1 files changed, 148 insertions, 0 deletions
diff --git a/libm/double/ellie.c b/libm/double/ellie.c
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+/*							ellie.c
+ *
+ *	Incomplete elliptic integral of the second kind
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * double phi, m, y, ellie();
+ *
+ * y = ellie( phi, m );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Approximates the integral
+ *
+ *
+ *                phi
+ *                 -
+ *                | |
+ *                |                   2
+ * E(phi_\m)  =    |    sqrt( 1 - m sin t ) dt
+ *                |
+ *              | |    
+ *               -
+ *                0
+ *
+ * of amplitude phi and modulus m, using the arithmetic -
+ * geometric mean algorithm.
+ *
+ *
+ *
+ * ACCURACY:
+ *
+ * Tested at random arguments with phi in [-10, 10] and m in
+ * [0, 1].
+ *                      Relative error:
+ * arithmetic   domain     # trials      peak         rms
+ *    DEC        0,2         2000       1.9e-16     3.4e-17
+ *    IEEE     -10,10      150000       3.3e-15     1.4e-16
+ *
+ *
+ */
+
+
+/*
+Cephes Math Library Release 2.8:  June, 2000
+Copyright 1984, 1987, 1993, 2000 by Stephen L. Moshier
+*/
+
+/*	Incomplete elliptic integral of second kind	*/
+#include <math.h>
+extern double PI, PIO2, MACHEP;
+#ifdef ANSIPROT
+extern double sqrt ( double );
+extern double fabs ( double );
+extern double log ( double );
+extern double sin ( double x );
+extern double tan ( double x );
+extern double atan ( double );
+extern double floor ( double );
+extern double ellpe ( double );
+extern double ellpk ( double );
+double ellie ( double, double );
+#else
+double sqrt(), fabs(), log(), sin(), tan(), atan(), floor();
+double ellpe(), ellpk(), ellie();
+#endif
+
+double ellie( phi, m )
+double phi, m;
+{
+double a, b, c, e, temp;
+double lphi, t, E;
+int d, mod, npio2, sign;
+
+if( m == 0.0 )
+	return( phi );
+lphi = phi;
+npio2 = floor( lphi/PIO2 );
+if( npio2 & 1 )
+	npio2 += 1;
+lphi = lphi - npio2 * PIO2;
+if( lphi < 0.0 )
+	{
+	lphi = -lphi;
+	sign = -1;
+	}
+else
+	{
+	sign = 1;
+	}
+a = 1.0 - m;
+E = ellpe( a );
+if( a == 0.0 )
+	{
+	temp = sin( lphi );
+	goto done;
+	}
+t = tan( lphi );
+b = sqrt(a);
+/* Thanks to Brian Fitzgerald <fitzgb@mml0.meche.rpi.edu>
+   for pointing out an instability near odd multiples of pi/2.  */
+if( fabs(t) > 10.0 )
+	{
+	/* Transform the amplitude */
+	e = 1.0/(b*t);
+	/* ... but avoid multiple recursions.  */
+	if( fabs(e) < 10.0 )
+		{
+		e = atan(e);
+		temp = E + m * sin( lphi ) * sin( e ) - ellie( e, m );
+		goto done;
+		}
+	}
+c = sqrt(m);
+a = 1.0;
+d = 1;
+e = 0.0;
+mod = 0;
+
+while( fabs(c/a) > MACHEP )
+	{
+	temp = b/a;
+	lphi = lphi + atan(t*temp) + mod * PI;
+	mod = (lphi + PIO2)/PI;
+	t = t * ( 1.0 + temp )/( 1.0 - temp * t * t );
+	c = ( a - b )/2.0;
+	temp = sqrt( a * b );
+	a = ( a + b )/2.0;
+	b = temp;
+	d += d;
+	e += c * sin(lphi);
+	}
+
+temp = E / ellpk( 1.0 - m );
+temp *= (atan(t) + mod * PI)/(d * a);
+temp += e;
+
+done:
+
+if( sign < 0 )
+	temp = -temp;
+temp += npio2 * E;
+return( temp );
+}