x - 1) < 0.5) {
return log(x) + lgam1p_taylor(x - 1);
} else {
return lgam(x + 1);
}
}
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