Sergey B Kirpichev
1211 lines
40 KiB
Python
1211 lines
40 KiB
Python
import numbers
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import sys
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from . import function_docs
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from .libmp import (MPQ, MPZ, ComplexResult, dps_to_prec, finf, fnan, fninf,
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format_mpc, format_mpf, from_Decimal, from_float, from_int,
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from_man_exp, from_npfloat, from_rational, from_str, fzero,
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int_types, mpc_abs, mpc_add, mpc_add_mpf, mpc_conjugate,
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mpc_div, mpc_div_mpf, mpc_hash, mpc_is_inf, mpc_is_nonzero,
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mpc_mpf_div, mpc_mpf_sub, mpc_mul, mpc_mul_int,
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mpc_mul_mpf, mpc_neg, mpc_pos, mpc_pow, mpc_pow_int,
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mpc_pow_mpf, mpc_sub, mpc_sub_mpf, mpc_to_complex,
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mpc_to_str, mpf_abs, mpf_add, mpf_cmp, mpf_div, mpf_eq,
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mpf_ge, mpf_gt, mpf_hash, mpf_le, mpf_lt, mpf_mod, mpf_mul,
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mpf_mul_int, mpf_neg, mpf_pos, mpf_pow, mpf_pow_int,
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mpf_rdiv_int, mpf_sub, mpf_sum, normalize, prec_to_dps,
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round_nearest, to_fixed, to_float, to_int, to_man_exp,
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to_rational, to_str)
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new = object.__new__
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class mpnumeric:
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"""Base class for mpf and mpc."""
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__slots__ = []
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def __new__(cls, val):
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raise NotImplementedError
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# pickling support
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def _make_mpf(x):
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from mpmath import mp
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return mp.mpf(x)
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def _make_mpc(x, y):
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from mpmath import mp
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return mp.mpc(x, y)
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class _mpf(mpnumeric):
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"""
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An mpf instance holds a real-valued floating-point number. mpf:s
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work analogously to Python floats, but support arbitrary-precision
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arithmetic.
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"""
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__slots__ = ['_mpf_']
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def __new__(cls, val=fzero, **kwargs):
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"""A new mpf can be created from a Python float, an int, a
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or a decimal string representing a number in floating-point
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format."""
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prec, rounding = cls.context._prec_rounding
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base = 0
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if kwargs:
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prec = kwargs.get('prec', prec)
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if 'dps' in kwargs:
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prec = dps_to_prec(kwargs['dps'])
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rounding = kwargs.get('rounding', rounding)
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base = kwargs.get('base', base)
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v = new(cls)
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if type(val) is cls:
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val = val._mpf_
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elif type(val) is tuple:
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if len(val) == 4:
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val = val[0], MPZ(val[1]), *val[2:]
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elif len(val) == 2:
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v._mpf_ = from_man_exp(val[0], val[1], prec, rounding)
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return v
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else:
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raise ValueError
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elif isinstance(val, str):
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val = from_str(val, prec, rounding, base)
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else:
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val = cls.mpf_convert_arg(val, prec, rounding)
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v._mpf_ = mpf_pos(val, prec, rounding)
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return v
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@classmethod
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def mpf_convert_arg(cls, x, prec, rounding):
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if isinstance(x, int_types): return from_int(x)
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if isinstance(x, float): return from_float(x)
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if isinstance(x, cls.context.constant): return x.func(prec, rounding)
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if hasattr(x, '_mpf_'): return x._mpf_
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if hasattr(x, '_mpmath_'):
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t = cls.context.convert(x._mpmath_(prec, rounding))
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if hasattr(t, '_mpf_'):
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return t._mpf_
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if hasattr(x, '_mpi_'):
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a, b = x._mpi_
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if a == b:
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return a
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raise ValueError("can only create mpf from zero-width interval")
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if isinstance(x, numbers.Rational): return from_rational(x.numerator,
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x.denominator,
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prec, rounding)
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if type(x).__module__ == 'decimal':
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return from_Decimal(x, prec, rounding)
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raise TypeError("cannot create mpf from " + repr(x))
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@classmethod
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def mpf_convert_rhs(cls, x):
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if isinstance(x, int_types): return from_int(x)
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if isinstance(x, float): return from_float(x)
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if isinstance(x, complex_types): return cls.context.mpc(x)
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if isinstance(x, MPQ):
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p, q = x.numerator, x.denominator
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return from_rational(p, q, cls.context.prec)
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if hasattr(x, '_mpf_'): return x._mpf_
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if hasattr(x, '_mpmath_'):
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t = cls.context.convert(x._mpmath_(*cls.context._prec_rounding))
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if hasattr(t, '_mpf_'):
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return t._mpf_
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return t
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return NotImplemented
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@classmethod
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def mpf_convert_lhs(cls, x):
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x = cls.mpf_convert_rhs(x)
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if type(x) is tuple:
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return cls.context.make_mpf(x)
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return x
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man_exp = property(lambda self: to_man_exp(self._mpf_, signed=False))
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man = property(lambda self: self.man_exp[0])
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exp = property(lambda self: self.man_exp[1])
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bc = property(lambda self: self.man.bit_length())
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real = property(lambda self: self)
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imag = property(lambda self: self.context.zero)
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conjugate = lambda self: self
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def as_integer_ratio(self):
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return to_rational(self._mpf_)
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def __reduce__(self): return _make_mpf, (self._mpf_,)
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def __repr__(s):
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if s.context.pretty:
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return str(s)
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return "mpf('%s')" % to_str(s._mpf_, s.context._repr_digits)
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def __str__(s): return to_str(s._mpf_, s.context._str_digits)
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def __hash__(s): return mpf_hash(s._mpf_)
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def __int__(s): return int(to_int(s._mpf_))
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def __float__(s): return to_float(s._mpf_, rnd=s.context._prec_rounding[1])
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def __bool__(s): return s._mpf_ != fzero
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def __abs__(s):
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cls, new, (prec, rounding) = s._ctxdata
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v = new(cls)
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v._mpf_ = mpf_abs(s._mpf_, prec, rounding)
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return v
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def __pos__(s):
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cls, new, (prec, rounding) = s._ctxdata
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v = new(cls)
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v._mpf_ = mpf_pos(s._mpf_, prec, rounding)
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return v
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def __neg__(s):
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cls, new, (prec, rounding) = s._ctxdata
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v = new(cls)
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v._mpf_ = mpf_neg(s._mpf_, prec, rounding)
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return v
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def _cmp(s, t, func):
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if hasattr(t, '_mpf_'):
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t = t._mpf_
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else:
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t = s.mpf_convert_rhs(t)
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if t is NotImplemented:
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return t
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return func(s._mpf_, t)
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def __lt__(s, t): return s._cmp(t, mpf_lt)
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def __gt__(s, t): return s._cmp(t, mpf_gt)
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def __le__(s, t): return s._cmp(t, mpf_le)
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def __ge__(s, t): return s._cmp(t, mpf_ge)
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def __eq__(self, other):
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mpf, new, (prec, rounding) = self._ctxdata
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sval = self._mpf_
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if hasattr(other, '_mpf_'):
