Coverage for pygeodesy/fmath.py: 91%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''Utilities for precision floating point summation, multiplication, 

5C{fused-multiply-add}, polynomials, roots, etc. 

6''' 

7# make sure int/int division yields float quotient, see .basics 

8from __future__ import division as _; del _ # PYCHOK semicolon 

9 

10from pygeodesy.basics import _copysign, copysign0, isbool, isint, isscalar, \ 

11 len2, map1, _xiterable, typename 

12from pygeodesy.constants import EPS0, EPS02, EPS1, NAN, PI, PI_2, PI_4, \ 

13 _0_0, _0_125, _1_6th, _0_25, _1_3rd, _0_5, _1_0, \ 

14 _1_5, _copysign_0_0, isfinite, remainder 

15from pygeodesy.errors import _IsnotError, LenError, _TypeError, _ValueError, \ 

16 _xError, _xkwds, _xkwds_pop2, _xsError 

17from pygeodesy.fsums import _2float, Fsum, fsum, _isFsum_2Tuple, Fmt, unstr 

18# from pygeodesy.internals import typename # from .basics 

19from pygeodesy.interns import MISSING, _negative_, _not_scalar_ 

20from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS 

21# from pygeodesy.streprs import Fmt, unstr # from .fsums 

22from pygeodesy.units import Int_, _isHeight, _isRadius, Float_ # PYCHOK for .heights 

23 

24from math import fabs, sqrt # pow 

25import operator as _operator # in .datums, .trf, .utm 

26 

27__all__ = _ALL_LAZY.fmath 

28__version__ = '25.04.18' 

29 

30# sqrt(2) - 1 <https://WikiPedia.org/wiki/Square_root_of_2> 

31_0_4142 = 0.41421356237309504880 # ~ 3_730_904_090_310_553 / 9_007_199_254_740_992 

32_2_3rd = _1_3rd * 2 

33_h_lt_b_ = 'abs(h) < abs(b)' 

34 

35 

36class Fdot(Fsum): 

37 '''Precision dot product. 

38 ''' 

39 def __init__(self, a, *b, **start_name_f2product_nonfinites_RESIDUAL): 

40 '''New L{Fdot} precision dot product M{sum(a[i] * b[i] for i=0..len(a)-1)}. 

41 

42 @arg a: Iterable of values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

43 @arg b: Other values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all 

44 positional. 

45 @kwarg start_name_f2product_nonfinites_RESIDUAL: Optional bias C{B{start}=0} 

46 (C{scalar}, an L{Fsum} or L{Fsum2Tuple}), C{B{name}=NN} (C{str}) 

47 and other settings, see class L{Fsum<Fsum.__init__>}. 

48 

49 @raise LenError: Unequal C{len(B{a})} and C{len(B{b})}. 

50 

51 @raise OverflowError: Partial C{2sum} overflow. 

52 

53 @raise TypeError: Invalid B{C{x}}. 

54 

55 @raise ValueError: Non-finite B{C{x}}. 

56 

57 @see: Function L{fdot} and method L{Fsum.fadd}. 

58 ''' 

59 s, kwds = _xkwds_pop2(start_name_f2product_nonfinites_RESIDUAL, start=_0_0) 

60 Fsum.__init__(self, **kwds) 

61 self(s) 

62 

63 n = len(b) 

64 if len(a) != n: # PYCHOK no cover 

65 raise LenError(Fdot, a=len(a), b=n) 

66 self._facc_dot(n, a, b, **kwds) 

67 

68 

69class Fhorner(Fsum): 

70 '''Precision polynomial evaluation using the Horner form. 

71 ''' 

72 def __init__(self, x, *cs, **incx_name_f2product_nonfinites_RESIDUAL): 

73 '''New L{Fhorner} form evaluation of polynomial M{sum(cs[i] * x**i for 

74 i=0..n)} with in- or decreasing exponent M{sum(... i=n..0)}, where C{n 

75 = len(cs) - 1}. 

76 

77 @arg x: Polynomial argument (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

78 @arg cs: Polynomial coeffients (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), 

79 all positional. 

80 @kwarg incx_name_f2product_nonfinites_RESIDUAL: Optional C{B{name}=NN} (C{str}), 

81 C{B{incx}=True} for in-/decreasing exponents (C{bool}) and other 

82 settings, see class L{Fsum<Fsum.__init__>}. 

83 

84 @raise OverflowError: Partial C{2sum} overflow. 

85 

86 @raise TypeError: Invalid B{C{x}}. 

87 

88 @raise ValueError: Non-finite B{C{x}}. 

89 

90 @see: Function L{fhorner} and methods L{Fsum.fadd} and L{Fsum.fmul}. 

91 ''' 

92 incx, kwds = _xkwds_pop2(incx_name_f2product_nonfinites_RESIDUAL, incx=True) 

93 Fsum.__init__(self, **kwds) 

94 self._fhorner(x, cs, Fhorner, incx=incx) 

95 

96 

97class Fhypot(Fsum): 

98 '''Precision summation and hypotenuse, default C{root=2}. 

99 ''' 

100 def __init__(self, *xs, **root_name_f2product_nonfinites_RESIDUAL_raiser): 

101 '''New L{Fhypot} hypotenuse of (the I{root} of) several components (raised 

102 to the power I{root}). 

103 

104 @arg xs: Components (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all 

105 positional. 

