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# -*- coding: utf-8 -*- 

 

u'''Precision floating point functions, classes and utilities. 

''' 

# make sure int/int division yields float quotient, see .basics 

from __future__ import division 

 

from pygeodesy.basics import copysign0, isfinite, isint, isnear0, \ 

isscalar, len2, _xcopy 

from pygeodesy.errors import _IsnotError, LenError, _OverflowError, \ 

_TypeError, _ValueError 

from pygeodesy.interns import EPS0, EPS1, MISSING, NN, PI, PI_2, PI_4, \ 

_EPS0__2, _finite_, _few_, _h_, _negative_,\ 

_not_, _singular_, _SPACE_, _too_, \ 

_0_0, _1_0, _1_5 as _3_2nd, _2_0, _3_0 

from pygeodesy.lazily import _ALL_LAZY, _sys_version_info2 

from pygeodesy.streprs import Fmt, unstr 

from pygeodesy.units import Int_ 

 

from math import sqrt # pow 

from operator import mul as _mul 

 

__all__ = _ALL_LAZY.fmath 

__version__ = '21.06.30' 

 

# sqrt(2) <https://WikiPedia.org/wiki/Square_root_of_2> 

_0_4142 = 0.414213562373095 # sqrt(_2_0) - _1_0 

_1_3rd = _1_0 / _3_0 

_2_3rd = _2_0 / _3_0 

 

 

def _2even(s, r, p): 

'''(INTERNAL) Half-even rounding. 

''' 

if (r > 0 and p > 0) or \ 

(r < 0 and p < 0): # signs match 

t, p = _2sum(s, p * 2) 

if not p: 

s = t 

return s 

 

 

def _2sum(a, b): # by .testFmath 

'''(INTERNAL) Precision C{2sum} of M{a + b}. 

''' 

s = a + b 

if not isfinite(s): 

raise _OverflowError(unstr(_2sum.__name__, a, b), txt=str(s)) 

if abs(a) < abs(b): 

a, b = b, a 

return s, b - (s - a) 

 

 

class Fsum(object): 

'''Precision summation similar to standard Python function C{math.fsum}. 

 

Unlike C{math.fsum}, this class accumulates the values and provides 

intermediate, precision running sums. Accumulation may continue 

after intermediate summations. 

 

@note: Handling of exceptions, C{inf}, C{INF}, C{nan} and C{NAN} 

values is different from C{math.fsum}. 

 

@see: U{Hettinger<https://GitHub.com/ActiveState/code/blob/master/recipes/Python/ 

393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}, 

U{Kahan<https://WikiPedia.org/wiki/Kahan_summation_algorithm>}, 

U{Klein<https://Link.Springer.com/article/10.1007/s00607-005-0139-x>}, 

Python 2.6+ file I{Modules/mathmodule.c} and the issue log 

U{Full precision summation<https://Bugs.Python.org/issue2819>}. 

''' 

_fsum2_ = None 

_n = 0 

_ps = [] 

 

def __init__(self, *starts): 

'''Initialize a new accumulator with one or more start values. 

 

@arg starts: No, one or more start values (C{scalar}s). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise TypeError: Non-scalar B{C{starts}} value. 

 

@raise ValueError: Invalid or non-finite B{C{starts}} value. 

''' 

self._n = 0 

self._ps = [] 

if starts: 

self.fadd(starts) 

 

def __add__(self, other): 

'''Sum of this and an other instance or a scalar. 

 

@arg other: L{Fsum} instance or C{scalar}. 

 

@return: The sum, a new instance (L{Fsum}). 

 

@see: Method L{Fsum.__iadd__}. 

''' 

f = self.fcopy() 

f += other 

return f # self.fcopy().__iadd__(other) 

 

def __iadd__(self, other): 

'''Add a scalar or an other instance to this instance. 

 

@arg other: L{Fsum} instance or C{scalar}. 

 

@return: This instance, updated (L{Fsum}). 

 

@raise TypeError: Invalid B{C{other}} type. 

 

@see: Method L{Fsum.fadd}. 

''' 

if isscalar(other): 

self.fadd_(other) 

elif other is self: 

self.fmul(2) 

elif isinstance(other, Fsum): 

self.fadd(other._ps) 

else: 

raise _TypeError(_SPACE_(self, '+=', repr(other))) 

return self 

 

def __imul__(self, other): 

'''Multiply this instance by a scalar or an other instance. 

