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"""
Implement the cmath module functions.
"""
import cmath
import math
import llvmlite.llvmpy.core as lc
from llvmlite.llvmpy.core import Type
from numba.core.imputils import Registry, impl_ret_untracked
from numba.core import types, cgutils
from numba.core.typing import signature
from numba.cpython import builtins, mathimpl
registry = Registry()
lower = registry.lower
def is_nan(builder, z):
return builder.fcmp_unordered('uno', z.real, z.imag)
def is_inf(builder, z):
return builder.or_(mathimpl.is_inf(builder, z.real),
mathimpl.is_inf(builder, z.imag))
def is_finite(builder, z):
return builder.and_(mathimpl.is_finite(builder, z.real),
mathimpl.is_finite(builder, z.imag))
@lower(cmath.isnan, types.Complex)
def isnan_float_impl(context, builder, sig, args):
[typ] = sig.args
[value] = args
z = context.make_complex(builder, typ, value=value)
res = is_nan(builder, z)
return impl_ret_untracked(context, builder, sig.return_type, res)
@lower(cmath.isinf, types.Complex)
def isinf_float_impl(context, builder, sig, args):
[typ] = sig.args
[value] = args
z = context.make_complex(builder, typ, value=value)
res = is_inf(builder, z)
return impl_ret_untracked(context, builder, sig.return_type, res)
@lower(cmath.isfinite, types.Complex)
def isfinite_float_impl(context, builder, sig, args):
[typ] = sig.args
[value] = args
z = context.make_complex(builder, typ, value=value)
res = is_finite(builder, z)
return impl_ret_untracked(context, builder, sig.return_type, res)
@lower(cmath.rect, types.Float, types.Float)
def rect_impl(context, builder, sig, args):
[r, phi] = args
# We can't call math.isfinite() inside rect() below because it
# only exists on 3.2+.
phi_is_finite = mathimpl.is_finite(builder, phi)
def rect(r, phi, phi_is_finite):
if not phi_is_finite:
if not r:
# cmath.rect(0, phi={inf, nan}) = 0
return abs(r)
if math.isinf(r):
# cmath.rect(inf, phi={inf, nan}) = inf + j phi
return complex(r, phi)
real = math.cos(phi)
imag = math.sin(phi)
if real == 0. and math.isinf(r):
# 0 * inf would return NaN, we want to keep 0 but xor the sign
real /= r
else:
real *= r
if imag == 0. and math.isinf(r):
# ditto
imag /= r
else:
imag *= r
return complex(real, imag)
inner_sig = signature(sig.return_type, *sig.args + (types.boolean,))
res = context.compile_internal(builder, rect, inner_sig,
args + [phi_is_finite])
return impl_ret_untracked(context, builder, sig, res)
def intrinsic_complex_unary(inner_func):
def wrapper(context, builder, sig, args):
[typ] = sig.args
[value] = args
z = context.make_complex(builder, typ, value=value)
x = z.real
y = z.imag
# Same as above: math.isfinite() is unavailable on 2.x so we precompute
# its value and pass it to the pure Python implementation.
x_is_finite = mathimpl.is_finite(builder, x)
y_is_finite = mathimpl.is_finite(builder, y)
inner_sig = signature(sig.return_type,
*(typ.underlying_float,) * 2 + (types.boolean,) * 2)
res = context.compile_internal(builder, inner_func, inner_sig,
(x, y, x_is_finite, y_is_finite))
return impl_ret_untracked(context, builder, sig, res)
return wrapper
NAN = float('nan')
INF = float('inf')
@lower(cmath.exp, types.Complex)
@intrinsic_complex_unary
def exp_impl(x, y, x_is_finite, y_is_finite):
"""cmath.exp(x + y j)"""
if x_is_finite:
if y_is_finite:
c = math.cos(y)
s = math.sin(y)
r = math.exp(x)
return complex(r * c, r * s)
else:
return complex(NAN, NAN)
elif math.isnan(x):
if y:
return complex(x, x) # nan + j nan
else:
return complex(x, y) # nan + 0j
elif x > 0.0:
# x == +inf
if y_is_finite:
real = math.cos(y)
imag = math.sin(y)
# Avoid NaNs if math.cos(y) or math.sin(y) == 0
# (e.g. cmath.exp(inf + 0j) == inf + 0j)
if real != 0:
real *= x
if imag != 0:
imag *= x
return complex(real, imag)
else:
return complex(x, NAN)
else:
# x == -inf
if y_is_finite:
r = math.exp(x)
c = math.cos(y)
s = math.sin(y)
return complex(r * c, r * s)
else:
r = 0
return complex(r, r)
@lower(cmath.log, types.Complex)
@intrinsic_complex_unary
def log_impl(x, y, x_is_finite, y_is_finite):
"""cmath.log(x + y j)"""
a = math.log(math.hypot(x, y))
b = math.atan2(y, x)
return complex(a, b)
@lower(cmath.log, types.Complex, types.Complex)
def log_base_impl(context, builder, sig, args):
"""cmath.log(z, base)"""
[z, base] = args
def log_base(z, base):
return cmath.log(z) / cmath.log(base)
res = context.compile_internal(builder, log_base, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.log10, types.Complex)
def log10_impl(context, builder, sig, args):
LN_10 = 2.302585092994045684
def log10_impl(z):
"""cmath.log10(z)"""
z = cmath.log(z)
# This formula gives better results on +/-inf than cmath.log(z, 10)
# See http://bugs.python.org/issue22544
return complex(z.real / LN_10, z.imag / LN_10)
res = context.compile_internal(builder, log10_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.phase, types.Complex)
@intrinsic_complex_unary
def phase_impl(x, y, x_is_finite, y_is_finite):
"""cmath.phase(x + y j)"""
return math.atan2(y, x)
@lower(cmath.polar, types.Complex)
@intrinsic_complex_unary
def polar_impl(x, y, x_is_finite, y_is_finite):
"""cmath.polar(x + y j)"""
return math.hypot(x, y), math.atan2(y, x)
@lower(cmath.sqrt, types.Complex)
def sqrt_impl(context, builder, sig, args):
# We risk spurious overflow for components >= FLT_MAX / (1 + sqrt(2)).
SQRT2 = 1.414213562373095048801688724209698079E0
ONE_PLUS_SQRT2 = (1. + SQRT2)
theargflt = sig.args[0].underlying_float
# Get a type specific maximum value so scaling for overflow is based on that
MAX = mathimpl.DBL_MAX if theargflt.bitwidth == 64 else mathimpl.FLT_MAX
# THRES will be double precision, should not impact typing as it's just
# used for comparison, there *may* be a few values near THRES which
# deviate from e.g. NumPy due to rounding that occurs in the computation
# of this value in the case of a 32bit argument.
THRES = MAX / ONE_PLUS_SQRT2
def sqrt_impl(z):
"""cmath.sqrt(z)"""
# This is NumPy's algorithm, see npy_csqrt() in npy_math_complex.c.src
a = z.real
b = z.imag
if a == 0.0 and b == 0.0:
return complex(abs(b), b)
if math.isinf(b):
return complex(abs(b), b)
if math.isnan(a):
return complex(a, a)
if math.isinf(a):
if a < 0.0:
return complex(abs(b - b), math.copysign(a, b))
else:
return complex(a, math.copysign(b - b, b))
# The remaining special case (b is NaN) is handled just fine by
# the normal code path below.
# Scale to avoid overflow
if abs(a) >= THRES or abs(b) >= THRES:
a *= 0.25
b *= 0.25
scale = True
else:
scale = False
# Algorithm 312, CACM vol 10, Oct 1967
if a >= 0:
t = math.sqrt((a + math.hypot(a, b)) * 0.5)
real = t
imag = b / (2 * t)
else:
t = math.sqrt((-a + math.hypot(a, b)) * 0.5)
real = abs(b) / (2 * t)
imag = math.copysign(t, b)
# Rescale
if scale:
return complex(real * 2, imag)
else:
return complex(real, imag)
res = context.compile_internal(builder, sqrt_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.cos, types.Complex)
def cos_impl(context, builder, sig, args):
def cos_impl(z):
"""cmath.cos(z) = cmath.cosh(z j)"""
return cmath.cosh(complex(-z.imag, z.real))
res = context.compile_internal(builder, cos_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.cosh, types.Complex)
def cosh_impl(context, builder, sig, args):
def cosh_impl(z):
"""cmath.cosh(z)"""
x = z.real
y = z.imag
if math.isinf(x):
if math.isnan(y):
# x = +inf, y = NaN => cmath.cosh(x + y j) = inf + Nan * j
real = abs(x)
imag = y
elif y == 0.0:
# x = +inf, y = 0 => cmath.cosh(x + y j) = inf + 0j
real = abs(x)
imag = y
else:
real = math.copysign(x, math.cos(y))
imag = math.copysign(x, math.sin(y))
if x < 0.0:
# x = -inf => negate imaginary part of result
imag = -imag
return complex(real, imag)
return complex(math.cos(y) * math.cosh(x),
math.sin(y) * math.sinh(x))
res = context.compile_internal(builder, cosh_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.sin, types.Complex)
def sin_impl(context, builder, sig, args):
def sin_impl(z):
"""cmath.sin(z) = -j * cmath.sinh(z j)"""
r = cmath.sinh(complex(-z.imag, z.real))
return complex(r.imag, -r.real)
res = context.compile_internal(builder, sin_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.sinh, types.Complex)
def sinh_impl(context, builder, sig, args):
def sinh_impl(z):
"""cmath.sinh(z)"""
x = z.real
y = z.imag
if math.isinf(x):
if math.isnan(y):
# x = +/-inf, y = NaN => cmath.sinh(x + y j) = x + NaN * j
real = x
imag = y
else:
real = math.cos(y)
imag = math.sin(y)
if real != 0.:
real *= x
if imag != 0.:
imag *= abs(x)
return complex(real, imag)
return complex(math.cos(y) * math.sinh(x),
math.sin(y) * math.cosh(x))
res = context.compile_internal(builder, sinh_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.tan, types.Complex)
def tan_impl(context, builder, sig, args):
def tan_impl(z):
"""cmath.tan(z) = -j * cmath.tanh(z j)"""
r = cmath.tanh(complex(-z.imag, z.real))
return complex(r.imag, -r.real)
res = context.compile_internal(builder, tan_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.tanh, types.Complex)
def tanh_impl(context, builder, sig, args):
def tanh_impl(z):
"""cmath.tanh(z)"""
x = z.real
y = z.imag
if math.isinf(x):
real = math.copysign(1., x)
if math.isinf(y):
imag = 0.
else:
imag = math.copysign(0., math.sin(2. * y))
return complex(real, imag)
# This is CPython's algorithm (see c_tanh() in cmathmodule.c).
# XXX how to force float constants into single precision?
tx = math.tanh(x)
ty = math.tan(y)
cx = 1. / math.cosh(x)
txty = tx * ty
denom = 1. + txty * txty
return complex(
tx * (1. + ty * ty) / denom,
((ty / denom) * cx) * cx)
res = context.compile_internal(builder, tanh_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.acos, types.Complex)
def acos_impl(context, builder, sig, args):
LN_4 = math.log(4)
THRES = mathimpl.FLT_MAX / 4
def acos_impl(z):
"""cmath.acos(z)"""
# CPython's algorithm (see c_acos() in cmathmodule.c)
if abs(z.real) > THRES or abs(z.imag) > THRES:
# Avoid unnecessary overflow for large arguments
# (also handles infinities gracefully)
real = math.atan2(abs(z.imag), z.real)
imag = math.copysign(
math.log(math.hypot(z.real * 0.5, z.imag * 0.5)) + LN_4,
-z.imag)
return complex(real, imag)
else:
s1 = cmath.sqrt(complex(1. - z.real, -z.imag))
s2 = cmath.sqrt(complex(1. + z.real, z.imag))
real = 2. * math.atan2(s1.real, s2.real)
imag = math.asinh(s2.real * s1.imag - s2.imag * s1.real)
return complex(real, imag)
res = context.compile_internal(builder, acos_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.acosh, types.Complex)
def acosh_impl(context, builder, sig, args):
LN_4 = math.log(4)
THRES = mathimpl.FLT_MAX / 4
def acosh_impl(z):
"""cmath.acosh(z)"""
# CPython's algorithm (see c_acosh() in cmathmodule.c)
if abs(z.real) > THRES or abs(z.imag) > THRES:
# Avoid unnecessary overflow for large arguments
# (also handles infinities gracefully)
real = math.log(math.hypot(z.real * 0.5, z.imag * 0.5)) + LN_4
imag = math.atan2(z.imag, z.real)
return complex(real, imag)
else:
s1 = cmath.sqrt(complex(z.real - 1., z.imag))
s2 = cmath.sqrt(complex(z.real + 1., z.imag))
real = math.asinh(s1.real * s2.real + s1.imag * s2.imag)
imag = 2. * math.atan2(s1.imag, s2.real)
return complex(real, imag)
# Condensed formula (NumPy)
#return cmath.log(z + cmath.sqrt(z + 1.) * cmath.sqrt(z - 1.))
