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 (CMPLXF(atanhf(x), y));
if (x == 0)
return (CMPLXF(x, atanf(y)));
if (isnan(x) || isnan(y)) {
if (isinf(x))
return (CMPLXF(copysignf(0, x), y + y));
if (isinf(y))
return (CMPLXF(copysignf(0, x),
copysignf(pio2_hi + pio2_lo, y)));
return (CMPLXF(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
}
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
return (CMPLXF(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 (CMPLXF(copysignf(rx, x), copysignf(ry, y)));
}
long double complex
catanhl(long double complex z)
{
long double x, y, ax, ay, rx, ry;
x = creall(z);
y = cimagl(z);
ax = fabsl(x);
ay = fabsl(y);
if (y == 0 && ax <= 1)
return (CMPLXL(atanhl(x), y));
if (x == 0)
return (CMPLXL(x, atanl(y)));
if (isnan(x) || isnan(y)) {
if (isinf(x))
return (CMPLXL(copysignl(0, x), y + y));
if (isinf(y))
return (CMPLXL(copysignl(0, x),
copysignl(pio2_hi + pio2_lo, y)));
return (CMPLXL(nan_mix(x, y), nan_mix(x, y)));
}
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
return (CMPLXL(real_part_reciprocal(x, y),
copysignl(pio2_hi + pio2_lo, y)));
if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
raise_inexact();
return (z);
}
if (ax == 1 && ay < LDBL_EPSILON)
rx = (m_ln2 - logl(ay)) / 2;
else
rx = log1pl(4 * ax / sum_squares(ax - 1, ay)) / 4;
if (ax == 1)
ry = atan2l(2, -ay) / 2;
else if (ay < LDBL_EPSILON)
ry = atan2l(2 * ay, (1 - ax) * (1 + ax)) / 2;
else
ry = atan2l(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
return (CMPLXL(copysignl(rx, x), copysignl(ry, y)));
}
/*
* 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)));
}
