CocoRoboDesktop/pdf1.js/node_modules/cephes/cephes/ellpj.c

/*							ellpj.c
 *
 *	Jacobian Elliptic Functions
 *
 *
 *
 * SYNOPSIS:
 *
 * double u, m, sn, cn, dn, phi;
 * int ellpj();
 *
 * ellpj( u, m, _&sn, _&cn, _&dn, _&phi );
 *
 *
 *
 * DESCRIPTION:
 *
 *
 * Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
 * and dn(u|m) of parameter m between 0 and 1, and real
 * argument u.
 *
 * These functions are periodic, with quarter-period on the
 * real axis equal to the complete elliptic integral
 * ellpk(1.0-m).
 *
 * Relation to incomplete elliptic integral:
 * If u = ellik(phi,m), then sn(u|m) = sin(phi),
 * and cn(u|m) = cos(phi).  Phi is called the amplitude of u.
 *
 * Computation is by means of the arithmetic-geometric mean
 * algorithm, except when m is within 1e-9 of 0 or 1.  In the
 * latter case with m close to 1, the approximation applies
 * only for phi < pi/2.
 *
 * ACCURACY:
 *
 * Tested at random points with u between 0 and 10, m between
 * 0 and 1.
 *
 *            Absolute error (* = relative error):
 * arithmetic   function   # trials      peak         rms
 *    DEC       sn           1800       4.5e-16     8.7e-17
 *    IEEE      phi         10000       9.2e-16*    1.4e-16*
 *    IEEE      sn          50000       4.1e-15     4.6e-16
 *    IEEE      cn          40000       3.6e-15     4.4e-16
 *    IEEE      dn         100000       3.9e-15     1.7e-16
 *
 * Larger errors occur for m near 1.
 * Peak error observed in consistency check using addition
 * theorem for sn(u+v) was 4e-16 (absolute).  Also tested by
 * the above relation to the incomplete elliptic integral.
 * Accuracy deteriorates when u is large.
 *
 */

/*							ellpj.c		*/

/*
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000, 2014 by Stephen L. Moshier
*/

#include "mconf.h"
#ifdef ANSIPROT
extern double sqrt(double);
extern double fabs(double);
extern double sin(double);
extern double cos(double);
extern double asin(double);
extern double tanh(double);
extern double sinh(double);
extern double cosh(double);
extern double atan(double);
extern double exp(double);
#else
double sqrt(), fabs(), sin(), cos(), asin(), tanh();
double sinh(), cosh(), atan(), exp();
#endif
extern double PIO2, MACHEP;

int ellpj(u, m, sn, cn, dn, ph) double u, m;
double *sn, *cn, *dn, *ph;
{
  double ai, b, phi, t, twon;
  double a[9], c[9];
  int i;

  /* Check for special cases */

  if (m < 0.0 || m > 1.0) {
    mtherr("ellpj", DOMAIN);
    *sn = 0.0;
    *cn = 0.0;
    *ph = 0.0;
    *dn = 0.0;
    return (-1);
  }
  if (m < 1.0e-9) {
    t = sin(u);
    b = cos(u);
    ai = 0.25 * m * (u - t * b);
    *sn = t - ai * b;
    *cn = b + ai * t;
    *ph = u - ai;
    *dn = 1.0 - 0.5 * m * t * t;
    return (0);
  }

  if (m >= 0.9999999999) {
    ai = 0.25 * (1.0 - m);
    b = cosh(u);
    t = tanh(u);
    phi = 1.0 / b;
    twon = b * sinh(u);
    *sn = t + ai * (twon - u) / (b * b);
    *ph = 2.0 * atan(exp(u)) - PIO2 + ai * (twon - u) / b;
    ai *= t * phi;
    *cn = phi - ai * (twon - u);
    *dn = phi + ai * (twon + u);
    return (0);
  }

  /*	A. G. M. scale		*/
  a[0] = 1.0;
  b = sqrt(1.0 - m);
  c[0] = sqrt(m);
  twon = 1.0;
  i = 0;

  while (fabs(c[i] / a[i]) > MACHEP) {
    if (i > 7) {
      mtherr("ellpj", OVERFLOW);
      goto done;
    }
    ai = a[i];
    ++i;
    c[i] = (ai - b) / 2.0;
    t = sqrt(ai * b);
    a[i] = (ai + b) / 2.0;
    b = t;
    twon *= 2.0;
  }

done:

  /* backward recurrence */
  phi = twon * a[i] * u;
  do {
    t = c[i] * sin(phi) / a[i];
    b = phi;
    phi = (asin(t) + phi) / 2.0;
  } while (--i);

  t = sin(phi);
  *sn = t;
  *cn = cos(phi);
  /* Thanks to Hartmut Henkel for reporting a bug here:  */
  *dn = sqrt(1.0 - m * t * t);
  *ph = phi;
  return (0);
}