
# -*- 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_RESIDUAL} to a C{float} string greater than
C{"0.0"} as the threshold to throw a L{ResidualError} for a division, power or
root operation of an L{Fsum} instance with a C{residual} I{ratio} 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 isbool, iscomplex, isint, isscalar, \
                            _signOf, itemsorted, signOf, _xiterable, \
                            _xiterablen,  _enquote
from pygeodesy.constants import INT0, _isfinite, NEG0, _pos_self, \
                               _0_0, _1_0, _N_1_0,  Float, Int
from pygeodesy.errors import _OverflowError, _TypeError, _UnexpectedError, \
                             _ValueError, _xError, _xError2, _xkwds_get1, \
                             _xkwds_pop2
# from pygeodesy.internals import _enquote  # from .basics
from pygeodesy.interns import NN, _arg_, _COMMASPACE_, _DASH_, _DOT_, \
                             _EQUAL_, _from_, _LANGLE_, _NOTEQUAL_, \
                             _not_finite_, _PERCENT_, _PLUS_, \
                             _RANGLE_, _SLASH_, _SPACE_, _STAR_, _UNDER_
from pygeodesy.lazily import _ALL_LAZY, _getenv, _sys_version_info2
from pygeodesy.named import _name__, _name2__, _Named, _NamedTuple, \
                            _NotImplemented
from pygeodesy.props import _allPropertiesOf_n, deprecated_property_RO, \
                             Property, Property_RO, property_RO
from pygeodesy.streprs import Fmt, fstr, unstr
# from pygeodesy.units import Float, Int  # from .constants

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

__all__ = _ALL_LAZY.fsums
__version__ = '24.06.11'

_add_op_      = _PLUS_  # in .auxilats.auxAngle
_eq_op_       = _EQUAL_ * 2  # _DEQUAL_
_div_         = 'div'
_floordiv_op_ = _SLASH_ * 2  # _DSLASH_
_fset_op_     = _EQUAL_
_ge_op_       = _RANGLE_ + _EQUAL_
_gt_op_       = _RANGLE_
_iadd_op_     = _add_op_ + _EQUAL_  # in .auxilats.auxAngle, .fstats
_integer_     = 'integer'
_le_op_       = _LANGLE_ + _EQUAL_
_lt_op_       = _LANGLE_
_mod_         = 'mod'
_mod_op_      = _PERCENT_
_mul_op_      = _STAR_
_ne_op_       = _NOTEQUAL_
_non_zero_    = 'non-zero'
_pow_op_      = _STAR_ * 2  # _DSTAR_
_significant_ = 'significant'
_sub_op_      = _DASH_  # in .auxilats.auxAngle
_threshold_   = 'threshold'
_truediv_op_  = _SLASH_
_divmod_op_   = _floordiv_op_ + _mod_op_
_isub_op_     = _sub_op_ + _fset_op_  # in .auxilats.auxAngle


def _2delta(*ab):
    '''(INTERNAL) Helper for C{Fsum._fsum2}.
    '''
    try:
        a, b = _2sum(*ab)
    except _OverflowError:
        a, b =  ab
    return float(a if fabs(a) > fabs(b) else b)


def _2error(unused):  # in .fstats
    '''(INTERNAL) Throw a C{not-finite} exception.
    '''
    raise ValueError(_not_finite_)


def _2finite(x):
    '''(INTERNAL) return C{float(x)} if finite.
    '''
    x = float(x)
    return x if _isfinite(x) else _2error(x)


def _2float(index=None, **name_value):  # in .fmath, .fstats
    '''(INTERNAL) Raise C{TypeError} or C{ValueError} if not scalar or infinite.
    '''
    n, v = name_value.popitem()  # _xkwds_item2(name_value)
    try:
        return _2finite(v)
    except Exception as X:
        raise _xError(X, Fmt.INDEX(n, index), v)


def _X_ps(X):  # for _2floats only
    return X._ps


def _2floats(xs, origin=0, _X=_X_ps, _x=float):
    '''(INTERNAL) Yield each B{C{xs}} as a C{float}.
    '''
    try:
        i, x =  origin, _X
        _fin = _isfinite
        _FsT = _Fsum_Fsum2Tuple_types
        _isa =  isinstance
        for x in _xiterable(xs):
            if _isa(x, _FsT):
                for p in _X(x._Fsum):
                    yield p
            else:
                f = _x(x)
                yield f if _fin(f) else _2error(f)
            i += 1
    except Exception as X:
        raise _xError(X, xs=xs) if x is _X else \
              _xError(X, Fmt.INDEX(xs=i), x)


def _Fsumf_(*xs):  # floats=True, in .auxLat, ...
    '''(INTERNAL) An C{Fsum} of I{known scalars}.
    '''
    return Fsum()._facc_scalar(xs, up=False)


def _Fsum1f_(*xs):  # floats=True, in .albers, ...
    '''(INTERNAL) An C{Fsum} of I{known scalars}, 1-primed.
    '''
    return Fsum()._facc_scalar(_1primed(xs), up=False)


def _2halfeven(s, r, p):
    '''(INTERNAL) 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
    return s


def _isFsum(x):  # in .fmath
    '''(INTERNAL) Is C{x} an C{Fsum} instance?
    '''
    return isinstance(x, Fsum)


def _isFsumTuple(x):  # in .fmath
    '''(INTERNAL) Is C{x} an C{Fsum} or C{Fsum2Tuple} instance?
    '''
    return isinstance(x, _Fsum_Fsum2Tuple_types)


def _1_Over(x, op, **raiser_RESIDUAL):  # vs _1_over
    '''(INTERNAL) Return C{Fsum(1) / B{x}}.
    '''
    return _Psum_(_1_0)._ftruediv(x, op, **raiser_RESIDUAL)


def _1primed(xs):  # in .fmath
    '''(INTERNAL) 1-Primed summation of iterable C{xs}
       items, all I{known} to be C{scalar}.
    '''
    yield _1_0
    for x in xs:
        yield x
    yield _N_1_0


def _psum(ps):  # PYCHOK used!
    '''(INTERNAL) Partials summation, updating C{ps}.
    '''
    # assert isinstance(ps, list)
    i   =  len(ps) - 1
    s   = _0_0 if i < 0 else ps[i]
    _2s = _2sum
    while i > 0:
        i -= 1
        s, r = _2s(s, ps[i])
        if r:  # sum(ps) became inexact
            if s:
                ps[i:] = r, s
                if i > 0:
                    s = _2halfeven(s, r, ps[i-1])
                break  # return s
            s = r  # PYCHOK no cover
        ps[i:] = s,
    return s


def _Psum(ps, **name_RESIDUAL):
    '''(INTERNAL) Return an C{Fsum} from I{ordered} partials C{ps}.
    '''
    f = Fsum(**name_RESIDUAL) if name_RESIDUAL else Fsum()
    if ps:
        f._ps[:] = ps
        f._n = len(f._ps)
    return f


def _Psum_(*ps, **name_RESIDUAL):
    '''(INTERNAL) Return an C{Fsum} from 1 or 2 known scalar(s) C{ps}.
    '''
    return _Psum(ps, **name_RESIDUAL)


def _2scalar2(other):
    '''(INTERNAL) Return 2-tuple C{(other, r)} with C{other} as C{int},
       C{float} or C{as-is} and C{r} the residual of C{as-is}.
    '''
    if _isFsumTuple(other):
        s, r = other._fint2
        if r:
            s, r = other._fprs2
            if r:  # PYCHOK no cover
                s = other  # L{Fsum} as-is
    else:
        r = 0
        s = other  # C{type} as-is
        if isint(s, both=True):
            s = int(s)
    return s, r


def _s_r(s, r):
    '''(INTERNAL) Return C{(s, r)}, I{ordered}.
    '''
    if r:
        if fabs(s) < fabs(r):
            s, r = r, (s or INT0)
    else:
        r = INT0
    return s, r


def _strcomplex(s, *args):
    '''(INTERNAL) C{Complex} 2- or 3-arg C{pow} error as C{str}.
    '''
    c = _strcomplex.__name__[4:]
    n = _DASH_(len(args), _arg_)
    t =  unstr(pow, *args)
    return _SPACE_(c, s, _from_, n, t)


