/*-
* 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.
*/
/*
* 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.
*
* See catrig.c for complete comments.
*
* XXX comments were removed automatically, and even short ones on the right
* of statements were removed (all of them), contrary to normal style. Only
* a few comments on the right of declarations remain.
*/
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
#include <complex.h>
#include <float.h>
#include "math.h"
#include "math_private.h"
#undef isinf
#define isinf(x) (fabsf(x) == INFINITY)
#undef isnan
#define isnan(x) ((x) != (x))
#define raise_inexact() do { volatile float junk = 1 + tiny; } while(0)
#undef signbit
#define signbit(x) (__builtin_signbitf(x))
static const float
A_crossover = 10,
B_crossover = 0.6417,
FOUR_SQRT_MIN = 0x1p-61,
QUARTER_SQRT_MAX = 0x1p61,
m_e = 2.7182818285e0, /* 0xadf854.0p-22 */
m_ln2 = 6.9314718056e-1, /* 0xb17218.0p-24 */
pio2_hi = 1.5707962513e0, /* 0xc90fda.0p-23 */
RECIP_EPSILON = 1 / FLT_EPSILON,
SQRT_3_EPSILON = 5.9801995673e-4, /* 0x9cc471.0p-34 */
SQRT_6_EPSILON = 8.4572793338e-4, /* 0xddb3d7.0p-34 */
SQRT_MIN = 0x1p-63;
static const volatile float
pio2_lo = 7.5497899549e-8, /* 0xa22169.0p-47 */
tiny = 0x1p-100;
static float complex clog_for_large_values(float complex z);
static inline float
f(float a, float b, float 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);
}
static inline void
do_hard_work(float x, float y, float *rx, int *B_is_usable, float *B,
float *sqrt_A2my2, float *new_y)
{
float R, S, A;
float Am1, Amy;
R = hypotf(x, y + 1);
S = hypotf(x, y - 1);
A = (R + S) / 2;
if (A < 1)
A = 1;
if (A < A_crossover) {
if (y == 1 && x < FLT_EPSILON * FLT_EPSILON / 128) {
*rx = sqrtf(x);
} else if (x >= FLT_EPSILON * fabsf(y - 1)) {
Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
*rx = log1pf(Am1 + sqrtf(Am1 * (A + 1)));
} else if (y < 1) {
*rx = x / sqrtf((1 - y) * (1 + y));
} else {
*rx = log1pf((y - 1) + sqrtf((y - 1) * (y + 1)));
}
} else {
*rx = logf(A + sqrtf(A * A - 1));
}
*new_y = y;
if (y < FOUR_SQRT_MIN) {
*B_is_usable = 0;
*sqrt_A2my2 = A * (2 / FLT_EPSILON);
*new_y = y * (2 / FLT_EPSILON);
return;
}
*B = y / A;
*B_is_usable = 1;
if (*B > B_crossover) {
*B_is_usable = 0;
if (y == 1 && x < FLT_EPSILON / 128) {
*sqrt_A2my2 = sqrtf(x) * sqrtf((A + y) / 2);
} else if (x >= FLT_EPSILON * fabsf(y - 1)) {
Amy = f(x, y + 1, R) + f(x, y - 1, S);
*sqrt_A2my2 = sqrtf(Amy * (A + y));
} else if (y > 1) {
*sqrt_A2my2 = x * (4 / FLT_EPSILON / FLT_EPSILON) * y /
sqrtf((y + 1) * (y - 1));
*new_y = y * (4 / FLT_EPSILON / FLT_EPSILON);
} else {
*sqrt_A2my2 = sqrtf((1 - y) * (1 + y));
}
}
}
float complex
casinhf(float complex z)
{
float x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
int B_is_usable;
float complex w;
x = crealf(z);
y = cimagf(z);
ax = fabsf(x);
ay = fabsf(y);
if (isnan(x) || isnan(y)) {
if (isinf(x))
return (cpackf(x, y + y));
if (isinf(y))
return (cpackf(y, x + x));
if (y == 0)
return (cpackf(x + x, y));
return (cpackf(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
}
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
if (signbit(x) == 0)
w = clog_for_large_values(z) + m_ln2;
else
w = clog_for_large_values(-z) + m_ln2;
return (cpackf(copysignf(crealf(w), x),
copysignf(cimagf(w), y)));
}
if (x == 0 && y == 0)
return (z);
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 = asinf(B);
else
ry = atan2f(new_y, sqrt_A2my2);
return (cpackf(copysignf(rx, x), copysignf(ry, y)));
}
float complex
casinf(float complex z)
{
float complex w = casinhf(cpackf(cimagf(z), crealf(z)));
return (cpackf(cimagf(w), crealf(w)));
}
float complex
