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+/* clog.c
+ *
+ * Complex natural logarithm
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void clog();
+ * cmplx z, w;
+ *
+ * clog( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Returns complex logarithm to the base e (2.718...) of
+ * the complex argument x.
+ *
+ * If z = x + iy, r = sqrt( x**2 + y**2 ),
+ * then
+ * w = log(r) + i arctan(y/x).
+ *
+ * The arctangent ranges from -PI to +PI.
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 7000 8.5e-17 1.9e-17
+ * IEEE -10,+10 30000 5.0e-15 1.1e-16
+ *
+ * Larger relative error can be observed for z near 1 +i0.
+ * In IEEE arithmetic the peak absolute error is 5.2e-16, rms
+ * absolute error 1.0e-16.
+ */
+
+/*
+Cephes Math Library Release 2.8: June, 2000
+Copyright 1984, 1995, 2000 by Stephen L. Moshier
+*/
+#include <math.h>
+#ifdef ANSIPROT
+static void cchsh ( double x, double *c, double *s );
+static double redupi ( double x );
+static double ctans ( cmplx *z );
+/* These are supposed to be in some standard place. */
+double fabs (double);
+double sqrt (double);
+double pow (double, double);
+double log (double);
+double exp (double);
+double atan2 (double, double);
+double cosh (double);
+double sinh (double);
+double asin (double);
+double sin (double);
+double cos (double);
+double cabs (cmplx *);
+void cadd ( cmplx *, cmplx *, cmplx * );
+void cmul ( cmplx *, cmplx *, cmplx * );
+void csqrt ( cmplx *, cmplx * );
+static void cchsh ( double, double *, double * );
+static double redupi ( double );
+static double ctans ( cmplx * );
+void clog ( cmplx *, cmplx * );
+void casin ( cmplx *, cmplx * );
+void cacos ( cmplx *, cmplx * );
+void catan ( cmplx *, cmplx * );
+#else
+static void cchsh();
+static double redupi();
+static double ctans();
+double cabs(), fabs(), sqrt(), pow();
+double log(), exp(), atan2(), cosh(), sinh();
+double asin(), sin(), cos();
+void cadd(), cmul(), csqrt();
+void clog(), casin(), cacos(), catan();
+#endif
+
+
+extern double MAXNUM, MACHEP, PI, PIO2;
+
+void clog( z, w )
+register cmplx *z, *w;
+{
+double p, rr;
+
+/*rr = sqrt( z->r * z->r + z->i * z->i );*/
+rr = cabs(z);
+p = log(rr);
+#if ANSIC
+rr = atan2( z->i, z->r );
+#else
+rr = atan2( z->r, z->i );
+if( rr > PI )
+ rr -= PI + PI;
+#endif
+w->i = rr;
+w->r = p;
+}
+ /* cexp()
+ *
+ * Complex exponential function
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void cexp();
+ * cmplx z, w;
+ *
+ * cexp( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Returns the exponential of the complex argument z
+ * into the complex result w.
+ *
+ * If
+ * z = x + iy,
+ * r = exp(x),
+ *
+ * then
+ *
+ * w = r cos y + i r sin y.
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 8700 3.7e-17 1.1e-17
+ * IEEE -10,+10 30000 3.0e-16 8.7e-17
+ *
+ */
+
+void cexp( z, w )
+register cmplx *z, *w;
+{
+double r;
+
+r = exp( z->r );
+w->r = r * cos( z->i );
+w->i = r * sin( z->i );
+}
+ /* csin()
+ *
+ * Complex circular sine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void csin();
+ * cmplx z, w;
+ *
+ * csin( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * If
+ * z = x + iy,
+ *
+ * then
+ *
+ * w = sin x cosh y + i cos x sinh y.
+ *
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 8400 5.3e-17 1.3e-17
+ * IEEE -10,+10 30000 3.8e-16 1.0e-16
+ * Also tested by csin(casin(z)) = z.
