/*
* CDDL HEADER START
*
* The contents of this file are subject to the terms of the
* Common Development and Distribution License (the "License").
* You may not use this file except in compliance with the License.
*
* You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
* or http://www.opensolaris.org/os/licensing.
* See the License for the specific language governing permissions
* and limitations under the License.
*
* When distributing Covered Code, include this CDDL HEADER in each
* file and include the License file at usr/src/OPENSOLARIS.LICENSE.
* If applicable, add the following below this CDDL HEADER, with the
* fields enclosed by brackets "[]" replaced with your own identifying
* information: Portions Copyright [yyyy] [name of copyright owner]
*
* CDDL HEADER END
*/
/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
#pragma weak __catan = catan
/* INDENT OFF */
/*
* dcomplex catan(dcomplex z);
*
* If
* z = x + iy,
*
* then
* 1 ( 2x ) 1 2 2
* Re w = - arctan(-----------) = - ATAN2(2x, 1 - x - y )
* 2 ( 2 2) 2
* (1 - x - y )
*
* ( 2 2)
* 1 (x + (y+1) ) 1 4y
* Im w = - log(------------) .= --- log [ 1 + ------------- ]
* 4 ( 2 2) 4 2 2
* (x + (y-1) ) x + (y-1)
*
* 2 16 3 y
* = t - 2t + -- t - ..., where t = -----------------
* 3 x*x + (y-1)*(y-1)
*
* Note that: if catan( x, y) = ( u, v), then
* catan(-x, y) = (-u, v)
* catan( x,-y) = ( u,-v)
*
* Also, catan(x,y) = -i*catanh(-y,x), or
* catanh(x,y) = i*catan(-y,x)
* So, if catanh(y,x) = (v,u), then catan(x,y) = -i*(-v,u) = (u,v), i.e.,
* catan(x,y) = (u,v)
*
* EXCEPTION CASES (conform to ISO/IEC 9899:1999(E)):
* catan( 0 , 0 ) = (0 , 0 )
* catan( NaN, 0 ) = (NaN , 0 )
* catan( 0 , 1 ) = (0 , +inf) with divide-by-zero
* catan( inf, y ) = (pi/2 , 0 ) for finite +y
* catan( NaN, y ) = (NaN , NaN ) with invalid for finite y != 0
* catan( x , inf ) = (pi/2 , 0 ) for finite +x
* catan( inf, inf ) = (pi/2 , 0 )
* catan( NaN, inf ) = (NaN , 0 )
* catan( x , NaN ) = (NaN , NaN ) with invalid for finite x
* catan( inf, NaN ) = (pi/2 , +-0 )
*/
/* INDENT ON */
#include "libm.h" /* atan/atan2/fabs/log/log1p */
#include "complex_wrapper.h"
/* INDENT OFF */
static const double
pi_2 = 1.570796326794896558e+00,
zero = 0.0,
half = 0.5,
two = 2.0,
ln2 = 6.931471805599453094172321214581765680755e-0001,
one = 1.0;
/* INDENT ON */
dcomplex
catan(dcomplex z) {
dcomplex ans;
double x, y, ax, ay, t;
int hx, hy, ix, iy;
unsigned lx, ly;
x = D_RE(z);
y = D_IM(z);
ax = fabs(x);
ay = fabs(y);
hx = HI_WORD(x);
lx = LO_WORD(x);
hy = HI_WORD(y);
ly = LO_WORD(y);
ix = hx & 0x7fffffff;
iy = hy & 0x7fffffff;
/* x is inf or NaN */
if (ix >= 0x7ff00000) {
if (ISINF(ix, lx)) {
D_RE(ans) = pi_2;
D_IM(ans) = zero;
} else {
D_RE(ans) = x + x;
if ((iy | ly) == 0 || (ISINF(iy, ly)))
D_IM(ans) = zero;
else
D_IM(ans) = (fabs(y) - ay) / (fabs(y) - ay);
}
} else if (iy >= 0x7ff00000) {
/* y is inf or NaN */
if (ISINF(iy, ly)) {
D_RE(ans) = pi_2;
D_IM(ans) = zero;
} else {
D_RE(ans) = (fabs(x) - ax) / (fabs(x) - ax);
D_IM(ans) = y;
}
} else if ((ix | lx) == 0) {
/* INDENT OFF */
/*
* x = 0
* 1 1
* A = --- * atan2(2x, 1-x*x-y*y) = --- atan2(0,1-|y|)
* 2 2
*
* 1 [ (y+1)*(y+1) ] 1 2 1 2y
* B = - log [ ------------ ] = - log (1+ ---) or - log(1+ ----)
* 4 [ (y-1)*(y-1) ] 2 y-1 2 1-y
*/
/* INDENT ON */
t = one - ay;
if (((iy - 0x3ff00000) | ly) == 0) {
/* y=1: catan(0,1)=(0,+inf) with 1/0 signal */
