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/* $NetBSD: catrig.c,v 1.2 2016/09/20 18:25:20 christos Exp $ */
/*-
* Copyright (c) 2012 Stephen Montgomery-Smith <stephen@FreeBSD.ORG>
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#include <sys/cdefs.h>
#if 0
__FBSDID("$FreeBSD: head/lib/msun/src/catrig.c 275819 2014-12-16 09:21:56Z ed $");
#endif
__RCSID("$NetBSD: catrig.c,v 1.2 2016/09/20 18:25:20 christos Exp $");
#include "namespace.h"
#ifdef __weak_alias
__weak_alias(casin, _casin)
#endif
#ifdef __weak_alias
__weak_alias(catan, _catan)
#endif
#include <complex.h>
#include <float.h>
#include "math.h"
#include "math_private.h"
#undef isinf
#define isinf(x) (fabs(x) == INFINITY)
#undef isnan
#define isnan(x) ((x) != (x))
#define raise_inexact() do { volatile float junk __unused = /*LINTED*/1 + tiny; } while(/*CONSTCOND*/0)
#undef signbit
#define signbit(x) (__builtin_signbit(x))
/* We need that DBL_EPSILON^2/128 is larger than FOUR_SQRT_MIN. */
static const double
A_crossover = 10, /* Hull et al suggest 1.5, but 10 works better */
B_crossover = 0.6417, /* suggested by Hull et al */
m_e = 2.7182818284590452e0, /* 0x15bf0a8b145769.0p-51 */
m_ln2 = 6.9314718055994531e-1, /* 0x162e42fefa39ef.0p-53 */
pio2_hi = 1.5707963267948966e0, /* 0x1921fb54442d18.0p-52 */
RECIP_EPSILON = 1 / DBL_EPSILON,
SQRT_3_EPSILON = 2.5809568279517849e-8, /* 0x1bb67ae8584caa.0p-78 */
SQRT_6_EPSILON = 3.6500241499888571e-8, /* 0x13988e1409212e.0p-77 */
#if DBL_MAX_EXP == 1024 /* IEEE */
FOUR_SQRT_MIN = 0x1p-509, /* >= 4 * sqrt(DBL_MIN) */
QUARTER_SQRT_MAX = 0x1p509, /* <= sqrt(DBL_MAX) / 4 */
SQRT_MIN = 0x1p-511; /* >= sqrt(DBL_MIN) */
#elif DBL_MAX_EXP == 127 /* VAX */
FOUR_SQRT_MIN = 0x1p-62, /* >= 4 * sqrt(DBL_MIN) */
QUARTER_SQRT_MAX = 0x1p62, /* <= sqrt(DBL_MAX) / 4 */
SQRT_MIN = 0x1p-64; /* >= sqrt(DBL_MIN) */
#else
#error "unsupported floating point format"
#endif
static const volatile double
pio2_lo = 6.1232339957367659e-17; /* 0x11a62633145c07.0p-106 */
static const volatile float
tiny = 0x1p-100;
static double complex clog_for_large_values(double complex z);
/*
* Testing indicates that all these functions are accurate up to 4 ULP.
* The functions casin(h) and cacos(h) are about 2.5 times slower than asinh.
* The functions catan(h) are a little under 2 times slower than atanh.
*
* The code for casinh, casin, cacos, and cacosh comes first. The code is
* rather complicated, and the four functions are highly interdependent.
*
* The code for catanh and catan comes at the end. It is much simpler than
* the other functions, and the code for these can be disconnected from the
* rest of the code.
*/
/*
* ================================
* | casinh, casin, cacos, cacosh |
* ================================
*/
/*
* The algorithm is very close to that in "Implementing the complex arcsine
* and arccosine functions using exception handling" by T. E. Hull, Thomas F.
* Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on
* Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335,
* http://dl.acm.org/citation.cfm?id=275324.
*
* Throughout we use the convention z = x + I*y.
*
* casinh(z) = sign(x)*log(A+sqrt(A*A-1)) + I*asin(B)
* where
* A = (|z+I| + |z-I|) / 2
* B = (|z+I| - |z-I|) / 2 = y/A
*
* These formulas become numerically unstable:
* (a) for Re(casinh(z)) when z is close to the line segment [-I, I] (that
* is, Re(casinh(z)) is close to 0);
* (b) for Im(casinh(z)) when z is close to either of the intervals
* [I, I*infinity) or (-I*infinity, -I] (that is, |Im(casinh(z))| is
* close to PI/2).