106 @kwarg root_name_f2product_nonfinites_RESIDUAL_raiser: Optional, exponent 

107 and C{B{root}=2} order (C{scalar}), C{B{name}=NN} (C{str}), 

108 C{B{raiser}=True} (C{bool}) for raising L{ResidualError}s and 

109 other settings, see class L{Fsum<Fsum.__init__>} and method 

110 L{root<Fsum.root>}. 

111 ''' 

112 def _r_X_kwds(power=None, raiser=True, root=2, **kwds): 

113 # DEPRECATED keyword argument C{power=2}, use C{root=2} 

114 return (root if power is None else power), raiser, kwds 

115 

116 r = None # _xkwds_pop2 error 

117 try: 

118 r, X, kwds = _r_X_kwds(**root_name_f2product_nonfinites_RESIDUAL_raiser) 

119 Fsum.__init__(self, **kwds) 

120 self(_0_0) 

121 if xs: 

122 self._facc_power(r, xs, Fhypot, raiser=X) 

123 self._fset(self.root(r, raiser=X)) 

124 except Exception as X: 

125 raise self._ErrorXs(X, xs, root=r) 

126 

127 

128class Fpolynomial(Fsum): 

129 '''Precision polynomial evaluation. 

130 ''' 

131 def __init__(self, x, *cs, **name_f2product_nonfinites_RESIDUAL): 

132 '''New L{Fpolynomial} evaluation of the polynomial M{sum(cs[i] * x**i for 

133 i=0..len(cs)-1)}. 

134 

135 @arg x: Polynomial argument (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

136 @arg cs: Polynomial coeffients (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), 

137 all positional. 

138 @kwarg name_f2product_nonfinites_RESIDUAL: Optional C{B{name}=NN} (C{str}) 

139 and other settings, see class L{Fsum<Fsum.__init__>}. 

140 

141 @raise OverflowError: Partial C{2sum} overflow. 

142 

143 @raise TypeError: Invalid B{C{x}}. 

144 

145 @raise ValueError: Non-finite B{C{x}}. 

146 

147 @see: Class L{Fhorner}, function L{fpolynomial} and method L{Fsum.fadd}. 

148 ''' 

149 Fsum.__init__(self, **name_f2product_nonfinites_RESIDUAL) 

150 n = len(cs) - 1 

151 self(_0_0 if n < 0 else cs[0]) 

152 self._facc_dot(n, cs[1:], _powers(x, n), **name_f2product_nonfinites_RESIDUAL) 

153 

154 

155class Fpowers(Fsum): 

156 '''Precision summation of powers, optimized for C{power=2, 3 and 4}. 

157 ''' 

158 def __init__(self, power, *xs, **name_f2product_nonfinites_RESIDUAL_raiser): 

159 '''New L{Fpowers} sum of (the I{power} of) several bases. 

160 

161 @arg power: The exponent (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

162 @arg xs: One or more bases (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all 

163 positional. 

164 @kwarg name_f2product_nonfinites_RESIDUAL_raiser: Optional C{B{name}=NN} 

165 (C{str}), C{B{raiser}=True} (C{bool}) for raising L{ResidualError}s 

166 and other settings, see class L{Fsum<Fsum.__init__>} and method 

167 L{fpow<Fsum.fpow>}. 

168 ''' 

169 try: 

170 X, kwds = _xkwds_pop2(name_f2product_nonfinites_RESIDUAL_raiser, raiser=True) 

171 Fsum.__init__(self, **kwds) 

172 self(_0_0) 

173 if xs: 

174 self._facc_power(power, xs, Fpowers, raiser=X) # x**0 == 1 

175 except Exception as X: 

176 raise self._ErrorXs(X, xs, power=power) 

177 

178 

179class Froot(Fsum): 

180 '''The root of a precision summation. 

181 ''' 

182 def __init__(self, root, *xs, **name_f2product_nonfinites_RESIDUAL_raiser): 

183 '''New L{Froot} root of a precision sum. 

184 

185 @arg root: The order (C{scalar}, an L{Fsum} or L{Fsum2Tuple}), non-zero. 

186 @arg xs: Items to summate (each a C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all 

187 positional. 

188 @kwarg name_f2product_nonfinites_RESIDUAL_raiser: Optional C{B{name}=NN} 

189 (C{str}), C{B{raiser}=True} (C{bool}) for raising L{ResidualError}s 

190 and other settings, see class L{Fsum<Fsum.__init__>} and method 

191 L{fpow<Fsum.fpow>}. 

192 ''' 

193 try: 

194 X, kwds = _xkwds_pop2(name_f2product_nonfinites_RESIDUAL_raiser, raiser=True) 

195 Fsum.__init__(self, **kwds) 

196 self(_0_0) 

197 if xs: 

198 self.fadd(xs) 

199 self(self.root(root, raiser=X)) 

200 except Exception as X: 

201 raise self._ErrorXs(X, xs, root=root) 

202 

203 

204class Fcbrt(Froot): 

205 '''Cubic root of a precision summation. 

206 ''' 

207 def __init__(self, *xs, **name_f2product_nonfinites_RESIDUAL_raiser): 

208 '''New L{Fcbrt} cubic root of a precision sum. 

209 

210 @see: Class L{Froot<Froot.__init__>} for further details. 

211 ''' 

212 Froot.__init__(self, 3, *xs, **name_f2product_nonfinites_RESIDUAL_raiser) 

213 

214 

215class Fsqrt(Froot): 

216 '''Square root of a precision summation. 

217 ''' 

218 def __init__(self, *xs, **name_f2product_nonfinites_RESIDUAL_raiser): 

219 '''New L{Fsqrt} square root of a precision sum. 

220 

221 @see: Class L{Froot<Froot.__init__>} for further details. 

222 ''' 

223 Froot.__init__(self, 2, *xs, **name_f2product_nonfinites_RESIDUAL_raiser) 

224 

225 

226def bqrt(x): 

227 '''Return the 4-th, I{bi-quadratic} or I{quartic} root, M{x**(1 / 4)}, 

228 preserving C{type(B{x})}. 

