Coverage for pygeodesy/fmath.py: 90%
323 statements
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« prev ^ index » next coverage.py v7.2.2, created at 2024-05-25 12:04 -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, \
10 len2, map1, _xiterable
11from pygeodesy.constants import EPS0, EPS02, EPS1, NAN, PI, PI_2, PI_4, \
12 _0_0, _0_125, _1_6th, _0_25, _1_3rd, _0_5, _1_0, \
13 _N_1_0, _1_5, _copysign_0_0, _isfinite, remainder
14from pygeodesy.errors import _IsnotError, LenError, _TypeError, _ValueError, \
15 _xError, _xkwds_get, _xkwds_pop2
16from pygeodesy.fsums import _2float, Fsum, fsum, fsum1_, _isFsumTuple, _1primed, \
17 Fmt, unstr
18from pygeodesy.interns import MISSING, _negative_, _not_scalar_
19from pygeodesy.lazily import _ALL_LAZY, _sys_version_info2
20# from pygeodesy.streprs import Fmt, unstr # from .fsums
21from pygeodesy.units import Int_, _isHeight, _isRadius, Float_ # PYCHOK for .heights
23from math import fabs, sqrt # pow
24import operator as _operator # in .datums, .trf, .utm
26__all__ = _ALL_LAZY.fmath
27__version__ = '24.05.24'
29# sqrt(2) <https://WikiPedia.org/wiki/Square_root_of_2>
30_0_4142 = 0.41421356237309504880 # ... sqrt(2) - 1
31_2_3rd = _1_3rd * 2
32_h_lt_b_ = 'abs(h) < abs(b)'
35class Fdot(Fsum):
36 '''Precision dot product.
37 '''
38 def __init__(self, a, *b, **name_RESIDUAL):
39 '''New L{Fdot} precision dot product M{sum(a[i] * b[i] for
40 i=0..len(a)-1)}.
42 @arg a: Iterable of values (each C{scalar} or an L{Fsum} or L{Fsum2Tuple}
43 instance).
44 @arg b: Other values (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} instance),
45 all positional.
46 @kwarg name_RESIDUAL: Optional C{B{name}=NN} (C{str}) and the C{B{RESIDUAL}=0.0}
47 threshold (C{scalar}) for raising L{ResidualError}s, see class
48 L{Fsum<Fsum.__init__>}.
50 @raise LenError: Unequal C{len(B{a})} and C{len(B{b})}.
52 @raise OverflowError: Partial C{2sum} overflow.
54 @raise TypeError: Invalid B{C{x}}.
56 @raise ValueError: Non-finite B{C{x}}.
58 @see: Function L{fdot} and method L{Fsum.fadd}.
59 '''
60 Fsum.__init__(self, **name_RESIDUAL)
61 self.fadd(_map_mul(a, b, Fdot))
64class Fhorner(Fsum):
65 '''Precision polynomial evaluation using the Horner form.
66 '''
67 def __init__(self, x, *cs, **name_RESIDUAL):
68 '''New L{Fhorner} evaluation of polynomial M{sum(cs[i] * x**i for
69 i=0..len(cs)-1)}.
71 @arg x: Polynomial argument (C{scalar} or an L{Fsum} or L{Fsum2Tuple}).
72 @arg cs: Polynomial coeffients (each C{scalar} or an L{Fsum} or L{Fsum2Tuple}
73 instance), all positional.
74 @kwarg name_RESIDUAL: Optional C{B{name}=NN} (C{str}) and the C{B{RESIDUAL}=0.0}
75 threshold (C{scalar}) for raising L{ResidualError}s, see class
76 L{Fsum<Fsum.__init__>}.
78 @raise OverflowError: Partial C{2sum} overflow.
80 @raise TypeError: Invalid B{C{x}}.
82 @raise ValueError: Non-finite B{C{x}}.
84 @see: Function L{fhorner} and methods L{Fsum.fadd} and L{Fsum.fmul}.
85 '''
86 Fsum.__init__(self, **name_RESIDUAL)
87 if cs:
88 if _isFsumTuple(x):
89 _mul = self._mul_Fsum
90 else:
91 _mul = self._mul_scalar
92 x = _2float(x=x)
93 op = Fhorner.__name__
94 if len(cs) > 1 and x:
95 for c in reversed(cs):
96 self._fset_ps(_mul(x, op))
97 self._fadd(c, op, up=False)
98 self._update()
99 else: # x == 0
100 self._fadd(cs[0], op)
101 else:
102 self._fset_ps(_0_0)
105class Fhypot(Fsum):
106 '''Precision summation and hypotenuse, default C{root=2}.
107 '''
108 def __init__(self, *xs, **root_name_RESIDUAL_raiser):
109 '''New L{Fhypot} hypotenuse of (the I{root} of) several components
110 (raised to the power I{root}).
112 @arg xs: Components (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} instance),
113 all positional.