def _stresidual(prefix, residual, R=0, **mod_ratio):
    '''(INTERNAL) Residual error txt C{str}.
    '''
    p = _stresidual.__name__[3:]
    t =  Fmt.PARENSPACED(p, Fmt(residual))
    for n, v in itemsorted(mod_ratio):
        p =  Fmt.PARENSPACED(n, Fmt(v))
        t = _COMMASPACE_(t, p)
    return _SPACE_(prefix, t, Fmt.exceeds_R(R), _threshold_)


def _2sum(a, b):  # by .testFmath
    '''(INTERNAL) Return C{a + b} as 2-tuple (sum, residual).
    '''
    # Neumaier, A. U{Rundungsfehleranalyse einiger Verfahren zur Summation endlicher
    # Summen<https://OnlineLibrary.Wiley.com/doi/epdf/10.1002/zamm.19740540106>},
    # 1974, Zeitschrift für Angewandte Mathmatik und Mechanik, vol 51, nr 1, p 39-51
    # <https://StackOverflow.com/questions/78633770/can-neumaier-summation-be-sped-up>
    s = a + b
    if _isfinite(s):
        if fabs(a) < fabs(b):
            r = (b - s) + a
        else:
            r = (a - s) + b
        return s, r
    u = unstr(_2sum, a, b)
    t = Fmt.PARENSPACED(_not_finite_, s)
    raise _OverflowError(u, txt=t)


def _threshold(threshold=_0_0, **kwds):
    '''(INTERNAL) Get the L{ResidualError}s threshold,
       optionally from single kwds C{B{RESIDUAL}=scalar}.
    '''
    if kwds:
        threshold, kwds = _xkwds_pop2(kwds, RESIDUAL=threshold)
#       threshold = kwds.pop('RESIDUAL', threshold)
        if kwds:
            raise _UnexpectedError(**kwds)
    try:
        return _2finite(threshold)  # PYCHOK None
    except Exception as x:
        raise ResidualError(threshold=threshold, cause=x)


class Fsum(_Named):  # sync __methods__ with .vector3dBase.Vector3dBase
    '''Precision floating point summation and I{running} summation.

       Unlike Python's C{math.fsum}, this class accumulates values and provides intermediate,
       I{running}, precision floating point summations.  Accumulation may continue after any
       intermediate, I{running} summuation.

       @note: Values may be L{Fsum}, L{Fsum2Tuple}, C{int}, C{float} or C{scalar} instances,
              any C{type} having method C{__float__} to convert the C{scalar} to a single
              C{float}, except C{complex}.

       @note: Handling of exceptions and C{inf}, C{INF}, C{nan} and C{NAN} differs from
              Python's C{math.fsum}.

       @see: U{Hettinger<https://GitHub.com/ActiveState/code/tree/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
#   _ps_max    =  0  # max(Fsum._ps_max, len(Fsum._ps))
    _RESIDUAL  = _threshold(_getenv('PYGEODESY_FSUM_RESIDUAL', _0_0))

    def __init__(self, *xs, **name_RESIDUAL):
        '''New L{Fsum} for I{running} precision floating point summation.

           @arg xs: No, one or more initial items to add (each C{scalar} or
                    an L{Fsum} or L{Fsum2Tuple} instance), all positional.
           @kwarg name_RESIDUAL: Optional C{B{name}=NN} (C{str}) for this
                       L{Fsum} and the C{B{RESIDUAL}=0.0} threshold for
                       L{ResidualError}s (C{scalar}).

           @see: Methods L{Fsum.fadd} and L{Fsum.RESIDUAL}.
        '''
        if name_RESIDUAL:
            n, kwds = _name2__(**name_RESIDUAL)
            if kwds:
                R =  Fsum._RESIDUAL
                t = _threshold(R, **kwds)
                if t != R:
                    self._RESIDUAL = t
            if n:
                self.name = n

        self._ps = []  # [_0_0], see L{Fsum._fprs}
        if xs:
            self._facc_1(xs, up=False)

    def __abs__(self):
        '''Return this instance' absolute value as an L{Fsum}.
        '''
        s = self.signOf()  # == self._cmp_0(0)
        return (-self) if s < 0 else self._copy_2(self.__abs__)

    def __add__(self, other):
        '''Return C{B{self} + B{other}} as an L{Fsum}.

           @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar}.

           @return: The sum (L{Fsum}).

           @see: Methods L{Fsum.fadd_} and L{Fsum.fadd}.
        '''
        f = self._copy_2(self.__add__)
        return f._fadd(other, _add_op_)

    def __bool__(self):  # PYCHOK Python 3+
        '''Return C{True} if this instance is I{exactly} non-zero.
        '''
        s, r = self._fprs2
        return bool(s or r) and s != -r  # == 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 __cmp__(self, other):  # PYCHOK no cover
        '''Compare this with an other instance or C{scalar}, Python 2-.

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

           @raise TypeError: Incompatible B{C{other}} C{type}.
        '''
        s = self._cmp_0(other, self.cmp.__name__)
        return _signOf(s, 0)

    def __divmod__(self, other, **raiser_RESIDUAL):
        '''Return C{divmod(B{self}, B{other})} as a L{DivMod2Tuple}
           with quotient C{div} an C{int} in Python 3+ or C{float}
           in Python 2- and remainder C{mod} an L{Fsum} instance.

           @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} modulus.
           @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore
                         L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}
                         to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

           @raise ResidualError: Non-zero, significant residual or invalid
                                 B{C{RESIDUAL}}.

           @see: Method L{Fsum.fdiv}.
        '''
        f = self._copy_2(self.__divmod__)
        return f._fdivmod2(other, _divmod_op_, **raiser_RESIDUAL)

    def __eq__(self, other):
        '''Compare this with an other instance or C{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}, L{Fsum2Tuple} 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 C{scalar}.
        '''
        return self._cmp_0(other, _ge_op_) >= 0

    def __gt__(self, other):
        '''Compare this with an other instance or C{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}, L{Fsum2Tuple} or C{scalar} value or
                       an iterable of several of the former.

           @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}.
        '''
        try:
            return self._fadd(other, _iadd_op_)
        except TypeError:
            return self._facc_inplace(other, _iadd_op_, self._facc)

    def __ifloordiv__(self, other):
        '''Apply C{B{self} //= B{other}} to this instance.

           @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} divisor.

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

           @raise ResidualError: Non-zero, significant 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}, L{Fsum2Tuple} or C{scalar} modulus.

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

           @see: Method L{Fsum.__divmod__}.
        '''
        return self._fdivmod2(other, _mod_op_ + _fset_op_).mod

    def __imul__(self, other):
        '''Apply C{B{self} *= B{other}} to this instance.

           @arg other: An L{Fsum}, L{Fsum2Tuple} 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: Method L{Fsum.int_float} and properties L{Fsum.ceil}
              and L{Fsum.floor}.
        '''
        i, _ = self._fint2
        return i

    def __invert__(self):  # PYCHOK no cover
        '''Not implemented.'''
        # Luciano Ramalho, "Fluent Python", O'Reilly, 2nd Ed, 2022 p. 567
        return _NotImplemented(self)

    def __ipow__(self, other, *mod, **raiser_RESIDUAL):  # PYCHOK 2 vs 3 args
        '''Apply C{B{self} **= B{other}} to this instance.

           @arg other: The exponent (C{scalar}, L{Fsum} or L{Fsum2Tuple}).
           @arg mod: Optional modulus (C{int} or C{None}) for the 3-argument
                     C{pow(B{self}, B{other}, B{mod})} version.
           @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore
                         L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}
                         to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

           @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} and B{C{mod}} is given and C{None}.

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

           @raise ResidualError: Invalid B{C{RESIDUAL}} or the residual
                                 is non-zero and significant and either
                                 B{C{other}} is a fractional or negative
                                 C{scalar} or B{C{mod}} is given and not
                                 C{None}.

           @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 and significant residual or B{C{mod}}
                              is given as 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, **raiser_RESIDUAL)

    def __isub__(self, other):
        '''Apply C{B{self} -= B{other}} to this instance.