229 

230 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

231 

232 @return: I{Quartic} root (C{float} or an L{Fsum}). 

233 

234 @raise TypeeError: Invalid B{C{x}}. 

235 

236 @raise ValueError: Negative B{C{x}}. 

237 

238 @see: Functions L{zcrt} and L{zqrt}. 

239 ''' 

240 return _root(x, _0_25, bqrt) 

241 

242 

243try: 

244 from math import cbrt as _cbrt # Python 3.11+ 

245 

246except ImportError: # Python 3.10- 

247 

248 def _cbrt(x): 

249 '''(INTERNAL) Compute the I{signed}, cube root M{x**(1/3)}. 

250 ''' 

251 # <https://archive.lib.MSU.edu/crcmath/math/math/r/r021.htm> 

252 # simpler and more accurate than Ken Turkowski's CubeRoot, see 

253 # <https://People.FreeBSD.org/~lstewart/references/apple_tr_kt32_cuberoot.pdf> 

254 return _copysign(pow(fabs(x), _1_3rd), x) # to avoid complex 

255 

256 

257def cbrt(x): 

258 '''Compute the cube root M{x**(1/3)}, preserving C{type(B{x})}. 

259 

260 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

261 

262 @return: Cubic root (C{float} or L{Fsum}). 

263 

264 @see: Functions L{cbrt2} and L{sqrt3}. 

265 ''' 

266 if _isFsum_2Tuple(x): 

267 r = abs(x).fpow(_1_3rd) 

268 if x.signOf() < 0: 

269 r = -r 

270 else: 

271 r = _cbrt(x) 

272 return r # cbrt(-0.0) == -0.0 

273 

274 

275def cbrt2(x): # PYCHOK attr 

276 '''Compute the cube root I{squared} M{x**(2/3)}, preserving C{type(B{x})}. 

277 

278 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

279 

280 @return: Cube root I{squared} (C{float} or L{Fsum}). 

281 

282 @see: Functions L{cbrt} and L{sqrt3}. 

283 ''' 

284 return abs(x).fpow(_2_3rd) if _isFsum_2Tuple(x) else _cbrt(x**2) 

285 

286 

287def euclid(x, y): 

288 '''I{Appoximate} the norm M{sqrt(x**2 + y**2)} by M{max(abs(x), 

289 abs(y)) + min(abs(x), abs(y)) * 0.4142...}. 

290 

291 @arg x: X component (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

292 @arg y: Y component (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

293 

294 @return: Appoximate norm (C{float} or L{Fsum}). 

295 

296 @see: Function L{euclid_}. 

297 ''' 

298 x, y = abs(x), abs(y) # NOT fabs! 

299 return (x + y * _0_4142) if x > y else \ 

300 (y + x * _0_4142) # * _0_5 before 20.10.02 

301 

302 

303def euclid_(*xs): 

304 '''I{Appoximate} the norm M{sqrt(sum(x**2 for x in xs))} by cascaded 

305 L{euclid}. 

306 

307 @arg xs: X arguments (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), 

308 all positional. 

309 

310 @return: Appoximate norm (C{float} or L{Fsum}). 

311 

312 @see: Function L{euclid}. 

313 ''' 

314 e = _0_0 

315 for x in sorted(map(abs, xs)): # NOT fabs, reverse=True! 

316 # e = euclid(x, e) 

317 if e < x: 

318 e, x = x, e 

319 if x: 

320 e += x * _0_4142 

321 return e 

322 

323 

324def facos1(x): 

325 '''Fast approximation of L{pygeodesy.acos1}C{(B{x})}, scalar. 

326 

327 @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/ 

328 ShaderFastLibs/blob/master/ShaderFastMathLib.h>}. 

329 ''' 

330 a = fabs(x) 

331 if a < EPS0: 

332 r = PI_2 

333 elif a < EPS1: 

334 r = _fast(-a, 1.5707288, 0.2121144, 0.0742610, 0.0187293) 

335 r *= sqrt(_1_0 - a) 

336 if x < 0: 

337 r = PI - r 

338 else: 

339 r = PI if x < 0 else _0_0 

340 return r 

341 

342 

343def fasin1(x): # PYCHOK no cover 

344 '''Fast approximation of L{pygeodesy.asin1}C{(B{x})}, scalar. 

345 

346 @see: L{facos1}. 

347 ''' 

348 return PI_2 - facos1(x) 

349 

350 

351def _fast(x, *cs): 

352 '''(INTERNAL) Horner form for C{facos1} and C{fatan1}. 

353 ''' 

354 h = 0 

355 for c in reversed(cs): 

356 h = _fma(x, h, c) if h else c 

357 return h 

358 

359 

360def fatan(x): 

361 '''Fast approximation of C{atan(B{x})}, scalar. 

362 ''' 

363 a = fabs(x) 

364 if a < _1_0: 

365 r = fatan1(a) if a else _0_0 

366 elif a > _1_0: 

367 r = PI_2 - fatan1(_1_0 / a) # == fatan2(a, _1_0) 

368 else: 

369 r = PI_4 

370 if x < 0: # copysign0(r, x) 

371 r = -r 

372 return r 

373 

374 

375def fatan1(x): 

376 '''Fast approximation of C{atan(B{x})} for C{0 <= B{x} < 1}, I{unchecked}. 

