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-rw-r--r--libm/ldouble/clogl.c720
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diff --git a/libm/ldouble/clogl.c b/libm/ldouble/clogl.c
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-/* clogl.c
- *
- * Complex natural logarithm
- *
- *
- *
- * SYNOPSIS:
- *
- * void clogl();
- * cmplxl z, w;
- *
- * clogl( &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.
- */
-
-#include <math.h>
-#ifdef ANSIPROT
-static void cchshl ( long double x, long double *c, long double *s );
-static long double redupil ( long double x );
-static long double ctansl ( cmplxl *z );
-long double cabsl ( cmplxl *x );
-void csqrtl ( cmplxl *x, cmplxl *y );
-void caddl ( cmplxl *x, cmplxl *y, cmplxl *z );
-extern long double fabsl ( long double );
-extern long double sqrtl ( long double );
-extern long double logl ( long double );
-extern long double expl ( long double );
-extern long double atan2l ( long double, long double );
-extern long double coshl ( long double );
-extern long double sinhl ( long double );
-extern long double asinl ( long double );
-extern long double sinl ( long double );
-extern long double cosl ( long double );
-void clogl ( cmplxl *, cmplxl *);
-void casinl ( cmplxl *, cmplxl *);
-#else
-static void cchshl();
-static long double redupil();
-static long double ctansl();
-long double cabsl(), fabsl(), sqrtl();
-lnog double logl(), expl(), atan2l(), coshl(), sinhl();
-long double asinl(), sinl(), cosl();
-void caddl(), csqrtl(), clogl(), casinl();
-#endif
-
-extern long double MAXNUML, MACHEPL, PIL, PIO2L;
-
-void clogl( z, w )
-register cmplxl *z, *w;
-{
-long double p, rr;
-
-/*rr = sqrt( z->r * z->r + z->i * z->i );*/
-rr = cabsl(z);
-p = logl(rr);
-#if ANSIC
-rr = atan2l( z->i, z->r );
-#else
-rr = atan2l( z->r, z->i );
-if( rr > PIL )
- rr -= PIL + PIL;
-#endif
-w->i = rr;
-w->r = p;
-}
- /* cexpl()
- *
- * Complex exponential function
- *
- *
- *
- * SYNOPSIS:
- *
- * void cexpl();
- * cmplxl z, w;
- *
- * cexpl( &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 cexpl( z, w )
-register cmplxl *z, *w;
-{
-long double r;
-
-r = expl( z->r );
-w->r = r * cosl( z->i );
-w->i = r * sinl( z->i );
-}
- /* csinl()
- *
- * Complex circular sine
- *
- *
- *
- * SYNOPSIS:
- *
- * void csinl();
- * cmplxl z, w;
- *
- * csinl( &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 csinl( z, w )
-register cmplxl *z, *w;
-{
-long double ch, sh;
-
-cchshl( z->i, &ch, &sh );
-w->r = sinl( z->r ) * ch;
-w->i = cosl( z->r ) * sh;
-}
-
-
-
-/* calculate cosh and sinh */
-
-static void cchshl( x, c, s )
-long double x, *c, *s;
-{
-long double e, ei;
-
-if( fabsl(x) <= 0.5L )
- {
- *c = coshl(x);
- *s = sinhl(x);
- }
-else
- {
- e = expl(x);
- ei = 0.5L/e;
- e = 0.5L * e;
- *s = e - ei;
- *c = e + ei;
- }
-}
-
- /* ccosl()
- *
- * Complex circular cosine
- *
- *
- *
- * SYNOPSIS:
- *
- * void ccosl();
- * cmplxl z, w;
- *
- * ccosl( &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 ccosl( z, w )
-register cmplxl *z, *w;
-{
-long double ch, sh;
-
-cchshl( z->i, &ch, &sh );
-w->r = cosl( z->r ) * ch;
-w->i = -sinl( z->r ) * sh;
-}
- /* ctanl()
- *
- * Complex circular tangent
- *
- *
- *
- * SYNOPSIS:
- *
- * void ctanl();
- * cmplxl z, w;
- *
- * ctanl( &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 ctanl( z, w )
-register cmplxl *z, *w;
-{
-long double d;
-
-d = cosl( 2.0L * z->r ) + coshl( 2.0L * z->i );
-
-if( fabsl(d) < 0.25L )
- d = ctansl(z);
-
-if( d == 0.0L )
- {
- mtherr( "ctan", OVERFLOW );