           @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} value or
                       an iterable of several of the former.

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

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

           @see: Methods L{Fsum.fsub_} and L{Fsum.fsub}.
        '''
        try:
            return self._fsub(other, _isub_op_)
        except TypeError:
            return self._facc_inplace(other, _isub_op_, self._facc_neg)

    def __iter__(self):
        '''Return an C{iter}ator over a C{partials} duplicate.
        '''
        return iter(self.partials)

    def __itruediv__(self, other, **raiser_RESIDUAL):
        '''Apply C{B{self} /= B{other}} to this instance.

           @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} divisor.
           @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore
                         L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}
                         to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

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

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

           @raise ResidualError: Non-zero, significant residual or invalid
                                 B{C{RESIDUAL}}.

           @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_, **raiser_RESIDUAL)

    def __le__(self, other):
        '''Compare this with an other instance or C{scalar}.
        '''
        return self._cmp_0(other, _le_op_) <= 0

    def __len__(self):
        '''Return the number of values accumulated (C{int}).
        '''
        return self._n

    def __lt__(self, other):
        '''Compare this with an other instance or C{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_).mod

    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 C{scalar}.
        '''
        return self._cmp_0(other, _ne_op_) != 0

    def __neg__(self):
        '''Return I{a copy of} this instance, I{negated}.
        '''
        f = self._copy_2(self.__neg__)
        return f._fset(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_2r(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_2r(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_2r(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_2r(other, self.__rmod__)
        return f._fdivmod2(self, _mod_op_).mod

    def __rmul__(self, other):
        '''Return C{B{other} * B{self}} as an L{Fsum}.

           @see: Method L{Fsum.__imul__}.
        '''
        f = self._copy_2r(other, self.__rmul__)
        return f._fmul(self, _mul_op_)

    def __round__(self, *ndigits):  # PYCHOK Python 3+
        '''Return C{round(B{self}, *B{ndigits}} as an L{Fsum}.

           @arg ndigits: Optional number of digits (C{int}).
        '''
        f = self._copy_2(self.__round__)
        # <https://docs.Python.org/3.12/reference/datamodel.html?#object.__round__>
        return f._fset(round(float(self), *ndigits))  # can be C{int}

    def __rpow__(self, other, *mod):
        '''Return C{B{other}**B{self}} as an L{Fsum}.

           @see: Method L{Fsum.__ipow__}.
        '''
        f = self._copy_2r(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_2r(other, self.__rsub__)
        return f._fsub(self, _sub_op_)

    def __rtruediv__(self, other, **raiser_RESIDUAL):
        '''Return C{B{other} / B{self}} as an L{Fsum}.

           @see: Method L{Fsum.__itruediv__}.
        '''
        f = self._copy_2r(other, self.__rtruediv__)
        return f._ftruediv(self, _truediv_op_, **raiser_RESIDUAL)

    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}, L{Fsum2Tuple} 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, **raiser_RESIDUAL):
        '''Return C{B{self} / B{other}} as an L{Fsum}.

           @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} divisor.
           @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore
                         L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}
                         to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

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

           @raise ResidualError: Non-zero, significant residual or invalid
                                 B{C{RESIDUAL}}.

           @see: Method L{Fsum.__itruediv__}.
        '''
        return self._truediv(other, _truediv_op_, **raiser_RESIDUAL)

    __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}
                    with C{numerator} signed and C{denominator}
                    non-zero, positive.

           @see: Standard C{float.as_integer_ratio} in Python 2.7+.
        '''
        n, r = self._fint2
        if r:
            i, d = float(r).as_integer_ratio()
            n *= d
            n += i
        else:  # PYCHOK no cover
            d = 1
        return n, d

    @property_RO
    def as_iscalar(self):
        '''Get this instance I{as-is} (L{Fsum} or C{scalar}), the
           latter only if the C{residual} equals C{zero}.
        '''
        s, r = self._fprs2
        return self if r else s

    @property_RO
    def ceil(self):
        '''Get this instance' C{ceil} value (C{int} in Python 3+, but
           C{float} in Python 2-).

           @note: This 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  # _ceil(self._n_d)

    cmp = __cmp__

    def _cmp_0(self, other, op):
        '''(INTERNAL) Return C{scalar(self - B{other})} for 0-comparison.
        '''
        if _isFsumTuple(other):
            s = self._ps_1sum(*other._ps)
        elif self._scalar(other, op):
            s = self._ps_1sum(other)
        else:
            s = self.signOf()  # res=True
        return s

    def copy(self, deep=False, **name):
        '''Copy this instance, C{shallow} or B{C{deep}}.

           @kwarg name: Optional, overriding C{B{name}='"copy"} (C{str}).

           @return: The copy (L{Fsum}).
         '''
        n = _name__(name, name__=self.copy)
        f = _Named.copy(self, deep=deep, name=n)
        if f._ps is self._ps:
            f._ps = list(self._ps)  # separate list
        if not deep:
            f._n = 1
        # assert f._Fsum is f
        return f

    def _copy_2(self, which, name=NN):
        '''(INTERNAL) Copy for I{dyadic} operators.
        '''
        n =  name or which.__name__  # _dunder_nameof
        # NOT .classof due to .Fdot(a, *b) args, etc.
        f = _Named.copy(self, deep=False, name=n)
        f._ps = list(self._ps)  # separate list
        # assert f._n == self._n
        # assert f._Fsum is f
        return f

    def _copy_2r(self, other, which):
        '''(INTERNAL) Copy for I{reverse-dyadic} operators.
        '''
        return other._copy_2(which) if _isFsum(other) else \
                self._copy_2(which)._fset(other)

#   def _copy_RESIDUAL(self, other):
#       '''(INTERNAL) Copy C{other._RESIDUAL}.
#       '''
#       R = other._RESIDUAL
#       if R is not Fsum._RESIDUAL:
#           self._RESIDUAL = R

    divmod = __divmod__

    def _Error(self, op, other, Error, **txt_cause):
        '''(INTERNAL) Format an B{C{Error}} for C{{self} B{op} B{other}}.
        '''
        return Error(_SPACE_(self.as_iscalar, op, other), **txt_cause)

    def _ErrorX(self, X, op, other, *mod):
        '''(INTERNAL) Format the caught exception C{X}.
        '''
        E, t = _xError2(X)
        if mod:
            t = _COMMASPACE_(Fmt.PARENSPACED(mod=mod[0]), t)
        return self._Error(op, other, E, txt=t, cause=X)

    def _ErrorXs(self, X, xs, **kwds):  # in .fmath
        '''(INTERNAL) Format the caught exception C{X}.
        '''
        E, t = _xError2(X)
        u = unstr(self.named3, *xs[:3], _ELLIPSIS=len(xs) > 3, **kwds)
        return E(u, txt=t, cause=X)

    def _facc(self, xs, up=True, **origin_X_x):
        '''(INTERNAL) Accumulate more C{scalars} or L{Fsum}s.
        '''
        if xs:
            _xs   = _2floats(xs, **origin_X_x)  # PYCHOK yield
            ps    =  self._ps
            ps[:] =  self._ps_acc(list(ps), _xs, up=up)
        return self

    def _facc_1(self, xs, **up):
        '''(INTERNAL) Accumulate 0, 1 or more C{scalars} or L{Fsum}s,
           all positional C{xs} in the caller of this method.
        '''
        return self._fadd(xs[0], _add_op_, **up) if len(xs) == 1 else \
               self._facc(xs, origin=1, **up)

    def _facc_inplace(self, other, op, _facc):
        '''(INTERNAL) Accumulate from an iterable.
        '''
        try:
            return _facc(other, origin=1) if _xiterable(other) else self
        except Exception as X:
            raise self._ErrorX(X, op, other)

    def _facc_neg(self, xs, **up_origin):
        '''(INTERNAL) Accumulate more C{scalars} or L{Fsum}s, negated.
        '''
        def _N(X):
            return X._ps_neg

        def _n(x):
            return -float(x)

        return self._facc(xs, _X=_N, _x=_n, **up_origin)

    def _facc_power(self, power, xs, which, **raiser_RESIDUAL):  # in .fmath
        '''(INTERNAL) Add each C{xs} as C{float(x**power)}.
        '''
        def _Pow4(p):
            r = 0
            if _isFsumTuple(p):
                s, r = p._fprs2
                if r:
                    m = Fsum._pow
                else:  # scalar
                    return _Pow4(s)
            elif isint(p, both=True) and int(p) >= 0:
                p = s = int(p)
                m = Fsum._pow_int
            else:
                p = s = _2float(power=p)
                m = Fsum._pow_scalar
            return m, p, s, r