 

@arg other: L{Fsum} instance or C{scalar}. 

 

@return: This instance, updated (L{Fsum}). 

 

@raise TypeError: Invalid B{C{other}} type. 

 

@see: Method L{Fsum.fmul}. 

''' 

if isscalar(other): 

self.fmul(other) 

elif isinstance(other, Fsum): 

ps = list(other._ps) # copy 

if ps: 

s = self.fcopy() 

self.fmul(ps.pop()) 

while ps: # self += s * ps.pop() 

p = s.fcopy() 

p.fmul(ps.pop()) 

self.fadd(p._ps) 

else: 

self._ps = [] # zero 

self._fsum2_ = None 

else: 

raise _TypeError(_SPACE_(self, '*=', repr(other))) 

return self 

 

def __isub__(self, other): 

'''Subtract a scalar or an other instance from this instance. 

 

@arg other: L{Fsum} instance or C{scalar}. 

 

@return: This instance, updated (L{Fsum}). 

 

@raise TypeError: Invalid B{C{other}} type. 

 

@see: Method L{Fsum.fadd}. 

''' 

if isscalar(other): 

self.fadd_(-other) 

elif other is self: 

self._ps = [] # zero 

self._fsum2_ = None 

elif isinstance(other, Fsum): 

self.fadd(-p for p in other._ps) 

else: 

raise _TypeError(_SPACE_(self, '-=', repr(other))) 

return self 

 

def __len__(self): 

'''Return the number of accumulated values. 

''' 

return self._n 

 

def __mul__(self, other): 

'''Product of this and an other instance or a scalar. 

 

@arg other: L{Fsum} instance or C{scalar}. 

 

@return: The product, a new instance (L{Fsum}). 

 

@see: Method L{Fsum.__imul__}. 

''' 

f = self.fcopy() 

f *= other 

return f 

 

def __str__(self): 

from pygeodesy.named import classname 

return Fmt.PAREN(classname(self, prefixed=True), NN) 

 

def __sub__(self, other): 

'''Difference of this and an other instance or a scalar. 

 

@arg other: L{Fsum} instance or C{scalar}. 

 

@return: The difference, a new instance (L{Fsum}). 

 

@see: Method L{Fsum.__isub__}. 

''' 

f = self.fcopy() 

f -= other 

return f 

 

def fadd(self, xs): 

'''Accumulate more values from an iterable. 

 

@arg xs: Iterable, list, tuple, etc. (C{scalar}s). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise TypeError: Non-scalar B{C{xs}} value. 

 

@raise ValueError: Invalid or non-finite B{C{xs}} value. 

''' 

if isscalar(xs): # for backward compatibility 

xs = (xs,) 

 

ps = self._ps 

for n, x in enumerate(map(float, xs)): # _iter() 

if not isfinite(x): 

n = Fmt.SQUARE(xs=n) 

raise _ValueError(n, x, txt=_not_(_finite_)) 

i = 0 

for p in ps: 

x, p = _2sum(x, p) 

if p: 

ps[i] = p 

i += 1 

ps[i:] = [x] 

# assert self._ps is ps 

self._n += n + 1 

self._fsum2_ = None 

 

def fadd_(self, *xs): 

'''Accumulate more values from positional arguments. 

 

@arg xs: Values to add (C{scalar}s), all positional. 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise TypeError: Non-scalar B{C{xs}} value. 

 

@raise ValueError: Invalid or non-finite B{C{xs}} value. 

''' 

self.fadd(xs) 

 

def fcopy(self, deep=False): 

'''Copy this instance, C{shallow} or B{C{deep}}. 

 

@return: The copy, a new instance (L{Fsum}). 

''' 

f = _xcopy(self, deep=deep) 

# f._fsum2_ = self._fsum2_ 

# f._n = self._n 

f._ps = list(self._ps) # separate copy 

return f 

 

copy = fcopy 

 

def fmul(self, factor): 

'''Multiple the current, partial sum by a factor. 

 

@arg factor: The multiplier (C{scalar}). 

 

@raise TypeError: Non-scalar B{C{factor}}. 

 

@raise ValueError: Invalid or non-finite B{C{factor}}. 

 

@see: Method L{Fsum.fadd}. 

