def c_exp(x, y):
if not isfinite(x) or not isfinite(y):
if isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = copysign(INF, math.cos(y))
imag = copysign(INF, math.sin(y))
else:
real = copysign(0., math.cos(y))
imag = copysign(0., math.sin(y))
r = (real, imag)
else:
r = exp_special_values[special_type(x)][special_type(y)]
# need to raise ValueError if y is +/- infinity and x is not
# a NaN and not -infinity
if isinf(y) and (isfinite(x) or (isinf(x) and x > 0)):
raise ValueError("math domain error")
return r
if x > CM_LOG_LARGE_DOUBLE:
l = math.exp(x - 1.)
real = l * math.cos(y) * math.e
imag = l * math.sin(y) * math.e
else:
l = math.exp(x)
real = l * math.cos(y)
imag = l * math.sin(y)
if isinf(real) or isinf(imag):
raise OverflowError("math range error")
return real, imag
def c_rect(r, phi):
if not isfinite(r) or not isfinite(phi):
# if r is +/-infinity and phi is finite but nonzero then
# result is (+-INF +-INF i), but we need to compute cos(phi)
# and sin(phi) to figure out the signs.
if isinf(r) and isfinite(phi) and phi != 0.:
if r > 0:
real = copysign(INF, math.cos(phi))
imag = copysign(INF, math.sin(phi))
else:
real = -copysign(INF, math.cos(phi))
imag = -copysign(INF, math.sin(phi))
z = (real, imag)
else:
z = rect_special_values[special_type(r)][special_type(phi)]
# need to raise ValueError if r is a nonzero number and phi
# is infinite
if r != 0. and not isnan(r) and isinf(phi):
raise ValueError("math domain error")
return z
real = r * math.cos(phi)
imag = r * math.sin(phi)
return real, imag
def c_cosh(x, y):
if not isfinite(x) or not isfinite(y):
if isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = copysign(INF, math.cos(y))
imag = copysign(INF, math.sin(y))
else:
real = copysign(INF, math.cos(y))
imag = -copysign(INF, math.sin(y))
r = (real, imag)
else:
r = cosh_special_values[special_type(x)][special_type(y)]
# need to raise ValueError if y is +/- infinity and x is not
# a NaN
if isinf(y) and not isnan(x):
raise ValueError("math domain error")
return r
if fabs(x) > CM_LOG_LARGE_DOUBLE:
# deal correctly with cases where cosh(x) overflows but
# cosh(z) does not.
x_minus_one = x - copysign(1., x)
real = math.cos(y) * math.cosh(x_minus_one) * math.e
imag = math.sin(y) * math.sinh(x_minus_one) * math.e
else:
real = math.cos(y) * math.cosh(x)
imag = math.sin(y) * math.sinh(x)
if isinf(real) or isinf(imag):
raise OverflowError("math range error")
return real, imag
def c_sinh(x, y):
# special treatment for sinh(+/-inf + iy) if y is finite and nonzero
if not isfinite(x) or not isfinite(y):
if isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = copysign(INF, math.cos(y))
imag = copysign(INF, math.sin(y))
else:
real = -copysign(INF, math.cos(y))
imag = copysign(INF, math.sin(y))
r = (real, imag)
else:
r = sinh_special_values[special_type(x)][special_type(y)]
# need to raise ValueError if y is +/- infinity and x is not
# a NaN
if isinf(y) and not isnan(x):
raise ValueError("math domain error")
return r
if fabs(x) > CM_LOG_LARGE_DOUBLE:
x_minus_one = x - copysign(1., x)
real = math.cos(y) * math.sinh(x_minus_one) * math.e
imag = math.sin(y) * math.cosh(x_minus_one) * math.e
else:
real = math.cos(y) * math.sinh(x)
imag = math.sin(y) * math.cosh(x)
if isinf(real) or isinf(imag):
raise OverflowError("math range error")
return real, imag
def c_atanh(x, y):
if not isfinite(x) or not isfinite(y):
return atanh_special_values[special_type(x)][special_type(y)]
