Coverage for pygeodesy/fmath.py: 96%
275 statements
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2# -*- coding: utf-8 -*-
4u'''Utilities using precision floating point summation.
5'''
6# make sure int/int division yields float quotient, see .basics
7from __future__ import division as _; del _ # PYCHOK semicolon
9from pygeodesy.basics import _copysign, copysign0, isint, len2
10from pygeodesy.constants import EPS0, EPS02, EPS1, NAN, PI, PI_2, PI_4, \
11 _0_0, _0_125, _0_25, _0_5, _1_0, _N_1_0, \
12 _1_3rd, _1_5, _1_6th, _2_0, _2_3rd, _3_0, \
13 _isfinite, isnear0, isnear1, _over, remainder
14from pygeodesy.errors import _IsnotError, LenError, _TypeError, _ValueError, \
15 _xError, _xkwds_get, _xkwds_pop
16from pygeodesy.fsums import _2float, _Powers, Fsum, _fsum, fsum, fsum1_, \
17 _pow_op_, Fmt, unstr
18from pygeodesy.interns import MISSING, _few_, _h_, _invokation_, _negative_, \
19 _not_scalar_, _SPACE_, _too_
20from pygeodesy.lazily import _ALL_LAZY, _sys_version_info2
21# from pygeodesy.streprs import Fmt, unstr # from .fsums
22from pygeodesy.units import Int_, _isHeight, _isRadius, Float_ # PYCHOK for .heights
24from math import fabs, sqrt # pow
25from operator import mul as _mul # in .triaxials
27__all__ = _ALL_LAZY.fmath
28__version__ = '23.12.03'
30# sqrt(2) <https://WikiPedia.org/wiki/Square_root_of_2>
31_0_4142 = 0.414213562373095 # sqrt(_2_0) - _1_0
34class Fdot(Fsum):
35 '''Precision dot product.
36 '''
37 def __init__(self, a, *b, **name):
38 '''New L{Fdot} precision dot product M{sum(a[i] * b[i]
39 for i=0..len(a))}.
41 @arg a: Iterable, list, tuple, etc. (C{scalar}s).
42 @arg b: Other values (C{scalar}s), all positional.
43 @kwarg name: Optional name (C{str}).
45 @raise OverflowError: Partial C{2sum} overflow.
47 @raise LenError: Unequal C{len(B{a})} and C{len(B{b})}.
49 @see: Function L{fdot} and method L{Fsum.fadd}.
50 '''
51 Fsum.__init__(self, **name)
52 self.fadd(_map_mul(a, b, Fdot))
55class Fhorner(Fsum):
56 '''Precision polynomial evaluation using the Horner form.
57 '''
58 def __init__(self, x, *cs, **name):
59 '''New L{Fhorner} evaluation of the polynomial
60 M{sum(cs[i] * x**i for i=0..len(cs))}.
62 @arg x: Polynomial argument (C{scalar}).
63 @arg cs: Polynomial coeffients (C{scalar} or C{Fsum}
64 instances), all positional.
65 @kwarg name: Optional name (C{str}).
67 @raise OverflowError: Partial C{2sum} overflow.
69 @raise TypeError: Non-scalar B{C{x}}.
71 @raise ValueError: Non-finite B{C{x}}.
73 @see: Function L{fhorner} and methods L{Fsum.fadd} and L{Fsum.fmul}.
74 '''
75 Fsum.__init__(self, *cs[-1:], **name)
76 if len(cs) > 1:
77 x = _2float(x=x)
78 _a = self._fadd # (other, op)
79 _f = self._finite # (other, op)
80 op = Fhorner.__name__
81 ps = self._ps
82 for c in reversed(cs[:-1]): # multiply-accumulate
83 ps[:] = [_f(p * x, op) for p in ps]
84 _a(c, op)
85 # assert self._ps is ps
88class Fhypot(Fsum):
89 '''Precision hypotenuse of summation.
90 '''
91 def __init__(self, *xs, **power_name_RESIDUAL):
92 '''New L{Fhypot} hypotenuse of (the I{power} of) several
93 C{scalar} or C{Fsum} values.
95 @arg xs: One or more values to include (each C{scalar}
96 or an C{Fsum} instance).