        _Pow, p, s, r = _Pow4(power)
        if p:  # and xs:
            op   =  which.__name__
            _flt =  float
            _Fs  =  Fsum
            _isa =  isinstance
            _pow =  self._pow_2_3

            def _P(X):
                f = _Pow(X, p, power, op, **raiser_RESIDUAL)
                return f._ps if _isa(f, _Fs) else (f,)

            def _p(x):
                x = _flt(x)
                f = _pow(x, s, power, op, **raiser_RESIDUAL)
                if f and r:
                    f *= _pow(x, r, power, op, **raiser_RESIDUAL)
                return f

            f = self._facc(xs, origin=1, _X=_P, _x=_p)
        else:
            f = self._facc_scalar_(float(len(xs)))  # x**0 == 1
        return f

    def _facc_scalar(self, xs, **up):
        '''(INTERNAL) Accumulate all C{xs}, known to be scalar.
        '''
        if xs:
            _ = self._ps_acc(self._ps, xs, **up)
        return self

    def _facc_scalar_(self, *xs, **up):
        '''(INTERNAL) Accumulate all positional C{xs}, known to be scalar.
        '''
        if xs:
            _ = self._ps_acc(self._ps, xs, **up)
        return self

#   def _facc_up(self, up=True):
#       '''(INTERNAL) Update the C{partials}, by removing
#          and re-accumulating the final C{partial}.
#       '''
#       ps = self._ps
#       while len(ps) > 1:
#           p = ps.pop()
#           if p:
#               n = self._n
#               _ = self._ps_acc(ps, (p,), up=False)
#               self._n = n
#               break
#       return self._update() if up else self

    def fadd(self, xs=()):
        '''Add an iterable's items to this instance.

           @arg xs: Iterable of items to add (each C{scalar}
                    or an L{Fsum} or L{Fsum2Tuple} instance).

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

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

           @raise TypeError: An invalid B{C{xs}} item.

           @raise ValueError: Invalid or non-finite B{C{xs}} value.
        '''
        if _isFsumTuple(xs):
            self._facc_scalar(xs._ps)
        elif isscalar(xs):  # for backward compatibility
            self._facc_scalar_(_2float(x=xs))  # PYCHOK no cover
        elif xs:  # _xiterable(xs)
            self._facc(xs)
        return self

    def fadd_(self, *xs):
        '''Add all positional items to this instance.

           @arg xs: Values to add (each C{scalar} or an L{Fsum}
                    or L{Fsum2Tuple} instance), all positional.

           @see: Method L{Fsum.fadd} for further details.
        '''
        return self._facc_1(xs)

    def _fadd(self, other, op, **up):  # in .fmath.Fhorner
        '''(INTERNAL) Apply C{B{self} += B{other}}.
        '''
        if not self._ps:  # new Fsum(x)
            self._fset(other, op=op, **up)
        elif _isFsumTuple(other):
            self._facc_scalar(other._ps, **up)
        elif self._scalar(other, op):
            self._facc_scalar_(other, **up)
        return self

    fcopy   =   copy  # for backward compatibility
    fdiv    = __itruediv__
    fdivmod = __divmod__

    def _fdivmod2(self, other, op, **raiser_RESIDUAL):
        '''(INTERNAL) Apply C{B{self} %= B{other}} and return a L{DivMod2Tuple}.
        '''
        # 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._truediv(other, op, **raiser_RESIDUAL).floor
        if q:  # == float // other == floor(float / other)
            self -= Fsum(q) * other  # NOT 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:
#               raise self._Error(op, other, _AssertionError, txt__=signOf)
        return DivMod2Tuple(q, self)  # q is C{int} in Python 3+, but C{float} in Python 2-

    def _fhorner(self, x, cs, op):  # in .fmath
        '''(INTERNAL) Add an L{Fhorner} evaluation of polynomial
           M{sum(cs[i] * x**i for i=0..len(cs)-1)}.
        '''
        if _xiterablen(cs):
            H = Fsum(name__=self._fhorner)
            if _isFsumTuple(x):
                _mul = H._mul_Fsum
            else:
                _mul = H._mul_scalar
                x = _2float(x=x)
            if len(cs) > 1 and x:
                for c in reversed(cs):
                    H._fset_ps(_mul(x, op))
                    H._fadd(c, op, up=False)
            else:  # x == 0
                H = cs[0]
            self._fadd(H, op)
        return self

    def _finite(self, other, op=None):
        '''(INTERNAL) Return B{C{other}} if C{finite}.
        '''
        if _isfinite(other):
            return other
        raise ValueError(_not_finite_) if op is None else \
             self._Error(op, other, _ValueError, txt=_not_finite_)

    def fint(self, name=NN, **raiser_RESIDUAL):
        '''Return this instance' current running sum as C{integer}.

           @kwarg name: Optional, overriding C{B{name}="fint"} (C{str}).
           @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore
                         L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}
                         to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

           @return: The C{integer} sum (L{Fsum}) if this instance C{is_integer}
                    with a zero or insignificant I{integer} residual.

           @raise ResidualError: Non-zero, significant residual or invalid
                                 B{C{RESIDUAL}}.

           @see: Methods L{Fsum.fint2}, L{Fsum.int_float} and L{Fsum.is_integer}.
        '''
        i, r = self._fint2
        if r:
            R = self._raiser(r, i, **raiser_RESIDUAL)
            if R:
                t = _stresidual(_integer_, r, **R)
                raise ResidualError(_integer_, i, txt=t)
        return _Psum_(i, name=_name__(name, name__=self.fint))

    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
    def _fint2(self):  # see ._fset
        '''(INTERNAL) Get 2-tuple (C{int}, I{integer} residual).
        '''
        s, r = self._fprs2
        i = int(s)
        n = len(self._ps)
        r = self._ps_1sum(i) if r and n > 1 else float(s - i)
        return i, (r or INT0)  # Fsum2Tuple?

    @_fint2.setter_  # PYCHOK setter_underscore!
    def _fint2(self, s):
        '''(INTERNAL) Replace the C{_fint2} value.
        '''
        i = int(s)
        return i, ((s - i) 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: This 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 (f - s) > r:  # f > (s + r)
            f -= 1
        return f  # _floor(self._n_d)

#   ffloordiv = __ifloordiv__  # for naming consistency
#   floordiv  = __floordiv__   # for naming consistency

    def _floordiv(self, other, op, **raiser_RESIDUAL):  # rather _ffloordiv?
        '''Apply C{B{self} //= B{other}}.
        '''
        q = self._ftruediv(other, op, **raiser_RESIDUAL)  # == self
        return self._fset(q.floor)  # floor(q)

    fmul = __imul__

    def _fmul(self, other, op):
        '''(INTERNAL) Apply C{B{self} *= B{other}}.
        '''
        if _isFsumTuple(other):
            if len(self._ps) != 1:
                f = self._mul_Fsum(other, op)
            elif len(other._ps) != 1:  # and len(self._ps) == 1
                f = other._mul_scalar(self._ps[0], op)
            else:  # len(other._ps) == len(self._ps) == 1
                f = self._finite(self._ps[0] * other._ps[0])
        else:
            s = self._scalar(other, op)
            f = self._mul_scalar(s, op)
        return self._fset(f)  # n=len(self) + 1

    def fover(self, over, **raiser_RESIDUAL):
        '''Apply C{B{self} /= B{over}} and summate.

           @arg over: An L{Fsum} or C{scalar} denominator.
           @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore
                         L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}
                         to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

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

           @raise ResidualError: Non-zero, significant residual or invalid
                                 B{C{RESIDUAL}}.

