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-rw-r--r--libm/float/clogf.c669
1 files changed, 669 insertions, 0 deletions
diff --git a/libm/float/clogf.c b/libm/float/clogf.c
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+++ b/libm/float/clogf.c
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+/* clogf.c
+ *
+ * Complex natural logarithm
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void clogf();
+ * cmplxf z, w;
+ *
+ * clogf( &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
+ * IEEE -10,+10 30000 1.9e-6 6.2e-8
+ *
+ * Larger relative error can be observed for z near 1 +i0.
+ * In IEEE arithmetic the peak absolute error is 3.1e-7.
+ *
+ */
+
+#include <math.h>
+extern float MAXNUMF, MACHEPF, PIF, PIO2F;
+#ifdef ANSIC
+float cabsf(cmplxf *), sqrtf(float), logf(float), atan2f(float, float);
+float expf(float), sinf(float), cosf(float);
+float coshf(float), sinhf(float), asinf(float);
+float ctansf(cmplxf *), redupif(float);
+void cchshf( float, float *, float * );
+void caddf( cmplxf *, cmplxf *, cmplxf * );
+void csqrtf( cmplxf *, cmplxf * );
+#else
+float cabsf(), sqrtf(), logf(), atan2f();
+float expf(), sinf(), cosf();
+float coshf(), sinhf(), asinf();
+float ctansf(), redupif();
+void cchshf(), csqrtf(), caddf();
+#endif
+
+#define fabsf(x) ( (x) < 0 ? -(x) : (x) )
+
+void clogf( z, w )
+register cmplxf *z, *w;
+{
+float p, rr;
+
+/*rr = sqrtf( z->r * z->r + z->i * z->i );*/
+rr = cabsf(z);
+p = logf(rr);
+#if ANSIC
+rr = atan2f( z->i, z->r );
+#else
+rr = atan2f( z->r, z->i );
+if( rr > PIF )
+ rr -= PIF + PIF;
+#endif
+w->i = rr;
+w->r = p;
+}
+ /* cexpf()
+ *
+ * Complex exponential function
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void cexpf();
+ * cmplxf z, w;
+ *
+ * cexpf( &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
+ * IEEE -10,+10 30000 1.4e-7 4.5e-8
+ *
+ */
+
+void cexpf( z, w )
+register cmplxf *z, *w;
+{
+float r;
+
+r = expf( z->r );
+w->r = r * cosf( z->i );
+w->i = r * sinf( z->i );
+}
+ /* csinf()
+ *
+ * Complex circular sine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void csinf();
+ * cmplxf z, w;
+ *
+ * csinf( &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
+ * IEEE -10,+10 30000 1.9e-7 5.5e-8
+ *
+ */
+
+void csinf( z, w )
+register cmplxf *z, *w;
+{
+float ch, sh;
+
+cchshf( z->i, &ch, &sh );
+w->r = sinf( z->r ) * ch;
+w->i = cosf( z->r ) * sh;
+}
+
+
+
+/* calculate cosh and sinh */
+
+void cchshf( float xx, float *c, float *s )
+{
+float x, e, ei;
+
+x = xx;
+if( fabsf(x) <= 0.5f )
+ {
+ *c = coshf(x);
+ *s = sinhf(x);
+ }
+else
+ {
+ e = expf(x);
+ ei = 0.5f/e;
+ e = 0.5f * e;
+ *s = e - ei;
+ *c = e + ei;
+ }
+}
+
+ /* ccosf()
+ *
+ * Complex circular cosine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void ccosf();
+ * cmplxf z, w;
+ *
+ * ccosf( &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
+ * IEEE -10,+10 30000 1.8e-7 5.5e-8
+ */
+
+void ccosf( z, w )
+register cmplxf *z, *w;
+{
+float ch, sh;
+
+cchshf( z->i, &ch, &sh );
+w->r = cosf( z->r ) * ch;
+w->i = -sinf( z->r ) * sh;
+}
+ /* ctanf()
+ *
+ * Complex circular tangent
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void ctanf();
+ * cmplxf z, w;
+ *
+ * ctanf( &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
+ * IEEE -10,+10 30000 3.3e-7 5.1e-8
+ */
+
+void ctanf( z, w )
+register cmplxf *z, *w;
+{
+float d;
+
+d = cosf( 2.0f * z->r ) + coshf( 2.0f * z->i );
+
+if( fabsf(d) < 0.25f )
+ d = ctansf(z);
+
+if( d == 0.0f )
+ {
+ mtherr( "ctanf", OVERFLOW );
+ w->r = MAXNUMF;
+ w->i = MAXNUMF;
+ return;
+ }
+
+w->r = sinf( 2.0f * z->r ) / d;
+w->i = sinhf( 2.0f * z->i ) / d;
+}
+ /* ccotf()
+ *
+ * Complex circular cotangent
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void ccotf();
+ * cmplxf z, w;
+ *
+ * ccotf( &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
+ * IEEE -10,+10 30000 3.6e-7 5.7e-8
+ * Also tested by ctan * ccot = 1 + i0.