+ *
+ */
+
+void csin( z, w )
+register cmplx *z, *w;
+{
+double ch, sh;
+
+cchsh( z->i, &ch, &sh );
+w->r = sin( z->r ) * ch;
+w->i = cos( z->r ) * sh;
+}
+
+
+
+/* calculate cosh and sinh */
+
+static void cchsh( x, c, s )
+double x, *c, *s;
+{
+double e, ei;
+
+if( fabs(x) <= 0.5 )
+ {
+ *c = cosh(x);
+ *s = sinh(x);
+ }
+else
+ {
+ e = exp(x);
+ ei = 0.5/e;
+ e = 0.5 * e;
+ *s = e - ei;
+ *c = e + ei;
+ }
+}
+
+ /* ccos()
+ *
+ * Complex circular cosine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void ccos();
+ * cmplx z, w;
+ *
+ * ccos( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * If
+ * z = x + iy,
+ *
+ * then
+ *
+ * w = cos x cosh y - i sin x sinh y.
+ *
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 8400 4.5e-17 1.3e-17
+ * IEEE -10,+10 30000 3.8e-16 1.0e-16
+ */
+
+void ccos( z, w )
+register cmplx *z, *w;
+{
+double ch, sh;
+
+cchsh( z->i, &ch, &sh );
+w->r = cos( z->r ) * ch;
+w->i = -sin( z->r ) * sh;
+}
+ /* ctan()
+ *
+ * Complex circular tangent
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void ctan();
+ * cmplx z, w;
+ *
+ * ctan( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * If
+ * z = x + iy,
+ *
+ * then
+ *
+ * sin 2x + i sinh 2y
+ * w = --------------------.
+ * cos 2x + cosh 2y
+ *
+ * On the real axis the denominator is zero at odd multiples
+ * of PI/2. The denominator is evaluated by its Taylor
+ * series near these points.
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 5200 7.1e-17 1.6e-17
+ * IEEE -10,+10 30000 7.2e-16 1.2e-16
+ * Also tested by ctan * ccot = 1 and catan(ctan(z)) = z.
+ */
+
+void ctan( z, w )
+register cmplx *z, *w;
+{
+double d;
+
+d = cos( 2.0 * z->r ) + cosh( 2.0 * z->i );
+
+if( fabs(d) < 0.25 )
+ d = ctans(z);
+
+if( d == 0.0 )
+ {
+ mtherr( "ctan", OVERFLOW );
+ w->r = MAXNUM;
+ w->i = MAXNUM;
+ return;
+ }
+
+w->r = sin( 2.0 * z->r ) / d;
+w->i = sinh( 2.0 * z->i ) / d;
+}
+ /* ccot()
+ *
+ * Complex circular cotangent
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void ccot();
+ * cmplx z, w;
+ *
+ * ccot( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * If
+ * z = x + iy,
+ *
+ * then
+ *
+ * sin 2x - i sinh 2y
+ * w = --------------------.
+ * cosh 2y - cos 2x
+ *
+ * On the real axis, the denominator has zeros at even
+ * multiples of PI/2. Near these points it is evaluated
+ * by a Taylor series.
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 3000 6.5e-17 1.6e-17
+ * IEEE -10,+10 30000 9.2e-16 1.2e-16
+ * Also tested by ctan * ccot = 1 + i0.
