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# -*- coding: utf-8 -*- 

 

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

 

Set env variable C{PYGEODESY_FSUM_RESIDUAL} to nay non-empty string 

to throw a L{ResidualError} for division or exponention by an L{Fsum} 

instance with a non-zero C{residual}, see methoda L{Fsum.fdiv} and 

L{Fsum.fpow}. 

''' 

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

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

 

from pygeodesy.basics import _isfinite, isint, isscalar, map1, \ 

neg, signOf 

from pygeodesy.errors import _NotImplementedError, _OverflowError, \ 

_TypeError, _ValueError, _xkwds_get, \ 

_ZeroDivisionError 

from pygeodesy.interns import NN, _COMMASPACE_, _DASH_, _EQUAL_, \ 

_finite_, _iadd_, _negative_, _not_, \ 

_PERCENT_, _scalar_, _SLASH_, _SPACE_, \ 

_STAR_, _supported_, _0_0, _1_0, _N_1_0 

from pygeodesy.lazily import _ALL_LAZY, _getenv, _sys_version_info2 

from pygeodesy.named import _Named, _NamedTuple, _NotImplemented 

# from pygeodesy.props import Property_RO, property_RO 

from pygeodesy.streprs import Fmt, fstr, pairs, unstr 

from pygeodesy.units import Float, Int, Property_RO, property_RO 

 

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

 

__all__ = _ALL_LAZY.fsums 

__version__ = '22.02.09' 

 

_eq_ = _EQUAL_ * 2 

_ge_ = '>=' 

_gt_ = '>' # _RANGLE_ 

_le_ = '<=' 

_lt_ = '<' # _LANGLE_ 

_ne_ = '!=' 

 

_floordiv_ = _SLASH_ * 2 

_fset_ = _EQUAL_ 

_mod_ = _PERCENT_ 

_divmod_ = _floordiv_ + _mod_ 

_mul_ = _STAR_ 

_non_zero_ = 'non-zero' 

_pow_ = _STAR_ * 2 

_residual_ = 'residual' 

_sub_ = _DASH_ 

_truediv_ = _SLASH_ 

 

_int0 = 0 

_1p0iself = _1_0.__pos__() is _1_0 

 

 

def _2even(s, r, p): 

'''(INTERNAL) Half-even rounding. 

''' 

if (r > 0 and p > 0) or \ 

(r < 0 and p < 0): # signs match 

r, p = _2sum(s, p * 2) 

if not p: 

s = r 

return s 

 

 

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

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

''' 

n, v = name_value.popitem() 

try: 

if _isfinite(v): 

return v if isinstance(v, float) else float(v) 

X, t = _ValueError, _not_(_finite_) 

except TypeError as x: 

X, t = _TypeError, str(x) 

except ValueError as x: 

X, t = _ValueError, str(x) 

except Exception as x: 

X, t = _NotImplementedError, repr(x) 

if index is not None: 

n = Fmt.SQUARE(n, index) 

raise X(n, v, txt=t) 

 

 

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

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

''' 

_2f = _2float 

if prime: 

yield _1_0 

i = origin 

for x in xs: 

if 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 prime: 

yield _N_1_0 

 

 

def _2Fsum(other, name=NN): 

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

''' 

if not isinstance(other, Fsum): 

other = _2float(other=other) 

return Fsum(name=name)._fset(other) 

 

 

def _2sum(a, b): # by .testFmath 

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

''' 

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)) # abs(b) <= abs(a) 

 

 

class ResidualError(_ValueError): 

'''Error raised for an operation involving an L{Fsum} with a non-zero residual. 

''' 

pass 

 

 

class Fsum(_Named): 

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

 

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

I{running} precision floating point summation. Accumulation may continue 

after intermediate, I{running} summations. 

 

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

from standard Python's C{math.fsum}. 

 

@note: Values to be accumulated are C{scalar} or L{fsum} instances with C{scalar} 

meaning type C{float}, C{int} or any type convertible to C{float}. 

 

@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 = [] # partials 

_Rx = bool(_getenv('PYGEODESY_FSUM_RESIDUAL', NN)) 

 

def __init__(self, *xs, **name_NN): 

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

 

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

L{Fsum} instances). 