114 @kwarg root_name_RESIDUAL_raiser: Optional, exponent and C{B{root}=2} order
115 (C{scalar}), C{B{name}=NN} (C{str}), the C{B{RESIDUAL}=0.0}
116 threshold (C{scalar}) and C{B{raiser}=True} (C{bool}) for
117 raising L{ResidualError}s, see class L{Fsum<Fsum.__init__>} and
118 method L{root<Fsum.root>}.
119 '''
120 r = None # _xkwds_pop2 error
121 try:
122 r, kwds = _xkwds_pop2(root_name_RESIDUAL_raiser, root=2)
123 r, kwds = _xkwds_pop2(kwds, power=r) # for backward compatibility
124 raiser = _Fsum__init__(self, **kwds)
125 if xs:
126 self._facc_power(r, xs, Fhypot, **raiser)
127 self._fset(self.root(r, **raiser))
128 except Exception as X:
129 raise self._ErrorXs(X, xs, root=r)
132class Fpolynomial(Fsum):
133 '''Precision polynomial evaluation.
134 '''
135 def __init__(self, x, *cs, **name_RESIDUAL):
136 '''New L{Fpolynomial} evaluation of the polynomial
137 M{sum(cs[i] * x**i for i=0..len(cs)-1)}.
139 @arg x: Polynomial argument (C{scalar} or an L{Fsum} or L{Fsum2Tuple}).
140 @arg cs: Polynomial coeffients (each C{scalar} or an L{Fsum} or L{Fsum2Tuple}
141 instance), all positional.
142 @kwarg name_RESIDUAL: Optional C{B{name}=NN} (C{str}) and the C{B{RESIDUAL}=0.0}
143 threshold (C{scalar}) for raising L{ResidualError}s, see class
144 L{Fsum<Fsum.__init__>}.
146 @raise OverflowError: Partial C{2sum} overflow.
148 @raise TypeError: Invalid B{C{x}}.
150 @raise ValueError: Non-finite B{C{x}}.
152 @see: Class L{Fhorner}, function L{fpolynomial} and method L{Fsum.fadd}.
153 '''
154 Fsum.__init__(self, *cs[:1], **name_RESIDUAL)
155 n = len(cs) - 1
156 if n > 0:
157 self.fadd(_1map_mul(cs[1:], _powers(x, n)))
158 elif n < 0:
159 self._fset_ps(_0_0)
162class Fpowers(Fsum):
163 '''Precision summation of powers, optimized for C{power=2, 3 and 4}.
164 '''
165 def __init__(self, power, *xs, **name_RESIDUAL_raiser):
166 '''New L{Fpowers} sum of (the I{power} of) several bases.
168 @arg power: The exponent (C{scalar} or an L{Fsum} or L{Fsum2Tuple}).
169 @arg xs: One or more bases (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} instance),
170 all positional.
171 @kwarg name_RESIDUAL_raiser: Optional C{B{name}=NN} (C{str}), the C{B{RESIDUAL}=0.0}
172 threshold (C{scalar}) and C{B{raiser}=True} (C{bool}) for raising
173 L{ResidualError}s, see class L{Fsum<Fsum.__init__>} and method
174 L{fpow<Fsum.fpow>}.
175 '''
176 try:
177 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser)
178 if xs:
179 self._facc_power(power, xs, Fpowers, **raiser) # x**0 == 1
180 except Exception as X:
181 raise self._ErrorXs(X, xs, power=power)
184class Froot(Fsum):
185 '''The root of a precision summation.
186 '''
187 def __init__(self, root, *xs, **name_RESIDUAL_raiser):
188 '''New L{Froot} root of a precision sum.
190 @arg root: The order (C{scalar} or an L{Fsum} or L{Fsum2Tuple}), non-zero.
191 @arg xs: Items to summate (each a C{scalar} or an L{Fsum} or L{Fsum2Tuple} instance),
192 all positional.
193 @kwarg name_RESIDUAL_raiser: Optional C{B{name}=NN} (C{str}), the C{B{RESIDUAL}=0.0}
194 threshold (C{scalar}) and C{B{raiser}=True} (C{bool}) for raising
195 L{ResidualError}s, see class L{Fsum<Fsum.__init__>} and method
196 L{fpow<Fsum.fpow>}.
197 '''
198 try:
199 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser)
200 if xs:
201 self.fadd(xs)
202 self._fset(self.root(root, **raiser))
203 except Exception as X:
204 raise self._ErrorXs(X, xs, root=root)
207class Fcbrt(Froot):
208 '''Cubic root of a precision summation.
209 '''
210 def __init__(self, *xs, **name_RESIDUAL_raiser):
211 '''New L{Fcbrt} cubic root of a precision sum.
213 @see: Class L{Froot} for further details.
214 '''
215 Froot.__init__(self, 3, *xs, **name_RESIDUAL_raiser)
218class Fsqrt(Froot):
219 '''Square root of a precision summation.
220 '''
221 def __init__(self, *xs, **name_RESIDUAL_raiser):
222 '''New L{Fsqrt} square root of a precision sum.
224 @see: Class L{Froot} for further details.