*
* These numerical problems are overcome by defining
* f(a, b) = (hypot(a, b) - b) / 2 = a*a / (hypot(a, b) + b) / 2
* Then if A < A_crossover, we use
* log(A + sqrt(A*A-1)) = log1p((A-1) + sqrt((A-1)*(A+1)))
* A-1 = f(x, 1+y) + f(x, 1-y)
* and if B > B_crossover, we use
* asin(B) = atan2(y, sqrt(A*A - y*y)) = atan2(y, sqrt((A+y)*(A-y)))
* A-y = f(x, y+1) + f(x, y-1)
* where without loss of generality we have assumed that x and y are
* non-negative.
*
* Much of the difficulty comes because the intermediate computations may
* produce overflows or underflows. This is dealt with in the paper by Hull
* et al by using exception handling. We do this by detecting when
* computations risk underflow or overflow. The hardest part is handling the
* underflows when computing f(a, b).
*
* Note that the function f(a, b) does not appear explicitly in the paper by
* Hull et al, but the idea may be found on pages 308 and 309. Introducing the
* function f(a, b) allows us to concentrate many of the clever tricks in this
* paper into one function.
*/
/*
* Function f(a, b, hypot_a_b) = (hypot(a, b) - b) / 2.
* Pass hypot(a, b) as the third argument.
*/
static inline double
f(double a, double b, double hypot_a_b)
{
if (b < 0)
return ((hypot_a_b - b) / 2);
if (b == 0)
return (a / 2);
return (a * a / (hypot_a_b + b) / 2);
}
/*
* All the hard work is contained in this function.
* x and y are assumed positive or zero, and less than RECIP_EPSILON.
* Upon return:
* rx = Re(casinh(z)) = -Im(cacos(y + I*x)).
* B_is_usable is set to 1 if the value of B is usable.
* If B_is_usable is set to 0, sqrt_A2my2 = sqrt(A*A - y*y), and new_y = y.
* If returning sqrt_A2my2 has potential to result in an underflow, it is
* rescaled, and new_y is similarly rescaled.
*/
static inline void
do_hard_work(double x, double y, double *rx, int *B_is_usable, double *B,
double *sqrt_A2my2, double *new_y)
{
double R, S, A; /* A, B, R, and S are as in Hull et al. */
double Am1, Amy; /* A-1, A-y. */
R = hypot(x, y + 1); /* |z+I| */
S = hypot(x, y - 1); /* |z-I| */
/* A = (|z+I| + |z-I|) / 2 */
A = (R + S) / 2;
/*
* Mathematically A >= 1. There is a small chance that this will not
* be so because of rounding errors. So we will make certain it is
* so.
*/
if (A < 1)
A = 1;
if (A < A_crossover) {
/*
* Am1 = fp + fm, where fp = f(x, 1+y), and fm = f(x, 1-y).
* rx = log1p(Am1 + sqrt(Am1*(A+1)))
*/
if (y == 1 && x < DBL_EPSILON * DBL_EPSILON / 128) {
/*
* fp is of order x^2, and fm = x/2.
* A = 1 (inexactly).
*/
*rx = sqrt(x);
} else if (x >= DBL_EPSILON * fabs(y - 1)) {
/*
* Underflow will not occur because
* x >= DBL_EPSILON^2/128 >= FOUR_SQRT_MIN
*/
Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
*rx = log1p(Am1 + sqrt(Am1 * (A + 1)));
} else if (y < 1) {
/*
* fp = x*x/(1+y)/4, fm = x*x/(1-y)/4, and
* A = 1 (inexactly).
*/
*rx = x / sqrt((1 - y) * (1 + y));
} else { /* if (y > 1) */
/*
* A-1 = y-1 (inexactly).
*/
*rx = log1p((y - 1) + sqrt((y - 1) * (y + 1)));
}
} else {
*rx = log(A + sqrt(A * A - 1));
}
*new_y = y;
if (y < FOUR_SQRT_MIN) {
/*
* Avoid a possible underflow caused by y/A. For casinh this
* would be legitimate, but will be picked up by invoking atan2
* later on. For cacos this would not be legitimate.
