# Coverage for pygeodesy/fmath.py : 96%

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-*- coding: utf-8 -*- u'''Precision floating point functions, utilities and constants. @newfield example: Example, Examples ''' # make sure int/int division yields float quotient from __future__ import division division = 1 / 2 # double check int division, see .datum.py, .utily.py if not division: raise ImportError('%s 1/2 == %d' % ('division', division)) del division from pygeodesy.lazily import _ALL_LAZY, _xcopy from math import acos, copysign, hypot, isinf, isnan, sqrt # pow from operator import mul from sys import float_info as _float_info # all public contants, classes and functions __all__ = _ALL_LAZY.fmath __version__ = '20.02.19' try: # Luciano Ramalho, "Fluent Python", page 395, O'Reilly, 2016 from numbers import Integral as _Ints #: (INTERNAL) Int objects except ImportError: # PYCHOK no cover try: # _Ints imported by .utily _Ints = int, long #: (INTERNAL) Int objects (C{tuple}) except NameError: # Python 3+ _Ints = int, #: (INTERNAL) Int objects (C{tuple}) try: # similarly ... from numbers import Real as _Scalars #: (INTERNAL) Scalar objects except ImportError: # PYCHOK no cover try: _Scalars = int, long, float #: (INTERNAL) Scalar objects (C{tuple}) except NameError: _Scalars = int, float #: (INTERNAL) Scalar objects (C{tuple}) try: # _Seqs imported by .utily from collections import Sequence as _Seqs #: (INTERNAL) incl MutableSequence except ImportError: # PYCHOK no cover _Seqs = list, tuple, range # XXX also set? try: EPS = _float_info.epsilon #: System's epsilon (C{float}) MANTIS = _float_info.mant_dig #: System's mantissa bits (C{int}) MAX = _float_info.max #: System's float max (C{float}) MIN = _float_info.min #: System's float min (C{float}) except AttributeError: # PYCHOK no cover EPS = 2.220446049250313e-16 #: Epsilon (C{float}) 2**-52? MANTIS = 53 #: Mantissa bits ≈53 (C{int}) MAX = pow(2.0, 1023) * (2 - EPS) #: Float max (C{float}) ≈10**308, 2**1024? MIN = pow(2.0, -1021) # Float min (C{float}) ≈10**-308, 2**-1021? EPS_2 = EPS / 2 #: M{EPS / 2} ≈1.110223024625e-16 (C{float}) EPS1 = 1.0 - EPS #: M{1 - EPS} ≈0.9999999999999998 (C{float}) EPS1_2 = 1.0 - EPS_2 #: M{1 - EPS_2} ≈0.9999999999999999 (C{float}) # _1EPS = 1.0 + EPS #: M{1 + EPS} ≈1.0000000000000002 (C{float}) INF = float('inf') #: Infinity (C{float}), see C{isinf}, C{isfinite} NAN = float('nan') #: Not-A-Number (C{float}), see C{isnan} NEG0 = -0.0 #: Negative 0.0 (C{float}), see C{isneg0} _1_3rd = 1.0 / 3.0 #: (INTERNAL) One third (C{float}) _2_3rd = 2.0 / 3.0 #: (INTERNAL) Two thirds (C{float}) _3_2nd = 3.0 / 2.0 #: (INTERNAL) Three halfs (C{float}) def _IsNotError(*names, **pair): # Error=TypeError, name=value '''(INTERNAL) Format a C{TypeError} for a C{name=value} pair. ''' Error = pair.pop('Error', TypeError) for n, v in pair.items(): break else: n, v = 'pair', 'N/A' t = ' or ' .join(names) return Error('%s not %s: %r' % (n, t, v)) def _2even(s, r, p): '''(INTERNAL) Half-even rounding. ''' if (r > 0 and p > 0) or \ (r < 0 and p < 0): # signs match t, p = _2sum(s, p * 2) if not p: s = t return s def _2sum(a, b): '''(INTERNAL) Precision C{2sum} of M{a + b}. ''' s = a + b if not isfinite(s): raise OverflowError('%s: %r' % (_2sum.