# Coverage for pygeodesy/fmath.py : 94%

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''' # make sure int/int division yields float quotient, see .basics from __future__ import division as _; del _ # PYCHOK semicolon from pygeodesy.basics import copysign0, isint, isnear0, len2 from pygeodesy.errors import _IsnotError, LenError, _TypeError, _ValueError from pygeodesy.fsums import _2float, Fmt, Fsum, fsum, fsum1_ from pygeodesy.interns import EPS0, EPS02, EPS1, MISSING, PI, PI_2, PI_4, \ _few_, _h_, _negative_, \ _singular_, _too_, _0_0, _0_5, \ _1_0, _N_1_0, _1_5, _2_0, _3_0 from pygeodesy.lazily import _ALL_LAZY, _sys_version_info2 # from pygeodesy.streprs import Fmt # from .fsums from pygeodesy.units import Int_ from math import sqrt # pow from operator import mul as _mul __all__ = _ALL_LAZY.fmath __version__ = '22.02.04' # sqrt(2) <https://WikiPedia.org/wiki/Square_root_of_2> _0_4142 = 0.414213562373095 # sqrt(_2_0) - _1_0 _1_3rd = _1_0 / _3_0 _2_3rd = _2_0 / _3_0 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))}. @arg a: Iterable, list, tuple, etc. (C{scalar}s). @arg b: Other values (C{scalar}s), all positional. @raise OverflowError: Partial C{2sum} overflow. @raise LenError: Unequal C{len(B{a})} and C{len(B{b})}. @see: Function L{fdot} and method L{Fsum.fadd}. ''' Fsum.__init__(self) self.fadd(_map_a_x_b(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))}. @arg x: Polynomial argument (C{scalar}). @arg cs: Polynomial coeffients (C{scalar}s), all positional. @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: Non-finite B{C{x}}. @see: Function L{fhorner} and methods L{Fsum.fadd} and L{Fsum.fmul}. ''' Fsum.__init__(self, *cs[-1:]) if len(cs) > 1: x = _2float(x=x) a_ = self.fadd_ ps = self._ps for c in reversed(cs[:-1]): # multiply-accumulate ps[:] = [p * x for p in ps] a_(c) # assert self._ps is ps 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))}. @arg x: Polynomial argument (C{scalar}). @arg cs: Polynomial coeffients (C{scalar}s), all positional. @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: Non-finite B{C{x}}. @see: Function L{fpolynomial} and method L{Fsum.fadd}. ''' Fsum.__init__(self, *cs[:1]) n = len(cs) - 1 if n > 0: self.fadd(_map_a_x_b(cs[1:], fpowers(x, n))) def cbrt(x3): '''Compute the cube root M{x3**(1/3)}. @arg x3: Value (C{scalar}). @return: Cubic root (C{float}). @see: Functions L{cbrt2} and L{sqrt3}. ''' # <https://archive.lib.MSU.edu/crcmath/math/math/r/r021.htm> # simpler and more accurate than Ken Turkowski's CubeRoot, see # <https://People.FreeBSD.org/~lstewart/references/apple_tr_kt32_cuberoot.pdf> return copysign0(pow(abs(x3), _1_3rd), x3) def cbrt2(x3): '''Compute the cube root I{squared} M{x3**(2/3)}. @arg x3: Value (C{scalar}). @return: Cube root I{squared} (C{float}). @see: Functions L{cbrt} and L{sqrt3}. ''' return pow(abs(x3), _2_3rd) # XXX pow(abs(x3), _1_3rd)**2 def euclid(x, y): '''I{Appoximate} the norm M{sqrt(x**2 + y**2)} by M{max(abs(x), abs(y)) + min(abs(x), abs(y)) * 0.4142...}. @arg x: X component (C{scalar}). @arg y: Y component (C{scalar}). @return: Appoximate norm (C{float}). @see: Function L{euclid_}. ''' x, y = abs(x), abs(y) if y > x: x, y = y, x return x + y * _0_4142 # XXX * _0_5 before 20.10.02 def euclid_(*xs): '''I{Appoximate} the norm M{sqrt(sum(x**2 for x in xs))} by cascaded L{euclid}. @arg xs: X arguments, positional (C{scalar}s). @return: Appoximate norm (C{float}). @see: Function L{euclid}. ''' e = _0_0 for x in sorted(map(abs, xs)): # XXX not reverse=True # e = euclid(x, e) if x > e: e, x = x, e if x: e += x * _0_4142 return e def facos1(x): '''Fast approximation of L{pygeodesy.acos1}C{(B{x})}. @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/ ShaderFastLibs/blob/master/ShaderFastMathLib.h>}. ''' a = abs(x) if a < EPS0: r = PI_2 elif a < EPS1: H = Fhorner(-a, 1.5707288, 0.2121144, 0.0742610, 0.0187293) r = H.fmul(sqrt(_1_0 - a)).fsum() if x < 0: r = PI - r else: r = PI if x < 0 else _0_0 return r def fasin1(x): # PYCHOK no cover '''Fast approximation of L{pygeodesy.asin1}C{(B{x})}. @see: L{facos1}. ''' return PI_2 - facos1(x) def fatan(x): '''Fast approximation of C{atan(B{x})}. ''' a = abs(x) if a < _1_0: r = fatan1(a) if a else _0_0 elif a > _1_0: r = PI_2 - fatan1(_1_0 / a) # == fatan2(a, _1_0) else: r = PI_4 if x < 0: # copysign0(r, x) r = -r return r def fatan1(x): '''Fast approximation of C{atan(B{x})} for C{0 <= B{x} <= 1}, I{unchecked}. @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/ ShaderFastLibs/blob/master/ShaderFastMathLib.h>} and U{Efficient approximations for the arctangent function <http://www-Labs.IRO.UMontreal.Ca/~mignotte/IFT2425/Documents/ EfficientApproximationArctgFunction.pdf>}, IEEE Signal Processing Magazine, 111, May 2006. ''' # Eq (9): PI_4 * x - x * (x - 1) * (0.2447 + 0.0663 * x**2) # == x * (1.0300982 + x * (-0.2447 + x * 0.0663 * (1 - x))) return x * fhorner(x, 1.0300982, -0.2447, 0.0663, -0.0663) def fatan2(y, x): '''Fast approximation of C{atan2(B{y}, B{x})}. @see: U{fastApproximateAtan(x, y)<https://GitHub.com/CesiumGS/cesium/blob/ master/Source/Shaders/Builtin/Functions/fastApproximateAtan.glsl>} and L{fatan1}. ''' b, a = abs(y), abs(x) if a < b: r = (PI_2 - fatan1(a / b)) if a else PI_2 elif b < a: r = fatan1(b / a) if b else _0_0 elif a: # == b != 0 r = PI_4 else: # a == b == 0 return _0_0 if x < 0: r = PI - r if y < 0: # copysign0(r, y) r = -r return r def favg(v1, v2, f=_0_5): '''Return the average of two values. @arg v1: One value (C{scalar}). @arg v2: Other value (C{scalar}). @kwarg 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(fraction=f) # v1 + f * (v2 - v1) == v1 * (1 - f) + v2 * f return fsum1_(v1, -f * v1, f * v2) def _map_a_x_b(a, b): '''(INTERNAL) Yield B{C{a * b}}. ''' n = len(b) if len(a) != n: from pygeodesy.fmath import fdot raise LenError(fdot, a=len(a), b=n) return map(_mul, a, b) if n > 3 else _map_a_x_b1(a, b) def _map_a_x_b1(a, b): '''(INTERNAL) Yield B{C{a * b}}, primed with C{1.0}. ''' yield _1_0 for ab in map(_mul, a, b): if ab: yield ab yield _N_1_0 def fdot(a, *b): '''Return the precision dot product M{sum(a[i] * b[i] for i=0..len(a))}. @arg a: Iterable, list, tuple, etc. (C{scalar}s). @arg b: All positional arguments (C{scalar}s). @return: Dot product (C{float}). @raise LenError: Unequal C{len(B{a})} and C{len(B{b})}. @see: Class L{Fdot} and U{Algorithm 5.10 B{DotK} <https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>}. ''' return fsum(_map_a_x_b(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))}. @arg a: Iterable, list, tuple, etc. (C{scalar}s). @arg b: Iterable, list, tuple, etc. (C{scalar}s). @arg c: Iterable, list, tuple, etc. (C{scalar}s). @kwarg start: Optional bias (C{scalar}). @return: Dot product (C{float}). @raise LenError: Unequal C{len(B{a})}, C{len(B{b})} and/or C{len(B{c})}. @raise OverflowError: Partial C{2sum} overflow. ''' def _mul3(a, b, c): # map function return a * b * c def _muly(a, b, c, start): yield start for abc in map(_mul3, a, b, c): yield abc if not len(a) == len(b) == len(c): raise LenError(fdot3, a=len(a), b=len(b), c=len(c)) return fsum(_muly(a, b, c, start) if start else 