377 

378 @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/ShaderFastLibs/ 

379 blob/master/ShaderFastMathLib.h>} and U{Efficient approximations 

380 for the arctangent function<http://www-Labs.IRO.UMontreal.CA/ 

381 ~mignotte/IFT2425/Documents/EfficientApproximationArctgFunction.pdf>}, 

382 IEEE Signal Processing Magazine, 111, May 2006. 

383 ''' 

384 # Eq (9): PI_4 * x - x * (abs(x) - 1) * (0.2447 + 0.0663 * abs(x)), for -1 < x < 1 

385 # == PI_4 * x - (x**2 - x) * (0.2447 + 0.0663 * x), for 0 < x < 1 

386 # == x * (1.0300981633974482 + x * (-0.1784 - x * 0.0663)) 

387 return _fast(x, _0_0, 1.0300981634, -0.1784, -0.0663) 

388 

389 

390def fatan2(y, x): 

391 '''Fast approximation of C{atan2(B{y}, B{x})}, scalar. 

392 

393 @see: U{fastApproximateAtan(x, y)<https://GitHub.com/CesiumGS/cesium/blob/ 

394 master/Source/Shaders/Builtin/Functions/fastApproximateAtan.glsl>} 

395 and L{fatan1}. 

396 ''' 

397 a, b = fabs(x), fabs(y) 

398 if b > a: 

399 r = (PI_2 - fatan1(a / b)) if a else PI_2 

400 elif a > b: 

401 r = fatan1(b / a) if b else _0_0 

402 elif a: # a == b != 0 

403 r = PI_4 

404 else: # a == b == 0 

405 return _0_0 

406 if x < 0: 

407 r = PI - r 

408 if y < 0: # copysign0(r, y) 

409 r = -r 

410 return r 

411 

412 

413def favg(a, b, f=_0_5, nonfinites=True): 

414 '''Return the precise average of two values. 

415 

416 @arg a: One (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

417 @arg b: Other (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

418 @kwarg f: Optional fraction (C{float}). 

419 @kwarg nonfinites: Optional setting, see function L{fma}. 

420 

421 @return: M{a + f * (b - a)} (C{float}). 

422 ''' 

423 F = fma(f, (b - a), a, nonfinites=nonfinites) 

424 return float(F) 

425 

426 

427def fdot(xs, *ys, **start_f2product_nonfinites): 

428 '''Return the precision dot product M{sum(xs[i] * ys[i] for i in range(len(xs)))}. 

429 

430 @arg xs: Iterable of values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

431 @arg ys: Other values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all positional. 

432 @kwarg start_f2product_nonfinites: Optional bias C{B{start}=0} (C{scalar}, an 

433 L{Fsum} or L{Fsum2Tuple}) and settings C{B{f2product}=None} (C{bool}) 

434 and C{B{nonfinites=True}} (C{bool}), see class L{Fsum<Fsum.__init__>}. 

435 

436 @return: Dot product (C{float}). 

437 

438 @raise LenError: Unequal C{len(B{xs})} and C{len(B{ys})}. 

439 

440 @see: Class L{Fdot}, U{Algorithm 5.10 B{DotK} 

441 <https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>} and function 

442 C{math.sumprod} in Python 3.12 and later. 

443 ''' 

444 D = Fdot(xs, *ys, **_xkwds(start_f2product_nonfinites, nonfinites=True)) 

445 return float(D) 

446 

447 

448def fdot_(*xys, **start_f2product_nonfinites): 

449 '''Return the (precision) dot product M{sum(xys[i] * xys[i+1] for i in range(0, len(xys), B{2}))}. 

450 

451 @arg xys: Pairwise values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all positional. 

452 

453 @see: Function L{fdot} for further details. 

454 

455 @return: Dot product (C{float}). 

456 ''' 

457 return fdot(xys[0::2], *xys[1::2], **start_f2product_nonfinites) 

458 

459 

460def fdot3(xs, ys, zs, **start_f2product_nonfinites): 

461 '''Return the (precision) dot product M{start + sum(xs[i] * ys[i] * zs[i] for i in range(len(xs)))}. 

462 

463 @arg xs: X values iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

464 @arg ys: Y values iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

465 @arg zs: Z values iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

466 

467 @see: Function L{fdot} for further details. 

468 

469 @return: Dot product (C{float}). 

470 

471 @raise LenError: Unequal C{len(B{xs})}, C{len(B{ys})} and/or C{len(B{zs})}. 

472 ''' 

473 n = len(xs) 

474 if not n == len(ys) == len(zs): 

475 raise LenError(fdot3, xs=n, ys=len(ys), zs=len(zs)) 

476 

477 D = Fdot((), **_xkwds(start_f2product_nonfinites, nonfinites=True)) 

478 kwds = dict(f2product=D.f2product(), nonfinites=D.nonfinites()) 

479 _f = Fsum(**kwds) 

480 D = D._facc(_f(x).f2mul_(y, z, **kwds) for x, y, z in zip(xs, ys, zs)) 

481 return float(D) 

482 

483 

484def fhorner(x, *cs, **incx): 

485 '''Horner form evaluation of polynomial M{sum(cs[i] * x**i for i=0..n)} as 

486 in- or decreasing exponent M{sum(... i=n..0)}, where C{n = len(cs) - 1}. 

487 

488 @return: Horner sum (C{float}). 

489 

490 @see: Class L{Fhorner<Fhorner.__init__>} for further details. 

491 ''' 

492 H = Fhorner(x, *cs, **incx) 

493 return float(H) 

494 

495 

496def fidw(xs, ds, beta=2): 

497 '''Interpolate using U{Inverse Distance Weighting 

498 <https://WikiPedia.org/wiki/Inverse_distance_weighting>} (IDW). 

499 

500 @arg xs: Known values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

501 @arg ds: Non-negative distances (each C{scalar}, an L{Fsum} or 

502 L{Fsum2Tuple}). 

503 @kwarg beta: Inverse distance power (C{int}, 0, 1, 2, or 3). 

504 

505 @return: Interpolated value C{x} (C{float}). 

506 

507 @raise LenError: Unequal or zero C{len(B{ds})} and C{len(B{xs})}. 