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tval = other._mpf_
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return mpf_eq(sval, tval)
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if hasattr(other, '_mpc_'):
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tval = other._mpc_
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mpc = type(other)
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return (tval[1] == fzero) and mpf_eq(tval[0], sval)
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try:
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other = mpf.context.convert(other, strings=False)
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except TypeError:
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return NotImplemented
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return self.__eq__(other)
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def __add__(self, other):
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mpf, new, (prec, rounding) = self._ctxdata
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sval = self._mpf_
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if hasattr(other, '_mpf_'):
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tval = other._mpf_
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val = mpf_add(sval, tval, prec, rounding)
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obj = new(mpf)
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obj._mpf_ = val
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return obj
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if hasattr(other, '_mpc_'):
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tval = other._mpc_
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mpc = type(other)
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val = mpc_add_mpf(tval, sval, prec, rounding)
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obj = new(mpc)
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obj._mpc_ = val
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return obj
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try:
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other = mpf.context.convert(other, strings=False)
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except TypeError:
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return NotImplemented
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return self.__add__(other)
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__radd__ = __add__
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def __sub__(self, other):
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mpf, new, (prec, rounding) = self._ctxdata
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sval = self._mpf_
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if hasattr(other, '_mpf_'):
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tval = other._mpf_
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val = mpf_sub(sval, tval, prec, rounding)
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obj = new(mpf)
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obj._mpf_ = val
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return obj
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if hasattr(other, '_mpc_'):
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tval = other._mpc_
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mpc = type(other)
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val = mpc_mpf_sub(sval, tval, prec, rounding)
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obj = new(mpc)
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obj._mpc_ = val
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return obj
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try:
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other = mpf.context.convert(other, strings=False)
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except TypeError:
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return NotImplemented
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return self.__sub__(other)
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def __rsub__(s, t):
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t = s.mpf_convert_lhs(t)
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if t is NotImplemented:
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return t
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return t - s
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def __mul__(self, other):
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mpf, new, (prec, rounding) = self._ctxdata
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sval = self._mpf_
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if hasattr(other, '_mpf_'):
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tval = other._mpf_
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val = mpf_mul(sval, tval, prec, rounding)
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obj = new(mpf)
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obj._mpf_ = val
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return obj
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if hasattr(other, '_mpc_'):
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tval = other._mpc_
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mpc = type(other)
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val = mpc_mul_mpf(tval, sval, prec, rounding)
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obj = new(mpc)
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obj._mpc_ = val
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return obj
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try:
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other = mpf.context.convert(other, strings=False)
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except TypeError:
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return NotImplemented
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return self.__mul__(other)
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__rmul__ = __mul__
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def __truediv__(self, other):
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mpf, new, (prec, rounding) = self._ctxdata
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sval = self._mpf_
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if hasattr(other, '_mpf_'):
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tval = other._mpf_
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val = mpf_div(sval, tval, prec, rounding)
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obj = new(mpf)
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obj._mpf_ = val
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return obj
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if hasattr(other, '_mpc_'):
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tval = other._mpc_
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mpc = type(other)
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val = mpc_mpf_div(sval, tval, prec, rounding)
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obj = new(mpc)
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obj._mpc_ = val
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return obj
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try:
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other = mpf.context.convert(other, strings=False)
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except TypeError:
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return NotImplemented
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return self.__truediv__(other)
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def __rtruediv__(s, t):
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t = s.mpf_convert_lhs(t)
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if t is NotImplemented:
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return t
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return t / s
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def __mod__(self, other):
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mpf, new, (prec, rounding) = self._ctxdata
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sval = self._mpf_
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if hasattr(other, '_mpf_'):
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tval = other._mpf_
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val = mpf_mod(sval, tval, prec, rounding)
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obj = new(mpf)
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obj._mpf_ = val
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return obj
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if hasattr(other, '_mpc_'):
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return NotImplemented
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try:
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other = mpf.context.convert(other, strings=False)
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except TypeError:
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return NotImplemented
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return self.__mod__(other)
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def __rmod__(s, t):
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t = s.mpf_convert_lhs(t)
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if t is NotImplemented:
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return t
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return t % s
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def __pow__(self, other):
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mpf, new, (prec, rounding) = self._ctxdata
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sval = self._mpf_
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if hasattr(other, '_mpf_'):
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tval = other._mpf_
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try:
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val = mpf_pow(sval, tval, prec, rounding)
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obj = new(mpf)
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obj._mpf_ = val
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return obj
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except ComplexResult:
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if mpf.context.trap_complex:
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raise
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mpc = mpf.context.mpc
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val = mpc_pow((sval, fzero), (tval, fzero), prec, rounding)
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obj = new(mpc)
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obj._mpc_ = val
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return obj
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if hasattr(other, '_mpc_'):
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tval = other._mpc_
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mpc = type(other)
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val = mpc_pow((sval, fzero), tval, prec, rounding)
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obj = new(mpc)
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obj._mpc_ = val
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return obj
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try:
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other = mpf.context.convert(other, strings=False)
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except TypeError:
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return NotImplemented
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return self.__pow__(other)
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def __rpow__(s, t):
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t = s.mpf_convert_lhs(t)
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if t is NotImplemented:
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return t
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return t ** s
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def __format__(s, format_spec):
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"""
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``mpf`` objects allow for formatting similar to Python floats:
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>>> from mpmath import fp, mp, pi
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>>> mp.dps = 50
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>>> format(pi, '*^60.50f')
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'****3.14159265358979323846264338327950288419716939937511****'
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>>> f'{10*pi:.20e}'
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'3.14159265358979323846e+01'
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The format specification adopts the same general form as Python's
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:external:ref:`formatspec`. All of Python's format types are
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supported, with the exception of ``n``.
