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import math
from math import fabs
from rpython.rlib.rfloat import asinh, log1p, isfinite
from rpython.rlib.constant import DBL_MIN, CM_SCALE_UP, CM_SCALE_DOWN
from rpython.rlib.constant import CM_LARGE_DOUBLE, DBL_MANT_DIG
from rpython.rlib.constant import M_LN2, M_LN10
from rpython.rlib.constant import CM_SQRT_LARGE_DOUBLE, CM_SQRT_DBL_MIN
from rpython.rlib.constant import CM_LOG_LARGE_DOUBLE
from rpython.rlib.special_value import special_type, INF, NAN
from rpython.rlib.special_value import sqrt_special_values
from rpython.rlib.special_value import acos_special_values
from rpython.rlib.special_value import acosh_special_values
from rpython.rlib.special_value import asinh_special_values
from rpython.rlib.special_value import atanh_special_values
from rpython.rlib.special_value import log_special_values
from rpython.rlib.special_value import exp_special_values
from rpython.rlib.special_value import cosh_special_values
from rpython.rlib.special_value import sinh_special_values
from rpython.rlib.special_value import tanh_special_values
from rpython.rlib.special_value import rect_special_values
#binary
def c_add(x, y):
(r1, i1), (r2, i2) = x, y
r = r1 + r2
i = i1 + i2
return (r, i)
def c_sub(x, y):
(r1, i1), (r2, i2) = x, y
r = r1 - r2
i = i1 - i2
return (r, i)
def c_mul(x, y):
(r1, i1), (r2, i2) = x, y
r = r1 * r2 - i1 * i2
i = r1 * i2 + i1 * r2
return (r, i)
def c_div(x, y): #x/y
(r1, i1), (r2, i2) = x, y
if r2 < 0:
abs_r2 = -r2
else:
abs_r2 = r2
if i2 < 0:
abs_i2 = -i2
else:
abs_i2 = i2
if abs_r2 >= abs_i2:
if abs_r2 == 0.0:
raise ZeroDivisionError
else:
ratio = i2 / r2
denom = r2 + i2 * ratio
rr = (r1 + i1 * ratio) / denom
ir = (i1 - r1 * ratio) / denom
elif math.isnan(r2):
rr = NAN
ir = NAN
else:
ratio = r2 / i2
denom = r2 * ratio + i2
assert i2 != 0.0
rr = (r1 * ratio + i1) / denom
ir = (i1 * ratio - r1) / denom
return (rr, ir)
def c_pow(x, y):
(r1, i1), (r2, i2) = x, y
if i1 == 0 and i2 == 0 and r1 > 0:
rr = math.pow(r1, r2)
ir = 0.
elif r2 == 0.0 and i2 == 0.0:
rr, ir = 1, 0
elif r1 == 1.0 and i1 == 0.0:
rr, ir = (1.0, 0.0)
elif r1 == 0.0 and i1 == 0.0:
if i2 != 0.0 or r2 < 0.0:
raise ZeroDivisionError
rr, ir = (0.0, 0.0)
else:
vabs = math.hypot(r1,i1)
len = math.pow(vabs,r2)
at = math.atan2(i1,r1)
phase = at * r2
if i2 != 0.0:
len /= math.exp(at * i2)
phase += i2 * math.log(vabs)
try:
rr = len * math.cos(phase)
ir = len * math.sin(phase)
except ValueError:
rr = NAN
ir = NAN
return (rr, ir)
#unary
def c_neg(r, i):
return (-r, -i)
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, math.copysign(d, y))
else:
return (d, math.copysign(s, y))
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 = -math.copysign(math.log(math.hypot(x/2., y/2.)) +
M_LN2*2., y)
else:
imag = math.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_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_asin(x, y):
# asin(z) = -i asinh(iz)
sx, sy = c_asinh(-y, x)
return (sy, -sx)
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 = math.copysign(math.log(math.hypot(x/2., y/2.)) +
M_LN2*2., x)
else:
real = -math.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_atan(x, y):
# atan(z) = -i atanh(iz)
sx, sy = c_atanh(-y, x)
return (sy, -sx)
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 = -math.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 = math.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_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_log10(x, y):
rx, ry = c_log(x, y)
return (rx / M_LN10, ry / M_LN10)
def c_exp(x, y):
if not isfinite(x) or not isfinite(y):
if math.isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = math.copysign(INF, math.cos(y))
imag = math.copysign(INF, math.sin(y))
else:
real = math.copysign(0., math.cos(y))
imag = math.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 math.isinf(y) and (isfinite(x) or (math.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 math.isinf(real) or math.isinf(imag):
raise OverflowError("math range error")
return real, imag
def c_cosh(x, y):
if not isfinite(x) or not isfinite(y):
if math.isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = math.copysign(INF, math.cos(y))
imag = math.copysign(INF, math.sin(y))
else:
real = math.copysign(INF, math.cos(y))
imag = -math.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 math.isinf(y) and not math.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 - math.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 math.isinf(real) or math.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 math.isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = math.copysign(INF, math.cos(y))
imag = math.copysign(INF, math.sin(y))
else:
real = -math.copysign(INF, math.cos(y))
imag = math.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 math.isinf(y) and not math.isnan(x):
raise ValueError("math domain error")
return r
if fabs(x) > CM_LOG_LARGE_DOUBLE:
x_minus_one = x - math.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 math.isinf(real) or math.isinf(imag):
raise OverflowError("math range error")
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 math.isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = 1.0 # vv XXX why is the 2. there?
imag = math.copysign(0., 2. * math.sin(y) * math.cos(y))
else:
real = -1.0
imag = math.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 math.isinf(y) and isfinite(x):
raise ValueError("math domain error")
return r
if fabs(x) > CM_LOG_LARGE_DOUBLE:
real = math.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_cos(r, i):
# cos(z) = cosh(iz)
return c_cosh(-i, r)
def c_sin(r, i):
# sin(z) = -i sinh(iz)
sr, si = c_sinh(-i, r)
return si, -sr
def c_tan(r, i):
# tan(z) = -i tanh(iz)
sr, si = c_tanh(-i, r)
return si, -sr
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 math.isinf(r) and isfinite(phi) and phi != 0.:
if r > 0:
real = math.copysign(INF, math.cos(phi))
imag = math.copysign(INF, math.sin(phi))
else:
real = -math.copysign(INF, math.cos(phi))
imag = -math.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 math.isnan(r) and math.isinf(phi):
raise ValueError("math domain error")
return z
real = r * math.cos(phi)
imag = r * math.sin(phi)
return real, imag
def c_phase(x, y):
# Windows screws up atan2 for inf and nan, and alpha Tru64 5.1 doesn't
# follow C99 for atan2(0., 0.).
if math.isnan(x) or math.isnan(y):
return NAN
if math.isinf(y):
if math.isinf(x):
if math.copysign(1., x) == 1.:
# atan2(+-inf, +inf) == +-pi/4
return math.copysign(0.25 * math.pi, y)
else:
# atan2(+-inf, -inf) == +-pi*3/4
return math.copysign(0.75 * math.pi, y)
# atan2(+-inf, x) == +-pi/2 for finite x
return math.copysign(0.5 * math.pi, y)
if math.isinf(x) or y == 0.:
if math.copysign(1., x) == 1.:
# atan2(+-y, +inf) = atan2(+-0, +x) = +-0.
return math.copysign(0., y)
else:
# atan2(+-y, -inf) = atan2(+-0., -x) = +-pi.
return math.copysign(math.pi, y)
return math.atan2(y, x)
def c_abs(r, i):
if not isfinite(r) or not isfinite(i):
# C99 rules: if either the real or the imaginary part is an
# infinity, return infinity, even if the other part is a NaN.
if math.isinf(r):
return INF
if math.isinf(i):
return INF
# either the real or imaginary part is a NaN,
# and neither is infinite. Result should be NaN.
return NAN
result = math.hypot(r, i)
if not isfinite(result):
raise OverflowError("math range error")
return result
def c_polar(r, i):
real = c_abs(r, i)
phi = c_phase(r, i)
return real, phi
def c_isinf(r, i):
return math.isinf(r) or math.isinf(i)
def c_isnan(r, i):
return math.isnan(r) or math.isnan(i)
def c_isfinite(r, i):
return isfinite(r) and isfinite(i)
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