__name__, s)) if abs(a) < abs(b): a, b = b, a return s, b - (s - a) class Fsum(object): '''Precision summation similar to standard Python function C{math.fsum}. Unlike C{math.fsum}, this class accumulates the values repeatedly and provides intermediate, precision running sums. Accumulation may continue after intermediate summations. @note: Handling of exceptions, C{nan} and C{finite} values is different from C{math.fsum}. @see: U{Hettinger<https://code.ActiveState.com/recipes/393090>}, U{Kahan<https://WikiPedia.org/wiki/Kahan_summation_algorithm>}, U{Klein<https://Link.Springer.com/article/10.1007/s00607-005-0139-x>}, Python 2.6+ file I{Modules/mathmodule.c} and the issue log U{Full precision summation<https://Bugs.Python.org/issue2819>}. ''' _fsum2_ = None _n = 0 _ps = [] def __init__(self, *starts): '''Initialize a new accumulator with one or more start values. @param starts: No, one or more start values (C{scalar}s). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{starts}} value. @raise ValueError: Invalid or non-finite B{C{starts}} value. ''' self._n = 0 self._ps = [] if starts: self.fadd(starts) def __add__(self, other): '''Sum of this and an other instance or a scalar. @param other: L{Fsum} instance or C{scalar}. @return: The sum, a new instance (L{Fsum}). @see: Method L{Fsum.__iadd__}. ''' f = self.fcopy() f += other return f # self.fcopy().__iadd__(other) def __iadd__(self, other): '''Add a scalar or an other instance to this instance. @param other: L{Fsum} instance or C{scalar}. @return: This instance, updated (L{Fsum}). @raise TypeError: Invalid B{C{other}} type. @see: Method L{Fsum.fadd}. ''' if isscalar(other): self.fadd_(other) elif other is self: self.fmul(2) elif isinstance(other, Fsum): self.fadd(other._ps) else: raise TypeError('%s += %r' % (self, other)) return self def __imul__(self, other): '''Multiply this instance by a scalar or an other instance. @param other: L{Fsum} instance or C{scalar}. @return: This instance, updated (L{Fsum}). @raise TypeError: Invalid B{C{other}} type. @see: Method L{Fsum.fmul}. ''' if isscalar(other): self.fmul(other) elif isinstance(other, Fsum): ps = list(other._ps) # copy if ps: s = self.fcopy() self.fmul(ps.pop()) while ps: # self += s * ps.pop() p = s.fcopy() p.fmul(ps.pop()) self.fadd(p._ps) else: self._ps = [] # zero self._fsum2_ = None else: raise TypeError('%s *= %r' % (self, other)) return self def __isub__(self, other): '''Subtract a scalar or an other instance from this instance. @param other: L{Fsum} instance or C{scalar}. @return: This instance, updated (L{Fsum}). @raise TypeError: Invalid B{C{other}} type. @see: Method L{Fsum.fadd}. ''' if isscalar(other): self.fadd_(-other) elif other is self: self._ps = [] # zero self._fsum2_ = None elif isinstance(other, Fsum): self.fadd(-p for p in other._ps) else: raise TypeError('%s -= %r' % (self, other)) return self def __len__(self): '''Return the number of accumulated values. ''' return self._n def __mul__(self, other): '''Product of this and an other instance or a scalar. @param other: L{Fsum} instance or C{scalar}. @return: The product, a new instance (L{Fsum}). @see: Method L{Fsum.__imul__}. ''' f = self.fcopy() f *= other return f def __str__(self): # m = self.__module__.split('.')[-1] return '%s.%s()' % (self.__module__, self.__class__.__name__) def __sub__(self, other): '''Difference of this and an other instance or a scalar. @param other: L{Fsum} instance or C{scalar}. @return: The difference, a new instance (L{Fsum}). @see: Method L{Fsum.__isub__}. ''' f = self.fcopy() f -= other return f def fadd(self, iterable): '''Accumulate more values from an iterable. @param iterable: Sequence, list, tuple, etc. (C{scalar}s). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{iterable}} value. @raise ValueError: Invalid or non-finite B{C{iterable}} value. ''' if isscalar(iterable): # for backward compatibility iterable = tuple(iterable) # def _iter(): # for a in iterable: # if isinstance(a, Fsum): # if a is self: # self.fmul(2) # else: # for a in a._ps: # yield a # else: # yield a ps = self._ps for a in iterable: # _iter() if not isfinite(a): raise ValueError('%s, not %s: %r' % (self, 'finite', a)) i = 0 for p in ps: a, p = _2sum(a, p) if p: ps[i] = p i += 1 ps[i:] = [a] self._n += 1 # assert self._ps is ps self._fsum2_ = None def fadd_(self, *xs): '''Accumulate more values from positional arguments. @param xs: Values to add (C{scalar}s), all positional. @see: Method L{Fsum.fadd}. ''' self.fadd(xs) def fcopy(self, deep=False): '''Copy this instance, shallow or deep. @return: The copy, a new instance (L{Fsum}). ''' f = _xcopy(self, deep=deep) # f._fsum2_ = self._fsum2_ # f._n = self._n f._ps = list(self._ps) # separate copy return f copy = fcopy def fmul(self, factor): '''Multiple the current, partial sum by a factor. @param factor: The multiplier (C{scalar}). @raise TypeError: Non-scalar B{C{factor}}. @raise ValueError: Invalid or non-finite B{C{factor}}. @see: Method L{Fsum.fadd}. ''' if not isfinite(factor): raise ValueError('%s, not %s: %r' % (self, 'finite', factor)) ps = self._ps if ps: # multiply and adjust partial sums ps[:] = [p * factor for p in ps] self.fadd_(ps.pop()) self._n -= 1 # assert self._ps is ps def fsub(self, iterable): '''Accumulate more values from an iterable. @param iterable: Sequence, list, tuple, etc. (C{scalar}s). @see: Method L{Fsum.fadd}. ''' if iterable: self.fadd(-s for s in iterable) def fsub_(self, *xs): '''Accumulate more values from positional arguments. @param xs: Values to subtract (C{scalar}s), all positional. @see: Method L{Fsum.fadd}. ''' self.fsub(xs) def fsum(self, iterable=()): '''Accumulate more values from an iterable and sum all. @keyword iterable: Sequence, list, tuple, etc. (C{scalar}s), optional. @return: Accurate, running sum (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{iterable}} value. @raise ValueError: Invalid or non-finite B{C{iterable}} value. @note: Accumulation can continue after summation. ''' if iterable: self.fadd(iterable) ps = self._ps i = len(ps) - 1 if i < 0: s = 0.0 else: s = ps[i] while i > 0: i -= 1 s, p = _2sum(s, ps[i]) ps[i:] = [s] if p: # sum(ps) became inexact ps.append(p) if i > 0: # half-even round if signs match s = _2even(s, ps[i-1], p) break # assert self._ps is ps self._fsum2_ = s return s def fsum_(self, *xs): '''Accumulate more values from positional arguments and sum all. @param xs: Values to add (C{scalar}s), all positional. @return: Accurate, running sum (C{float}). @see: Method L{Fsum.fsum}. @note: Accumulation can continue after summation. ''' return self.fsum(xs) def fsum2_(self, *xs): '''Accumulate more values from positional arguments, sum all and provide the sum and delta. @param xs: Values to add (C{scalar}s), all positional. @return: 2-Tuple C{(sum, delta)} with the accurate, running C{sum} and the C{delta} with the previous running C{sum}, both (C{float}). @see: Method L{Fsum.fsum_}. @note: Accumulation can continue after summation. ''' p = self._fsum2_ if p is None: p = self.fsum() s = self.fsum(xs) # if xs else self._fsum2_ return s, s - p class Fdot(Fsum): '''Precision dot product. ''' def __init__(self, a, *b): '''New L{Fdot} precision dot product M{sum(a[i] * b[i] for i=0..len(a))}. @param a: List, sequence, tuple, etc. (C{scalar}s). @param b: All positional arguments (C{scalar}s). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: Unequal C{len}(B{a}) and C{len}(B{b}). @see: Function L{fdot} and method L{Fsum.fadd}. ''' if not len(a) == len(b): raise ValueError('%s, %s: %s vs %s' % (self, 'len', len(a), len(b))) Fsum.__init__(self) self.fadd(map(mul, a, b)) class Fhorner(Fsum): '''Precision polynomial evaluation using the Horner form. ''' def __init__(self, x, *cs): '''New L{Fhorner} evaluation of the polynomial M{sum(cs[i] * x**i for i=0..len(cs))}. @param x: Polynomial argument (C{scalar}). @param cs: Polynomial coeffients (C{scalar}[]). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. @see: Function L{fhorner} and methods L{Fsum.fadd} and L{Fsum.fmul}. ''' if not isfinite(x): raise ValueError('%s, not %s: %r' % (self, 'finite', x)) if not cs: raise ValueError('%s, no %s: %r' % (self, 'coefficents', cs)) x, cs = float(x), list(cs) Fsum.__init__(self, cs.pop()) while cs: self.fmul(x) self.fadd_(cs.pop()) class Fpolynomial(Fsum): '''Precision polynomial evaluation. ''' def __init__(self, x, *cs): '''New L{Fpolynomial} evaluation of the polynomial M{sum(cs[i] * x**i for i=0..len(cs))}. @param x: Polynomial argument (C{scalar}). @param cs: Polynomial coeffients (C{scalar}[]). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. @see: Function L{fpolynomial} and method L{Fsum.fadd}. ''' if not isfinite(x): raise ValueError('%s, not %s: %r' % (self, 'finite', x)) if not cs: raise ValueError('%s, no %s: %r' % (self, 'coefficents', cs)) x, cs, xp = float(x), list(cs), 1 Fsum.__init__(self, cs.pop(0)) while cs: xp *= x self.fadd_(xp * cs.pop(0)) def acos1(x): '''Return M{math.acos(max(-1, min(1, x)))}. ''' return acos(max(-1.0, min(1.0, x))) def cbrt(x): '''Compute the cubic root M{x**(1/3)}. @param x: Value (C{scalar}). @return: Cubic root (C{float}). @see: Functions L{cbrt2} and L{sqrt3}. ''' # simpler and more accurate than Ken Turkowski's CubeRoot, see # <https://People.FreeBSD.org/~lstewart/references/apple_tr_kt32_cuberoot.pdf> return copysign(pow(abs(x), _1_3rd), x) def cbrt2(x): '''Compute the cubic root squared M{x**(2/3)}. @param x: Value (C{scalar}). @return: Cubic root squared (C{float}). @see: Functions L{cbrt} and L{sqrt3}. ''' return pow(abs(x), _2_3rd) def favg(v1, v2, f=0.5): '''Return the weighted average of two values. @param v1: One value (C{scalar}). @param v2: Other value (C{scalar}). @keyword f: Optional fraction (C{float}). @return: M{v1 + f * (v2 - v1)} (C{float}). ''' # @raise ValueError: Fraction out of range. # ''' # if not 0 <= f <= 1: # XXX restrict fraction? # raise ValueError('%s invalid: %r' % ('fraction', f)) return v1 + f * (v2 - v1) # v1 * (1 - f) + v2 * f def fdot(a, *b): '''Return the precision dot product M{sum(a[i] * b[i] for i=0..len(a))}. @param a: List, sequence, tuple, etc. (C{scalar}s). @param b: All positional arguments (C{scalar}s). @return: Dot product (C{float}). @raise ValueError: Unequal C{len(B{a})} and C{len(B{b})}. @see: Class L{Fdot}. ''' if not len(a) == len(b): raise ValueError('%s(%s): %s vs %s' % (fdot.__name__, 'len', len(a), len(b))) return fsum(map(mul, a, b)) def fdot3(a, b, c, start=0): '''Return the precision dot product M{start + sum(a[i] * b[i] * c[i] for i=0..len(a))}. @param a: List, sequence, tuple, etc. (C{scalar}[]). @param b: List, sequence, tuple, etc. (C{scalar}[]). @param c: List, sequence, tuple, etc. (C{scalar}[]). @keyword start: Optional bias (C{scalar}). @return: Dot product (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: Unequal C{len(B{a})}, C{len(B{b})} and/or C{len(B{c})}. ''' def _mul3(a, b, c): # map function return a * b * c # PYCHOK returns if not len(a) == len(b) == len(c): raise ValueError('%s(%s): %s vs %s vs %s' % (fdot3.__name__, 'len', len(a), len(b), len(c))) if start: f = Fsum(start) return f.fsum(map(_mul3, a, b, c)) else: return fsum(map(_mul3, a, b, c)) def fhorner(x, *cs): '''Evaluate the polynomial M{sum(cs[i] * x**i for i=0..len(cs))} using the Horner form. @param x: Polynomial argument (C{scalar}). @param cs: Polynomial coeffients (C{scalar}[]). @return: Horner value (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. @see: Function L{fpolynomial} and class L{Fhorner}. ''' h = Fhorner(x, *cs) return h.fsum() def fidw(xs, ds, beta=2): '''Interpolate using using U{Inverse Distance Weighting <https://WikiPedia.org/wiki/Inverse_distance_weighting>} (IDW). @param xs: Known values (C{scalar}[]). @param ds: Non-negative distances (C{scalar}[]). @keyword beta: Inverse distance power (C{int}, 0, 1, 2, or 3). @return: Interpolated value C{x} (C{float}). @raise ValueError: Invalid B{C{beta}}, negative B{C{ds}} value, weighted B{C{ds}} below L{EPS} or unequal C{len(B{ds})} and C{len(B{xs})}. @note: Using C{B{beta}=0} returns the mean of B{C{xs}}. ''' n, xs = len2(xs) d, ds = len2(ds) if n != d or n < 1: raise ValueError('%s(%s): %s vs %s' % (fidw.__name, 'len', n, d)) d, x = min(zip(ds, xs)) if d > EPS and n > 1: b = -int(beta) if -4 < b < 0: # and b == -beta ds = tuple(d**b for d in ds) d = fsum(ds) if d < EPS: raise ValueError('%s(%s[%s]) invalid: %r' % (fidw.__name, 'ds', '', d)) x = fdot(xs, *ds) / d elif b == 0: x = fmean(xs) else: raise ValueError('%s(%s=%r) invalid' % (fidw.__name, 'beta', beta)) elif d < 0: i = ds.index(d) raise ValueError('%s(%s[%s]) invalid: %r' % (fidw.__name, 'ds', i, d)) return x def fmean(xs): '''Compute the accurate mean M{sum(xs[i] for i=0..len(xs)) / len(xs)}. @param xs: Values (C{scalar}s). @return: Mean value (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: No B{C{xs}} values. ''' n, xs = len2(xs) if n > 0: return fsum(xs) / n raise ValueError('%s(%r)' % (fmean.__name__, xs)) def fpolynomial(x, *cs): '''Evaluate the polynomial M{sum(cs[i] * x**i for i=0..len(cs))}. @param x: Polynomial argument (C{scalar}). @param cs: Polynomial coeffients (C{scalar}[]). @return: Polynomial value (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. @see: Function L{fhorner} and class L{Fpolynomial}. ''' p = Fpolynomial(x, *cs) return p.fsum() def fpowers(x, n, alts=0): '''Return a series of powers M{[x**i for i=1..n]}. @param x: Value (C{scalar}). @param n: Highest exponent (C{int}). @keyword alts: Only alternating powers, starting with this exponent (C{int}). @return: Powers of B{C{x}} (C{float}[]). @raise TypeError: Non-scalar B{C{x}} or B{C{n}} not C{int}. @raise ValueError: Non-finite B{C{x}} or non-positive B{C{n}}. ''' if not isfinite(x): raise ValueError('not %s: %r' %('finite', x)) if not isinstance(n, _Ints): raise _IsNotError(int.__name_, n=n) elif n < 1: raise ValueError('%s invalid: %r' % ('n', n)) xs = [x] for _ in range(1, n): xs.append(xs[-1] * x) if alts > 0: # x**2, x**4, ... # XXX PyChecker chokes on xs[alts-1::2] xs = xs[slice(alts-1, None, 2)] # XXX PyChecker claims result is None return xs try: from math import prod as fprod # Python 3.8 except ImportError: def fprod(iterable, start=1.0): '''Iterable product, like C{math.prod} or C{numpy.prod}. @param iterable: Values to be multiplied (C{scalar}[]). @keyword start: Initial product, also the value returned for an empty iterable (C{scalar}). @return: The product (C{float}). @see: U{NumPy.prod<https://docs.SciPy.org/doc/ numpy/reference/generated/numpy.prod.html>}. ''' return freduce(mul, iterable, start) def frange(start, number, step=1): '''Generate a range of C{float}s. @param start: First value (C{float}). @param number: The number of C{float}s to generate (C{int}). @keyword step: Increment value (C{float}). @return: A generator (C{float}s). @see: U{NumPy.prod<https://docs.SciPy.org/doc/ numpy/reference/generated/numpy.arange.html>}. ''' if not isinstance(number, _Ints): raise _IsNotError(int.