*/
*B_is_usable = 0;
*sqrt_A2my2 = A * (2 / DBL_EPSILON);
*new_y = y * (2 / DBL_EPSILON);
return;
}
/* B = (|z+I| - |z-I|) / 2 = y/A */
*B = y / A;
*B_is_usable = 1;
if (*B > B_crossover) {
*B_is_usable = 0;
/*
* Amy = fp + fm, where fp = f(x, y+1), and fm = f(x, y-1).
* sqrt_A2my2 = sqrt(Amy*(A+y))
*/
if (y == 1 && x < DBL_EPSILON / 128) {
/*
* fp is of order x^2, and fm = x/2.
* A = 1 (inexactly).
*/
*sqrt_A2my2 = sqrt(x) * sqrt((A + y) / 2);
} else if (x >= DBL_EPSILON * fabs(y - 1)) {
/*
* Underflow will not occur because
* x >= DBL_EPSILON/128 >= FOUR_SQRT_MIN
* and
* x >= DBL_EPSILON^2 >= FOUR_SQRT_MIN
*/
Amy = f(x, y + 1, R) + f(x, y - 1, S);
*sqrt_A2my2 = sqrt(Amy * (A + y));
} else if (y > 1) {
/*
* fp = x*x/(y+1)/4, fm = x*x/(y-1)/4, and
* A = y (inexactly).
*
* y < RECIP_EPSILON. So the following
* scaling should avoid any underflow problems.
*/
*sqrt_A2my2 = x * (4 / DBL_EPSILON / DBL_EPSILON) * y /
sqrt((y + 1) * (y - 1));
*new_y = y * (4 / DBL_EPSILON / DBL_EPSILON);
} else { /* if (y < 1) */
/*
* fm = 1-y >= DBL_EPSILON, fp is of order x^2, and
* A = 1 (inexactly).
*/
*sqrt_A2my2 = sqrt((1 - y) * (1 + y));
}
}
}
/*
* casinh(z) = z + O(z^3) as z -> 0
*
* casinh(z) = sign(x)*clog(sign(x)*z) + O(1/z^2) as z -> infinity
* The above formula works for the imaginary part as well, because
* Im(casinh(z)) = sign(x)*atan2(sign(x)*y, fabs(x)) + O(y/z^3)
* as z -> infinity, uniformly in y
*/
double complex
casinh(double complex z)
{
double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
int B_is_usable;
double complex w;
x = creal(z);
y = cimag(z);
ax = fabs(x);
ay = fabs(y);
if (isnan(x) || isnan(y)) {
/* casinh(+-Inf + I*NaN) = +-Inf + I*NaN */
if (isinf(x))
return (CMPLX(x, y + y));
/* casinh(NaN + I*+-Inf) = opt(+-)Inf + I*NaN */
if (isinf(y))
return (CMPLX(y, x + x));
/* casinh(NaN + I*0) = NaN + I*0 */
if (y == 0)
return (CMPLX(x + x, y));
/*
* All other cases involving NaN return NaN + I*NaN.
* C99 leaves it optional whether to raise invalid if one of
* the arguments is not NaN, so we opt not to raise it.
*/
return (CMPLX(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
}
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
/* clog...() will raise inexact unless x or y is infinite. */
if (signbit(x) == 0)
w = clog_for_large_values(z) + m_ln2;
else
w = clog_for_large_values(-z) + m_ln2;
return (CMPLX(copysign(creal(w), x), copysign(cimag(w), y)));
}
/* Avoid spuriously raising inexact for z = 0. */
if (x == 0 && y == 0)
return (z);
/* All remaining cases are inexact. */
raise_inexact();
if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
return (z);
do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
if (B_is_usable)
ry = asin(B);
else
ry = atan2(new_y, sqrt_A2my2);
return (CMPLX(copysign(rx, x), copysign(ry, y)));
}
/*
* casin(z) = reverse(casinh(reverse(z)))
* where reverse(x + I*y) = y + I*x = I*conj(z).
*/
double complex
casin(double complex z)
{
double complex w = casinh(CMPLX(cimag(z), creal(z)));
return (CMPLX(cimag(w), creal(w)));
}
/*
* cacos(z) = PI/2 - casin(z)
* but do the computation carefully so cacos(z) is accurate when z is
* close to 1.