# Reduce to case where x >= 0., using atanh(z) = -atanh(-z).
if x < 0.:
return c_neg(*c_atanh(*c_neg(x, y)))
ay = fabs(y)
if x > CM_SQRT_LARGE_DOUBLE or ay > CM_SQRT_LARGE_DOUBLE:
# if abs(z) is large then we use the approximation
# atanh(z) ~ 1/z +/- i*pi/2 (+/- depending on the sign
# of y
h = math.hypot(x/2., y/2.) # safe from overflow
real = x/4./h/h
# the two negations in the next line cancel each other out
# except when working with unsigned zeros: they're there to
# ensure that the branch cut has the correct continuity on
# systems that don't support signed zeros
imag = -copysign(math.pi/2., -y)
elif x == 1. and ay < CM_SQRT_DBL_MIN:
# C99 standard says: atanh(1+/-0.) should be inf +/- 0i
if ay == 0.:
raise ValueError("math domain error")
#real = INF
#imag = y
else:
real = -math.log(math.sqrt(ay)/math.sqrt(math.hypot(ay, 2.)))
imag = copysign(math.atan2(2., -ay) / 2, y)
else:
real = log1p(4.*x/((1-x)*(1-x) + ay*ay))/4.
imag = -math.atan2(-2.*y, (1-x)*(1+x) - ay*ay) / 2.
return (real, imag)
def c_exp(x, y):
if not isfinite(x) or not isfinite(y):
if isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = copysign(INF, math.cos(y))
imag = copysign(INF, math.sin(y))
else:
real = copysign(0., math.cos(y))
imag = copysign(0., math.sin(y))
r = (real, imag)
else:
r = exp_special_values[special_type(x)][special_type(y)]
# need to raise ValueError if y is +/- infinity and x is not
# a NaN and not -infinity
if isinf(y) and (isfinite(x) or (isinf(x) and x > 0)):
raise ValueError("math domain error")
return r
if x > CM_LOG_LARGE_DOUBLE:
l = math.exp(x-1.)
real = l * math.cos(y) * math.e
imag = l * math.sin(y) * math.e
else:
l = math.exp(x)
real = l * math.cos(y)
imag = l * math.sin(y)
if isinf(real) or isinf(imag):
raise OverflowError("math range error")
return real, imag
def c_cosh(x, y):
if not isfinite(x) or not isfinite(y):
if isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = copysign(INF, math.cos(y))
imag = copysign(INF, math.sin(y))
else:
real = copysign(INF, math.cos(y))
imag = -copysign(INF, math.sin(y))
r = (real, imag)
else:
r = cosh_special_values[special_type(x)][special_type(y)]
# need to raise ValueError if y is +/- infinity and x is not
# a NaN
if isinf(y) and not isnan(x):
raise ValueError("math domain error")
return r
if fabs(x) > CM_LOG_LARGE_DOUBLE:
# deal correctly with cases where cosh(x) overflows but
# cosh(z) does not.
x_minus_one = x - copysign(1., x)
real = math.cos(y) * math.cosh(x_minus_one) * math.e
imag = math.sin(y) * math.sinh(x_minus_one) * math.e
else:
real = math.cos(y) * math.cosh(x)
imag = math.sin(y) * math.sinh(x)
if isinf(real) or isinf(imag):
raise OverflowError("math range error")
return real, imag
def c_sinh(x, y):
# special treatment for sinh(+/-inf + iy) if y is finite and nonzero
if not isfinite(x) or not isfinite(y):
if isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = copysign(INF, math.cos(y))
imag = copysign(INF, math.sin(y))
else:
real = -copysign(INF, math.cos(y))
imag = copysign(INF, math.sin(y))
r = (real, imag)
else:
r = sinh_special_values[special_type(x)][special_type(y)]
# need to raise ValueError if y is +/- infinity and x is not
# a NaN
if isinf(y) and not isnan(x):
raise ValueError("math domain error")
return r
if fabs(x) > CM_LOG_LARGE_DOUBLE:
x_minus_one = x - copysign(1., x)
real = math.cos(y) * math.sinh(x_minus_one) * math.e
imag = math.sin(y) * math.cosh(x_minus_one) * math.e
else:
real = math.cos(y) * math.sinh(x)
imag = math.sin(y) * math.cosh(x)
if isinf(real) or isinf(imag):
raise OverflowError("math range error")
return real, imag
def c_atanh(x, y):
if not isfinite(x) or not isfinite(y):
return atanh_special_values[special_type(x)][special_type(y)]