97 @kwarg power_name_RESIDUAL: Optional exponent and root
98 order C{B{power}=2}, C{B{name}=NN} and
99 C{B{RESIDUAL}=None}, see L{Fsum.__init__}.
100 '''
101 try:
102 p = _xkwds_pop(power_name_RESIDUAL, power=2)
103 Fsum.__init__(self, **power_name_RESIDUAL)
104 if xs:
105 self._facc(_Powers(p, xs), up=False) # PYCHOK yield
106 self._fset(self._fpow(_1_0 / p, _pow_op_), asis=True)
107 except Exception as X:
108 raise self._ErrorX(X, xs, power=p)
111class Fpolynomial(Fsum):
112 '''Precision polynomial evaluation.
113 '''
114 def __init__(self, x, *cs, **name):
115 '''New L{Fpolynomial} evaluation of the polynomial
116 M{sum(cs[i] * x**i for i=0..len(cs))}.
118 @arg x: Polynomial argument (C{scalar}).
119 @arg cs: Polynomial coeffients (C{scalar}s), all
120 positional.
121 @kwarg name: Optional name (C{str}).
123 @raise OverflowError: Partial C{2sum} overflow.
125 @raise TypeError: Non-scalar B{C{x}}.
127 @raise ValueError: Non-finite B{C{x}}.
129 @see: Function L{fpolynomial} and method L{Fsum.fadd}.
130 '''
131 Fsum.__init__(self, *cs[:1], **name)
132 n = len(cs) - 1
133 if n > 0:
134 self.fadd(_map_mul(cs[1:], fpowers(x, n), Fpolynomial))
137class Fpowers(Fsum):
138 '''Precision summation or powers, optimized for C{power=2}.
139 '''
140 def __init__(self, power, *xs, **name_RESIDUAL):
141 '''New L{Fpowers} sum of (the I{power} of) several C{scalar}
142 or C{Fsum} values.
144 @arg power: The exponent (C{scalar} or C{Fsum}).
145 @arg xs: One or more values to include (each C{scalar}
146 or an C{Fsum} instance).
147 @kwarg power_name_RESIDUAL: Optional exponent and root
148 order C{B{power}=2}, C{B{name}=NN} and
149 C{B{RESIDUAL}=None}, see L{Fsum.__init__}.
150 '''
151 try:
152 Fsum.__init__(self, **name_RESIDUAL)
153 if xs:
154 self._facc(_Powers(power, xs), up=False) # PYCHOK yield
155 except Exception as X:
156 raise self._ErrorX(X, xs, power=power)
159class Fn_rt(Fsum):
160 '''Precision n-th root of summation.
161 '''
162 def __init__(self, root, *xs, **name_RESIDUAL):
163 '''New L{Fn_rt} root of the precision sum of several
164 C{scalar} or C{Fsum} values.
166 @arg root: The order (C{scalar} or C{Fsum}).
167 @arg xs: Values to include (each C{scalar} or an
168 C{Fsum} instance).
169 @kwarg name_RESIDUAL: See L{Fsum.__init__}.
170 '''
171 try:
172 Fsum.__init__(self, *xs, **name_RESIDUAL)
173 self._fset(self._fpow(_1_0 / root, _pow_op_), asis=True)
174 except Exception as X:
175 raise self._ErrorX(X, xs, root=root)
178class Fcbrt(Fn_rt):
179 '''Precision cubic root of summation.
180 '''
181 def __init__(self, *xs, **name_RESIDUAL):
182 '''New L{Fcbrt} cubic root of the precision sum of
183 several C{scalar} or C{Fsum} values.
185 @arg xs: Values to include (each C{scalar} or an
186 C{Fsum} instance).
187 @kwarg name_RESIDUAL: See L{Fsum.__init__}.
188 '''
189 Fn_rt.__init__(self, _3_0, *xs, **name_RESIDUAL)
192class Fsqrt(Fn_rt):
193 '''Precision square root of summation.
194 '''
195 def __init__(self, *xs, **name_RESIDUAL):
196 '''New L{Fsqrt} square root of the precision sum of
197 several C{scalar} or C{Fsum} values.
199 @arg xs: Values to include (each C{scalar} or an
200 C{Fsum} instance).
201 @kwarg name_RESIDUAL: See L{Fsum.__init__}.
202 '''
203 Fn_rt.__init__(self, _2_0, *xs, **name_RESIDUAL)
206def bqrt(x):
207 '''Return the 4-th, I{bi-quadratic} or I{quartic} root, M{x**(1 / 4)}.
209 @arg x: Value (C{scalar}).
211 @return: I{Quartic} root (C{float}).
213 @raise ValueError: Negative B{C{x}}.
215 @see: Functions L{zcrt} and L{zqrt}.
