Coverage for pygeodesy/fmath.py: 91%
303 statements
« prev ^ index » next coverage.py v7.2.2, created at 2024-05-02 14:35 -0400
« prev ^ index » next coverage.py v7.2.2, created at 2024-05-02 14:35 -0400
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, isbool, isint, isscalar, len2
10from pygeodesy.constants import EPS0, EPS02, EPS1, NAN, PI, PI_2, PI_4, \
11 _0_0, _0_125, _1_6th, _0_25, _1_3rd, _0_5, _1_0, \
12 _1_5, _copysign_0_0, _isfinite, _over, remainder
13from pygeodesy.errors import _IsnotError, LenError, _TypeError, _ValueError, \
14 _xError, _xkwds_get, _xkwds_pop2
15from pygeodesy.fsums import _2float, Fsum, fsum, fsum1_, _1primed, Fmt, unstr
16from pygeodesy.interns import MISSING, _few_, _negative_, _not_scalar_, _too_
17from pygeodesy.lazily import _ALL_LAZY, _sys_version_info2
18# from pygeodesy.streprs import Fmt, unstr # from .fsums
19from pygeodesy.units import Int_, _isHeight, _isRadius, Float_ # PYCHOK for .heights
21from math import fabs, sqrt # pow
22import operator as _operator # in .datums, .trf, .utm
24__all__ = _ALL_LAZY.fmath
25__version__ = '24.04.24'
27# sqrt(2) <https://WikiPedia.org/wiki/Square_root_of_2>
28_0_4142 = 0.41421356237309504880 # ... sqrt(2) - 1
29_2_3rd = _1_3rd * 2
30_h_lt_b_ = 'abs(h) < abs(b)'
33class Fdot(Fsum):
34 '''Precision dot product.
35 '''
36 def __init__(self, a, *b, **name_RESIDUAL):
37 '''New L{Fdot} precision dot product M{sum(a[i] * b[i]
38 for i=0..len(a)-1)}.
40 @arg a: Iterable, list, tuple, etc. (C{scalar}s).
41 @arg b: Other values (C{scalar}s), all positional.
42 @kwarg name_RESIDUAL: Optional C{B{name}=NN} and C{B{RESIDUAL}=None},
43 see L{Fsum<Fsum.__init__>}.
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_RESIDUAL)
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_RESIDUAL):
59 '''New L{Fhorner} evaluation of polynomial M{sum(cs[i] * x**i
60 for i=0..len(cs)-1)}.
62 @arg x: Polynomial argument (C{scalar} or C{Fsum} instance).
63 @arg cs: Polynomial coeffients (C{scalar} or C{Fsum}
64 instances), all positional.
65 @kwarg name_RESIDUAL: Optional C{B{name}=NN} and C{B{RESIDUAL}=None},
66 see L{Fsum<Fsum.__init__>}.
68 @raise OverflowError: Partial C{2sum} overflow.
70 @raise TypeError: Non-scalar B{C{x}}.
72 @raise ValueError: Non-finite B{C{x}}.
74 @see: Function L{fhorner} and methods L{Fsum.fadd} and
75 L{Fsum.fmul}.
76 '''
77 Fsum.__init__(self, **name_RESIDUAL)
78 if cs:
79 if isinstance(x, Fsum):
80 _mul = self._mul_Fsum
81 else:
82 _mul = self._mul_scalar
83 x = _2float(x=x)
84 op = Fhorner.__name__
85 if len(cs) > 1 and x:
86 for c in reversed(cs):
87 self._fset_ps(_mul(x, op))
88 self._fadd(c, op, up=False)
89 self._update()
90 else: # x == 0
91 self._fadd(cs[0], op)
92 else:
93 self._fset_ps(_0_0)
96class Fhypot(Fsum):
97 '''Precision summation and hypotenuse, default C{root=2}.
98 '''
99 def __init__(self, *xs, **root_name_RESIDUAL_raiser):
100 '''New L{Fhypot} hypotenuse of (the I{root} of) several components.
102 @arg xs: One or more components (each a C{scalar} or an C{Fsum} instance).
103 @kwarg root_name_RESIDUAL_raiser: Optional, exponent and C{B{root}=2} order,
104 C{B{name}=NN}, C{B{RESIDUAL}=None} and C{B{raiser}=True}, see
105 class L{Fsum<Fsum.__init__>} and method L{root<Fsum.root>}.
106 '''
107 r = None # _xkwds_pop2 error
108 try:
109 r, kwds = _xkwds_pop2(root_name_RESIDUAL_raiser, root=2)
110 r, kwds = _xkwds_pop2(kwds, power=r) # for backward compatibility
111 raiser = _Fsum__init__(self, **kwds)
112 if xs:
113 self._facc_power(r, xs, Fhypot, **raiser)
114 self._fset(self.root(r, **raiser))
115 except Exception as X:
116 raise self._ErrorXs(X, xs, root=r)
119class Fpolynomial(Fsum):
120 '''Precision polynomial evaluation.