508 

509 @raise TypeError: An invalid B{C{ds}} or B{C{xs}}. 

510 

511 @raise ValueError: Invalid B{C{beta}}, negative B{C{ds}} or 

512 weighted B{C{ds}} below L{EPS}. 

513 

514 @note: Using C{B{beta}=0} returns the mean of B{C{xs}}. 

515 ''' 

516 n, xs = len2(xs) 

517 if n > 1: 

518 b = -Int_(beta=beta, low=0, high=3) 

519 if b < 0: 

520 try: # weighted 

521 _d, W, X = (Fsum() for _ in range(3)) 

522 for i, d in enumerate(_xiterable(ds)): 

523 x = xs[i] 

524 D = _d(d) 

525 if D < EPS0: 

526 if D < 0: 

527 raise ValueError(_negative_) 

528 x = float(x) 

529 i = n 

530 break 

531 if D.fpow(b): 

532 W += D 

533 X += D.fmul(x) 

534 else: 

535 x = X.fover(W, raiser=False) 

536 i += 1 # len(xs) >= len(ds) 

537 except IndexError: 

538 i += 1 # len(xs) < i < len(ds) 

539 except Exception as X: 

540 _I = Fmt.INDEX 

541 raise _xError(X, _I(xs=i), x, 

542 _I(ds=i), d) 

543 else: # b == 0 

544 x = fsum(xs) / n # fmean(xs) 

545 i = n 

546 elif n: 

547 x = float(xs[0]) 

548 i = n 

549 else: 

550 x = _0_0 

551 i, _ = len2(ds) 

552 if i != n: 

553 raise LenError(fidw, xs=n, ds=i) 

554 return x 

555 

556 

557try: 

558 from math import fma as _fma 

559except ImportError: # PYCHOK DSPACE! 

560 

561 def _fma(x, y, z): # no need for accuracy 

562 return x * y + z 

563 

564 

565def fma(x, y, z, **nonfinites): # **raiser 

566 '''Fused-multiply-add, using C{math.fma(x, y, z)} in Python 3.13+ 

567 or an equivalent implementation. 

568 

569 @arg x: Multiplicand (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

570 @arg y: Multiplier (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

571 @arg z: Addend (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

572 @kwarg nonfinites: Use C{B{nonfinites}=True} or C{=False}, 

573 to override default L{nonfiniterrors} 

574 (C{bool}), see method L{Fsum.fma}. 

575 

576 @return: C{(x * y) + z} (C{float} or L{Fsum}). 

577 ''' 

578 F, raiser = _Fm2(x, **nonfinites) 

579 return F.fma(y, z, **raiser).as_iscalar 

580 

581 

582def _Fm2(x, nonfinites=None, **raiser): 

583 '''(INTERNAL) Handle C{fma} and C{f2mul} DEPRECATED C{raiser=False}. 

584 ''' 

585 return Fsum(x, nonfinites=nonfinites), raiser 

586 

587 

588def fmean(xs): 

589 '''Compute the accurate mean M{sum(xs) / len(xs)}. 

590 

591 @arg xs: Values (each C{scalar}, or L{Fsum} or L{Fsum2Tuple}). 

592 

593 @return: Mean value (C{float}). 

594 

595 @raise LenError: No B{C{xs}} values. 

596 

597 @raise OverflowError: Partial C{2sum} overflow. 

598 ''' 

599 n, xs = len2(xs) 

600 if n < 1: 

601 raise LenError(fmean, xs=xs) 

602 M = Fsum(*xs, nonfinites=True) 

603 return M.fover(n) if n > 1 else float(M) 

604 

605 

606def fmean_(*xs, **nonfinites): 

607 '''Compute the accurate mean M{sum(xs) / len(xs)}. 

608 

609 @see: Function L{fmean} for further details. 

610 ''' 

611 return fmean(xs, **nonfinites) 

612 

613 

614def f2mul_(x, *ys, **nonfinites): # **raiser 

615 '''Cascaded, accurate multiplication C{B{x} * B{y} * B{y} ...} for all B{C{ys}}. 

616 

617 @arg x: Multiplicand (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

618 @arg ys: Multipliers (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all 

619 positional. 

620 @kwarg nonfinites: Use C{B{nonfinites}=True} or C{=False}, to override default 

621 L{nonfiniterrors} (C{bool}), see method L{Fsum.f2mul_}. 

622 

623 @return: The cascaded I{TwoProduct} (C{float}, C{int} or L{Fsum}). 

624 

625 @see: U{Equations 2.3<https://www.TUHH.De/ti3/paper/rump/OzOgRuOi06.pdf>} 

626 ''' 

627 F, raiser = _Fm2(x, **nonfinites) 

628 return F.f2mul_(*ys, **raiser).as_iscalar 

629 

630 

631def fpolynomial(x, *cs, **over_f2product_nonfinites): 

632 '''Evaluate the polynomial M{sum(cs[i] * x**i for i=0..len(cs)) [/ over]}. 

633 

634 @kwarg over_f2product_nonfinites: Optional final divisor C{B{over}=None} 

635 (I{non-zero} C{scalar}) and other settings, see class 

636 L{Fpolynomial<Fpolynomial.__init__>}. 

637 

638 @return: Polynomial value (C{float} or L{Fpolynomial}). 

639 ''' 

640 d, kwds = _xkwds_pop2(over_f2product_nonfinites, over=0) 

641 P = Fpolynomial(x, *cs, **kwds) 

642 return P.fover(d) if d else float(P) 

643 

644 

645def fpowers(x, n, alts=0): 

646 '''Return a series of powers M{[x**i for i=1..n]}, note I{1..!} 

647 

648 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

649 @arg n: Highest exponent (C{int}). 