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If precision is left as default, the resulting string is exactly the
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same as if printing a regular :external:class:`float`:
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>>> mp.dps = fp.dps
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>>> f"{mp.mpf('1.22'):.25f}"
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'1.2199999999999999733546474'
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>>> f'{1.22:.25f}'
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'1.2199999999999999733546474'
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>>> mp.dps = 50
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>>> f"{mp.mpf('1.22'):.25f}"
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'1.2200000000000000000000000'
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In addition to the normal Python features, four different kinds of
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rounding are supported:
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* 'U': rounding towards plus infinity
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* 'D': rounding towards minus infinity
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* 'Y': rounding away from zero
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* 'Z': rounding towards zero
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* 'N': rounding to nearest (default)
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The rounding option must be set right before the presentation type:
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>>> x = mp.mpf('-1.2345678')
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>>> f'{x:.5Uf}'
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'-1.23456'
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>>> f'{x:.5Df}'
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'-1.23457'
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Format types 'a' and 'A' (use uppercase digits) allow to represent
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floating-point number as a C99-style hexadecimal string
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``[±][0x]h[.hhh]p±d``, where there is one hexadecimal digit before the
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dot and the fractional part either is exact or the number of its
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hexadecimal digits is equal to the specified precision. The exponent
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``d`` is written in decimal, it always contains at least one digit, and
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it gives the power of 2 by which to multiply the coefficient. If no
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digits follow the decimal point, the decimal point is also removed
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unless the ``#`` option is specified.
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>>> f'{x:a}'
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'-0x1.3c0ca2a5b1d5d0818d3359c99ff1a26f2b31063249p+0'
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>>> f'{x:.10a}'
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'-0x1.3c0ca2a5b2p+0'
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"""
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_, _, (prec, _) = s._ctxdata
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return format_mpf(s._mpf_, format_spec, prec)
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def sqrt(s):
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return s.context.sqrt(s)
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def ae(s, t, rel_eps=None, abs_eps=None):
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return s.context.almosteq(s, t, rel_eps, abs_eps)
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def to_fixed(self, prec):
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return to_fixed(self._mpf_, prec)
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def __round__(self, ndigits=0):
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ctx = self.context
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if ctx.isfinite(self):
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frac = MPQ(*self.as_integer_ratio())
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res = round(frac, ndigits)
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return ctx.convert(res)
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return self
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class _constant(_mpf):
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"""Represents a mathematical constant with dynamic precision.
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When printed or used in an arithmetic operation, a constant
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is converted to a regular mpf at the working precision. A
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regular mpf can also be obtained using the operation +x."""
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def __new__(cls, func, name, docname='', _reprdps_getter=lambda: 15):
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a = object.__new__(cls)
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a.name = name
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a.func = func
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a._reprdps_getter = _reprdps_getter
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a.__doc__ = getattr(function_docs, docname, '')
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return a
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def __call__(self, prec=None, dps=None, rounding=None):
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prec2, rounding2 = self.context._prec_rounding
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if not prec: prec = prec2
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if not rounding: rounding = rounding2
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if dps: prec = dps_to_prec(dps)
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return self.context.make_mpf(self.func(prec, rounding))
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@property
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def _mpf_(self):
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prec, rounding = self.context._prec_rounding
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return self.func(prec, rounding)
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def __repr__(self):
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return "<%s: %s~>" % (self.name, self.context.nstr(self(dps=self._reprdps_getter())))
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class _mpc(mpnumeric):
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"""
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An mpc represents a complex number using a pair of mpf's (one
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for the real part and another for the imaginary part.) The mpc
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class behaves fairly similarly to Python's complex type.