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. @arg x: Polynomial argument (C{scalar}). @arg cs: Polynomial coeffients (C{scalar}s). @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). @arg xs: Known values (C{scalar}s). @arg ds: Non-negative distances (C{scalar}s). @kwarg beta: Inverse distance power (C{int}, 0, 1, 2, or 3). @return: Interpolated value C{x} (C{float}). @raise LenError: Unequal or zero C{len(B{ds})} and C{len(B{xs})}. @raise ValueError: Invalid B{C{beta}}, negative B{C{ds}} value, weighted B{C{ds}} below L{EPS}. @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 LenError(fidw, xs=n, ds=d) d, x = min(zip(ds, xs)) if d > EPS0 and n > 1: b = -Int_(beta=beta, low=0, high=3) if b < 0: ds = tuple(d**b for d in ds) d = fsum(ds) if isnear0(d): # PYCHOK no cover n = Fmt.PAREN(fsum='ds') raise _ValueError(n, d, txt=_singular_) x = fdot(xs, *ds) / d else: # b == 0 x = fsum(xs) / n # fmean(xs) elif d < 0: # PYCHOK no cover n = Fmt.SQUARE(ds=ds.index(d)) raise _ValueError(n, d, txt=_negative_) return x def fmean(xs): '''Compute the accurate mean M{sum(xs[i] for i=0..len(xs)) / len(xs)}. @arg xs: Values (C{scalar} or L{Fsum} instances). @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(xs=xs) def fmean_(*xs): '''Compute the accurate mean M{sum(xs[i] for i=0..len(xs)) / len(xs)}. @arg xs: Values (C{scalar} or L{Fsum} instances). @return: Mean value (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: No B{C{xs}} values. ''' return fmean(xs) def fpolynomial(x, *cs): '''Evaluate the polynomial M{sum(cs[i] * x**i for i=0..len(cs))}. @arg x: Polynomial argument (C{scalar}). @arg cs: Polynomial coeffients (C{scalar}s), all positional. @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]}. @arg x: Value (C{scalar}). @arg n: Highest exponent (C{int}). @kwarg alts: Only alternating powers, starting with this exponent (C{int}). @return: Powers of B{C{x}} (C{float}s or C{int}s). @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 isint(n): raise _IsnotError(int.__name__, n=n) elif n < 1: raise _ValueError(n=n) p = t = x if isint(x) else _2float(x=x) ps = [p] a_ = ps.append for _ in range(1, n): p *= t a_(p) if alts > 0: # x**2, x**4, ... # ps[alts-1::2] chokes PyChecker ps = ps[slice(alts-1, None, 2)] return ps try: from math import prod as fprod # Python 3.8 except ImportError: def fprod(xs, start=_1_0): '''Iterable product, like C{math.prod} or C{numpy.prod}. @arg xs: Terms to be multiplied as an iterable, list, tuple, etc. (C{scalar}s). @kwarg start: Initial term, also the value returned for an empty B{C{xs}} (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, xs, start) def frange(start, number, step=1): '''Generate a range of C{float}s. @arg start: First value (C{float}). @arg number: The number of C{float}s to generate (C{int}). @kwarg 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 isint(number): raise _IsnotError(int.__name__, number=number) for i in range(number): yield start + i * step try: from functools import reduce as freduce except ImportError: try: freduce = reduce # PYCHOK expected except NameError: # Python 3+ def freduce(f, xs, *start): '''For missing C{functools.reduce}. ''' if start: r = v = start[0] else: r, v = 0, MISSING for v in xs: r = f(r, v) if v is MISSING: raise _TypeError(xs=(), start=MISSING) return r if _sys_version_info2 < (3, 8): from math import hypot # OK in Python 3.7- def hypot_(*xs): '''Compute the norm M{sqrt(sum(x**2 for x in xs))}. Similar to Python 3.8+ n-dimension U{math.hypot <https://docs.Python.org/3.8/library/math.html#math.hypot>}, but