@kwarg name_NN: Optional name (C{str}). 

 

@see: Method L{Fsum.fadd}. 

''' 

# self._n = 0 

self._ps = [] 

if name_NN: 

self.name = _xkwds_get(name_NN, name=NN) 

if xs: 

self._fadd(_2floats(xs, origin=1)) 

 

def __abs__(self): 

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

''' 

f = self.fcopy(name=self.__abs__.__name__) 

return f._fneg() if f < 0 else f 

 

def __add__(self, other): 

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

 

@arg other: An L{Fsum} or C{scalar}. 

 

@return: The sum (L{Fsum}). 

 

@see: Method L{Fsum.__iadd__}. 

''' 

f = self.fcopy(name=self.__add__.__name__) 

f += other 

return f 

 

def as_integer_ratio(self): 

'''Return this instance as the integer ratio. 

 

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

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

 

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

''' 

s, r = self._fsum2 

n, d = (int(s), 1) if isint(s) else s.as_integer_ratio() 

if r: 

if isint(r): # PYCHOK no cover 

n += int(r) * d 

else: 

rn, rd = r.as_integer_ratio() 

n = (n * rd) + (rn * d) 

d *= rd 

return n, d 

 

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

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

''' 

s, r = self._fsum2 

return bool(s or r) 

 

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+, 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.fcopy(name=self.__divmod__.__name__) 

return f._fdivmod(other, _divmod_) 

 

def __eq__(self, other): 

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

''' 

s, r = self._cmp2(other, _eq_) 

return not bool(s or r) 

 

def __float__(self): 

'''Return this instance' current precision running sum 

as C{float} or C{int}. 

 

@see: Method L{Fsum.fsum}. 

''' 

return self._fsum0 

 

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+, 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.fcopy(name=self.__floordiv__.__name__) 

return f._floordiv(other, _floordiv_) 

 

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. 

''' 

s, r = self._cmp2(other, _ge_) 

return r >= -s 

 

def __gt__(self, other): 

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

''' 

s, r = self._cmp2(other, _gt_) 

return r > -s 

 

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}. 

 

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

 

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

 

@see: Method L{Fsum.fadd}. 

''' 

if isinstance(other, Fsum): 

if other is self: # or other._fsum2 == self._fsum2: 

self._fmul(2) 

elif other._ps: 

self._fadd(other._ps) 

elif not isscalar(other): 

raise self._TypeError(_iadd_, other) 

elif other: 

self._fadd_(other) 

return self 

 

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_ + _fset_) 

 

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._fdivmod(other, _mod_ + _fset_) 

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. 

 

@see: Method L{Fsum.fmul}. 

''' 

if isinstance(other, Fsum): 

a = False 

if len(self._ps) != 1: 

f = self._mul_Fsum(other) 

elif len(other._ps) != 1: 

f = other._copy()._fmul(self._ps[0]) 

f._n = self._n 

else: 

f = other._ps[0] * self._ps[0] 

a = isint(f) 

self._fset(f, asis=a) 

elif isscalar(other): 

self._fmul(other) 

else: 

raise self._TypeError(_mul_ + _fset_, other) 

return self 

 

def __int__(self): 

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

 

@see: Methods L{Fsum.__ceil__} and L{Fsum.__floor__}. 

''' 

return int(self._fsum0) 

 

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

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

 

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

@arg mod: Optional modulus (C{int}) for the 3-argument 

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

 

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

 

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

 

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

a negative or fractional C{scalar} 

B{C{other}} and this instance has 

a non-zero residual. 

 

@raise TypeError: Invalid B{C{other}} type or the 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{other}} is an L{Fsum} 

with a non-zero residual. 

 

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

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

''' 

return self._fpow(other, _pow_ + _fset_, *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}. 

''' 

if isinstance(other, Fsum): 

if other is self: # or other._fsum2 == self._fsum2: 

self._fset(_0_0) 

elif other._ps: 

self._fadd(map(neg, other._ps)) 

elif not isscalar(other): 

raise self._TypeError(_sub_ + _fset_, other) 

elif other: 

self._fadd_(neg(other)) 

return self 

 

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}}. 