# Reduce to case where x >= 0., using atanh(z) = -atanh(-z).
if x < 0.:
return c_neg(*c_atanh(*c_neg(x, y)))
ay = fabs(y)
if x > CM_SQRT_LARGE_DOUBLE or ay > CM_SQRT_LARGE_DOUBLE:
# if abs(z) is large then we use the approximation
# atanh(z) ~ 1/z +/- i*pi/2 (+/- depending on the sign
# of y
h = math.hypot(x / 2., y / 2.) # safe from overflow
real = x / 4. / h / h
# the two negations in the next line cancel each other out
# except when working with unsigned zeros: they're there to
# ensure that the branch cut has the correct continuity on
# systems that don't support signed zeros
imag = -copysign(math.pi / 2., -y)
elif x == 1. and ay < CM_SQRT_DBL_MIN:
# C99 standard says: atanh(1+/-0.) should be inf +/- 0i
if ay == 0.:
raise ValueError("math domain error")
#real = INF
#imag = y
else:
real = -math.log(math.sqrt(ay) / math.sqrt(math.hypot(ay, 2.)))
imag = copysign(math.atan2(2., -ay) / 2, y)
else:
real = log1p(4. * x / ((1 - x) * (1 - x) + ay * ay)) / 4.
imag = -math.atan2(-2. * y, (1 - x) * (1 + x) - ay * ay) / 2.
return (real, imag)
def c_rect(r, phi):
if not isfinite(r) or not isfinite(phi):
# if r is +/-infinity and phi is finite but nonzero then
# result is (+-INF +-INF i), but we need to compute cos(phi)
# and sin(phi) to figure out the signs.
if isinf(r) and isfinite(phi) and phi != 0.:
if r > 0:
real = copysign(INF, math.cos(phi))
imag = copysign(INF, math.sin(phi))
else:
real = -copysign(INF, math.cos(phi))
imag = -copysign(INF, math.sin(phi))
z = (real, imag)
else:
z = rect_special_values[special_type(r)][special_type(phi)]
# need to raise ValueError if r is a nonzero number and phi
# is infinite
if r != 0. and not isnan(r) and isinf(phi):
raise ValueError("math domain error")
return z
real = r * math.cos(phi)
imag = r * math.sin(phi)
return real, imag
def c_log(x, y):
# The usual formula for the real part is log(hypot(z.real, z.imag)).
# There are four situations where this formula is potentially
# problematic:
#
# (1) the absolute value of z is subnormal. Then hypot is subnormal,
# so has fewer than the usual number of bits of accuracy, hence may
# have large relative error. This then gives a large absolute error
# in the log. This can be solved by rescaling z by a suitable power
# of 2.
#
# (2) the absolute value of z is greater than DBL_MAX (e.g. when both
# z.real and z.imag are within a factor of 1/sqrt(2) of DBL_MAX)
# Again, rescaling solves this.
#
# (3) the absolute value of z is close to 1. In this case it's
# difficult to achieve good accuracy, at least in part because a
# change of 1ulp in the real or imaginary part of z can result in a
# change of billions of ulps in the correctly rounded answer.
#
# (4) z = 0. The simplest thing to do here is to call the
# floating-point log with an argument of 0, and let its behaviour
# (returning -infinity, signaling a floating-point exception, setting
# errno, or whatever) determine that of c_log. So the usual formula
# is fine here.
# XXX the following two lines seem unnecessary at least on Linux;
# the tests pass fine without them
if not isfinite(x) or not isfinite(y):
return log_special_values[special_type(x)][special_type(y)]
ax = fabs(x)
ay = fabs(y)
if ax > CM_LARGE_DOUBLE or ay > CM_LARGE_DOUBLE:
real = math.log(math.hypot(ax / 2., ay / 2.)) + M_LN2
elif ax < DBL_MIN and ay < DBL_MIN:
if ax > 0. or ay > 0.:
# catch cases where hypot(ax, ay) is subnormal
real = math.log(
math.hypot(math.ldexp(ax, DBL_MANT_DIG),
math.ldexp(ay, DBL_MANT_DIG)))
real -= DBL_MANT_DIG * M_LN2
else:
# log(+/-0. +/- 0i)
raise ValueError("math domain error")
#real = -INF
#imag = atan2(y, x)
else:
h = math.hypot(ax, ay)
if 0.71 <= h and h <= 1.73:
am = max(ax, ay)
an = min(ax, ay)
real = log1p((am - 1) * (am + 1) + an * an) / 2.