D_IM(ans) = ay / ax;
D_RE(ans) = zero;
} else if (iy >= 0x3ff00000) { /* y>1 */
D_IM(ans) = half * log1p(two / (-t));
D_RE(ans) = pi_2;
} else { /* y<1 */
D_IM(ans) = half * log1p((ay + ay) / t);
D_RE(ans) = zero;
}
} else if (iy < 0x3e200000 || ((ix - iy) >> 20) >= 30) {
/* INDENT OFF */
/*
* Tiny y (relative to 1+|x|)
* |y| < E*(1+|x|)
* where E=2**-29, -35, -60 for double, double extended, quad precision
*
* 1 [ x<=1: atan(x)
* A = --- * atan2(2x, 1-x*x-y*y) ~ [ 1 1+x
* 2 [ x>=1: - atan2(2,(1-x)*(-----))
* 2 x
*
* y/x
* B ~ t*(1-2t), where t = ----------------- is tiny
* x + (y-1)*(y-1)/x
*/
/* INDENT ON */
if (ix < 0x3ff00000)
D_RE(ans) = atan(ax);
else
D_RE(ans) = half * atan2(two, (one - ax) * (one +
one / ax));
if ((iy | ly) == 0) {
D_IM(ans) = ay;
} else {
if (ix < 0x3e200000)
t = ay / ((ay - one) * (ay - one));
else if (ix > 0x41c00000)
t = (ay / ax) / ax;
else
t = ay / (ax * ax + (ay - one) * (ay - one));
D_IM(ans) = t * (one - (t + t));
}
} else if (iy >= 0x41c00000 && ((iy - ix) >> 20) >= 30) {
/* INDENT OFF */
/*
* Huge y relative to 1+|x|
* |y| > Einv*(1+|x|), where Einv~2**(prec/2+3),
* 1
* A ~ --- * atan2(2x, -y*y) ~ pi/2
* 2
* y
* B ~ t*(1-2t), where t = --------------- is tiny
* (y-1)*(y-1)
*/
/* INDENT ON */
D_RE(ans) = pi_2;
t = (ay / (ay - one)) / (ay - one);
D_IM(ans) = t * (one - (t + t));
} else if (((iy - 0x3ff00000) | ly) == 0) {
/* INDENT OFF */
/*
* y = 1
* 1 1
* A = --- * atan2(2x, -x*x) = --- atan2(2,-x)
* 2 2
*
* 1 [x*x + 4] 1 4 [ 0.5(log2-logx) if
* B = - log [-------] = - log (1+ ---) = [ |x|<E, else 0.25*
* 4 [ x*x ] 4 x*x [ log1p((2/x)*(2/x))
*/
/* INDENT ON */
D_RE(ans) = half * atan2(two, -ax);
if (ix < 0x3e200000)
D_IM(ans) = half * (ln2 - log(ax));
else {
t = two / ax;
D_IM(ans) = 0.25 * log1p(t * t);
}
} else if (ix >= 0x43900000) {
/* INDENT OFF */
/*
* Huge x:
* when |x| > 1/E^2,
* 1 pi
* A ~ --- * atan2(2x, -x*x-y*y) ~ ---
* 2 2
* y y/x
* B ~ t*(1-2t), where t = --------------- = (-------------- )/x
* x*x+(y-1)*(y-1) 1+((y-1)/x)^2
*/
/* INDENT ON */
D_RE(ans) = pi_2;
t = ((ay / ax) / (one + ((ay - one) / ax) * ((ay - one) /
ax))) / ax;
D_IM(ans) = t * (one - (t + t));
} else if (ix < 0x38b00000) {
/* INDENT OFF */
/*
* Tiny x:
* when |x| < E^4, (note that y != 1)
* 1 1
* A = --- * atan2(2x, 1-x*x-y*y) ~ --- * atan2(2x,(1-y)*(1+y))
* 2 2
*
* 1 [(y+1)*(y+1)] 1 2 1 2y
* B = - log [-----------] = - log (1+ ---) or - log(1+ ----)
* 4 [(y-1)*(y-1)] 2 y-1 2 1-y
*/
/* INDENT ON */
D_RE(ans) = half * atan2(ax + ax, (one - ay) * (one + ay));
if (iy >= 0x3ff00000)
D_IM(ans) = half * log1p(two / (ay - one));
else
D_IM(ans) = half * log1p((ay + ay) / (one - ay));
} else {
/* INDENT OFF */
/*
* normal x,y
* 1
* A = --- * atan2(2x, 1-x*x-y*y)
* 2
*
* 1 [x*x+(y+1)*(y+1)] 1 4y
* B = - log [---------------] = - log (1+ -----------------)
* 4 [x*x+(y-1)*(y-1)] 4 x*x + (y-1)*(y-1)
*/
/* INDENT ON */
t = one - ay;
if (iy >= 0x3fe00000 && iy < 0x40000000) {
/* y close to 1 */
D_RE(ans) = half * (atan2((ax + ax), (t * (one + ay) -
ax * ax)));
} else if (ix >= 0x3fe00000 && ix < 0x40000000) {
/* x close to 1 */
D_RE(ans) = half * atan2((ax + ax), ((one - ax) *
(one + ax) - ay * ay));
} else
D_RE(ans) = half * atan2((ax + ax), ((one - ax * ax) -
ay * ay));
D_IM(ans) = 0.25 * log1p((4.0 * ay) / (ax * ax + t * t));
}
if (hx < 0)
D_RE(ans) = -D_RE(ans);
if (hy < 0)
D_IM(ans) = -D_IM(ans);
return (ans);
}