225 '''
226 Froot.__init__(self, 2, *xs, **name_RESIDUAL_raiser)
229def _Fsum__init__(inst, raiser=MISSING, **name_RESIDUAL):
230 '''(INTERNAL) Init an C{F...} instance above.
231 '''
232 Fsum.__init__(inst, **name_RESIDUAL) # PYCHOK self
233 inst._fset_ps(_0_0)
234 return {} if raiser is MISSING else dict(raiser=raiser)
237def bqrt(x):
238 '''Return the 4-th, I{bi-quadratic} or I{quartic} root, M{x**(1 / 4)},
239 preserving C{type(B{x})}.
241 @arg x: Value (C{scalar} or an L{Fsum} or L{Fsum2Tuple}).
243 @return: I{Quartic} root (C{float} or an L{Fsum}).
245 @raise TypeeError: Invalid B{C{x}}.
247 @raise ValueError: Negative B{C{x}}.
249 @see: Functions L{zcrt} and L{zqrt}.
250 '''
251 return _root(x, _0_25, bqrt)
254try:
255 from math import cbrt as _cbrt # Python 3.11+
257except ImportError: # Python 3.10-
259 def _cbrt(x):
260 '''(INTERNAL) Compute the I{signed}, cube root M{x**(1/3)}.
261 '''
262 # <https://archive.lib.MSU.edu/crcmath/math/math/r/r021.htm>
263 # simpler and more accurate than Ken Turkowski's CubeRoot, see
264 # <https://People.FreeBSD.org/~lstewart/references/apple_tr_kt32_cuberoot.pdf>
265 return _copysign(pow(fabs(x), _1_3rd), x) # to avoid complex
268def cbrt(x):
269 '''Compute the cube root M{x**(1/3)}, preserving C{type(B{x})}.
271 @arg x: Value (C{scalar} or an L{Fsum} or L{Fsum2Tuple}).
273 @return: Cubic root (C{float} or L{Fsum}).
275 @see: Functions L{cbrt2} and L{sqrt3}.
276 '''
277 if _isFsumTuple(x):
278 r = abs(x).fpow(_1_3rd)
279 if x.signOf() < 0:
280 r = -r
281 else:
282 r = _cbrt(x)
283 return r # cbrt(-0.0) == -0.0
286def cbrt2(x): # PYCHOK attr
287 '''Compute the cube root I{squared} M{x**(2/3)}, preserving C{type(B{x})}.
289 @arg x: Value (C{scalar} or an L{Fsum} or L{Fsum2Tuple}).
291 @return: Cube root I{squared} (C{float} or L{Fsum}).
293 @see: Functions L{cbrt} and L{sqrt3}.
294 '''
295 return abs(x).fpow(_2_3rd) if _isFsumTuple(x) else _cbrt(x**2)
298def euclid(x, y):
299 '''I{Appoximate} the norm M{sqrt(x**2 + y**2)} by
300 M{max(abs(x), abs(y)) + min(abs(x), abs(y)) * 0.4142...}.
302 @arg x: X component (C{scalar} or L{Fsum} instance).
303 @arg y: Y component (C{scalar} or L{Fsum} instance).
305 @return: Appoximate norm (C{float} or L{Fsum}).
307 @see: Function L{euclid_}.
308 '''
309 x, y = abs(x), abs(y) # NOT fabs!
310 if y > x:
311 x, y = y, x
312 return x + y * _0_4142 # XXX * _0_5 before 20.10.02
315def euclid_(*xs):
316 '''I{Appoximate} the norm M{sqrt(sum(x**2 for x in xs))} by
317 cascaded L{euclid}.
319 @arg xs: X arguments (each C{scalar} or an L{Fsum}
320 instance), all positional.
322 @return: Appoximate norm (C{float} or L{Fsum}).
324 @see: Function L{euclid}.
325 '''
326 e = _0_0
327 for x in sorted(map(abs, xs)): # NOT fabs, reverse=True!
328 # e = euclid(x, e)
329 if e < x:
330 e, x = x, e
331 if x:
332 e += x * _0_4142
333 return e
336def facos1(x):
337 '''Fast approximation of L{pygeodesy.acos1}C{(B{x})}.
339 @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/
340 ShaderFastLibs/blob/master/ShaderFastMathLib.h>}.
341 '''
342 a = fabs(x)
343 if a < EPS0:
344 r = PI_2
345 elif a < EPS1:
346 H = Fhorner(-a, 1.5707288, 0.2121144, 0.0742610, 0.0187293)
347 r = float(H * sqrt(_1_0 - a))
348 if x < 0:
349 r = PI - r
350 else:
351 r = PI if x < 0 else _0_0
352 return r
355def fasin1(x): # PYCHOK no cover
356 '''Fast approximation of L{pygeodesy.asin1}C{(B{x})}.
358 @see: L{facos1}.
359 '''
360 return PI_2 - facos1(x)
363def fatan(x):
364 '''Fast approximation of C{atan(B{x})}.