__name_, number=number) for i in range(number): yield start + i * step try: from functools import reduce as freduce except ImportError: # PYCHOK no cover try: freduce = reduce # PYCHOK expected except NameError: # Python 3+ _EMPTY = object() def freduce(f, iterable, *start): '''For missing C{functools.reduce}. ''' if start: r = v = start[0] else: r, v = 0, _EMPTY for v in iterable: r = f(r, v) if v is _EMPTY: raise TypeError('%s() empty, no start' % (freduce.__name__,)) return r def fStr(floats, prec=6, sep=', ', fmt='%.*f', ints=False): '''Convert floats to string, optionally with trailing zero decimals stripped. @param floats: List, sequence, tuple, etc. (C{scalar}s). @keyword prec: Optional precision, number of decimal digits (0..9). Trailing zero decimals are stripped for B{C{prec}} values of 1 and above, but kept for negative B{C{prec}} values. @keyword sep: Optional, separator to join (string). @keyword fmt: Optional, float format (string). @keyword ints: Optionally, remove decimal dot (C{bool}). @return: The floats as 'f, f, ... f' (string). ''' def _fstr(p, f): t = fmt % (abs(p), float(f)) if ints and (isint(f, both=True) or # for ... # corner case testLcc lon0=-96.0 t.rstrip('0').endswith('.')): t = t.split('.')[0] elif p > 1: t = fStrzs(t) return t # PYCHOK returns if isscalar(floats): return _fstr(prec, floats) else: return sep.join(_fstr(prec, f) for f in floats) def fStrzs(fstr): '''Strip trailing zero decimals from a float string. @param fstr: Float (string). @return: Float (string). ''' if fstr.endswith('0'): z = fstr.find('.') + 2 # keep 1st zero decimal if z > 1 and fstr[z:].isdigit(): # don't strip 'e+0..' fstr = fstr[:z] + fstr[z:].rstrip('0') return fstr def fsum_(*xs): '''Precision summation of the positional argument vulues. @param xs: Values to be added (C{scalar}[]). @return: Accurate L{fsum} (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{xs}} value. @raise ValueError: Invalid or non-finite B{C{xs}} value. ''' return fsum(xs) try: from math import fsum # precision IEEE-754 sum, Python 2.6+ # make sure fsum works as expected (XXX check # float.__getformat__('float')[:4] == 'IEEE'?) if fsum_(1, 1e101, 1, -1e101) != 2: del fsum # no, remove fsum ... raise ImportError # ... use fsum below except ImportError: # PYCHOK no cover def fsum(iterable): '''Precision summation similar to standard Python function C{math.fsum}. Exception and I{non-finite} handling differs from C{math.fsum}. @param iterable: Values to be added (C{scalar}[]). @return: Accurate C{sum} (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{iterable}} value. @raise ValueError: Invalid or non-finite B{C{iterable}} value. @see: Class L{Fsum}. ''' f = Fsum() return f.fsum(iterable) try: _ = hypot(1, 2, 3) # new in Python 3.8+ hypot_ = hypot del _ except TypeError: # Python 3.7- def hypot_(*xs): '''Compute the norm M{sqrt(sum(xs[i]**2)) for i=0..len(xs)}. @param xs: X arguments, positional (C{scalar}[]). @return: Norm (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: Invalid or no B{C{xs}} value. @see: Similar to Python 3.8+ U{math.hypot <https://docs.Python.org/3.8/library/math.html#math.hypot>}, but handling of exceptions, C{nan} and C{infinite} values is different. @note: The Python 3.8+ U{math.dist <https://docs.Python.org/3.8/library/math.html#math.dist>} Euclidian distance between 2 I{n}-dimensional points I{p1} and I{p2} can be computed as M{hypot_(*((c1 - c2) for c1, c2 in zip(p1, p2)))}, provided I{p1} and I{p2} have the same, non-zero length I{n}. ''' if xs: n, xs = len2(xs) if n > 0: h = float(max(abs(x) for x in xs)) if h > 0 and n > 1: X = Fsum(1.0) X.fadd((x / h)**2 for x in xs) h *= sqrt(X.fsum_(-1.0)) return h else: n = '' raise ValueError('%s(): %r[%s]' % (hypot_.