216 '''
217 return _root(x, _0_25, bqrt)
220try:
221 from math import cbrt # Python 3.11+
223 def cbrt2(x):
224 '''Compute the cube root I{squared} M{x**(2/3)}.
225 '''
226 return cbrt(x)**2 # cbrt(-0.0*2) == -0.0
228except ImportError: # Python 3.10-
230 def cbrt(x):
231 '''Compute the cube root M{x**(1/3)}.
233 @arg x: Value (C{scalar}).
235 @return: Cubic root (C{float}).
237 @see: Functions L{cbrt2} and L{sqrt3}.
238 '''
239 # <https://archive.lib.MSU.edu/crcmath/math/math/r/r021.htm>
240 # simpler and more accurate than Ken Turkowski's CubeRoot, see
241 # <https://People.FreeBSD.org/~lstewart/references/apple_tr_kt32_cuberoot.pdf>
242 return _copysign(pow(fabs(x), _1_3rd), x) # cbrt(-0.0) == -0.0
244 def cbrt2(x): # PYCHOK attr
245 '''Compute the cube root I{squared} M{x**(2/3)}.
247 @arg x: Value (C{scalar}).
249 @return: Cube root I{squared} (C{float}).
251 @see: Functions L{cbrt} and L{sqrt3}.
252 '''
253 return pow(fabs(x), _2_3rd) # XXX pow(fabs(x), _1_3rd)**2
256def euclid(x, y):
257 '''I{Appoximate} the norm M{sqrt(x**2 + y**2)} by
258 M{max(abs(x), abs(y)) + min(abs(x), abs(y)) * 0.4142...}.
260 @arg x: X component (C{scalar}).
261 @arg y: Y component (C{scalar}).
263 @return: Appoximate norm (C{float}).
265 @see: Function L{euclid_}.
266 '''
267 x, y = fabs(x), fabs(y)
268 if x < y:
269 x, y = y, x
270 return x + y * _0_4142 # XXX * _0_5 before 20.10.02
273def euclid_(*xs):
274 '''I{Appoximate} the norm M{sqrt(sum(x**2 for x in xs))}
275 by cascaded L{euclid}.
277 @arg xs: X arguments, positional (C{scalar}s).
279 @return: Appoximate norm (C{float}).
281 @see: Function L{euclid}.
282 '''
283 e = _0_0
284 for x in sorted(map(fabs, xs)): # XXX not reverse=True
285 # e = euclid(x, e)
286 if e < x:
287 e, x = x, e
288 if x:
289 e += x * _0_4142
290 return e
293def facos1(x):
294 '''Fast approximation of L{pygeodesy.acos1}C{(B{x})}.
296 @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/
297 ShaderFastLibs/blob/master/ShaderFastMathLib.h>}.
298 '''
299 a = fabs(x)
300 if a < EPS0:
301 r = PI_2
302 elif a < EPS1:
303 H = Fhorner(-a, 1.5707288, 0.2121144, 0.0742610, 0.0187293)
304 r = float(H * sqrt(_1_0 - a))
305 if x < 0:
306 r = PI - r
307 else:
308 r = PI if x < 0 else _0_0
309 return r
312def fasin1(x): # PYCHOK no cover
313 '''Fast approximation of L{pygeodesy.asin1}C{(B{x})}.
315 @see: L{facos1}.
316 '''
317 return PI_2 - facos1(x)
320def fatan(x):
321 '''Fast approximation of C{atan(B{x})}.
322 '''
323 a = fabs(x)
324 if a < _1_0:
325 r = fatan1(a) if a else _0_0
326 elif a > _1_0:
327 r = PI_2 - fatan1(_1_0 / a) # == fatan2(a, _1_0)
328 else:
329 r = PI_4
330 if x < 0: # copysign0(r, x)
331 r = -r
332 return r
335def fatan1(x):
336 '''Fast approximation of C{atan(B{x})} for C{0 <= B{x} <= 1}, I{unchecked}.
338 @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/ShaderFastLibs/
339 blob/master/ShaderFastMathLib.h>} and U{Efficient approximations
340 for the arctangent function<http://www-Labs.IRO.UMontreal.CA/
341 ~mignotte/IFT2425/Documents/EfficientApproximationArctgFunction.pdf>},
342 IEEE Signal Processing Magazine, 111, May 2006.
343 '''
344 # Eq (9): PI_4 * x - x * (abs(x) - 1) * (0.2447 + 0.0663 * abs(x)), for -1 < x < 1
345 # PI_4 * x - (x**2 - x) * (0.2447 + 0.0663 * x), for 0 < x - 1
346 # x * (1.0300981633974482 + x * (-0.1784 - x * 0.0663))
347 H = Fhorner(x, _0_0, 1.0300982, -0.1784, -0.0663)
348 return float(H)
351def fatan2(y, x):
352 '''Fast approximation of C{atan2(B{y}, B{x})}.
