xref: /illumos-gate/usr/src/lib/libm/common/complex/cacos.c (revision ddc0e0b5)

/*
 * 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 __cacos = cacos

/* INDENT OFF */
/*
 * dcomplex cacos(dcomplex z);
 *
 * Alogrithm
 * (based on T.E.Hull, Thomas F. Fairgrieve and Ping Tak Peter Tang's
 * paper "Implementing the Complex Arcsine and Arccosine Functins Using
 * Exception Handling", ACM TOMS, Vol 23, pp 299-335)
 *
 * The principal value of complex inverse cosine function cacos(z),
 * where z = x+iy, can be defined by
 *
 * 	cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)),
 *
 * where the log function is the natural log, and
 *             ____________           ____________
 *       1    /     2    2      1    /     2    2
 *  A = ---  / (x+1)  + y   +  ---  / (x-1)  + y
 *       2 \/                   2 \/
 *             ____________           ____________
 *       1    /     2    2      1    /     2    2
 *  B = ---  / (x+1)  + y   -  ---  / (x-1)  + y   .
 *       2 \/                   2 \/
 *
 * The Branch cuts are on the real line from -inf to -1 and from 1 to inf.
 * The real and imaginary parts are based on Abramowitz and Stegun
 * [Handbook of Mathematic Functions, 1972].  The sign of the imaginary
 * part is chosen to be the generally considered the principal value of
 * this function.
 *
 * Notes:1. A is the average of the distances from z to the points (1,0)
 *          and (-1,0) in the complex z-plane, and in particular A>=1.
 *       2. B is in [-1,1], and A*B = x
 *
 * Basic relations
 *    cacos(conj(z)) = conj(cacos(z))
 *    cacos(-z)      = pi   - cacos(z)
 *    cacos( z)      = pi/2 - casin(z)
 *
 * Special cases (conform to ISO/IEC 9899:1999(E)):
 *    cacos(+-0  + i y  ) = pi/2 - i y for y is +-0, +-inf, NaN
 *    cacos( x   + i inf) = pi/2 - i inf for all x
 *    cacos( x   + i NaN) = NaN  + i NaN with invalid for non-zero finite x
 *    cacos(-inf + i y  ) = pi   - i inf for finite +y
 *    cacos( inf + i y  ) = 0    - i inf for finite +y
 *    cacos(-inf + i inf) = 3pi/4- i inf
 *    cacos( inf + i inf) = pi/4 - i inf
 *    cacos(+-inf+ i NaN) = NaN  - i inf (sign of imaginary is unspecified)
 *    cacos(NaN  + i y  ) = NaN  + i NaN with invalid for finite y
 *    cacos(NaN  + i inf) = NaN  - i inf
 *    cacos(NaN  + i NaN) = NaN  + i NaN
 *
 * Special Regions (better formula for accuracy and for avoiding spurious
 * overflow or underflow) (all x and y are assumed nonnegative):
 *  case 1: y = 0
 *  case 2: tiny y relative to x-1: y <= ulp(0.5)*|x-1|
 *  case 3: tiny y: y < 4 sqrt(u), where u = minimum normal number
 *  case 4: huge y relative to x+1: y >= (1+x)/ulp(0.5)
 *  case 5: huge x and y: x and y >= sqrt(M)/8, where M = maximum normal number
 *  case 6: tiny x: x < 4 sqrt(u)
 *  --------
 *  case	1 & 2. y=0 or y/|x-1| is tiny. We have
 *             ____________              _____________
 *            /      2    2             /       y    2
 *           / (x+-1)  + y   =  |x+-1| / 1 + (------)
 *         \/                        \/       |x+-1|
 *
 *                                            1     y    2
 *                           ~  |x+-1| ( 1 + --- (------)  )
 *                                            2   |x+-1|
 *
 *                                          2
 *                                         y
 *                           = |x+-1| + --------.
 *                                      2|x+-1|
