Coverage for pygeodesy/fmath.py: 91%
325 statements
« prev ^ index » next coverage.py v7.6.1, created at 2025-01-10 16:55 -0500
« prev ^ index » next coverage.py v7.6.1, created at 2025-01-10 16:55 -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__ = '25.01.09'
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 def _r_X_kwds(power=None, raiser=True, root=2, **kwds):
111 # DEPRECATED keyword argument C{power=2}, use C{root=2}
112 return (root if power is None else power), raiser, kwds
114 r = None # _xkwds_pop2 error
115 try:
116 r, X, kwds = _r_X_kwds(**root_name_f2product_nonfinites_RESIDUAL_raiser)
117 Fsum.__init__(self, **kwds)
118 self(_0_0)
119 if xs:
120 self._facc_power(r, xs, Fhypot, raiser=X)
121 self._fset(self.root(r, raiser=X))
122 except Exception as X:
123 raise self._ErrorXs(X, xs, root=r)
126class Fpolynomial(Fsum):
127 '''Precision polynomial evaluation.
128 '''
129 def __init__(self, x, *cs, **name_f2product_nonfinites_RESIDUAL):
130 '''New L{Fpolynomial} evaluation of the polynomial M{sum(cs[i] * x**i for
131 i=0..len(cs)-1)}.
133 @arg x: Polynomial argument (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
134 @arg cs: Polynomial coeffients (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}),
135 all positional.
136 @kwarg name_f2product_nonfinites_RESIDUAL: Optional C{B{name}=NN} (C{str})
137 and other settings, see class L{Fsum<Fsum.__init__>}.
139 @raise OverflowError: Partial C{2sum} overflow.
141 @raise TypeError: Invalid B{C{x}}.
143 @raise ValueError: Non-finite B{C{x}}.
145 @see: Class L{Fhorner}, function L{fpolynomial} and method L{Fsum.fadd}.
146 '''
147 Fsum.__init__(self, **name_f2product_nonfinites_RESIDUAL)
148 n = len(cs) - 1
149 self(_0_0 if n < 0 else cs[0])
150 self._facc_dot(n, cs[1:], _powers(x, n), **name_f2product_nonfinites_RESIDUAL)
153class Fpowers(Fsum):
154 '''Precision summation of powers, optimized for C{power=2, 3 and 4}.
155 '''
156 def __init__(self, power, *xs, **name_f2product_nonfinites_RESIDUAL_raiser):
157 '''New L{Fpowers} sum of (the I{power} of) several bases.
159 @arg power: The exponent (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
160 @arg xs: One or more bases (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all
161 positional.
162 @kwarg name_f2product_nonfinites_RESIDUAL_raiser: Optional C{B{name}=NN}
163 (C{str}), C{B{raiser}=True} (C{bool}) for raising L{ResidualError}s
164 and other settings, see class L{Fsum<Fsum.__init__>} and method
165 L{fpow<Fsum.fpow>}.
166 '''
167 try:
168 X, kwds = _xkwds_pop2(name_f2product_nonfinites_RESIDUAL_raiser, raiser=True)
169 Fsum.__init__(self, **kwds)
170 self(_0_0)
171 if xs:
172 self._facc_power(power, xs, Fpowers, raiser=X) # x**0 == 1
173 except Exception as X:
174 raise self._ErrorXs(X, xs, power=power)
177class Froot(Fsum):
178 '''The root of a precision summation.
179 '''
180 def __init__(self, root, *xs, **name_f2product_nonfinites_RESIDUAL_raiser):
181 '''New L{Froot} root of a precision sum.
183 @arg root: The order (C{scalar}, an L{Fsum} or L{Fsum2Tuple}), non-zero.
184 @arg xs: Items to summate (each a C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all
185 positional.
186 @kwarg name_f2product_nonfinites_RESIDUAL_raiser: Optional C{B{name}=NN}
187 (C{str}), C{B{raiser}=True} (C{bool}) for raising L{ResidualError}s
188 and other settings, see class L{Fsum<Fsum.__init__>} and method
189 L{fpow<Fsum.fpow>}.
190 '''
191 try:
192 X, kwds = _xkwds_pop2(name_f2product_nonfinites_RESIDUAL_raiser, raiser=True)
193 Fsum.__init__(self, **kwds)
194 self(_0_0)
195 if xs:
196 self.fadd(xs)
197 self(self.root(root, raiser=X))
198 except Exception as X:
199 raise self._ErrorXs(X, xs, root=root)
202class Fcbrt(Froot):
203 '''Cubic root of a precision summation.