+ */
+
+void ccotf( z, w )
+register cmplxf *z, *w;
+{
+float d;
+
+
+d = coshf(2.0f * z->i) - cosf(2.0f * z->r);
+
+if( fabsf(d) < 0.25f )
+ d = ctansf(z);
+
+if( d == 0.0f )
+ {
+ mtherr( "ccotf", OVERFLOW );
+ w->r = MAXNUMF;
+ w->i = MAXNUMF;
+ return;
+ }
+
+d = 1.0f/d;
+w->r = sinf( 2.0f * z->r ) * d;
+w->i = -sinhf( 2.0f * z->i ) * d;
+}
+
+/* Program to subtract nearest integer multiple of PI */
+/* extended precision value of PI: */
+
+static float DP1 = 3.140625;
+static float DP2 = 9.67502593994140625E-4;
+static float DP3 = 1.509957990978376432E-7;
+
+
+float redupif(float xx)
+{
+float x, t;
+long i;
+
+x = xx;
+t = x/PIF;
+if( t >= 0.0f )
+ t += 0.5f;
+else
+ t -= 0.5f;
+
+i = t; /* the multiple */
+t = i;
+t = ((x - t * DP1) - t * DP2) - t * DP3;
+return(t);
+}
+
+/* Taylor series expansion for cosh(2y) - cos(2x) */
+
+float ctansf(z)
+cmplxf *z;
+{
+float f, x, x2, y, y2, rn, t, d;
+
+x = fabsf( 2.0f * z->r );
+y = fabsf( 2.0f * z->i );
+
+x = redupif(x);
+
+x = x * x;
+y = y * y;
+x2 = 1.0f;
+y2 = 1.0f;
+f = 1.0f;
+rn = 0.0f;
+d = 0.0f;
+do
+ {
+ rn += 1.0f;
+ f *= rn;
+ rn += 1.0f;
+ f *= rn;
+ x2 *= x;
+ y2 *= y;
+ t = y2 + x2;
+ t /= f;
+ d += t;
+
+ rn += 1.0f;
+ f *= rn;
+ rn += 1.0f;
+ f *= rn;
+ x2 *= x;
+ y2 *= y;
+ t = y2 - x2;
+ t /= f;
+ d += t;
+ }
+while( fabsf(t/d) > MACHEPF );
+return(d);
+}
+ /* casinf()
+ *
+ * Complex circular arc sine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void casinf();
+ * cmplxf z, w;
+ *
+ * casinf( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Inverse complex sine:
+ *
+ * 2
+ * w = -i clog( iz + csqrt( 1 - z ) ).
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * IEEE -10,+10 30000 1.1e-5 1.5e-6
+ * Larger relative error can be observed for z near zero.
+ *
+ */
+
+void casinf( z, w )
+cmplxf *z, *w;
+{
+float x, y;
+static cmplxf ca, ct, zz, z2;
+/*
+float cn, n;
+static float a, b, s, t, u, v, y2;
+static cmplxf sum;
+*/
+
+x = z->r;
+y = z->i;
+
+if( y == 0.0f )
+ {
+ if( fabsf(x) > 1.0f )
+ {
+ w->r = PIO2F;
+ w->i = 0.0f;
+ mtherr( "casinf", DOMAIN );
+ }
+ else
+ {
+ w->r = asinf(x);
+ w->i = 0.0f;
+ }
+ return;
+ }
+
+/* Power series expansion */
+/*
+b = cabsf(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 = fabsf(ct.r) + fabsf(ct.i);
+ }
+while( b > MACHEPF );
+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 = 2.0f * ca.r * ca.i;
+
+zz.r = 1.0f - zz.r;
+zz.i = -zz.i;
+csqrtf( &zz, &z2 );
+
+caddf( &z2, &ct, &zz );
+clogf( &zz, &zz );
+w->r = zz.i; /* mult by 1/i = -i */
+w->i = -zz.r;
+return;
+}
+ /* cacosf()
+ *
+ * Complex circular arc cosine
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void cacosf();
+ * cmplxf z, w;
+ *
+ * cacosf( &z, &w );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ *
+ * w = arccos z = PI/2 - arcsin z.
+ *
+ *
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * IEEE -10,+10 30000 9.2e-6 1.2e-6
+ *
+ */
+
+void cacosf( z, w )
+cmplxf *z, *w;
+{
+
+casinf( z, w );
+w->r = PIO2F - w->r;
+w->i = -w->i;
+}
+ /* catan()
+ *
+ * Complex circular arc tangent
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * void catan();
+ * cmplxf 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
+ * IEEE -10,+10 30000 2.3e-6 5.2e-8
+ *
+ */
+
+void catanf( z, w )
+cmplxf *z, *w;
+{
+float a, t, x, x2, y;
+
+x = z->r;
+y = z->i;
+
+if( (x == 0.0f) && (y > 1.0f) )
+ goto ovrf;
+
+x2 = x * x;
+a = 1.0f - x2 - (y * y);
+if( a == 0.0f )
+ goto ovrf;
+
+#if ANSIC
+t = 0.5f * atan2f( 2.0f * x, a );
+#else
+t = 0.5f * atan2f( a, 2.0f * x );
+#endif
+w->r = redupif( t );
+
+t = y - 1.0f;
+a = x2 + (t * t);
+if( a == 0.0f )
+ goto ovrf;
+
+t = y + 1.0f;
+a = (x2 + (t * t))/a;
+w->i = 0.25f*logf(a);
+return;
+
+ovrf:
+mtherr( "catanf", OVERFLOW );
+w->r = MAXNUMF;
+w->i = MAXNUMF;
+}
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