354 @see: U{fastApproximateAtan(x, y)<https://GitHub.com/CesiumGS/cesium/blob/
355 master/Source/Shaders/Builtin/Functions/fastApproximateAtan.glsl>}
356 and L{fatan1}.
357 '''
358 b, a = fabs(y), fabs(x)
359 if a < b:
360 r = (PI_2 - fatan1(a / b)) if a else PI_2
361 elif b < a:
362 r = fatan1(b / a) if b else _0_0
363 elif a: # == b != 0
364 r = PI_4
365 else: # a == b == 0
366 return _0_0
367 if x < 0:
368 r = PI - r
369 if y < 0: # copysign0(r, y)
370 r = -r
371 return r
374def favg(v1, v2, f=_0_5):
375 '''Return the average of two values.
377 @arg v1: One value (C{scalar}).
378 @arg v2: Other value (C{scalar}).
379 @kwarg f: Optional fraction (C{float}).
381 @return: M{v1 + f * (v2 - v1)} (C{float}).
382 '''
383# @raise ValueError: Fraction out of range.
384# '''
385# if not 0 <= f <= 1: # XXX restrict fraction?
386# raise _ValueError(fraction=f)
387 # v1 + f * (v2 - v1) == v1 * (1 - f) + v2 * f
388 return fsum1_(v1, -f * v1, f * v2)
391def fdot(a, *b):
392 '''Return the precision dot product M{sum(a[i] * b[i] for
393 i=0..len(a))}.
395 @arg a: Iterable, list, tuple, etc. (C{scalar}s).
396 @arg b: All positional arguments (C{scalar}s).
398 @return: Dot product (C{float}).
400 @raise LenError: Unequal C{len(B{a})} and C{len(B{b})}.
402 @see: Class L{Fdot} and U{Algorithm 5.10 B{DotK}
403 <https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>}.
404 '''
405 return fsum(_map_mul(a, b, fdot))
408def fdot3(a, b, c, start=0):
409 '''Return the precision dot product M{start +
410 sum(a[i] * b[i] * c[i] for i=0..len(a))}.
412 @arg a: Iterable, list, tuple, etc. (C{scalar}s).
413 @arg b: Iterable, list, tuple, etc. (C{scalar}s).
414 @arg c: Iterable, list, tuple, etc. (C{scalar}s).
415 @kwarg start: Optional bias (C{scalar}).
417 @return: Dot product (C{float}).
419 @raise LenError: Unequal C{len(B{a})}, C{len(B{b})}
420 and/or C{len(B{c})}.
422 @raise OverflowError: Partial C{2sum} overflow.
423 '''
424 def _mul3(a, b, c): # map function
425 return a * b * c
427 def _muly(a, b, c, start):
428 yield start
429 for abc in map(_mul3, a, b, c):
430 yield abc
432 if not len(a) == len(b) == len(c):
433 raise LenError(fdot3, a=len(a), b=len(b), c=len(c))
435 return fsum(_muly(a, b, c, start) if start else map(_mul3, a, b, c))
438def fhorner(x, *cs):
439 '''Evaluate the polynomial M{sum(cs[i] * x**i for
440 i=0..len(cs))} using the Horner form.
442 @arg x: Polynomial argument (C{scalar}).
443 @arg cs: Polynomial coeffients (C{scalar}s).
445 @return: Horner value (C{float}).
447 @raise OverflowError: Partial C{2sum} overflow.
449 @raise TypeError: Non-scalar B{C{x}}.
451 @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite.
453 @see: Function L{fpolynomial} and class L{Fhorner}.
454 '''
455 H = Fhorner(x, *cs)
456 return float(H)
459def fidw(xs, ds, beta=2):
460 '''Interpolate using U{Inverse Distance Weighting
461 <https://WikiPedia.org/wiki/Inverse_distance_weighting>} (IDW).
463 @arg xs: Known values (C{scalar}s).
464 @arg ds: Non-negative distances (C{scalar}s).
465 @kwarg beta: Inverse distance power (C{int}, 0, 1, 2, or 3).
467 @return: Interpolated value C{x} (C{float}).
469 @raise LenError: Unequal or zero C{len(B{ds})} and C{len(B{xs})}.
471 @raise ValueError: Invalid B{C{beta}}, negative B{C{ds}} value,
472 weighted B{C{ds}} below L{EPS}.
474 @note: Using C{B{beta}=0} returns the mean of B{C{xs}}.
475 '''
476 n, xs = len2(xs)
477 d, ds = len2(ds)
478 if n != d or n < 1:
479 raise LenError(fidw, xs=n, ds=d)
481 d, x = min(zip(ds, xs))
482 if d > EPS0 and n > 1:
483 b = -Int_(beta=beta, low=0, high=3)
484 if b < 0:
485 ws = tuple(float(d)**b for d in ds)
486 t = fsum(_map_mul1(xs, ws)) # fdot(xs, *ws)
487 x = _over(t, fsum(ws, floats=True))
488 else: # b == 0
489 x = fsum(xs) / n # fmean(xs)
490 elif d < 0: # PYCHOK no cover
491 n = Fmt.SQUARE(ds=ds.index(d))
492 raise _ValueError(n, d, txt=_negative_)
493 return x
496def fmean(xs):
497 '''Compute the accurate mean M{sum(xs[i] for
498 i=0..len(xs)) / len(xs)}.
500 @arg xs: Values (C{scalar} or L{Fsum} instances).
502 @return: Mean value (C{float}).
504 @raise OverflowError: Partial C{2sum} overflow.
506 @raise ValueError: No B{C{xs}} values.
507 '''
508 n, xs = len2(xs)
509 if n > 0:
510 return fsum(xs) / n # if n > 1 else _2float(index=0, xs=xs[0])
511 raise _ValueError(xs=xs)
514def fmean_(*xs):
515 '''Compute the accurate mean M{sum(xs[i] for
516 i=0..len(xs)) / len(xs)}.
518 @arg xs: Values (C{scalar} or L{Fsum} instances).
520 @return: Mean value (C{float}).
522 @raise OverflowError: Partial C{2sum} overflow.
524 @raise ValueError: No B{C{xs}} values.
525 '''
526 return fmean(xs)
529def fpolynomial(x, *cs, **over):
530 '''Evaluate the polynomial M{sum(cs[i] * x**i for
531 i=0..len(cs)) [/ over]}.
533 @arg x: Polynomial argument (C{scalar}).
534 @arg cs: Polynomial coeffients (C{scalar}s), all
535 positional.
536 @kwarg over: Optional, final divisor (C{scalar}
538 @return: Polynomial value (C{float}).
540 @raise OverflowError: Partial C{2sum} overflow.
542 @raise TypeError: Non-scalar B{C{x}}.
544 @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite.
546 @see: Function L{fhorner} and class L{Fpolynomial}.
547 '''
548 P = Fpolynomial(x, *cs)
549 d = _xkwds_get(over, over=0) if over else 0
550 return P.fover(d) if d else float(P)
553def fpowers(x, n, alts=0):
554 '''Return a series of powers M{[x**i for i=1..n]}.
556 @arg x: Value (C{scalar}).
557 @arg n: Highest exponent (C{int}).
558 @kwarg alts: Only alternating powers, starting with
559 this exponent (C{int}).
561 @return: Powers of B{C{x}} (C{float}s or C{int}s).
563 @raise TypeError: Non-scalar B{C{x}} or B{C{n}} not C{int}.
565 @raise ValueError: Non-finite B{C{x}} or non-positive B{C{n}}.
566 '''
567 if not isint(n):
568 raise _IsnotError(int.__name__, n=n)
569 elif n < 1:
570 raise _ValueError(n=n)
572 p = t = x if isint(x) else _2float(x=x)
573 ps = [p]
574 _a = ps.append
575 for _ in range(1, n):
576 p *= t
577 _a(p)
579 if alts > 0: # x**2, x**4, ...
580 # ps[alts-1::2] chokes PyChecker
581 ps = ps[slice(alts-1, None, 2)]
583 return ps
586try:
587 from math import prod as fprod # Python 3.8
588except ImportError:
590 def fprod(xs, start=1):
591 '''Iterable product, like C{math.prod} or C{numpy.prod}.
593 @arg xs: Terms to be multiplied, an iterable, list,
594 tuple, etc. (C{scalar}s).
595 @kwarg start: Initial term, also the value returned
596 for an empty B{C{xs}} (C{scalar}).