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"""
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__slots__ = ['_mpc_']
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def __new__(cls, real=0, imag=0):
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s = object.__new__(cls)
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if isinstance(real, str):
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real = cls.context.convert(real)
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if isinstance(real, complex_types):
|
|
r_real, r_imag = real.real, real.imag
|
|
elif hasattr(real, '_mpc_'):
|
|
r_real, r_imag = real._mpc_
|
|
else:
|
|
r_real, r_imag = real, 0
|
|
if isinstance(imag, complex_types):
|
|
i_real, i_imag = imag.real, imag.imag
|
|
elif hasattr(imag, '_mpc_'):
|
|
i_real, i_imag = imag._mpc_
|
|
else:
|
|
i_real, i_imag = imag, 0
|
|
r_real, r_imag = map(cls.context.mpf, [r_real, r_imag])
|
|
i_real, i_imag = map(cls.context.mpf, [i_real, i_imag])
|
|
real = r_real - i_imag
|
|
imag = r_imag + i_real
|
|
s._mpc_ = (real._mpf_, imag._mpf_)
|
|
return s
|
|
|
|
real = property(lambda self: self.context.make_mpf(self._mpc_[0]))
|
|
imag = property(lambda self: self.context.make_mpf(self._mpc_[1]))
|
|
|
|
def __reduce__(self): return _make_mpc, self._mpc_
|
|
|
|
def __repr__(s):
|
|
if s.context.pretty:
|
|
return str(s)
|
|
r = repr(s.real)[4:-1]
|
|
i = repr(s.imag)[4:-1]
|
|
return "%s(real=%s, imag=%s)" % (type(s).__name__, r, i)
|
|
|
|
def __str__(s):
|
|
return "(%s)" % mpc_to_str(s._mpc_, s.context._str_digits)
|
|
|
|
def __complex__(s):
|
|
return mpc_to_complex(s._mpc_, rnd=s.context._prec_rounding[1])
|
|
|
|
def __pos__(s):
|
|
cls, new, (prec, rounding) = s._ctxdata
|
|
v = new(cls)
|
|
v._mpc_ = mpc_pos(s._mpc_, prec, rounding)
|
|
return v
|
|
|
|
def __abs__(s):
|
|
prec, rounding = s.context._prec_rounding
|
|
v = new(s.context.mpf)
|
|
v._mpf_ = mpc_abs(s._mpc_, prec, rounding)
|
|
return v
|
|
|
|
def __neg__(s):
|
|
cls, new, (prec, rounding) = s._ctxdata
|
|
v = new(cls)
|
|
v._mpc_ = mpc_neg(s._mpc_, prec, rounding)
|
|
return v
|
|
|
|
def conjugate(s):
|
|
cls, new, (prec, rounding) = s._ctxdata
|
|
v = new(cls)
|
|
v._mpc_ = mpc_conjugate(s._mpc_, prec, rounding)
|
|
return v
|
|
|
|
def __bool__(s):
|
|
return mpc_is_nonzero(s._mpc_)
|
|
|
|
def __hash__(s):
|
|
return mpc_hash(s._mpc_)
|
|
|
|
@classmethod
|
|
def mpc_convert_lhs(cls, x):
|
|
try:
|
|
y = cls.context.convert(x)
|
|
return y
|
|
except TypeError:
|
|
return NotImplemented
|
|
|
|
def __eq__(s, t):
|
|
if not hasattr(t, '_mpc_'):
|
|
if isinstance(t, str):
|
|
return False
|
|
t = s.mpc_convert_lhs(t)
|
|
if t is NotImplemented:
|
|
return t
|
|
return s.real == t.real and s.imag == t.imag
|
|
|
|
def __add__(s, t):
|
|
cls, new, (prec, rounding) = s._ctxdata
|
|
if not hasattr(t, '_mpc_'):
|
|
t = s.mpc_convert_lhs(t)
|
|
if t is NotImplemented:
|
|
return t
|
|
if hasattr(t, '_mpf_'):
|
|
v = new(cls)
|
|
v._mpc_ = mpc_add_mpf(s._mpc_, t._mpf_, prec, rounding)
|
|
return v
|
|
v = new(cls)
|
|
v._mpc_ = mpc_add(s._mpc_, t._mpc_, prec, rounding)
|
|
return v
|
|
__radd__ = __add__
|
|
|
|
def __sub__(s, t):
|
|
cls, new, (prec, rounding) = s._ctxdata
|
|
if not hasattr(t, '_mpc_'):
|
|
t = s.mpc_convert_lhs(t)
|
|
if t is NotImplemented:
|
|
return t
|
|
if hasattr(t, '_mpf_'):
|
|
v = new(cls)
|
|
v._mpc_ = mpc_sub_mpf(s._mpc_, t._mpf_, prec, rounding)
|
|
return v
|
|
v = new(cls)
|
|
v._mpc_ = mpc_sub(s._mpc_, t._mpc_, prec, rounding)
|
|
return v
|
|
|
|
def __rsub__(s, t):
|
|
t = s.mpc_convert_lhs(t)
|
|
if t is NotImplemented:
|
|
return t
|
|
return t - s
|
|
|
|
def __mul__(s, t):
|
|
cls, new, (prec, rounding) = s._ctxdata
|
|
if not hasattr(t, '_mpc_'):
|
|
if isinstance(t, int_types):
|
|
v = new(cls)
|
|
v._mpc_ = mpc_mul_int(s._mpc_, t, prec, rounding)
|
|
return v
|
|
t = s.mpc_convert_lhs(t)
|
|
if t is NotImplemented:
|
|
return t
|
|
if hasattr(t, '_mpf_'):
|
|
v = new(cls)
|
|
v._mpc_ = mpc_mul_mpf(s._mpc_, t._mpf_, prec, rounding)
|
|
return v
|
|
t = s.mpc_convert_lhs(t)
|
|
v = new(cls)
|
|
v._mpc_ = mpc_mul(s._mpc_, t._mpc_, prec, rounding)
|
|
return v
|
|
|
|
def __rmul__(s, t):
|
|
cls, new, (prec, rounding) = s._ctxdata
|
|
if isinstance(t, int_types):
|
|
v = new(cls)
|
|
v._mpc_ = mpc_mul_int(s._mpc_, t, prec, rounding)
|
|
return v
|
|
t = s.mpc_convert_lhs(t)
|
|
if t is NotImplemented:
|
|
return t
|
|
return t * s
|
|
|
|
def __truediv__(s, t):
|
|
cls, new, (prec, rounding) = s._ctxdata
|
|
if not hasattr(t, '_mpc_'):
|
|
t = s.mpc_convert_lhs(t)
|
|
if t is NotImplemented:
|
|
return t
|
|
if hasattr(t, '_mpf_'):
|
|
v = new(cls)
|
|
v._mpc_ = mpc_div_mpf(s._mpc_, t._mpf_, prec, rounding)
|
|
return v
|
|
v = new(cls)
|
|
v._mpc_ = mpc_div(s._mpc_, t._mpc_, prec, rounding)
|
|
return v
|
|
|
|
def __rtruediv__(s, t):
|
|
t = s.mpc_convert_lhs(t)
|
|
if t is NotImplemented:
|
|
return t
|
|
return t / s
|
|
|
|
def __pow__(s, t):
|
|
cls, new, (prec, rounding) = s._ctxdata
|
|
if isinstance(t, int_types):
|
|
v = new(cls)
|
|
v._mpc_ = mpc_pow_int(s._mpc_, t, prec, rounding)
|
|
return v
|
|
t = s.mpc_convert_lhs(t)
|
|
if t is NotImplemented:
|
|
return t
|
|
v = new(cls)
|
|
if hasattr(t, '_mpf_'):
|
|
v._mpc_ = mpc_pow_mpf(s._mpc_, t._mpf_, prec, rounding)
|
|
else:
|
|
v._mpc_ = mpc_pow(s._mpc_, t._mpc_, prec, rounding)
|
|
return v
|
|
|
|
def __rpow__(s, t):
|
|
t = s.mpc_convert_lhs(t)
|
|
if t is NotImplemented:
|
|
return t
|
|
return t ** s
|
|
|
|
def ae(s, t, rel_eps=None, abs_eps=None):
|
|
return s.context.almosteq(s, t, rel_eps, abs_eps)
|
|
|
|
def __format__(s, format_spec):
|
|
"""
|
|
``mpc`` objects allow for formatting similar to Python
|
|
:external:class:`complex`, specified in :external:ref:`formatspec`.