__name__, xs, n)) def hypot1(x): '''Compute the norm M{sqrt(1 + x**2)}. @param x: Argument (C{scalar}). @return: Norm (C{float}). ''' return hypot(1.0, x) def hypot2(x, y): '''Compute the norm squared M{x**2 + y**2}. @param x: Argument (C{scalar}). @param y: Argument (C{scalar}). @return: C{B{x}**2 + B{y}**2} (C{float}). ''' x, y = x**2, y**2 if x < y: x, y = y, x if x: x *= 1 + y / x return x try: from math import isfinite # new in Python 3+ except ImportError: def isfinite(obj): '''Check for C{Inf} and C{NaN} values. @param obj: Value (C{scalar}). @return: C{False} if B{C{obj}} is C{INF} or C{NAN}, C{True} otherwise. @raise TypeError: Non-scalar B{C{obj}}. ''' if not isscalar(obj): raise _IsNotError(isscalar.__name__, obj=obj) return not (isinf(obj) or isnan(obj)) def isint(obj, both=False): '''Check for integer type or integer value. @param obj: The object (any C{type}). @keyword both: Optionally, check both type and value (C{bool}). @return: C{True} if B{C{obj}} is C{int}, C{False} otherwise. ''' if both and isinstance(obj, float): # NOT _Scalars! try: return obj.is_integer() except AttributeError: return False # XXX float(int(obj)) == obj? return isinstance(obj, _Ints) def isneg0(obj): '''Check for NEG0, negative 0.0. @param obj: Value (C{scalar}). @return: C{True} if B{C{obj}} is C{NEG0} or -0.0, C{False} otherwise. ''' return obj in (0.0, NEG0) and copysign(1, obj) < 0 # and str(obj).rstrip('0') == '-0.' def isscalar(obj): '''Check for scalar types. @param obj: The object (any C{type}). @return: C{True} if B{C{obj}} is C{scalar}, C{False} otherwise. ''' return isinstance(obj, _Scalars) def len2(seq): '''Make built-in function L{len} work for generators, iterators, etc. since those can only be started exactly once. @param seq: Generator, iterator, list, range, tuple, etc. @return: 2-Tuple (number, ...) of items (C{int}, C{list} or C{range} or C{tuple}). ''' if not isinstance(seq, _Seqs): # not hasattr(seq, '__len__'): seq = list(seq) return len(seq), seq def map1(func, *xs): '''Apply each argument to a single-argument function and return a tuple of results. @param func: Function to apply (C{callable}). @param xs: Arguments to apply (C{any positional}). @return: Function results (C{tuple}). ''' return tuple(map(func, xs)) def map2(func, *xs): '''Apply arguments to a function and return a tuple of results. Unlike Python 2's built-in L{map}, Python 3+ L{map} returns a L{map} object, an iterator-like object which generates the results only once. Converting the L{map} object to a tuple maintains Python 2 behavior. @param func: Function to apply (C{callable}). @param xs: Arguments to apply (C{list, tuple, ...}). @return: Function results (C{tuple}). ''' return tuple(map(func, *xs)) def scalar(value, low=EPS, high=1.0, name='scalar', Error=ValueError): '''Validate a scalar. @param value: The value (C{scalar}). @keyword low: Optional lower bound (C{scalar}). @keyword high: Optional upper bound (C{scalar}). @keyword name: Optional name of value (C{str}). @keyword Error: Exception to raise (C{ValueError}). @return: New value (C{type} of B{C{low}}). @raise TypeError: Non-scalar B{C{value}}. @raise Error: Out-of-bounds B{C{value}}. ''' if not isscalar(value): raise _IsNotError(scalar.__name__, **{name: value}) try: if low is None: v = float(value) else: v = type(low)(value) if v < low or v > high: raise ValueError except (TypeError, ValueError): raise _IsNotError('valid', Error=Error, **{name: value}) return v def sqrt3(x): '''Compute the square root cubed M{sqrt(x)**3} or M{sqrt(x**3)}. @param x: Value (C{scalar}). @return: Cubed square root (C{float}). @raise ValueError: Negative B{C{x}}. @see: Functions L{cbrt} and L{cbrt2}. ''' if x < 0: raise ValueError('%s(%r)' % (sqrt3.__name__, x)) return pow(x, _3_2nd) # **) MIT License # # Copyright (C) 2016-2020 -- 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.