int
gsl_sf_multiply_e(const double x, const double y, gsl_sf_result * result)
{
const double ax = fabs(x);
const double ay = fabs(y);
if(x == 0.0 || y == 0.0) {
/* It is necessary to eliminate this immediately.
*/
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if((ax <= 1.0 && ay >= 1.0) || (ay <= 1.0 && ax >= 1.0)) {
/* Straddling 1.0 is always safe.
*/
result->val = x*y;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
const double f = 1.0 - 2.0 * GSL_DBL_EPSILON;
const double min = GSL_MIN_DBL(fabs(x), fabs(y));
const double max = GSL_MAX_DBL(fabs(x), fabs(y));
if(max < 0.9 * GSL_SQRT_DBL_MAX || min < (f * DBL_MAX)/max) {
result->val = GSL_COERCE_DBL(x*y);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
else {
OVERFLOW_ERROR(result);
}
}
}
int
gsl_sf_hydrogenicR_e(const int n, const int l,
const double Z, const double r,
gsl_sf_result * result)
{
if(n < 1 || l > n-1 || Z <= 0.0 || r < 0.0) {
DOMAIN_ERROR(result);
}
else {
double A = 2.0*Z/n;
gsl_sf_result norm;
int stat_norm = R_norm(n, l, Z, &norm);
double rho = A*r;
double ea = exp(-0.5*rho);
double pp = gsl_sf_pow_int(rho, l);
gsl_sf_result lag;
int stat_lag = gsl_sf_laguerre_n_e(n-l-1, 2*l+1, rho, &lag);
double W_val = norm.val * ea * pp;
double W_err = norm.err * ea * pp;
W_err += norm.val * ((0.5*rho + 1.0) * GSL_DBL_EPSILON) * ea * pp;
W_err += norm.val * ea * ((l+1.0) * GSL_DBL_EPSILON) * pp;
result->val = W_val * lag.val;
result->err = W_val * lag.err + W_err * fabs(lag.val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
if ((l == 0 || (r > 0 && l > 0)) && lag.val != 0.0
&& stat_lag == GSL_SUCCESS && stat_norm == GSL_SUCCESS) {
CHECK_UNDERFLOW(result);
};
return GSL_ERROR_SELECT_2(stat_lag, stat_norm);
}
}
int gsl_sf_debye_6_e(const double x, gsl_sf_result * result)
{
const double val_infinity = 4356.06887828990661194792541535 ;
const double xcut = -GSL_LOG_DBL_MIN;
/* CHECK_POINTER(result) */
if(x < 0.0) {
DOMAIN_ERROR(result);
}
else if(x < 2.0*M_SQRT2*GSL_SQRT_DBL_EPSILON) {
result->val = 1.0 - 3.0*x/7.0 + x*x/16.0;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else if(x <= 4.0) {
const double t = x*x/8.0 - 1.0;
gsl_sf_result c;
cheb_eval_e(&adeb6_cs, t, &c);
result->val = c.val - 3.0*x/7.0;
result->err = c.err + GSL_DBL_EPSILON * 3.0*x/7.0;
return GSL_SUCCESS;
}
else if(x < -(M_LN2 + GSL_LOG_DBL_EPSILON)) {
const int nexp = floor(xcut/x);
const double ex = exp(-x);
double xk = nexp * x;
double rk = nexp;
double sum = 0.0;
int i;
for(i=nexp; i>=1; i--) {
double xk_inv = 1.0/xk;
sum *= ex;
sum += ((((((720.0*xk_inv + 720.0)*xk_inv + 360.0)*xk_inv + 120.0)*xk_inv + 30.0)*xk_inv+ 6.0)*xk_inv+ 1.0) / rk;
rk -= 1.0;
xk -= x;
}
result->val = val_infinity/(x*x*x*x*x*x) - 6.0 * sum * ex;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else if(x < xcut) {
const double x2 = x*x;
const double x4 = x2*x2;
const double x6 = x4*x2;
const double sum = 720.0 + 720.0*x + 360.0*x2 + 120.0*x2*x + 30.0*x4 + 6.0*x4*x +x6 ;
result->val = (val_infinity - 6.0 * sum * exp(-x)) / x6;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else {
result->val = (((((val_infinity/x)/x)/x)/x)/x)/x ;
result->err = GSL_DBL_EPSILON * result->val;
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
}
int gsl_sf_debye_5_e(const double x, gsl_sf_result * result)
{
const double val_infinity = 610.405837190669483828710757875 ;
const double xcut = -GSL_LOG_DBL_MIN;
/* CHECK_POINTER(result) */
if(x < 0.0) {
DOMAIN_ERROR(result);
}
else if(x < 2.0*M_SQRT2*GSL_SQRT_DBL_EPSILON) {
result->val = 1.0 - 5.0*x/12.0 + 5.0*x*x/84.0;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else if(x <= 4.0) {
const double t = x*x/8.0 - 1.0;
gsl_sf_result c;
cheb_eval_e(&adeb5_cs, t, &c);
result->val = c.val - 5.0*x/12.0;
result->err = c.err + GSL_DBL_EPSILON * 5.0*x/12.0;
return GSL_SUCCESS;
}
else if(x < -(M_LN2 + GSL_LOG_DBL_EPSILON)) {
const int nexp = floor(xcut/x);
const double ex = exp(-x);
double xk = nexp * x;
double rk = nexp;
double sum = 0.0;
int i;
for(i=nexp; i>=1; i--) {
double xk_inv = 1.0/xk;
sum *= ex;
sum += (((((120.0*xk_inv + 120.0)*xk_inv + 60.0)*xk_inv + 20.0)*xk_inv + 5.0)*xk_inv+ 1.0) / rk;
rk -= 1.0;
xk -= x;
}
result->val = val_infinity/(x*x*x*x*x) - 5.0 * sum * ex;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else if(x < xcut) {
const double x2 = x*x;
const double x4 = x2*x2;
const double x5 = x4*x;
const double sum = 120.0 + 120.0*x + 60.0*x2 + 20.0*x2*x + 5.0*x4 + x5;
result->val = (val_infinity - 5.0 * sum * exp(-x)) / x5;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else {
result->val = ((((val_infinity/x)/x)/x)/x)/x;
result->err = GSL_DBL_EPSILON * result->val;
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
}
int gsl_sf_debye_4_e(const double x, gsl_sf_result * result)
{
const double val_infinity = 99.5450644937635129;
const double xcut = -GSL_LOG_DBL_MIN;
/* CHECK_POINTER(result) */
if(x < 0.0) {
DOMAIN_ERROR(result);
}
else if(x < 2.0*M_SQRT2*GSL_SQRT_DBL_EPSILON) {
result->val = 1.0 - 2.0*x/5.0 + x*x/18.0;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else if(x <= 4.0) {
const double t = x*x/8.0 - 1.0;
gsl_sf_result c;
cheb_eval_e(&adeb4_cs, t, &c);
result->val = c.val - 2.0*x/5.0;
result->err = c.err + GSL_DBL_EPSILON * 2.0*x/5.0;
return GSL_SUCCESS;
}
else if(x < -(M_LN2 + GSL_LOG_DBL_EPSILON)) {
const int nexp = floor(xcut/x);
const double ex = exp(-x);
double xk = nexp * x;
double rk = nexp;
double sum = 0.0;
int i;
for(i=nexp; i>=1; i--) {
double xk_inv = 1.0/xk;
sum *= ex;
sum += ((((24.0*xk_inv + 24.0)*xk_inv + 12.0)*xk_inv + 4.0)*xk_inv + 1.0) / rk;
rk -= 1.0;
xk -= x;
}
result->val = val_infinity/(x*x*x*x) - 4.0 * sum * ex;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else if(x < xcut) {
const double x2 = x*x;
const double x4 = x2*x2;
const double sum = 24.0 + 24.0*x + 12.0*x2 + 4.0*x2*x + x4;
result->val = (val_infinity - 4.0 * sum * exp(-x)) / x4;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else {
result->val = (((val_infinity/x)/x)/x)/x;
result->err = GSL_DBL_EPSILON * result->val;
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
}
int gsl_sf_debye_3_e(const double x, gsl_sf_result * result)
{
const double val_infinity = 19.4818182068004875;
const double xcut = -GSL_LOG_DBL_MIN;
/* CHECK_POINTER(result) */
if(x < 0.0) {
