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__ = '25.04.30' 

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 Fdot_(Fdot): 

70 '''Precision dot product. 

71 ''' 

72 def __init__(self, *xys, **start_name_f2product_nonfinites_RESIDUAL): 

73 '''New L{Fdot_} precision dot product M{sum(xys[i] * xys[i+1] for i in 

74 range(0, len(xys), B{2}))}. 

75 

76 @arg xys: Pairwise values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), 

77 all positional. 

78 

79 @see: Class L{Fdot<Fdot.__init__>} for further details. 

80 ''' 

81 Fdot.__init__(self, xys[0::2], *xys[1::2], **start_name_f2product_nonfinites_RESIDUAL) 

82 

83 

84class Fhorner(Fsum): 

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

86 ''' 

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

88 '''New L{Fhorner} form evaluation of polynomial M{sum(cs[i] * x**i for i=0..n)} 

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

90 

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

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

93 all positional. 

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

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

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

97 

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

99 

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

101 

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

103 

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

105 ''' 

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

107 Fsum.__init__(self, **kwds) 

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

109 

110 

111class Fhypot(Fsum): 

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

113 ''' 

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

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

116 to the power I{root}). 

117 

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

119 positional. 

120 @kwarg root_name_f2product_nonfinites_RESIDUAL_raiser: Optional, exponent 

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

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

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

124 L{root<Fsum.root>}. 

125 ''' 

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

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

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

129 

130 r = None # _xkwds_pop2 error 

131 try: 

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

133 Fsum.__init__(self, **kwds) 

134 self(_0_0) 

135 if xs: 

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

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

138 except Exception as X: 

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

140 

141 

142class Fpolynomial(Fsum): 

143 '''Precision polynomial evaluation. 

144 ''' 

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

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

147 i=0..len(cs)-1)}. 

148 

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

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

151 all positional. 

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

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

154 

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

156 

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

158 

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

160 

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

162 ''' 

163 Fsum.__init__(self, **name_f2product_nonfinites_RESIDUAL) 

164 n = len(cs) - 1 

165 self(_0_0 if n < 0 else cs[0]) 

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

167 

168 

169class Fpowers(Fsum): 

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

171 ''' 

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

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

174 

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

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

177 positional. 

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

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

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

181 L{fpow<Fsum.fpow>}. 

182 ''' 

183 try: 

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

185 Fsum.__init__(self, **kwds) 

186 self(_0_0) 

187 if xs: 

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

189 except Exception as X: 

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

191 

192 

193class Froot(Fsum): 

194 '''The root of a precision summation. 

195 ''' 

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

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

198 

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

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

201 positional. 

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

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

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

205 L{fpow<Fsum.fpow>}. 

206 ''' 

207 try: 

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

209 Fsum.__init__(self, **kwds) 

210 self(_0_0) 

211 if xs: 

212 self.fadd(xs) 

213 self(self.root(root, raiser=X)) 

214 except Exception as X: 

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

216 

217 

218class Fcbrt(Froot): 

219 '''Cubic root of a precision summation. 

220 ''' 

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

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

223 

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

225 ''' 

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

227 

228 

229class Fsqrt(Froot): 

230 '''Square root of a precision summation. 

231 ''' 

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

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

234 

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

236 ''' 

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

238 

239 

240def bqrt(x): 

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

242 preserving C{type(B{x})}. 

243 

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

245 

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

247 

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

249 

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

251 

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

253 ''' 

254 return _root(x, _0_25, bqrt) 

255 

256 

257try: 

258 from math import cbrt as _cbrt # Python 3.11+ 

259 

260except ImportError: # Python 3.10- 

261 

262 def _cbrt(x): 

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

264 ''' 

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

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

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

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

269 

270 

271def cbrt(x): 

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

273 

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

275 

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

277 

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

279 ''' 

280 if _isFsum_2Tuple(x): 

281 r = abs(x).fpow(_1_3rd) 

282 if x.signOf() < 0: 

283 r = -r 

284 else: 

285 r = _cbrt(x) 

286 return r # cbrt(-0.0) == -0.0 

287 

288 

289def cbrt2(x): # PYCHOK attr 

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

291 

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

293 

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

295 

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

297 ''' 

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

299 

300 

301def euclid(x, y): 

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

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

304 

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

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

307 

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

309 

310 @see: Function L{euclid_}. 

311 ''' 

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

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

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

315 

316 

317def euclid_(*xs): 

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

319 L{euclid}. 

320 

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

322 all positional. 

323 

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

325 

326 @see: Function L{euclid}. 

327 ''' 

328 e = _0_0 

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

330 # e = euclid(x, e) 

331 if e < x: 

332 e, x = x, e 

333 if x: 

334 e += x * _0_4142 

335 return e 

336 

337 

338def facos1(x): 

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

340 

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

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

343 ''' 

344 a = fabs(x) 

345 if a < EPS0: 

346 r = PI_2 

347 elif a < EPS1: 

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

349 r *= sqrt(_1_0 - a) 

350 if x < 0: 

351 r = PI - r 

352 else: 

353 r = PI if x < 0 else _0_0 

354 return r 

355 

356 

357def fasin1(x): # PYCHOK no cover 

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

359 

360 @see: L{facos1}. 

361 ''' 

362 return PI_2 - facos1(x) 

363 

364 

365def _fast(x, *cs): 

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

367 ''' 

368 h = 0 

369 for c in reversed(cs): 

370 h = _fma(x, h, c) if h else c 

371 return h 

372 

373 

374def fatan(x): 

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

376 ''' 

377 a = fabs(x) 

378 if a < _1_0: 

379 r = fatan1(a) if a else _0_0 

380 elif a > _1_0: 

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

382 else: 

383 r = PI_4 

384 if x < 0: # copysign0(r, x) 

385 r = -r 

386 return r 

387 

388 

389def fatan1(x): 

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

391 

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

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

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

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

396 IEEE Signal Processing Magazine, 111, May 2006. 

397 ''' 

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

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

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

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

402 

403 

404def fatan2(y, x): 

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

406 

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

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

409 and L{fatan1}. 

410 ''' 

411 a, b = fabs(x), fabs(y) 

412 if b > a: 

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

414 elif a > b: 

415 r = fatan1(b / a) if b else _0_0 

416 elif a: # a == b != 0 

417 r = PI_4 

418 else: # a == b == 0 

419 return _0_0 

420 if x < 0: 

421 r = PI - r 

422 if y < 0: # copysign0(r, y) 

423 r = -r 

424 return r 

425 

426 

427def favg(a, b, f=_0_5, nonfinites=True): 

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

429 

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

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

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

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

434 

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

436 ''' 

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

438 return float(F) 

439 

440 

441def fdot(xs, *ys, **start_f2product_nonfinites): 

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

443 

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

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

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

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

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

449 

450 @return: Dot product (C{float}). 

451 

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

453 

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

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

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

457 ''' 

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

459 return float(D) 

460 

461 

462def fdot_(*xys, **start_f2product_nonfinites): 

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

464 

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

466 

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

468 

469 @return: Dot product (C{float}). 

470 ''' 

471 D = Fdot_(*xys, **_xkwds(start_f2product_nonfinites, nonfinites=True)) 

472 return float(D) 

473 

474 

475def fdot3(xs, ys, zs, **start_f2product_nonfinites): 

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

477 

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

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

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

481 

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

483 

484 @return: Dot product (C{float}). 

485 

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

487 ''' 

488 n = len(xs) 

489 if not n == len(ys) == len(zs): 

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

491 

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

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

494 _f = Fsum(**kwds) 

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

496 return float(D) 

497 

498 

499def fhorner(x, *cs, **incx): 

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

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

502 

503 @return: Horner sum (C{float}). 

504 

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

506 ''' 

507 H = Fhorner(x, *cs, **incx) 

508 return float(H) 

509 

510 

511def fidw(xs, ds, beta=2): 

512 '''Interpolate using U{Inverse Distance Weighting 

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

514 

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

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

517 L{Fsum2Tuple}). 

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

519 

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

521 

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

523 

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

525 

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

527 weighted B{C{ds}} below L{EPS}. 

