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1862

 

# -*- coding: utf-8 -*- 

 

u'''Class L{Fsum} for precision floating point summation and I{running} 

summation, based on respectively similar to Python's C{math.fsum}. 

 

Generally, an L{Fsum} instance is considered a C{float} plus a small or zero 

C{residual} value, see property L{Fsum.residual}. However, there are several 

C{integer} L{Fsum} cases, for example the result of C{ceil}, C{floor}, 

C{Fsum.__floordiv__} and methods L{Fsum.fint} and L{Fsum.fint2}. 

 

Also, L{Fsum} methods L{Fsum.pow}, L{Fsum.__ipow__}, L{Fsum.__pow__} and 

L{Fsum.__rpow__} return a (very long) C{int} if invoked with optional argument 

C{mod} set to C{None}. The C{residual} of an C{integer} L{Fsum} may be between 

C{-1.0} and C{+1.0}, including C{INT0} if considered to be I{exact}. 

 

Set env variable C{PYGEODESY_FSUM_PARTIALS} to an empty string (or anything 

other than C{"fsum"}) for backward compatible summation of L{Fsum} partials. 

 

Set env variable C{PYGEODESY_FSUM_RESIDUAL} to a C{float} string for the 

threshold to throw a L{ResidualError} in division or exponention of an L{Fsum} 

instance with a I{relative} C{residual} exceeding the threshold, see methods 

L{Fsum.RESIDUAL}, L{Fsum.pow}, L{Fsum.__ipow__} and L{Fsum.__itruediv__}. 

''' 

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

from __future__ import division as _; del _ # PYCHOK semicolon 

 

from pygeodesy.basics import iscomplex, isint, isscalar, map1, neg, \ 

signOf, _signOf 

from pygeodesy.constants import INT0, _isfinite, isinf, isnan, \ 

_0_0, _1_0, _N_1_0 

from pygeodesy.errors import itemsorted, _OverflowError, _TypeError, \ 

_ValueError, _xError2, _xkwds_get, _xkwds_get_, \ 

_ZeroDivisionError 

from pygeodesy.interns import NN, _arg_, _BANG_, _COMMASPACE_, _DASH_, \ 

_EQUAL_, _exceeds_, _from_, _iadd_, _LANGLE_, \ 

_negative_, _not_finite_, _not_scalar_, \ 

_PERCENT_, _PLUS_, _R_, _RANGLE_, _SLASH_, \ 

_SPACE_, _STAR_, _UNDER_ 

from pygeodesy.lazily import _ALL_LAZY, _getenv, _sys, _sys_version_info2 

from pygeodesy.named import _Named, _NamedTuple, _NotImplemented, Fmt, unstr 

from pygeodesy.props import _allPropertiesOf_n, deprecated_property_RO, \ 

Property_RO, property_RO 

# from pygeodesy.streprs import Fmt, unstr # from .named 

from pygeodesy.units import Float, Int 

 

from math import ceil as _ceil, fabs, floor as _floor # PYCHOK used! 

 

__all__ = _ALL_LAZY.fsums 

__version__ = '22.09.14' 

 

_add_op_ = _PLUS_ 

_eq_op_ = _EQUAL_ * 2 # _DEQUAL_ 

_COMMASPACE_R_ = _COMMASPACE_ + _R_ 

_exceeds_R_ = _SPACE_ + _exceeds_(_R_) 

_floordiv_op_ = _SLASH_ * 2 # _DSLASH_ 

_fset_op_ = _EQUAL_ 

_ge_op_ = _RANGLE_ + _EQUAL_ 

_gt_op_ = _RANGLE_ 

_integer_ = 'integer' 

_le_op_ = _LANGLE_ + _EQUAL_ 

_lt_op_ = _LANGLE_ 

_mod_op_ = _PERCENT_ 

_mul_op_ = _STAR_ 

_ne_op_ = _BANG_ + _EQUAL_ 

_non_zero_ = 'non-zero' 

_pow_op_ = _STAR_ * 2 # _DSTAR_, in .fmath 

_sub_op_ = _DASH_ 

_truediv_op_ = _SLASH_ 

_divmod_op_ = _floordiv_op_ + _mod_op_ 

 

_pos_self = _1_0.__pos__() is _1_0 

 

 

def _2float(index=None, **name_value): 

'''(INTERNAL) Raise C{TypeError} or C{ValueError} if not scalar or infinite. 

''' 

n, v = name_value.popitem() # _xkwds_popitem(name_value) 

try: 

v = float(v) 

if _isfinite(v): 

return v 

E, t = _ValueError, _not_finite_ 

except Exception as X: 

E, t = _xError2(X) 

if index is not None: 

n = Fmt.SQUARE(n, index) 

raise E(n, v, txt=t) 

 

 

def _2floats(xs, origin=0, primed=False, sub=False, floats=False): 

'''(INTERNAL) Yield all B{C{xs}} as C{float}s. 

''' 

_2f = _2float 

if primed: 

yield _1_0 

i = origin 

for x in xs: 

if floats: 

yield x 

elif isinstance(x, Fsum): 

ps = x._ps 

if ps: 

if sub: 

ps = map(neg, ps) 

for x in ps: 

yield x 

else: 

x = _2f(index=i, xs=x) 

if x: 

yield (-x) if sub else x 

i += 1 

if primed: 

yield _N_1_0 

 

 

def _Powers(power, xs, origin=1): # in .fmath 

'''(INTERNAL) Yield each C{xs} as C{float(x**power)}. 

''' 

i = None 

s = 2 == power 

_2f = _2float 

try: 

for i, x in enumerate(xs): 

if isinstance(x, Fsum): 

T = x._mul_Fsum(x) if s else x.pow(power) 

for t in T._ps: 

yield t 

else: 

t = _2f(_=x) 

if t: 

yield _2f(__=t**power) 

except Exception as X: 

E, t = _xError2(X) 

if i is None: 

i = Fmt.PARENSPACED(xs=xs) 

else: 

i = Fmt.SQUARE(xs=i + origin) 

i = Fmt.PARENSPACED(i, x) 

raise E(i, txt=t) 

 

 

def _psum(ps): # PYCHOK used! 

'''(INTERNAL) Partials summation updating C{ps}, I{overridden below}. 

''' 

i = len(ps) - 1 # len(ps) > 2 

s = ps[i] 

_2s = _2sum 

while i > 0: 

i -= 1 

s, r = _2s(s, ps[i]) 

if r: # sum(ps) became inexact 

ps[i:] = [s, r] if s else [r] 

if i > 0: 

p = ps[i-1] # round half-even 

if (p > 0 and r > 0) or \ 

(p < 0 and r < 0): # signs match 

r *= 2 

t = s + r 

if r == (t - s): 

s = t 

break 

ps[i:] = [s] 

return s 

 

 

def _2scalar(other, _raiser=None): 

'''(INTERNAL) Return B{C{other}} as C{int}, C{float} or C{as-is}. 

''' 

if isinstance(other, Fsum): 

s, r = other._fint2 

if r: 

s, r = other._fprs2 

if r: # PYCHOK no cover 

if _raiser and _raiser(r, s): 

raise ValueError(_stresidual(_non_zero_, r)) 

s = other # L{Fsum} as-is 

else: 

s = other # C{type} as-is 

if isint(s, both=True): 

s = int(s) 

return s 

 

 

def _strcomplex(s, *args): 

'''(INTERNAL) C{Complex} 2- or 3-arg C{pow} error C{str}. 

''' 

c = iscomplex.__name__[2:] 

n = _DASH_(len(args), _arg_) 

t = _SPACE_(c, s, _from_, n, pow.__name__) 

return unstr(t, *args) 

 

 

def _stresidual(prefix, residual, **name_values): 

'''(INTERNAL) Residual error C{str}. 

''' 

p = _SPACE_(prefix, Fsum.residual.name) 

t = Fmt.PARENSPACED(p, Fmt(residual)) 

for n, v in itemsorted(name_values): 

n = n.replace(_UNDER_, _SPACE_) 

p = Fmt.PARENSPACED(n, Fmt(v)) 

t = _COMMASPACE_(t, p) 

return t 

 

 

def _2sum(a, b): # by .testFmath 

'''(INTERNAL) Return C{a + b} as 2-tuple (sum, residual). 

''' 

s = a + b 

if not _isfinite(s): 

u = unstr(_2sum.__name__, a, b) 

t = Fmt.PARENSPACED(_not_finite_, s) 

raise _OverflowError(u, txt=t) 

r = (a - (s - b)) if abs(a) < abs(b) else \ 

(b - (s - a)) # abs(a) >= abs(b) 

return s, r 

 

 

class Fsum(_Named): 

'''Precision floating point I{running} summation similar to standard Python's C{math.fsum}. 

 

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

precision floating point summation. Accumulation may continue after I{intermediate} 

summuation, aka I{running} summation. 

 

@note: Accumulated values may be L{Fsum} or C{scalar} instances with C{scalar} meaning 

type C{float}, C{int} or any C{type} convertible to a single C{float}, having 

method C{__float__}. 

 

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

from standard Python's 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>}. 

''' 

_math_fsum = None 

_n = 0 

# _ps = [] # partial sums 

# _px = 0 

_ratio = None 

_RESIDUAL = max(float(_getenv('PYGEODESY_FSUM_RESIDUAL', _0_0)), _0_0) 

 

def __init__(self, *xs, **name_RESIDUAL): 

'''New L{Fsum} for precision floating point I{running} summation. 

 

@arg xs: No, one or more initial values (each C{scalar} or an 

L{Fsum} instance). 

@kwarg name_RESIDUAL: Optional C{B{name}=NN} for this L{Fsum} 

(C{str}) and C{B{RESIDUAL}=None} for the 

L{ResidualError} threshold. 

 

@see: Methods L{Fsum.fadd} and L{Fsum.RESIDUAL}. 

''' 

if name_RESIDUAL: 

n, r = _xkwds_get_(name_RESIDUAL, name=NN, RESIDUAL=None) 

if n: # set name ... 

self.name = n 

if r is not None: 

self.RESIDUAL(r) # ... for ResidualError 

# self._n = 0 

self._ps = [] # [_0_0], see L{Fsum._fprs} 

if len(xs) > 1: 

self._facc(_2floats(xs, origin=1), up=False) 

elif xs: # len(xs) == 1 

self._ps = [_2float(x=xs[0])] 

self._n = 1 

 

def __abs__(self): 

'''Return this instance' absolute value as an L{Fsum}. 

''' 

s = _fsum(self._ps_1()) # == self._cmp_0(0, ...) 

return self._copy_n(self.__abs__) if s < 0 else \ 

self._copy_2(self.__abs__) 

 

def __add__(self, other): 

'''Return the C{Fsum(B{self}, B{other})}. 

 

@arg other: An L{Fsum} or C{scalar}. 

 

@return: The sum (L{Fsum}). 

 

@see: Method L{Fsum.__iadd__}. 

''' 

f = self._copy_2(self.__add__) 

return f._fadd(other, _add_op_) 

 

def __bool__(self): # PYCHOK not special in Python 2- 

'''Return C{True} if this instance is non-zero. 

''' 

return self != 0 

 

def __ceil__(self): # PYCHOK not special in Python 2- 

'''Return this instance' C{math.ceil} as C{int} or C{float}. 

 

@return: An C{int} in Python 3+, but C{float} in Python 2-. 

 

@see: Methods L{Fsum.__floor__} and property L{Fsum.ceil}. 

''' 

return self.ceil 

 

def __divmod__(self, other): 

'''Return C{divmod(B{self}, B{other})} as 2-tuple C{(quotient, 

remainder)}, an C{int} in Python 3+ or C{float} in Python 2- 

and an L{Fsum}. 

 

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

 

@see: Method L{Fsum.__itruediv__}. 

''' 

f = self._copy_2(self.__divmod__) 

return f._fdivmod2(other, _divmod_op_) 

 

def __eq__(self, other): 

'''Compare this with an other instance or scalar. 

''' 

return self._cmp_0(other, _eq_op_) == 0 

 

def __float__(self): 

'''Return this instance' current precision running sum as C{float}. 

 

@see: Methods L{Fsum.fsum} and L{Fsum.int_float}. 

''' 

return float(self._fprs) 

 

def __floor__(self): # PYCHOK not special in Python 2- 

'''Return this instance' C{math.floor} as C{int} or C{float}. 

 

@return: An C{int} in Python 3+, but C{float} in Python 2-. 

 

@see: Methods L{Fsum.__ceil__} and property L{Fsum.floor}. 

''' 

return self.floor 

 

def __floordiv__(self, other): 

'''Return C{B{self} // B{other}} as an L{Fsum}. 

 

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

 

@return: The C{floor} quotient (L{Fsum}). 

 

@see: Methods L{Fsum.__ifloordiv__}. 

''' 

f = self._copy_2(self.__floordiv__) 

return f._floordiv(other, _floordiv_op_) 

 

def __format__(self, *other): # PYCHOK no cover 

'''Not implemented.''' 

return _NotImplemented(self, *other) 

 

def __ge__(self, other): 

'''Compare this with an other instance or scalar. 

''' 

return self._cmp_0(other, _ge_op_) >= 0 

 

def __gt__(self, other): 

'''Compare this with an other instance or scalar. 

''' 

return self._cmp_0(other, _gt_op_) > 0 

 

def __hash__(self): # PYCHOK no cover 

'''Return this instance' C{hash}. 

''' 

return hash(self._ps) # XXX id(self)? 

 

def __iadd__(self, other): 

'''Apply C{B{self} += B{other}} to this instance. 

 

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

 

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

 

@raise TypeError: Invalid B{C{other}}, not 

C{scalar} nor L{Fsum}. 

 

@see: Methods L{Fsum.fadd} and L{Fsum.fadd_}. 

''' 

return self._fadd(other, _iadd_) 

 

def __ifloordiv__(self, other): 

'''Apply C{B{self} //= B{other}} to this instance. 

 

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

 

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

 

@raise ResidualError: Non-zero residual in B{C{other}}. 

 

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

 

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

 

@raise ZeroDivisionError: Zero B{C{other}}. 

 

@see: Methods L{Fsum.__itruediv__}. 

''' 

return self._floordiv(other, _floordiv_op_ + _fset_op_) 

 

def __imatmul__(self, other): # PYCHOK no cover 

'''Not implemented.''' 

return _NotImplemented(self, other) 

 

def __imod__(self, other): 

'''Apply C{B{self} %= B{other}} to this instance. 

 

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

 

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

 

@see: Method L{Fsum.__divmod__}. 

''' 

self._fdivmod2(other, _mod_op_ + _fset_op_) 

return self 

 

def __imul__(self, other): 

'''Apply C{B{self} *= B{other}} to this instance. 

 

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

 

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

 

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

 

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

 

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

''' 

return self._fmul(other, _mul_op_ + _fset_op_) 

 

def __int__(self): 

'''Return this instance as an C{int}. 

 

@see: Methods L{Fsum.int_float}, L{Fsum.__ceil__} 

and L{Fsum.__floor__} and properties 

L{Fsum.ceil} and L{Fsum.floor}. 

''' 

i, _ = self._fint2 

return i 

 

def __ipow__(self, other, *mod): # PYCHOK 2 vs 3 args 

'''Apply C{B{self} **= B{other}} to this instance. 

 

@arg other: The exponent (L{Fsum} or C{scalar}). 

@arg mod: Optional modulus (C{int} or C{None}) for the 

3-argument C{pow(B{self}, B{other}, B{mod})} 

version. 

 

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

 

@note: If B{C{mod}} is given, the result will be an C{integer} 

L{Fsum} in Python 3+ if this instance C{is_integer} or 

set to C{as_integer} if B{C{mod}} given as C{None}. 

 

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

 

@raise ResidualError: Non-zero residual in B{C{other}} and 

env var C{PYGEODESY_FSUM_RESIDUAL} 

set or this instance has a non-zero 

residual and either B{C{mod}} is 

given and non-C{None} or B{C{other}} 

is a negative or fractional C{scalar}. 