 *
 *	Consequently, it is not difficult to see that
 *                                 2
 *                                y
 *                    [ 1 + ------------ ,     if x < 1,
 *                    [      2(1+x)(1-x)
 *                    [
 *                    [
 *                    [ x,                     if x = 1 (y = 0),
 *                    [
 *		A ~=  [             2
 *                    [        x * y
 *                    [ x + ------------ ~ x,  if x > 1
 *                    [      2(x+1)(x-1)
 *
 *	and hence
 *                      ______                                 2
 *                     / 2                    y               y
 *               A + \/ A  - 1  ~  1 + ---------------- + -----------, if x < 1,
 *                                     sqrt((x+1)(1-x))   2(x+1)(1-x)
 *
 *
 *			        ~  x + sqrt((x-1)*(x+1)),             if x >= 1.
 *
 *                                         2
 *                                        y
 *                          [ x(1 - -----------) ~ x,  if x < 1,
 *                          [       2(1+x)(1-x)
 *		B = x/A  ~  [
 *                          [ 1,                       if x = 1,
 *			    [
 *                          [           2
 *                          [          y
 *                          [ 1 - ------------ ,       if x > 1,
 *                          [      2(x+1)(x-1)
 *	Thus
 *                            [ acos(x) - i y/sqrt((x-1)*(x+1)),      if x < 1,
 *                            [
 *		cacos(x+i*y)~ [ 0 - i 0,                              if x = 1,
 *                            [
 *                            [ y/sqrt(x*x-1) - i log(x+sqrt(x*x-1)), if x > 1.
 *
 *      Note: y/sqrt(x*x-1) ~ y/x when x >= 2**26.
 *  case 3. y < 4 sqrt(u), where u = minimum normal x.
 *	After case 1 and 2, this will only occurs when x=1. When x=1, we have
 *	   A = (sqrt(4+y*y)+y)/2 ~ 1 + y/2 + y^2/8 + ...
 *	and
 *	   B = 1/A = 1 - y/2 + y^2/8 + ...
 * 	Since
 *         cos(sqrt(y)) ~ 1 - y/2 + ...
 *      we have, for the real part,
 *         acos(B) ~ acos(1 - y/2) ~ sqrt(y)
 *	For the imaginary part,
 *	   log(A+sqrt(A*A-1)) ~ log(1+y/2+sqrt(2*y/2))
 *	                      = log(1+y/2+sqrt(y))
 *	                      = (y/2+sqrt(y)) - (y/2+sqrt(y))^2/2 + ...
 *	                      ~ sqrt(y) - y*(sqrt(y)+y/2)/2
 *	                      ~ sqrt(y)
 *
 *  case 4. y >= (x+1)/ulp(0.5). In this case, A ~ y and B ~ x/y. Thus
 *	   real part = acos(B) ~ pi/2
 * 	and
 *	   imag part = log(y+sqrt(y*y-one))
 *
 *  case 5. Both x and y are large: x and y > sqrt(M)/8, where M = maximum x
 *	In this case,
 *	   A ~ sqrt(x*x+y*y)
 *	   B ~ x/sqrt(x*x+y*y).
 *	Thus
 *	   real part = acos(B) = atan(y/x),
 *	   imag part = log(A+sqrt(A*A-1)) ~ log(2A)
 *	             = log(2) + 0.5*log(x*x+y*y)
 *	             = log(2) + log(y) + 0.5*log(1+(x/y)^2)
 *
 *  case 6. x < 4 sqrt(u). In this case, we have
 *	    A ~ sqrt(1+y*y), B = x/sqrt(1+y*y).
 *	Since B is tiny, we have
 *	    real part = acos(B) ~ pi/2
 *	    imag part = log(A+sqrt(A*A-1)) = log (A+sqrt(y*y))
 *	              = log(y+sqrt(1+y*y))
 *	              = 0.5*log(y^2+2ysqrt(1+y^2)+1+y^2)
 *	              = 0.5*log(1+2y(y+sqrt(1+y^2)));
 *	              = 0.5*log1p(2y(y+A));
 *
 * 	cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)),
 */
/* INDENT ON */

#include "libm.h"
#include "complex_wrapper.h"