528 

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

530 ''' 

531 n, xs = len2(xs) 

532 if n > 1: 

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

534 if b < 0: 

535 try: # weighted 

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

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

538 x = xs[i] 

539 D = _d(d) 

540 if D < EPS0: 

541 if D < 0: 

542 raise ValueError(_negative_) 

543 x = float(x) 

544 i = n 

545 break 

546 if D.fpow(b): 

547 W += D 

548 X += D.fmul(x) 

549 else: 

550 x = X.fover(W, raiser=False) 

551 i += 1 # len(xs) >= len(ds) 

552 except IndexError: 

553 i += 1 # len(xs) < i < len(ds) 

554 except Exception as X: 

555 _I = Fmt.INDEX 

556 raise _xError(X, _I(xs=i), x, 

557 _I(ds=i), d) 

558 else: # b == 0 

559 x = fsum(xs) / n # fmean(xs) 

560 i = n 

561 elif n: 

562 x = float(xs[0]) 

563 i = n 

564 else: 

565 x = _0_0 

566 i, _ = len2(ds) 

567 if i != n: 

568 raise LenError(fidw, xs=n, ds=i) 

569 return x 

570 

571 

572try: 

573 from math import fma as _fma # in .resections 

574except ImportError: # PYCHOK DSPACE! 

575 

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

577 return x * y + z 

578 

579 

580def fma(x, y, z, **nonfinites): # **raiser 

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

582 or an equivalent implementation. 

583 

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

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

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

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

588 to override default L{nonfiniterrors} 

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

590 

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

592 ''' 

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

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

595 

596 

597def _Fm2(x, nonfinites=None, **raiser): 

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

599 ''' 

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

601 

602 

603def fmean(xs): 

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

605 

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

607 

608 @return: Mean value (C{float}). 

609 

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

611 

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

613 ''' 

614 n, xs = len2(xs) 

615 if n < 1: 

616 raise LenError(fmean, xs=xs) 

617 M = Fsum(*xs, nonfinites=True) 

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

619 

620 

621def fmean_(*xs, **nonfinites): 

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

623 

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

625 ''' 

626 return fmean(xs, **nonfinites) 

627 

628 

629def f2mul_(x, *ys, **nonfinites): # **raiser 

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

631 

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

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

634 positional. 

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

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

637 

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

639 

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

641 ''' 

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

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

644 

645 

646def fpolynomial(x, *cs, **over_f2product_nonfinites): 

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

648 

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

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

651 L{Fpolynomial<Fpolynomial.__init__>}. 

652 

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

654 ''' 

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

656 P = Fpolynomial(x, *cs, **kwds) 

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

658 

659 

660def fpowers(x, n, alts=0): 

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

662 

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

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

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

666 exponent (C{int}). 

667 

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

669 

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

671 

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

673 ''' 

674 if not isint(n): 

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

676 elif n < 1: 

677 raise _ValueError(n=n) 

678 

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

680 ps = tuple(_powers(p, n)) 

681 

682 if alts > 0: # x**2, x**4, ... 

683 # ps[alts-1::2] chokes PyChecker 

684 ps = ps[slice(alts-1, None, 2)] 

685 

686 return ps 

687 

688 

689try: 

690 from math import prod as fprod # Python 3.8 

691except ImportError: 

692 

693 def fprod(xs, start=1): 

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

695 

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

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

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

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

700 

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

702 

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

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

705 ''' 

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

707 

708 

709def frandoms(n, seeded=None): 

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

711 

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

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

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

715 

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

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

718 ''' 

719 from random import gauss, random, seed, shuffle 

720 

721 if seeded is None: 

722 pass 

723 elif seeded and isbool(seeded): 

724 from time import localtime 

725 seed(localtime().tm_yday) 

726 elif isscalar(seeded): 

727 seed(seeded) 

728 

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

730 for _ in range(n): 

731 s = 0 

732 t = list(c) 

733 _a = t.append 

734 for _ in range(n * 8): 

735 v = gauss(0, random())**7 - s 

736 _a(v) 

737 s += v 

738 shuffle(t) 

739 yield t 

740 

741 

742def frange(start, number, step=1): 

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

744 

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

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

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

748 

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

750 

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

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

753 ''' 

754 if not isint(number): 

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

756 for i in range(number): 

757 yield start + (step * i) 

758 

759 

760try: 

761 from functools import reduce as freduce 

762except ImportError: 

763 try: 

764 freduce = reduce # PYCHOK expected 

765 except NameError: # Python 3+ 

766 

767 def freduce(f, xs, *start): 

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

769 ''' 

770 if start: 

771 r = v = start[0] 

772 else: 

773 r, v = 0, MISSING 

774 for v in xs: 

775 r = f(r, v) 

776 if v is MISSING: 

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

778 return r 

779 

780 

781def fremainder(x, y): 

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

783 

784 @arg x: Numerator (C{scalar}). 

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

786 

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

788 0.0) or C{NAN} for any non-finite B{C{x}}. 

