long double
__lgamma_negl (long double x, int *signgamp)
{
/* Determine the half-integer region X lies in, handle exact
integers and determine the sign of the result. */
int i = floorl (-2 * x);
if ((i & 1) == 0 && i == -2 * x)
return 1.0L / 0.0L;
long double xn = ((i & 1) == 0 ? -i / 2 : (-i - 1) / 2);
i -= 4;
*signgamp = ((i & 2) == 0 ? -1 : 1);
SET_RESTORE_ROUNDL (FE_TONEAREST);
/* Expand around the zero X0 = X0_HI + X0_LO. */
long double x0_hi = lgamma_zeros[i][0], x0_lo = lgamma_zeros[i][1];
long double xdiff = x - x0_hi - x0_lo;
/* For arguments in the range -3 to -2, use polynomial
approximations to an adjusted version of the gamma function. */
if (i < 2)
{
int j = floorl (-8 * x) - 16;
long double xm = (-33 - 2 * j) * 0.0625L;
long double x_adj = x - xm;
size_t deg = poly_deg[j];
size_t end = poly_end[j];
long double g = poly_coeff[end];
for (size_t j = 1; j <= deg; j++)
g = g * x_adj + poly_coeff[end - j];
return __log1pl (g * xdiff / (x - xn));
}
/* The result we want is log (sinpi (X0) / sinpi (X))
+ log (gamma (1 - X0) / gamma (1 - X)). */
long double x_idiff = fabsl (xn - x), x0_idiff = fabsl (xn - x0_hi - x0_lo);
long double log_sinpi_ratio;
if (x0_idiff < x_idiff * 0.5L)
/* Use log not log1p to avoid inaccuracy from log1p of arguments
close to -1. */
log_sinpi_ratio = __ieee754_logl (lg_sinpi (x0_idiff)
/ lg_sinpi (x_idiff));
else
{
/* Use log1p not log to avoid inaccuracy from log of arguments
close to 1. X0DIFF2 has positive sign if X0 is further from
XN than X is from XN, negative sign otherwise. */
long double x0diff2 = ((i & 1) == 0 ? xdiff : -xdiff) * 0.5L;
long double sx0d2 = lg_sinpi (x0diff2);
long double cx0d2 = lg_cospi (x0diff2);
log_sinpi_ratio = __log1pl (2 * sx0d2
* (-sx0d2 + cx0d2 * lg_cotpi (x_idiff)));
}
long double log_gamma_ratio;
long double y0 = 1 - x0_hi;
long double y0_eps = -x0_hi + (1 - y0) - x0_lo;
long double y = 1 - x;
long double y_eps = -x + (1 - y);
/* We now wish to compute LOG_GAMMA_RATIO
= log (gamma (Y0 + Y0_EPS) / gamma (Y + Y_EPS)). XDIFF
accurately approximates the difference Y0 + Y0_EPS - Y -
Y_EPS. Use Stirling's approximation. First, we may need to
adjust into the range where Stirling's approximation is
sufficiently accurate. */
long double log_gamma_adj = 0;
if (i < 18)
{
int n_up = (19 - i) / 2;
long double ny0, ny0_eps, ny, ny_eps;
ny0 = y0 + n_up;
ny0_eps = y0 - (ny0 - n_up) + y0_eps;
y0 = ny0;
y0_eps = ny0_eps;
ny = y + n_up;
ny_eps = y - (ny - n_up) + y_eps;
y = ny;
y_eps = ny_eps;
long double prodm1 = __lgamma_productl (xdiff, y - n_up, y_eps, n_up);
log_gamma_adj = -__log1pl (prodm1);
}
long double log_gamma_high
= (xdiff * __log1pl ((y0 - e_hi - e_lo + y0_eps) / e_hi)
+ (y - 0.5L + y_eps) * __log1pl (xdiff / y) + log_gamma_adj);
/* Compute the sum of (B_2k / 2k(2k-1))(Y0^-(2k-1) - Y^-(2k-1)). */
long double y0r = 1 / y0, yr = 1 / y;
long double y0r2 = y0r * y0r, yr2 = yr * yr;
long double rdiff = -xdiff / (y * y0);
long double bterm[NCOEFF];
long double dlast = rdiff, elast = rdiff * yr * (yr + y0r);
bterm[0] = dlast * lgamma_coeff[0];
for (size_t j = 1; j < NCOEFF; j++)
{
long double dnext = dlast * y0r2 + elast;
long double enext = elast * yr2;
bterm[j] = dnext * lgamma_coeff[j];