650 @kwarg alts: Only alternating powers, starting with this 

651 exponent (C{int}). 

652 

653 @return: Tuple of powers of B{C{x}} (each C{type(B{x})}). 

654 

655 @raise TypeError: Invalid B{C{x}} or B{C{n}} not C{int}. 

656 

657 @raise ValueError: Non-finite B{C{x}} or invalid B{C{n}}. 

658 ''' 

659 if not isint(n): 

660 raise _IsnotError(typename(int), n=n) 

661 elif n < 1: 

662 raise _ValueError(n=n) 

663 

664 p = x if isscalar(x) or _isFsum_2Tuple(x) else _2float(x=x) 

665 ps = tuple(_powers(p, n)) 

666 

667 if alts > 0: # x**2, x**4, ... 

668 # ps[alts-1::2] chokes PyChecker 

669 ps = ps[slice(alts-1, None, 2)] 

670 

671 return ps 

672 

673 

674try: 

675 from math import prod as fprod # Python 3.8 

676except ImportError: 

677 

678 def fprod(xs, start=1): 

679 '''Iterable product, like C{math.prod} or C{numpy.prod}. 

680 

681 @arg xs: Iterable of values to be multiplied (each 

682 C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

683 @kwarg start: Initial value, also the value returned 

684 for an empty B{C{xs}} (C{scalar}). 

685 

686 @return: The product (C{float} or L{Fsum}). 

687 

688 @see: U{NumPy.prod<https://docs.SciPy.org/doc/ 

689 numpy/reference/generated/numpy.prod.html>}. 

690 ''' 

691 return freduce(_operator.mul, xs, start) 

692 

693 

694def frandoms(n, seeded=None): 

695 '''Generate C{n} (long) lists of random C{floats}. 

696 

697 @arg n: Number of lists to generate (C{int}, non-negative). 

698 @kwarg seeded: If C{scalar}, use C{random.seed(B{seeded})} or 

699 if C{True}, seed using today's C{year-day}. 

700 

701 @see: U{Hettinger<https://GitHub.com/ActiveState/code/tree/master/recipes/ 

702 Python/393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}. 

703 ''' 

704 from random import gauss, random, seed, shuffle 

705 

706 if seeded is None: 

707 pass 

708 elif seeded and isbool(seeded): 

709 from time import localtime 

710 seed(localtime().tm_yday) 

711 elif isscalar(seeded): 

712 seed(seeded) 

713 

714 c = (7, 1e100, -7, -1e100, -9e-20, 8e-20) * 7 

715 for _ in range(n): 

716 s = 0 

717 t = list(c) 

718 _a = t.append 

719 for _ in range(n * 8): 

720 v = gauss(0, random())**7 - s 

721 _a(v) 

722 s += v 

723 shuffle(t) 

724 yield t 

725 

726 

727def frange(start, number, step=1): 

728 '''Generate a range of C{float}s. 

729 

730 @arg start: First value (C{float}). 

731 @arg number: The number of C{float}s to generate (C{int}). 

732 @kwarg step: Increment value (C{float}). 

733 

734 @return: A generator (C{float}s). 

735 

736 @see: U{NumPy.prod<https://docs.SciPy.org/doc/ 

737 numpy/reference/generated/numpy.arange.html>}. 

738 ''' 

739 if not isint(number): 

740 raise _IsnotError(typename(int), number=number) 

741 for i in range(number): 

742 yield start + (step * i) 

743 

744 

745try: 

746 from functools import reduce as freduce 

747except ImportError: 

748 try: 

749 freduce = reduce # PYCHOK expected 

750 except NameError: # Python 3+ 

751 

752 def freduce(f, xs, *start): 

753 '''For missing C{functools.reduce}. 

754 ''' 

755 if start: 

756 r = v = start[0] 

757 else: 

758 r, v = 0, MISSING 

759 for v in xs: 

760 r = f(r, v) 

761 if v is MISSING: 

762 raise _TypeError(xs=(), start=MISSING) 

763 return r 

764 

765 

766def fremainder(x, y): 

767 '''Remainder in range C{[-B{y / 2}, B{y / 2}]}. 

768 

769 @arg x: Numerator (C{scalar}). 

770 @arg y: Modulus, denominator (C{scalar}). 

771 

772 @return: Remainder (C{scalar}, preserving signed 

773 0.0) or C{NAN} for any non-finite B{C{x}}. 

774 

775 @raise ValueError: Infinite or near-zero B{C{y}}. 

776 

777 @see: I{Karney}'s U{Math.remainder<https://PyPI.org/ 

778 project/geographiclib/>} and Python 3.7+ 

779 U{math.remainder<https://docs.Python.org/3/ 

780 library/math.html#math.remainder>}. 

781 ''' 

782 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and 

783 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native 

784 # fmod( 0, 360) == 0.0 

785 # fmod( 360, 360) == 0.0 

786 # fmod(-0, 360) == 0.0 

787 # fmod(-0.0, 360) == -0.0 

788 # fmod(-360, 360) == -0.0 

789 # however, using the % operator ... 

790 # 0 % 360 == 0 

791 # 360 % 360 == 0 

792 # 360.0 % 360 == 0.0 

793 # -0 % 360 == 0 

794 # -360 % 360 == 0 == (-360) % 360 

795 # -0.0 % 360 == 0.0 == (-0.0) % 360 

796 # -360.0 % 360 == 0.0 == (-360.0) % 360 

797 

798 # On Windows 32-bit with python 2.7, math.fmod(-0.0, 360) 

799 # == +0.0. This fixes this bug. See also Math::AngNormalize 

800 # in the C++ library, Math.sincosd has a similar fix. 