+ */
+
+void ccot( z, w )
+register cmplx *z, *w;
+{
+double d;
+
+d = cosh(2.0 * z->i) - cos(2.0 * z->r);
+
+if( fabs(d) < 0.25 )
+ d = ctans(z);
+
+if( d == 0.0 )
+ {
+ mtherr( "ccot", OVERFLOW );
+ w->r = MAXNUM;
+ w->i = MAXNUM;
+ return;
+ }
+
+w->r = sin( 2.0 * z->r ) / d;
+w->i = -sinh( 2.0 * z->i ) / d;
+}
+
+/* Program to subtract nearest integer multiple of PI */
+/* extended precision value of PI: */
+#ifdef UNK
+static double DP1 = 3.14159265160560607910E0;
+static double DP2 = 1.98418714791870343106E-9;
+static double DP3 = 1.14423774522196636802E-17;
+#endif
+
+#ifdef DEC
+static unsigned short P1[] = {0040511,0007732,0120000,0000000,};
+static unsigned short P2[] = {0031010,0055060,0100000,0000000,};
+static unsigned short P3[] = {0022123,0011431,0105056,0001560,};
+#define DP1 *(double *)P1
+#define DP2 *(double *)P2
+#define DP3 *(double *)P3
+#endif
+
+#ifdef IBMPC
+static unsigned short P1[] = {0x0000,0x5400,0x21fb,0x4009};
+static unsigned short P2[] = {0x0000,0x1000,0x0b46,0x3e21};
+static unsigned short P3[] = {0xc06e,0x3145,0x6263,0x3c6a};
+#define DP1 *(double *)P1
+#define DP2 *(double *)P2
+#define DP3 *(double *)P3
+#endif
+
+#ifdef MIEEE
+static unsigned short P1[] = {
+0x4009,0x21fb,0x5400,0x0000
+};
+static unsigned short P2[] = {
+0x3e21,0x0b46,0x1000,0x0000
+};
+static unsigned short P3[] = {
+0x3c6a,0x6263,0x3145,0xc06e
+};
+#define DP1 *(double *)P1
+#define DP2 *(double *)P2
+#define DP3 *(double *)P3
+#endif
+
+static double redupi(x)
+double x;
+{
+double t;
+long i;
+
+t = x/PI;
+if( t >= 0.0 )
+ t += 0.5;
+else
+ t -= 0.5;
+
+i = t; /* the multiple */
+t = i;
+t = ((x - t * DP1) - t * DP2) - t * DP3;
+return(t);
+}
+
+/* Taylor series expansion for cosh(2y) - cos(2x) */
+
+static double ctans(z)
+cmplx *z;
+{
+double f, x, x2, y, y2, rn, t;
+double d;
+
+x = fabs( 2.0 * z->r );
+y = fabs( 2.0 * z->i );
+
+x = redupi(x);
+
+x = x * x;
+y = y * y;
+x2 = 1.0;
+y2 = 1.0;
+f = 1.0;
+rn = 0.0;
+d = 0.0;
+do
+ {
+ rn += 1.0;
+ f *= rn;
+ rn += 1.0;
+ f *= rn;
+ x2 *= x;
+ y2 *= y;
+ t = y2 + x2;
+ t /= f;
+ d += t;
+
+ rn += 1.0;
+ f *= rn;
+ rn += 1.0;
+ f *= rn;
+ x2 *= x;
+ y2 *= y;
+ t = y2 - x2;
+ t /= f;
+ d += t;
+ }
+while( fabs(t/d) > MACHEP );
+return(d);
+}
+ /* casin()
+ *
+ * Complex circular arc sine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void casin();
+ * cmplx z, w;
+ *
+ * casin( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Inverse complex sine:
+ *
+ * 2
+ * w = -i clog( iz + csqrt( 1 - z ) ).
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 10100 2.1e-15 3.4e-16
+ * IEEE -10,+10 30000 2.2e-14 2.7e-15
+ * Larger relative error can be observed for z near zero.
+ * Also tested by csin(casin(z)) = z.
+ */
+
+void casin( z, w )
+cmplx *z, *w;
+{
+static cmplx ca, ct, zz, z2;
+double x, y;
+
+x = z->r;
+y = z->i;
+
+if( y == 0.0 )
+ {
+ if( fabs(x) > 1.0 )
+ {
+ w->r = PIO2;
+ w->i = 0.0;
+ mtherr( "casin", DOMAIN );
+ }
+ else
+ {
+ w->r = asin(x);
+ w->i = 0.0;
+ }
+ return;
+ }
+
+/* Power series expansion */
+/*
+b = cabs(z);
+if( b < 0.125 )
+{
+z2.r = (x - y) * (x + y);
+z2.i = 2.0 * x * y;
+
+cn = 1.0;
+n = 1.0;
+ca.r = x;
+ca.i = y;
+sum.r = x;
+sum.i = y;
+do
+ {
+ ct.r = z2.r * ca.r - z2.i * ca.i;
+ ct.i = z2.r * ca.i + z2.i * ca.r;
+ ca.r = ct.r;
+ ca.i = ct.i;
+
+ cn *= n;
+ n += 1.0;
+ cn /= n;
+ n += 1.0;
+ b = cn/n;
+
+ ct.r *= b;
+ ct.i *= b;
+ sum.r += ct.r;
+ sum.i += ct.i;
+ b = fabs(ct.r) + fabs(ct.i);
+ }
+while( b > MACHEP );
+w->r = sum.r;
+w->i = sum.i;
+return;
+}
+*/
+
+
+ca.r = x;
+ca.i = y;
+
+ct.r = -ca.i; /* iz */
+ct.i = ca.r;
+
+ /* sqrt( 1 - z*z) */
+/* cmul( &ca, &ca, &zz ) */
+zz.r = (ca.r - ca.i) * (ca.r + ca.i); /*x * x - y * y */
+zz.i = 2.0 * ca.r * ca.i;
+
+zz.r = 1.0 - zz.r;
+zz.i = -zz.i;
+csqrt( &zz, &z2 );
+
+cadd( &z2, &ct, &zz );
+clog( &zz, &zz );
+w->r = zz.i; /* mult by 1/i = -i */
+w->i = -zz.r;
+return;
+}
+ /* cacos()
+ *
+ * Complex circular arc cosine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void cacos();
+ * cmplx z, w;
+ *
+ * cacos( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ *
+ * w = arccos z = PI/2 - arcsin z.
