text: /*-
   * 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.
   *
   * $FreeBSD: head/lib/msun/src/catrigf.c 251404 2013-06-05 05:33:01Z das $
   */
  
  /*
   * 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 <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)));
  }
  