801 if isfinite(x): 

802 try: 

803 r = remainder(x, y) if x else x 

804 except Exception as e: 

805 raise _xError(e, unstr(fremainder, x, y)) 

806 else: # handle x INF and NINF as NAN 

807 r = NAN 

808 return r 

809 

810 

811if _MODS.sys_version_info2 < (3, 8): # PYCHOK no cover 

812 from math import hypot # OK in Python 3.7- 

813 

814 def hypot_(*xs): 

815 '''Compute the norm M{sqrt(sum(x**2 for x in xs))}. 

816 

817 Similar to Python 3.8+ n-dimension U{math.hypot 

818 <https://docs.Python.org/3.8/library/math.html#math.hypot>}, 

819 but exceptions, C{nan} and C{infinite} values are 

820 handled differently. 

821 

822 @arg xs: X arguments (C{scalar}s), all positional. 

823 

824 @return: Norm (C{float}). 

825 

826 @raise OverflowError: Partial C{2sum} overflow. 

827 

828 @raise ValueError: Invalid or no B{C{xs}} values. 

829 

830 @note: The Python 3.8+ Euclidian distance U{math.dist 

831 <https://docs.Python.org/3.8/library/math.html#math.dist>} 

832 between 2 I{n}-dimensional points I{p1} and I{p2} can be 

833 computed as M{hypot_(*((c1 - c2) for c1, c2 in zip(p1, p2)))}, 

834 provided I{p1} and I{p2} have the same, non-zero length I{n}. 

835 ''' 

836 return float(_Hypot(*xs)) 

837 

838elif _MODS.sys_version_info2 < (3, 10): # PYCHOK no cover 

839 # In Python 3.8 and 3.9 C{math.hypot} is inaccurate, see 

840 # U{agdhruv<https://GitHub.com/geopy/geopy/issues/466>}, 

841 # U{cffk<https://Bugs.Python.org/issue43088>} and module 

842 # U{geomath.py<https://PyPI.org/project/geographiclib/1.52>} 

843 

844 def hypot(x, y): 

845 '''Compute the norm M{sqrt(x**2 + y**2)}. 

846 

847 @arg x: X argument (C{scalar}). 

848 @arg y: Y argument (C{scalar}). 

849 

850 @return: C{sqrt(B{x}**2 + B{y}**2)} (C{float}). 

851 ''' 

852 return float(_Hypot(x, y)) 

853 

854 from math import hypot as hypot_ # PYCHOK in Python 3.8 and 3.9 

855else: 

856 from math import hypot # PYCHOK in Python 3.10+ 

857 hypot_ = hypot 

858 

859 

860def _Hypot(*xs): 

861 '''(INTERNAL) Substitute for inaccurate C{math.hypot}. 

862 ''' 

863 return Fhypot(*xs, nonfinites=True, raiser=False) # f2product=True 

864 

865 

866def hypot1(x): 

867 '''Compute the norm M{sqrt(1 + x**2)}. 

868 

869 @arg x: Argument (C{scalar} or L{Fsum} or L{Fsum2Tuple}). 

870 

871 @return: Norm (C{float} or L{Fhypot}). 

872 ''' 

873 h = _1_0 

874 if x: 

875 if _isFsum_2Tuple(x): 

876 h = _Hypot(h, x) 

877 h = float(h) 

878 else: 

879 h = hypot(h, x) 

880 return h 

881 

882 

883def hypot2(x, y): 

884 '''Compute the I{squared} norm M{x**2 + y**2}. 

885 

886 @arg x: X (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

887 @arg y: Y (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

888 

889 @return: C{B{x}**2 + B{y}**2} (C{float}). 

890 ''' 

891 x, y = map1(abs, x, y) # NOT fabs! 

892 if y > x: 

893 x, y = y, x 

894 h2 = x**2 

895 if h2 and y: 

896 h2 *= (y / x)**2 + _1_0 

897 return float(h2) 

898 

899 

900def hypot2_(*xs): 

901 '''Compute the I{squared} norm C{fsum(x**2 for x in B{xs})}. 

902 

903 @arg xs: Components (each C{scalar}, an L{Fsum} or 

904 L{Fsum2Tuple}), all positional. 

905 

906 @return: Squared norm (C{float}). 

907 

908 @see: Class L{Fpowers} for further details. 

909 ''' 

910 h2 = float(max(map(abs, xs))) if xs else _0_0 

911 if h2: # and isfinite(h2) 

912 _h = _1_0 / h2 

913 xs = ((x * _h) for x in xs) 

914 H2 = Fpowers(2, *xs, nonfinites=True) # f2product=True 

915 h2 = H2.fover(_h**2) 

916 return h2 

917 

918 

919def norm2(x, y): 

920 '''Normalize a 2-dimensional vector. 

921 

922 @arg x: X component (C{scalar}). 

923 @arg y: Y component (C{scalar}). 

924 

925 @return: 2-Tuple C{(x, y)}, normalized. 

926 

927 @raise ValueError: Invalid B{C{x}} or B{C{y}} 

928 or zero norm. 

929 ''' 

930 try: 

931 h = None 

932 h = hypot(x, y) 

933 if h: 

934 x, y = (x / h), (y / h) 

935 else: 

936 x = _copysign_0_0(x) # pass? 

937 y = _copysign_0_0(y) 

938 except Exception as e: 

939 raise _xError(e, x=x, y=y, h=h) 

940 return x, y 

941 

942 

943def norm_(*xs): 

944 '''Normalize the components of an n-dimensional vector. 

945 

946 @arg xs: Components (each C{scalar}, an L{Fsum} or 

947 L{Fsum2Tuple}), all positional. 

948 

949 @return: Yield each component, normalized. 