           @see: Methods L{Fsum.fsum} and L{Fsum.__itruediv__}.
        '''
        return float(self.fdiv(over, **raiser_RESIDUAL)._fprs)

    fpow = __ipow__

    def _fpow(self, other, op, *mod, **raiser_RESIDUAL):
        '''Apply C{B{self} **= B{other}}, optional B{C{mod}} or C{None}.
        '''
        if mod:
            if mod[0] is not None:  # == 3-arg C{pow}
                f = self._pow_2_3(self, other, other, op, *mod, **raiser_RESIDUAL)
            elif self.is_integer():
                # return an exact C{int} for C{int}**C{int}
                i, _ = self._fint2  # assert _ == 0
                x, r = _2scalar2(other)  # C{int}, C{float} or other
                f = _Psum_(i)._pow_Fsum(other, op, **raiser_RESIDUAL) if r else \
                    self._pow_2_3(i, x, other, op, **raiser_RESIDUAL)
            else:  # mod[0] is None, power(self, other)
                f =  self._pow(other, other, op, **raiser_RESIDUAL)
        else:  # pow(self, other)
            f = self._pow(other, other, op, **raiser_RESIDUAL)
        return self._fset(f)  # n=max(len(self), 1)

    @Property
    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+.
        '''
        s, _ = self._fprs2
        return s  # ._fprs2.fsum

    @_fprs.setter_  # PYCHOK setter_underscore!
    def _fprs(self, s):
        '''(INTERNAL) Replace the C{_fprs} value.
        '''
        return s

    @Property
    def _fprs2(self):
        '''(INTERNAL) Get and cache this instance' precision
           running sum and residual (L{Fsum2Tuple}).
        '''
        ps = self._ps
        n  = len(ps) - 2
        if n > 0:  # len(ps) > 2
            s = _psum(ps)
            n =  len(ps) - 2
            if n > 0:
                r = self._ps_1sum(s)
                return Fsum2Tuple(*_s_r(s, r))
        if n == 0:  # len(ps) == 2
            s, r  = _s_r(*_2sum(*ps))
            ps[:] = (r, s) if r else (s,)
        elif ps:  # len(ps) == 1
            s, r = ps[0], INT0
        else:  # len(ps) == 0
            s, r = _0_0, INT0
            ps[:] = s,
        # assert self._ps is ps
        return Fsum2Tuple(s, r)

    @_fprs2.setter_  # PYCHOK setter_underscore!
    def _fprs2(self, s_r):
        '''(INTERNAL) Replace the C{_fprs2} value.
        '''
        return Fsum2Tuple(s_r)

    def fset_(self, *xs):
        '''Replace this instance' value with all positional items.

           @arg xs: Optional, new values (each C{scalar} or
                    an L{Fsum} or L{Fsum2Tuple} instance),
                    all positional.

           @return: This instance, replaced (C{Fsum}).

           @see: Method L{Fsum.fadd} for further details.
        '''
        f = xs[0] if len(xs) == 1 else (
            Fsum(*xs) if xs else _0_0)
        return self._fset(f)

    def _fset(self, other, n=0, up=True, **op):
        '''(INTERNAL) Overwrite this instance with an other or a C{scalar}.
        '''
        if other is self:
            pass  # from ._fmul, ._ftruediv and ._pow_0_1
        elif _isFsumTuple(other):
            self._ps[:] = other._ps
            self._n     = n or other._n
#           self._copy_RESIDUAL(other)
            if up:  # 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 = float(self._finite(other, **op)) if op else other
            self._ps[:] = s,
            self._n     = n or 1
            if up:  # Property _fint2, _fprs and _fprs2 all have
                # @.setter_underscore and NOT @.setter because the
                # latter's _fset zaps the value set by @.setter
                self._fint2 = s
                self._fprs  = s
                self._fprs2 = s, INT0
                # assert self._fprs is s
        else:  # PYCHOK no cover
            op = _xkwds_get1(op, op=_fset_op_)
            raise self._Error(op, other, _TypeError)
        return self

    def _fset_ps(self, other):  # in .fmath
        '''(INTERNAL) Set partials from a known C{scalar}, L{Fsum} or L{Fsum2Tuple}.
        '''
        return self._fset(other, up=False)

    def fsub(self, xs=()):
        '''Subtract an iterable's items from this instance.

           @see: Method L{Fsum.fadd} for further details.
        '''
        return self._facc_neg(xs)

    def fsub_(self, *xs):
        '''Subtract all positional items from this instance.

           @see: Method L{Fsum.fadd_} for further details.
        '''
        return self._fsub(xs[0], _sub_op_) if len(xs) == 1 else \
               self._facc_neg(xs, origin=1)

    def _fsub(self, other, op):
        '''(INTERNAL) Apply C{B{self} -= B{other}}.
        '''
        if _isFsumTuple(other):
            if other is self:  # or other._fprs2 == self._fprs2:
                self._fset(_0_0, n=len(self) * 2)
            elif other._ps:
                self._facc_scalar(other._ps_neg)
        elif self._scalar(other, op):
            self._facc_scalar_(-other)
        return self

    def fsum(self, xs=()):
        '''Add an iterable's items, summate and return the
           current precision running sum.

           @arg xs: Iterable of items to add (each item C{scalar}
                    or an L{Fsum} or L{Fsum2Tuple} instance).

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

           @see: Method L{Fsum.fadd}.

           @note: Accumulation can continue after summation.
        '''
        return self._facc(xs)._fprs

    def fsum_(self, *xs):
        '''Add any positional items, summate and return the
           current precision running sum.

           @arg xs: Items to add (each C{scalar} or an L{Fsum}
                    or L{Fsum2Tuple} instance), all positional.

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

           @see: Methods L{Fsum.fsum}, L{Fsum.Fsum_} and L{Fsum.fsumf_}.
        '''
        return self._facc_1(xs)._fprs

    @property_RO
    def _Fsum(self):  # like L{Fsum2Tuple._Fsum}, for C{_2floats}, .fstats
        return self  # NOT @Property_RO, see .copy and ._copy_2

    def Fsum_(self, *xs, **name):
        '''Like method L{Fsum.fsum_} but returning a named L{Fsum}.

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

           @return: Copy of this updated instance (L{Fsum}).
        '''
        return self._facc_1(xs)._copy_2(self.Fsum_, **name)

    def Fsum2Tuple_(self, *xs, **name):
        '''Like method L{Fsum.fsum_} but returning a named L{Fsum2Tuple}.

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

           @return: Precision running sum (L{Fsum2Tuple}).
        '''
        return Fsum2Tuple(self._facc_1(xs)._fprs2, **name)

    def fsum2(self, xs=(), **name):
        '''Add an iterable's items, summate and return the
           current precision running sum I{and} the C{residual}.

           @arg xs: Iterable of items to add (each item C{scalar}
                    or an L{Fsum} or L{Fsum2Tuple} instance).
           @kwarg name: Optional C{B{name}=NN} (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_}
        '''
        t = self._facc(xs)._fprs2
        return t.dup(name=name) if name else t

    def fsum2_(self, *xs):
        '''Add any positional items, summate and return the current
           precision running sum and the I{differential}.

           @arg xs: Values to add (each C{scalar} or an L{Fsum} or
                    L{Fsum2Tuple} instance), all positional.

           @return: 2Tuple C{(fsum, delta)} with the current, precision
                    running C{fsum} like method L{Fsum.fsum} and C{delta},
                    the difference with previous running C{fsum}, C{float}.