res = context.compile_internal(builder, acosh_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.asinh, types.Complex)
def asinh_impl(context, builder, sig, args):
LN_4 = math.log(4)
THRES = mathimpl.FLT_MAX / 4
def asinh_impl(z):
"""cmath.asinh(z)"""
# CPython's algorithm (see c_asinh() in cmathmodule.c)
if abs(z.real) > THRES or abs(z.imag) > THRES:
real = math.copysign(
math.log(math.hypot(z.real * 0.5, z.imag * 0.5)) + LN_4,
z.real)
imag = math.atan2(z.imag, abs(z.real))
return complex(real, imag)
else:
s1 = cmath.sqrt(complex(1. + z.imag, -z.real))
s2 = cmath.sqrt(complex(1. - z.imag, z.real))
real = math.asinh(s1.real * s2.imag - s2.real * s1.imag)
imag = math.atan2(z.imag, s1.real * s2.real - s1.imag * s2.imag)
return complex(real, imag)
res = context.compile_internal(builder, asinh_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.asin, types.Complex)
def asin_impl(context, builder, sig, args):
def asin_impl(z):
"""cmath.asin(z) = -j * cmath.asinh(z j)"""
r = cmath.asinh(complex(-z.imag, z.real))
return complex(r.imag, -r.real)
res = context.compile_internal(builder, asin_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.atan, types.Complex)
def atan_impl(context, builder, sig, args):
def atan_impl(z):
"""cmath.atan(z) = -j * cmath.atanh(z j)"""
r = cmath.atanh(complex(-z.imag, z.real))
if math.isinf(z.real) and math.isnan(z.imag):
# XXX this is odd but necessary
return complex(r.imag, r.real)
else:
return complex(r.imag, -r.real)
res = context.compile_internal(builder, atan_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
@lower(cmath.atanh, types.Complex)
def atanh_impl(context, builder, sig, args):
LN_4 = math.log(4)
THRES_LARGE = math.sqrt(mathimpl.FLT_MAX / 4)
THRES_SMALL = math.sqrt(mathimpl.FLT_MIN)
PI_12 = math.pi / 2
def atanh_impl(z):
"""cmath.atanh(z)"""
# CPython's algorithm (see c_atanh() in cmathmodule.c)
if z.real < 0.:
# Reduce to case where z.real >= 0., using atanh(z) = -atanh(-z).
negate = True
z = -z
else:
negate = False
ay = abs(z.imag)
if math.isnan(z.real) or z.real > THRES_LARGE or ay > THRES_LARGE:
if math.isinf(z.imag):
real = math.copysign(0., z.real)
elif math.isinf(z.real):
real = 0.
else:
# may be safe from overflow, depending on hypot's implementation...
h = math.hypot(z.real * 0.5, z.imag * 0.5)
real = z.real/4./h/h
imag = -math.copysign(PI_12, -z.imag)
elif z.real == 1. and ay < THRES_SMALL:
# C99 standard says: atanh(1+/-0.) should be inf +/- 0j
if ay == 0.:
real = INF
imag = z.imag
else:
real = -math.log(math.sqrt(ay) /
math.sqrt(math.hypot(ay, 2.)))
imag = math.copysign(math.atan2(2., -ay) / 2, z.imag)
else:
sqay = ay * ay
zr1 = 1 - z.real
real = math.log1p(4. * z.real / (zr1 * zr1 + sqay)) * 0.25
imag = -math.atan2(-2. * z.imag,
zr1 * (1 + z.real) - sqay) * 0.5
if math.isnan(z.imag):
imag = NAN
if negate:
return complex(-real, -imag)
else:
return complex(real, imag)
res = context.compile_internal(builder, atanh_impl, sig, args)
return impl_ret_untracked(context, builder, sig, res)
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