/* INDENT OFF */
static const double
	zero = 0.0,
	one = 1.0,
	E = 1.11022302462515654042e-16,			/* 2**-53 */
	ln2 = 6.93147180559945286227e-01,
	pi = 3.1415926535897931159979634685,
	pi_l = 1.224646799147353177e-16,
	pi_2 = 1.570796326794896558e+00,
	pi_2_l = 6.123233995736765886e-17,
	pi_4 = 0.78539816339744827899949,
	pi_4_l = 3.061616997868382943e-17,
	pi3_4 = 2.356194490192344836998,
	pi3_4_l = 9.184850993605148829195e-17,
	Foursqrtu = 5.96667258496016539463e-154,	/* 2**(-509) */
	Acrossover = 1.5,
	Bcrossover = 0.6417,
	half = 0.5;
/* INDENT ON */

dcomplex
cacos(dcomplex z) {
	double x, y, t, R, S, A, Am1, B, y2, xm1, xp1, Apx;
	int ix, iy, hx, hy;
	unsigned lx, ly;
	dcomplex ans;

	x = D_RE(z);
	y = D_IM(z);
	hx = HI_WORD(x);
	lx = LO_WORD(x);
	hy = HI_WORD(y);
	ly = LO_WORD(y);
	ix = hx & 0x7fffffff;
	iy = hy & 0x7fffffff;

	/* x is 0 */
	if ((ix | lx) == 0) {
		if (((iy | ly) == 0) || (iy >= 0x7ff00000)) {
			D_RE(ans) = pi_2;
			D_IM(ans) = -y;
			return (ans);
		}
	}

	/* |y| is inf or NaN */
	if (iy >= 0x7ff00000) {
		if (ISINF(iy, ly)) {	/* cacos(x + i inf) = pi/2  - i inf */
			D_IM(ans) = -y;
			if (ix < 0x7ff00000) {
				D_RE(ans) = pi_2 + pi_2_l;
			} else if (ISINF(ix, lx)) {
				if (hx >= 0)
					D_RE(ans) = pi_4 + pi_4_l;
				else
					D_RE(ans) = pi3_4 + pi3_4_l;
			} else {
				D_RE(ans) = x;
			}
		} else {		/* cacos(x + i NaN) = NaN  + i NaN */
			D_RE(ans) = y + x;
			if (ISINF(ix, lx))
				D_IM(ans) = -fabs(x);
			else
				D_IM(ans) = y;
		}
		return (ans);
	}

	x = fabs(x);
	y = fabs(y);

	/* x is inf or NaN */
	if (ix >= 0x7ff00000) {	/* x is inf or NaN */
		if (ISINF(ix, lx)) {	/* x is INF */
			D_IM(ans) = -x;
			if (iy >= 0x7ff00000) {
				if (ISINF(iy, ly)) {
					/* INDENT OFF */
					/* cacos(inf + i inf) = pi/4 - i inf */
					/* cacos(-inf+ i inf) =3pi/4 - i inf */
					/* INDENT ON */
					if (hx >= 0)
						D_RE(ans) = pi_4 + pi_4_l;
					else
						D_RE(ans) = pi3_4 + pi3_4_l;
				} else
					/* INDENT OFF */
					/* cacos(inf + i NaN) = NaN  - i inf  */
					/* INDENT ON */
					D_RE(ans) = y + y;
			} else
				/* INDENT OFF */
				/* cacos(inf + iy ) = 0  - i inf */
				/* cacos(-inf+ iy  ) = pi - i inf */
				/* INDENT ON */
			if (hx >= 0)
				D_RE(ans) = zero;
			else
				D_RE(ans) = pi + pi_l;
		} else {		/* x is NaN */
			/* INDENT OFF */
			/*
			 * cacos(NaN + i inf) = NaN  - i inf
			 * cacos(NaN + i y  ) = NaN  + i NaN
			 * cacos(NaN + i NaN) = NaN  + i NaN
			 */
			/* INDENT ON */
			D_RE(ans) = x + y;
			if (iy >= 0x7ff00000) {
				D_IM(ans) = -y;
			} else {
				D_IM(ans) = x;
			}
		}
		if (hy < 0)
			D_IM(ans) = -D_IM(ans);
		return (ans);
	}