365 '''
366 a = fabs(x)
367 if a < _1_0:
368 r = fatan1(a) if a else _0_0
369 elif a > _1_0:
370 r = PI_2 - fatan1(_1_0 / a) # == fatan2(a, _1_0)
371 else:
372 r = PI_4
373 if x < 0: # copysign0(r, x)
374 r = -r
375 return r
378def fatan1(x):
379 '''Fast approximation of C{atan(B{x})} for C{0 <= B{x} <= 1}, I{unchecked}.
381 @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/ShaderFastLibs/
382 blob/master/ShaderFastMathLib.h>} and U{Efficient approximations
383 for the arctangent function<http://www-Labs.IRO.UMontreal.CA/
384 ~mignotte/IFT2425/Documents/EfficientApproximationArctgFunction.pdf>},
385 IEEE Signal Processing Magazine, 111, May 2006.
386 '''
387 # Eq (9): PI_4 * x - x * (abs(x) - 1) * (0.2447 + 0.0663 * abs(x)), for -1 < x < 1
388 # PI_4 * x - (x**2 - x) * (0.2447 + 0.0663 * x), for 0 < x - 1
389 # x * (1.0300981633974482 + x * (-0.1784 - x * 0.0663))
390 H = Fhorner(x, _0_0, 1.0300981634, -0.1784, -0.0663)
391 return float(H)
394def fatan2(y, x):
395 '''Fast approximation of C{atan2(B{y}, B{x})}.
397 @see: U{fastApproximateAtan(x, y)<https://GitHub.com/CesiumGS/cesium/blob/
398 master/Source/Shaders/Builtin/Functions/fastApproximateAtan.glsl>}
399 and L{fatan1}.
400 '''
401 a, b = fabs(x), fabs(y)
402 if b > a:
403 r = (PI_2 - fatan1(a / b)) if a else PI_2
404 elif a > b:
405 r = fatan1(b / a) if b else _0_0
406 elif a: # a == b != 0
407 r = PI_4
408 else: # a == b == 0
409 return _0_0
410 if x < 0:
411 r = PI - r
412 if y < 0: # copysign0(r, y)
413 r = -r
414 return r
417def favg(a, b, f=_0_5):
418 '''Return the precision average of two values.
420 @arg a: One (C{scalar} or an L{Fsum} or L{Fsum2Tuple}).
421 @arg b: Other (C{scalar} or an L{Fsum} or L{Fsum2Tuple}).
422 @kwarg f: Optional fraction (C{float}).
424 @return: M{a + f * (b - a)} (C{float}).
425 '''
426# @raise ValueError: Fraction out of range.
427# '''
428# if not 0 <= f <= 1: # XXX restrict fraction?
429# raise _ValueError(fraction=f)
430 # a + f * (b - a) == a * (1 - f) + b * f
431 return fsum1_(a, a * (-f), b * f)
434def fdot(a, *b):
435 '''Return the precision dot product M{sum(a[i] * b[i] for
436 i=0..len(a))}.
438 @arg a: Iterable of values (each C{scalar}).
439 @arg b: Other values (each C{scalar}), all positional.
441 @return: Dot product (C{float}).
443 @raise LenError: Unequal C{len(B{a})} and C{len(B{b})}.
445 @see: Class L{Fdot} and U{Algorithm 5.10 B{DotK}
446 <https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>}.
447 '''
448 return fsum(_map_mul(a, b, fdot))
451def fdot3(xs, ys, zs, start=0):
452 '''Return the precision dot product M{start +
453 sum(a[i] * b[i] * c[i] for i=0..len(a)-1)}.
455 @arg xs: Iterable (each C{scalar} or an L{Fsum} or
456 L{Fsum2Tuple} instance).
457 @arg ys: Iterable (each C{scalar} or an L{Fsum} or
458 L{Fsum2Tuple} instance).
459 @arg zs: Iterable (each C{scalar} or an L{Fsum} or
460 L{Fsum2Tuple} instance).
461 @kwarg start: Optional bias (C{scalar} or an L{Fsum}
462 or L{Fsum2Tuple}).
464 @return: Dot product (C{float}).
466 @raise LenError: Unequal C{len(B{xs})}, C{len(B{ys})}
467 and/or C{len(B{zs})}.
469 @raise OverflowError: Partial C{2sum} overflow.
470 '''
471 def _mul3(xs, ys, zs, s, p):
472 if s:
473 yield s
474 if p:
475 yield _1_0
476 _F = Fsum
477 for x, y, z in zip(xs, ys, zs):
478 yield (_F(x) * y) * z
479 if p:
480 yield _N_1_0
482 n = len(xs)
483 if not n == len(ys) == len(zs):
484 raise LenError(fdot3, xs=n, ys=len(ys), zs=len(zs))
486 return fsum(_mul3(xs, ys, zs, start, n < 4))
489def fhorner(x, *cs):
490 '''Evaluate the polynomial M{sum(cs[i] * x**i for
491 i=0..len(cs)-1)} using the Horner form.
493 @return: Horner sum (C{float}).
495 @see: Class L{Fhorner} for further details.
496 '''
497 H = Fhorner(x, *cs)
498 return float(H)
501def fidw(xs, ds, beta=2):
502 '''Interpolate using U{Inverse Distance Weighting
503 <https://WikiPedia.org/wiki/Inverse_distance_weighting>} (IDW).
505 @arg xs: Known values (each C{scalar} or an L{Fsum} or
506 L{Fsum2Tuple} instance).
507 @arg ds: Non-negative distances (each C{scalar} or an L{Fsum}
508 or L{Fsum2Tuple} instance).
509 @kwarg beta: Inverse distance power (C{int}, 0, 1, 2, or 3).
511 @return: Interpolated value C{x} (C{float}).
513 @raise LenError: Unequal or zero C{len(B{ds})} and C{len(B{xs})}.
515 @raise TypeError: An invalid B{C{ds}} or B{C{xs}}.
517 @raise ValueError: Invalid B{C{beta}}, negative B{C{ds}} or
518 weighted B{C{ds}} below L{EPS}.
520 @note: Using C{B{beta}=0} returns the mean of B{C{xs}}.
521 '''
522 n, xs = len2(xs)
523 if n > 1:
524 b = -Int_(beta=beta, low=0, high=3)
525 if b < 0:
526 try: # weighted
527 _F = Fsum
528 W = _F()
529 X = _F()
530 for i, d in enumerate(_xiterable(ds)):
531 x = xs[i]
532 D = _F(d)
533 if D < EPS0:
534 if D < 0:
535 raise ValueError(_negative_)
536 x = float(x)
537 i = n
538 break
539 if D.fpow(b):
540 W += D
541 X += D.fmul(x)
542 else:
543 x = X.fover(W, raiser=False)
544 i += 1 # len(xs) >= len(ds)
545 except IndexError:
546 i += 1 # len(xs) < i < len(ds)
547 except Exception as X:
548 _I = Fmt.INDEX
549 raise _xError(X, _I(xs=i), x, _I(ds=i), d)
550 else: # b == 0
551 x = fsum(xs) / n # fmean(xs)
552 i = n
553 elif n:
554 x = float(xs[0])
555 i = n
556 else:
557 x = _0_0
558 i, _ = len2(ds)
559 if i != n:
560 raise LenError(fidw, xs=n, ds=i)
561 return x
564def fmean(xs):
565 '''Compute the accurate mean M{sum(xs) / len(xs)}.
567 @arg xs: Values (C{scalar} or L{Fsum} instances).
569 @return: Mean value (C{float}).
571 @raise LenError: No B{C{xs}} values.
573 @raise OverflowError: Partial C{2sum} overflow.
574 '''
575 n, xs = len2(xs)
576 if n < 1:
577 raise LenError(fmean, xs=xs)
578 return Fsum(*xs).fover(n) if n > 1 else _2float(index=0, xs=xs[0])
581def fmean_(*xs):
582 '''Compute the accurate mean M{sum(xs) / len(xs)}.
584 @see: Function L{fmean} for further details.
585 '''
586 return fmean(xs)
589def fpolynomial(x, *cs, **over):
590 '''Evaluate the polynomial M{sum(cs[i] * x**i for
591 i=0..len(cs)) [/ over]}.
593 @kwarg over: Optional final, I{non-zero} divisor (C{scalar}).
595 @return: Polynomial value (C{float}).
597 @see: Class L{Fpolynomial} for further details.
598 '''
599 P = Fpolynomial(x, *cs)
600 d = _xkwds_get(over, over=0) if over else 0
601 return P.fover(d) if d else float(P)
604def fpowers(x, n, alts=0):
605 '''Return a series of powers M{[x**i for i=1..n]}.
607 @arg x: Value (C{scalar} or an L{Fsum} or L{Fsum2Tuple}).
608 @arg n: Highest exponent (C{int}).
609 @kwarg alts: Only alternating powers, starting with this
610 exponent (C{int}).
612 @return: Tuple of powers of B{C{x}} (each C{type(B{x})}).
614 @raise TypeError: Invalid B{C{x}} or B{C{n}} not C{int}.
616 @raise ValueError: Non-finite B{C{x}} or invalid B{C{n}}.
617 '''
618 if not isint(n):
619 raise _IsnotError(int.__name__, n=n)
620 elif n < 1:
621 raise _ValueError(n=n)
623 p = x if isint(x) or _isFsumTuple(x) else _2float(x=x)
624 ps = tuple(_powers(p, n))
626 if alts > 0: # x**2, x**4, ...
627 # ps[alts-1::2] chokes PyChecker
628 ps = ps[slice(alts-1, None, 2)]
630 return ps
633try:
634 from math import prod as fprod # Python 3.8
635except ImportError:
637 def fprod(xs, start=1):
638 '''Iterable product, like C{math.prod} or C{numpy.prod}.