789 

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

791 

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

793 project/geographiclib/>} and Python 3.7+ 

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

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

796 ''' 

797 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and 

798 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native 

799 # fmod( 0, 360) == 0.0 

800 # fmod( 360, 360) == 0.0 

801 # fmod(-0, 360) == 0.0 

802 # fmod(-0.0, 360) == -0.0 

803 # fmod(-360, 360) == -0.0 

804 # however, using the % operator ... 

805 # 0 % 360 == 0 

806 # 360 % 360 == 0 

807 # 360.0 % 360 == 0.0 

808 # -0 % 360 == 0 

809 # -360 % 360 == 0 == (-360) % 360 

810 # -0.0 % 360 == 0.0 == (-0.0) % 360 

811 # -360.0 % 360 == 0.0 == (-360.0) % 360 

812 

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

814 # == +0.0. This fixes this bug. See also Math::AngNormalize 

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

816 if isfinite(x): 

817 try: 

818 r = remainder(x, y) if x else x 

819 except Exception as e: 

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

821 else: # handle x INF and NINF as NAN 

822 r = NAN 

823 return r 

824 

825 

826if _MODS.sys_version_info2 < (3, 8): # PYCHOK no cover 

827 from math import hypot # OK in Python 3.7- 

828 

829 def hypot_(*xs): 

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

831 

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

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

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

835 handled differently. 

836 

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

838 

839 @return: Norm (C{float}). 

840 

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

842 

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

844 

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

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

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

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

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

850 ''' 

851 return float(_Hypot(*xs)) 

852 

853elif _MODS.sys_version_info2 < (3, 10): # PYCHOK no cover 

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

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

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

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

858 

859 def hypot(x, y): 

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

861 

862 @arg x: X argument (C{scalar}). 

863 @arg y: Y argument (C{scalar}). 

864 

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

866 ''' 

867 return float(_Hypot(x, y)) 

868 

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

870else: 

871 from math import hypot # PYCHOK in Python 3.10+ 

872 hypot_ = hypot 

873 

874 

875def _Hypot(*xs): 

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

877 ''' 

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

879 

880 

881def hypot1(x): 

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

883 

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

885 

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

887 ''' 

888 h = _1_0 

889 if x: 

890 if _isFsum_2Tuple(x): 

891 h = _Hypot(h, x) 

892 h = float(h) 

893 else: 

894 h = hypot(h, x) 

895 return h 

896 

897 

898def hypot2(x, y): 

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

900 

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

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

903 

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

905 ''' 

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

907 if y > x: 

908 x, y = y, x 

909 h2 = x**2 

910 if h2 and y: 

911 h2 *= (y / x)**2 + _1_0 

912 return float(h2) 

913 

914 

915def hypot2_(*xs): 

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

917 

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

919 L{Fsum2Tuple}), all positional. 

920 

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

922 

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

924 ''' 

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

926 if h2: # and isfinite(h2) 

927 _h = _1_0 / h2 

928 xs = ((x * _h) for x in xs) 

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

930 h2 = H2.fover(_h**2) 

931 return h2 

932 

933 

934def norm2(x, y): 

935 '''Normalize a 2-dimensional vector. 

936 

937 @arg x: X component (C{scalar}). 

938 @arg y: Y component (C{scalar}). 

939 

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

941 

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

943 or zero norm. 

944 ''' 

945 try: 

946 h = None 

947 h = hypot(x, y) 

948 if h: 

949 x, y = (x / h), (y / h) 

950 else: 

951 x = _copysign_0_0(x) # pass? 

952 y = _copysign_0_0(y) 

953 except Exception as e: 

954 raise _xError(e, x=x, y=y, h=h) 

955 return x, y 

956 

957 

958def norm_(*xs): 

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

960 

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

962 L{Fsum2Tuple}), all positional. 

963 

964 @return: Yield each component, normalized. 

965 

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

967 or zero norm. 

968 ''' 

969 try: 

970 i = h = None 

971 x = xs 

972 h = hypot_(*xs) 

973 _h = (_1_0 / h) if h else _0_0 

974 for i, x in enumerate(xs): 

975 yield x * _h 

976 except Exception as X: 

977 raise _xsError(X, xs, i, x, h=h) 

978 

979 

980def _powers(x, n): 

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

982 ''' 

983 p = 1 # type(p) == type(x) 

984 for _ in range(n): 

985 p *= x 

986 yield p 

987 

988 

989def _root(x, p, where): 

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

991 ''' 

992 try: 

993 if x > 0: 

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

995 return r.fpow(p).as_iscalar 

996 elif x < 0: 

997 raise ValueError(_negative_) 

998 except Exception as X: 

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

1000 return _0_0 if p else _1_0 

1001 

1002 

1003def sqrt0(x, Error=None): 

1004 '''Return the square root C{sqrt(B{x})} iff C{B{x} > }L{EPS02}, 

1005 preserving C{type(B{x})}. 