204 '''
205 def __init__(self, *xs, **name_f2product_nonfinites_RESIDUAL_raiser):
206 '''New L{Fcbrt} cubic root of a precision sum.
208 @see: Class L{Froot<Froot.__init__>} for further details.
209 '''
210 Froot.__init__(self, 3, *xs, **name_f2product_nonfinites_RESIDUAL_raiser)
213class Fsqrt(Froot):
214 '''Square root of a precision summation.
215 '''
216 def __init__(self, *xs, **name_f2product_nonfinites_RESIDUAL_raiser):
217 '''New L{Fsqrt} square root of a precision sum.
219 @see: Class L{Froot<Froot.__init__>} for further details.
220 '''
221 Froot.__init__(self, 2, *xs, **name_f2product_nonfinites_RESIDUAL_raiser)
224def bqrt(x):
225 '''Return the 4-th, I{bi-quadratic} or I{quartic} root, M{x**(1 / 4)},
226 preserving C{type(B{x})}.
228 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
230 @return: I{Quartic} root (C{float} or an L{Fsum}).
232 @raise TypeeError: Invalid B{C{x}}.
234 @raise ValueError: Negative B{C{x}}.
236 @see: Functions L{zcrt} and L{zqrt}.
237 '''
238 return _root(x, _0_25, bqrt)
241try:
242 from math import cbrt as _cbrt # Python 3.11+
244except ImportError: # Python 3.10-
246 def _cbrt(x):
247 '''(INTERNAL) Compute the I{signed}, cube root M{x**(1/3)}.
248 '''
249 # <https://archive.lib.MSU.edu/crcmath/math/math/r/r021.htm>
250 # simpler and more accurate than Ken Turkowski's CubeRoot, see
251 # <https://People.FreeBSD.org/~lstewart/references/apple_tr_kt32_cuberoot.pdf>
252 return _copysign(pow(fabs(x), _1_3rd), x) # to avoid complex
255def cbrt(x):
256 '''Compute the cube root M{x**(1/3)}, preserving C{type(B{x})}.
258 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
260 @return: Cubic root (C{float} or L{Fsum}).
262 @see: Functions L{cbrt2} and L{sqrt3}.
263 '''
264 if _isFsum_2Tuple(x):
265 r = abs(x).fpow(_1_3rd)
266 if x.signOf() < 0:
267 r = -r
268 else:
269 r = _cbrt(x)
270 return r # cbrt(-0.0) == -0.0
273def cbrt2(x): # PYCHOK attr
274 '''Compute the cube root I{squared} M{x**(2/3)}, preserving C{type(B{x})}.
276 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
278 @return: Cube root I{squared} (C{float} or L{Fsum}).
280 @see: Functions L{cbrt} and L{sqrt3}.
281 '''
282 return abs(x).fpow(_2_3rd) if _isFsum_2Tuple(x) else _cbrt(x**2)
285def euclid(x, y):
286 '''I{Appoximate} the norm M{sqrt(x**2 + y**2)} by M{max(abs(x),
287 abs(y)) + min(abs(x), abs(y)) * 0.4142...}.
289 @arg x: X component (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
290 @arg y: Y component (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
292 @return: Appoximate norm (C{float} or L{Fsum}).
294 @see: Function L{euclid_}.
295 '''
296 x, y = abs(x), abs(y) # NOT fabs!
297 if y > x:
298 x, y = y, x
299 return x + y * _0_4142 # * _0_5 before 20.10.02
302def euclid_(*xs):
303 '''I{Appoximate} the norm M{sqrt(sum(x**2 for x in xs))} by cascaded
304 L{euclid}.
306 @arg xs: X arguments (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}),
307 all positional.
309 @return: Appoximate norm (C{float} or L{Fsum}).
311 @see: Function L{euclid}.
312 '''
313 e = _0_0
314 for x in sorted(map(abs, xs)): # NOT fabs, reverse=True!
315 # e = euclid(x, e)
316 if e < x:
317 e, x = x, e
318 if x:
319 e += x * _0_4142
320 return e
323def facos1(x):
324 '''Fast approximation of L{pygeodesy.acos1}C{(B{x})}, scalar.
326 @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/
327 ShaderFastLibs/blob/master/ShaderFastMathLib.h>}.
328 '''
329 a = fabs(x)
330 if a < EPS0:
331 r = PI_2
332 elif a < EPS1:
333 r = _fast(-a, 1.5707288, 0.2121144, 0.0742610, 0.0187293)
334 r *= sqrt(_1_0 - a)
335 if x < 0:
336 r = PI - r
337 else:
338 r = PI if x < 0 else _0_0
339 return r
342def fasin1(x): # PYCHOK no cover
343 '''Fast approximation of L{pygeodesy.asin1}C{(B{x})}, scalar.