640 @arg xs: Iterable of values to be multiplied (each
641 C{scalar} or an L{Fsum}).
642 @kwarg start: Initial value, also the value returned
643 for an empty B{C{xs}} (C{scalar}).
645 @return: The product (C{float} or an L{Fsum}).
647 @see: U{NumPy.prod<https://docs.SciPy.org/doc/
648 numpy/reference/generated/numpy.prod.html>}.
649 '''
650 return freduce(_operator.mul, xs, start)
653def frandoms(n, seeded=None):
654 '''Generate C{n} (long) lists of random C{floats}.
656 @arg n: Number of lists to generate (C{int}, non-negative).
657 @kwarg seeded: If C{scalar}, use C{random.seed(B{seeded})} or
658 if C{True}, seed using today's C{year-day}.
660 @see: U{Hettinger<https://GitHub.com/ActiveState/code/tree/master/recipes/
661 Python/393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}.
662 '''
663 from random import gauss, random, seed, shuffle
665 if seeded is None:
666 pass
667 elif seeded and isbool(seeded):
668 from time import localtime
669 seed(localtime().tm_yday)
670 elif isscalar(seeded):
671 seed(seeded)
673 c = (7, 1e100, -7, -1e100, -9e-20, 8e-20) * 7
674 for _ in range(n):
675 s = 0
676 t = list(c)
677 _a = t.append
678 for _ in range(n * 8):
679 v = gauss(0, random())**7 - s
680 _a(v)
681 s += v
682 shuffle(t)
683 yield t
686def frange(start, number, step=1):
687 '''Generate a range of C{float}s.
689 @arg start: First value (C{float}).
690 @arg number: The number of C{float}s to generate (C{int}).
691 @kwarg step: Increment value (C{float}).
693 @return: A generator (C{float}s).
695 @see: U{NumPy.prod<https://docs.SciPy.org/doc/
696 numpy/reference/generated/numpy.arange.html>}.
697 '''
698 if not isint(number):
699 raise _IsnotError(int.__name__, number=number)
700 for i in range(number):
701 yield start + (step * i)
704try:
705 from functools import reduce as freduce
706except ImportError:
707 try:
708 freduce = reduce # PYCHOK expected
709 except NameError: # Python 3+
711 def freduce(f, xs, *start):
712 '''For missing C{functools.reduce}.
713 '''
714 if start:
715 r = v = start[0]
716 else:
717 r, v = 0, MISSING
718 for v in xs:
719 r = f(r, v)
720 if v is MISSING:
721 raise _TypeError(xs=(), start=MISSING)
722 return r
725def fremainder(x, y):
726 '''Remainder in range C{[-B{y / 2}, B{y / 2}]}.
728 @arg x: Numerator (C{scalar}).
729 @arg y: Modulus, denominator (C{scalar}).
731 @return: Remainder (C{scalar}, preserving signed
732 0.0) or C{NAN} for any non-finite B{C{x}}.
734 @raise ValueError: Infinite or near-zero B{C{y}}.
736 @see: I{Karney}'s U{Math.remainder<https://PyPI.org/
737 project/geographiclib/>} and Python 3.7+
738 U{math.remainder<https://docs.Python.org/3/
739 library/math.html#math.remainder>}.
740 '''
741 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and
742 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native
743 # fmod( 0, 360) == 0.0
744 # fmod( 360, 360) == 0.0
745 # fmod(-0, 360) == 0.0
746 # fmod(-0.0, 360) == -0.0
747 # fmod(-360, 360) == -0.0
748 # however, using the % operator ...
749 # 0 % 360 == 0
750 # 360 % 360 == 0
751 # 360.0 % 360 == 0.0
752 # -0 % 360 == 0
753 # -360 % 360 == 0 == (-360) % 360
754 # -0.0 % 360 == 0.0 == (-0.0) % 360
755 # -360.0 % 360 == 0.0 == (-360.0) % 360
757 # On Windows 32-bit with python 2.7, math.fmod(-0.0, 360)
758 # == +0.0. This fixes this bug. See also Math::AngNormalize
759 # in the C++ library, Math.sincosd has a similar fix.
760 if _isfinite(x):
761 try:
762 r = remainder(x, y) if x else x
763 except Exception as e:
764 raise _xError(e, unstr(fremainder, x, y))
765 else: # handle x INF and NINF as NAN
766 r = NAN
767 return r
770if _sys_version_info2 < (3, 8): # PYCHOK no cover
771 from math import hypot # OK in Python 3.7-
773 def hypot_(*xs):
774 '''Compute the norm M{sqrt(sum(x**2 for x in xs))}.
776 Similar to Python 3.8+ n-dimension U{math.hypot
777 <https://docs.Python.org/3.8/library/math.html#math.hypot>},
778 but exceptions, C{nan} and C{infinite} values are
779 handled differently.
781 @arg xs: X arguments (C{scalar}s), all positional.
783 @return: Norm (C{float}).
785 @raise OverflowError: Partial C{2sum} overflow.
787 @raise ValueError: Invalid or no B{C{xs}} values.