 

@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_ + _fset_) 

 

def __le__(self, other): 

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

''' 

s, r = self._cmp2(other, _le_) 

return r <= -s 

 

def __len__(self): 

'''Return the I{total} number of values accumulated (C{int}). 

''' 

return self._n 

 

def __lt__(self, other): 

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

''' 

s, r = self._cmp2(other, _lt_) 

return r < -s 

 

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.fcopy(name=self.__mod__.__name__) 

return f._fdivmod(other, _mod_)[1] 

 

def __mul__(self, other): 

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

 

@see: Method L{Fsum.__imul__}. 

''' 

f = self.fcopy(name=self.__mul__.__name__) 

f *= other 

return f 

 

def __ne__(self, other): 

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

''' 

s, r = self._cmp2(other, _ne_) 

return bool(s or r) 

 

def __neg__(self): 

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

''' 

f = self.fcopy(name=self.__neg__.__name__) 

return f._fneg() 

 

def __pos__(self): 

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

''' 

return self if _1p0iself else self.fcopy(name=self.__pos__.__name__) 

 

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.fcopy(name=self.__pow__.__name__) 

return f._fpow(other, _pow_, *mod) 

 

def __radd__(self, other): 

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

 

@see: Method L{Fsum.__iadd__}. 

''' 

f = _2Fsum(other, name=self.__radd__.__name__) 

return f._fadd(self._ps) 

 

def __rdivmod__(self, other): 

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

remainder)}. 

 

@see: Method L{Fsum.__divmod__}. 

''' 

f = _2Fsum(other, name=self.__rdivmod__.__name__) 

return f._fdivmod(self, _divmod_) 

 

# def __repr__(self): 

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

# ''' 

# return self.toRepr() 

 

def __rfloordiv__(self, other): 

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

 

@see: Method L{Fsum.__ifloordiv__}. 

''' 

f = _2Fsum(other, name=self.__rfloordiv__.__name__) 

return f._floordiv(self, _floordiv_) 

 

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 = _2Fsum(other, name=self.__rmod__.__name__) 

return f._fdivmod(self, _mod_)[1] 

 

def __rmul__(self, other): 

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

 

@see: Method L{Fsum.__imul__}. 

''' 

f = _2Fsum(other, name=self.__rmul__.__name__) 

f *= self 

return f 

 

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 = _2Fsum(other, name=self.__rpow__.__name__) 

return f._fpow(self, _pow_, *mod) 

 

def __rsub__(self, other): 

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

 

@see: Method L{Fsum.__isub__}. 

''' 

f = _2Fsum(other, name=self.__rsub__.__name__) 

f -= self 

return f 

 

def __rtruediv__(self, other): 

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

 

@see: Method L{Fsum.__itruediv__}. 

''' 

f = _2Fsum(other, name=self.__rtruediv__.__name__) 

return f._ftruediv(self, _truediv_) 

 

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

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

''' 

from sys import getsizeof 

return sum(map1(getsizeof, self._fsum0, 

self._fsum2, 

self._fsum2.fsum, 

self._fsum2.residual, 

self._n, 

self._ps, *self._ps)) 

 

def __str__(self): 

'''Return the default C{str(this)}. 

''' 

return self.toStr() 

 

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.fcopy(name=self.__sub__.__name__) 

f -= other 

return f 

 

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.fcopy(name=self.__truediv__.__name__) 

return f._ftruediv(other, _truediv_) 

 

__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__ 

 

@property_RO 

def ceil(self): 

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

C{float} in Python 2-). 

 

@note: The C{ceil} accounts for the C{residual}. 

 

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

''' 

s, r = self._fsum2 

c = _ceil(s) + int(r) - 1 

while r > (c - s): # (s + r) > c 

c += 1 

return c 

 

def _cmp2(self, other, cop): 

'''(INTERNAL) Subtract an B{C{other}} instance or scalar and 

return an L{Fsum2Tuple}C{(fsum, residual)} for comparison 

operator B{C{cop}}. 

''' 

if other: 

f = self._copy() # fast fcopy, no caches 

if isinstance(other, Fsum): 

f._fadd(map(neg, other._ps)) # fast f -= other 

elif isscalar(other): 

f._fadd_(-other) # fast f -= other 

else: 

raise self._TypeError(cop, other) 

else: 

f = self 

return f._fsum2 

 

def _copy(self, _fsum2=False): 

'''(INTERNAL) Make a fast, un-named copy. 

''' 

f = Fsum() # NN 

f._n = self._n 

f._ps[:] = self._ps 

if _fsum2: 

Fsum._fsum2._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__}. 

''' 

return self.fcopy(name=self.divmod.__name__)._fdivmod(other, _divmod_) 

 

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 fadd(self, xs): 

'''Accumulate more scalar values from an iterable. 

 

@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: Non-scalar B{C{xs}} value. 

 

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

''' 

if isinstance(xs, Fsum): 

self._fadd(xs._ps) 

elif not isscalar(xs): # for ... 

self._fadd(_2floats(xs)) 

else: # ... backward compatibility 

self._fadd_(_2float(xs=xs)) # PYCHOK no cover 

return self 

 

def fadd_(self, *xs): 

'''Accumulate more I{scalar} values from positional arguments. 

 

@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: Non-scalar B{C{xs}} value. 

 

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

''' 

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

 

def _fadd(self, xs): 

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

''' 

ps, n, _2s = self._ps, 0, _2sum 

for x in xs: # _iter() 

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 

self._update() 

return self 

 

def _fadd_(self, *xs): 

'''(INTERNAL) Add all positional, I{known} C{scalar}s. 

''' 

return self._fadd(xs) 

 

def _fadd_1(self): 

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

and re-adding the final C{partial}. 

''' 

while len(self._ps) > 1: 

p = self._ps.pop() 

if p: 

self._fadd_(p) 

self._n -= 1 

break 

else: # must zap 

self._update() 

return self 

 

def fcopy(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._update(other=self) 

f._n = self._n if deep else 1 

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

return f 

 

copy = fcopy 

fdiv = __itruediv__ # for backward compatibility 

fdivmod = __divmod__ # for backward compatibility 

 

def _fdivmod(self, other, op): 

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

''' 

# mostly like like 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(_fsum2=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: 

self += other 

q -= 1 

 

# t = f.signOf() 

# if t and t != s: 

# from pygeodesy.errors import _AssertionError 

# raise f._Error(_dimo_, other, _AssertionError, txt=signOf.__name__) 

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

 

@property_RO 

def floor(self): 

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

C{float} in Python 2-). 

 

@note: The C{floor} accounts for the C{residual}. 

 

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

''' 

s, r = self._fsum2 

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): 

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

''' 

if isinstance(other, Fsum): # PYCHOK no cover 

self._fset(self._mul_Fsum(other)) 

elif other and self._ps: 

if other != _1_0: 

# multiply and adjust partials 

self._ps[:] = self._ps_x(other) 

self._fadd_1() 

else: 

self._fset(_0_0) 

# assert self._ps is ps 

return self 

 

def _fneg(self): 

'''(INTERNAL) Negate this instance. 

''' 

if self._ps: 

self._ps[:] = map(neg, self._ps) 

self._fadd_1() 

return self 

 

fpow = __ipow__ # for backward compatibility 

 

def _fpow(self, other, op, *mod): 

'''Apply C{B{self} **= B{other}}. 

''' 

if mod: 

try: 

f = int(self) if self.is_integer() else self._fsum0 

# throw the same TypeError as C{pow} if not all 

# self._fsum0, other and mod are C{int} and map 

# the ValueError for mod==0 into TypeError 

s = pow(f, other, *mod) 

a = isint(s) 

except (TypeError, ValueError) as x: 

t = str(x) 

t = _COMMASPACE_(Fmt.PARENSPACED(mod=mod[0]), t) 

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

else: 

x, p = other, _1_0 

if isinstance(x, Fsum): 

x, r = x._fsum2 

if r: 

if Fsum._Rx: 

raise self._ResidualError(op, other, r) 

p = self._pow_scalar(r, other, op) 

s = self._pow_scalar(x, other, op) 

if p != _1_0: 

s *= p # C{scalar}s or L{Fsum}s 

a = False 

return self._fset(s, asis=a) 