121 '''
122 def __init__(self, x, *cs, **name_RESIDUAL):
123 '''New L{Fpolynomial} evaluation of the polynomial
124 M{sum(cs[i] * x**i for i=0..len(cs)-1)}.
126 @arg x: Polynomial argument (C{scalar} or L{Fsum}).
127 @arg cs: Polynomial coeffients (each a C{scalar} or an L{Fsum} instance),
128 all positional.
129 @kwarg name_RESIDUAL: Optional C{B{name}=NN} and C{B{RESIDUAL}=None},
130 see L{Fsum<Fsum.__init__>}.
132 @raise OverflowError: Partial C{2sum} overflow.
134 @raise TypeError: Non-scalar B{C{x}}.
136 @raise ValueError: Non-finite B{C{x}}.
138 @see: Class L{Fhorner}, function L{fpolynomial} and
139 method L{Fsum.fadd}.
140 '''
141 Fsum.__init__(self, *cs[:1], **name_RESIDUAL)
142 n = len(cs) - 1
143 if n > 0:
144 self.fadd(_1map_mul(cs[1:], _powers(x, n)))
145 elif n < 0:
146 self._fset_ps(_0_0)
149class Fpowers(Fsum):
150 '''Precision summation of powers, optimized for C{power=2, 3 and 4}.
151 '''
152 def __init__(self, power, *xs, **name_RESIDUAL_raiser):
153 '''New L{Fpowers} sum of (the I{power} of) several values.
155 @arg power: The exponent (C{scalar} or L{Fsum}).
156 @arg xs: One or more values (each a C{scalar} or an C{Fsum} instance).
157 @kwarg name_RESIDUAL_raiser: Optional C{B{name}=NN}, C{B{RESIDUAL}=None} and
158 C{B{raiser}=True}, see L{Fsum<Fsum.__init__>} and L{fpow<Fsum.fpow>}.
159 '''
160 try:
161 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser)
162 if xs:
163 self._facc_power(power, xs, Fpowers, **raiser) # x**0 == 1
164 except Exception as X:
165 raise self._ErrorXs(X, xs, power=power)
168class Froot(Fsum):
169 '''The root of a precision summation.
170 '''
171 def __init__(self, root, *xs, **name_RESIDUAL_raiser):
172 '''New L{Froot} root of a precision sum.
174 @arg root: The order (C{scalar} or C{Fsum}), non-zero.
175 @arg xs: Values to summate (each a C{scalar} or an C{Fsum} instance).
176 @kwarg name_RESIDUAL_raiser: Optional C{B{name}=NN}, C{B{RESIDUAL}=None} and
177 C{B{raiser}=True}, see L{Fsum<Fsum.__init__>} and L{fpow<Fsum.fpow>}.
178 '''
179 try:
180 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser)
181 if xs:
182 self.fadd(xs)
183 self._fset(self.root(root, **raiser))
184 except Exception as X:
185 raise self._ErrorXs(X, xs, root=root)
188class Fcbrt(Froot):
189 '''Cubic root of a precision summation.
190 '''
191 def __init__(self, *xs, **name_RESIDUAL_raiser):
192 '''New L{Fcbrt} cubic root of a precision sum.
194 @see: Class L{Froot} for further details.
195 '''
196 Froot.__init__(self, 3, *xs, **name_RESIDUAL_raiser)
199class Fsqrt(Froot):
200 '''Square root of a precision summation.
201 '''
202 def __init__(self, *xs, **name_RESIDUAL_raiser):
203 '''New L{Fsqrt} square root of a precision sum.
205 @see: Class L{Froot} for further details.
206 '''
207 Froot.__init__(self, 2, *xs, **name_RESIDUAL_raiser)
210def _Fsum__init__(inst, raiser=MISSING, **name_RESIDUAL):
211 '''(INTERNAL) Init an C{F...} instance above.
212 '''
213 Fsum.__init__(inst, **name_RESIDUAL) # PYCHOK self
214 inst._fset_ps(_0_0)
215 return {} if raiser is MISSING else dict(raiser=raiser)
218def bqrt(x):
219 '''Return the 4-th, I{bi-quadratic} or I{quartic} root, M{x**(1 / 4)},
220 preserving C{type(B{x})}.
222 @arg x: Value (C{scalar} or L{Fsum} instance).
224 @return: I{Quartic} root (C{float} or L{Fsum}).
226 @raise ValueError: Negative B{C{x}}.
228 @see: Functions L{zcrt} and L{zqrt}.