DOMAIN_ERROR(result);
}
else if(x < 2.0*M_SQRT2*GSL_SQRT_DBL_EPSILON) {
result->val = 1.0 - 3.0*x/8.0 + x*x/20.0;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else if(x <= 4.0) {
const double t = x*x/8.0 - 1.0;
gsl_sf_result c;
cheb_eval_e(&adeb3_cs, t, &c);
result->val = c.val - 0.375*x;
result->err = c.err + GSL_DBL_EPSILON * 0.375*x;
return GSL_SUCCESS;
}
else if(x < -(M_LN2 + GSL_LOG_DBL_EPSILON)) {
const int nexp = floor(xcut/x);
const double ex = exp(-x);
double xk = nexp * x;
double rk = nexp;
double sum = 0.0;
int i;
for(i=nexp; i>=1; i--) {
double xk_inv = 1.0/xk;
sum *= ex;
sum += (((6.0*xk_inv + 6.0)*xk_inv + 3.0)*xk_inv + 1.0) / rk;
rk -= 1.0;
xk -= x;
}
result->val = val_infinity/(x*x*x) - 3.0 * sum * ex;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else if(x < xcut) {
const double x3 = x*x*x;
const double sum = 6.0 + 6.0*x + 3.0*x*x + x3;
result->val = (val_infinity - 3.0 * sum * exp(-x)) / x3;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else {
result->val = ((val_infinity/x)/x)/x;
result->err = GSL_DBL_EPSILON * result->val;
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
}
int gsl_sf_debye_2_e(const double x, gsl_sf_result * result)
{
const double val_infinity = 4.80822761263837714;
const double xcut = -GSL_LOG_DBL_MIN;
/* CHECK_POINTER(result) */
if(x < 0.0) {
DOMAIN_ERROR(result);
}
else if(x < 2.0*M_SQRT2*GSL_SQRT_DBL_EPSILON) {
result->val = 1.0 - x/3.0 + x*x/24.0;
result->err = GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else if(x <= 4.0) {
const double t = x*x/8.0 - 1.0;
gsl_sf_result c;
cheb_eval_e(&adeb2_cs, t, &c);
result->val = c.val - x/3.0;
result->err = c.err + GSL_DBL_EPSILON * x/3.0;
return GSL_SUCCESS;
}
else if(x < -(M_LN2 + GSL_LOG_DBL_EPSILON)) {
const int nexp = floor(xcut/x);
const double ex = exp(-x);
double xk = nexp * x;
double rk = nexp;
double sum = 0.0;
int i;
for(i=nexp; i>=1; i--) {
sum *= ex;
sum += (1.0 + 2.0/xk + 2.0/(xk*xk)) / rk;
rk -= 1.0;
xk -= x;
}
result->val = val_infinity/(x*x) - 2.0 * sum * ex;
result->err = GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < xcut) {
const double x2 = x*x;
const double sum = 2.0 + 2.0*x + x2;
result->val = (val_infinity - 2.0 * sum * exp(-x)) / x2;
result->err = GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
result->val = (val_infinity/x)/x;
result->err = GSL_DBL_EPSILON * result->val;
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
}
static
int expint_En_impl(const int n, const double x, gsl_sf_result * result, const int scale)
{
if (n < 0) {
DOMAIN_ERROR(result);
} else if (n == 0) {
if (x == 0) {
DOMAIN_ERROR(result);
} else {
result->val = (scale ? 1.0 : exp(-x)) / x;
result->err = 2 * GSL_DBL_EPSILON * fabs(result->val);
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
} else if (n == 1) {
return expint_E1_impl(x, result, scale);
} else if (n == 2) {
return expint_E2_impl(x, result, scale);
} else {
if(x < 0) {
DOMAIN_ERROR(result);
}
if (x == 0) {
result->val = (scale ? exp(x) : 1 ) * (1/(n-1.0));
result->err = 2 * GSL_DBL_EPSILON * fabs(result->val);
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
} else {