''' 

if not isfinite(factor): 

raise _ValueError(factor=factor, txt=_not_(_finite_)) 

 

f, ps = float(factor), self._ps 

if ps: # multiply and adjust partial sums 

ps[:] = [p * f for p in ps] 

self.fadd_(ps.pop()) 

self._n -= 1 

# assert self._ps is ps 

 

def fsub(self, iterable): 

'''Accumulate more values from an iterable. 

 

@arg iterable: Sequence, list, tuple, etc. (C{scalar}s). 

 

@see: Method L{Fsum.fadd}. 

''' 

if iterable: 

self.fadd(-s for s in iterable) 

 

def fsub_(self, *xs): 

'''Accumulate more values from positional arguments. 

 

@arg xs: Values to subtract (C{scalar}s), all positional. 

 

@see: Method L{Fsum.fadd}. 

''' 

self.fsub(xs) 

 

def fsum(self, iterable=()): 

'''Accumulate more values from an iterable and sum all. 

 

@kwarg iterable: Sequence, list, tuple, etc. (C{scalar}s), optional. 

 

@return: Accurate, running sum (C{float}). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise TypeError: Non-scalar B{C{iterable}} value. 

 

@raise ValueError: Invalid or non-finite B{C{iterable}} value. 

 

@note: Accumulation can continue after summation. 

''' 

if iterable: 

self.fadd(iterable) 

 

ps = self._ps 

i = len(ps) - 1 

if i < 0: 

s = _0_0 

else: 

s = ps[i] 

while i > 0: 

i -= 1 

s, p = _2sum(s, ps[i]) 

ps[i:] = [s] 

if p: # sum(ps) became inexact 

ps.append(p) 

if i > 0: # half-even round if signs match 

s = _2even(s, ps[i-1], p) 

break 

# assert self._ps is ps 

self._fsum2_ = s 

return s 

 

def fsum_(self, *xs): 

'''Accumulate more values from positional arguments and sum all. 

 

@arg xs: Values to add (C{scalar}s), all positional. 

 

@return: Accurate, running sum (C{float}). 

 

@see: Method L{Fsum.fsum}. 

 

@note: Accumulation can continue after summation. 

''' 

return self.fsum(xs) 

 

def fsum2_(self, *xs): 

'''Accumulate more values from positional arguments, sum all 

and provide the sum and delta. 

 

@arg xs: Values to add (C{scalar}s), all positional. 

 

@return: 2-Tuple C{(sum, delta)} with the accurate, 

running C{sum} and the C{delta} with the 

previous running C{sum}, both (C{float}). 

 

@see: Method L{Fsum.fsum_}. 

 

@note: Accumulation can continue after summation. 

''' 

p = self._fsum2_ 

if p is None: 

p = self.fsum() 

s = self.fsum(xs) # if xs else self._fsum2_ 

return s, s - p 

 

 

class Fdot(Fsum): 

'''Precision dot product. 

''' 

def __init__(self, a, *b): 

'''New L{Fdot} precision dot product M{sum(a[i] * b[i] 

for i=0..len(a))}. 

 

@arg a: List, sequence, tuple, etc. (C{scalar}s). 

@arg b: All positional arguments (C{scalar}s). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise LenError: Unequal C{len(B{a})} and C{len(B{b})}. 

 

@see: Function L{fdot} and method L{Fsum.fadd}. 

''' 

if len(a) != len(b): 

raise LenError(Fdot, a=len(a), b=len(b)) 

 

Fsum.__init__(self) 

self.fadd(map(_mul, a, b)) 

 

 

class Fhorner(Fsum): 

'''Precision polynomial evaluation using the Horner form. 

''' 

def __init__(self, x, *cs): 

'''New L{Fhorner} evaluation of the polynomial 

M{sum(cs[i] * x**i for i=0..len(cs))}. 

 

@arg x: Polynomial argument (C{scalar}). 

@arg cs: Polynomial coeffients (C{scalar}[]). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise TypeError: Non-scalar B{C{x}}. 

 

@raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. 

 

@see: Function L{fhorner} and methods L{Fsum.fadd} and L{Fsum.fmul}. 

''' 

if not isfinite(x): 

raise _ValueError(x=x, txt=_not_(_finite_)) 

if not cs: 

raise _ValueError(cs=cs, txt=MISSING) 

 

x, cs = float(x), list(cs) 

 

Fsum.__init__(self, cs.pop()) 

while cs: 

self.fmul(x) 

c = cs.pop() 

if c: 

self.fadd_(c) 

 

 

class Fpolynomial(Fsum): 

'''Precision polynomial evaluation. 