950 

951 @raise ValueError: Invalid or insufficent B{C{xs}} 

952 or zero norm. 

953 ''' 

954 try: 

955 i = h = None 

956 x = xs 

957 h = hypot_(*xs) 

958 _h = (_1_0 / h) if h else _0_0 

959 for i, x in enumerate(xs): 

960 yield x * _h 

961 except Exception as X: 

962 raise _xsError(X, xs, i, x, h=h) 

963 

964 

965def _powers(x, n): 

966 '''(INTERNAL) Yield C{x**i for i=1..n}. 

967 ''' 

968 p = 1 # type(p) == type(x) 

969 for _ in range(n): 

970 p *= x 

971 yield p 

972 

973 

974def _root(x, p, where): 

975 '''(INTERNAL) Raise C{x} to power C{0 < p < 1}. 

976 ''' 

977 try: 

978 if x > 0: 

979 r = Fsum(f2product=True, nonfinites=True)(x) 

980 return r.fpow(p).as_iscalar 

981 elif x < 0: 

982 raise ValueError(_negative_) 

983 except Exception as X: 

984 raise _xError(X, unstr(where, x)) 

985 return _0_0 if p else _1_0 

986 

987 

988def sqrt0(x, Error=None): 

989 '''Return the square root C{sqrt(B{x})} iff C{B{x} > }L{EPS02}, 

990 preserving C{type(B{x})}. 

991 

992 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

993 @kwarg Error: Error to raise for negative B{C{x}}. 

994 

995 @return: Square root (C{float} or L{Fsum}) or C{0.0}. 

996 

997 @raise TypeeError: Invalid B{C{x}}. 

998 

999 @note: Any C{B{x} < }L{EPS02} I{including} C{B{x} < 0} 

1000 returns C{0.0}. 

1001 ''' 

1002 if Error and x < 0: 

1003 raise Error(unstr(sqrt0, x)) 

1004 return _root(x, _0_5, sqrt0) if x > EPS02 else ( 

1005 _0_0 if x < EPS02 else EPS0) 

1006 

1007 

1008def sqrt3(x): 

1009 '''Return the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)}, 

1010 preserving C{type(B{x})}. 

1011 

1012 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1013 

1014 @return: Square root I{cubed} (C{float} or L{Fsum}). 

1015 

1016 @raise TypeeError: Invalid B{C{x}}. 

1017 

1018 @raise ValueError: Negative B{C{x}}. 

1019 

1020 @see: Functions L{cbrt} and L{cbrt2}. 

1021 ''' 

1022 return _root(x, _1_5, sqrt3) 

1023 

1024 

1025def sqrt_a(h, b): 

1026 '''Compute C{I{a}} side of a right-angled triangle from 

1027 C{sqrt(B{h}**2 - B{b}**2)}. 

1028 

1029 @arg h: Hypotenuse or outer annulus radius (C{scalar}). 

1030 @arg b: Triangle side or inner annulus radius (C{scalar}). 

1031 

1032 @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}). 

1033 

1034 @raise TypeError: Non-scalar B{C{h}} or B{C{b}}. 

1035 

1036 @raise ValueError: If C{abs(B{h}) < abs(B{b})}. 

1037 

1038 @see: Inner tangent chord B{I{d}} of an U{annulus 

1039 <https://WikiPedia.org/wiki/Annulus_(mathematics)>} 

1040 and function U{annulus_area<https://People.SC.FSU.edu/ 

1041 ~jburkardt/py_src/geometry/geometry.py>}. 

1042 ''' 

1043 try: 

1044 if not (_isHeight(h) and _isRadius(b)): 

1045 raise TypeError(_not_scalar_) 

1046 c = fabs(h) 

1047 if c > EPS0: 

1048 s = _1_0 - (b / c)**2 

1049 if s < 0: 

1050 raise ValueError(_h_lt_b_) 

1051 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0) 

1052 else: # PYCHOK no cover 

1053 b = fabs(b) 

1054 d = c - b 

1055 if d < 0: 

1056 raise ValueError(_h_lt_b_) 

1057 d *= c + b 

1058 a = sqrt(d) if d else _0_0 

1059 except Exception as x: 

1060 raise _xError(x, h=h, b=b) 

1061 return copysign0(a, h) 

1062 

1063 

1064def zcrt(x): 

1065 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)}, 

1066 preserving C{type(B{x})}. 

1067 

1068 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1069 

1070 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}). 

1071 

1072 @see: Functions L{bqrt} and L{zqrt}. 

1073 

1074 @raise TypeeError: Invalid B{C{x}}. 

1075 

1076 @raise ValueError: Negative B{C{x}}. 

1077 ''' 

1078 return _root(x, _1_6th, zcrt) 

1079 

1080 

1081def zqrt(x): 

1082 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root, 

1083 M{x**(1 / 8)}, preserving C{type(B{x})}. 

1084 

1085 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1086 

1087 @return: I{Zenzi-quartic} root (C{float} or L{Fsum}). 

1088 

1089 @see: Functions L{bqrt} and L{zcrt}. 

1090 

1091 @raise TypeeError: Invalid B{C{x}}. 

1092 

1093 @raise ValueError: Negative B{C{x}}. 

1094 ''' 

1095 return _root(x, _0_125, zqrt) 

1096 

1097# **) MIT License 

1098# 

1099# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved. 

1100# 

1101# Permission is hereby granted, free of charge, to any person obtaining a 

1102# copy of this software and associated documentation files (the "Software"), 

1103# to deal in the Software without restriction, including without limitation 

1104# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1105# and/or sell copies of the Software, and to permit persons to whom the 

1106# Software is furnished to do so, subject to the following conditions: 

1107# 

1108# The above copyright notice and this permission notice shall be included 

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1110# 

1111# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

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