else:
real = math.log(h)
imag = math.atan2(y, x)
return (real, imag)
def c_log(x, y):
# The usual formula for the real part is log(hypot(z.real, z.imag)).
# There are four situations where this formula is potentially
# problematic:
#
# (1) the absolute value of z is subnormal. Then hypot is subnormal,
# so has fewer than the usual number of bits of accuracy, hence may
# have large relative error. This then gives a large absolute error
# in the log. This can be solved by rescaling z by a suitable power
# of 2.
#
# (2) the absolute value of z is greater than DBL_MAX (e.g. when both
# z.real and z.imag are within a factor of 1/sqrt(2) of DBL_MAX)
# Again, rescaling solves this.
#
# (3) the absolute value of z is close to 1. In this case it's
# difficult to achieve good accuracy, at least in part because a
# change of 1ulp in the real or imaginary part of z can result in a
# change of billions of ulps in the correctly rounded answer.
#
# (4) z = 0. The simplest thing to do here is to call the
# floating-point log with an argument of 0, and let its behaviour
# (returning -infinity, signaling a floating-point exception, setting
# errno, or whatever) determine that of c_log. So the usual formula
# is fine here.
# XXX the following two lines seem unnecessary at least on Linux;
# the tests pass fine without them
if not isfinite(x) or not isfinite(y):
return log_special_values[special_type(x)][special_type(y)]
ax = fabs(x)
ay = fabs(y)
if ax > CM_LARGE_DOUBLE or ay > CM_LARGE_DOUBLE:
real = math.log(math.hypot(ax/2., ay/2.)) + M_LN2
elif ax < DBL_MIN and ay < DBL_MIN:
if ax > 0. or ay > 0.:
# catch cases where hypot(ax, ay) is subnormal
real = math.log(math.hypot(math.ldexp(ax, DBL_MANT_DIG),
math.ldexp(ay, DBL_MANT_DIG)))
real -= DBL_MANT_DIG*M_LN2
else:
# log(+/-0. +/- 0i)
raise ValueError("math domain error")
#real = -INF
#imag = atan2(y, x)
else:
h = math.hypot(ax, ay)
if 0.71 <= h and h <= 1.73:
am = max(ax, ay)
an = min(ax, ay)
real = log1p((am-1)*(am+1) + an*an) / 2.
else:
real = math.log(h)
imag = math.atan2(y, x)
return (real, imag)
def c_sqrt(x, y):
'''
Method: use symmetries to reduce to the case when x = z.real and y
= z.imag are nonnegative. Then the real part of the result is
given by
s = sqrt((x + hypot(x, y))/2)
and the imaginary part is
d = (y/2)/s
If either x or y is very large then there's a risk of overflow in
computation of the expression x + hypot(x, y). We can avoid this
by rewriting the formula for s as:
s = 2*sqrt(x/8 + hypot(x/8, y/8))
This costs us two extra multiplications/divisions, but avoids the
overhead of checking for x and y large.
If both x and y are subnormal then hypot(x, y) may also be
subnormal, so will lack full precision. We solve this by rescaling
x and y by a sufficiently large power of 2 to ensure that x and y
are normal.
'''
if not isfinite(x) or not isfinite(y):
return sqrt_special_values[special_type(x)][special_type(y)]
if x == 0. and y == 0.:
return (0., y)
ax = fabs(x)
ay = fabs(y)
if ax < DBL_MIN and ay < DBL_MIN and (ax > 0. or ay > 0.):
# here we catch cases where hypot(ax, ay) is subnormal
ax = math.ldexp(ax, CM_SCALE_UP)
ay1= math.ldexp(ay, CM_SCALE_UP)
s = math.ldexp(math.sqrt(ax + math.hypot(ax, ay1)),
CM_SCALE_DOWN)
else:
ax /= 8.
s = 2.*math.sqrt(ax + math.hypot(ax, ay/8.))
d = ay/(2.*s)
if x >= 0.:
return (s, copysign(d, y))
else:
return (d, copysign(s, y))
def c_sqrt(x, y):
'''
Method: use symmetries to reduce to the case when x = z.real and y
= z.imag are nonnegative. Then the real part of the result is
given by
s = sqrt((x + hypot(x, y))/2)
and the imaginary part is
d = (y/2)/s
If either x or y is very large then there's a risk of overflow in
computation of the expression x + hypot(x, y). We can avoid this
by rewriting the formula for s as:
s = 2*sqrt(x/8 + hypot(x/8, y/8))
This costs us two extra multiplications/divisions, but avoids the
overhead of checking for x and y large.
If both x and y are subnormal then hypot(x, y) may also be
subnormal, so will lack full precision. We solve this by rescaling
x and y by a sufficiently large power of 2 to ensure that x and y
are normal.