345 @see: L{facos1}.
346 '''
347 return PI_2 - facos1(x)
350def _fast(x, *cs):
351 '''(INTERNAL) Horner form for C{facos1} and C{fatan1}.
352 '''
353 h = 0
354 for c in reversed(cs):
355 h = _fma(x, h, c) if h else c
356 return h
359def fatan(x):
360 '''Fast approximation of C{atan(B{x})}, scalar.
361 '''
362 a = fabs(x)
363 if a < _1_0:
364 r = fatan1(a) if a else _0_0
365 elif a > _1_0:
366 r = PI_2 - fatan1(_1_0 / a) # == fatan2(a, _1_0)
367 else:
368 r = PI_4
369 if x < 0: # copysign0(r, x)
370 r = -r
371 return r
374def fatan1(x):
375 '''Fast approximation of C{atan(B{x})} for C{0 <= B{x} < 1}, I{unchecked}.
377 @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/ShaderFastLibs/
378 blob/master/ShaderFastMathLib.h>} and U{Efficient approximations
379 for the arctangent function<http://www-Labs.IRO.UMontreal.CA/
380 ~mignotte/IFT2425/Documents/EfficientApproximationArctgFunction.pdf>},
381 IEEE Signal Processing Magazine, 111, May 2006.
382 '''
383 # Eq (9): PI_4 * x - x * (abs(x) - 1) * (0.2447 + 0.0663 * abs(x)), for -1 < x < 1
384 # == PI_4 * x - (x**2 - x) * (0.2447 + 0.0663 * x), for 0 < x < 1
385 # == x * (1.0300981633974482 + x * (-0.1784 - x * 0.0663))
386 return _fast(x, _0_0, 1.0300981634, -0.1784, -0.0663)
389def fatan2(y, x):
390 '''Fast approximation of C{atan2(B{y}, B{x})}, scalar.
392 @see: U{fastApproximateAtan(x, y)<https://GitHub.com/CesiumGS/cesium/blob/
393 master/Source/Shaders/Builtin/Functions/fastApproximateAtan.glsl>}
394 and L{fatan1}.
395 '''
396 a, b = fabs(x), fabs(y)
397 if b > a:
398 r = (PI_2 - fatan1(a / b)) if a else PI_2
399 elif a > b:
400 r = fatan1(b / a) if b else _0_0
401 elif a: # a == b != 0
402 r = PI_4
403 else: # a == b == 0
404 return _0_0
405 if x < 0:
406 r = PI - r
407 if y < 0: # copysign0(r, y)
408 r = -r
409 return r
412def favg(a, b, f=_0_5, nonfinites=True):
413 '''Return the precise average of two values.
415 @arg a: One (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
416 @arg b: Other (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
417 @kwarg f: Optional fraction (C{float}).
418 @kwarg nonfinites: Optional setting, see function L{fma}.
420 @return: M{a + f * (b - a)} (C{float}).
421 '''
422 F = fma(f, (b - a), a, nonfinites=nonfinites)
423 return float(F)
426def fdot(xs, *ys, **start_f2product_nonfinites):
427 '''Return the precision dot product M{sum(xs[i] * ys[i] for i in range(len(xs)))}.
429 @arg xs: Iterable of values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
430 @arg ys: Other values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all positional.
431 @kwarg start_f2product_nonfinites: Optional bias C{B{start}=0} (C{scalar}, an
432 L{Fsum} or L{Fsum2Tuple}) and settings C{B{f2product}=None} (C{bool})
433 and C{B{nonfinites=True}} (C{bool}), see class L{Fsum<Fsum.__init__>}.
435 @return: Dot product (C{float}).
437 @raise LenError: Unequal C{len(B{xs})} and C{len(B{ys})}.
439 @see: Class L{Fdot}, U{Algorithm 5.10 B{DotK}
440 <https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>} and function
441 C{math.sumprod} in Python 3.12 and later.
442 '''
443 D = Fdot(xs, *ys, **_xkwds(start_f2product_nonfinites, nonfinites=True))
444 return float(D)
447def fdot_(*xys, **start_f2product_nonfinites):
448 '''Return the (precision) dot product M{sum(xys[i] * xys[i+1] for i in range(0, len(xys), B{2}))}.
450 @arg xys: Pairwise values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all positional.
452 @see: Function L{fdot} for further details.
454 @return: Dot product (C{float}).