 

def _fset(self, other, asis=False): 

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

''' 

if other is self: 

pass # from ._fpow_scalar 

elif isinstance(other, Fsum): 

self._n = other._n 

self._ps[:] = other._ps 

self._update(other=other) 

elif isscalar(other): 

f = other if asis else float(other) 

self._n = 1 

self._ps[:] = [f] 

self._update(_fsum0=f, _fsum2=Fsum2Tuple(f, _int0)) 

else: # PYCHOK no cover 

raise self._TypeError(_fset_, other) 

return self 

 

def fsub(self, xs): 

'''Subtract several values. 

 

@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._fadd(_2floats(xs, sub=True)) if xs else self 

 

def fsub_(self, *xs): 

'''Subtract any positional value. 

 

@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._fadd(_2floats(xs, sub=True)) if xs else self 

 

def fsum(self, xs=None): 

'''Add C{None} or several values and sum all. 

 

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

L{Fsum} instances). 

 

@return: Precision running sum (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: Accumulation can continue after summation. 

''' 

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

return f._fsum0 

 

def fsum_(self, *xs): 

'''Add any positional value and sum all. 

 

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

all positional. 

 

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

 

@see: Method L{Fsum.fsum}. 

''' 

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

return f._fsum0 

 

@Property_RO 

def _fsum0(self): 

'''(INTERNAL) Get this instance', memoized precision 

running sum (C{float} or C{int}), without C{residual}. 

 

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

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

 

@see: Method L{Fsum.fsum2} and property L{Fsum.residual}. 

''' 

ps = self._ps 

i = len(ps) - 1 

if i < 0: 

s = _0_0 # XXX 0? 

else: 

s, _2s = ps[i], _2sum 

while i > 0: 

i -= 1 

s, p = _2s(s, ps[i]) 

ps[i:] = [s] 

if p: # sum(ps) became inexact 

if s: 

ps.append(p) 

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

s = _2even(s, ps[i-1], p) 

break 

else: # PYCHOK no cover 

ps[i] = s = p # swap s and p and continue 

# else: 

# s = float(s) 

# assert self._ps is ps 

# assert self.__dict__.get(Fsum._fsum2.name, None) is None 

# Fsum._fsum2._update(self) 

return s 

 

def fsum2(self, xs=None): 

'''Add C{None} or several values and return the current 

precision running sum and residual. 

 

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

L{Fsum} instances). 

 

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

the current precision running and C{residual}, 

the sum of the remaining C{partials}. 

 

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

''' 

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

return f._fsum2 

 

@Property_RO 

def _fsum2(self): 

'''(INTERNAL) Get this instance', memoized current 

precision running sum and residual (L{Fsum2Tuple}). 

 

@note: If the C{residual} is C{int(0)}, the precision 

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

 

@see: Method L{Fsum.fsum} and property L{Fsum.residual}. 

''' 

s = self._fsum0 

r = _fsum(self._ps1(s)) if len(self._ps) > 1 else _int0 

t = Fsum2Tuple(s, r) if s else Fsum2Tuple((r or _0_0), _int0) 

return t 

 

def fsum2_(self, *xs): 

'''Add any positional value and return the current precision 

running sum and C{delta}, the increment. 

 

@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 prior running C{fsum} 

(C{float}s). 

 

@see: Method L{Fsum.fsum_}. 

''' 

p, r = self._fsum2 

s, t = self._fadd(_2floats(xs, origin=1))._fsum2 if xs else (p, r) 

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

 

# ftruediv = __itruediv__ # for naming consistency 

 

def _ftruediv(self, other, op): 

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

''' 

if isinstance(other, Fsum): 

if other is self or other._fsum2 == self._fsum2: 

return self._fset(_1_0) 

d, r = other._fsum2 

if r: 

if Fsum._Rx: 

raise self._ResidualError(op, other, r) 

if d: 

# self / (d + r) == self * n / d 

# n = d / (d + r) = 1 / (1 + r / d) 

# d' = d / n = d * (1 + r / d), but 

# is pointless if (1 + r / d) == 1 

d *= r / d + _1_0 

else: # PYCHOK no cover 

d = r 

elif isscalar(other): 

d = other 

else: 

raise self._TypeError(op, other) 

try: 

self._fmul(_1_0 / d) 

except ZeroDivisionError as x: 

raise self._ZeroDivisionError(op, d, txt=str(x)) 

except (TypeError, ValueError) as x: 

raise self._ValueError(op, d, txt=str(x)) 

return self 

 

@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 is_exact(self): 

'''Is this instance' precision running C{fsum} considered to be exact? (C{bool}). 

