casin.c revision 25c28e83beb90e7c80452a7c818c5e6f73a07dc8
/*
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*
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* When distributing Covered Code, include this CDDL HEADER in each
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* If applicable, add the following below this CDDL HEADER, with the
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*/
/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
/* INDENT OFF */
/*
* dcomplex casin(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 sine function casin(z),
* where z = x+iy, can be defined by
*
* casin(z) = asin(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.
*
* Special notes: if casin( x, y) = ( u, v), then
* casin(-x, y) = (-u, v),
* casin( x,-y) = ( u,-v),
* in general, we have casin(conj(z)) = conj(casin(z))
* casin(-z) = -casin(z)
* casin(z) = pi/2 - cacos(z)
*
* casin( 0 + i 0 ) = 0 + i 0
* casin( 0 + i NaN ) = 0 + i NaN
* casin( x + i inf ) = 0 + i inf for finite x
* casin( x + i NaN ) = NaN + i NaN with invalid for finite x != 0
* casin(inf + iy ) = pi/2 + i inf finite y
* casin(inf + i inf) = pi/4 + i inf
* casin(inf + i NaN) = NaN + i inf
* casin(NaN + i y ) = NaN + i NaN for finite y
* casin(NaN + i inf) = NaN + i inf
* casin(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 + ------------ , if x > 1
* [ 2(1+x)(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 - ------------), if x < 1,
* [ 2(1+x)(1-x)
* B = x/A ~ [
* [ 1, if x = 1,
* [
* [ 2
* [ y
* [ 1 - ------------ , if x > 1,
* [ 2(1+x)(1-x)
* Thus
* casin(x+i*y)=[
* [ pi/2 + i log(x+sqrt(x*x-1)), if x >= 1
*
* 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
* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
* asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
* we have, for the real part asin(B),
* asin(1-y/2) ~ pi/2 - 2 asin(sqrt(y/4))
* ~ pi/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 = asin(B) ~ x/y (be careful, x/y may underflow)
* 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)
* Thus
* real part = asin(B) = atan(x/y),
* 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
* Since B is tiny, we have
* 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));
*
* casin(z) = asin(B) + i sign(y) log (A + sqrt(A*A-1)),
*/
/* INDENT ON */
#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_2 = 1.570796326794896558e+00,
pi_2_l = 6.123233995736765886e-17,
pi_4 = 7.85398163397448278999e-01,
Acrossover = 1.5,
Bcrossover = 0.6417,
half = 0.5;
/* INDENT ON */
x = D_RE(z);
y = D_IM(z);
x = fabs(x);
y = fabs(y);
/* special cases */
/* x is inf or NaN */
if (iy >= 0x7ff00000) {
/* casin(inf + i inf) = pi/4 + i inf */
else /* casin(inf + i NaN) = NaN + i inf */
} else /* casin(inf + iy) = pi/2 + i inf */
} else { /* x is NaN */
if (iy >= 0x7ff00000) {
/* INDENT OFF */
/*
* casin(NaN + i inf) = NaN + i inf
* casin(NaN + i NaN) = NaN + i NaN
*/
/* INDENT ON */
} else {
/* casin(NaN + i y ) = NaN + i NaN */
}
}
if (hx < 0)
if (hy < 0)
return (ans);
}
/* casin(+0 + i 0 ) = 0 + i 0. */
return (z);
} else { /* casin(x + i NaN) = NaN + i NaN */
else
}
if (hx < 0)
if (hy < 0)
return (ans);
}
} else {
one)));
else {
}
}
} else {
one)));
else
(x + one)));
}
} else if (y < Foursqrtu) { /* region 3 */
t = sqrt(y);
} else if (E * y - one >= x) { /* region 4 */
/* region 5: x+1 or y is very large (>= sqrt(max)/8) */
t = x / y;
} else if (x < Foursqrtu) {
/* region 6: x is very small, < 4sqrt(min) */
else
} else { /* safe region */
y2 = y * y;
A = half * (R + S);
B = x / A;
if (B <= Bcrossover)
else { /* use atan and an accurate approx to a-x */
Apx = A + x;
if (x <= one)
else
}
if (A <= Acrossover) {
/* use log1p and an accurate approx to A-1 */
if (x < one)
else
} else {
}
}
if (hx < 0)
if (hy < 0)
return (ans);
}