229 '''
230 return _root(x, _0_25, bqrt)
233try:
234 from math import cbrt as _cbrt # Python 3.11+
236except ImportError: # Python 3.10-
238 def _cbrt(x):
239 '''(INTERNAL) Compute the I{signed}, cube root M{x**(1/3)}.
240 '''
241 # <https://archive.lib.MSU.edu/crcmath/math/math/r/r021.htm>
242 # simpler and more accurate than Ken Turkowski's CubeRoot, see
243 # <https://People.FreeBSD.org/~lstewart/references/apple_tr_kt32_cuberoot.pdf>
244 return _copysign(pow(fabs(x), _1_3rd), x) # to avoid complex
247def cbrt(x):
248 '''Compute the cube root M{x**(1/3)}, preserving C{type(B{x})}.
250 @arg x: Value (C{scalar} or L{Fsum} instance).
252 @return: Cubic root (C{float} or L{Fsum}).
254 @see: Functions L{cbrt2} and L{sqrt3}.
255 '''
256 if isinstance(x, Fsum):
257 r = (-(-x).pow(_1_3rd)) if x < 0 else x.pow(_1_3rd)
258 else:
259 r = _cbrt(x)
260 return r # cbrt(-0.0) == -0.0
263def cbrt2(x): # PYCHOK attr
264 '''Compute the cube root I{squared} M{x**(2/3)}, preserving C{type(B{x})}.
266 @arg x: Value (C{scalar} or L{Fsum} instance).
268 @return: Cube root I{squared} (C{float} or L{Fsum}).
270 @see: Functions L{cbrt} and L{sqrt3}.
271 '''
272 return abs(x).pow(_2_3rd) if isinstance(x, Fsum) else _cbrt(x**2)
275def euclid(x, y):
276 '''I{Appoximate} the norm M{sqrt(x**2 + y**2)} by
277 M{max(abs(x), abs(y)) + min(abs(x), abs(y)) * 0.4142...}.
279 @arg x: X component (C{scalar} or L{Fsum} instance).
280 @arg y: Y component (C{scalar} or L{Fsum} instance).
282 @return: Appoximate norm (C{float} or L{Fsum}).
284 @see: Function L{euclid_}.
285 '''
286 x, y = abs(x), abs(y) # NOT fabs!
287 if x < y:
288 x, y = y, x
289 return x + y * _0_4142 # XXX * _0_5 before 20.10.02
292def euclid_(*xs):
293 '''I{Appoximate} the norm M{sqrt(sum(x**2 for x in xs))} by
294 cascaded L{euclid}.
296 @arg xs: X arguments, positional (C{scalar}s or L{Fsum} instances).
298 @return: Appoximate norm (C{float} or L{Fsum}).
300 @see: Function L{euclid}.
301 '''
302 e = _0_0
303 for x in sorted(map(abs, xs)): # NOT fabs, reverse=True!
304 # e = euclid(x, e)
305 if e < x:
306 e, x = x, e
307 if x:
308 e += x * _0_4142
309 return e
312def facos1(x):
313 '''Fast approximation of L{pygeodesy.acos1}C{(B{x})}.
315 @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/
316 ShaderFastLibs/blob/master/ShaderFastMathLib.h>}.
317 '''
318 a = fabs(x)
319 if a < EPS0:
320 r = PI_2
321 elif a < EPS1:
322 H = Fhorner(-a, 1.5707288, 0.2121144, 0.0742610, 0.0187293)
323 r = float(H * sqrt(_1_0 - a))
324 if x < 0:
325 r = PI - r
326 else:
327 r = PI if x < 0 else _0_0
328 return r
331def fasin1(x): # PYCHOK no cover
332 '''Fast approximation of L{pygeodesy.asin1}C{(B{x})}.
334 @see: L{facos1}.
335 '''
336 return PI_2 - facos1(x)
339def fatan(x):
340 '''Fast approximation of C{atan(B{x})}.
341 '''
342 a = fabs(x)
343 if a < _1_0:
344 r = fatan1(a) if a else _0_0
345 elif a > _1_0:
346 r = PI_2 - fatan1(_1_0 / a) # == fatan2(a, _1_0)
347 else:
348 r = PI_4
349 if x < 0: # copysign0(r, x)
350 r = -r
351 return r
354def fatan1(x):
355 '''Fast approximation of C{atan(B{x})} for C{0 <= B{x} <= 1}, I{unchecked}.
357 @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/ShaderFastLibs/
358 blob/master/ShaderFastMathLib.h>} and U{Efficient approximations
359 for the arctangent function<http://www-Labs.IRO.UMontreal.CA/
360 ~mignotte/IFT2425/Documents/EfficientApproximationArctgFunction.pdf>},
361 IEEE Signal Processing Magazine, 111, May 2006.
362 '''
363 # Eq (9): PI_4 * x - x * (abs(x) - 1) * (0.2447 + 0.0663 * abs(x)), for -1 < x < 1
364 # PI_4 * x - (x**2 - x) * (0.2447 + 0.0663 * x), for 0 < x - 1
365 # x * (1.0300981633974482 + x * (-0.1784 - x * 0.0663))
366 H = Fhorner(x, _0_0, 1.0300981634, -0.1784, -0.0663)
367 return float(H)
370def fatan2(y, x):
371 '''Fast approximation of C{atan2(B{y}, B{x})}.
373 @see: U{fastApproximateAtan(x, y)<https://GitHub.com/CesiumGS/cesium/blob/
374 master/Source/Shaders/Builtin/Functions/fastApproximateAtan.glsl>}
375 and L{fatan1}.
376 '''
377 a, b = fabs(x), fabs(y)
378 if b > a:
379 r = (PI_2 - fatan1(a / b)) if a else PI_2
380 elif a > b:
381 r = fatan1(b / a) if b else _0_0
382 elif a: # a == b != 0
383 r = PI_4
384 else: # a == b == 0
385 return _0_0
386 if x < 0:
387 r = PI - r
388 if y < 0: # copysign0(r, y)
389 r = -r
390 return r
393def favg(v1, v2, f=_0_5):
394 '''Return the average of two values.
396 @arg v1: One value (C{scalar} or L{Fsum} instance).
397 @arg v2: Other value (C{scalar} or L{Fsum} instance).
398 @kwarg f: Optional fraction (C{float}).
400 @return: M{v1 + f * (v2 - v1)} (C{float}).
401 '''
402# @raise ValueError: Fraction out of range.
403# '''
404# if not 0 <= f <= 1: # XXX restrict fraction?
405# raise _ValueError(fraction=f)
406 # v1 + f * (v2 - v1) == v1 * (1 - f) + v2 * f
407 return fsum1_(v1, -f * v1, f * v2)
410def fdot(a, *b):
411 '''Return the precision dot product M{sum(a[i] * b[i] for
412 i=0..len(a))}.
414 @arg a: Iterable, list, tuple, etc. (C{scalar}s).
415 @arg b: All positional arguments (C{scalar}s).
417 @return: Dot product (C{float}).
419 @raise LenError: Unequal C{len(B{a})} and C{len(B{b})}.
421 @see: Class L{Fdot} and U{Algorithm 5.10 B{DotK}
422 <https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>}.
423 '''
424 return fsum(_map_mul(a, b, fdot))
427def fdot3(a, b, c, start=0):
428 '''Return the precision dot product M{start +
429 sum(a[i] * b[i] * c[i] for i=0..len(a)-1)}.
431 @arg a: Iterable, list, tuple, etc. (C{scalar}s).
432 @arg b: Iterable, list, tuple, etc. (C{scalar}s).
433 @arg c: Iterable, list, tuple, etc. (C{scalar}s).
434 @kwarg start: Optional bias (C{scalar}).
436 @return: Dot product (C{float}).
438 @raise LenError: Unequal C{len(B{a})}, C{len(B{b})}
439 and/or C{len(B{c})}.
441 @raise OverflowError: Partial C{2sum} overflow.
442 '''
443 def _mul3(a, b, c): # map function
444 return a * b * c
446 def _mul3_(a, b, c, start):
447 yield start
448 for abc in map(_mul3, a, b, c):
449 yield abc
451 if not len(a) == len(b) == len(c):
452 raise LenError(fdot3, a=len(a), b=len(b), c=len(c))
454 return fsum(_mul3_(a, b, c, start) if start else map(_mul3, a, b, c))
457def fhorner(x, *cs):
458 '''Evaluate the polynomial M{sum(cs[i] * x**i for
459 i=0..len(cs)-1)} using the Horner form.
461 @arg x: Polynomial argument (C{scalar}).
462 @arg cs: Polynomial coeffients (C{scalar}s).
464 @return: Horner value (C{float}).
466 @raise OverflowError: Partial C{2sum} overflow.
468 @raise TypeError: Non-scalar B{C{x}}.
470 @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite.
472 @see: Function L{fpolynomial} and class L{Fhorner}.
473 '''
474 H = Fhorner(x, *cs)
475 return float(H)
478def fidw(xs, ds, beta=2):
479 '''Interpolate using U{Inverse Distance Weighting
480 <https://WikiPedia.org/wiki/Inverse_distance_weighting>} (IDW).
482 @arg xs: Known values (C{scalar}s).
483 @arg ds: Non-negative distances (C{scalar}s).
484 @kwarg beta: Inverse distance power (C{int}, 0, 1, 2, or 3).