''' 

return self.residual is _int0 

 

def is_integer(self): 

'''Return C{True} if this instance is an integer, C{False} otherwise. 

''' 

s, r = self._fsum2 if len(self._ps) != 1 else (self._ps[0], 0) 

return (not r) and isint(s, both=True) 

 

def is_math_fsum(self): 

'''Return C{True} if functions C{fsum}, C{fsum}_, C{fsum1} and C{fsum1_} 

are all based on Python's C{math.fsum}, C{False} otherwise. 

''' 

return bool(Fsum._math_fsum) 

 

def _mul_Fsum(self, other): 

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

''' 

return Fsum(_0_0)._fadd(self._ps_x(*other._ps)) 

 

@property_RO 

def partials(self): 

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

''' 

return tuple(self._ps) 

 

def pow(self, x): 

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

 

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

 

@return: The C{pow(self, B{x})} (L{Fsum}). 

 

@raise ResidualError: This residual non-zero and negative 

or fractional B{C{x}}. 

 

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

 

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

 

@see: Method L{Fsum.__ipow__}. 

''' 

if isint(x, both=True) and int(x) >= 0: 

f = self._pow_int(int(x), _pow_) 

else: # negative or not int B{C{x}} 

f = self.fcopy(name=self.pow.__name__)._fpow(x, _pow_) # f **= x 

return f 

 

def _pow_int(self, x, op): 

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

''' 

if x < 0: # assert x >= 0 

raise self._ValueError(op, x, txt=_SPACE_(_negative_, int.__name__)) 

 

a = False 

if len(self._ps) > 1: 

if x > 2: 

m = 1 # single-bit mask 

p = self._copy() 

if x & m: 

x -= m # x ^= m 

f = p._copy() 

else: 

f = Fsum(_1_0) 

while x: 

p = p._mul_Fsum(p) # p **= 2 

m += m # m <<= 1 

if x & m: 

f = f._mul_Fsum(p) # f *= p 

x -= m # x ^= m 

elif x > 1: # self**2 

f = self._mul_Fsum(self) 

else: # self**1 or self**0 

f = self if x else _1_0 

elif self._ps: 

f = self._ps[0]**x 

a = True # preserve C{type(ps[0])} 

else: # 0**non0 == 0, but 0**0 == 1 

f = _0_0 if x else _1_0 

return Fsum(name=self.pow.__name__)._fset(f, asis=a) 

 

def _pow_scalar(self, x, other, op): # MCCABE 14 

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

''' 

s, r = self._fsum2 

if isint(x, both=True): 

x = int(x) # Fsum**int 

y = abs(x) 

if y > 1: 

if r: 

s = self._pow_int(y, op) 

if x < 0: 

s, r = s._fsum2 

if r: 

raise self._ResidualError(op, other, r) 

# use **= -1 for the CPython float_pow 

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

x = -1 

else: 

x = None 

elif x < 0: 

if r: # self**-1 == 1 / self 

s = Fsum(_1_0)._ftruediv(self, op) 

x = None 

else: # self**1 or self**0 

s = self if x else _1_0 

x = None 

elif not isscalar(x): # assert ... 

raise self._TypeError(op, other, txt=_not_(_scalar_)) 

elif r: # non0_residual**fractional 

raise self._ResidualError(op, other, r) 

elif s < 0: # neg**fractional yields complex 

raise self._ValueError(op, other, txt=_not_(_supported_)) 

if x not in (None, _1_0): 

try: 

s **= x 

except Exception as e: 

raise self._ValueError(op, other, txt=str(e)) 

return s # C{scalar} or an L{Fsum} 

 

def _ps1(self, less): 

'''(INTERNAL) Yield partials, pseudo-sorted, 1-primed 

minus C{less} if non-zero. 