dlast = dnext;
elast = enext;
}
long double log_gamma_low = 0;
for (size_t j = 0; j < NCOEFF; j++)
log_gamma_low += bterm[NCOEFF - 1 - j];
log_gamma_ratio = log_gamma_high + log_gamma_low;
return log_sinpi_ratio + log_gamma_ratio;
}
float
__lgamma_negf (float x, int *signgamp)
{
/* Determine the half-integer region X lies in, handle exact
integers and determine the sign of the result. */
int i = __floorf (-2 * x);
if ((i & 1) == 0 && i == -2 * x)
return 1.0f / 0.0f;
float xn = ((i & 1) == 0 ? -i / 2 : (-i - 1) / 2);
i -= 4;
*signgamp = ((i & 2) == 0 ? -1 : 1);
SET_RESTORE_ROUNDF (FE_TONEAREST);
/* Expand around the zero X0 = X0_HI + X0_LO. */
float x0_hi = lgamma_zeros[i][0], x0_lo = lgamma_zeros[i][1];
float xdiff = x - x0_hi - x0_lo;
/* For arguments in the range -3 to -2, use polynomial
approximations to an adjusted version of the gamma function. */
if (i < 2)
{
int j = __floorf (-8 * x) - 16;
float xm = (-33 - 2 * j) * 0.0625f;
float x_adj = x - xm;
size_t deg = poly_deg[j];
size_t end = poly_end[j];
float g = poly_coeff[end];
for (size_t j = 1; j <= deg; j++)
g = g * x_adj + poly_coeff[end - j];
return __log1pf (g * xdiff / (x - xn));
}
/* The result we want is log (sinpi (X0) / sinpi (X))
+ log (gamma (1 - X0) / gamma (1 - X)). */
float x_idiff = fabsf (xn - x), x0_idiff = fabsf (xn - x0_hi - x0_lo);
float log_sinpi_ratio;
if (x0_idiff < x_idiff * 0.5f)
/* Use log not log1p to avoid inaccuracy from log1p of arguments
close to -1. */
log_sinpi_ratio = __ieee754_logf (lg_sinpi (x0_idiff)
/ lg_sinpi (x_idiff));
else
{
/* Use log1p not log to avoid inaccuracy from log of arguments
close to 1. X0DIFF2 has positive sign if X0 is further from
XN than X is from XN, negative sign otherwise. */
float x0diff2 = ((i & 1) == 0 ? xdiff : -xdiff) * 0.5f;
float sx0d2 = lg_sinpi (x0diff2);
float cx0d2 = lg_cospi (x0diff2);
log_sinpi_ratio = __log1pf (2 * sx0d2
* (-sx0d2 + cx0d2 * lg_cotpi (x_idiff)));
}
float log_gamma_ratio;
float y0 = math_narrow_eval (1 - x0_hi);
float y0_eps = -x0_hi + (1 - y0) - x0_lo;
float y = math_narrow_eval (1 - x);
float y_eps = -x + (1 - y);
/* We now wish to compute LOG_GAMMA_RATIO
= log (gamma (Y0 + Y0_EPS) / gamma (Y + Y_EPS)). XDIFF
accurately approximates the difference Y0 + Y0_EPS - Y -
Y_EPS. Use Stirling's approximation. */
float log_gamma_high
= (xdiff * __log1pf ((y0 - e_hi - e_lo + y0_eps) / e_hi)
+ (y - 0.5f + y_eps) * __log1pf (xdiff / y));
/* Compute the sum of (B_2k / 2k(2k-1))(Y0^-(2k-1) - Y^-(2k-1)). */
float y0r = 1 / y0, yr = 1 / y;
float y0r2 = y0r * y0r, yr2 = yr * yr;
float rdiff = -xdiff / (y * y0);
float bterm[NCOEFF];
float dlast = rdiff, elast = rdiff * yr * (yr + y0r);
bterm[0] = dlast * lgamma_coeff[0];
for (size_t j = 1; j < NCOEFF; j++)
{
float dnext = dlast * y0r2 + elast;
float enext = elast * yr2;
bterm[j] = dnext * lgamma_coeff[j];
dlast = dnext;
elast = enext;
}
float log_gamma_low = 0;
for (size_t j = 0; j < NCOEFF; j++)
log_gamma_low += bterm[NCOEFF - 1 - j];
log_gamma_ratio = log_gamma_high + log_gamma_low;
return log_sinpi_ratio + log_gamma_ratio;
}
static long double
lg_cotpi (long double x)
{
return lg_cospi (x) / lg_sinpi (x);
}
static float
lg_cotpi (float x)
{
return lg_cospi (x) / lg_sinpi (x);
}