           @see: Methods L{Fsum.fsum_} and L{Fsum.fsum}.
        '''
        return self._fsum2(xs, self._facc_1)

    def _fsum2(self, xs, _facc, **origin):
        '''(INTERNAL) Helper for L{Fsum.fsum2_} and L{Fsum.fsum2f_}.
        '''
        p, q = self._fprs2
        if xs:
            s, r = _facc(xs, **origin)._fprs2
            return s, _2delta(s - p, r - q)  # _fsum(_1primed((s, -p, r, -q))
        else:
            return p, _0_0

    def fsumf_(self, *xs):
        '''Like method L{Fsum.fsum_} iff I{all} C{B{xs}} are I{known to be scalar}.
        '''
        return self._facc_scalar(xs)._fprs

    def Fsumf_(self, *xs):
        '''Like method L{Fsum.Fsum_} iff I{all} C{B{xs}} are I{known to be scalar}.
        '''
        return self._facc_scalar(xs)._copy_2(self.Fsumf_)

    def fsum2f_(self, *xs):
        '''Like method L{Fsum.fsum2_} iff I{all} C{B{xs}} are I{known to be scalar}.
        '''
        return self._fsum2(xs, self._facc_scalar, origin=1)

#   ftruediv = __itruediv__   # for naming consistency?

    def _ftruediv(self, other, op, **raiser_RESIDUAL):
        '''(INTERNAL) Apply C{B{self} /= B{other}}.
        '''
        n = _1_0
        if _isFsumTuple(other):
            if other is self or self == other:
                return self._fset(n, n=len(self))
            d, r = other._fprs2
            if r:
                R = self._raiser(r, d, **raiser_RESIDUAL)
                if R:
                    raise self._ResidualError(op, other, r, **R)
                d, n = other.as_integer_ratio()
        else:
            d = self._scalar(other, op)
        try:
            s = n / d
        except Exception as X:
            raise self._ErrorX(X, op, other)
        f = self._mul_scalar(s, _mul_op_)  # handles 0, INF, NAN
        return self._fset(f)

    @property_RO
    def imag(self):
        '''Get the C{imaginary} part of this instance (C{0.0}, always).

           @see: Property L{Fsum.real}.
        '''
        return _0_0

    def int_float(self, **raiser_RESIDUAL):
        '''Return this instance' current running sum as C{int} or C{float}.

           @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore
                         L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}
                         to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

           @return: This C{integer} sum if this instance C{is_integer},
                    otherwise return the C{float} sum if the residual is
                    zero or not significant.

           @raise ResidualError: Non-zero, significant residual or invalid
                                 B{C{RESIDUAL}}.

           @see: Methods L{Fsum.fint}, L{Fsum.fint2}, L{Fsum.RESIDUAL} and
                 property L{Fsum.as_iscalar}.
        '''
        s, r = self._fint2
        if r:
            s, r = self._fprs2
            if r:  # PYCHOK no cover
                R = self._raiser(r, s, **raiser_RESIDUAL)
                if R:
                    t = _stresidual(_non_zero_, r, **R)
                    raise ResidualError(int_float=s, txt=t)
            s = float(s)
        return s

    def is_exact(self):
        '''Is this instance' running C{fsum} considered to be exact?
           (C{bool}), C{True} only if the C{residual is }L{INT0}.
        '''
        return self.residual is INT0

    def is_integer(self):
        '''Is this instance' running sum C{integer}? (C{bool}).

           @see: Methods L{Fsum.fint}, L{Fsum.fint2} and L{Fsum.is_scalar}.
        '''
        _, 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 is_scalar(self, **raiser_RESIDUAL):
        '''Is this instance' running sum C{scalar} without residual or with
           a residual I{ratio} not exceeding the RESIDUAL threshold?

           @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore
                         L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}
                         to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

           @return: C{True} if this instance' non-zero residual C{ratio} exceeds
                    the L{RESIDUAL<Fsum.RESIDUAL>} threshold (C{bool}).

           @raise ResidualError: Non-zero, significant residual or invalid
                                 B{C{RESIDUAL}}.

           @see: Method L{Fsum.RESIDUAL}, L{Fsum.is_integer} and property
                 L{Fsum.as_iscalar}.
        '''
        s, r = self._fprs2
        return False if r and self._raiser(r, s, **raiser_RESIDUAL) else True

    def _mul_Fsum(self, other, op=_mul_op_):  # in .fmath.Fhorner
        '''(INTERNAL) Return C{B{self} * B{other}} as L{Fsum} or C{0}.
        '''
        # assert _isFsumTuple(other)
        if self._ps and other._ps:
            f =  self._ps_mul(op, *other._ps)  # NO .as_iscalar!
        else:
            f = _0_0
        return f

    def _mul_scalar(self, factor, op):  # in .fmath.Fhorner
        '''(INTERNAL) Return C{B{self} * scalar B{factor}} as L{Fsum}, C{0.0} or C{self}.
        '''
        # assert isscalar(factor)
        if self._ps and self._finite(factor, op):
            f =  self      if factor == _1_0   else (
                 self._neg if factor == _N_1_0 else
                 self._ps_mul(op, factor).as_iscalar)
        else:
            f = _0_0
        return f

#   @property_RO
#   def _n_d(self):
#       n, d = self.as_integer_ratio()
#       return n / d

    @property_RO
    def _neg(self):
        '''(INTERNAL) Return C{Fsum(-self)} or scalar C{NEG0}.
        '''
        return _Psum(self._ps_neg) if self._ps else NEG0

    @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, *mod, **raiser_RESIDUAL):
        '''Return C{B{self}**B{x}} as L{Fsum}.

           @arg x: The exponent (C{scalar} or L{Fsum}).
           @arg mod: Optional modulus (C{int} or C{None}) for the 3-argument
                     C{pow(B{self}, B{other}, B{mod})} version.
           @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore
                         L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}
                         to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

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

           @raise ResidualError: Non-zero, significant residual or invalid
                                 B{C{RESIDUAL}}.

           @note: If B{C{mod}} is given and C{None}, the result will be an
                  C{integer} L{Fsum} provided this instance C{is_integer}
                  or set to C{integer} by an L{Fsum.fint} call.

           @see: Methods L{Fsum.__ipow__}, L{Fsum.fint}, L{Fsum.is_integer}
                 and L{Fsum.root}.
        '''
        f = self._copy_2(self.pow)
        return f._fpow(x, _pow_op_, *mod, **raiser_RESIDUAL)  # f = pow(f, x, *mod)

    def _pow(self, other, unused, op, **raiser_RESIDUAL):
        '''Return C{B{self} ** B{other}}.
        '''
        if _isFsumTuple(other):
            f = self._pow_Fsum(other, op, **raiser_RESIDUAL)
        elif self._scalar(other, op):
            x = self._finite(other, op)
            f = self._pow_scalar(x, other, op, **raiser_RESIDUAL)
        else:
            f = self._pow_0_1(0, other)
        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 isint(other) and self.is_integer() else _1_0)

    def _pow_2_3(self, b, x, other, op, *mod, **raiser_RESIDUAL):
        '''(INTERNAL) 2-arg C{pow(B{b}, scalar B{x})} and 3-arg C{pow(B{b},
           B{x}, int B{mod} or C{None})}, embellishing errors.
        '''

        if mod:  # b, x, mod all C{int}, unless C{mod} is C{None}
            m = mod[0]
            # assert _isFsumTuple(b)

            def _s(s, r):
                R = self._raiser(r, s, **raiser_RESIDUAL)
                if R:
                    raise self._ResidualError(op, other, r, mod=m, **R)
                return s

            b = _s(*(b._fprs2 if m is None else b._fint2))
            x = _s(*_2scalar2(x))

        try:
            # 0**INF == 0.0, 1**INF == 1.0, -1**2.3 == -(1**2.3)
            s = pow(b, x, *mod)
            if iscomplex(s):
                # neg**frac == complex in Python 3+, but ValueError in 2-
                raise ValueError(_strcomplex(s, b, x, *mod))
            return self._finite(s)
        except Exception as X:
            raise self._ErrorX(X, op, other, *mod)

    def _pow_Fsum(self, other, op, **raiser_RESIDUAL):
        '''(INTERNAL) Return C{B{self} **= B{other}} for C{_isFsumTuple(other)}.
        '''
        # assert _isFsumTuple(other)
        x, r = other._fprs2
        f = self._pow_scalar(x, other, op, **raiser_RESIDUAL)
        if f and r:
            f *= self._pow_scalar(r, other, op, **raiser_RESIDUAL)
        return f