598 @return: The product (C{float}).
600 @see: U{NumPy.prod<https://docs.SciPy.org/doc/
601 numpy/reference/generated/numpy.prod.html>}.
602 '''
603 return freduce(_mul, xs, start)
606def frange(start, number, step=1):
607 '''Generate a range of C{float}s.
609 @arg start: First value (C{float}).
610 @arg number: The number of C{float}s to generate (C{int}).
611 @kwarg step: Increment value (C{float}).
613 @return: A generator (C{float}s).
615 @see: U{NumPy.prod<https://docs.SciPy.org/doc/
616 numpy/reference/generated/numpy.arange.html>}.
617 '''
618 if not isint(number):
619 raise _IsnotError(int.__name__, number=number)
620 for i in range(number):
621 yield start + i * step
624try:
625 from functools import reduce as freduce
626except ImportError:
627 try:
628 freduce = reduce # PYCHOK expected
629 except NameError: # Python 3+
631 def freduce(f, xs, *start):
632 '''For missing C{functools.reduce}.
633 '''
634 if start:
635 r = v = start[0]
636 else:
637 r, v = 0, MISSING
638 for v in xs:
639 r = f(r, v)
640 if v is MISSING:
641 raise _TypeError(xs=(), start=MISSING)
642 return r
645def fremainder(x, y):
646 '''Remainder in range C{[-B{y / 2}, B{y / 2}]}.
648 @arg x: Numerator (C{scalar}).
649 @arg y: Modulus, denominator (C{scalar}).
651 @return: Remainder (C{scalar}, preserving signed
652 0.0) or C{NAN} for any non-finite B{C{x}}.
654 @raise ValueError: Infinite or near-zero B{C{y}}.
656 @see: I{Karney}'s U{Math.remainder<https://PyPI.org/
657 project/geographiclib/>} and Python 3.7+
658 U{math.remainder<https://docs.Python.org/3/
659 library/math.html#math.remainder>}.
660 '''
661 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and
662 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native
663 # fmod( 0, 360) == 0.0
664 # fmod( 360, 360) == 0.0
665 # fmod(-0, 360) == 0.0
666 # fmod(-0.0, 360) == -0.0
667 # fmod(-360, 360) == -0.0
668 # however, using the % operator ...
669 # 0 % 360 == 0
670 # 360 % 360 == 0
671 # 360.0 % 360 == 0.0
672 # -0 % 360 == 0
673 # -360 % 360 == 0 == (-360) % 360
674 # -0.0 % 360 == 0.0 == (-0.0) % 360
675 # -360.0 % 360 == 0.0 == (-360.0) % 360
677 # On Windows 32-bit with python 2.7, math.fmod(-0.0, 360)
678 # == +0.0. This fixes this bug. See also Math::AngNormalize
679 # in the C++ library, Math.sincosd has a similar fix.
680 if _isfinite(x):
681 try:
682 r = remainder(x, y) if x else x
683 except Exception as e:
684 raise _xError(e, unstr(fremainder, x, y))
685 else: # handle x INF and NINF as NAN
686 r = NAN
687 return r
690if _sys_version_info2 < (3, 8): # PYCHOK no cover
691 from math import hypot # OK in Python 3.7-
693 def hypot_(*xs):
694 '''Compute the norm M{sqrt(sum(x**2 for x in xs))}.
696 Similar to Python 3.8+ n-dimension U{math.hypot
697 <https://docs.Python.org/3.8/library/math.html#math.hypot>},
698 but exceptions, C{nan} and C{infinite} values are
699 handled differently.
701 @arg xs: X arguments (C{scalar}s), all positional.
703 @return: Norm (C{float}).
705 @raise OverflowError: Partial C{2sum} overflow.
707 @raise ValueError: Invalid or no B{C{xs}} values.
709 @note: The Python 3.8+ Euclidian distance U{math.dist
710 <https://docs.Python.org/3.8/library/math.html#math.dist>}
711 between 2 I{n}-dimensional points I{p1} and I{p2} can be
712 computed as M{hypot_(*((c1 - c2) for c1, c2 in zip(p1, p2)))},
713 provided I{p1} and I{p2} have the same, non-zero length I{n}.