'''
if not isfinite(x) or not isfinite(y):
return sqrt_special_values[special_type(x)][special_type(y)]
if x == 0. and y == 0.:
return (0., y)
ax = fabs(x)
ay = fabs(y)
if ax < DBL_MIN and ay < DBL_MIN and (ax > 0. or ay > 0.):
# here we catch cases where hypot(ax, ay) is subnormal
ax = math.ldexp(ax, CM_SCALE_UP)
ay1 = math.ldexp(ay, CM_SCALE_UP)
s = math.ldexp(math.sqrt(ax + math.hypot(ax, ay1)), CM_SCALE_DOWN)
else:
ax /= 8.
s = 2. * math.sqrt(ax + math.hypot(ax, ay / 8.))
d = ay / (2. * s)
if x >= 0.:
return (s, copysign(d, y))
else:
return (d, copysign(s, y))
def c_acosh(x, y):
# XXX the following two lines seem unnecessary at least on Linux;
# the tests pass fine without them
if not isfinite(x) or not isfinite(y):
return acosh_special_values[special_type(x)][special_type(y)]
if fabs(x) > CM_LARGE_DOUBLE or fabs(y) > CM_LARGE_DOUBLE:
# avoid unnecessary overflow for large arguments
real = math.log(math.hypot(x / 2., y / 2.)) + M_LN2 * 2.
imag = math.atan2(y, x)
else:
s1x, s1y = c_sqrt(x - 1., y)
s2x, s2y = c_sqrt(x + 1., y)
real = asinh(s1x * s2x + s1y * s2y)
imag = 2. * math.atan2(s1y, s2x)
return (real, imag)
def c_acosh(x, y):
# XXX the following two lines seem unnecessary at least on Linux;
# the tests pass fine without them
if not isfinite(x) or not isfinite(y):
return acosh_special_values[special_type(x)][special_type(y)]
if fabs(x) > CM_LARGE_DOUBLE or fabs(y) > CM_LARGE_DOUBLE:
# avoid unnecessary overflow for large arguments
real = math.log(math.hypot(x/2., y/2.)) + M_LN2*2.
imag = math.atan2(y, x)
else:
s1x, s1y = c_sqrt(x - 1., y)
s2x, s2y = c_sqrt(x + 1., y)
real = asinh(s1x*s2x + s1y*s2y)
imag = 2.*math.atan2(s1y, s2x)
return (real, imag)
def c_asinh(x, y):
if not isfinite(x) or not isfinite(y):
return asinh_special_values[special_type(x)][special_type(y)]
if fabs(x) > CM_LARGE_DOUBLE or fabs(y) > CM_LARGE_DOUBLE:
if y >= 0.:
real = copysign(math.log(math.hypot(x/2., y/2.)) +
M_LN2*2., x)
else:
real = -copysign(math.log(math.hypot(x/2., y/2.)) +
M_LN2*2., -x)
imag = math.atan2(y, fabs(x))
else:
s1x, s1y = c_sqrt(1.+y, -x)
s2x, s2y = c_sqrt(1.-y, x)
real = asinh(s1x*s2y - s2x*s1y)
imag = math.atan2(y, s1x*s2x - s1y*s2y)
return (real, imag)
def c_asinh(x, y):
if not isfinite(x) or not isfinite(y):
return asinh_special_values[special_type(x)][special_type(y)]
if fabs(x) > CM_LARGE_DOUBLE or fabs(y) > CM_LARGE_DOUBLE:
if y >= 0.:
real = copysign(
math.log(math.hypot(x / 2., y / 2.)) + M_LN2 * 2., x)
else:
real = -copysign(
math.log(math.hypot(x / 2., y / 2.)) + M_LN2 * 2., -x)
imag = math.atan2(y, fabs(x))
else:
s1x, s1y = c_sqrt(1. + y, -x)
s2x, s2y = c_sqrt(1. - y, x)
real = asinh(s1x * s2y - s2x * s1y)
imag = math.atan2(y, s1x * s2x - s1y * s2y)
return (real, imag)
def c_acos(x, y):
if not isfinite(x) or not isfinite(y):
return acos_special_values[special_type(x)][special_type(y)]
if fabs(x) > CM_LARGE_DOUBLE or fabs(y) > CM_LARGE_DOUBLE:
# avoid unnecessary overflow for large arguments
real = math.atan2(fabs(y), x)
# split into cases to make sure that the branch cut has the
# correct continuity on systems with unsigned zeros
if x < 0.:
imag = -copysign(math.log(math.hypot(x/2., y/2.)) +