455 '''
456 return fdot(xys[0::2], *xys[1::2], **start_f2product_nonfinites)
459def fdot3(xs, ys, zs, **start_f2product_nonfinites):
460 '''Return the (precision) dot product M{start + sum(xs[i] * ys[i] * zs[i] for i in range(len(xs)))}.
462 @arg xs: X values iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
463 @arg ys: Y values iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
464 @arg zs: Z values iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
466 @see: Function L{fdot} for further details.
468 @return: Dot product (C{float}).
470 @raise LenError: Unequal C{len(B{xs})}, C{len(B{ys})} and/or C{len(B{zs})}.
471 '''
472 n = len(xs)
473 if not n == len(ys) == len(zs):
474 raise LenError(fdot3, xs=n, ys=len(ys), zs=len(zs))
476 D = Fdot((), **_xkwds(start_f2product_nonfinites, nonfinites=True))
477 kwds = dict(f2product=D.f2product(), nonfinites=D.nonfinites())
478 _f = Fsum(**kwds)
479 D = D._facc(_f(x).f2mul_(y, z, **kwds) for x, y, z in zip(xs, ys, zs))
480 return float(D)
483def fhorner(x, *cs, **incx):
484 '''Horner form evaluation of polynomial M{sum(cs[i] * x**i for i=0..n)} as
485 in- or decreasing exponent M{sum(... i=n..0)}, where C{n = len(cs) - 1}.
487 @return: Horner sum (C{float}).
489 @see: Class L{Fhorner<Fhorner.__init__>} for further details.
490 '''
491 H = Fhorner(x, *cs, **incx)
492 return float(H)
495def fidw(xs, ds, beta=2):
496 '''Interpolate using U{Inverse Distance Weighting
497 <https://WikiPedia.org/wiki/Inverse_distance_weighting>} (IDW).
499 @arg xs: Known values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
500 @arg ds: Non-negative distances (each C{scalar}, an L{Fsum} or
501 L{Fsum2Tuple}).
502 @kwarg beta: Inverse distance power (C{int}, 0, 1, 2, or 3).
504 @return: Interpolated value C{x} (C{float}).
506 @raise LenError: Unequal or zero C{len(B{ds})} and C{len(B{xs})}.
508 @raise TypeError: An invalid B{C{ds}} or B{C{xs}}.
510 @raise ValueError: Invalid B{C{beta}}, negative B{C{ds}} or
511 weighted B{C{ds}} below L{EPS}.
513 @note: Using C{B{beta}=0} returns the mean of B{C{xs}}.
514 '''
515 n, xs = len2(xs)
516 if n > 1:
517 b = -Int_(beta=beta, low=0, high=3)
518 if b < 0:
519 try: # weighted
520 _d, W, X = (Fsum() for _ in range(3))
521 for i, d in enumerate(_xiterable(ds)):
522 x = xs[i]
523 D = _d(d)
524 if D < EPS0:
525 if D < 0:
526 raise ValueError(_negative_)
527 x = float(x)
528 i = n
529 break
530 if D.fpow(b):
531 W += D
532 X += D.fmul(x)
533 else:
534 x = X.fover(W, raiser=False)
535 i += 1 # len(xs) >= len(ds)
536 except IndexError:
537 i += 1 # len(xs) < i < len(ds)
538 except Exception as X:
539 _I = Fmt.INDEX
540 raise _xError(X, _I(xs=i), x,
541 _I(ds=i), d)
542 else: # b == 0
543 x = fsum(xs) / n # fmean(xs)
544 i = n
545 elif n:
546 x = float(xs[0])
547 i = n
548 else:
549 x = _0_0
550 i, _ = len2(ds)
551 if i != n:
552 raise LenError(fidw, xs=n, ds=i)
553 return x
556try:
557 from math import fma as _fma
558except ImportError: # PYCHOK DSPACE!
560 def _fma(x, y, z): # no need for accuracy
561 return x * y + z
564def fma(x, y, z, **nonfinites): # **raiser
565 '''Fused-multiply-add, using C{math.fma(x, y, z)} in Python 3.13+
566 or an equivalent implementation.
568 @arg x: Multiplicand (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
569 @arg y: Multiplier (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
570 @arg z: Addend (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
571 @kwarg nonfinites: Use C{B{nonfinites}=True} or C{=False},
572 to override default L{nonfiniterrors}
573 (C{bool}), see method L{Fsum.fma}.
575 @return: C{(x * y) + z} (C{float} or L{Fsum}).
576 '''
577 F, raiser = _Fm2(x, **nonfinites)
578 return F.fma(y, z, **raiser).as_iscalar
581def _Fm2(x, nonfinites=None, **raiser):
582 '''(INTERNAL) Handle C{fma} and C{f2mul} DEPRECATED C{raiser=False}.
583 '''
584 return Fsum(x, nonfinites=nonfinites), raiser
587def fmean(xs):
588 '''Compute the accurate mean M{sum(xs) / len(xs)}.
590 @arg xs: Values (each C{scalar}, or L{Fsum} or L{Fsum2Tuple}).
592 @return: Mean value (C{float}).
594 @raise LenError: No B{C{xs}} values.
596 @raise OverflowError: Partial C{2sum} overflow.
597 '''
598 n, xs = len2(xs)
599 if n < 1:
600 raise LenError(fmean, xs=xs)
601 M = Fsum(*xs, nonfinites=True)
602 return M.fover(n) if n > 1 else float(M)
605def fmean_(*xs, **nonfinites):
606 '''Compute the accurate mean M{sum(xs) / len(xs)}.
608 @see: Function L{fmean} for further details.
609 '''
610 return fmean(xs, **nonfinites)
613def f2mul_(x, *ys, **nonfinites): # **raiser
614 '''Cascaded, accurate multiplication C{B{x} * B{y} * B{y} ...} for all B{C{ys}}.
616 @arg x: Multiplicand (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
617 @arg ys: Multipliers (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all
618 positional.
619 @kwarg nonfinites: Use C{B{nonfinites}=True} or C{=False}, to override default
620 L{nonfiniterrors} (C{bool}), see method L{Fsum.f2mul_}.
622 @return: The cascaded I{TwoProduct} (C{float}, C{int} or L{Fsum}).
624 @see: U{Equations 2.3<https://www.TUHH.De/ti3/paper/rump/OzOgRuOi06.pdf>}
625 '''
626 F, raiser = _Fm2(x, **nonfinites)
627 return F.f2mul_(*ys, **raiser).as_iscalar
630def fpolynomial(x, *cs, **over_f2product_nonfinites):
631 '''Evaluate the polynomial M{sum(cs[i] * x**i for i=0..len(cs)) [/ over]}.
633 @kwarg over_f2product_nonfinites: Optional final divisor C{B{over}=None}
634 (I{non-zero} C{scalar}) and other settings, see class
635 L{Fpolynomial<Fpolynomial.__init__>}.
637 @return: Polynomial value (C{float} or L{Fpolynomial}).
638 '''
639 d, kwds = _xkwds_pop2(over_f2product_nonfinites, over=0)
640 P = Fpolynomial(x, *cs, **kwds)
641 return P.fover(d) if d else float(P)
644def fpowers(x, n, alts=0):
645 '''Return a series of powers M{[x**i for i=1..n]}, note I{1..!}
647 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
648 @arg n: Highest exponent (C{int}).
649 @kwarg alts: Only alternating powers, starting with this
650 exponent (C{int}).
652 @return: Tuple of powers of B{C{x}} (each C{type(B{x})}).
654 @raise TypeError: Invalid B{C{x}} or B{C{n}} not C{int}.
656 @raise ValueError: Non-finite B{C{x}} or invalid B{C{n}}.
657 '''
658 if not isint(n):
659 raise _IsnotError(int.__name__, n=n)
660 elif n < 1:
661 raise _ValueError(n=n)
663 p = x if isscalar(x) or _isFsum_2Tuple(x) else _2float(x=x)
664 ps = tuple(_powers(p, n))
666 if alts > 0: # x**2, x**4, ...
667 # ps[alts-1::2] chokes PyChecker
668 ps = ps[slice(alts-1, None, 2)]
670 return ps
673try:
674 from math import prod as fprod # Python 3.8
675except ImportError:
677 def fprod(xs, start=1):
678 '''Iterable product, like C{math.prod} or C{numpy.prod}.
680 @arg xs: Iterable of values to be multiplied (each
681 C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
682 @kwarg start: Initial value, also the value returned
683 for an empty B{C{xs}} (C{scalar}).
685 @return: The product (C{float} or L{Fsum}).
687 @see: U{NumPy.prod<https://docs.SciPy.org/doc/
688 numpy/reference/generated/numpy.prod.html>}.
689 '''
690 return freduce(_operator.mul, xs, start)
693def frandoms(n, seeded=None):
694 '''Generate C{n} (long) lists of random C{floats}.
696 @arg n: Number of lists to generate (C{int}, non-negative).
697 @kwarg seeded: If C{scalar}, use C{random.seed(B{seeded})} or
698 if C{True}, seed using today's C{year-day}.
700 @see: U{Hettinger<https://GitHub.com/ActiveState/code/tree/master/recipes/
701 Python/393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}.
702 '''
703 from random import gauss, random, seed, shuffle
705 if seeded is None:
706 pass
707 elif seeded and isbool(seeded):
708 from time import localtime
709 seed(localtime().tm_yday)
710 elif isscalar(seeded):
711 seed(seeded)
713 c = (7, 1e100, -7, -1e100, -9e-20, 8e-20) * 7
714 for _ in range(n):
715 s = 0
716 t = list(c)
717 _a = t.append
718 for _ in range(n * 8):
719 v = gauss(0, random())**7 - s
720 _a(v)
721 s += v
722 shuffle(t)
723 yield t
726def frange(start, number, step=1):
727 '''Generate a range of C{float}s.
729 @arg start: First value (C{float}).
730 @arg number: The number of C{float}s to generate (C{int}).
731 @kwarg step: Increment value (C{float}).
733 @return: A generator (C{float}s).
735 @see: U{NumPy.prod<https://docs.SciPy.org/doc/
736 numpy/reference/generated/numpy.arange.html>}.
737 '''
738 if not isint(number):
739 raise _IsnotError(int.__name__, number=number)
740 for i in range(number):
741 yield start + (step * i)
744try:
745 from functools import reduce as freduce
746except ImportError:
747 try:
748 freduce = reduce # PYCHOK expected
749 except NameError: # Python 3+
751 def freduce(f, xs, *start):
752 '''For missing C{functools.reduce}.
753 '''
754 if start:
755 r = v = start[0]
756 else:
757 r, v = 0, MISSING
758 for v in xs:
759 r = f(r, v)
760 if v is MISSING:
761 raise _TypeError(xs=(), start=MISSING)
762 return r
765def fremainder(x, y):
766 '''Remainder in range C{[-B{y / 2}, B{y / 2}]}.
768 @arg x: Numerator (C{scalar}).
769 @arg y: Modulus, denominator (C{scalar}).
771 @return: Remainder (C{scalar}, preserving signed
772 0.0) or C{NAN} for any non-finite B{C{x}}.
774 @raise ValueError: Infinite or near-zero B{C{y}}.
776 @see: I{Karney}'s U{Math.remainder<https://PyPI.org/
777 project/geographiclib/>} and Python 3.7+
778 U{math.remainder<https://docs.Python.org/3/
779 library/math.html#math.remainder>}.
780 '''
781 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and
782 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native
783 # fmod( 0, 360) == 0.0
784 # fmod( 360, 360) == 0.0
785 # fmod(-0, 360) == 0.0
786 # fmod(-0.0, 360) == -0.0
787 # fmod(-360, 360) == -0.0
788 # however, using the % operator ...
789 # 0 % 360 == 0
790 # 360 % 360 == 0
791 # 360.0 % 360 == 0.0
792 # -0 % 360 == 0
793 # -360 % 360 == 0 == (-360) % 360
794 # -0.0 % 360 == 0.0 == (-0.0) % 360
795 # -360.0 % 360 == 0.0 == (-360.0) % 360
797 # On Windows 32-bit with python 2.7, math.fmod(-0.0, 360)
798 # == +0.0. This fixes this bug. See also Math::AngNormalize
799 # in the C++ library, Math.sincosd has a similar fix.
800 if isfinite(x):
801 try:
802 r = remainder(x, y) if x else x
803 except Exception as e:
804 raise _xError(e, unstr(fremainder, x, y))
805 else: # handle x INF and NINF as NAN
806 r = NAN
807 return r
810if _MODS.sys_version_info2 < (3, 8): # PYCHOK no cover
811 from math import hypot # OK in Python 3.7-
813 def hypot_(*xs):
814 '''Compute the norm M{sqrt(sum(x**2 for x in xs))}.
816 Similar to Python 3.8+ n-dimension U{math.hypot
817 <https://docs.Python.org/3.8/library/math.html#math.hypot>},
818 but exceptions, C{nan} and C{infinite} values are
819 handled differently.
821 @arg xs: X arguments (C{scalar}s), all positional.
823 @return: Norm (C{float}).
825 @raise OverflowError: Partial C{2sum} overflow.
827 @raise ValueError: Invalid or no B{C{xs}} values.
829 @note: The Python 3.8+ Euclidian distance U{math.dist
830 <https://docs.Python.org/3.8/library/math.html#math.dist>}
831 between 2 I{n}-dimensional points I{p1} and I{p2} can be
832 computed as M{hypot_(*((c1 - c2) for c1, c2 in zip(p1, p2)))},
833 provided I{p1} and I{p2} have the same, non-zero length I{n}.
834 '''
835 return float(_Hypot(*xs))
837elif _MODS.sys_version_info2 < (3, 10): # PYCHOK no cover
838 # In Python 3.8 and 3.9 C{math.hypot} is inaccurate, see
839 # U{agdhruv<https://GitHub.com/geopy/geopy/issues/466>},
840 # U{cffk<https://Bugs.Python.org/issue43088>} and module
841 # U{geomath.py<https://PyPI.org/project/geographiclib/1.52>}
843 def hypot(x, y):
844 '''Compute the norm M{sqrt(x**2 + y**2)}.
846 @arg x: X argument (C{scalar}).
847 @arg y: Y argument (C{scalar}).
849 @return: C{sqrt(B{x}**2 + B{y}**2)} (C{float}).
850 '''
851 return float(_Hypot(x, y))
853 from math import hypot as hypot_ # PYCHOK in Python 3.8 and 3.9
854else:
855 from math import hypot # PYCHOK in Python 3.10+
856 hypot_ = hypot
859def _Hypot(*xs):
860 '''(INTERNAL) Substitute for inaccurate C{math.hypot}.
861 '''
862 return Fhypot(*xs, nonfinites=True, raiser=False) # f2product=True
865def hypot1(x):
866 '''Compute the norm M{sqrt(1 + x**2)}.
868 @arg x: Argument (C{scalar} or L{Fsum} or L{Fsum2Tuple}).
870 @return: Norm (C{float} or L{Fhypot}).
871 '''
872 h = _1_0
873 if x:
874 if _isFsum_2Tuple(x):
875 h = _Hypot(h, x)
876 h = float(h)
877 else:
878 h = hypot(h, x)
879 return h
882def hypot2(x, y):
883 '''Compute the I{squared} norm M{x**2 + y**2}.
885 @arg x: X (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
886 @arg y: Y (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
888 @return: C{B{x}**2 + B{y}**2} (C{float}).
889 '''
890 x, y = map1(abs, x, y) # NOT fabs!
891 if y > x:
892 x, y = y, x
893 h2 = x**2
894 if h2 and y:
895 h2 *= (y / x)**2 + _1_0
896 return float(h2)
899def hypot2_(*xs):
900 '''Compute the I{squared} norm C{fsum(x**2 for x in B{xs})}.
902 @arg xs: Components (each C{scalar}, an L{Fsum} or
903 L{Fsum2Tuple}), all positional.
905 @return: Squared norm (C{float}).
907 @see: Class L{Fpowers} for further details.
908 '''
909 h2 = float(max(map(abs, xs))) if xs else _0_0
910 if h2: # and isfinite(h2)
911 _h = _1_0 / h2
912 xs = ((x * _h) for x in xs)
913 H2 = Fpowers(2, *xs, nonfinites=True) # f2product=True
914 h2 = H2.fover(_h**2)
915 return h2
918def norm2(x, y):
919 '''Normalize a 2-dimensional vector.
921 @arg x: X component (C{scalar}).
922 @arg y: Y component (C{scalar}).
924 @return: 2-Tuple C{(x, y)}, normalized.
926 @raise ValueError: Invalid B{C{x}} or B{C{y}}
927 or zero norm.
928 '''
929 try:
930 h = None
931 h = hypot(x, y)
932 if h:
933 x, y = (x / h), (y / h)
934 else:
935 x = _copysign_0_0(x) # pass?
936 y = _copysign_0_0(y)
937 except Exception as e:
938 raise _xError(e, x=x, y=y, h=h)
939 return x, y
942def norm_(*xs):
943 '''Normalize the components of an n-dimensional vector.
945 @arg xs: Components (each C{scalar}, an L{Fsum} or
946 L{Fsum2Tuple}), all positional.
948 @return: Yield each component, normalized.
950 @raise ValueError: Invalid or insufficent B{C{xs}}
951 or zero norm.
952 '''
953 try:
954 i = h = None
955 x = xs
956 h = hypot_(*xs)
957 _h = (_1_0 / h) if h else _0_0
958 for i, x in enumerate(xs):
959 yield x * _h
960 except Exception as X:
961 raise _xsError(X, xs, i, x, h=h)
964def _powers(x, n):
965 '''(INTERNAL) Yield C{x**i for i=1..n}.
966 '''
967 p = 1 # type(p) == type(x)
968 for _ in range(n):
969 p *= x
970 yield p
973def _root(x, p, where):
974 '''(INTERNAL) Raise C{x} to power C{0 < p < 1}.
975 '''
976 try:
977 if x > 0:
978 r = Fsum(f2product=True, nonfinites=True)(x)
979 return r.fpow(p).as_iscalar
980 elif x < 0:
981 raise ValueError(_negative_)
982 except Exception as X:
983 raise _xError(X, unstr(where, x))
984 return _0_0 if p else _1_0
987def sqrt0(x, Error=None):
988 '''Return the square root C{sqrt(B{x})} iff C{B{x} > }L{EPS02},
989 preserving C{type(B{x})}.
991 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
992 @kwarg Error: Error to raise for negative B{C{x}}.
994 @return: Square root (C{float} or L{Fsum}) or C{0.0}.
996 @raise TypeeError: Invalid B{C{x}}.
998 @note: Any C{B{x} < }L{EPS02} I{including} C{B{x} < 0}
999 returns C{0.0}.
1000 '''
1001 if Error and x < 0:
1002 raise Error(unstr(sqrt0, x))
1003 return _root(x, _0_5, sqrt0) if x > EPS02 else (
1004 _0_0 if x < EPS02 else EPS0)
1007def sqrt3(x):
1008 '''Return the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)},
1009 preserving C{type(B{x})}.
1011 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
1013 @return: Square root I{cubed} (C{float} or L{Fsum}).
1015 @raise TypeeError: Invalid B{C{x}}.
1017 @raise ValueError: Negative B{C{x}}.
1019 @see: Functions L{cbrt} and L{cbrt2}.
1020 '''
1021 return _root(x, _1_5, sqrt3)
1024def sqrt_a(h, b):
1025 '''Compute C{I{a}} side of a right-angled triangle from
1026 C{sqrt(B{h}**2 - B{b}**2)}.
1028 @arg h: Hypotenuse or outer annulus radius (C{scalar}).
1029 @arg b: Triangle side or inner annulus radius (C{scalar}).
1031 @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}).
1033 @raise TypeError: Non-scalar B{C{h}} or B{C{b}}.
1035 @raise ValueError: If C{abs(B{h}) < abs(B{b})}.
1037 @see: Inner tangent chord B{I{d}} of an U{annulus
1038 <https://WikiPedia.org/wiki/Annulus_(mathematics)>}
1039 and function U{annulus_area<https://People.SC.FSU.edu/
1040 ~jburkardt/py_src/geometry/geometry.py>}.
1041 '''
1042 try:
1043 if not (_isHeight(h) and _isRadius(b)):
1044 raise TypeError(_not_scalar_)
1045 c = fabs(h)
1046 if c > EPS0:
1047 s = _1_0 - (b / c)**2
1048 if s < 0:
1049 raise ValueError(_h_lt_b_)
1050 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0)
1051 else: # PYCHOK no cover
1052 b = fabs(b)
1053 d = c - b
1054 if d < 0:
1055 raise ValueError(_h_lt_b_)
1056 d *= c + b
1057 a = sqrt(d) if d else _0_0
1058 except Exception as x:
1059 raise _xError(x, h=h, b=b)
1060 return copysign0(a, h)
1063def zcrt(x):
1064 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)},
1065 preserving C{type(B{x})}.
1067 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
1069 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}).
1071 @see: Functions L{bqrt} and L{zqrt}.
1073 @raise TypeeError: Invalid B{C{x}}.
1075 @raise ValueError: Negative B{C{x}}.
1076 '''
1077 return _root(x, _1_6th, zcrt)
1080def zqrt(x):
1081 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root,
1082 M{x**(1 / 8)}, preserving C{type(B{x})}.
1084 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).
1086 @return: I{Zenzi-quartic} root (C{float} or L{Fsum}).
1088 @see: Functions L{bqrt} and L{zcrt}.
1090 @raise TypeeError: Invalid B{C{x}}.
1092 @raise ValueError: Negative B{C{x}}.
1093 '''
1094 return _root(x, _0_125, zqrt)
1096# **) MIT License
1097#
1098# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.
1099#
1100# Permission is hereby granted, free of charge, to any person obtaining a
1101# copy of this software and associated documentation files (the "Software"),
1102# to deal in the Software without restriction, including without limitation
1103# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1104# and/or sell copies of the Software, and to permit persons to whom the
1105# Software is furnished to do so, subject to the following conditions:
1106#
1107# The above copyright notice and this permission notice shall be included
1108# in all copies or substantial portions of the Software.
1109#
1110# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1111# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1112# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1113# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1114# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1115# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1116# OTHER DEALINGS IN THE SOFTWARE.