486 @return: Interpolated value C{x} (C{float}).
488 @raise LenError: Unequal or zero C{len(B{ds})} and C{len(B{xs})}.
490 @raise ValueError: Invalid B{C{beta}}, negative B{C{ds}} value,
491 weighted B{C{ds}} below L{EPS}.
493 @note: Using C{B{beta}=0} returns the mean of B{C{xs}}.
494 '''
495 n, xs = len2(xs)
496 d, ds = len2(ds)
497 if n != d or n < 1:
498 raise LenError(fidw, xs=n, ds=d)
500 d, x = min(zip(ds, xs))
501 if d > EPS0 and n > 1:
502 b = -Int_(beta=beta, low=0, high=3)
503 if b < 0:
504 ws = tuple(float(d)**b for d in ds)
505 t = fsum(_1map_mul(xs, ws)) # Fdot(xs, *ws)
506 x = _over(t, fsum(ws, floats=True))
507 else: # b == 0
508 x = fsum(xs) / n # fmean(xs)
509 elif d < 0: # PYCHOK no cover
510 n = Fmt.SQUARE(distance=ds.index(d))
511 raise _ValueError(n, d, txt=_negative_)
512 return x
515def fmean(xs):
516 '''Compute the accurate mean M{sum(xs) / len(xs)}.
518 @arg xs: Values (C{scalar} or L{Fsum} instances).
520 @return: Mean value (C{float}).
522 @raise LenError: No B{C{xs}} values.
524 @raise OverflowError: Partial C{2sum} overflow.
525 '''
526 n, xs = len2(xs)
527 if n < 1:
528 raise LenError(fmean, xs=xs)
529 return Fsum(*xs).fover(n) if n > 1 else _2float(index=0, xs=xs[0])
532def fmean_(*xs):
533 '''Compute the accurate mean M{sum(xs) / len(xs)}.
535 @see: Function L{fmean} for further details.
536 '''
537 return fmean(xs)
540def fpolynomial(x, *cs, **over):
541 '''Evaluate the polynomial M{sum(cs[i] * x**i for
542 i=0..len(cs)) [/ over]}.
544 @arg x: Polynomial argument (C{scalar}).
545 @arg cs: Polynomial coeffients (C{scalar}s), all
546 positional.
547 @kwarg over: Optional final, I{non-zero} divisor
548 (C{scalar}).
550 @return: Polynomial value (C{float}).
552 @raise OverflowError: Partial C{2sum} overflow.
554 @raise TypeError: Non-scalar B{C{x}}.
556 @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite.
558 @see: Function L{fhorner} and class L{Fpolynomial}.
559 '''
560 P = Fpolynomial(x, *cs)
561 d = _xkwds_get(over, over=0) if over else 0
562 return P.fover(d) if d else float(P)
565def fpowers(x, n, alts=0):
566 '''Return a series of powers M{[x**i for i=1..n]}.
568 @arg x: Value (C{scalar} or L{Fsum}).
569 @arg n: Highest exponent (C{int}).
570 @kwarg alts: Only alternating powers, starting with this
571 exponent (C{int}).
573 @return: Tuple of powers of B{C{x}} (C{type(B{x})}).
575 @raise TypeError: Invalid B{C{x}} or B{C{n}} not C{int}.
577 @raise ValueError: Non-finite B{C{x}} or invalid B{C{n}}.
578 '''
579 if not isint(n):
580 raise _IsnotError(int.__name__, n=n)
581 elif n < 1:
582 raise _ValueError(n=n)
584 p = x if isint(x) or isinstance(x, Fsum) else _2float(x=x)
585 ps = tuple(_powers(p, n))
587 if alts > 0: # x**2, x**4, ...
588 # ps[alts-1::2] chokes PyChecker
589 ps = ps[slice(alts-1, None, 2)]
591 return ps
594try:
595 from math import prod as fprod # Python 3.8
596except ImportError:
598 def fprod(xs, start=1):
599 '''Iterable product, like C{math.prod} or C{numpy.prod}.
601 @arg xs: Terms to be multiplied, an iterable, list,
602 tuple, etc. (C{scalar}s).
603 @kwarg start: Initial term, also the value returned
604 for an empty B{C{xs}} (C{scalar}).
606 @return: The product (C{float}).
608 @see: U{NumPy.prod<https://docs.SciPy.org/doc/
609 numpy/reference/generated/numpy.prod.html>}.
610 '''
611 return freduce(_operator.mul, xs, start)
614def frandoms(n, seeded=None):
615 '''Generate C{n} (long) lists of random C{floats}.
617 @arg n: Number of lists to generate (C{int}, non-negative).
618 @kwarg seeded: If C{scalar}, use C{random.seed(B{seeded})} or
619 if C{True}, seed using today's C{year-day}.