gsl_sf_result result_g;
double prefactor = pow(x, n-1);
int status = gsl_sf_gamma_inc_e (1-n, x, &result_g);
double scale_factor = ( scale ? exp(x) : 1.0 );
result->val = scale_factor * prefactor * result_g.val;
result->err = 2 * GSL_DBL_EPSILON * fabs(result->val);
result->err += 2 * fabs(scale_factor * prefactor * result_g.err);
if (status == GSL_SUCCESS) CHECK_UNDERFLOW(result);
return status;
}
}
}
int gsl_sf_bessel_k0_scaled_e(const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0) {
DOMAIN_ERROR(result);
}
else {
result->val = M_PI/(2.0*x);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
}
int
gsl_sf_hydrogenicR_1_e(const double Z, const double r, gsl_sf_result * result)
{
if(Z > 0.0 && r >= 0.0) {
double A = 2.0*Z;
double norm = A*sqrt(Z);
double ea = exp(-Z*r);
result->val = norm*ea;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) * fabs(Z*r);
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
else {
DOMAIN_ERROR(result);
}
}
int gsl_sf_bessel_k1_scaled_e(const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0) {
DOMAIN_ERROR(result);
}
else if(x < (M_SQRTPI+1.0)/(M_SQRT2*GSL_SQRT_DBL_MAX)) {
OVERFLOW_ERROR(result);
}
else {
result->val = M_PI/(2.0*x) * (1.0 + 1.0/x);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
}
int
gsl_sf_beta_inc_e(
const double a,
const double b,
const double x,
gsl_sf_result * result
)
{
if(a <= 0.0 || b <= 0.0 || x < 0.0 || x > 1.0) {
DOMAIN_ERROR(result);
}
else if(x == 0.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(x == 1.0) {
result->val = 1.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
gsl_sf_result ln_beta;
gsl_sf_result ln_x;
gsl_sf_result ln_1mx;
gsl_sf_result prefactor;
const int stat_ln_beta = gsl_sf_lnbeta_e(a, b, &ln_beta);
const int stat_ln_1mx = gsl_sf_log_1plusx_e(-x, &ln_1mx);
const int stat_ln_x = gsl_sf_log_e(x, &ln_x);
const int stat_ln = GSL_ERROR_SELECT_3(stat_ln_beta, stat_ln_1mx, stat_ln_x);
const double ln_pre_val = -ln_beta.val + a * ln_x.val + b * ln_1mx.val;
const double ln_pre_err = ln_beta.err + fabs(a*ln_x.err) + fabs(b*ln_1mx.err);
const int stat_exp = gsl_sf_exp_err_e(ln_pre_val, ln_pre_err, &prefactor);
if(stat_ln != GSL_SUCCESS) {
result->val = 0.0;
result->err = 0.0;
GSL_ERROR ("error", GSL_ESANITY);
}
if(x < (a + 1.0)/(a+b+2.0)) {
/* Apply continued fraction directly. */
gsl_sf_result cf;
const int stat_cf = beta_cont_frac(a, b, x, &cf);
int stat;
result->val = prefactor.val * cf.val / a;
result->err = (fabs(prefactor.err * cf.val) + fabs(prefactor.val * cf.err))/a;
stat = GSL_ERROR_SELECT_2(stat_exp, stat_cf);
if(stat == GSL_SUCCESS) {
CHECK_UNDERFLOW(result);
}
return stat;
}
else {
/* Apply continued fraction after hypergeometric transformation. */
gsl_sf_result cf;
const int stat_cf = beta_cont_frac(b, a, 1.0-x, &cf);
int stat;
const double term = prefactor.val * cf.val / b;
result->val = 1.0 - term;
result->err = fabs(prefactor.err * cf.val)/b;
result->err += fabs(prefactor.val * cf.err)/b;
result->err += 2.0 * GSL_DBL_EPSILON * (1.0 + fabs(term));
stat = GSL_ERROR_SELECT_2(stat_exp, stat_cf);
if(stat == GSL_SUCCESS) {
CHECK_UNDERFLOW(result);
}
return stat;
}
}
}