''' 

def __init__(self, x, *cs): 

'''New L{Fpolynomial} evaluation of the polynomial 

M{sum(cs[i] * x**i for i=0..len(cs))}. 

 

@arg x: Polynomial argument (C{scalar}). 

@arg cs: Polynomial coeffients (C{scalar}[]). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise TypeError: Non-scalar B{C{x}}. 

 

@raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. 

 

@see: Function L{fpolynomial} and method L{Fsum.fadd}. 

''' 

if not isfinite(x): 

raise _ValueError(x=x, txt=_not_(_finite_)) 

if not cs: 

raise _ValueError(cs=cs, txt=MISSING) 

 

x, cs, xp = float(x), list(cs), _1_0 

 

Fsum.__init__(self, cs.pop(0)) 

while cs: 

xp *= x 

c = cs.pop(0) 

if c: 

self.fadd_(xp * c) 

 

 

def cbrt(x): 

'''Compute the cubic root M{x**(1/3)}. 

 

@arg x: Value (C{scalar}). 

 

@return: Cubic root (C{float}). 

 

@see: Functions L{cbrt2} and L{sqrt3}. 

''' 

# simpler and more accurate than Ken Turkowski's CubeRoot, see 

# <https://People.FreeBSD.org/~lstewart/references/apple_tr_kt32_cuberoot.pdf> 

return copysign0(pow(abs(x), _1_3rd), x) 

 

 

def cbrt2(x): 

'''Compute the cubic root squared M{x**(2/3)}. 

 

@arg x: Value (C{scalar}). 

 

@return: Cubic root squared (C{float}). 

 

@see: Functions L{cbrt} and L{sqrt3}. 

''' 

return pow(abs(x), _2_3rd) # XXX pow(abs(x), _1_3rd)**2 

 

 

def euclid(x, y): 

'''I{Appoximate} the norm M{sqrt(x**2 + y**2)} by 

M{max(abs(x), abs(y)) + min(abs(x), abs(y)) * 0.4142...}. 

 

@arg x: X component (C{scalar}). 

@arg y: Y component (C{scalar}). 

 

@return: Appoximate norm (C{float}). 

 

@see: Function L{euclid_}. 

''' 

x, y = abs(x), abs(y) 

if y > x: 

x, y = y, x 

return x + y * _0_4142 # XXX _0_5 before 20.10.02 

 

 

def euclid_(*xs): 

'''I{Appoximate} the norm M{sqrt(sum(x**2 for x in xs))} 

by cascaded L{euclid}. 

 

@arg xs: X arguments, positional (C{scalar}[]). 

 

@return: Appoximate norm (C{float}). 

 

@see: Function L{euclid}. 

''' 

e = _0_0 

for x in sorted(map(abs, xs)): 

# e = euclid(x, e) 

if x > e: 

e, x = x, e 

e += x * _0_4142 

return e 

 

 

def facos1(x): 

'''Fast approximation of L{acos1}C{(B{x})}. 

 

@see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/ 

ShaderFastLibs/blob/master/ShaderFastMathLib.h>}. 

''' 

a = abs(x) 

if a < EPS0: 

r = PI_2 

elif a < EPS1: 

r = 1.5707288 - a * (0.2121144 - a * (0.0742610 - a * 0.0187293)) 

r *= sqrt(_1_0 - a) 

else: 

r = _0_0 

return (PI - r) if x < 0 else r 

 

 

def fasin1(x): # PYCHOK no cover 

'''Fast approximation of L{asin1}C{(B{x})}. 

 

@see: L{facos1}. 

''' 

return PI_2 - facos1(x) 

 

 

def fatan(x): 

'''Fast approximation of C{atan(B{x})}. 

''' 

a = abs(x) 

if a < _1_0: 

r = fatan1(a) if a else _0_0 

elif a > _1_0: 

r = PI_2 - fatan1(_1_0 / a) # == fatan2(a, _1_0) 

else: 

r = PI_4 

return -r if x < 0 else r 

 

 

def fatan1(x): 

'''Fast approximation of C{atan(B{x})} for C{0 <= B{x} <= 1}, I{unchecked}. 