621 @see: U{Hettinger<https://GitHub.com/ActiveState/code/tree/master/recipes/
622 Python/393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}.
623 '''
624 from random import gauss, random, seed, shuffle
626 if seeded is None:
627 pass
628 elif seeded and isbool(seeded):
629 from time import localtime
630 seed(localtime().tm_yday)
631 elif isscalar(seeded):
632 seed(seeded)
634 c = (7, 1e100, -7, -1e100, -9e-20, 8e-20) * 7
635 for _ in range(n):
636 s = 0
637 t = list(c)
638 _a = t.append
639 for _ in range(n * 8):
640 v = gauss(0, random())**7 - s
641 _a(v)
642 s += v
643 shuffle(t)
644 yield t
647def frange(start, number, step=1):
648 '''Generate a range of C{float}s.
650 @arg start: First value (C{float}).
651 @arg number: The number of C{float}s to generate (C{int}).
652 @kwarg step: Increment value (C{float}).
654 @return: A generator (C{float}s).
656 @see: U{NumPy.prod<https://docs.SciPy.org/doc/
657 numpy/reference/generated/numpy.arange.html>}.
658 '''
659 if not isint(number):
660 raise _IsnotError(int.__name__, number=number)
661 for i in range(number):
662 yield start + (step * i)
665try:
666 from functools import reduce as freduce
667except ImportError:
668 try:
669 freduce = reduce # PYCHOK expected
670 except NameError: # Python 3+
672 def freduce(f, xs, *start):
673 '''For missing C{functools.reduce}.
674 '''
675 if start:
676 r = v = start[0]
677 else:
678 r, v = 0, MISSING
679 for v in xs:
680 r = f(r, v)
681 if v is MISSING:
682 raise _TypeError(xs=(), start=MISSING)
683 return r
686def fremainder(x, y):
687 '''Remainder in range C{[-B{y / 2}, B{y / 2}]}.
689 @arg x: Numerator (C{scalar}).
690 @arg y: Modulus, denominator (C{scalar}).
692 @return: Remainder (C{scalar}, preserving signed
693 0.0) or C{NAN} for any non-finite B{C{x}}.
695 @raise ValueError: Infinite or near-zero B{C{y}}.
697 @see: I{Karney}'s U{Math.remainder<https://PyPI.org/
698 project/geographiclib/>} and Python 3.7+
699 U{math.remainder<https://docs.Python.org/3/
700 library/math.html#math.remainder>}.
701 '''
702 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and
703 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native
704 # fmod( 0, 360) == 0.0
705 # fmod( 360, 360) == 0.0
706 # fmod(-0, 360) == 0.0
707 # fmod(-0.0, 360) == -0.0
708 # fmod(-360, 360) == -0.0
709 # however, using the % operator ...
710 # 0 % 360 == 0
711 # 360 % 360 == 0
712 # 360.0 % 360 == 0.0
713 # -0 % 360 == 0
714 # -360 % 360 == 0 == (-360) % 360
715 # -0.0 % 360 == 0.0 == (-0.0) % 360
716 # -360.0 % 360 == 0.0 == (-360.0) % 360
718 # On Windows 32-bit with python 2.7, math.fmod(-0.0, 360)
719 # == +0.0. This fixes this bug. See also Math::AngNormalize
720 # in the C++ library, Math.sincosd has a similar fix.
721 if _isfinite(x):
722 try:
723 r = remainder(x, y) if x else x
724 except Exception as e:
725 raise _xError(e, unstr(fremainder, x, y))
726 else: # handle x INF and NINF as NAN
727 r = NAN
728 return r
731if _sys_version_info2 < (3, 8): # PYCHOK no cover
732 from math import hypot # OK in Python 3.7-
734 def hypot_(*xs):
735 '''Compute the norm M{sqrt(sum(x**2 for x in xs))}.
737 Similar to Python 3.8+ n-dimension U{math.hypot
738 <https://docs.Python.org/3.8/library/math.html#math.hypot>},
739 but exceptions, C{nan} and C{infinite} values are
740 handled differently.
742 @arg xs: X arguments (C{scalar}s), all positional.
744 @return: Norm (C{float}).
746 @raise OverflowError: Partial C{2sum} overflow.
748 @raise ValueError: Invalid or no B{C{xs}} values.
750 @note: The Python 3.8+ Euclidian distance U{math.dist
751 <https://docs.Python.org/3.8/library/math.html#math.dist>}
752 between 2 I{n}-dimensional points I{p1} and I{p2} can be
753 computed as M{hypot_(*((c1 - c2) for c1, c2 in zip(p1, p2)))},
754 provided I{p1} and I{p2} have the same, non-zero length I{n}.
755 '''
756 _, R = _h_xs2(xs, True, hypot_)
757 return float(R)
759elif _sys_version_info2 < (3, 10):
760 # In Python 3.8 and 3.9 C{math.hypot} is inaccurate, see
761 # U{agdhruv<https://GitHub.com/geopy/geopy/issues/466>},
762 # U{cffk<https://Bugs.Python.org/issue43088>} and module
763 # U{geomath.py<https://PyPI.org/project/geographiclib/1.52>}
765 def hypot(x, y):
766 '''Compute the norm M{sqrt(x**2 + y**2)}.
768 @arg x: X argument (C{scalar}).
769 @arg y: Y argument (C{scalar}).
771 @return: C{sqrt(B{x}**2 + B{y}**2)} (C{float}).
772 '''
773 return (float(Fhypot(x, y, raiser=False)) if y else
774 fabs(x)) if x else fabs(y)
776 from math import hypot as hypot_ # PYCHOK in Python 3.8 and 3.9
777else:
778 from math import hypot # PYCHOK in Python 3.10+
779 hypot_ = hypot
782def _h_xs2(xs, pot_, which):
783 '''(INTERNAL) Helper for L{hypot_} and L{hypot2_}.
784 '''
785 n, xs = len2(xs)
786 if n > 0:
787 h = float(max(map(abs, xs))) # NOT fabs!
788 if h < EPS0:
789 R = _0_0
790 elif n > 1:
791 if pot_:
792 if h != _1_0:
793 xs = ((x / h) for x in xs)
794 R = Fhypot(*xs, raiser=False) * h
795 else:
796 R = Fpowers(2, *xs)
797 else:
798 R = h if pot_ else (h**2)
799 return h, R
801 raise _ValueError(unstr(which, xs), txt=_too_(_few_))
804def hypot1(x):
805 '''Compute the norm M{sqrt(1 + x**2)}.
807 @arg x: Argument (C{scalar} or L{Fsum} instance).
809 @return: Norm (C{float}).
810 '''
811 return hypot(_1_0, x) if x else _1_0
814def hypot2(x, y):
815 '''Compute the I{squared} norm M{x**2 + y**2}.
817 @arg x: X argument (C{scalar} or L{Fsum} instance).
818 @arg y: Y argument (C{scalar} or L{Fsum} instance).
820 @return: C{B{x}**2 + B{y}**2} (C{float} or L{Fsum}).
821 '''
822 if abs(x) < abs(y): # NOT fabs!
823 x, y = y, x
824 if x:
825 h2 = x**2
826 if y:
827 h2 *= (y / x)**2 + _1_0
828 else:
829 h2 = _0_0
830 return h2
833def hypot2_(*xs):
834 '''Compute the I{squared} norm C{fsum(x**2 for x in B{xs})}.
836 @arg xs: X arguments (C{scalar}s or L{Fsum} instances),
837 all positional.
839 @return: Squared norm (C{float}).
841 @raise OverflowError: Partial C{2sum} overflow.
843 @raise ValueError: Invalid or no B{C{xs}} value.
845 @see: Function L{hypot_}.
846 '''
847 _, R = _h_xs2(xs, False, hypot2_)
848 return float(R)
851def _map_mul(a, b, where):
852 '''(INTERNAL) Yield each B{C{a * b}}.
853 '''
854 n = len(b)
855 if len(a) != n: # PYCHOK no cover
856 raise LenError(where, a=len(a), b=n)
857 return _1map_mul(a, b) if n < 4 else map(_operator.mul, a, b)
860def _1map_mul(a, b):
861 '''(INTERNAL) Yield each B{C{a * b}}, 1-primed.
862 '''
863 return _1primed(map(_operator.mul, a, b))
866def norm2(x, y):
867 '''Normalize a 2-dimensional vector.
869 @arg x: X component (C{scalar}).
870 @arg y: Y component (C{scalar}).
872 @return: 2-Tuple C{(x, y)}, normalized.
874 @raise ValueError: Invalid B{C{x}} or B{C{y}}
875 or zero norm.
876 '''
877 try:
878 h = hypot(x, y)
879 if h:
880 x, y = (x / h), (y / h)
881 else:
882 x = _copysign_0_0(x) # pass?
883 y = _copysign_0_0(y)
884 except Exception as e:
885 raise _xError(e, x=x, y=y, h=h)
886 return x, y
889def norm_(*xs):
890 '''Normalize all n-dimensional vector components.
892 @arg xs: Components (C{scalar}s), all positional.
894 @return: Yield each component, normalized.
896 @raise ValueError: Invalid or insufficent B{C{xs}}
897 or zero norm.
898 '''
899 try:
900 i = x = h = None
901 h = hypot_(*xs)
902 _h = (_1_0 / h) if h else _0_0
903 for i, x in enumerate(xs):
904 yield x * _h
905 except Exception as X:
906 raise _xError(X, Fmt.SQUARE(xs=i), x, h=h)
909def _powers(x, n):
910 '''(INTERNAL) Yield C{x**i for i=1..n}.
911 '''
912 p = 1 # type(p) == type(x)
913 for _ in range(n):
914 p *= x
915 yield p
918def _root(x, p, where):
919 '''(INTERNAL) Raise C{x} to power C{0 < p < 1}.
920 '''
921 try:
922 if x > 0:
923 return pow(x, p)
924 elif x < 0:
925 raise ValueError(_negative_)
926 except Exception as X:
927 raise _xError(X, unstr(where, x))
928 return _0_0
931def sqrt0(x, Error=None):
932 '''Return the square root C{sqrt(B{x})} iff C{B{x} > }L{EPS02},
933 preserving C{type(B{x})}.
935 @arg x: Value (C{scalar} or L{Fsum} instance).
936 @kwarg Error: Error to raise for negative B{C{x}}.
938 @return: Square root (C{float} or L{Fsum}) or C{0.0}.
940 @note: Any C{B{x} < }L{EPS02} I{including} C{B{x} < 0}
941 returns C{0.0}.
942 '''
943 if Error and x < 0:
944 raise Error(unstr(sqrt0, x))
945 return _root(x, _0_5, sqrt0) if x > EPS02 else (_0_0 if x < EPS02 else EPS0)
948def sqrt3(x):
949 '''Return the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)},
950 preserving C{type(B{x})}.
952 @arg x: Value (C{scalar} or L{Fsum} instance).
954 @return: Square root I{cubed} (C{float} or L{Fsum}).
956 @raise ValueError: Negative B{C{x}}.
958 @see: Functions L{cbrt} and L{cbrt2}.
959 '''
960 return _root(x, _1_5, sqrt3)
963def sqrt_a(h, b):
964 '''Compute C{I{a}} side of a right-angled triangle from
965 C{sqrt(B{h}**2 - B{b}**2)}.
967 @arg h: Hypotenuse or outer annulus radius (C{scalar}).
968 @arg b: Triangle side or inner annulus radius (C{scalar}).
970 @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}).
972 @raise TypeError: Non-scalar B{C{h}} or B{C{b}}.
974 @raise ValueError: If C{abs(B{h}) < abs(B{b})}.
976 @see: Inner tangent chord B{I{d}} of an U{annulus
977 <https://WikiPedia.org/wiki/Annulus_(mathematics)>}
978 and function U{annulus_area<https://People.SC.FSU.edu/
979 ~jburkardt/py_src/geometry/geometry.py>}.
980 '''
981 try:
982 if not (_isHeight(h) and _isRadius(b)):
983 raise TypeError(_not_scalar_)
984 c = fabs(h)
985 if c > EPS0:
986 s = _1_0 - (b / c)**2
987 if s < 0:
988 raise ValueError(_h_lt_b_)
989 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0)
990 else: # PYCHOK no cover
991 b = fabs(b)
992 d = c - b
993 if d < 0:
994 raise ValueError(_h_lt_b_)
995 d *= c + b
996 a = sqrt(d) if d else _0_0
997 except Exception as x:
998 raise _xError(x, h=h, b=b)
999 return copysign0(a, h)
1002def zcrt(x):
1003 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)},
1004 preserving C{type(B{x})}.
1006 @arg x: Value (C{scalar} or L{Fsum} instance).
1008 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}).
1010 @see: Functions L{bqrt} and L{zqrt}.
1012 @raise ValueError: Negative B{C{x}}.
1013 '''
1014 return _root(x, _1_6th, zcrt)
1017def zqrt(x):
1018 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root,
1019 M{x**(1 / 8)}, preserving C{type(B{x})}.
1021 @arg x: Value (C{scalar} or L{Fsum} instance).
1023 @return: I{Zenzi-quartic} root (C{float} or L{Fsum}).
1025 @see: Functions L{bqrt} and L{zcrt}.
1027 @raise ValueError: Negative B{C{x}}.
1028 '''
1029 return _root(x, _0_125, zqrt)
1031# **) MIT License
1032#
1033# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
1034#
1035# Permission is hereby granted, free of charge, to any person obtaining a
1036# copy of this software and associated documentation files (the "Software"),
1037# to deal in the Software without restriction, including without limitation
1038# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1039# and/or sell copies of the Software, and to permit persons to whom the
1040# Software is furnished to do so, subject to the following conditions:
1041#
1042# The above copyright notice and this permission notice shall be included
1043# in all copies or substantial portions of the Software.
1044#
1045# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1046# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1047# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1048# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1049# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1050# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1051# OTHER DEALINGS IN THE SOFTWARE.