cacosf(float complex z)
{
float x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
int sx, sy;
int B_is_usable;
float complex w;
x = crealf(z);
y = cimagf(z);
sx = signbit(x);
sy = signbit(y);
ax = fabsf(x);
ay = fabsf(y);
if (isnan(x) || isnan(y)) {
if (isinf(x))
return (cpackf(y + y, -INFINITY));
if (isinf(y))
return (cpackf(x + x, -y));
if (x == 0)
return (cpackf(pio2_hi + pio2_lo, y + y));
return (cpackf(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
}
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
w = clog_for_large_values(z);
rx = fabsf(cimagf(w));
ry = crealf(w) + m_ln2;
if (sy == 0)
ry = -ry;
return (cpackf(rx, ry));
}
if (x == 1 && y == 0)
return (cpackf(0, -y));
raise_inexact();
if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
return (cpackf(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 = acosf(B);
else
rx = acosf(-B);
} else {
if (sx == 0)
rx = atan2f(sqrt_A2mx2, new_x);
else
rx = atan2f(sqrt_A2mx2, -new_x);
}
if (sy == 0)
ry = -ry;
return (cpackf(rx, ry));
}
float complex
cacoshf(float complex z)
{
float complex w;
float rx, ry;
w = cacosf(z);
rx = crealf(w);
ry = cimagf(w);
if (isnan(rx) && isnan(ry))
return (cpackf(ry, rx));
if (isnan(rx))
return (cpackf(fabsf(ry), rx));
if (isnan(ry))
return (cpackf(ry, ry));
return (cpackf(fabsf(ry), copysignf(rx, cimagf(z))));
}
static float complex
clog_for_large_values(float complex z)
{
float x, y;
float ax, ay, t;
x = crealf(z);
y = cimagf(z);
ax = fabsf(x);
ay = fabsf(y);
if (ax < ay) {
t = ax;
ax = ay;
ay = t;
}
if (ax > FLT_MAX / 2)
return (cpackf(logf(hypotf(x / m_e, y / m_e)) + 1,
atan2f(y, x)));
if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
return (cpackf(logf(hypotf(x, y)), atan2f(y, x)));
return (cpackf(logf(ax * ax + ay * ay) / 2, atan2f(y, x)));
}
static inline float
sum_squares(float x, float y)
{
if (y < SQRT_MIN)
return (x * x);
return (x * x + y * y);
}
static inline float
real_part_reciprocal(float x, float y)
{
float scale;
uint32_t hx, hy;
int32_t ix, iy;
GET_FLOAT_WORD(hx, x);
ix = hx & 0x7f800000;
GET_FLOAT_WORD(hy, y);
iy = hy & 0x7f800000;
#define BIAS (FLT_MAX_EXP - 1)
#define CUTOFF (FLT_MANT_DIG / 2 + 1)
if (ix - iy >= CUTOFF << 23 || isinf(x))
return (1 / x);
if (iy - ix >= CUTOFF << 23)
return (x / y / y);
if (ix <= (BIAS + FLT_MAX_EXP / 2 - CUTOFF) << 23)
return (x / (x * x + y * y));
SET_FLOAT_WORD(scale, 0x7f800000 - ix);
x *= scale;
y *= scale;
return (x / (x * x + y * y) * scale);
}
float complex
catanhf(float complex z)
{
float x, y, ax, ay, rx, ry;
x = crealf(z);
y = cimagf(z);
ax = fabsf(x);
ay = fabsf(y);
if (y == 0 && ax <= 1)
return (cpackf(atanhf(x), y));
if (x == 0)
return (cpackf(x, atanf(y)));
if (isnan(x) || isnan(y)) {
if (isinf(x))
return (cpackf(copysignf(0, x), y + y));
if (isinf(y))
return (cpackf(copysignf(0, x),
copysignf(pio2_hi + pio2_lo, y)));
return (cpackf(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
}
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
return (cpackf(real_part_reciprocal(x, y),
copysignf(pio2_hi + pio2_lo, y)));
if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
raise_inexact();
return (z);
}
if (ax == 1 && ay < FLT_EPSILON)
rx = (m_ln2 - logf(ay)) / 2;
else
rx = log1pf(4 * ax / sum_squares(ax - 1, ay)) / 4;
if (ax == 1)
ry = atan2f(2, -ay) / 2;
else if (ay < FLT_EPSILON)
ry = atan2f(2 * ay, (1 - ax) * (1 + ax)) / 2;
else
ry = atan2f(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
return (cpackf(copysignf(rx, x), copysignf(ry, y)));
}
float complex
catanf(float complex z)
{
float complex w = catanhf(cpackf(cimagf(z), crealf(z)));
return (cpackf(cimagf(w), crealf(w)));
}