exceptions, C{nan} and C{infinite} values are handled differently. @arg xs: X arguments (C{scalar}s), all positional. @return: Norm (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: Invalid or no B{C{xs}} values. @note: The Python 3.8+ Euclidian distance U{math.dist <https://docs.Python.org/3.8/library/math.html#math.dist>} 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}. ''' h, x2 = _h_x2(xs) return (h * sqrt(x2)) if x2 else _0_0 elif _sys_version_info2 < (3, 10): # In Python 3.8 and 3.9 C{math.hypot} is inaccurate, see # agdhruv <https://GitHub.com/geopy/geopy/issues/466>, # cffk <https://Bugs.Python.org/issue43088> and module # geomath.py <https://PyPI.org/project/geographiclib/1.52> def hypot(x, y): '''Compute the norm M{sqrt(x**2 + y**2)}. @arg x: X argument (C{scalar}). @arg y: Y argument (C{scalar}). @return: C{sqrt(B{x}**2 + B{y}**2)} (C{float}). ''' if x: h = sqrt(fsum1_(x**2, y**2)) if y else abs(x) elif y: h = abs(y) else: h = _0_0 return h from math import hypot as hypot_ # PYCHOK in Python 3.8 and 3.9 else: from math import hypot # PYCHOK in Python 3.10+ hypot_ = hypot def _h_x2(xs): '''(INTERNAL) Helper for L{hypot_} and L{hypot2_}. ''' def _x2s(xs, h): yield _1_0 for x in xs: if x: yield (x / h)**2 yield _N_1_0 if xs: n, xs = len2(xs) if n > 0: h = float(max(map(abs, xs))) x2 = fsum(_x2s(xs, h)) if h else _0_0 return h, x2 raise _ValueError(xs=xs, txt=_too_(_few_)) def hypot1(x): '''Compute the norm M{sqrt(1 + x**2)}. @arg x: Argument (C{scalar}). @return: Norm (C{float}). ''' return hypot(_1_0, x) if x else _1_0 def hypot2(x, y): '''Compute the I{squared} norm M{x**2 + y**2}. @arg x: X argument (C{scalar}). @arg y: Y argument (C{scalar}). @return: C{B{x}**2 + B{y}**2} (C{float}). ''' if x: x *= x h2 = fsum1_(x, y**2) if y else x elif y: h2 = y**2 else: h2 = _0_0 return h2 def hypot2_(*xs): '''Compute the I{squared} norm C{sum(x**2 for x in B{xs})}. @arg xs: X arguments (C{scalar}s), all positional. @return: Squared norm (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: Invalid or no B{C{xs}} value. @see: Function L{hypot_}. ''' h, x2 = _h_x2(xs) return (h**2 * x2) if x2 else _0_0 def norm2(x, y): '''Normalize a 2-dimensional vector. @arg x: X component (C{scalar}). @arg y: Y component (C{scalar}). @return: 2-Tuple C{(x, y)}, normalized. @raise ValueError: Invalid B{C{x}} or B{C{y}} or zero norm. ''' h = hypot(x, y) try: return x / h, y / h except (TypeError, ValueError) as X: raise _ValueError(x=x, y=y, h=h, txt=str(X)) def norm_(*xs): '''Normalize all n-dimensional vector components. @arg xs: Components (C{scalar}s), all positional. @return: Yield each component, normalized. @raise ValueError: Invalid or insufficent B{C{xs}} or zero norm. ''' h = hypot_(*xs) try: for i, x in enumerate(xs): yield x / h except (TypeError, ValueError) as X: raise _ValueError(Fmt.SQUARE(xs=i), x, _h_, h, txt=str(X)) def sqrt0(x2): '''Compute the square root iff C{B{x2} >} L{EPS02}. @arg x2: Value (C{scalar}). @return: Square root (C{float}) or C{0.0}. @note: Any C{B{x2} <} L{EPS02} I{including} C{B{x2} < 0} returns C{0.0}. ''' return sqrt(x2) if x2 > EPS02 else (_0_0 if x2 < EPS02 else EPS0) def sqrt3(x2): '''Compute the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)}. @arg x2: Value (C{scalar}). @return: Cubed square root (C{float}). @raise ValueError: Negative B{C{x2}}. @see: Functions L{cbrt} and L{cbrt2}. ''' if x2 < 0: raise _ValueError(x2=x2, txt=_negative_) return pow(x2, _1_5) if x2 else _0_0 # **) MIT License # # Copyright (C) 2016-2022 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE.