Coverage for pygeodesy/fmath.py: 90%
329 statements
« prev ^ index » next coverage.py v7.6.1, created at 2024-11-12 16:17 -0500
« prev ^ index » next coverage.py v7.6.1, created at 2024-11-12 16:17 -0500
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 _1_5, _copysign_0_0, isfinite, remainder
14from pygeodesy.errors import _IsnotError, LenError, _TypeError, _ValueError, \
15 _xError, _xkwds, _xkwds_pop2, _xsError
16from pygeodesy.fsums import _2float, Fsum, fsum, _isFsum_2Tuple, Fmt, unstr
17from pygeodesy.interns import MISSING, _negative_, _not_scalar_
18from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS
19# from pygeodesy.streprs import Fmt, unstr # from .fsums
20from pygeodesy.units import Int_, _isHeight, _isRadius, Float_ # PYCHOK for .heights
22from math import fabs, sqrt # pow
23import operator as _operator # in .datums, .trf, .utm
25__all__ = _ALL_LAZY.fmath
26__version__ = '24.11.08'
28# sqrt(2) - 1 <https://WikiPedia.org/wiki/Square_root_of_2>
29_0_4142 = 0.41421356237309504880 # ... ~ 3730904090310553 / 9007199254740992
30_2_3rd = _1_3rd * 2
31_h_lt_b_ = 'abs(h) < abs(b)'
34class Fdot(Fsum):
35 '''Precision dot product.
36 '''
37 def __init__(self, a, *b, **start_name_f2product_nonfinites_RESIDUAL):
38 '''New L{Fdot} precision dot product M{sum(a[i] * b[i] for i=0..len(a)-1)}.
40 @arg a: Iterable of values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
41 @arg b: Other values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all
42 positional.
43 @kwarg start_name_f2product_nonfinites_RESIDUAL: Optional bias C{B{start}=0}
44 (C{scalar}, an L{Fsum} or L{Fsum2Tuple}), C{B{name}=NN} (C{str})
45 and other settings, see class L{Fsum<Fsum.__init__>}.
47 @raise LenError: Unequal C{len(B{a})} and C{len(B{b})}.
49 @raise OverflowError: Partial C{2sum} overflow.
51 @raise TypeError: Invalid B{C{x}}.
53 @raise ValueError: Non-finite B{C{x}}.
55 @see: Function L{fdot} and method L{Fsum.fadd}.
56 '''
57 s, kwds = _xkwds_pop2(start_name_f2product_nonfinites_RESIDUAL, start=_0_0)
58 Fsum.__init__(self, **kwds)
59 self(s)
61 n = len(b)
62 if len(a) != n: # PYCHOK no cover
63 raise LenError(Fdot, a=len(a), b=n)
64 self._facc_dot(n, a, b, **kwds)
67class Fhorner(Fsum):
68 '''Precision polynomial evaluation using the Horner form.
69 '''
70 def __init__(self, x, *cs, **incx_name_f2product_nonfinites_RESIDUAL):
71 '''New L{Fhorner} form evaluation of polynomial M{sum(cs[i] * x**i for
72 i=0..n)} with in- or decreasing exponent M{sum(... i=n..0)}, where C{n
73 = len(cs) - 1}.
75 @arg x: Polynomial argument (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
76 @arg cs: Polynomial coeffients (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}),
77 all positional.
78 @kwarg incx_name_f2product_nonfinites_RESIDUAL: Optional C{B{name}=NN} (C{str}),
79 C{B{incx}=True} for in-/decreasing exponents (C{bool}) and other
80 settings, see class L{Fsum<Fsum.__init__>}.
82 @raise OverflowError: Partial C{2sum} overflow.
84 @raise TypeError: Invalid B{C{x}}.
86 @raise ValueError: Non-finite B{C{x}}.
88 @see: Function L{fhorner} and methods L{Fsum.fadd} and L{Fsum.fmul}.
89 '''
90 incx, kwds = _xkwds_pop2(incx_name_f2product_nonfinites_RESIDUAL, incx=True)
91 Fsum.__init__(self, **kwds)
92 self._fhorner(x, cs, Fhorner, incx=incx)
95class Fhypot(Fsum):
96 '''Precision summation and hypotenuse, default C{root=2}.
97 '''
98 def __init__(self, *xs, **root_name_f2product_nonfinites_RESIDUAL_raiser):
99 '''New L{Fhypot} hypotenuse of (the I{root} of) several components (raised
100 to the power I{root}).
102 @arg xs: Components (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all
103 positional.
104 @kwarg root_name_f2product_nonfinites_RESIDUAL_raiser: Optional, exponent
105 and C{B{root}=2} order (C{scalar}), C{B{name}=NN} (C{str}),
106 C{B{raiser}=True} (C{bool}) for raising L{ResidualError}s and
107 other settings, see class L{Fsum<Fsum.__init__>} and method
108 L{root<Fsum.root>}.
109 '''
110 r = None # _xkwds_pop2 error
111 try:
112 r, kwds = _xkwds_pop2(root_name_f2product_nonfinites_RESIDUAL_raiser, root=2)
113 r, kwds = _xkwds_pop2(kwds, power=r) # for backward compatibility
114 t, kwds = _xkwds_pop2(kwds, raiser=True)
115 Fsum.__init__(self, **kwds)
116 self(_0_0)
117 if xs:
118 self._facc_power(r, xs, Fhypot, raiser=t)
119 self._fset(self.root(r, raiser=t))
120 except Exception as X:
121 raise self._ErrorXs(X, xs, root=r)
124class Fpolynomial(Fsum):
125 '''Precision polynomial evaluation.
126 '''
127 def __init__(self, x, *cs, **name_f2product_nonfinites_RESIDUAL):
128 '''New L{Fpolynomial} evaluation of the polynomial M{sum(cs[i] * x**i for
129 i=0..len(cs)-1)}.
131 @arg x: Polynomial argument (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
132 @arg cs: Polynomial coeffients (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}),
133 all positional.
134 @kwarg name_f2product_nonfinites_RESIDUAL: Optional C{B{name}=NN} (C{str})
135 and other settings, see class L{Fsum<Fsum.__init__>}.
137 @raise OverflowError: Partial C{2sum} overflow.
139 @raise TypeError: Invalid B{C{x}}.
141 @raise ValueError: Non-finite B{C{x}}.
143 @see: Class L{Fhorner}, function L{fpolynomial} and method L{Fsum.fadd}.
144 '''
145 Fsum.__init__(self, **name_f2product_nonfinites_RESIDUAL)
146 n = len(cs) - 1
147 self(_0_0 if n < 0 else cs[0])
148 self._facc_dot(n, cs[1:], _powers(x, n), **name_f2product_nonfinites_RESIDUAL)
151class Fpowers(Fsum):
152 '''Precision summation of powers, optimized for C{power=2, 3 and 4}.
153 '''
154 def __init__(self, power, *xs, **name_f2product_nonfinites_RESIDUAL_raiser):
155 '''New L{Fpowers} sum of (the I{power} of) several bases.
157 @arg power: The exponent (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
158 @arg xs: One or more bases (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all
159 positional.
160 @kwarg name_f2product_nonfinites_RESIDUAL_raiser: Optional C{B{name}=NN}
161 (C{str}), C{B{raiser}=True} (C{bool}) for raising L{ResidualError}s
162 and other settings, see class L{Fsum<Fsum.__init__>} and method
163 L{fpow<Fsum.fpow>}.
164 '''
165 try:
166 t, kwds = _xkwds_pop2(name_f2product_nonfinites_RESIDUAL_raiser, raiser=True)
167 Fsum.__init__(self, **kwds)
168 self(_0_0)
169 if xs:
170 self._facc_power(power, xs, Fpowers, raiser=t) # x**0 == 1
171 except Exception as X:
172 raise self._ErrorXs(X, xs, power=power)
175class Froot(Fsum):
176 '''The root of a precision summation.
177 '''
178 def __init__(self, root, *xs, **name_f2product_nonfinites_RESIDUAL_raiser):
179 '''New L{Froot} root of a precision sum.
181 @arg root: The order (C{scalar}, an L{Fsum} or L{Fsum2Tuple}), non-zero.
182 @arg xs: Items to summate (each a C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all
183 positional.
184 @kwarg name_f2product_nonfinites_RESIDUAL_raiser: Optional C{B{name}=NN}
185 (C{str}), C{B{raiser}=True} (C{bool}) for raising L{ResidualError}s
186 and other settings, see class L{Fsum<Fsum.__init__>} and method
187 L{fpow<Fsum.fpow>}.
188 '''
189 try:
190 raiser, kwds = _xkwds_pop2(name_f2product_nonfinites_RESIDUAL_raiser, raiser=True)
191 Fsum.__init__(self, **kwds)
192 self(_0_0)
193 if xs:
194 self.fadd(xs)
195 self(self.root(root, raiser=raiser))
196 except Exception as X:
197 raise self._ErrorXs(X, xs, root=root)
200class Fcbrt(Froot):
201 '''Cubic root of a precision summation.
202 '''
203 def __init__(self, *xs, **name_f2product_nonfinites_RESIDUAL_raiser):
204 '''New L{Fcbrt} cubic root of a precision sum.
206 @see: Class L{Froot<Froot.__init__>} for further details.
207 '''
208 Froot.__init__(self, 3, *xs, **name_f2product_nonfinites_RESIDUAL_raiser)
211class Fsqrt(Froot):
212 '''Square root of a precision summation.
213 '''
214 def __init__(self, *xs, **name_f2product_nonfinites_RESIDUAL_raiser):
215 '''New L{Fsqrt} square root of a precision sum.
217 @see: Class L{Froot<Froot.__init__>} for further details.
218 '''
219 Froot.__init__(self, 2, *xs, **name_f2product_nonfinites_RESIDUAL_raiser)
222def bqrt(x):
223 '''Return the 4-th, I{bi-quadratic} or I{quartic} root, M{x**(1 / 4)},
224 preserving C{type(B{x})}.
226 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
228 @return: I{Quartic} root (C{float} or an L{Fsum}).
230 @raise TypeeError: Invalid B{C{x}}.
232 @raise ValueError: Negative B{C{x}}.
234 @see: Functions L{zcrt} and L{zqrt}.
235 '''
236 return _root(x, _0_25, bqrt)
239try:
240 from math import cbrt as _cbrt # Python 3.11+
242except ImportError: # Python 3.10-
244 def _cbrt(x):
245 '''(INTERNAL) Compute the I{signed}, cube root M{x**(1/3)}.
246 '''
247 # <https://archive.lib.MSU.edu/crcmath/math/math/r/r021.htm>
248 # simpler and more accurate than Ken Turkowski's CubeRoot, see
249 # <https://People.FreeBSD.org/~lstewart/references/apple_tr_kt32_cuberoot.pdf>
250 return _copysign(pow(fabs(x), _1_3rd), x) # to avoid complex
253def cbrt(x):
254 '''Compute the cube root M{x**(1/3)}, preserving C{type(B{x})}.
256 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
258 @return: Cubic root (C{float} or L{Fsum}).
260 @see: Functions L{cbrt2} and L{sqrt3}.
261 '''
262 if _isFsum_2Tuple(x):
263 r = abs(x).fpow(_1_3rd)
264 if x.signOf() < 0:
265 r = -r
266 else:
267 r = _cbrt(x)
268 return r # cbrt(-0.0) == -0.0
271def cbrt2(x): # PYCHOK attr
272 '''Compute the cube root I{squared} M{x**(2/3)}, preserving C{type(B{x})}.
274 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
276 @return: Cube root I{squared} (C{float} or L{Fsum}).
278 @see: Functions L{cbrt} and L{sqrt3}.
279 '''
280 return abs(x).fpow(_2_3rd) if _isFsum_2Tuple(x) else _cbrt(x**2)
283def euclid(x, y):
284 '''I{Appoximate} the norm M{sqrt(x**2 + y**2)} by M{max(abs(x),
285 abs(y)) + min(abs(x), abs(y)) * 0.4142...}.
287 @arg x: X component (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
288 @arg y: Y component (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
290 @return: Appoximate norm (C{float} or L{Fsum}).
292 @see: Function L{euclid_}.
293 '''
294 x, y = abs(x), abs(y) # NOT fabs!
295 if y > x:
296 x, y = y, x
297 return x + y * _0_4142 # * _0_5 before 20.10.02
300def euclid_(*xs):
301 '''I{Appoximate} the norm M{sqrt(sum(x**2 for x in xs))} by cascaded
302 L{euclid}.
304 @arg xs: X arguments (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}),
305 all positional.
307 @return: Appoximate norm (C{float} or L{Fsum}).
309 @see: Function L{euclid}.
310 '''
311 e = _0_0
312 for x in sorted(map(abs, xs)): # NOT fabs, reverse=True!
313 # e = euclid(x, e)
314 if e < x:
315 e, x = x, e
316 if x:
317 e += x * _0_4142
318 return e
321def facos1(x):
322 '''Fast approximation of L{pygeodesy.acos1}C{(B{x})}, scalar.
324 @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/
325 ShaderFastLibs/blob/master/ShaderFastMathLib.h>}.
326 '''
327 a = fabs(x)
328 if a < EPS0:
329 r = PI_2
330 elif a < EPS1:
331 r = _fast(-a, 1.5707288, 0.2121144, 0.0742610, 0.0187293)
332 r *= sqrt(_1_0 - a)
333 if x < 0:
334 r = PI - r
335 else:
336 r = PI if x < 0 else _0_0
337 return r
340def fasin1(x): # PYCHOK no cover
341 '''Fast approximation of L{pygeodesy.asin1}C{(B{x})}, scalar.
343 @see: L{facos1}.
344 '''
345 return PI_2 - facos1(x)
348def _fast(x, *cs):
349 '''(INTERNAL) Horner form for C{facos1} and C{fatan1}.
350 '''
351 h = 0
352 for c in reversed(cs):
353 h = _fma(x, h, c) if h else c
354 return h
357def fatan(x):
358 '''Fast approximation of C{atan(B{x})}, scalar.
359 '''
360 a = fabs(x)
361 if a < _1_0:
362 r = fatan1(a) if a else _0_0
363 elif a > _1_0:
364 r = PI_2 - fatan1(_1_0 / a) # == fatan2(a, _1_0)
365 else:
366 r = PI_4
367 if x < 0: # copysign0(r, x)
368 r = -r
369 return r
372def fatan1(x):
373 '''Fast approximation of C{atan(B{x})} for C{0 <= B{x} < 1}, I{unchecked}.
375 @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/ShaderFastLibs/
376 blob/master/ShaderFastMathLib.h>} and U{Efficient approximations
377 for the arctangent function<http://www-Labs.IRO.UMontreal.CA/
378 ~mignotte/IFT2425/Documents/EfficientApproximationArctgFunction.pdf>},
379 IEEE Signal Processing Magazine, 111, May 2006.
380 '''
381 # Eq (9): PI_4 * x - x * (abs(x) - 1) * (0.2447 + 0.0663 * abs(x)), for -1 < x < 1
382 # == PI_4 * x - (x**2 - x) * (0.2447 + 0.0663 * x), for 0 < x < 1
383 # == x * (1.0300981633974482 + x * (-0.1784 - x * 0.0663))
384 return _fast(x, _0_0, 1.0300981634, -0.1784, -0.0663)
387def fatan2(y, x):
388 '''Fast approximation of C{atan2(B{y}, B{x})}, scalar.
390 @see: U{fastApproximateAtan(x, y)<https://GitHub.com/CesiumGS/cesium/blob/
391 master/Source/Shaders/Builtin/Functions/fastApproximateAtan.glsl>}
392 and L{fatan1}.
393 '''
394 a, b = fabs(x), fabs(y)
395 if b > a:
396 r = (PI_2 - fatan1(a / b)) if a else PI_2
397 elif a > b:
398 r = fatan1(b / a) if b else _0_0
399 elif a: # a == b != 0
400 r = PI_4
401 else: # a == b == 0
402 return _0_0
403 if x < 0:
404 r = PI - r
405 if y < 0: # copysign0(r, y)
406 r = -r
407 return r
410def favg(a, b, f=_0_5, nonfinites=True):
411 '''Return the precise average of two values.
413 @arg a: One (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
414 @arg b: Other (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
415 @kwarg f: Optional fraction (C{float}).
416 @kwarg nonfinites: Optional setting, see function L{fma}.
418 @return: M{a + f * (b - a)} (C{float}).
419 '''
420 F = fma(f, (b - a), a, nonfinites=nonfinites)
421 return float(F)
424def fdot(xs, *ys, **start_f2product_nonfinites):
425 '''Return the precision dot product M{sum(xs[i] * ys[i] for i in range(len(xs)))}.
427 @arg xs: Iterable of values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
428 @arg ys: Other values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all
429 positional.
430 @kwarg start_f2product_nonfinites: Optional bias C{B{start}=0} (C{scalar}, an
431 L{Fsum} or L{Fsum2Tuple}) and settings C{B{f2product}=None} (C{bool})
432 and C{B{nonfinites=True}} (C{bool}), see class L{Fsum<Fsum.__init__>}.
434 @return: Dot product (C{float}).
436 @raise LenError: Unequal C{len(B{xs})} and C{len(B{ys})}.
438 @see: Class L{Fdot}, U{Algorithm 5.10 B{DotK}
439 <https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>} and function
440 C{math.sumprod} in Python 3.12 and later.
441 '''
442 D = Fdot(xs, *ys, **_xkwds(start_f2product_nonfinites, nonfinites=True))
443 return float(D)
446def fdot_(*xys, **start_f2product_nonfinites):
447 '''Return the (precision) dot product M{sum(xys[i] * xys[i+1] for i in range(0, len(xys), B{2}))}.
449 @arg xys: Pairwise values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all positional.
451 @see: Function L{fdot} for further details.
453 @return: Dot product (C{float}).
454 '''
455 return fdot(xys[0::2], *xys[1::2], **start_f2product_nonfinites)
458def fdot3(xs, ys, zs, **start_f2product_nonfinites):
459 '''Return the (precision) dot product M{start + sum(xs[i] * ys[i] * zs[i] for i in range(len(xs)))}.
461 @arg xs: Iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
462 @arg ys: Iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
463 @arg zs: Iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
465 @see: Function L{fdot} for further details.
467 @return: Dot product (C{float}).
469 @raise LenError: Unequal C{len(B{xs})}, C{len(B{ys})} and/or C{len(B{zs})}.
470 '''
471 n = len(xs)
472 if not n == len(ys) == len(zs):
473 raise LenError(fdot3, xs=n, ys=len(ys), zs=len(zs))
475 D = Fdot((), **_xkwds(start_f2product_nonfinites, nonfinites=True))
476 kwds = dict(f2product=D.f2product(), nonfinites=D.nonfinites())
477 _f = Fsum(**kwds)
478 D = D._facc(_f(x).f2mul_(y, z, **kwds) for x, y, z in zip(xs, ys, zs))
479 return float(D)
482def fhorner(x, *cs, **incx):
483 '''Horner form evaluation of polynomial M{sum(cs[i] * x**i for i=0..n)} as
484 in- or decreasing exponent M{sum(... i=n..0)}, where C{n = len(cs) - 1}.
486 @return: Horner sum (C{float}).
488 @see: Class L{Fhorner<Fhorner.__init__>} for further details.
489 '''
490 H = Fhorner(x, *cs, **incx)
491 return float(H)
494def fidw(xs, ds, beta=2):
495 '''Interpolate using U{Inverse Distance Weighting
496 <https://WikiPedia.org/wiki/Inverse_distance_weighting>} (IDW).
498 @arg xs: Known values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
499 @arg ds: Non-negative distances (each C{scalar}, an L{Fsum} or
500 L{Fsum2Tuple}).
501 @kwarg beta: Inverse distance power (C{int}, 0, 1, 2, or 3).
503 @return: Interpolated value C{x} (C{float}).
505 @raise LenError: Unequal or zero C{len(B{ds})} and C{len(B{xs})}.
507 @raise TypeError: An invalid B{C{ds}} or B{C{xs}}.
509 @raise ValueError: Invalid B{C{beta}}, negative B{C{ds}} or
510 weighted B{C{ds}} below L{EPS}.
512 @note: Using C{B{beta}=0} returns the mean of B{C{xs}}.
513 '''
514 n, xs = len2(xs)
515 if n > 1:
516 b = -Int_(beta=beta, low=0, high=3)
517 if b < 0:
518 try: # weighted
519 _d, W, X = (Fsum() for _ in range(3))
520 for i, d in enumerate(_xiterable(ds)):
521 x = xs[i]
522 D = _d(d)
523 if D < EPS0:
524 if D < 0:
525 raise ValueError(_negative_)
526 x = float(x)
527 i = n
528 break
529 if D.fpow(b):
530 W += D
531 X += D.fmul(x)
532 else:
533 x = X.fover(W, raiser=False)
534 i += 1 # len(xs) >= len(ds)
535 except IndexError:
536 i += 1 # len(xs) < i < len(ds)
537 except Exception as X:
538 _I = Fmt.INDEX
539 raise _xError(X, _I(xs=i), x,
540 _I(ds=i), d)
541 else: # b == 0
542 x = fsum(xs) / n # fmean(xs)
543 i = n
544 elif n:
545 x = float(xs[0])
546 i = n
547 else:
548 x = _0_0
549 i, _ = len2(ds)
550 if i != n:
551 raise LenError(fidw, xs=n, ds=i)
552 return x
555try:
556 from math import fma as _fma
557except ImportError: # PYCHOK DSPACE!
559 def _fma(x, y, z): # no need for accuracy
560 return x * y + z
563def fma(x, y, z, **nonfinites): # **raiser
564 '''Fused-multiply-add, using C{math.fma(x, y, z)} in Python 3.13+
565 or an equivalent implementation.
567 @arg x: Multiplicand (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
568 @arg y: Multiplier (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
569 @arg z: Addend (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
570 @kwarg nonfinites: Use C{B{nonfinites}=True} or C{=False},
571 to override default L{nonfiniterrors}
572 (C{bool}), see method L{Fsum.fma}.
574 @return: C{(x * y) + z} (C{float} or L{Fsum}).
575 '''
576 F, raiser = _Fm2(x, **nonfinites)
577 return F.fma(y, z, **raiser).as_iscalar
580def _Fm2(x, nonfinites=None, **raiser):
581 '''(INTERNAL) Handle C{fma} and C{f2mul} DEPRECATED C{raiser=False}.
582 '''
583 return Fsum(x, nonfinites=nonfinites), raiser
586def fmean(xs):
587 '''Compute the accurate mean M{sum(xs) / len(xs)}.
589 @arg xs: Values (each C{scalar}, or L{Fsum} or L{Fsum2Tuple}).
591 @return: Mean value (C{float}).
593 @raise LenError: No B{C{xs}} values.
595 @raise OverflowError: Partial C{2sum} overflow.
596 '''
597 n, xs = len2(xs)
598 if n < 1:
599 raise LenError(fmean, xs=xs)
600 M = Fsum(*xs, nonfinites=True)
601 return M.fover(n) if n > 1 else float(M)
604def fmean_(*xs, **nonfinites):
605 '''Compute the accurate mean M{sum(xs) / len(xs)}.
607 @see: Function L{fmean} for further details.
608 '''
609 return fmean(xs, **nonfinites)
612def f2mul_(x, *ys, **nonfinites): # **raiser
613 '''Cascaded, accurate multiplication C{B{x} * B{y} * B{y} ...} for all B{C{ys}}.
615 @arg x: Multiplicand (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
616 @arg ys: Multipliers (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all
617 positional.
618 @kwarg nonfinites: Use C{B{nonfinites}=True} or C{=False}, to override default
619 L{nonfiniterrors} (C{bool}), see method L{Fsum.f2mul_}.
621 @return: The cascaded I{TwoProduct} (C{float}, C{int} or L{Fsum}).
623 @see: U{Equations 2.3<https://www.TUHH.De/ti3/paper/rump/OzOgRuOi06.pdf>}
624 '''
625 F, raiser = _Fm2(x, **nonfinites)
626 return F.f2mul_(*ys, **raiser).as_iscalar
629def fpolynomial(x, *cs, **over_f2product_nonfinites):
630 '''Evaluate the polynomial M{sum(cs[i] * x**i for i=0..len(cs)) [/ over]}.
632 @kwarg over_f2product_nonfinites: Optional final divisor C{B{over}=None}
633 (I{non-zero} C{scalar}) and other settings, see class
634 L{Fpolynomial<Fpolynomial.__init__>}.
636 @return: Polynomial value (C{float} or L{Fpolynomial}).
637 '''
638 d, kwds = _xkwds_pop2(over_f2product_nonfinites, over=0)
639 P = Fpolynomial(x, *cs, **kwds)
640 return P.fover(d) if d else float(P)
643def fpowers(x, n, alts=0):
644 '''Return a series of powers M{[x**i for i=1..n]}, note I{1..!}
646 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
647 @arg n: Highest exponent (C{int}).
648 @kwarg alts: Only alternating powers, starting with this
649 exponent (C{int}).
651 @return: Tuple of powers of B{C{x}} (each C{type(B{x})}).
653 @raise TypeError: Invalid B{C{x}} or B{C{n}} not C{int}.
655 @raise ValueError: Non-finite B{C{x}} or invalid B{C{n}}.
656 '''
657 if not isint(n):
658 raise _IsnotError(int.__name__, n=n)
659 elif n < 1:
660 raise _ValueError(n=n)
662 p = x if isscalar(x) or _isFsum_2Tuple(x) else _2float(x=x)
663 ps = tuple(_powers(p, n))
665 if alts > 0: # x**2, x**4, ...
666 # ps[alts-1::2] chokes PyChecker
667 ps = ps[slice(alts-1, None, 2)]
669 return ps
672try:
673 from math import prod as fprod # Python 3.8
674except ImportError:
676 def fprod(xs, start=1):
677 '''Iterable product, like C{math.prod} or C{numpy.prod}.
679 @arg xs: Iterable of values to be multiplied (each
680 C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
681 @kwarg start: Initial value, also the value returned
682 for an empty B{C{xs}} (C{scalar}).
684 @return: The product (C{float} or L{Fsum}).
686 @see: U{NumPy.prod<https://docs.SciPy.org/doc/
687 numpy/reference/generated/numpy.prod.html>}.
688 '''
689 return freduce(_operator.mul, xs, start)
692def frandoms(n, seeded=None):
693 '''Generate C{n} (long) lists of random C{floats}.
695 @arg n: Number of lists to generate (C{int}, non-negative).
696 @kwarg seeded: If C{scalar}, use C{random.seed(B{seeded})} or
697 if C{True}, seed using today's C{year-day}.
699 @see: U{Hettinger<https://GitHub.com/ActiveState/code/tree/master/recipes/
700 Python/393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}.
701 '''
702 from random import gauss, random, seed, shuffle
704 if seeded is None:
705 pass
706 elif seeded and isbool(seeded):
707 from time import localtime
708 seed(localtime().tm_yday)
709 elif isscalar(seeded):
710 seed(seeded)
712 c = (7, 1e100, -7, -1e100, -9e-20, 8e-20) * 7
713 for _ in range(n):
714 s = 0
715 t = list(c)
716 _a = t.append
717 for _ in range(n * 8):
718 v = gauss(0, random())**7 - s
719 _a(v)
720 s += v
721 shuffle(t)
722 yield t
725def frange(start, number, step=1):
726 '''Generate a range of C{float}s.
728 @arg start: First value (C{float}).
729 @arg number: The number of C{float}s to generate (C{int}).
730 @kwarg step: Increment value (C{float}).
732 @return: A generator (C{float}s).
734 @see: U{NumPy.prod<https://docs.SciPy.org/doc/
735 numpy/reference/generated/numpy.arange.html>}.
736 '''
737 if not isint(number):
738 raise _IsnotError(int.__name__, number=number)
739 for i in range(number):
740 yield start + (step * i)
743try:
744 from functools import reduce as freduce
745except ImportError:
746 try:
747 freduce = reduce # PYCHOK expected
748 except NameError: # Python 3+
750 def freduce(f, xs, *start):
751 '''For missing C{functools.reduce}.
752 '''
753 if start:
754 r = v = start[0]
755 else:
756 r, v = 0, MISSING
757 for v in xs:
758 r = f(r, v)
759 if v is MISSING:
760 raise _TypeError(xs=(), start=MISSING)
761 return r
764def fremainder(x, y):
765 '''Remainder in range C{[-B{y / 2}, B{y / 2}]}.
767 @arg x: Numerator (C{scalar}).
768 @arg y: Modulus, denominator (C{scalar}).
770 @return: Remainder (C{scalar}, preserving signed
771 0.0) or C{NAN} for any non-finite B{C{x}}.
773 @raise ValueError: Infinite or near-zero B{C{y}}.
775 @see: I{Karney}'s U{Math.remainder<https://PyPI.org/
776 project/geographiclib/>} and Python 3.7+
777 U{math.remainder<https://docs.Python.org/3/
778 library/math.html#math.remainder>}.
779 '''
780 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and
781 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native
782 # fmod( 0, 360) == 0.0
783 # fmod( 360, 360) == 0.0
784 # fmod(-0, 360) == 0.0
785 # fmod(-0.0, 360) == -0.0
786 # fmod(-360, 360) == -0.0
787 # however, using the % operator ...
788 # 0 % 360 == 0
789 # 360 % 360 == 0
790 # 360.0 % 360 == 0.0
791 # -0 % 360 == 0
792 # -360 % 360 == 0 == (-360) % 360
793 # -0.0 % 360 == 0.0 == (-0.0) % 360
794 # -360.0 % 360 == 0.0 == (-360.0) % 360
796 # On Windows 32-bit with python 2.7, math.fmod(-0.0, 360)
797 # == +0.0. This fixes this bug. See also Math::AngNormalize
798 # in the C++ library, Math.sincosd has a similar fix.
799 if isfinite(x):
800 try:
801 r = remainder(x, y) if x else x
802 except Exception as e:
803 raise _xError(e, unstr(fremainder, x, y))
804 else: # handle x INF and NINF as NAN
805 r = NAN
806 return r
809if _MODS.sys_version_info2 < (3, 8): # PYCHOK no cover
810 from math import hypot # OK in Python 3.7-
812 def hypot_(*xs):
813 '''Compute the norm M{sqrt(sum(x**2 for x in xs))}.
815 Similar to Python 3.8+ n-dimension U{math.hypot
816 <https://docs.Python.org/3.8/library/math.html#math.hypot>},
817 but exceptions, C{nan} and C{infinite} values are
818 handled differently.
820 @arg xs: X arguments (C{scalar}s), all positional.
822 @return: Norm (C{float}).
824 @raise OverflowError: Partial C{2sum} overflow.
826 @raise ValueError: Invalid or no B{C{xs}} values.
828 @note: The Python 3.8+ Euclidian distance U{math.dist
829 <https://docs.Python.org/3.8/library/math.html#math.dist>}
830 between 2 I{n}-dimensional points I{p1} and I{p2} can be
831 computed as M{hypot_(*((c1 - c2) for c1, c2 in zip(p1, p2)))},
832 provided I{p1} and I{p2} have the same, non-zero length I{n}.
833 '''
834 return float(_Hypot(*xs))
836elif _MODS.sys_version_info2 < (3, 10):
837 # In Python 3.8 and 3.9 C{math.hypot} is inaccurate, see
838 # U{agdhruv<https://GitHub.com/geopy/geopy/issues/466>},
839 # U{cffk<https://Bugs.Python.org/issue43088>} and module
840 # U{geomath.py<https://PyPI.org/project/geographiclib/1.52>}
842 def hypot(x, y):
843 '''Compute the norm M{sqrt(x**2 + y**2)}.
845 @arg x: X argument (C{scalar}).
846 @arg y: Y argument (C{scalar}).
848 @return: C{sqrt(B{x}**2 + B{y}**2)} (C{float}).
849 '''
850 return float(_Hypot(x, y))
852 from math import hypot as hypot_ # PYCHOK in Python 3.8 and 3.9
853else:
854 from math import hypot # PYCHOK in Python 3.10+
855 hypot_ = hypot
858def _Hypot(*xs):
859 '''(INTERNAL) Substitute for inaccurate C{math.hypot}.
860 '''
861 return Fhypot(*xs, nonfinites=True, raiser=False) # f2product=True
864def hypot1(x):
865 '''Compute the norm M{sqrt(1 + x**2)}.
867 @arg x: Argument (C{scalar} or L{Fsum} or L{Fsum2Tuple}).
869 @return: Norm (C{float} or L{Fhypot}).
870 '''
871 h = _1_0
872 if x:
873 if _isFsum_2Tuple(x):
874 h = _Hypot(h, x)
875 h = float(h)
876 else:
877 h = hypot(h, x)
878 return h
881def hypot2(x, y):
882 '''Compute the I{squared} norm M{x**2 + y**2}.
884 @arg x: X (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
885 @arg y: Y (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
887 @return: C{B{x}**2 + B{y}**2} (C{float}).
888 '''
889 x, y = map1(abs, x, y) # NOT fabs!
890 if y > x:
891 x, y = y, x
892 h2 = x**2
893 if h2 and y:
894 h2 *= (y / x)**2 + _1_0
895 return float(h2)
898def hypot2_(*xs):
899 '''Compute the I{squared} norm C{fsum(x**2 for x in B{xs})}.
901 @arg xs: Components (each C{scalar}, an L{Fsum} or
902 L{Fsum2Tuple}), all positional.
904 @return: Squared norm (C{float}).
906 @see: Class L{Fpowers} for further details.
907 '''
908 h2 = float(max(map(abs, xs))) if xs else _0_0
909 if h2: # and isfinite(h2)
910 _h = _1_0 / h2
911 xs = ((x * _h) for x in xs)
912 H2 = Fpowers(2, *xs, nonfinites=True) # f2product=True
913 h2 = H2.fover(_h**2)
914 return h2
917def norm2(x, y):
918 '''Normalize a 2-dimensional vector.
920 @arg x: X component (C{scalar}).
921 @arg y: Y component (C{scalar}).
923 @return: 2-Tuple C{(x, y)}, normalized.
925 @raise ValueError: Invalid B{C{x}} or B{C{y}}
926 or zero norm.
927 '''
928 try:
929 h = None
930 h = hypot(x, y)
931 if h:
932 x, y = (x / h), (y / h)
933 else:
934 x = _copysign_0_0(x) # pass?
935 y = _copysign_0_0(y)
936 except Exception as e:
937 raise _xError(e, x=x, y=y, h=h)
938 return x, y
941def norm_(*xs):
942 '''Normalize the components of an n-dimensional vector.
944 @arg xs: Components (each C{scalar}, an L{Fsum} or
945 L{Fsum2Tuple}), all positional.
947 @return: Yield each component, normalized.
949 @raise ValueError: Invalid or insufficent B{C{xs}}
950 or zero norm.
951 '''
952 try:
953 i = h = None
954 x = xs
955 h = hypot_(*xs)
956 _h = (_1_0 / h) if h else _0_0
957 for i, x in enumerate(xs):
958 yield x * _h
959 except Exception as X:
960 raise _xsError(X, xs, i, x, h=h)
963def _powers(x, n):
964 '''(INTERNAL) Yield C{x**i for i=1..n}.
965 '''
966 p = 1 # type(p) == type(x)
967 for _ in range(n):
968 p *= x
969 yield p
972def _root(x, p, where):
973 '''(INTERNAL) Raise C{x} to power C{0 < p < 1}.
974 '''
975 try:
976 if x > 0:
977 r = Fsum(f2product=True, nonfinites=True)(x)
978 return r.fpow(p).as_iscalar
979 elif x < 0:
980 raise ValueError(_negative_)
981 except Exception as X:
982 raise _xError(X, unstr(where, x))
983 return _0_0
986def sqrt0(x, Error=None):
987 '''Return the square root C{sqrt(B{x})} iff C{B{x} > }L{EPS02},
988 preserving C{type(B{x})}.
990 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
991 @kwarg Error: Error to raise for negative B{C{x}}.
993 @return: Square root (C{float} or L{Fsum}) or C{0.0}.
995 @raise TypeeError: Invalid B{C{x}}.
997 @note: Any C{B{x} < }L{EPS02} I{including} C{B{x} < 0}
998 returns C{0.0}.
999 '''
1000 if Error and x < 0:
1001 raise Error(unstr(sqrt0, x))
1002 return _root(x, _0_5, sqrt0) if x > EPS02 else (
1003 _0_0 if x < EPS02 else EPS0)
1006def sqrt3(x):
1007 '''Return the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)},
1008 preserving C{type(B{x})}.
1010 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
1012 @return: Square root I{cubed} (C{float} or L{Fsum}).
1014 @raise TypeeError: Invalid B{C{x}}.
1016 @raise ValueError: Negative B{C{x}}.
1018 @see: Functions L{cbrt} and L{cbrt2}.
1019 '''
1020 return _root(x, _1_5, sqrt3)
1023def sqrt_a(h, b):
1024 '''Compute C{I{a}} side of a right-angled triangle from
1025 C{sqrt(B{h}**2 - B{b}**2)}.
1027 @arg h: Hypotenuse or outer annulus radius (C{scalar}).
1028 @arg b: Triangle side or inner annulus radius (C{scalar}).
1030 @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}).
1032 @raise TypeError: Non-scalar B{C{h}} or B{C{b}}.
1034 @raise ValueError: If C{abs(B{h}) < abs(B{b})}.
1036 @see: Inner tangent chord B{I{d}} of an U{annulus
1037 <https://WikiPedia.org/wiki/Annulus_(mathematics)>}
1038 and function U{annulus_area<https://People.SC.FSU.edu/
1039 ~jburkardt/py_src/geometry/geometry.py>}.
1040 '''
1041 try:
1042 if not (_isHeight(h) and _isRadius(b)):
1043 raise TypeError(_not_scalar_)
1044 c = fabs(h)
1045 if c > EPS0:
1046 s = _1_0 - (b / c)**2
1047 if s < 0:
1048 raise ValueError(_h_lt_b_)
1049 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0)
1050 else: # PYCHOK no cover
1051 b = fabs(b)
1052 d = c - b
1053 if d < 0:
1054 raise ValueError(_h_lt_b_)
1055 d *= c + b
1056 a = sqrt(d) if d else _0_0
1057 except Exception as x:
1058 raise _xError(x, h=h, b=b)
1059 return copysign0(a, h)
1062def zcrt(x):
1063 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)},
1064 preserving C{type(B{x})}.
1066 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
1068 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}).
1070 @see: Functions L{bqrt} and L{zqrt}.
1072 @raise TypeeError: Invalid B{C{x}}.
1074 @raise ValueError: Negative B{C{x}}.
1075 '''
1076 return _root(x, _1_6th, zcrt)
1079def zqrt(x):
1080 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root,
1081 M{x**(1 / 8)}, preserving C{type(B{x})}.
1083 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
1085 @return: I{Zenzi-quartic} root (C{float} or L{Fsum}).
1087 @see: Functions L{bqrt} and L{zcrt}.
1089 @raise TypeeError: Invalid B{C{x}}.
1091 @raise ValueError: Negative B{C{x}}.
1092 '''
1093 return _root(x, _0_125, zqrt)
1095# **) MIT License
1096#
1097# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
1098#
1099# Permission is hereby granted, free of charge, to any person obtaining a
1100# copy of this software and associated documentation files (the "Software"),
1101# to deal in the Software without restriction, including without limitation
1102# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1103# and/or sell copies of the Software, and to permit persons to whom the
1104# Software is furnished to do so, subject to the following conditions:
1105#
1106# The above copyright notice and this permission notice shall be included
1107# in all copies or substantial portions of the Software.
1108#
1109# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1110# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1111# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1112# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1113# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1114# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1115# OTHER DEALINGS IN THE SOFTWARE.