- w->r = MAXNUML;
- w->i = MAXNUML;
- return;
- }
-
-w->r = sinl( 2.0L * z->r ) / d;
-w->i = sinhl( 2.0L * z->i ) / d;
-}
- /* ccotl()
- *
- * Complex circular cotangent
- *
- *
- *
- * SYNOPSIS:
- *
- * void ccotl();
- * cmplxl z, w;
- *
- * ccotl( &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 ccotl( z, w )
-register cmplxl *z, *w;
-{
-long double d;
-
-d = coshl(2.0L * z->i) - cosl(2.0L * z->r);
-
-if( fabsl(d) < 0.25L )
- d = ctansl(z);
-
-if( d == 0.0L )
- {
- mtherr( "ccot", OVERFLOW );
- w->r = MAXNUML;
- w->i = MAXNUML;
- return;
- }
-
-w->r = sinl( 2.0L * z->r ) / d;
-w->i = -sinhl( 2.0L * 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 long double redupil(x)
-long double x;
-{
-long double t;
-long i;
-
-t = x/PIL;
-if( t >= 0.0L )
- t += 0.5L;
-else
- t -= 0.5L;
-
-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 long double ctansl(z)
-cmplxl *z;
-{
-long double f, x, x2, y, y2, rn, t;
-long double d;
-
-x = fabsl( 2.0L * z->r );
-y = fabsl( 2.0L * z->i );
-
-x = redupil(x);
-
-x = x * x;
-y = y * y;
-x2 = 1.0L;
-y2 = 1.0L;
-f = 1.0L;
-rn = 0.0;
-d = 0.0;
-do
- {
- rn += 1.0L;
- f *= rn;
- rn += 1.0L;
- f *= rn;
- x2 *= x;
- y2 *= y;
- t = y2 + x2;
- t /= f;
- d += t;
-
- rn += 1.0L;
- f *= rn;
- rn += 1.0L;
- f *= rn;
- x2 *= x;
- y2 *= y;
- t = y2 - x2;
- t /= f;
- d += t;
- }
-while( fabsl(t/d) > MACHEPL );
-return(d);
-}
- /* casinl()
- *
- * Complex circular arc sine
- *
- *
- *
- * SYNOPSIS:
- *
- * void casinl();
- * cmplxl z, w;
- *
- * casinl( &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 casinl( z, w )
-cmplxl *z, *w;
-{
-static cmplxl ca, ct, zz, z2;
-long double x, y;
-
-x = z->r;
-y = z->i;
-
-if( y == 0.0L )
- {
- if( fabsl(x) > 1.0L )
- {
- w->r = PIO2L;
- w->i = 0.0L;
- mtherr( "casinl", DOMAIN );
- }
- else
- {
- w->r = asinl(x);
- w->i = 0.0L;
- }
- return;
- }
-
-/* Power series expansion */
-/*
-b = cabsl(z);
-if( b < 0.125L )
-{
-z2.r = (x - y) * (x + y);
-z2.i = 2.0L * x * y;
-
-cn = 1.0L;
-n = 1.0L;
-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 += 1.0L;
- cn /= n;
- n += 1.0L;
- b = cn/n;
-
- ct.r *= b;
- ct.i *= b;
- sum.r += ct.r;
- sum.i += ct.i;
- b = fabsl(ct.r) + fabs(ct.i);
- }
-while( b > MACHEPL );
-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 = 2.0L * ca.r * ca.i;
-
-zz.r = 1.0L - zz.r;
-zz.i = -zz.i;
-csqrtl( &zz, &z2 );
-
-caddl( &z2, &ct, &zz );
-clogl( &zz, &zz );
-w->r = zz.i; /* mult by 1/i = -i */
-w->i = -zz.r;
-return;
-}
- /* cacosl()
- *
- * Complex circular arc cosine
- *
- *
- *
- * SYNOPSIS:
- *
- * void cacosl();
- * cmplxl z, w;
- *
- * cacosl( &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 cacosl( z, w )
-cmplxl *z, *w;
-{
-
-casinl( z, w );
-w->r = PIO2L - w->r;
-w->i = -w->i;
-}
- /* catanl()
- *
- * Complex circular arc tangent
- *
- *
- *
- * SYNOPSIS:
- *
- * void catanl();
- * cmplxl z, w;
- *
- * catanl( &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 catanl( z, w )
-cmplxl *z, *w;
-{
-long double a, t, x, x2, y;
-
-x = z->r;
-y = z->i;
-
-if( (x == 0.0L) && (y > 1.0L) )
- goto ovrf;
-
-x2 = x * x;
-a = 1.0L - x2 - (y * y);
-if( a == 0.0L )
- goto ovrf;
-
-#if ANSIC
-t = atan2l( 2.0L * x, a ) * 0.5L;
-#else
-t = atan2l( a, 2.0 * x ) * 0.5L;
-#endif
-w->r = redupil( t );
-
-t = y - 1.0L;
-a = x2 + (t * t);
-if( a == 0.0L )
- goto ovrf;
-
-t = y + 1.0L;
-a = (x2 + (t * t))/a;
-w->i = logl(a)/4.0;
-return;
-
-ovrf:
-mtherr( "catanl", OVERFLOW );
-w->r = MAXNUML;
-w->i = MAXNUML;
-}
generated by cgit v1.2.3 (git 2.25.1) at 2024-04-18 12:09:54 +0000