 

@raise TypeError: Invalid B{C{other}} type or 3-argument 

C{pow} invocation failed. 

 

@raise ValueError: If B{C{other}} is a negative C{scalar} 

and this instance is C{0} or B{C{other}} 

is a fractional C{scalar} and this 

instance is negative or has a non-zero 

residual or B{C{mod}} is given and C{0}. 

 

@see: CPython function U{float_pow<https://GitHub.com/ 

python/cpython/blob/main/Objects/floatobject.c>}. 

''' 

return self._fpow(other, _pow_op_ + _fset_op_, *mod) 

 

def __isub__(self, other): 

'''Apply C{B{self} -= B{other}} to this instance. 

 

@arg other: An L{Fsum} or C{scalar}. 

 

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

 

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

 

@see: Method L{Fsum.fadd}. 

''' 

return self._fsub(other, _sub_op_ + _fset_op_) 

 

def __iter__(self): 

'''Return an C{iter}ator over a C{partials} duplicate. 

''' 

return iter(self.partials) 

 

def __itruediv__(self, other): 

'''Apply C{B{self} /= B{other}} to this instance. 

 

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

 

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

 

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

 

@raise ResidualError: Non-zero residual in B{C{other}} and 

env var C{PYGEODESY_FSUM_RESIDUAL} set. 

 

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

 

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

 

@raise ZeroDivisionError: Zero B{C{other}}. 

 

@see: Method L{Fsum.__ifloordiv__}. 

''' 

return self._ftruediv(other, _truediv_op_ + _fset_op_) 

 

def __le__(self, other): 

'''Compare this with an other instance or scalar. 

''' 

return self._cmp_0(other, _le_op_) <= 0 

 

def __len__(self): 

'''Return the I{total} number accumulations (C{int}). 

''' 

return self._n 

 

def __lt__(self, other): 

'''Compare this with an other instance or scalar. 

''' 

return self._cmp_0(other, _lt_op_) < 0 

 

def __matmul__(self, other): # PYCHOK no cover 

'''Not implemented.''' 

return _NotImplemented(self, other) 

 

def __mod__(self, other): 

'''Return C{B{self} % B{other}} as an L{Fsum}. 

 

@see: Method L{Fsum.__imod__}. 

''' 

f = self._copy_2(self.__mod__) 

return f._fdivmod2(other, _mod_op_)[1] 

 

def __mul__(self, other): 

'''Return C{B{self} * B{other}} as an L{Fsum}. 

 

@see: Method L{Fsum.__imul__}. 

''' 

f = self._copy_2(self.__mul__) 

return f._fmul(other, _mul_op_) 

 

def __ne__(self, other): 

'''Compare this with an other instance or scalar. 

''' 

return self._cmp_0(other, _ne_op_) != 0 

 

def __neg__(self): 

'''Return I{a copy of} this instance, negated. 

''' 

return self._copy_n(self.__neg__) 

 

def __pos__(self): 

'''Return this instance I{as-is}, like C{float.__pos__()}. 

''' 

return self if _pos_self else self._copy_2(self.__pos__) 

 

def __pow__(self, other, *mod): # PYCHOK 2 vs 3 args 

'''Return C{B{self}**B{other}} as an L{Fsum}. 

 

@see: Method L{Fsum.__ipow__}. 

''' 

f = self._copy_2(self.__pow__) 

return f._fpow(other, _pow_op_, *mod) 

 

def __radd__(self, other): 

'''Return C{B{other} + B{self}} as an L{Fsum}. 

 

@see: Method L{Fsum.__iadd__}. 

''' 

f = self._copy_r2(other, self.__radd__) 

return f._fadd(self, _add_op_) 

 

def __rdivmod__(self, other): 

'''Return C{divmod(B{other}, B{self})} as 2-tuple C{(quotient, 

remainder)}. 

 

@see: Method L{Fsum.__divmod__}. 

''' 

f = self._copy_r2(other, self.__rdivmod__) 

return f._fdivmod2(self, _divmod_op_) 

 

# def __repr__(self): 

# '''Return the default C{repr(this)}. 

# ''' 

# return self.toRepr(lenc=True) 

 

def __rfloordiv__(self, other): 

'''Return C{B{other} // B{self}} as an L{Fsum}. 

 

@see: Method L{Fsum.__ifloordiv__}. 

''' 

f = self._copy_r2(other, self.__rfloordiv__) 

return f._floordiv(self, _floordiv_op_) 

 

def __rmatmul__(self, other): # PYCHOK no cover 

'''Not implemented.''' 

return _NotImplemented(self, other) 

 

def __rmod__(self, other): 

'''Return C{B{other} % B{self}} as an L{Fsum}. 

 

@see: Method L{Fsum.__imod__}. 

''' 

f = self._copy_r2(other, self.__rmod__) 

return f._fdivmod2(self, _mod_op_)[1] 

 

def __rmul__(self, other): 

'''Return C{B{other} * B{self}} as an L{Fsum}. 

 

@see: Method L{Fsum.__imul__}. 

''' 

f = self._copy_r2(other, self.__rmul__) 

return f._fmul(self, _mul_op_) 

 

def __round__(self, ndigits=None): # PYCHOK no cover 

'''Not implemented.''' 

return _NotImplemented(self, ndigits=ndigits) 

 

def __rpow__(self, other, *mod): 

'''Return C{B{other}**B{self}} as an L{Fsum}. 

 

@see: Method L{Fsum.__ipow__}. 

''' 

f = self._copy_r2(other, self.__rpow__) 

return f._fpow(self, _pow_op_, *mod) 

 

def __rsub__(self, other): 

'''Return C{B{other} - B{self}} as L{Fsum}. 

 

@see: Method L{Fsum.__isub__}. 

''' 

f = self._copy_r2(other, self.__rsub__) 

return f._fsub(self, _sub_op_) 

 

def __rtruediv__(self, other): 

'''Return C{B{other} / B{self}} as an L{Fsum}. 

 

@see: Method L{Fsum.__itruediv__}. 

''' 

f = self._copy_r2(other, self.__rtruediv__) 

return f._ftruediv(self, _truediv_op_) 

 

def __sizeof__(self): # PYCHOK not special in Python 2- 

'''Return the current size of this instance in C{bytes}. 

''' 

return sum(map1(_sys.getsizeof, self._fint2, 

self._fint2[0], 

self._fint2[1], 

self._fprs, 

self._fprs2, 

self._fprs2.fsum, 

self._fprs2.residual, 

self._n, 

self._ps, *self._ps)) 

 

def __str__(self): 

'''Return the default C{str(self)}. 

''' 

return self.toStr(lenc=True) 

 

def __sub__(self, other): 

'''Return C{B{self} - B{other}} as an L{Fsum}. 

 

@arg other: An L{Fsum} or C{scalar}. 

 

@return: The difference (L{Fsum}). 

 

@see: Method L{Fsum.__isub__}. 

''' 

f = self._copy_2(self.__sub__) 

return f._fsub(other, _sub_op_) 

 

def __truediv__(self, other): 

'''Return C{B{self} / B{other}} as an L{Fsum}. 

 

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

 

@return: The quotient (L{Fsum}). 

 

@see: Method L{Fsum.__itruediv__}. 

''' 

f = self._copy_2(self.__truediv__) 

return f._ftruediv(other, _truediv_op_) 

 

__trunc__ = __int__ 

 

if _sys_version_info2 < (3, 0): # PYCHOK no cover 

# <https://docs.Python.org/2/library/operator.html#mapping-operators-to-functions> 

__div__ = __truediv__ 

__idiv__ = __itruediv__ 

__long__ = __int__ 

__nonzero__ = __bool__ 

__rdiv__ = __rtruediv__ 

 

def as_integer_ratio(self): 

'''Return this instance as the ratio of 2 integers. 

 

@return: 2-Tuple C{(numerator, denominator)} both 

C{int} and with positive C{denominator}. 

 

@see: Standard C{float.as_integer_ratio} in Python 3+. 