    def _pow_int(self, x, other, op, **raiser_RESIDUAL):
        '''(INTERNAL) Return C{B{self} **= B{x}} for C{int B{x} >= 0}.
        '''
        # assert isint(x) and x >= 0
        ps = self._ps
        if len(ps) > 1:
            _mul_Fsum = Fsum._mul_Fsum
            if x > 4:
                p = self
                f = self if (x & 1) else _Psum_(_1_0)
                m = x >> 1  # // 2
                while m:
                    p = _mul_Fsum(p, p, op)  # p **= 2
                    if (m & 1):
                        f = _mul_Fsum(f, p, op)  # f *= p
                    m >>= 1  # //= 2
            elif x > 1:  # self**2, 3, or 4
                f = _mul_Fsum(self, self, op)
                if x > 2:  # self**3 or 4
                    p =  self if x < 4 else f
                    f = _mul_Fsum(f, p, op)
            else:  # self**1 or self**0 == 1 or _1_0
                f = self._pow_0_1(x, other)
        elif ps:  # self._ps[0]**x
            f = self._pow_2_3(ps[0], x, other, op, **raiser_RESIDUAL)
        else:  # PYCHOK no cover
            # 0**pos_int == 0, but 0**0 == 1
            f = 0 if x else 1
        return f

    def _pow_scalar(self, x, other, op, **raiser_RESIDUAL):
        '''(INTERNAL) Return C{self**B{x}} for C{scalar B{x}}.
        '''
        s, r = self._fprs2
        if r:
            # assert s != 0
            if isint(x, both=True):  # self**int
                x = int(x)
                y = abs(x)
                if y > 1:
                    f = self._pow_int(y, other, op, **raiser_RESIDUAL)
                    if x > 0:  # i.e. > 1
                        return f  # Fsum or scalar
                    # assert x < 0  # i.e. < -1
                    if _isFsum(f):
                        s, r = f._fprs2
                        if r:
                            return _1_Over(f, op, **raiser_RESIDUAL)
                    else:  # scalar
                        s = f
                    # use s**(-1) to get the CPython
                    # float_pow error iff s is zero
                    x = -1
                elif x < 0:  # self**(-1)
                    return _1_Over(self, op, **raiser_RESIDUAL)  # 1 / self
                else:  # self**1 or self**0
                    return self._pow_0_1(x, other)  # self, 1 or 1.0
            else:  # self**fractional
                R = self._raiser(r, s, **raiser_RESIDUAL)
                if R:
                    raise self._ResidualError(op, other, r, **R)
                n, d = self.as_integer_ratio()
                if abs(n) > abs(d):
                    n, d, x = d, n, (-x)
                s = n / d
        # assert isscalar(s) and isscalar(x)
        return self._pow_2_3(s, x, other, op, **raiser_RESIDUAL)

    def _ps_acc(self, ps, xs, up=True, **unused):
        '''(INTERNAL) Accumulate C{xs} known scalars into list C{ps}.
        '''
        n   =  0
        _2s = _2sum
        for x in (tuple(xs) if xs is ps else xs):
            # assert isscalar(x) and _isfinite(x)
            if x:
                i = 0
                for p in ps:
                    x, p = _2s(x, p)
                    if p:
                        ps[i] = p
                        i += 1
                ps[i:] = (x,) if x else ()
                n += 1
        if n:
            self._n += n
            # Fsum._ps_max = max(Fsum._ps_max, len(ps))
            if up:
                self._update()
        return ps

    def _ps_mul(self, op, *factors):
        '''(INTERNAL) Multiply this instance' C{partials} with
           each scalar C{factor} and accumulate into an C{Fsum}.
        '''
        def _pfs(ps, fs):
            if len(ps) < len(fs):
                ps, fs = fs, ps
            _fin = _isfinite
            for f in fs:
                for p in ps:
                    p *= f
                    yield p if _fin(p) else self._finite(p, op)

        return Fsum()._facc_scalar(_pfs(self._ps, factors), up=False)

    @property_RO
    def _ps_neg(self):
        '''(INTERNAL) Yield the partials, I{negated}.
        '''
        for p in self._ps:
            yield -p

    def _ps_1sum(self, *less):
        '''(INTERNAL) Return the partials sum, 1-primed C{less} some scalars.
        '''
        def _1pls(ps, ls):
            yield _1_0
            for p in ps:
                yield  p
            for p in ls:
                yield -p
            yield _N_1_0

        return _fsum(_1pls(self._ps, less))

    def _raiser(self, r, s, raiser=True, **RESIDUAL):
        '''(INTERNAL) Does ratio C{r / s} exceed the RESIDUAL threshold
           I{and} is residual C{r} I{non-zero} or I{significant} (for a
           negative respectively positive C{RESIDUAL} threshold)?
        '''
        if r and raiser:
            t = self._RESIDUAL
            if RESIDUAL:
                t = _threshold(t, **RESIDUAL)
            if t < 0 or (s + r) != s:
                q = (r / s) if s else s  # == 0.
                if fabs(q) > fabs(t):
                    return dict(ratio=q, R=t)
        return {}

    rdiv = __rtruediv__

    @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: The C{residual is INT0} iff 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 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}), default C{0.0}.

           @raise ResidualError: Invalid B{C{threshold}}.

           @note: L{ResidualError}s may be thrown if (1) the non-zero I{ratio}
                  C{residual / fsum} exceeds the given B{C{threshold}} and (2)
                  the C{residual} is non-zero and (3) I{significant} vs the
                  C{fsum}, i.e. C{(fsum + residual) != fsum} and (4) optional
                  keyword argument C{raiser=False} is missing.  Specify a
                  negative B{C{threshold}} for only non-zero C{residual}
                  testing without I{significant}.
        '''
        r = self._RESIDUAL
        if threshold:
            t = threshold[0]
            self._RESIDUAL = Fsum._RESIDUAL if t is None else (  # for ...
                           (_0_0 if t else _1_0) if isbool(t) else
                           _threshold(t))  # ... backward compatibility
        return r

    def _ResidualError(self, op, other, residual, **mod_R):
        '''(INTERNAL) Non-zero B{C{residual}} etc.
        '''
        def _p(mod=None, R=0, **unused):  # ratio=0
            return (_non_zero_ if R < 0 else _significant_) \
                               if mod is None else _integer_

        t = _stresidual(_p(**mod_R), residual, **mod_R)
        return self._Error(op, other, ResidualError, txt=t)

    def root(self, root, **raiser_RESIDUAL):
        '''Return C{B{self}**(1 / B{root})} as L{Fsum}.

           @arg root: The order (C{scalar} or L{Fsum}), non-zero.
           @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore
                         L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}
                         to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

           @return: The C{self ** (1 / B{root})} result (L{Fsum}).

           @raise ResidualError: Non-zero, significant residual or invalid
                                 B{C{RESIDUAL}}.

           @see: Method L{Fsum.pow}.
        '''
        x = _1_Over(root, _truediv_op_, **raiser_RESIDUAL)
        f =  self._copy_2(self.root)
        return f._fpow(x, f.name, **raiser_RESIDUAL)  # == pow(f, x)

    def _scalar(self, other, op, **txt):
        '''(INTERNAL) Return scalar C{other}.
        '''
        if isscalar(other):
            return other
        raise self._Error(op, other, _TypeError, **txt)  # _invalid_

    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
        r = (-r) if res else 0
        return _signOf(s, r)

    def toRepr(self, **lenc_prec_sep_fmt):  # PYCHOK signature
        '''Return this C{Fsum} instance as representation.

           @kwarg lenc_prec_sep_fmt: Optional keyword arguments
                       for method L{Fsum.toStr}.

           @return: This instance (C{repr}).
        '''
        return Fmt.repr_at(self, self.toStr(**lenc_prec_sep_fmt))

    def toStr(self, lenc=True, **prec_sep_fmt):  # PYCHOK signature
        '''Return this C{Fsum} instance as string.

           @kwarg lenc: If C{True} include the current C{[len]} of this
                        L{Fsum} enclosed in I{[brackets]} (C{bool}).
           @kwarg prec_sep_fmt: Optional keyword arguments for method
                       L{Fsum2Tuple.toStr}.