+ *
+ *
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 5200 1.6e-15 2.8e-16
+ * IEEE -10,+10 30000 1.8e-14 2.2e-15
+ */
+
+void cacos( z, w )
+cmplx *z, *w;
+{
+
+casin( z, w );
+w->r = PIO2 - w->r;
+w->i = -w->i;
+}
+ /* catan()
+ *
+ * Complex circular arc tangent
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void catan();
+ * cmplx z, w;
+ *
+ * catan( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * If
+ * z = x + iy,
+ *
+ * then
+ * 1 ( 2x )
+ * Re w = - arctan(-----------) + k PI
+ * 2 ( 2 2)
+ * (1 - x - y )
+ *
+ * ( 2 2)
+ * 1 (x + (y+1) )
+ * Im w = - log(------------)
+ * 4 ( 2 2)
+ * (x + (y-1) )
+ *
+ * Where k is an arbitrary integer.
+ *
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * DEC -10,+10 5900 1.3e-16 7.8e-18
+ * IEEE -10,+10 30000 2.3e-15 8.5e-17
+ * The check catan( ctan(z) ) = z, with |x| and |y| < PI/2,
+ * had peak relative error 1.5e-16, rms relative error
+ * 2.9e-17. See also clog().
+ */
+
+void catan( z, w )
+cmplx *z, *w;
+{
+double a, t, x, x2, y;
+
+x = z->r;
+y = z->i;
+
+if( (x == 0.0) && (y > 1.0) )
+ goto ovrf;
+
+x2 = x * x;
+a = 1.0 - x2 - (y * y);
+if( a == 0.0 )
+ goto ovrf;
+
+#if ANSIC
+t = atan2( 2.0 * x, a )/2.0;
+#else
+t = atan2( a, 2.0 * x )/2.0;
+#endif
+w->r = redupi( t );
+
+t = y - 1.0;
+a = x2 + (t * t);
+if( a == 0.0 )
+ goto ovrf;
+
+t = y + 1.0;
+a = (x2 + (t * t))/a;
+w->i = log(a)/4.0;
+return;
+
+ovrf:
+mtherr( "catan", OVERFLOW );
+w->r = MAXNUM;
+w->i = MAXNUM;
+}
+
+
+/* csinh
+ *
+ * Complex hyperbolic sine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void csinh();
+ * cmplx z, w;
+ *
+ * csinh( &z, &w );
+ *
+ *
+ * DESCRIPTION:
+ *
+ * csinh z = (cexp(z) - cexp(-z))/2
+ * = sinh x * cos y + i cosh x * sin y .
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * IEEE -10,+10 30000 3.1e-16 8.2e-17
+ *
+ */
+
+void
+csinh (z, w)
+ cmplx *z, *w;
+{
+ double x, y;
+
+ x = z->r;
+ y = z->i;
+ w->r = sinh (x) * cos (y);
+ w->i = cosh (x) * sin (y);
+}
+
+
+/* casinh
+ *
+ * Complex inverse hyperbolic sine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void casinh();
+ * cmplx z, w;
+ *
+ * casinh (&z, &w);
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * casinh z = -i casin iz .