''' 

yield _1_0 

if less: 

yield -less 

for p in self._ps: 

if p > 0: 

yield p 

for p in self._ps: 

if p < 0: 

yield p 

yield _N_1_0 

 

def _ps_x(self, *factors): 

'''(INTERNAL) Yield C{partials}, multiplied by each of the B{C{factors}}. 

''' 

ps = self._ps 

if len(ps) < len(factors): 

ps, factors = factors, ps 

for f in factors: 

for p in ps: 

p *= f 

if p: 

yield p 

 

@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._fsum0) 

 

@property_RO 

def residual(self): 

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

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

 

@note: If the C{residual} is C{int(0)}, 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._fsum2.residual 

 

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

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

''' 

t = _SPACE_(_non_zero_, _residual_) 

t = Fmt.PARENSPACED(t, fstr(residual, fmt=Fmt.e, prec=8)) 

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._fsum2 if res else (self._fsum0, 0) 

return signOf(r, off=-s) 

 

def toRepr(self, prec=6, sep=_COMMASPACE_, fmt=Fmt.g, **unused): # PYCHOK signature 

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

 

@kwarg prec: The C{float} precision, number of decimal digits (0..9). 

Trailing zero decimals are stripped for B{C{prec}} values 

of 1 and above, but kept for negative B{C{prec}} values. 

@kwarg sep: Optional separator to join (C{str}). 

@kwarg fmt: Optional, C{float} format (C{str}). 

 

@return: This instance (C{str}). 

''' 

t = sep.join(pairs(self._fsum2.items(), prec=prec, fmt=fmt)) 

return _SPACE_(Fmt.SQUARE(self.named3, len(self)), Fmt.PAREN(t)) 

 

def toStr(self, prec=6, sep=_COMMASPACE_, fmt=Fmt.g, **unused): # PYCHOK signature 

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

 

@kwarg prec: The C{float} precision, number of decimal digits (0..9). 

Trailing zero decimals are stripped for B{C{prec}} values 

of 1 and above, but kept for negative B{C{prec}} values. 

@kwarg sep: Optional separator to join (C{str}). 

@kwarg fmt: Optional, C{float} format (C{str}). 

 

@return: This instance (C{repr}). 

''' 

t = self._fsum2.toStr(prec=prec, sep=sep, fmt=fmt) 

return _SPACE_(Fmt.SQUARE(self.named3, len(self)), t) 

 

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

'''(INTERNAL) Operand C{TypeError}. 

''' 

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

 

def _update(self, other=None, **setters): 

'''(INTERNAL) Copy, set or zap all cached C{Property_RO} values. 

''' 

if other is None: # zap all 

Fsum._fsum0._update(self) 

Fsum._fsum2._update(self) 

else: # dup if present, otherwise zap 

Fsum._fsum0._update_from(self, other) 

Fsum._fsum2._update_from(self, other) 

if setters: 

# Property_RO ._fsum0 and ._fsum2 can't be a Property since 

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

self.__dict__.update(setters) 

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): 

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

''' 

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

 

 

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 if any. 

Each item is either C{float} or C{int}. 

''' 

_Names_ = ( Fsum.fsum.__name__, _residual_) 

_Units_ = (_Float_Int, _Float_Int) 

 

 

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 

 

except ImportError: 

 

def _fsum(xs): 

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

''' 

return Fsum(name=_fsum.__name__)._fadd(xs)._fsum0 

 

 

def fsum(xs): 

'''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). 

 

@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)) if xs else _0_0 

 

 

def fsum_(*xs): 

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

 

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

all positional. 

 

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

 

@see: Function C{fsum}. 

''' 

return _fsum(_2floats(xs, origin=1)) if xs else _0_0 

 

 

def fsum1(xs): 

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

 

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

L{Fsum} instances). 

 

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

 

@see: Function C{fsum}. 

''' 

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

 

 

def fsum1_(*xs): 

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

 

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

all positional. 

 

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

 

@see: Function C{fsum} 

''' 

return _fsum(_2floats(xs, origin=1, prime=True)) if xs else _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.