*
* cacos(z) = PI/2 - z + O(z^3) as z -> 0
*
* cacos(z) = -sign(y)*I*clog(z) + O(1/z^2) as z -> infinity
* The above formula works for the real part as well, because
* Re(cacos(z)) = atan2(fabs(y), x) + O(y/z^3)
* as z -> infinity, uniformly in y
*/
double complex
cacos(double complex z)
{
double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
int sx, sy;
int B_is_usable;
double complex w;
x = creal(z);
y = cimag(z);
sx = signbit(x);
sy = signbit(y);
ax = fabs(x);
ay = fabs(y);
if (isnan(x) || isnan(y)) {
/* cacos(+-Inf + I*NaN) = NaN + I*opt(-)Inf */
if (isinf(x))
return (CMPLX(y + y, -INFINITY));
/* cacos(NaN + I*+-Inf) = NaN + I*-+Inf */
if (isinf(y))
return (CMPLX(x + x, -y));
/* cacos(0 + I*NaN) = PI/2 + I*NaN with inexact */
if (x == 0)
return (CMPLX(pio2_hi + pio2_lo, y + y));
/*
* All other cases involving NaN return NaN + I*NaN.
* C99 leaves it optional whether to raise invalid if one of
* the arguments is not NaN, so we opt not to raise it.
*/
return (CMPLX(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
}
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
/* clog...() will raise inexact unless x or y is infinite. */
w = clog_for_large_values(z);
rx = fabs(cimag(w));
ry = creal(w) + m_ln2;
if (sy == 0)
ry = -ry;
return (CMPLX(rx, ry));
}
/* Avoid spuriously raising inexact for z = 1. */
if (x == 1 && y == 0)
return (CMPLX(0, -y));
/* All remaining cases are inexact. */
raise_inexact();
if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
return (CMPLX(pio2_hi - (x - pio2_lo), -y));
do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
if (B_is_usable) {
if (sx == 0)
rx = acos(B);
else
rx = acos(-B);
} else {
if (sx == 0)
rx = atan2(sqrt_A2mx2, new_x);
else
rx = atan2(sqrt_A2mx2, -new_x);
}
if (sy == 0)
ry = -ry;
return (CMPLX(rx, ry));
}
/*
* cacosh(z) = I*cacos(z) or -I*cacos(z)
* where the sign is chosen so Re(cacosh(z)) >= 0.
*/
double complex
cacosh(double complex z)
{
double complex w;
double rx, ry;
w = cacos(z);
rx = creal(w);
ry = cimag(w);
/* cacosh(NaN + I*NaN) = NaN + I*NaN */
if (isnan(rx) && isnan(ry))
return (CMPLX(ry, rx));
/* cacosh(NaN + I*+-Inf) = +Inf + I*NaN */
/* cacosh(+-Inf + I*NaN) = +Inf + I*NaN */
if (isnan(rx))
return (CMPLX(fabs(ry), rx));
/* cacosh(0 + I*NaN) = NaN + I*NaN */
if (isnan(ry))
return (CMPLX(ry, ry));
return (CMPLX(fabs(ry), copysign(rx, cimag(z))));
}
/*
* Optimized version of clog() for |z| finite and larger than ~RECIP_EPSILON.
*/
static double complex
clog_for_large_values(double complex z)
{
double x, y;
double ax, ay, t;
x = creal(z);
y = cimag(z);
ax = fabs(x);
ay = fabs(y);
if (ax < ay) {
t = ax;
ax = ay;
ay = t;
}
/*
* Avoid overflow in hypot() when x and y are both very large.
* Divide x and y by E, and then add 1 to the logarithm. This depends
* on E being larger than sqrt(2).
* Dividing by E causes an insignificant loss of accuracy; however
* this method is still poor since it is uneccessarily slow.
*/
if (ax > DBL_MAX / 2)
return (CMPLX(log(hypot(x / m_e, y / m_e)) + 1, atan2(y, x)));
/*
* Avoid overflow when x or y is large. Avoid underflow when x or
* y is small.
*/
if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
return (CMPLX(log(hypot(x, y)), atan2(y, x)));
return (CMPLX(log(ax * ax + ay * ay) / 2, atan2(y, x)));
}
/*
* =================
* | catanh, catan |
* =================
*/
/*
* sum_squares(x,y) = x*x + y*y (or just x*x if y*y would underflow).
* Assumes x*x and y*y will not overflow.
* Assumes x and y are finite.
* Assumes y is non-negative.
* Assumes fabs(x) >= DBL_EPSILON.
*/
static inline double
sum_squares(double x, double y)
{
/* Avoid underflow when y is small. */
if (y < SQRT_MIN)
return (x * x);
return (x * x + y * y);
}
/*
* real_part_reciprocal(x, y) = Re(1/(x+I*y)) = x/(x*x + y*y).
