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diff --git a/libm/double/clog.c b/libm/double/clog.c
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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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