|
|
All of Python's format types are supported, with the exception of
|
|
``n``. Additionally, as for :func:`mpmath.mpf.__format__`, four
|
|
different kinds of rounding are supported and format types 'a'
|
|
and 'A' (use uppercase digits) allow to represent components as
|
|
C99-style hexadecimal strings.
|
|
|
|
"""
|
|
|
|
_, _, (prec, _) = s._ctxdata
|
|
return format_mpc(s._mpc_, format_spec, prec)
|
|
|
|
|
|
complex_types = (complex, _mpc)
|
|
|
|
|
|
class PythonMPContext:
|
|
def __init__(ctx):
|
|
ctx._prec_rounding = [sys.float_info.mant_dig, round_nearest]
|
|
ctx.mpf = type('mpf', (_mpf,), {})
|
|
ctx.mpc = type('mpc', (_mpc,), {})
|
|
ctx.mpf._ctxdata = [ctx.mpf, new, ctx._prec_rounding]
|
|
ctx.mpc._ctxdata = [ctx.mpc, new, ctx._prec_rounding]
|
|
ctx.mpf.context = ctx
|
|
ctx.mpc.context = ctx
|
|
ctx.constant = type('constant', (_constant,), {})
|
|
ctx.constant._ctxdata = [ctx.mpf, new, ctx._prec_rounding]
|
|
ctx.constant.context = ctx
|
|
|
|
def make_mpf(ctx, v):
|
|
a = new(ctx.mpf)
|
|
a._mpf_ = v
|
|
return a
|
|
|
|
def make_mpc(ctx, v):
|
|
a = new(ctx.mpc)
|
|
a._mpc_ = v
|
|
return a
|
|
|
|
def default(ctx):
|
|
ctx._prec = ctx._prec_rounding[0] = sys.float_info.mant_dig
|
|
ctx._dps = sys.float_info.dig
|
|
ctx.trap_complex = False
|
|
|
|
def _set_prec(ctx, n):
|
|
ctx._prec = ctx._prec_rounding[0] = max(1, int(n))
|
|
ctx._dps = prec_to_dps(n)
|
|
|
|
def _set_dps(ctx, n):
|
|
ctx._prec = ctx._prec_rounding[0] = dps_to_prec(n)
|
|
ctx._dps = max(1, int(n))
|
|
|
|
prec = property(lambda ctx: ctx._prec, _set_prec)
|
|
dps = property(lambda ctx: ctx._dps, _set_dps)
|
|
|
|
def convert(ctx, x, strings=True):
|
|
"""
|
|
Converts *x* to an ``mpf`` or ``mpc``. If *x* is of type ``mpf``,
|
|
``mpc``, ``int``, ``float``, ``complex``, the conversion
|
|
will be performed losslessly.
|
|
|
|
If *x* is a string, the result will be rounded to the present
|
|
working precision. Strings representing fractions or complex
|
|
numbers are permitted.
|
|
|
|
>>> from mpmath import mpmathify
|
|
>>> mpmathify(3.5)
|
|
mpf('3.5')
|
|
>>> mpmathify('2.1')
|
|
mpf('2.1000000000000001')
|
|
>>> mpmathify('3/4')
|
|
mpf('0.75')
|
|
>>> mpmathify('2+3j')
|
|
mpc(real='2.0', imag='3.0')
|
|
|
|
"""
|
|
if type(x) in ctx.types: return x
|
|
if isinstance(x, int_types): return ctx.make_mpf(from_int(x))
|
|
if isinstance(x, float): return ctx.make_mpf(from_float(x))
|
|
if isinstance(x, complex):
|
|
return ctx.make_mpc((from_float(x.real), from_float(x.imag)))
|
|
if type(x).__module__ == 'numpy': return ctx.npconvert(x)
|
|
prec, rounding = ctx._prec_rounding
|
|
if hasattr(x, '_mpf_'): return ctx.make_mpf(x._mpf_)
|
|
if hasattr(x, '_mpc_'): return ctx.make_mpc(x._mpc_)
|
|
if hasattr(x, '_mpmath_'):
|
|
return ctx.convert(x._mpmath_(prec, rounding))
|
|
if isinstance(x, numbers.Rational):
|
|
p, q = x.numerator, x.denominator
|
|
return ctx.make_mpf(from_rational(p, q, prec, rounding))
|
|
if strings and isinstance(x, str):
|
|
try:
|
|
_mpf_ = from_str(x, prec, rounding)
|
|
return ctx.make_mpf(_mpf_)
|
|
except ValueError:
|
|
pass
|
|
if type(x).__module__ == 'decimal':
|
|
return ctx.make_mpf(from_Decimal(x, prec, rounding))
|
|
return ctx._convert_fallback(x, strings)
|
|
|
|
def npconvert(ctx, x):
|
|
"""
|
|
Converts *x* to an ``mpf`` or ``mpc``. *x* should be a numpy
|
|
scalar.
|
|
"""
|
|
import numpy as np
|
|
if isinstance(x, np.ndarray) and x.ndim == 0: x = x.item()
|
|
if isinstance(x, (np.integer, int)): return ctx.make_mpf(from_int(int(x)))
|
|
if isinstance(x, (np.floating, float)): return ctx.mpf(from_npfloat(x))
|
|
if isinstance(x, (np.complexfloating, complex)):
|
|
return ctx.make_mpc((from_npfloat(x.real), from_npfloat(x.imag)))
|
|
raise TypeError("cannot create mpf from " + repr(x))
|
|
|
|
def isinf(ctx, x):
|
|
"""
|
|
Return *True* if the absolute value of *x* is infinite;
|
|
otherwise return *False*::
|
|
|
|
>>> from mpmath import isinf, inf, mpc
|
|
>>> isinf(inf)
|
|
True
|
|
>>> isinf(-inf)
|
|
True
|
|
>>> isinf(3)
|
|
False
|
|
>>> isinf(3+4j)
|
|
False
|
|
>>> isinf(mpc(3,inf))
|
|
True
|
|
>>> isinf(mpc(inf,3))
|
|
True
|
|
|
|
"""
|
|
if hasattr(x, "_mpf_"):
|
|
return x._mpf_ in (finf, fninf)
|
|
if hasattr(x, "_mpc_"):
|
|
return mpc_is_inf(x._mpc_)
|
|
if isinstance(x, int_types) or isinstance(x, MPQ):
|
|
return False
|
|
x = ctx.convert(x)
|
|
return ctx.isinf(x)
|
|
|
|
def isnormal(ctx, x):
|
|
"""
|
|
Determine whether *x* is "normal" in the sense of floating-point
|
|
representation; that is, return *False* if *x* is zero, an
|
|
infinity or NaN; otherwise return *True*. By extension, a
|
|
complex number *x* is considered "normal" if its magnitude is
|
|
normal::
|
|
|
|
>>> from mpmath import isnormal, inf, nan, mpc
|
|
>>> isnormal(3)
|
|
True
|
|
>>> isnormal(0)
|
|
False
|
|
>>> isnormal(inf); isnormal(-inf); isnormal(nan)
|
|
False
|
|
False
|
|
False
|
|
>>> isnormal(0+0j)
|
|
False
|
|
>>> isnormal(0+3j)
|
|
True
|
|
>>> isnormal(mpc(2,nan))
|
|
False
|
|
"""
|
|
if hasattr(x, "_mpf_"):
|
|
if ctx.isfinite(x):
|
|
return bool(to_man_exp(x._mpf_, signed=True)[0])
|
|
return False
|
|
if hasattr(x, "_mpc_"):
|
|
re, im = x._mpc_
|
|
re_normal = bool(re[1])
|
|
im_normal = bool(im[1])
|
|
if re == fzero: return im_normal
|
|
if im == fzero: return re_normal
|
|
return re_normal and im_normal
|
|
if isinstance(x, int_types) or isinstance(x, MPQ):
|
|
return bool(x)
|
|
x = ctx.convert(x)
|
|
return ctx.isnormal(x)
|
|
|
|
def isint(ctx, x, gaussian=False):
|
|
"""
|
|
Return *True* if *x* is integer-valued; otherwise return
|
|
*False*::
|
|
|
|
>>> from mpmath import isint, mpf, inf
|
|
>>> isint(3)
|
|
True
|
|
>>> isint(mpf(3))
|
|
True
|
|
>>> isint(3.2)
|
|
False
|
|
>>> isint(inf)
|
|
False
|
|
|
|
Optionally, Gaussian integers can be checked for::
|
|
|
|
>>> isint(3+0j)
|
|
True
|
|
>>> isint(3+2j)
|
|
False
|
|
>>> isint(3+2j, gaussian=True)
|
|
True
|
|
|
|
"""
|
|
if isinstance(x, int_types):
|
|
return True
|
|
if hasattr(x, "_mpf_"):
|
|
if ctx.isfinite(x):
|
|
man, exp = to_man_exp(x._mpf_, signed=True)
|
|
return bool((man and exp >= 0) or x._mpf_ == fzero)
|
|
return False
|
|
if hasattr(x, "_mpc_"):
|
|
re, im = x._mpc_
|
|
if ctx.isfinite(x):
|
|
man, exp = to_man_exp(re, signed=True)
|
|
re_isint = bool((man and exp >= 0) or re == fzero)
|
|
man, exp = to_man_exp(im, signed=True)
|
|
im_isint = bool((man and exp >= 0) or im == fzero)
|
|
else:
|
|
return False
|
|
if gaussian:
|
|
return re_isint and im_isint
|
|
return re_isint and im == fzero
|
|
if isinstance(x, MPQ):
|
|
p, q = x.numerator, x.denominator
|
|
return p % q == 0
|
|
x = ctx.convert(x)
|
|
return ctx.isint(x, gaussian)
|
|
|
|
def fsum(ctx, terms, absolute=False, squared=False):
|
|
"""
|
|
Calculates a sum containing a finite number of terms (for infinite
|
|
series, see :func:`~mpmath.nsum`). The terms will be converted to
|
|
mpmath numbers. For len(terms) > 2, this function is generally
|
|
faster and produces more accurate results than the builtin
|
|
Python function :func:`sum`.
|
|
|
|
>>> from mpmath import fsum
|
|
>>> fsum([1, 2, 0.5, 7])
|
|
mpf('10.5')
|
|
|
|
With squared=True each term is squared, and with absolute=True
|
|
the absolute value of each term is used.
|
|
"""
|
|
prec, rnd = ctx._prec_rounding
|
|
real = []
|
|
imag = []
|
|
for term in terms:
|
|
reval = imval = 0
|
|
if hasattr(term, "_mpf_"):
|
|
reval = term._mpf_
|
|
elif hasattr(term, "_mpc_"):
|
|
reval, imval = term._mpc_
|
|
else:
|
|
term = ctx.convert(term)
|
|
if hasattr(term, "_mpf_"):
|
|
reval = term._mpf_
|
|
elif hasattr(term, "_mpc_"):
|
|
reval, imval = term._mpc_
|
|
else:
|
|
raise NotImplementedError
|
|
if imval:
|
|
if squared:
|
|
if absolute:
|
|
real.append(mpf_mul(reval,reval))
|
|
real.append(mpf_mul(imval,imval))
|
|
else:
|
|
reval, imval = mpc_pow_int((reval,imval),2,prec+10)
|
|
real.append(reval)
|
|
imag.append(imval)
|
|
elif absolute:
|
|
real.append(mpc_abs((reval,imval), prec))
|
|
else:
|
|
real.append(reval)
|
|
imag.append(imval)
|
|
else:
|
|
if squared:
|
|
reval = mpf_mul(reval, reval)
|
|
elif absolute:
|
|
reval = mpf_abs(reval)
|
|
real.append(reval)
|
|
s = mpf_sum(real, prec, rnd, absolute)
|
|
if imag:
|
|
s = ctx.make_mpc((s, mpf_sum(imag, prec, rnd)))
|
|
else:
|
|
s = ctx.make_mpf(s)
|
|
return s
|
|
|
|
def fdot(ctx, A, B=None, conjugate=False):
|
|
r"""
|
|
Computes the dot product of the iterables `A` and `B`,
|
|
|
|
.. math ::
|
|
|
|
\sum_{k=0} A_k B_k.
|
|
|
|
Alternatively, :func:`~mpmath.fdot` accepts a single iterable of pairs.
|
|
In other words, ``fdot(A,B)`` and ``fdot(zip(A,B))`` are equivalent.
|
|
The elements are automatically converted to mpmath numbers.
|
|
|
|
With ``conjugate=True``, the elements in the second vector
|
|
will be conjugated:
|
|
|
|
.. math ::
|
|
|
|
\sum_{k=0} A_k \overline{B_k}
|
|
|
|
**Examples**
|
|
|
|
>>> from mpmath import fdot, j
|
|
>>> A = [2, 1.5, 3]
|
|
>>> B = [1, -1, 2]
|
|
>>> fdot(A, B)
|
|
mpf('6.5')
|
|
>>> list(zip(A, B))
|
|
[(2, 1), (1.5, -1), (3, 2)]
|
|
>>> fdot(_)
|
|
mpf('6.5')
|
|
>>> A = [2, 1.5, 3j]
|
|
>>> B = [1+j, 3, -1-j]
|
|
>>> fdot(A, B)
|
|
mpc(real='9.5', imag='-1.0')
|
|
>>> fdot(A, B, conjugate=True)
|
|
mpc(real='3.5', imag='-5.0')
|
|
|
|
"""
|
|
if B is not None:
|
|
A = zip(A, B)
|
|
prec, rnd = ctx._prec_rounding
|
|
real = []
|
|
imag = []
|
|
hasattr_ = hasattr
|
|
types = (ctx.mpf, ctx.mpc)
|
|
for a, b in A:
|
|
if type(a) not in types: a = ctx.convert(a)
|
|
if type(b) not in types: b = ctx.convert(b)
|
|
a_real = hasattr_(a, "_mpf_")
|
|
b_real = hasattr_(b, "_mpf_")
|
|
if a_real and b_real:
|
|
real.append(mpf_mul(a._mpf_, b._mpf_))
|
|
continue
|
|
a_complex = hasattr_(a, "_mpc_")
|
|
b_complex = hasattr_(b, "_mpc_")
|
|
if a_real and b_complex:
|
|
aval = a._mpf_
|
|
bre, bim = b._mpc_
|
|
if conjugate:
|
|
bim = mpf_neg(bim)
|
|
real.append(mpf_mul(aval, bre))
|
|
imag.append(mpf_mul(aval, bim))
|
|
elif b_real and a_complex:
|
|
are, aim = a._mpc_
|
|
bval = b._mpf_
|
|
real.append(mpf_mul(are, bval))
|
|
imag.append(mpf_mul(aim, bval))
|
|
elif a_complex and b_complex:
|
|
#re, im = mpc_mul(a._mpc_, b._mpc_, prec+20)
|
|
are, aim = a._mpc_
|
|
bre, bim = b._mpc_
|
|
if conjugate:
|
|
bim = mpf_neg(bim)
|
|
real.append(mpf_mul(are, bre))
|
|
real.append(mpf_neg(mpf_mul(aim, bim)))
|
|
imag.append(mpf_mul(are, bim))
|
|
imag.append(mpf_mul(aim, bre))
|
|
else:
|
|
raise NotImplementedError
|
|
s = mpf_sum(real, prec, rnd)
|
|
if imag:
|
|
s = ctx.make_mpc((s, mpf_sum(imag, prec, rnd)))
|
|
else:
|
|
s = ctx.make_mpf(s)
|
|
return s
|
|
|
|
def _wrap_libmp_function(ctx, mpf_f, mpc_f=None, mpi_f=None, doc="<no doc>"):
|
|
"""
|
|
Given a low-level mpf_ function, and optionally similar functions
|
|
for mpc_ and mpi_, defines the function as a context method.
|
|
|
|
It is assumed that the return type is the same as that of
|
|
the input; the exception is that propagation from mpf to mpc is possible
|
|
by raising ComplexResult.