M_LN2*2., y)
else:
imag = copysign(math.log(math.hypot(x/2., y/2.)) +
M_LN2*2., -y)
else:
s1x, s1y = c_sqrt(1.-x, -y)
s2x, s2y = c_sqrt(1.+x, y)
real = 2.*math.atan2(s1x, s2x)
imag = asinh(s2x*s1y - s2y*s1x)
return (real, imag)
def c_acos(x, y):
if not isfinite(x) or not isfinite(y):
return acos_special_values[special_type(x)][special_type(y)]
if fabs(x) > CM_LARGE_DOUBLE or fabs(y) > CM_LARGE_DOUBLE:
# avoid unnecessary overflow for large arguments
real = math.atan2(fabs(y), x)
# split into cases to make sure that the branch cut has the
# correct continuity on systems with unsigned zeros
if x < 0.:
imag = -copysign(
math.log(math.hypot(x / 2., y / 2.)) + M_LN2 * 2., y)
else:
imag = copysign(
math.log(math.hypot(x / 2., y / 2.)) + M_LN2 * 2., -y)
else:
s1x, s1y = c_sqrt(1. - x, -y)
s2x, s2y = c_sqrt(1. + x, y)
real = 2. * math.atan2(s1x, s2x)
imag = asinh(s2x * s1y - s2y * s1x)
return (real, imag)
def c_tanh(x, y):
# Formula:
#
# tanh(x+iy) = (tanh(x)(1+tan(y)^2) + i tan(y)(1-tanh(x))^2) /
# (1+tan(y)^2 tanh(x)^2)
#
# To avoid excessive roundoff error, 1-tanh(x)^2 is better computed
# as 1/cosh(x)^2. When abs(x) is large, we approximate 1-tanh(x)^2
# by 4 exp(-2*x) instead, to avoid possible overflow in the
# computation of cosh(x).
if not isfinite(x) or not isfinite(y):
if isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = 1.0 # vv XXX why is the 2. there?
imag = copysign(0., 2. * math.sin(y) * math.cos(y))
else:
real = -1.0
imag = copysign(0., 2. * math.sin(y) * math.cos(y))
r = (real, imag)
else:
r = tanh_special_values[special_type(x)][special_type(y)]
# need to raise ValueError if y is +/-infinity and x is finite
if isinf(y) and isfinite(x):
raise ValueError("math domain error")
return r
if fabs(x) > CM_LOG_LARGE_DOUBLE:
real = copysign(1., x)
imag = 4. * math.sin(y) * math.cos(y) * math.exp(-2. * fabs(x))
else:
tx = math.tanh(x)
ty = math.tan(y)
cx = 1. / math.cosh(x)
txty = tx * ty
denom = 1. + txty * txty
real = tx * (1. + ty * ty) / denom
imag = ((ty / denom) * cx) * cx
return real, imag
def c_tanh(x, y):
# Formula:
#
# tanh(x+iy) = (tanh(x)(1+tan(y)^2) + i tan(y)(1-tanh(x))^2) /
# (1+tan(y)^2 tanh(x)^2)
#
# To avoid excessive roundoff error, 1-tanh(x)^2 is better computed
# as 1/cosh(x)^2. When abs(x) is large, we approximate 1-tanh(x)^2
# by 4 exp(-2*x) instead, to avoid possible overflow in the
# computation of cosh(x).
if not isfinite(x) or not isfinite(y):
if isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = 1.0 # vv XXX why is the 2. there?
imag = copysign(0., 2. * math.sin(y) * math.cos(y))
else:
real = -1.0
imag = copysign(0., 2. * math.sin(y) * math.cos(y))
r = (real, imag)
else:
r = tanh_special_values[special_type(x)][special_type(y)]
# need to raise ValueError if y is +/-infinity and x is finite
if isinf(y) and isfinite(x):
raise ValueError("math domain error")
return r
if fabs(x) > CM_LOG_LARGE_DOUBLE:
real = copysign(1., x)
imag = 4. * math.sin(y) * math.cos(y) * math.exp(-2.*fabs(x))
else:
tx = math.tanh(x)
ty = math.tan(y)
cx = 1. / math.cosh(x)
txty = tx * ty
denom = 1. + txty * txty
real = tx * (1. + ty*ty) / denom
imag = ((ty / denom) * cx) * cx
return real, imag