1006 

1007 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1008 @kwarg Error: Error to raise for negative B{C{x}}. 

1009 

1010 @return: Square root (C{float} or L{Fsum}) or C{0.0}. 

1011 

1012 @raise TypeeError: Invalid B{C{x}}. 

1013 

1014 @note: Any C{B{x} < }L{EPS02} I{including} C{B{x} < 0} 

1015 returns C{0.0}. 

1016 ''' 

1017 if Error and x < 0: 

1018 raise Error(unstr(sqrt0, x)) 

1019 return _root(x, _0_5, sqrt0) if x > EPS02 else ( 

1020 _0_0 if x < EPS02 else EPS0) 

1021 

1022 

1023def sqrt3(x): 

1024 '''Return the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)}, 

1025 preserving C{type(B{x})}. 

1026 

1027 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1028 

1029 @return: Square root I{cubed} (C{float} or L{Fsum}). 

1030 

1031 @raise TypeeError: Invalid B{C{x}}. 

1032 

1033 @raise ValueError: Negative B{C{x}}. 

1034 

1035 @see: Functions L{cbrt} and L{cbrt2}. 

1036 ''' 

1037 return _root(x, _1_5, sqrt3) 

1038 

1039 

1040def sqrt_a(h, b): 

1041 '''Compute C{I{a}} side of a right-angled triangle from 

1042 C{sqrt(B{h}**2 - B{b}**2)}. 

1043 

1044 @arg h: Hypotenuse or outer annulus radius (C{scalar}). 

1045 @arg b: Triangle side or inner annulus radius (C{scalar}). 

1046 

1047 @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}). 

1048 

1049 @raise TypeError: Non-scalar B{C{h}} or B{C{b}}. 

1050 

1051 @raise ValueError: If C{abs(B{h}) < abs(B{b})}. 

1052 

1053 @see: Inner tangent chord B{I{d}} of an U{annulus 

1054 <https://WikiPedia.org/wiki/Annulus_(mathematics)>} 

1055 and function U{annulus_area<https://People.SC.FSU.edu/ 

1056 ~jburkardt/py_src/geometry/geometry.py>}. 

1057 ''' 

1058 try: 

1059 if not (_isHeight(h) and _isRadius(b)): 

1060 raise TypeError(_not_scalar_) 

1061 c = fabs(h) 

1062 if c > EPS0: 

1063 s = _1_0 - (b / c)**2 

1064 if s < 0: 

1065 raise ValueError(_h_lt_b_) 

1066 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0) 

1067 else: # PYCHOK no cover 

1068 b = fabs(b) 

1069 d = c - b 

1070 if d < 0: 

1071 raise ValueError(_h_lt_b_) 

1072 d *= c + b 

1073 a = sqrt(d) if d else _0_0 

1074 except Exception as x: 

1075 raise _xError(x, h=h, b=b) 

1076 return copysign0(a, h) 

1077 

1078 

1079def zcrt(x): 

1080 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)}, 

1081 preserving C{type(B{x})}. 

1082 

1083 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1084 

1085 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}). 

1086 

1087 @see: Functions L{bqrt} and L{zqrt}. 

1088 

1089 @raise TypeeError: Invalid B{C{x}}. 

1090 

1091 @raise ValueError: Negative B{C{x}}. 

1092 ''' 

1093 return _root(x, _1_6th, zcrt) 

1094 

1095 

1096def zqrt(x): 

1097 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root, 

1098 M{x**(1 / 8)}, preserving C{type(B{x})}. 

1099 

1100 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1101 

1102 @return: I{Zenzi-quartic} root (C{float} or L{Fsum}). 

1103 

1104 @see: Functions L{bqrt} and L{zcrt}. 

1105 

1106 @raise TypeeError: Invalid B{C{x}}. 

1107 

1108 @raise ValueError: Negative B{C{x}}. 

1109 ''' 

1110 return _root(x, _0_125, zqrt) 

1111 

1112# **) MIT License 

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1114# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved. 

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