|
|
|
|
"""
|
|
def f(x, **kwargs):
|
|
if type(x) not in ctx.types:
|
|
x = ctx.convert(x)
|
|
prec, rounding = ctx._prec_rounding
|
|
if kwargs:
|
|
prec = kwargs.get('prec', prec)
|
|
if 'dps' in kwargs:
|
|
prec = dps_to_prec(kwargs['dps'])
|
|
rounding = kwargs.get('rounding', rounding)
|
|
if hasattr(x, '_mpf_'):
|
|
try:
|
|
return ctx.make_mpf(mpf_f(x._mpf_, prec, rounding))
|
|
except ComplexResult:
|
|
# Handle propagation to complex
|
|
if ctx.trap_complex:
|
|
raise
|
|
return ctx.make_mpc(mpc_f((x._mpf_, fzero), prec, rounding))
|
|
elif hasattr(x, '_mpc_'):
|
|
return ctx.make_mpc(mpc_f(x._mpc_, prec, rounding))
|
|
raise NotImplementedError("%s of a %s" % (name, type(x)))
|
|
name = mpf_f.__name__[4:]
|
|
f.__doc__ = function_docs.__dict__.get(name, "Computes the %s of x" % doc)
|
|
return f
|
|
|
|
# Called by SpecialFunctions.__init__()
|
|
@classmethod
|
|
def _wrap_specfun(cls, name, f, wrap):
|
|
if wrap:
|
|
def f_wrapped(ctx, *args, **kwargs):
|
|
convert = ctx.convert
|
|
args = [convert(a) for a in args]
|
|
prec = ctx.prec
|
|
try:
|
|
ctx.prec += 10
|
|
retval = f(ctx, *args, **kwargs)
|
|
finally:
|
|
ctx.prec = prec
|
|
return +retval
|
|
else:
|
|
f_wrapped = f
|
|
f_wrapped.__doc__ = function_docs.__dict__.get(name, f.__doc__)
|
|
setattr(cls, name, f_wrapped)
|
|
|
|
def _convert_param(ctx, x):
|
|
if hasattr(x, "_mpc_"):
|
|
v, im = x._mpc_
|
|
if im != fzero:
|
|
return x, 'C'
|
|
elif hasattr(x, "_mpf_"):
|
|
v = x._mpf_
|
|
else:
|
|
if type(x) in int_types:
|
|
return int(x), 'Z'
|
|
p = None
|
|
if isinstance(x, tuple):
|
|
p, q = x
|
|
elif isinstance(x, str) and '/' in x:
|
|
p, q = x.split('/')
|
|
p = int(p)
|
|
q = int(q)
|
|
if p is not None:
|
|
if not p % q:
|
|
return p // q, 'Z'
|
|
return MPQ(p,q), 'Q'
|
|
x = ctx.convert(x)
|
|
if hasattr(x, "_mpc_"):
|
|
v, im = x._mpc_
|
|
if im != fzero:
|
|
return x, 'C'
|
|
elif hasattr(x, "_mpf_"):
|
|
v = x._mpf_
|
|
else:
|
|
raise NotImplementedError
|
|
man, exp = to_man_exp(v, signed=True)
|
|
if man:
|
|
if exp >= -4:
|
|
if exp >= 0:
|
|
return int(man) << exp, 'Z'
|
|
p, q = int(man), (1<<(-exp))
|
|
return MPQ(p,q), 'Q'
|
|
x = ctx.make_mpf(v)
|
|
return x, 'R'
|
|
if not exp:
|
|
return 0, 'Z'
|
|
raise NotImplementedError
|
|
|
|
def _mpf_mag(ctx, x):
|
|
if x == fzero:
|
|
return ctx.ninf
|
|
if x in (finf, fninf, fnan):
|
|
return ctx.make_mpf(mpf_abs(x))
|
|
man, exp = to_man_exp(x, signed=True)
|
|
return exp+man.bit_length()
|
|
|
|
def mag(ctx, x):
|
|
"""
|
|
Quick logarithmic magnitude estimate of a number. Returns an
|
|
integer or infinity `m` such that `|x| <= 2^m`. It is not
|
|
guaranteed that `m` is an optimal bound, but it will never
|
|
be too large by more than 2 (and probably not more than 1).
|
|
|
|
**Examples**
|
|
|
|
>>> from mpmath import mp, mag, ceil, mpf, log, inf, nan
|
|
>>> mp.pretty = True
|
|
>>> mag(10), mag(10.0), mag(mpf(10)), int(ceil(log(10,2)))
|
|
(4, 4, 4, 4)
|
|
>>> mag(10j), mag(10+10j)
|
|
(4, 5)
|
|
>>> mag(0.01), int(ceil(log(0.01,2)))
|
|
(-6, -6)
|
|
>>> mag(0), mag(inf), mag(-inf), mag(nan)
|
|
(-inf, inf, inf, nan)
|
|
|
|
"""
|
|
if hasattr(x, "_mpf_"):
|
|
return ctx._mpf_mag(x._mpf_)
|
|
if hasattr(x, "_mpc_"):
|
|
r, i = x._mpc_
|
|
if r == fzero:
|
|
return ctx._mpf_mag(i)
|
|
if i == fzero:
|
|
return ctx._mpf_mag(r)
|
|
return 1+max(ctx._mpf_mag(r), ctx._mpf_mag(i))
|
|
if isinstance(x, int_types):
|
|
if x:
|
|
return x.bit_length()
|
|
return ctx.ninf
|
|
if isinstance(x, MPQ):
|
|
p, q = x.numerator, x.denominator
|
|
if p:
|
|
return 1 + p.bit_length() - q.bit_length()
|
|
return ctx.ninf
|
|
x = ctx.convert(x)
|
|
return ctx.mag(x)
|
|
|
|
|
|
# Register with "numbers" ABC
|
|
# We do not subclass, hence we do not use the @abstractmethod checks. While
|
|
# this is less invasive it may turn out that we do not actually support
|
|
# parts of the expected interfaces. See
|
|
# https://docs.python.org/3/library/numbers.html for list of abstract methods.
|
|
numbers.Complex.register(_mpc)
|
|
numbers.Real.register(_mpf)
|