''' 

n, r = self._fint2 

if r: 

i, d = r.as_integer_ratio() 

n *= d 

n += i 

else: # PYCHOK no cover 

d = 1 

return n, d 

 

@property_RO 

def ceil(self): 

'''Get this instance' C{ceil} value (C{int} in Python 3+, 

but C{float} in Python 2-). 

 

@note: The C{ceil} takes the C{residual} into account. 

 

@see: Method L{Fsum.int_float} and properties L{Fsum.floor}, 

L{Fsum.imag} and L{Fsum.real}. 

''' 

s, r = self._fprs2 

c = _ceil(s) + int(r) - 1 

while r > (c - s): # (s + r) > c 

c += 1 

return c 

 

def _cmp_0(self, other, op): 

'''(INTERNAL) Return C{scalar(self - B{other})} for 0-comparison. 

''' 

if isscalar(other): 

s = _fsum(self._ps_1(other)) 

elif isinstance(other, Fsum): 

s = _fsum(self._ps_1(*other._ps)) 

else: 

raise self._TypeError(op, other) # txt=_invalid_ 

return s 

 

def copy(self, deep=False, name=NN): 

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

 

@return: The copy (L{Fsum}). 

''' 

f = _Named.copy(self, deep=deep, name=name) 

f._n = self._n if deep else 1 

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

return f 

 

def _copy_0(self, *xs): 

'''(INTERNAL) Copy with/-out overriding C{partials}. 

''' 

# for x in xs: 

# assert isscalar(x) 

f = self._Fsum(self._n + len(xs), *xs) 

if self.name: 

f._name = self.name # .rename calls _update_attrs 

return f 

 

def _copy_2(self, which): 

'''(INTERNAL) Copy for I{dyadic} operators. 

''' 

# NOT .classof due to .Fdot(a, *b) args, etc. 

f = _Named.copy(self, deep=False, name=which.__name__) 

# assert f._n == self._n 

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

return f 

 

def _copy_n(self, which): 

'''(INTERNAL) Negated copy for I{monadic} C{__abs__} and C{__neg__}. 

''' 

if self._ps: 

f = self._Fsum(self._n) 

f._ps[:] = self._ps_n() 

# f._facc_up(up=False) 

else: 

f = self._Fsum(self._n, _0_0) 

f._name = which.__name__ # .rename calls _update_attrs 

return f 

 

def _copy_r2(self, other, which): 

'''(INTERNAL) Copy for I{reverse-dyadic} operators. 

''' 

return other._copy_2(which) if isinstance(other, Fsum) else \ 

Fsum(other, name=which.__name__) # see ._copy_2 

 

def _copy_RESIDUAL(self, other): 

'''(INTERNAL) Copy C{other._RESIDUAL}. 

''' 

R = other._RESIDUAL 

if R is not Fsum._RESIDUAL: 

self._RESIDUAL = R 

 

def _copy_up(self, _fprs2=False): 

'''(INTERNAL) Minimal, anonymous copy. 

''' 

f = self._Fsum(self._n, *self._ps) 

if _fprs2: # only the ._fprs2 2-tuple 

Fsum._fprs2._update_from(f, self) 

return f 

 

def divmod(self, other): 

'''Return C{divmod(B{self}, B{other})} as 2-tuple C{(quotient, 

remainder)}. 

 

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

 

@return: 2-Tuple C{(quotient, remainder)}, with the C{quotient} 

an C{int} in Python 3+ or a C{float} in Python 2- and 

the C{remainder} an L{Fsum} instance. 

 

@see: Method L{Fsum.__itruediv__}. 

''' 

f = self._copy_2(self.divmod) 

return f._fdivmod2(other, _divmod_op_) 

 

def _Error(self, op, other, Error, **txt): 

'''(INTERNAL) Format an B{C{Error}} for C{{self} B{op} B{other}}. 

''' 

return Error(_SPACE_(self.toRepr(), op, repr(other)), **txt) 

 

def _ErrorX(self, X, xs, **kwds): # in .fmath 

'''(INTERNAL) Format a caught exception. 

''' 

E, t = _xError2(X) 

n = unstr(self.named3, *xs[:3], _ELLIPSIS=len(xs) > 3, **kwds) 

return E(n, txt=t) 

 

def _facc(self, xs, up=True): # from .elliptic._Defer.Fsum 

'''(INTERNAL) Accumulate more known C{scalar}s. 

''' 

n, ps, _2s = 0, self._ps, _2sum 

for x in xs: # _iter() 

# assert isscalar(x) and isfinite(x) 

i = 0 

for p in ps: 

x, p = _2s(x, p) 

if p: 

ps[i] = p 

i += 1 

ps[i:] = [x] 

n += 1 

# assert self._ps is ps 

if n: 

self._n += n 

# Fsum._px = max(Fsum._px, len(ps)) 

if up: 

self._update() 

return self 

 

def _facc_(self, *xs, **up): 

'''(INTERNAL) Accumulate all positional C{scalar}s. 

''' 

return self._facc(xs, **up) if xs else self 

 

# def _facc_up(self, up=True): 

# '''(INTERNAL) Update the C{partials}, by removing 

# and re-accumulating the final C{partial}. 

# ''' 

# while len(self._ps) > 1: 

# p = self._ps.pop() 

# if p: 

# n = self._n 

# self._facc_(p, up=False) 

# self._n = n 

# break 

# return self._update() if up else self # ._fpsqz() 

 

def fadd(self, xs=()): 

'''Add an iterable of C{scalar} or L{Fsum} instances 

to this instance. 

 

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

L{Fsum} instances). 

 

@return: This instance (L{Fsum}). 

 

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

 

@raise TypeError: An invalid B{C{xs}} type, not C{scalar} 

nor L{Fsum}. 

 

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

''' 

if isinstance(xs, Fsum): 

self._facc(xs._ps) 

elif isscalar(xs): # for backward compatibility 

self._facc_(_2float(x=xs)) # PYCHOK no cover 

elif xs: 

self._facc(_2floats(xs)) 

return self 

 

def fadd_(self, *xs): 

'''Add all positional C{scalar} or L{Fsum} instances 

to this instance. 

 

@arg xs: Values to add (C{scalar} or L{Fsum} instances), 

all positional. 

 

@return: This instance (L{Fsum}). 

 

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

 

@raise TypeError: An invalid B{C{xs}} type, not C{scalar} 

nor L{Fsum}. 

 

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

''' 

return self._facc(_2floats(xs, origin=1)) 

 

def _fadd(self, other, op): 

'''(INTERNAL) Apply C{B{self} += B{other}}. 

''' 

if isinstance(other, Fsum): 

if other is self: 

self._facc_(*other._ps) # == ._facc(tuple(other._ps)) 

elif other._ps: 

self._facc(other._ps) 

elif not isscalar(other): 

raise self._TypeError(op, other) # txt=_invalid_ 

elif other: 

self._facc_(other) 

return self 

 

fcopy = copy # for backward compatibility 

fdiv = __itruediv__ # for backward compatibility 

fdivmod = __divmod__ # for backward compatibility 

 

def _fdivmod2(self, other, op): 

'''(INTERNAL) C{divmod(B{self}, B{other})} as 2-tuple 

(C{int} or C{float}, remainder C{self}). 

''' 

# result mostly follows CPython function U{float_divmod 

# <https://GitHub.com/python/cpython/blob/main/Objects/floatobject.c>}, 

# but at least divmod(-3, 2) equals Cpython's result (-2, 1). 

q = self._copy_up(_fprs2=True)._ftruediv(other, op).floor 

if q: # == float // other == floor(float / other) 

self -= other * q 

 

s = signOf(other) # make signOf(self) == signOf(other) 

if s and self.signOf() == -s: # PYCHOK no cover 

self += other 

q -= 1 

 

# t = self.signOf() 

# if t and t != s: 

# from pygeodesy.errors import _AssertionError 

# raise self._Error(op, other, _AssertionError, txt=signOf.__name__) 

return q, self # q is C{int} in Python 3+, but C{float} in Python 2- 

 

def _finite(self, other, op=None): 

'''(INTERNAL) Return B{C{other}} if C{finite}. 