           @return: This instance (C{str}).
        '''
        p = self.classname
        if lenc:
            p = Fmt.SQUARE(p, len(self))
        n = _enquote(self.name, white=_UNDER_)
        t =  self._fprs2.toStr(**prec_sep_fmt)
        return NN(p, _SPACE_, n, t)

    def _truediv(self, other, op, **raiser_RESIDUAL):
        '''(INTERNAL) Return C{B{self} / B{other}} as an L{Fsum}.
        '''
        f = self._copy_2(self.__truediv__)
        return f._ftruediv(other, op, **raiser_RESIDUAL)

    def _update(self, updated=True):  # see ._fset
        '''(INTERNAL) Zap all cached C{Property_RO} values.
        '''
        if updated:
            _pop = self.__dict__.pop
            for p in _ROs:
                _ = _pop(p, None)
#           Fsum._fint2._update(self)
#           Fsum._fprs ._update(self)
#           Fsum._fprs2._update(self)
        return self  # for .fset_

_ROs = _allPropertiesOf_n(3, Fsum, Property_RO)  # PYCHOK see Fsum._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 DivMod2Tuple(_NamedTuple):
    '''2-Tuple C{(div, mod)} with the quotient C{div} and remainder
       C{mod} results of a C{divmod} operation.

       @note: Quotient C{div} an C{int} in Python 3+ but a C{float}
              in Python 2-.  Remainder C{mod} an L{Fsum} instance.
    '''
    _Names_ = (_div_,     _mod_)
    _Units_ = (_Float_Int, Fsum)


class Fsum2Tuple(_NamedTuple):  # in .fstats
    '''2-Tuple C{(fsum, residual)} with the precision running C{fsum}
       and the C{residual}, the sum of the remaining partials.  Each
       item is 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)

    def __abs__(self):  # in .fmath
        return self._Fsum.__abs__()

    def __bool__(self):  # PYCHOK Python 3+
        return bool(self._Fsum)

    def __eq__(self, other):
        return self._other_op(other, self.__eq__)

    def __float__(self):
        return self._Fsum.__float__()

    def __ge__(self, other):
        return self._other_op(other, self.__ge__)

    def __gt__(self, other):
        return self._other_op(other, self.__gt__)

    def __le__(self, other):
        return self._other_op(other, self.__le__)

    def __lt__(self, other):
        return self._other_op(other, self.__lt__)

    def __int__(self):
        return self._Fsum.__int__()

    def __ne__(self, other):
        return self._other_op(other, self.__ne__)

    def __neg__(self):
        return self._Fsum.__neg__()

    __nonzero__ = __bool__  # Python 2-

    def __pos__(self):
        return self._Fsum.__pos__()

    def as_integer_ratio(self):
        '''Return this instance as the ratio of 2 integers.

           @see: Method L{Fsum.as_integer_ratio} for further details.
        '''
        return self._Fsum.as_integer_ratio()

    @property_RO
    def _fint2(self):
        return self._Fsum._fint2

    @property_RO
    def _fprs2(self):
        return self._Fsum._fprs2

    @Property_RO
    def _Fsum(self):  # this C{Fsum2Tuple} as L{Fsum}, in .fstats
        s, r = _s_r(*self)
        ps = (r, s) if r else (s,)
        return _Psum(ps, name=self.name)

    def Fsum_(self, *xs, **name_RESIDUAL):
        '''Return this C{Fsum2Tuple} as an L{Fsum} plus some C{xs}.
        '''
        f = _Psum(self._Fsum._ps, **name_RESIDUAL)
        return f._facc_1(xs, up=False) if xs else f

    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()

    def _mul_scalar(self, other, op):  # for Fsum._fmul
        return self._Fsum._mul_scalar(other, op)

    @property_RO
    def _n(self):
        return self._Fsum._n

    def _other_op(self, other, which):
        C, s = (tuple, self) if isinstance(other, tuple) else (Fsum, self._Fsum)
        return getattr(C, which.__name__)(s, other)

    @property_RO
    def _ps(self):
        return self._Fsum._ps

    @property_RO
    def _ps_neg(self):
        return self._Fsum._ps_neg

    def signOf(self, **res):
        '''Like method L{Fsum.signOf}.
        '''
        return self._Fsum.signOf(**res)

    def toStr(self, fmt=Fmt.g, **prec_sep):  # PYCHOK signature
        '''Return this L{Fsum2Tuple} as string (C{str}).

           @kwarg fmt: Optional C{float} format (C{letter}).
           @kwarg prec_sep: Optional keyword arguments for function
                       L{fstr<streprs.fstr>}.
        '''
        return Fmt.PAREN(fstr(self, fmt=fmt, strepr=str, force=False, **prec_sep))

_Fsum_Fsum2Tuple_types = Fsum, Fsum2Tuple  # PYCHOK lines


class ResidualError(_ValueError):
    '''Error raised for a division, power or root operation of
       an L{Fsum} instance with a C{residual} I{ratio} exceeding
       the L{RESIDUAL<Fsum.RESIDUAL>} threshold.

       @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 = _sum = _fsum  # PYCHOK exported
except ImportError:
    _sum = sum  # Fsum(NAN) exception fall-back, in .elliptic

    def _fsum(xs):
        '''(INTERNAL) Precision summation, Python 2.5-.
        '''
        F = Fsum()
        F.name = _fsum.__name__
        return F._facc(xs, up=False)._fprs2.fsum


def fsum(xs, floats=False):
    '''Precision floating point summation based on/like Python's C{math.fsum}.

       @arg xs: Iterable of items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple}
                instance).
       @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to
                      be scalar} (C{bool}).

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

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

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

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

       @note: Exception 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(xs if floats is True else _2floats(xs)) if xs else _0_0  # PYCHOK yield


def fsum_(*xs, **floats):
    '''Precision floating point summation of all positional items.

       @arg xs: Items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} instance),
                all positional.
       @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to
                      be scalar} (C{bool}).

       @see: Function L{fsum<fsums.fsum>} for further details.
    '''
    return _fsum(xs if _xkwds_get1(floats, floats=False) is True else
                _2floats(xs, origin=1)) if xs else _0_0  # PYCHOK yield


def fsumf_(*xs):
    '''Precision floating point summation iff I{all} C{B{xs}} items are I{known to be scalar}.

       @see: Function L{fsum_<fsums.fsum_>} for further details.
    '''
    return _fsum(xs) if xs else _0_0


def fsum1(xs, floats=False):
    '''Precision floating point summation, 1-primed.

       @arg xs: Iterable of items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple}
                instance).
       @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to
                      be scalar} (C{bool}).

       @see: Function L{fsum<fsums.fsum>} for further details.
    '''
    return _fsum(_1primed(xs if floats is True else _2floats(xs))) if xs else _0_0  # PYCHOK yield


def fsum1_(*xs, **floats):
    '''Precision floating point summation, 1-primed of all positional items.

       @arg xs: Items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} instance),
                all positional.
       @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to
                      be scalar} (C{bool}).

       @see: Function L{fsum_<fsums.fsum_>} for further details.
    '''
    return _fsum(_1primed(xs if _xkwds_get1(floats, floats=False) is True else
                         _2floats(xs, origin=1))) if xs else _0_0  # PYCHOK yield


def fsum1f_(*xs):
    '''Precision floating point summation iff I{all} C{B{xs}} items are I{known to be scalar}.

       @see: Function L{fsum_<fsums.fsum_>} for further details.
    '''
    return _fsum(_1primed(xs)) if xs else _0_0


if __name__ == '__main__':

    # usage: [env _psum=fsum] python3 -m pygeodesy.fsums

    if _getenv(_psum.__name__, NN) == _fsum.__name__:
        _psum = _fsum

    def _test(n):
        # copied from Hettinger, see L{Fsum} reference
        from pygeodesy import frandoms, printf

        printf(_fsum.__name__, end=_COMMASPACE_)
        printf(_psum.__name__, end=_COMMASPACE_)

        F = Fsum()
        if F.is_math_fsum():
            for t in frandoms(n, seeded=True):
                assert float(F.fset_(*t)) == _fsum(t)
                printf(_DOT_, end=NN)
        printf(NN)

    _test(128)

# **) MIT License
#
# Copyright (C) 2016-2024 -- 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.