	if ((iy | ly) == 0) {	/* region 1: y=0 */
		if (ix < 0x3ff00000) {	/* |x| < 1 */
			D_RE(ans) = acos(x);
			D_IM(ans) = zero;
		} else {
			D_RE(ans) = zero;
			if (ix >= 0x43500000)	/* |x| >= 2**54 */
				D_IM(ans) = ln2 + log(x);
			else if (ix >= 0x3ff80000)	/* x > Acrossover */
				D_IM(ans) = log(x + sqrt((x - one) * (x +
					one)));
			else {
				xm1 = x - one;
				D_IM(ans) = log1p(xm1 + sqrt(xm1 * (x + one)));
			}
		}
	} else if (y <= E * fabs(x - one)) {	/* region 2: y < tiny*|x-1| */
		if (ix < 0x3ff00000) {	/* x < 1 */
			D_RE(ans) = acos(x);
			D_IM(ans) = y / sqrt((one + x) * (one - x));
		} else if (ix >= 0x43500000) {	/* |x| >= 2**54 */
			D_RE(ans) = y / x;
			D_IM(ans) = ln2 + log(x);
		} else {
			t = sqrt((x - one) * (x + one));
			D_RE(ans) = y / t;
			if (ix >= 0x3ff80000)	/* x > Acrossover */
				D_IM(ans) = log(x + t);
			else
				D_IM(ans) = log1p((x - one) + t);
		}
	} else if (y < Foursqrtu) {	/* region 3 */
		t = sqrt(y);
		D_RE(ans) = t;
		D_IM(ans) = t;
	} else if (E * y - one >= x) {	/* region 4 */
		D_RE(ans) = pi_2;
		D_IM(ans) = ln2 + log(y);
	} else if (ix >= 0x5fc00000 || iy >= 0x5fc00000) {	/* x,y>2**509 */
		/* region 5: x+1 or y is very large (>= sqrt(max)/8) */
		t = x / y;
		D_RE(ans) = atan(y / x);
		D_IM(ans) = ln2 + log(y) + half * log1p(t * t);
	} else if (x < Foursqrtu) {
		/* region 6: x is very small, < 4sqrt(min) */
		D_RE(ans) = pi_2;
		A = sqrt(one + y * y);
		if (iy >= 0x3ff80000)	/* if y > Acrossover */
			D_IM(ans) = log(y + A);
		else
			D_IM(ans) = half * log1p((y + y) * (y + A));
	} else {	/* safe region */
		y2 = y * y;
		xp1 = x + one;
		xm1 = x - one;
		R = sqrt(xp1 * xp1 + y2);
		S = sqrt(xm1 * xm1 + y2);
		A = half * (R + S);
		B = x / A;
		if (B <= Bcrossover)
			D_RE(ans) = acos(B);
		else {		/* use atan and an accurate approx to a-x */
			Apx = A + x;
			if (x <= one)
				D_RE(ans) = atan(sqrt(half * Apx * (y2 / (R +
					xp1) + (S - xm1))) / x);
			else
				D_RE(ans) = atan((y * sqrt(half * (Apx / (R +
					xp1) + Apx / (S + xm1)))) / x);
		}
		if (A <= Acrossover) {
			/* use log1p and an accurate approx to A-1 */
			if (x < one)
				Am1 = half * (y2 / (R + xp1) + y2 / (S - xm1));
			else
				Am1 = half * (y2 / (R + xp1) + (S + xm1));
			D_IM(ans) = log1p(Am1 + sqrt(Am1 * (A + one)));
		} else {
			D_IM(ans) = log(A + sqrt(A * A - one));
		}
	}
	if (hx < 0)
		D_RE(ans) = pi - D_RE(ans);
	if (hy >= 0)
		D_IM(ans) = -D_IM(ans);
	return (ans);
}