''' 

if _isfinite(other): 

return other 

raise ValueError(_not_finite_) if not op else \ 

self._ValueError(op, other, txt=_not_finite_) 

 

def fint(self, raiser=True, name=NN): 

'''Return this instance' current running sum as C{integer}. 

 

@kwarg raiser: If C{True} throw a L{ResidualError} if the 

I{integer} residual is non-zero. 

@kwarg name: Optional name (C{str}), overriding C{"fint"}. 

 

@return: The C{integer} (L{Fsum}). 

 

@raise ResidualError: Non-zero I{integer} residual. 

 

@see: Methods L{Fsum.int_float} and L{Fsum.is_integer}. 

''' 

i, r = self._fint2 

if r and raiser: 

t = _stresidual(_integer_, r) 

raise ResidualError(_integer_, i, txt=t) 

n = name or self.fint.__name__ 

return Fsum(name=n)._fset(i, asis=True) 

 

def fint2(self, **name): 

'''Return this instance' current running sum as C{int} and 

the I{integer} residual. 

 

@kwarg name: Optional name (C{str}). 

 

@return: An L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} 

an C{int} and I{integer} C{residual} a C{float} or 

C{INT0} if the C{fsum} is considered to be I{exact}. 

''' 

return Fsum2Tuple(*self._fint2, **name) 

 

@Property_RO 

def _fint2(self): # see ._fset 

'''(INTERNAL) Get 2-tuple (C{int}, I{integer} residual). 

''' 

i = int(self._fprs) # int(self) 

r = _fsum(self._ps_1(i)) if len(self._ps) > 1 else ( 

(self._ps[0] - i) if self._ps else -i) 

return i, (r or INT0) 

 

@deprecated_property_RO 

def float_int(self): # PYCHOK no cover 

'''DEPRECATED, use method C{Fsum.int_float}.''' 

return self.int_float() # raiser=False 

 

@property_RO 

def floor(self): 

'''Get this instance' C{floor} (C{int} in Python 3+, but 

C{float} in Python 2-). 

 

@note: The C{floor} takes the C{residual} into account. 

 

@see: Method L{Fsum.int_float} and properties L{Fsum.ceil}, 

L{Fsum.imag} and L{Fsum.real}. 

''' 

s, r = self._fprs2 

f = _floor(s) + _floor(r) + 1 

while r < (f - s): # (s + r) < f 

f -= 1 

return f 

 

# floordiv = __floordiv__ # for naming consistency 

 

def _floordiv(self, other, op): # rather _ffloordiv? 

'''Apply C{B{self} //= B{other}}. 

''' 

q = self._ftruediv(other, op) # == self 

return self._fset(q.floor, asis=True) # floor(q) 

 

fmul = __imul__ # for backward compatibility 

 

def _fmul(self, other, op): 

'''(INTERNAL) Apply C{B{self} *= B{other}}. 

''' 

if isscalar(other): 

f = self._mul_scalar(other, op) 

elif not isinstance(other, Fsum): 

raise self._TypeError(op, other) # txt=_invalid_ 

elif len(self._ps) != 1: 

f = self._mul_Fsum(other, op) 

elif len(other._ps) != 1: # len(self._ps) == 1 

f = other._copy_up()._mul_scalar(self._ps[0], op) 

else: # len(other._ps) == len(self._ps) == 1 

s = self._finite(self._ps[0] * other._ps[0]) 

return self._fset(s, asis=True, n=len(self) + 1) 

return self._fset(f) 

 

def fover(self, over): 

'''Apply C{B{self} /= B{over}} and summate. 

 

@arg over: An L{Fsum} or C{scalar} denominator. 

 

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

 

@see: Methods L{Fsum.fsum} and L{Fsum.__itruediv__}. 

''' 

return float(self.fdiv(over)._fprs) 

 

fpow = __ipow__ # for backward compatibility 

 

def _fpow(self, other, op, *mod): 

'''Apply C{B{self} **= B{other}}, optional B{C{mod}} or C{None}. 

''' 

if mod and mod[0] is not None: # == 3-arg C{pow} 

s = self._pow_3(other, mod[0], op) 

elif mod and mod[0] is None and self.is_integer(): 

# return an exact C{int} for C{int}**C{int} 

i = self._copy_0(self._fint2[0]) # assert _fint2[1] == 0 

x = _2scalar(other) # C{int}, C{float} or other 

s = i._pow_2(x, other, op) if isscalar(x) else i._fpow(x, op) 

else: # pow(self, other) == pow(self, other, None) 

p = None 

if isinstance(other, Fsum): 

x, r = other._fprs2 

if r: 

if self._raiser(r, x): 

raise self._ResidualError(op, other, r) 

p = self._pow_scalar(r, other, op) 

# p = _2scalar(p) # _raiser = None 

elif not isscalar(other): 

raise self._TypeError(op, other) # txt=_invalid_ 

else: 

x = self._finite(other, op) 

s = self._pow_scalar(x, other, op) 

if p is not None: 

s *= p 

return self._fset(s, asis=isint(s), n=max(len(self), 1)) 

 

@Property_RO 

def _fprs(self): 

'''(INTERNAL) Get and cache this instance' precision 

running sum (C{float} or C{int}), ignoring C{residual}. 

 

@note: The precision running C{fsum} after a C{//=} or 

C{//} C{floor} division is C{int} in Python 3+. 

''' 

ps = self._ps 

n = len(ps) - 1 

if n > 1: 

s = _psum(ps) 

elif n > 0: # len(ps) == 2 

s, p = _2sum(*ps) if ps[1] else ps 

ps[:] = ([p, s] if s else [p]) if p else [s] 

elif n < 0: # see L{Fsum.__init__} 

s = _0_0 

ps[:] = [s] 

else: # len(ps) == 1 

s = ps[0] 

# assert self._ps is ps 

# assert Fsum._fprs2.name not in self.__dict__ 

return s 

 

@Property_RO 

def _fprs2(self): 

'''(INTERNAL) Get and cache this instance' precision 

running sum and residual (L{Fsum2Tuple}). 

''' 

s = self._fprs 

r = _fsum(self._ps_1(s)) if len(self._ps) > 1 else INT0 

return Fsum2Tuple(s, r) # name=Fsum.fsum2.__name__ 

 

# def _fpsqz(self): 

# '''(INTERNAL) Compress, squeeze the C{partials}. 

# ''' 

# if len(self._ps) > 2: 

# _ = self._fprs 

# return self 

 

def _fset(self, other, asis=False, n=1): 

'''(INTERNAL) Overwrite this instance with an other or a C{scalar}. 

''' 

if other is self: 

pass # from ._fmul, ._ftruediv and ._pow_scalar 

elif isinstance(other, Fsum): 

self._n = other._n 

self._ps[:] = other._ps 

self._copy_RESIDUAL(other) 

# use or zap the C{Property_RO} values 

Fsum._fint2._update_from(self, other) 

Fsum._fprs ._update_from(self, other) 

Fsum._fprs2._update_from(self, other) 

elif isscalar(other): 

s = other if asis else float(other) 

i = int(s) # see ._fint2 

t = i, ((s - i) or INT0) 

self._n = n 

self._ps[:] = [s] 

# Property_RO _fint2, _fprs and _fprs2 can't be a Property: 

# Property's _fset zaps the value just set by the @setter 

self.__dict__.update(_fint2=t, _fprs=s, _fprs2=Fsum2Tuple(s, INT0)) 

else: # PYCHOK no cover 

raise self._TypeError(_fset_op_, other) # txt=_invalid_ 

return self 

 

def fsub(self, xs=()): 

'''Subtract an iterable of C{scalar} or L{Fsum} instances 

from this instance. 

 

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

or L{Fsum} instances). 

 

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

 

@see: Method L{Fsum.fadd}. 

''' 

return self._facc(_2floats(xs, sub=True)) if xs else self 

 

def fsub_(self, *xs): 

'''Subtract all positional C{scalar} or L{Fsum} instances 

from this instance. 