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * IEEE -10,+10 30000 1.8e-14 2.6e-15
+ *
+ */
+
+void
+casinh (z, w)
+ cmplx *z, *w;
+{
+ cmplx u;
+
+ u.r = 0.0;
+ u.i = 1.0;
+ cmul( z, &u, &u );
+ casin( &u, w );
+ u.r = 0.0;
+ u.i = -1.0;
+ cmul( &u, w, w );
+}
+
+/* ccosh
+ *
+ * Complex hyperbolic cosine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void ccosh();
+ * cmplx z, w;
+ *
+ * ccosh (&z, &w);
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * ccosh(z) = cosh x cos y + i sinh x sin y .
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * IEEE -10,+10 30000 2.9e-16 8.1e-17
+ *
+ */
+
+void
+ccosh (z, w)
+ cmplx *z, *w;
+{
+ double x, y;
+
+ x = z->r;
+ y = z->i;
+ w->r = cosh (x) * cos (y);
+ w->i = sinh (x) * sin (y);
+}
+
+
+/* cacosh
+ *
+ * Complex inverse hyperbolic cosine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void cacosh();
+ * cmplx z, w;
+ *
+ * cacosh (&z, &w);
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * acosh z = i acos z .
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * IEEE -10,+10 30000 1.6e-14 2.1e-15
+ *
+ */
+
+void
+cacosh (z, w)
+ cmplx *z, *w;
+{
+ cmplx u;
+
+ cacos( z, w );
+ u.r = 0.0;
+ u.i = 1.0;
+ cmul( &u, w, w );
+}
+
+
+/* ctanh
+ *
+ * Complex hyperbolic tangent
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void ctanh();
+ * cmplx z, w;
+ *
+ * ctanh (&z, &w);
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * tanh z = (sinh 2x + i sin 2y) / (cosh 2x + cos 2y) .
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * IEEE -10,+10 30000 1.7e-14 2.4e-16
+ *
+ */
+
+/* 5.253E-02,1.550E+00 1.643E+01,6.553E+00 1.729E-14 21355 */
+
+void
+ctanh (z, w)
+ cmplx *z, *w;
+{
+ double x, y, d;
+
+ x = z->r;
+ y = z->i;
+ d = cosh (2.0 * x) + cos (2.0 * y);
+ w->r = sinh (2.0 * x) / d;
+ w->i = sin (2.0 * y) / d;
+ return;
+}
+
+
+/* catanh
+ *
+ * Complex inverse hyperbolic tangent
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void catanh();
+ * cmplx z, w;
+ *
+ * catanh (&z, &w);
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Inverse tanh, equal to -i catan (iz);
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * IEEE -10,+10 30000 2.3e-16 6.2e-17
+ *
+ */
+
+void
+catanh (z, w)
+ cmplx *z, *w;
+{
+ cmplx u;
+
+ u.r = 0.0;
+ u.i = 1.0;
+ cmul (z, &u, &u); /* i z */
+ catan (&u, w);
+ u.r = 0.0;
+ u.i = -1.0;
+ cmul (&u, w, w); /* -i catan iz */
+ return;
+}
+
+
+/* cpow
+ *
+ * Complex power function
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void cpow();
+ * cmplx a, z, w;
+ *
+ * cpow (&a, &z, &w);
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Raises complex A to the complex Zth power.
+ * Definition is per AMS55 # 4.2.8,
+ * analytically equivalent to cpow(a,z) = cexp(z clog(a)).
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * IEEE -10,+10 30000 9.4e-15 1.5e-15
+ *
+ */
+
+
+void
+cpow (a, z, w)
+ cmplx *a, *z, *w;
+{
+ double x, y, r, theta, absa, arga;
+
+ x = z->r;
+ y = z->i;
+ absa = cabs (a);
+ if (absa == 0.0)
+ {
+ w->r = 0.0;
+ w->i = 0.0;
+ return;
+ }
+ arga = atan2 (a->i, a->r);
+ r = pow (absa, x);
+ theta = x * arga;
+ if (y != 0.0)
+ {
+ r = r * exp (-y * arga);
+ theta = theta + y * log (absa);
+ }
+ w->r = r * cos (theta);
+ w->i = r * sin (theta);
+ return;
+}
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