 

@see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/ 

ShaderFastLibs/blob/master/ShaderFastMathLib.h>} and 

U{Efficient approximations for the arctangent function 

<http://www-Labs.IRO.UMontreal.CA/~mignotte/IFT2425/Documents/ 

EfficientApproximationArctgFunction.pdf>}, IEEE Signal 

Processing Magazine, 111, May 2006. 

''' 

# Eq (9): PI_4 * x - x * (x - 1) * (0.2447 + 0.0663 * x**2) 

return x * (1.0300982 - x * (0.1784 + 0.0663 * x)) # w/o x**4 

 

 

def fatan2(y, x): 

'''Fast approximation of C{atan2(B{y}, B{x})}. 

 

@see: U{fastApproximateAtan(x, y)<https://GitHub.com/CesiumGS/cesium/blob/ 

master/Source/Shaders/Builtin/Functions/fastApproximateAtan.glsl>} 

and L{fatan1}. 

''' 

b, a = abs(y), abs(x) 

if a < b: 

r = (PI_2 - fatan1(a / b)) if a else PI_2 

elif b < a: 

r = fatan1(b / a) if b else _0_0 

elif a: # == b != 0 

r = PI_4 

else: # a == b == 0 

return _0_0 

if x < 0: 

r = PI - r 

return -r if y < 0 else r 

 

 

def favg(v1, v2, f=0.5): 

'''Return the average of two values. 

 

@arg v1: One value (C{scalar}). 

@arg v2: Other value (C{scalar}). 

@kwarg f: Optional fraction (C{float}). 

 

@return: M{v1 + f * (v2 - v1)} (C{float}). 

''' 

# @raise ValueError: Fraction out of range. 

# ''' 

# if not 0 <= f <= 1: # XXX restrict fraction? 

# raise _ValueError(fraction=f) 

return v1 + f * (v2 - v1) # v1 * (1 - f) + v2 * f 

 

 

def fdot(a, *b): 

'''Return the precision dot product M{sum(a[i] * b[i] for 

i=0..len(a))}. 

 

@arg a: List, sequence, tuple, etc. (C{scalar}s). 

@arg b: All positional arguments (C{scalar}s). 

 

@return: Dot product (C{float}). 

 

@raise LenError: Unequal C{len(B{a})} and C{len(B{b})}. 

 

@see: Class L{Fdot}. 

''' 

if len(a) != len(b): 

raise LenError(fdot, a=len(a), b=len(b)) 

 

return fsum(map(_mul, a, b)) 

 

 

def fdot3(a, b, c, start=0): 

'''Return the precision dot product M{start + 

sum(a[i] * b[i] * c[i] for i=0..len(a))}. 

 

@arg a: List, sequence, tuple, etc. (C{scalar}[]). 

@arg b: List, sequence, tuple, etc. (C{scalar}[]). 

@arg c: List, sequence, tuple, etc. (C{scalar}[]). 

@kwarg start: Optional bias (C{scalar}). 

 

@return: Dot product (C{float}). 

 

@raise LenError: Unequal C{len(B{a})}, C{len(B{b})} 

and/or C{len(B{c})}. 

 

@raise OverflowError: Partial C{2sum} overflow. 

''' 

def _mul3(a, b, c): # map function 

return a * b * c # PYCHOK returns 

 

if not len(a) == len(b) == len(c): 

raise LenError(fdot3, a=len(a), b=len(b), c=len(c)) 

 

if start: 

f = Fsum(start) 

return f.fsum(map(_mul3, a, b, c)) 

else: 

return fsum(map(_mul3, a, b, c)) 

 

 

def fhorner(x, *cs): 

'''Evaluate the polynomial M{sum(cs[i] * x**i for 

i=0..len(cs))} using the Horner form. 

 

@arg x: Polynomial argument (C{scalar}). 

@arg cs: Polynomial coeffients (C{scalar}[]). 

 

@return: Horner value (C{float}). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise TypeError: Non-scalar B{C{x}}. 

 

@raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. 

 

@see: Function L{fpolynomial} and class L{Fhorner}. 

''' 

h = Fhorner(x, *cs) 

return h.fsum() 

 

 

def fidw(xs, ds, beta=2): 

'''Interpolate using using U{Inverse Distance Weighting 

<https://WikiPedia.org/wiki/Inverse_distance_weighting>} (IDW). 

 

@arg xs: Known values (C{scalar}[]). 

@arg ds: Non-negative distances (C{scalar}[]). 

@kwarg beta: Inverse distance power (C{int}, 0, 1, 2, or 3). 

 

@return: Interpolated value C{x} (C{float}). 

 

@raise LenError: Unequal or zero C{len(B{ds})} and C{len(B{xs})}. 

 

@raise ValueError: Invalid B{C{beta}}, negative B{C{ds}} value, 

weighted B{C{ds}} below L{EPS}. 

 

@note: Using C{B{beta}=0} returns the mean of B{C{xs}}. 

''' 

n, xs = len2(xs) 

d, ds = len2(ds) 

if n != d or n < 1: 

raise LenError(fidw, xs=n, ds=d) 

 

d, x = min(zip(ds, xs)) 

if d > EPS0 and n > 1: 

b = -Int_(beta=beta, low=0, high=3) 

if b < 0: 

ds = tuple(d**b for d in ds) 

d = fsum(ds) 

if isnear0(d): 

n = Fmt.PAREN(fsum='ds') 

raise _ValueError(n, d, txt=_singular_) 

x = fdot(xs, *ds) / d 

else: # b == 0 

x = fsum(xs) / n # fmean(xs) 

elif d < 0: 

n = Fmt.SQUARE(ds=ds.index(d)) 

raise _ValueError(n, d, txt=_negative_) 

return x 

 

 

def fmean(xs): 

'''Compute the accurate mean M{sum(xs[i] for 

i=0..len(xs)) / len(xs)}. 

 

@arg xs: Values (C{scalar}s). 

 

@return: Mean value (C{float}). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise ValueError: No B{C{xs}} values. 

''' 

n, xs = len2(xs) 

if n > 0: 

return fsum(xs) / n 

raise _ValueError(xs=xs) 

 

 

def fmean_(*xs): 

'''Compute the accurate mean M{sum(xs[i] for 

i=0..len(xs)) / len(xs)}. 

 

@arg xs: Values (C{scalar}s). 

 

@return: Mean value (C{float}). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise ValueError: No B{C{xs}} values. 

''' 

return fmean(xs) 

 

 

def fpolynomial(x, *cs): 

'''Evaluate the polynomial M{sum(cs[i] * x**i for 

i=0..len(cs))}. 

 

@arg x: Polynomial argument (C{scalar}). 

@arg cs: Polynomial coeffients (C{scalar}[]). 

 

@return: Polynomial value (C{float}). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise TypeError: Non-scalar B{C{x}}. 

 

@raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. 

 

@see: Function L{fhorner} and class L{Fpolynomial}. 

''' 

p = Fpolynomial(x, *cs) 

return p.fsum() 

 

 

def fpowers(x, n, alts=0): 

'''Return a series of powers M{[x**i for i=1..n]}. 

 

@arg x: Value (C{scalar}). 

@arg n: Highest exponent (C{int}). 

@kwarg alts: Only alternating powers, starting with 

this exponent (C{int}). 

 

@return: Powers of B{C{x}} (C{float}s or C{int}s). 

 

@raise TypeError: Non-scalar B{C{x}} or B{C{n}} not C{int}. 

 

@raise ValueError: Non-finite B{C{x}} or non-positive B{C{n}}. 

''' 

if not isfinite(x): 

raise _ValueError(x=x, txt=_not_(_finite_)) 

if not isint(n): 

raise _IsnotError(int.__name__, n=n) 

elif n < 1: 

raise _ValueError(n=n) 

 

xs = [x] 

for _ in range(1, n): 

xs.append(xs[-1] * x) 

 

if alts > 0: # x**2, x**4, ... 

# xs[alts-1::2] chokes PyChecker 

xs = xs[slice(alts-1, None, 2)] 

 

return xs 

 

 

try: 

from math import prod as fprod # Python 3.8 

except ImportError: 

 

def fprod(iterable, start=_1_0): 

'''Iterable product, like C{math.prod} or C{numpy.prod}. 

 

@arg iterable: Terms to be multiplied (C{scalar}[]). 

@kwarg start: Initial term, also the value returned 

for an empty iterable (C{scalar}). 

 

@return: The product (C{float}). 

 

@see: U{NumPy.prod<https://docs.SciPy.org/doc/ 

numpy/reference/generated/numpy.prod.html>}. 

''' 

return freduce(_mul, iterable, start) 

 

 

def frange(start, number, step=1): 

'''Generate a range of C{float}s. 