 

@arg xs: Values to subtract (C{scalar} or 

L{Fsum} instances), all positional. 

 

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

 

@see: Method L{Fsum.fadd}. 

''' 

return self._facc(_2floats(xs, origin=1, sub=True)) if xs else self 

 

def _fsub(self, other, op): 

'''(INTERNAL) Apply C{B{self} -= B{other}}. 

''' 

if isinstance(other, Fsum): 

if other is self: # or other._fprs2 == self._fprs2: 

self._fset(_0_0, asis=True, n=len(self) * 2) # self -= self 

elif other._ps: 

self._facc(other._ps_n()) 

elif not isscalar(other): 

raise self._TypeError(op, other) # txt=_invalid_ 

elif self._finite(other, op): 

self._facc_(-other) 

return self 

 

def _Fsum(self, n, *ps): 

'''(INTERNAL) New L{Fsum} instance. 

''' 

f = Fsum() 

f._n = n 

if ps: 

f._ps[:] = ps 

f._copy_RESIDUAL(self) 

return f 

 

def fsum(self, xs=()): 

'''Add more C{scalar} or L{Fsum} instances and summate. 

 

@kwarg xs: Iterable, list, tuple, etc. (C{scalar} or 

L{Fsum} instances). 

 

@return: Precision running sum (C{float} or C{int}). 

 

@see: Method L{Fsum.fadd}. 

 

@note: Accumulation can continue after summation. 

''' 

f = self._facc(_2floats(xs)) if xs else self 

return f._fprs 

 

def fsum_(self, *xs): 

'''Add all positional C{scalar} or L{Fsum} instances and summate. 

 

@arg xs: Values to add (C{scalar} or L{Fsum} instances), 

all positional. 

 

@return: Precision running sum (C{float} or C{int}). 

 

@see: Method L{Fsum.fsum}. 

''' 

f = self._facc(_2floats(xs, origin=1)) if xs else self 

return f._fprs 

 

def fsum2(self, xs=(), **name): 

'''Add more C{scalar} or L{Fsum} instances and return the 

current precision running sum and the C{residual}. 

 

@kwarg xs: Iterable, list, tuple, etc. (C{scalar} or 

L{Fsum} instances). 

@kwarg name: Optional name (C{str}). 

 

@return: L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} the 

current precision running sum and C{residual}, the 

(precision) sum of the remaining C{partials}. The 

C{residual is INT0} if the C{fsum} is considered 

to be I{exact}. 

 

@see: Methods L{Fsum.fint2}, L{Fsum.fsum} and L{Fsum.fsum2_} 

''' 

f = self._facc(_2floats(xs)) if xs else self 

t = f._fprs2 

if name: 

t = t.dup(name=_xkwds_get(name, name=NN)) 

return t 

 

def fsum2_(self, *xs): 

'''Add any positional C{scalar} or L{Fsum} instances and return 

the precision running sum and the C{differential}. 

 

@arg xs: Values to add (C{scalar} or L{Fsum} instances), 

all positional. 

 

@return: 2-Tuple C{(fsum, delta)} with the current precision 

running C{fsum} and C{delta}, the difference with 

the previous running C{fsum} (C{float}s). 

 

@see: Methods L{Fsum.fsum_} and L{Fsum.fsum}. 

''' 

p, r = self._fprs2 

if xs: 

s, t = self._facc(_2floats(xs, origin=1))._fprs2 

return s, _fsum((s, -p, r, -t)) # ((s - p) + (r - t)) 

else: # PYCHOK no cover 

return p, _0_0 

 

# ftruediv = __itruediv__ # for naming consistency 

 

def _ftruediv(self, other, op): 

'''(INTERNAL) Apply C{B{self} /= B{other}}. 

''' 

n = _1_0 

if isinstance(other, Fsum): 

if other is self or other._fprs2 == self._fprs2: 

return self._fset(_1_0, asis=True, n=len(self)) 

d, r = other._fprs2 

if r: 

if not d: # PYCHOK no cover 

d = r 

elif self._raiser(r, d): 

raise self._ResidualError(op, other, r) 

else: 

d, n = other.as_integer_ratio() 

elif isscalar(other): 

d = other 

else: # PYCHOK no cover 

raise self._TypeError(op, other) # txt=_invalid_ 

try: 

s = 0 if isinf(d) else ( 

d if isnan(d) else self._finite(n / d)) 

except Exception as x: 

E, t = _xError2(x) 

raise self._Error(op, other, E, txt=t) 

f = self._mul_scalar(s, _mul_op_) # handles 0, NAN, etc. 

return self._fset(f) 

 

@property_RO 

def imag(self): 

'''Get the C{imaginary} part of this instance (C{0.0}, always). 

 

@see: Properties L{Fsum.ceil}, L{Fsum.floor} and L{Fsum.real}. 

''' 

return _0_0 

 

def int_float(self, raiser=False): 

'''Return this instance' current running sum as C{int} or C{float}. 

 

@kwarg raiser: If C{True} throw a L{ResidualError} if the 

residual is non-zero. 

 

@return: This C{integer} sum if this instance C{is_integer}, 

otherwise return the C{float} sum if the residual 

is zero or if C{B{raiser}=False}. 

 

@raise ResidualError: Non-zero residual and C{B{raiser}=True}. 

 

@see: Methods L{Fsum.fint} and L{Fsum.fint2}. 

''' 

s, r = self._fint2 

if r: 

s, r = self._fprs2 

if r and raiser: # PYCHOK no cover 

t = _stresidual(_non_zero_, r) 

raise ResidualError(int_float=s, txt=t) 

s = float(s) # redundant 

return s 

 

def is_exact(self): 

'''Is this instance' current running C{fsum} considered to 

be exact? (C{bool}). 

''' 

return self.residual is INT0 

 

def is_integer(self): 

'''Is this instance' current running sum C{integer}? (C{bool}). 

 

@see: Methods L{Fsum.fint} and L{Fsum.fint2}. 

''' 

_, r = self._fint2 

return False if r else True 

 

def is_math_fsum(self): 

'''Return whether functions L{fsum}, L{fsum_}, L{fsum1} 

and L{fsum1_} plus partials summation are based on 

Python's C{math.fsum} or not. 

 

@return: C{2} if all functions and partials summation 

are based on C{math.fsum}, C{True} if only 

the functions are based on C{math.fsum} (and 

partials summation is not) or C{False} if 

none are. 

''' 

f = Fsum._math_fsum 

return 2 if _psum is f else bool(f) 

 

def _mul_Fsum(self, other, op=_mul_op_): 

'''(INTERNAL) Return C{B{self} * Fsum B{other}} as L{Fsum}. 

''' 

# assert isinstance(other, Fsum) 

return self._copy_0()._facc(self._ps_x(op, *other._ps), up=False) 

 

def _mul_scalar(self, factor, op): 

'''(INTERNAL) Return C{B{self} * scalar B{factor}} as L{Fsum} or C{0}. 

''' 

# assert isscalar(factor) 

if self._finite(factor, op) and self._ps: 

if factor == _1_0: 

return self 

f = self._copy_0()._facc(self._ps_x(op, factor), up=False) 

else: 

f = self._copy_0(_0_0) 

return f 

 

@property_RO 

def partials(self): 

'''Get this instance' current partial sums (C{tuple} of C{float}s and/or C{int}s). 

''' 

return tuple(self._ps) 

 

def pow(self, x, *mod): 

'''Return C{B{self}**B{x}} as L{Fsum}. 

 

@arg x: The exponent (L{Fsum} or C{scalar}). 

@arg mod: Optional modulus (C{int} or C{None}) for the 3-argument 

C{pow(B{self}, B{other}, B{mod})} version. 

 

@return: The C{pow(self, B{x})} or C{pow(self, B{x}, *B{mod})} 

result (L{Fsum}). 

 

@note: If B{C{mod}} is given as C{None}, the result will be an 

C{integer} L{Fsum} provided this instance C{is_integer} 

or set C{integer} with L{Fsum.fint}. 

