```"""geomath.py: transcription of GeographicLib::Math class."""
# geomath.py
#
# This is a rather literal translation of the GeographicLib::Math class to
# python.  See the documentation for the C++ class for more information at
#
#    https://geographiclib.sourceforge.io/html/annotated.html
#
# Copyright (c) Charles Karney (2011-2017) <charles@karney.com> and
# https://geographiclib.sourceforge.io/
######################################################################

import sys
import math

class Math(object):
"""

This defines constants:
epsilon, difference between 1 and the next bigger number
digits, the number of digits in the fraction of a real number
minval, minimum normalized positive number
maxval, maximum finite number
nan, not a number
inf, infinity
"""

digits = 53
epsilon = math.pow(2.0, 1-digits)
minval = math.pow(2.0, -1022)
maxval = math.pow(2.0, 1023) * (2 - epsilon)
inf = float("inf") if sys.version_info > (2, 6) else 2 * maxval
nan = float("nan") if sys.version_info > (2, 6) else inf - inf

def sq(x):
"""Square a number"""

return x * x
sq = staticmethod(sq)

def cbrt(x):
"""Real cube root of a number"""

y = math.pow(abs(x), 1/3.0)
return y if x >= 0 else -y
cbrt = staticmethod(cbrt)

def log1p(x):
"""log(1 + x) accurate for small x (missing from python 2.5.2)"""

if sys.version_info > (2, 6):
return math.log1p(x)

y = 1 + x
z = y - 1
# Here's the explanation for this magic: y = 1 + z, exactly, and z
# approx x, thus log(y)/z (which is nearly constant near z = 0) returns
# a good approximation to the true log(1 + x)/x.  The multiplication x *
# (log(y)/z) introduces little additional error.
return x if z == 0 else x * math.log(y) / z
log1p = staticmethod(log1p)

def atanh(x):
"""atanh(x) (missing from python 2.5.2)"""

if sys.version_info > (2, 6):
return math.atanh(x)

y = abs(x)                  # Enforce odd parity
y = Math.log1p(2 * y/(1 - y))/2
return -y if x < 0 else y
atanh = staticmethod(atanh)

def copysign(x, y):
"""return x with the sign of y (missing from python 2.5.2)"""

if sys.version_info > (2, 6):
return math.copysign(x, y)

return math.fabs(x) * (-1 if y < 0 or (y == 0 and 1/y < 0) else 1)
copysign = staticmethod(copysign)

def norm(x, y):
"""Private: Normalize a two-vector."""
r = math.hypot(x, y)
return x/r, y/r
norm = staticmethod(norm)

def sum(u, v):
"""Error free transformation of a sum."""
# Error free transformation of a sum.  Note that t can be the same as one
# of the first two arguments.
s = u + v
up = s - v
vpp = s - up
up -= u
vpp -= v
t = -(up + vpp)
# u + v =       s      + t
#       = round(u + v) + t
return s, t
sum = staticmethod(sum)

def polyval(N, p, s, x):
"""Evaluate a polynomial."""
y = float(0 if N < 0 else p[s]) # make sure the returned value is a float
while N > 0:
N -= 1; s += 1
y = y * x + p[s]
return y
polyval = staticmethod(polyval)

def AngRound(x):
"""Private: Round an angle so that small values underflow to zero."""
# The makes the smallest gap in x = 1/16 - nextafter(1/16, 0) = 1/2^57
# for reals = 0.7 pm on the earth if x is an angle in degrees.  (This
# is about 1000 times more resolution than we get with angles around 90
# degrees.)  We use this to avoid having to deal with near singular
# cases when x is non-zero but tiny (e.g., 1.0e-200).
z = 1/16.0
y = abs(x)
# The compiler mustn't "simplify" z - (z - y) to y
if y < z: y = z - (z - y)
return 0.0 if x == 0 else (-y if x < 0 else y)
AngRound = staticmethod(AngRound)

def AngNormalize(x):
"""reduce angle to (-180,180]"""

y = math.fmod(x, 360)
# On Windows 32-bit with python 2.7, math.fmod(-0.0, 360) = +0.0
# This fixes this bug.  See also Math::AngNormalize in the C++ library.
# sincosd has a similar fix.
y = x if x == 0 else y
return (y + 360 if y <= -180 else
(y if y <= 180 else y - 360))
AngNormalize = staticmethod(AngNormalize)

def LatFix(x):
"""replace angles outside [-90,90] by NaN"""

return Math.nan if abs(x) > 90 else x
LatFix = staticmethod(LatFix)

def AngDiff(x, y):
"""compute y - x and reduce to [-180,180] accurately"""

d, t = Math.sum(Math.AngNormalize(-x), Math.AngNormalize(y))
d = Math.AngNormalize(d)
return Math.sum(-180 if d == 180 and t > 0 else d, t)
AngDiff = staticmethod(AngDiff)

def sincosd(x):
"""Compute sine and cosine of x in degrees."""

r = math.fmod(x, 360)
q = Math.nan if Math.isnan(r) else int(math.floor(r / 90 + 0.5))
r -= 90 * q; r = math.radians(r)
s = math.sin(r); c = math.cos(r)
q = q % 4
if q == 1:
s, c =  c, -s
elif q == 2:
s, c = -s, -c
elif q == 3:
s, c = -c,  s
# Remove the minus sign on -0.0 except for sin(-0.0).
# On Windows 32-bit with python 2.7, math.fmod(-0.0, 360) = +0.0
# (x, c) here fixes this bug.  See also Math::sincosd in the C++ library.
# AngNormalize has a similar fix.
s, c = (x, c) if x == 0 else (0.0+s, 0.0+c)
return s, c
sincosd = staticmethod(sincosd)

def atan2d(y, x):
"""compute atan2(y, x) with the result in degrees"""

if abs(y) > abs(x):
q = 2; x, y = y, x
else:
q = 0
if x < 0:
q += 1; x = -x
ang = math.degrees(math.atan2(y, x))
if q == 1:
ang = (180 if y >= 0 else -180) - ang
elif q == 2:
ang =  90 - ang
elif q == 3:
ang = -90 + ang
return ang
atan2d = staticmethod(atan2d)

def isfinite(x):
"""Test for finiteness"""

return abs(x) <= Math.maxval
isfinite = staticmethod(isfinite)

def isnan(x):
"""Test if nan"""

return math.isnan(x) if sys.version_info > (2, 6) else x != x
isnan = staticmethod(isnan)
```