 

@arg start: First value (C{float}). 

@arg number: The number of C{float}s to generate (C{int}). 

@kwarg step: Increment value (C{float}). 

 

@return: A generator (C{float}s). 

 

@see: U{NumPy.prod<https://docs.SciPy.org/doc/ 

numpy/reference/generated/numpy.arange.html>}. 

''' 

if not isint(number): 

raise _IsnotError(int.__name__, number=number) 

for i in range(number): 

yield start + i * step 

 

 

try: 

from functools import reduce as freduce 

except ImportError: # PYCHOK no cover 

try: 

freduce = reduce # PYCHOK expected 

except NameError: # Python 3+ 

 

def freduce(f, iterable, *start): 

'''For missing C{functools.reduce}. 

''' 

if start: 

r = v = start[0] 

else: 

r, v = 0, MISSING 

for v in iterable: 

r = f(r, v) 

if v is MISSING: 

raise _TypeError(iterable=(), start=MISSING) 

return r 

 

 

def fsum_(*xs): 

'''Precision summation of all positional arguments. 

 

@arg xs: Values to be added (C{scalar}[]). 

 

@return: Accurate L{fsum} (C{float}). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise TypeError: Non-scalar B{C{xs}} value. 

 

@raise ValueError: Invalid or non-finite B{C{xs}} value. 

''' 

return fsum(map(float, xs)) 

 

 

try: 

from math import fsum # precision IEEE-754 sum, Python 2.6+ 

 

# make sure fsum works as expected (XXX check 

# float.__getformat__('float')[:4] == 'IEEE'?) 

if fsum_(1, 1e101, 1, -1e101) != 2: 

del fsum # nope, remove fsum ... 

raise ImportError # ... use fsum below 

 

except ImportError: # PYCHOK no cover 

 

def fsum(iterable): 

'''Precision summation similar to standard Python function C{math.fsum}. 

 

Exception and I{non-finite} handling differs from C{math.fsum}. 

 

@arg iterable: Values to be added (C{scalar}[]). 

 

@return: Accurate C{sum} (C{float}). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise TypeError: Non-scalar B{C{iterable}} value. 

 

@raise ValueError: Invalid or non-finite B{C{iterable}} value. 

 

@see: Class L{Fsum}. 

''' 

f = Fsum() 

return f.fsum(iterable) 

 

 

if _sys_version_info2 > (3, 9): 

from math import hypot # OK in Python 3.10+ 

hypot_ = hypot 

else: 

# Python 3.8 and 3.9 C{math.hypot} is inaccurate, see 

# agdhruv <https://GitHub.com/geopy/geopy/issues/466>, 

# cffk <https://Bugs.Python.org/issue43088> and module 

# geomath.py <https://PyPI.org/project/geographiclib/1.52> 

if _sys_version_info2 < (3, 8): 

from math import hypot # PYCHOK re-imported? 

else: 

def hypot(x, y): 

'''Compute the norm M{sqrt(x**2 + y**2)}. 

 

@arg x: Argument (C{scalar}). 

@arg y: Argument (C{scalar}). 

 

@return: C{sqrt(B{x}**2 + B{y}**2)} (C{float}). 

''' 

if x: 

h = sqrt(fsum_(_1_0, x**2, y**2, -_1_0)) if y else abs(x) 

elif y: 

h = abs(y) 

else: 

h = _0_0 

return h 

 

def hypot_(*xs): 

'''Compute the norm M{sqrt(sum(x**2 for x in xs))}. 

 

@arg xs: X arguments, positional (C{scalar}[]). 

 

@return: Norm (C{float}). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise ValueError: Invalid or no B{C{xs}} value. 

 

@see: Similar to Python 3.8+ n-dimension U{math.hypot 

<https://docs.Python.org/3.8/library/math.html#math.hypot>}, 

but handling of exceptions, C{nan} and C{infinite} values 

is different. 

 

@note: The Python 3.8+ Euclidian distance U{math.dist 

<https://docs.Python.org/3.8/library/math.html#math.dist>} 

between 2 I{n}-dimensional points I{p1} and I{p2} can be 

computed as M{hypot_(*((c1 - c2) for c1, c2 in zip(p1, p2)))}, 

provided I{p1} and I{p2} have the same, non-zero length I{n}. 

''' 

h, x2 = _h_x2(xs) 

return (h * sqrt(x2)) if x2 else _0_0 

 

 

def _h_x2(xs): 

'''(INTERNAL) Helper for L{hypot_} and L{hypot2_}. 

''' 

def _x2h(xs, h): 

yield _1_0 

for x in xs: 

yield (x / h)**2 

yield -_1_0 

 

if xs: 

n, xs = len2(xs) 

if n > 0: 

h = float(max(abs(x) for x in xs)) 

if h > 0: 

x2 = fsum(_x2h(xs, h)) if n > 1 else _1_0 

else: 

x2 = h = _0_0 

return h, x2 

raise _ValueError(xs=xs, txt=_too_(_few_)) 

 

 

def hypot1(x): 

'''Compute the norm M{sqrt(1 + x**2)}. 

 

@arg x: Argument (C{scalar}). 

 

@return: Norm (C{float}). 

''' 

return hypot(_1_0, x) if x else _1_0 

 

 

def hypot2(x, y): 

'''Compute the I{squared} norm M{x**2 + y**2}. 

 

@arg x: Argument (C{scalar}). 

@arg y: Argument (C{scalar}). 

 

@return: C{B{x}**2 + B{y}**2} (C{float}). 

''' 

if x: 

x *= x 

h2 = fsum_(_1_0, x, y**2, -_1_0) if y else x 

elif y: 

h2 = y**2 

else: 

h2 = _0_0 

return h2 

 

 

def hypot2_(*xs): 

'''Compute the I{squared} norm C{sum(x**2 for x in B{xs})}. 

 

@arg xs: X arguments, positional (C{scalar}[]). 

 

@return: Squared norm (C{float}). 

 

@raise OverflowError: Partial C{2sum} overflow. 

 

@raise ValueError: Invalid or no B{C{xs}} value. 

 

@see: Function L{hypot_}. 

''' 

h, x2 = _h_x2(xs) 

return (h**2 * x2) if x2 else _0_0 

 

 

def norm2(x, y): 

'''Normalize a 2-dimensional vector. 

 

@arg x: X component (C{scalar}). 

@arg y: Y component (C{scalar}). 

 

@return: 2-Tuple C{(x, y)}, normalized. 

 

@raise ValueError: Invalid B{C{x}} or B{C{y}} 

or zero norm. 

''' 

h = hypot(x, y) 

try: 

return x / h, y / h 

except (TypeError, ValueError) as X: 

raise _ValueError(x=x, y=y, h=h, txt=str(X)) 

 

 

def norm_(*xs): 

'''Normalize all n-dimensional vector components. 

 

@arg xs: The component (C{scalar}[]). 

 

@return: Yield each component, normalized. 

 

@raise ValueError: Invalid or insufficent B{C{xs}} 

or zero norm. 

''' 

h = hypot_(*xs) 

try: 

for i, x in enumerate(xs): 

yield x / h 

except (TypeError, ValueError) as X: 

raise _ValueError(Fmt.SQUARE(xs=i), x, _h_, h, txt=str(X)) 

 

 

def sqrt0(x): 

'''Compute the square root iff C{B{x} > EPS0**2}. 

 

@arg x: Value (C{scalar}). 

 

@return: Square root (C{float}) or C{0.0}. 

''' 

return sqrt(x) if x > _EPS0__2 else _0_0 

 

 

def sqrt3(x): 

'''Compute the square root, cubed M{sqrt(x)**3} or M{sqrt(x**3)}. 

 

@arg x: Value (C{scalar}). 

 

@return: Cubed square root (C{float}). 

 

@raise ValueError: Negative B{C{x}}. 

 

@see: Functions L{cbrt} and L{cbrt2}. 

''' 

if x < 0: 

raise _ValueError(x=x) 

return pow(x, _3_2nd) if x else _0_0 

 

# **) MIT License 

# 

# Copyright (C) 2016-2021 -- mrJean1 at Gmail -- All Rights Reserved. 

# 

# Permission is hereby granted, free of charge, to any person obtaining a 

# copy of this software and associated documentation files (the "Software"), 

# to deal in the Software without restriction, including without limitation 

# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

# and/or sell copies of the Software, and to permit persons to whom the 

# Software is furnished to do so, subject to the following conditions: 

# 

# The above copyright notice and this permission notice shall be included 

# in all copies or substantial portions of the Software. 

# 

# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

# OTHER DEALINGS IN THE SOFTWARE.