 

@see: Methods L{Fsum.__ipow__}, L{Fsum.fint} and L{Fsum.is_integer}. 

''' 

f = self._copy_2(self.pow) 

if f and isint(x) and x >= 0 and not mod: 

f._pow_int(x, x, _pow_op_) # f **= x 

else: 

f._fpow(x, _pow_op_, *mod) # f = pow(f, x, *mod) 

return f 

 

def _pow_0_1(self, x, other): 

'''(INTERNAL) Return B{C{self}**1} or C{B{self}**0 == 1.0}. 

''' 

return self if x else (1 if self.is_integer() and isint(other) else _1_0) 

 

def _pow_2(self, x, other, op): 

'''(INTERNAL) 2-arg C{pow(B{self}, scalar B{x})} embellishing errors. 

''' 

# assert len(self._ps) == 1 and isscalar(x) 

b = self._ps[0] # assert isscalar(b) 

try: # type(s) == type(x) if x in (_1_0, 1) 

s = pow(b, x) # -1**2.3 == -(1**2.3) 

if not iscomplex(s): 

return self._finite(s) # 0**INF == 0.0, 1**INF==1.0 

# neg**frac == complex in Python 3+, but ValueError in 2- 

E, t = _ValueError, _strcomplex(s, b, x) # PYCHOK no cover 

except Exception as x: 

E, t = _xError2(x) 

raise self._Error(op, other, E, txt=t) 

 

def _pow_3(self, other, mod, op): 

'''(INTERNAL) 3-arg C{pow(B{self}, B{other}, int B{mod} or C{None})}. 

''' 

b, r = self._fprs2 if mod is None else self._fint2 

if r and self._raiser(r, b): 

t = _non_zero_ if mod is None else _integer_ 

E, t = ResidualError, _stresidual(t, r, mod=mod) 

else: 

try: # b, other, mod all C{int}, unless C{mod} is C{None} 

x = _2scalar(other, _raiser=self._raiser) 

s = pow(b, x, mod) 

if not iscomplex(s): 

return self._finite(s) 

# neg**frac == complex in Python 3+, but ValueError in 2- 

E, t = _ValueError, _strcomplex(s, b, x, mod) # PYCHOK no cover 

except Exception as x: 

E, t = _xError2(x) 

t = _COMMASPACE_(Fmt.PARENSPACED(mod=mod), t) 

raise self._Error(op, other, E, txt=t) 

 

def _pow_int(self, x, other, op): 

'''(INTERNAL) Return C{B{self} **= B{x}} for C{int B{x} >= 0}. 

''' 

# assert isint(x) and x >= 0 

if len(self._ps) > 1: 

if x > 2: 

p = self._copy_up() 

m = 1 # single-bit mask 

if x & m: 

x -= m # x ^= m 

f = p._copy_up() 

else: 

f = self._copy_0(_1_0) 

while x: 

p = p._mul_Fsum(p, op) # p **= 2 

m += m # m <<= 1 

if x & m: 

x -= m # x ^= m 

f = f._mul_Fsum(p, op) # f *= p 

elif x > 1: # self**2 

f = self._mul_Fsum(self, op) 

else: # self**1 or self**0 

f = self._pow_0_1(x, other) 

elif self._ps: # self._ps[0]**x 

f = self._pow_2(x, other, op) 

else: # PYCHOK no cover 

# 0**pos_int == 0, but 0**0 == 1 

f = 0 if x else 1 # like ._fprs 

return self._fset(f, asis=isint(f), n=len(self)) 

 

def _pow_scalar(self, x, other, op): 

'''(INTERNAL) Return C{self**B{x}} for C{scalar B{x}}. 

''' 

s, r = self._fprs2 

if isint(x, both=True): 

x = int(x) # Fsum**int 

y = abs(x) 

if y > 1: 

if r: 

f = self._copy_up()._pow_int(y, other, op) 

if x > 0: # > 1 

return f 

# assert x < 0 # < -1 

s, r = f._fprs2 

if r: 

return self._copy_0(_1_0)._ftruediv(f, op) 

# use **= -1 for the CPython float_pow 

# error if s is zero, and not s = 1 / s 

x = -1 

# elif y > 1: # self**2 or self**-2 

# f = self._mul_Fsum(self, op) 

# if x < 0: 

# f = f._copy_0(_1_0)._ftruediv(f, op) 

# return f 

elif x < 0: # self**-1 == 1 / self 

if r: 

return self._copy_0(_1_0)._ftruediv(self, op) 

else: # self**1 or self**0 

return self._pow_0_1(x, other) # self or 0.0 

elif not isscalar(x): # assert ... 

raise self._TypeError(op, other, txt=_not_scalar_) 

elif r and self._raiser(r, s): # non-zero residual**fractional 

# raise self._ResidualError(op, other, r, fractional_power=x) 

t = _stresidual(_non_zero_, r, fractional_power=x) 

raise self._Error(op, other, ResidualError, txt=t) 

# assert isscalar(s) and isscalar(x) 

return self._copy_0(s)._pow_2(x, other, op) 

 

def _ps_1(self, *less): 

'''(INTERNAL) Yield partials, 1-primed and subtract any C{less}. 

''' 

yield _1_0 

for p in self._ps: 

if p: 

yield p 

for p in less: 

if p: 

yield -p 

yield _N_1_0 

 

def _ps_n(self): 

'''(INTERNAL) Yield partials, negated. 

''' 

for p in self._ps: 

if p: 

yield -p 

 

def _ps_x(self, op, *factors): # see .fmath.Fhorner 

'''(INTERNAL) Yield all C{partials} times each B{C{factor}}, 

in total, up to C{len(partials) * len(factors)} items. 

''' 

ps = self._ps 

if len(ps) < len(factors): 

ps, factors = factors, ps 

_f = _isfinite 

for f in factors: 

for p in ps: 

p *= f 

if _f(p): 

yield p 

else: # PYCHOK no cover 

self._finite(p, op) # throw ValueError 

 

@property_RO 

def real(self): 

'''Get the C{real} part of this instance (C{float}). 

 

@see: Methods L{Fsum.__float__} and L{Fsum.fsum} 

and properties L{Fsum.ceil}, L{Fsum.floor}, 

L{Fsum.imag} and L{Fsum.residual}. 

''' 

return float(self._fprs) 

 

@property_RO 

def residual(self): 

'''Get this instance' residual (C{float} or C{int}), the 

C{sum(partials)} less the precision running sum C{fsum}. 

 

@note: If the C{residual is INT0}, the precision running 

C{fsum} is considered to be I{exact}. 

 

@see: Methods L{Fsum.fsum}, L{Fsum.fsum2} and L{Fsum.is_exact}. 

''' 

return self._fprs2.residual 

 

def _raiser(self, r, s): 

'''(INTERNAL) Does the ratio C{r / s} exceed threshold? 

''' 

self._ratio = t = fabs((r / s) if s else r) 

return t > self._RESIDUAL 

 

def RESIDUAL(self, *threshold): 

'''Get and set this instance' I{ratio} for raising L{ResidualError}s, 

overriding the default from env variable C{PYGEODESY_FSUM_RESIDUAL}. 

 

@arg threshold: If C{scalar}, the I{ratio} to exceed for raising 

L{ResidualError}s in division and exponention, if 

C{None} restore the default set with env variable 

C{PYGEODESY_FSUM_RESIDUAL} or if omitted, keep the 

current setting. 

 

@return: The previous C{RESIDUAL} setting (C{float}). 

 

@raise ValueError: Negative B{C{threshold}}. 

 

@note: A L{ResidualError} is thrown if the non-zero I{ratio} 

C{residual} / C{fsum} exceeds the B{C{threshold}}. 

''' 

r = self._RESIDUAL 

if threshold: 

t = threshold[0] 

t = Fsum._RESIDUAL if t is None else ( 

float(t) if isscalar(t) else ( # for backward ... 