714 '''
715 h, x2 = _h_x2(xs)
716 return (h * sqrt(x2)) if x2 else _0_0
718elif _sys_version_info2 < (3, 10):
719 # In Python 3.8 and 3.9 C{math.hypot} is inaccurate, see
720 # U{agdhruv<https://GitHub.com/geopy/geopy/issues/466>},
721 # U{cffk<https://Bugs.Python.org/issue43088>} and module
722 # U{geomath.py<https://PyPI.org/project/geographiclib/1.52>}
724 def hypot(x, y):
725 '''Compute the norm M{sqrt(x**2 + y**2)}.
727 @arg x: X argument (C{scalar}).
728 @arg y: Y argument (C{scalar}).
730 @return: C{sqrt(B{x}**2 + B{y}**2)} (C{float}).
731 '''
732 if x:
733 h = sqrt(x**2 + y**2) if y else fabs(x)
734 elif y:
735 h = fabs(y)
736 else:
737 h = _0_0
738 return h
740 from math import hypot as hypot_ # PYCHOK in Python 3.8 and 3.9
741else:
742 from math import hypot # PYCHOK in Python 3.10+
743 hypot_ = hypot
746def _h_x2(xs):
747 '''(INTERNAL) Helper for L{hypot_} and L{hypot2_}.
748 '''
749 if xs:
750 n, xs = len2(xs)
751 if n > 0:
752 h = float(max(map(fabs, xs)))
753 if h < EPS0:
754 x2 = _0_0
755 else: # math.fsum, see C{_hypot21_} below
756 x2 = _fsum(_x2_h2(_1_0, xs, h, _N_1_0))
757 return h, x2
759 raise _ValueError(xs=xs, txt=_too_(_few_))
762def hypot1(x):
763 '''Compute the norm M{sqrt(1 + x**2)}.
765 @arg x: Argument (C{scalar}).
767 @return: Norm (C{float}).
768 '''
769 return hypot(_1_0, x) if x else _1_0
772def hypot2(x, y):
773 '''Compute the I{squared} norm M{x**2 + y**2}.
775 @arg x: X argument (C{scalar}).
776 @arg y: Y argument (C{scalar}).
778 @return: C{B{x}**2 + B{y}**2} (C{float}).
779 '''
780 if x:
781 if y:
782 if fabs(x) < fabs(y):
783 x, y = y, x
784 h2 = x**2 * ((y / x)**2 + _1_0)
785 else:
786 h2 = x**2
787 elif y:
788 h2 = y**2
789 else:
790 h2 = _0_0
791 return h2
794def hypot2_(*xs):
795 '''Compute the I{squared} norm C{sum(x**2 for x in B{xs})}.
797 @arg xs: X arguments (C{scalar}s), all positional.
799 @return: Squared norm (C{float}).
801 @raise OverflowError: Partial C{2sum} overflow.
803 @raise ValueError: Invalid or no B{C{xs}} value.
805 @see: Function L{hypot_}.
806 '''
807 h, x2 = _h_x2(xs)
808 return (h**2 * x2) if x2 else _0_0
811def _map_mul(a, b, where):
812 '''(INTERNAL) Yield each B{C{a * b}}.
813 '''
814 n = len(b)
815 if len(a) != n: # PYCHOK no cover
816 raise LenError(where, a=len(a), b=n)
817 return map(_mul, a, b) if n > 3 else _map_mul1(a, b)
820def _map_mul1(a, b):
821 '''(INTERNAL) Yield each B{C{a * b}}, 1-primed.
822 '''
823 yield _1_0
824 for ab in map(_mul, a, b):
825 if ab:
826 yield ab
827 yield _N_1_0
830def norm2(x, y):
831 '''Normalize a 2-dimensional vector.
833 @arg x: X component (C{scalar}).
834 @arg y: Y component (C{scalar}).
836 @return: 2-Tuple C{(x, y)}, normalized.
838 @raise ValueError: Invalid B{C{x}} or B{C{y}}
839 or zero norm.
840 '''
841 h = hypot(x, y)
842 if not h:
843 x = y = _0_0 # pass?
844 elif not isnear1(h):
845 try:
846 x, y = x / h, y / h
847 except Exception as e:
848 raise _xError(e, x=x, y=y, h=h)
849 return x, y
852def norm_(*xs):
853 '''Normalize all n-dimensional vector components.
855 @arg xs: Components (C{scalar}s), all positional.
857 @return: Yield each component, normalized.
859 @raise ValueError: Invalid or insufficent B{C{xs}}
860 or zero norm.
861 '''
862 h = hypot_(*xs)
863 if h:
864 try:
865 for i, x in enumerate(xs):
866 yield x / h
867 except Exception as e:
868 raise _xError(e, Fmt.SQUARE(xs=i), x, _h_, h)
869 else:
870 for _ in xs:
871 yield _0_0
874def _root(x, p, where):
875 '''(INTERNAL) Raise C{x} to power C{0 < p < 1}.
876 '''
877 if x < 0:
878 t = _SPACE_(_invokation_, where.__name__)
879 raise _ValueError(unstr(t, x), txt=_negative_)
880 return pow(x, p) if x else _0_0
883def sqrt0(x, Error=None):
884 '''Return the square root iff C{B{x} >} L{EPS02}.
886 @arg x: Value (C{scalar}).
887 @kwarg Error: Error to raise for negative B{C{x}}.
889 @return: Square root (C{float}) or C{0.0}.
891 @note: Any C{B{x} < }L{EPS02} I{including} C{B{x} < 0}
892 returns C{0.0}.
893 '''
894 if Error and x < 0:
895 raise Error(Fmt.PAREN(sqrt=x))
896 return sqrt(x) if x > EPS02 else (_0_0 if x < EPS02 else EPS0)
899def sqrt3(x):
900 '''Return the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)}.
902 @arg x: Value (C{scalar}).
904 @return: Square root I{cubed} (C{float}).
906 @raise ValueError: Negative B{C{x}}.
908 @see: Functions L{cbrt} and L{cbrt2}.
909 '''
910 return _root(x, _1_5, sqrt3)
913def sqrt_a(h, b):
914 '''Compute C{I{a}} side of a right-angled triangle from
915 C{sqrt(B{h}**2 - B{b}**2)}.
917 @arg h: Hypotenuse or outer annulus radius (C{scalar}).
918 @arg b: Triangle side or inner annulus radius (C{scalar}).
920 @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}).
922 @raise TypeError: Non-scalar B{C{h}} or B{C{b}}.
924 @raise ValueError: If C{abs(B{h}) < abs(B{b})}.
926 @see: Inner tangent chord B{I{d}} of an U{annulus
927 <https://WikiPedia.org/wiki/Annulus_(mathematics)>}
928 and function U{annulus_area<https://People.SC.FSU.edu/
929 ~jburkardt/py_src/geometry/geometry.py>}.
930 '''
931 try:
932 if not (_isHeight(h) and _isRadius(b)):
933 raise TypeError(_not_scalar_)
934 elif isnear0(h): # PYCHOK no cover
935 c, b = fabs(h), fabs(b)
936 d = c - b
937 if d < 0:
938 raise ValueError('abs(h) < abs(b)')
939 a = copysign0(sqrt((c + b) * d), h) if d > 0 else _0_0
940 else:
941 c = float(h)
942 s = _1_0 - (b / c)**2
943 if s < 0:
944 raise ValueError('abs(h) < abs(b)')
945 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0)
946 except Exception as x:
947 raise _xError(x, h=h, b=b)
948 return a
951def _x2_h2(s, xs, h, e):
952 '''(INTERNAL) Yield M{(x / h)**2 for x in xs}.
953 '''
954 yield s
955 if h in (_0_0, _1_0):
956 for x in xs:
957 if x:
958 yield x**2
959 else:
960 for x in xs:
961 if x:
962 yield (x / h)**2
963 yield e
966def zcrt(x):
967 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)}.
969 @arg x: Value (C{scalar}).
971 @return: I{Zenzi-cubic} root (C{float}).
973 @see: Functions L{bqrt} and L{zqrt}.
975 @raise ValueError: Negative B{C{x}}.
976 '''
977 return _root(x, _1_6th, zcrt)
980def zqrt(x):
981 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root, M{x**(1 / 8)}.
983 @arg x: Value (C{scalar}).
985 @return: I{Zenzi-quartic} root (C{float}).
987 @see: Functions L{bqrt} and L{zcrt}.
989 @raise ValueError: Negative B{C{x}}.
990 '''
991 return _root(x, _0_125, zqrt)
993# **) MIT License
994#
995# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
996#
997# Permission is hereby granted, free of charge, to any person obtaining a
998# copy of this software and associated documentation files (the "Software"),
999# to deal in the Software without restriction, including without limitation
1000# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1001# and/or sell copies of the Software, and to permit persons to whom the
1002# Software is furnished to do so, subject to the following conditions:
1003#
1004# The above copyright notice and this permission notice shall be included
1005# in all copies or substantial portions of the Software.
1006#
1007# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1008# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1009# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1010# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1011# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1012# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1013# OTHER DEALINGS IN THE SOFTWARE.