* Assumes x and y are not NaN, and one of x and y is larger than
* RECIP_EPSILON. We avoid unwarranted underflow. It is important to not use
* the code creal(1/z), because the imaginary part may produce an unwanted
* underflow.
* This is only called in a context where inexact is always raised before
* the call, so no effort is made to avoid or force inexact.
*/
static inline double
real_part_reciprocal(double x, double y)
{
double scale;
uint32_t hx, hy;
int32_t ix, iy;
/*
* This code is inspired by the C99 document n1124.pdf, Section G.5.1,
* example 2.
*/
GET_HIGH_WORD(hx, x);
ix = hx & 0x7ff00000;
GET_HIGH_WORD(hy, y);
iy = hy & 0x7ff00000;
#define BIAS (DBL_MAX_EXP - 1)
/* XXX more guard digits are useful iff there is extra precision. */
#define CUTOFF (DBL_MANT_DIG / 2 + 1) /* just half or 1 guard digit */
if (ix - iy >= CUTOFF << 20 || isinf(x))
return (1 / x); /* +-Inf -> +-0 is special */
if (iy - ix >= CUTOFF << 20)
return (x / y / y); /* should avoid double div, but hard */
if (ix <= (BIAS + DBL_MAX_EXP / 2 - CUTOFF) << 20)
return (x / (x * x + y * y));
scale = 1;
SET_HIGH_WORD(scale, 0x7ff00000 - ix); /* 2**(1-ilogb(x)) */
x *= scale;
y *= scale;
return (x / (x * x + y * y) * scale);
}
/*
* catanh(z) = log((1+z)/(1-z)) / 2
* = log1p(4*x / |z-1|^2) / 4
* + I * atan2(2*y, (1-x)*(1+x)-y*y) / 2
*
* catanh(z) = z + O(z^3) as z -> 0
*
* catanh(z) = 1/z + sign(y)*I*PI/2 + O(1/z^3) as z -> infinity
* The above formula works for the real part as well, because
* Re(catanh(z)) = x/|z|^2 + O(x/z^4)
* as z -> infinity, uniformly in x
*/
double complex
catanh(double complex z)
{
double x, y, ax, ay, rx, ry;
x = creal(z);
y = cimag(z);
ax = fabs(x);
ay = fabs(y);
/* This helps handle many cases. */
if (y == 0 && ax <= 1)
return (CMPLX(atanh(x), y));
/* To ensure the same accuracy as atan(), and to filter out z = 0. */
if (x == 0)
return (CMPLX(x, atan(y)));
if (isnan(x) || isnan(y)) {
/* catanh(+-Inf + I*NaN) = +-0 + I*NaN */
if (isinf(x))
return (CMPLX(copysign(0, x), y + y));
/* catanh(NaN + I*+-Inf) = sign(NaN)0 + I*+-PI/2 */
if (isinf(y))
return (CMPLX(copysign(0, x),
copysign(pio2_hi + pio2_lo, y)));
/*
* All other cases involving NaN return NaN + I*NaN.
* C99 leaves it optional whether to raise invalid if one of
* the arguments is not NaN, so we opt not to raise it.
*/
return (CMPLX(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
}
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
return (CMPLX(real_part_reciprocal(x, y),
copysign(pio2_hi + pio2_lo, y)));
if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
/*
* z = 0 was filtered out above. All other cases must raise
* inexact, but this is the only only that needs to do it
* explicitly.
*/
raise_inexact();
return (z);
}
if (ax == 1 && ay < DBL_EPSILON)
rx = (m_ln2 - log(ay)) / 2;
else
rx = log1p(4 * ax / sum_squares(ax - 1, ay)) / 4;
if (ax == 1)
ry = atan2(2, -ay) / 2;
else if (ay < DBL_EPSILON)
ry = atan2(2 * ay, (1 - ax) * (1 + ax)) / 2;
else
ry = atan2(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
return (CMPLX(copysign(rx, x), copysign(ry, y)));
}
/*
* catan(z) = reverse(catanh(reverse(z)))
* where reverse(x + I*y) = y + I*x = I*conj(z).
*/
double complex
catan(double complex z)
{
double complex w = catanh(CMPLX(cimag(z), creal(z)));
return (CMPLX(cimag(w), creal(w)));
}
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