_0_0 if bool(t) else _1_0)) # ... compatibility 

if t < 0: 

u = self._unstr(self.RESIDUAL, *threshold) 

raise _ValueError(u, RESIDUAL=t, txt=_negative_) 

self._RESIDUAL = t 

return r 

 

def _ResidualError(self, op, other, residual): 

'''(INTERNAL) Non-zero B{C{residual}} etc. 

''' 

t = _stresidual(_non_zero_, residual, ratio=self._ratio, 

RESIDUAL=self._RESIDUAL) 

t = t.replace(_COMMASPACE_R_, _exceeds_R_) 

return self._Error(op, other, ResidualError, txt=t) 

 

def signOf(self, res=True): 

'''Determine the sign of this instance. 

 

@kwarg res: If C{True} consider, otherwise 

ignore the residual (C{bool}). 

 

@return: The sign (C{int}, -1, 0 or +1). 

''' 

s, r = self._fprs2 if res else (self._fprs, 0) 

return _signOf(r, -s) 

 

def toRepr(self, **prec_sep_fmt_lenc): # PYCHOK signature 

'''Return this C{Fsum} instance as representation. 

 

@kwarg prec_sep_fmt_lenc: Optional keyword arguments for 

method L{Fsum2Tuple.toRepr} plus C{B{lenc}=True} 

(C{bool}) to in-/exclude the current C{[len]} 

of this L{Fsum} enclosed in I{[brackets]}. 

 

@return: This instance (C{repr}). 

''' 

return self._toT(self._fprs2.toRepr, **prec_sep_fmt_lenc) 

 

def toStr(self, **prec_sep_fmt_lenc): # PYCHOK signature 

'''Return this C{Fsum} instance as string. 

 

@kwarg prec_sep_fmt_lenc: Optional keyword arguments for 

method L{Fsum2Tuple.toStr} plus C{B{lenc}=True} 

(C{bool}) to in-/exclude the current C{[len]} 

of this L{Fsum} enclosed in I{[brackets]}. 

 

@return: This instance (C{str}). 

''' 

return self._toT(self._fprs2.toStr, **prec_sep_fmt_lenc) 

 

def _toT(self, toT, fmt=Fmt.g, lenc=True, **kwds): 

'''(INTERNAL) Helper for C{toRepr} and C{toStr}. 

''' 

n = self.named3 

if lenc: 

n = Fmt.SQUARE(n, len(self)) 

return _SPACE_(n, toT(fmt=fmt, **kwds)) 

 

def _TypeError(self, op, other, **txt): # PYCHOK no cover 

'''(INTERNAL) Return a C{TypeError}. 

''' 

return self._Error(op, other, _TypeError, **txt) 

 

def _update(self): # see ._fset 

'''(INTERNAL) Zap all cached C{Property_RO} values. 

''' 

Fsum._fint2._update(self) 

Fsum._fprs ._update(self) 

Fsum._fprs2._update(self) 

return self 

 

def _ValueError(self, op, other, **txt): # PYCHOK no cover 

'''(INTERNAL) Return a C{ValueError}. 

''' 

return self._Error(op, other, _ValueError, **txt) 

 

def _ZeroDivisionError(self, op, other, **txt): # PYCHOK no cover 

'''(INTERNAL) Return a C{ZeroDivisionError}. 

''' 

return self._Error(op, other, _ZeroDivisionError, **txt) 

 

_allPropertiesOf_n(3, Fsum, Property_RO) # PYCHOK assert, see Fsum._fset, -._update 

 

 

def _Float_Int(arg, **name_Error): 

'''(INTERNAL) Unit of L{Fsum2Tuple} items. 

''' 

U = Int if isint(arg) else Float 

return U(arg, **name_Error) 

 

 

class Fsum2Tuple(_NamedTuple): 

'''2-Tuple C{(fsum, residual)} with the precision running C{fsum} 

and the C{residual}, the sum of the remaining partials. Each 

item is either C{float} or C{int}. 

 

@note: If the C{residual is INT0}, the C{fsum} is considered 

to be I{exact}, see method L{Fsum2Tuple.is_exact}. 

''' 

_Names_ = ( Fsum.fsum.__name__, Fsum.residual.name) 

_Units_ = (_Float_Int, _Float_Int) 

 

@Property_RO 

def Fsum(self): 

'''Get this L{Fsum2Tuple} as an L{Fsum}. 

''' 

f = Fsum(name=self.name) 

return f._copy_0(*(s for s in reversed(self) if s)) 

 

def is_exact(self): 

'''Is this L{Fsum2Tuple} considered to be exact? (C{bool}). 

''' 

return self.Fsum.is_exact() 

 

def is_integer(self): 

'''Is this L{Fsum2Tuple} C{integer}? (C{bool}). 

''' 

return self.Fsum.is_integer() 

 

 

class ResidualError(_ValueError): 

'''Error raised for an operation involving a L{pygeodesy.sums.Fsum} 

instance with a non-zero C{residual}, I{integer} or otherwise. 

 

@see: Module L{pygeodesy.fsums} and method L{Fsum.RESIDUAL}. 

''' 

pass 

 

 

try: 

from math import fsum as _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: # PYCHOK no cover 

del _fsum # nope, remove _fsum ... 

raise ImportError # ... use _fsum below 

 

Fsum._math_fsum = _fsum 

 

if _getenv('PYGEODESY_FSUM_PARTIALS', _fsum.__name__) == _fsum.__name__: 

_psum = _fsum # PYCHOK redef 

 

except ImportError: 

 

def _fsum(xs): 

'''(INTERNAL) Precision summation, Python 2.5-. 

''' 

return Fsum(name=_fsum.__name__)._facc(xs, up=False)._fprs 

 

 

def fsum(xs, floats=False): 

'''Precision floating point summation based on or like Python's C{math.fsum}. 

 

@arg xs: Iterable, list, tuple, etc. of values (C{scalar} or 

L{Fsum} instances). 

@kwarg floats: Optionally, set C{B{floats}=True} iff I{all} 

B{C{xs}} are known to be C{float}. 

 

@return: Precision C{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. 

 

@note: Exceptions and I{non-finite} handling may differ if not 

based on Python's C{math.fsum}. 

 

@see: Class L{Fsum} and methods L{Fsum.fsum} and L{Fsum.fadd}. 

''' 

return _fsum(_2floats(xs, floats=floats)) if xs else _0_0 

 

 

def fsum_(*xs, **floats): 

'''Precision floating point summation of all positional arguments. 

 

@arg xs: Values to be added (C{scalar} or L{Fsum} instances), 

all positional. 

@kwarg floats: Optionally, set C{B{floats}=True} iff I{all} 

B{C{xs}} are known to be C{float}. 

 

@return: Precision C{fsum} (C{float}). 

 

@see: Function C{fsum}. 

''' 

if xs: 

f = _xkwds_get(floats, floats=False) 

return _fsum(_2floats(xs, origin=1, floats=f)) 

else: # PYCHOK no cover 

return _0_0 

 

 

def fsum1(xs, floats=False): 

'''Precision floating point summation of a few values, 1-primed. 

 

@arg xs: Iterable, list, tuple, etc. of values (C{scalar} or 

L{Fsum} instances). 

@kwarg floats: Optionally, set C{B{floats}=True} iff I{all} 

B{C{xs}} are known to be C{float}. 

 

@return: Precision C{fsum} (C{float}). 

 

@see: Function C{fsum}. 

''' 

return _fsum(_2floats(xs, primed=True, floats=floats)) if xs else _0_0 

 

 

def fsum1_(*xs, **floats): 

'''Precision floating point summation of a few arguments, 1-primed. 

 

@arg xs: Values to be added (C{scalar} or L{Fsum} instances), 

all positional. 

@kwarg floats: Optionally, set C{B{floats}=True} iff I{all} 

B{C{xs}} are known to be C{float}. 

 

@return: Precision C{fsum} (C{float}). 

 

@see: Function C{fsum} 

''' 

if xs: 

f = _xkwds_get(floats, floats=False) 

return _fsum(_2floats(xs, origin=1, primed=True, floats=f)) 

else: # PYCHOK no cover 

return _0_0 

 

# **) MIT License 

# 

# Copyright (C) 2016-2022 -- 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.