789 @note: The Python 3.8+ Euclidian distance U{math.dist
790 <https://docs.Python.org/3.8/library/math.html#math.dist>}
791 between 2 I{n}-dimensional points I{p1} and I{p2} can be
792 computed as M{hypot_(*((c1 - c2) for c1, c2 in zip(p1, p2)))},
793 provided I{p1} and I{p2} have the same, non-zero length I{n}.
794 '''
795 return float(Fhypot(*xs, raiser=False))
797elif _sys_version_info2 < (3, 10):
798 # In Python 3.8 and 3.9 C{math.hypot} is inaccurate, see
799 # U{agdhruv<https://GitHub.com/geopy/geopy/issues/466>},
800 # U{cffk<https://Bugs.Python.org/issue43088>} and module
801 # U{geomath.py<https://PyPI.org/project/geographiclib/1.52>}
803 def hypot(x, y):
804 '''Compute the norm M{sqrt(x**2 + y**2)}.
806 @arg x: X argument (C{scalar}).
807 @arg y: Y argument (C{scalar}).
809 @return: C{sqrt(B{x}**2 + B{y}**2)} (C{float}).
810 '''
811 return float(Fhypot(x, y, raiser=False))
813 from math import hypot as hypot_ # PYCHOK in Python 3.8 and 3.9
814else:
815 from math import hypot # PYCHOK in Python 3.10+
816 hypot_ = hypot
819def hypot1(x):
820 '''Compute the norm M{sqrt(1 + x**2)}.
822 @arg x: Argument (C{scalar} or L{Fsum} or L{Fsum2Tuple}).
824 @return: Norm (C{float}).
825 '''
826 if _isFsumTuple(x):
827 h = float(Fhypot(_1_0, x)) if x else _1_0
828 else:
829 h = hypot(_1_0, x) if x else _1_0
830 return h
833def hypot2(x, y):
834 '''Compute the I{squared} norm M{x**2 + y**2}.
836 @arg x: X (C{scalar} or L{Fsum} or L{Fsum2Tuple}).
837 @arg y: Y (C{scalar} or L{Fsum} or L{Fsum2Tuple}).
839 @return: C{B{x}**2 + B{y}**2} (C{float}).
840 '''
841 x, y = map1(abs, x, y) # NOT fabs!
842 if y > x:
843 x, y = y, x
844 if x:
845 h2 = x**2
846 if y:
847 h2 *= (y / x)**2 + _1_0
848 h2 = float(h2)
849 else:
850 h2 = _0_0
851 return h2
854def hypot2_(*xs):
855 '''Compute the I{squared} norm C{fsum(x**2 for x in B{xs})}.
857 @arg xs: Components (each C{scalar} or an L{Fsum} or
858 L{Fsum2Tuple} instance), all positional.
860 @return: Squared norm (C{float}).
862 @see: Class L{Fpowers} for further details.
863 '''
864 h2 = float(max(map(abs, xs))) if xs else _0_0
865 if h2:
866 _h = _1_0 / h2
867 h2 = Fpowers(2, *((x * _h) for x in xs))
868 h2 = h2.fover(_h**2)
869 return h2
872def _map_mul(xs, ys, where):
873 '''(INTERNAL) Yield each B{C{x * y}}.
874 '''
875 n = len(ys)
876 if len(xs) != n: # PYCHOK no cover
877 raise LenError(where, xs=len(xs), ys=n)
878 return _1map_mul(xs, ys) if n < 4 else map(
879 _operator.mul, map(Fsum, xs), ys)
882def _1map_mul(xs, ys):
883 '''(INTERNAL) Yield each B{C{x * y}}, 1-primed.
884 '''
885 return _1primed(map(_operator.mul, map(Fsum, xs), ys))
888def norm2(x, y):
889 '''Normalize a 2-dimensional vector.
891 @arg x: X component (C{scalar}).
892 @arg y: Y component (C{scalar}).
894 @return: 2-Tuple C{(x, y)}, normalized.
896 @raise ValueError: Invalid B{C{x}} or B{C{y}}
897 or zero norm.
898 '''
899 try:
900 h = hypot(x, y)
901 if h:
902 x, y = (x / h), (y / h)
903 else:
904 x = _copysign_0_0(x) # pass?
905 y = _copysign_0_0(y)
906 except Exception as e:
907 raise _xError(e, x=x, y=y, h=h)
908 return x, y
911def norm_(*xs):
912 '''Normalize all n-dimensional vector components.
914 @arg xs: Components (C{scalar}s), all positional.
916 @return: Yield each component, normalized.
918 @raise ValueError: Invalid or insufficent B{C{xs}}
919 or zero norm.
920 '''
921 try:
922 i = x = h = None
923 h = hypot_(*xs)
924 _h = (_1_0 / h) if h else _0_0
925 for i, x in enumerate(xs):
926 yield x * _h
927 except Exception as X:
928 raise _xError(X, Fmt.SQUARE(xs=i), x, h=h)
931def _powers(x, n):
932 '''(INTERNAL) Yield C{x**i for i=1..n}.
933 '''
934 p = 1 # type(p) == type(x)
935 for _ in range(n):
936 p *= x
937 yield p
940def _root(x, p, where):
941 '''(INTERNAL) Raise C{x} to power C{0 < p < 1}.
942 '''
943 try:
944 if x > 0:
945 return Fsum(x).fpow(p).as_iscalar
946 elif x < 0:
947 raise ValueError(_negative_)
948 except Exception as X:
949 raise _xError(X, unstr(where, x))
950 return _0_0
953def sqrt0(x, Error=None):
954 '''Return the square root C{sqrt(B{x})} iff C{B{x} > }L{EPS02},
955 preserving C{type(B{x})}.
957 @arg x: Value (C{scalar} or an L{Fsum} or L{Fsum2Tuple}).
958 @kwarg Error: Error to raise for negative B{C{x}}.
960 @return: Square root (C{float} or L{Fsum}) or C{0.0}.
962 @raise TypeeError: Invalid B{C{x}}.
964 @note: Any C{B{x} < }L{EPS02} I{including} C{B{x} < 0}
965 returns C{0.0}.
966 '''
967 if Error and x < 0:
968 raise Error(unstr(sqrt0, x))
969 return _root(x, _0_5, sqrt0) if x > EPS02 else (_0_0 if x < EPS02 else EPS0)
972def sqrt3(x):
973 '''Return the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)},
974 preserving C{type(B{x})}.
976 @arg x: Value (C{scalar} or an L{Fsum} or L{Fsum2Tuple}).
978 @return: Square root I{cubed} (C{float} or L{Fsum}).
980 @raise TypeeError: Invalid B{C{x}}.
982 @raise ValueError: Negative B{C{x}}.
984 @see: Functions L{cbrt} and L{cbrt2}.
985 '''
986 return _root(x, _1_5, sqrt3)
989def sqrt_a(h, b):
990 '''Compute C{I{a}} side of a right-angled triangle from
991 C{sqrt(B{h}**2 - B{b}**2)}.
993 @arg h: Hypotenuse or outer annulus radius (C{scalar}).
994 @arg b: Triangle side or inner annulus radius (C{scalar}).
996 @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}).
998 @raise TypeError: Non-scalar B{C{h}} or B{C{b}}.
1000 @raise ValueError: If C{abs(B{h}) < abs(B{b})}.
1002 @see: Inner tangent chord B{I{d}} of an U{annulus
1003 <https://WikiPedia.org/wiki/Annulus_(mathematics)>}
1004 and function U{annulus_area<https://People.SC.FSU.edu/
1005 ~jburkardt/py_src/geometry/geometry.py>}.
1006 '''
1007 try:
1008 if not (_isHeight(h) and _isRadius(b)):
1009 raise TypeError(_not_scalar_)
1010 c = fabs(h)
1011 if c > EPS0:
1012 s = _1_0 - (b / c)**2
1013 if s < 0:
1014 raise ValueError(_h_lt_b_)
1015 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0)
1016 else: # PYCHOK no cover
1017 b = fabs(b)
1018 d = c - b
1019 if d < 0:
1020 raise ValueError(_h_lt_b_)
1021 d *= c + b
1022 a = sqrt(d) if d else _0_0
1023 except Exception as x:
1024 raise _xError(x, h=h, b=b)
1025 return copysign0(a, h)
1028def zcrt(x):
1029 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)},
1030 preserving C{type(B{x})}.
1032 @arg x: Value (C{scalar} or an L{Fsum} or L{Fsum2Tuple}).
1034 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}).
1036 @see: Functions L{bqrt} and L{zqrt}.
1038 @raise TypeeError: Invalid B{C{x}}.
1040 @raise ValueError: Negative B{C{x}}.
1041 '''
1042 return _root(x, _1_6th, zcrt)
1045def zqrt(x):
1046 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root,
1047 M{x**(1 / 8)}, preserving C{type(B{x})}.
1049 @arg x: Value (C{scalar} or an L{Fsum} or L{Fsum2Tuple}).
1051 @return: I{Zenzi-quartic} root (C{float} or L{Fsum}).
1053 @see: Functions L{bqrt} and L{zcrt}.
1055 @raise TypeeError: Invalid B{C{x}}.
1057 @raise ValueError: Negative B{C{x}}.
1058 '''
1059 return _root(x, _0_125, zqrt)
1061# **) MIT License
1062#
1063# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
1064#
1065# Permission is hereby granted, free of charge, to any person obtaining a
1066# copy of this software and associated documentation files (the "Software"),
1067# to deal in the Software without restriction, including without limitation
1068# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1069# and/or sell copies of the Software, and to permit persons to whom the
1070# Software is furnished to do so, subject to the following conditions:
1071#
1072# The above copyright notice and this permission notice shall be included
1073# in all copies or substantial portions of the Software.
1074#
1075# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1076# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1077# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1078# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1079# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1080# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1081# OTHER DEALINGS IN THE SOFTWARE.