Coverage for pygeodesy/fmath.py: 91%

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 _ # noqa: E702 ; 

9 

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

11 isscalar, len2, map1, _xiterable, typename 

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

13 _0_0, _0_125, _0_25, _1_3rd, _0_5, _2_3rd, \ 

14 _1_0, _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 

23 

24from math import fabs, sqrt # pow 

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

26 

27__all__ = _ALL_LAZY.fmath 

28__version__ = '26.01.06' 

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_1_6th = 1 / 6 

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{start + 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): # in .elliptic 

70 '''Precision dot product. 

71 ''' 

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

73 '''New L{Fdot_} precision dot product M{start + 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 if isodd(len(xys)): 

82 raise LenError(Fdot_, xys=len(xys)) 

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

84 

85 

86class Fhorner(Fsum): 

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

88 ''' 

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

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

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

92 

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

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

95 all positional. 

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

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

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

99 

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

101 

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

103 

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

105 

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

107 ''' 

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

109 Fsum.__init__(self, **kwds) 

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

111 

112 

113class Fhypot(Fsum): 

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

115 ''' 

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

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

118 to the power I{root}). 

119 

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

121 positional. 

122 @kwarg root_name_f2product_nonfinites_RESIDUAL_raiser: Optional, exponent 

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

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

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

126 L{root<Fsum.root>}. 

127 ''' 

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

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

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

131 

132 r = None # _xkwds_pop2 error 

133 try: 

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

135 Fsum.__init__(self, **kwds) 

136 self(_0_0) 

137 if xs: 

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

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

140 except Exception as X: 

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

142 

143 

144class Fpolynomial(Fsum): 

145 '''Precision polynomial evaluation. 

146 ''' 

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

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

149 i=0..len(cs)-1)}. 

150 

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

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

153 all positional. 

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

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

156 

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

158 

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

160 

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

162 

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

164 ''' 

165 Fsum.__init__(self, **name_f2product_nonfinites_RESIDUAL) 

166 n = len(cs) - 1 

167 self(_0_0 if n < 0 else cs[0]) 

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

169 

170 

171class Fpowers(Fsum): 

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

173 ''' 

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

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

176 

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

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

179 positional. 

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

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

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

183 L{fpow<Fsum.fpow>}. 

184 ''' 

185 try: 

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

187 Fsum.__init__(self, **kwds) 

188 self(_0_0) 

189 if xs: 

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

191 except Exception as X: 

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

193 

194 

195class Froot(Fsum): 

196 '''The root of a precision summation. 

197 ''' 

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

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

200 

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

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

203 positional. 

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

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

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

207 L{fpow<Fsum.fpow>}. 

208 ''' 

209 try: 

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

211 Fsum.__init__(self, **kwds) 

212 self(_0_0) 

213 if xs: 

214 self.fadd(xs) 

215 self(self.root(root, raiser=X)) 

216 except Exception as X: 

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

218 

219 

220class Fcbrt(Froot): 

221 '''Cubic root of a precision summation. 

222 ''' 

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

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

225 

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

227 ''' 

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

229 

230 

231class Fsqrt(Froot): 

232 '''Square root of a precision summation. 

233 ''' 

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

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

236 

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

238 ''' 

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

240 

241 

242def bqrt(x): 

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

244 preserving C{type(B{x})}. 

245 

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

247 

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

249 

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

251 

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

253 

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

255 ''' 

256 return _root(x, _0_25, bqrt) 

257 

258 

259try: 

260 from math import cbrt as _cbrt # Python 3.11+ 

261except ImportError: # Python 3.10- 

262 

263 def _cbrt(x): 

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

265 ''' 

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

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

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

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

270 

271 

272def cbrt(x): 

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

274 

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

276 

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

278 

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

280 ''' 

281 if _isFsum_2Tuple(x): 

282 r = abs(x).fpow(_1_3rd) 

283 if x.signOf() < 0: 

284 r = -r 

285 else: 

286 r = _cbrt(x) 

287 return r # cbrt(-0.0) == -0.0 

288 

289 

290def cbrt2(x): # PYCHOK attr 

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

292 

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

294 

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

296 

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

298 ''' 

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

300 

301 

302def euclid(x, y): 

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

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

305 

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

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

308 

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

310 

311 @see: Function L{euclid_}. 

312 ''' 

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

314 if x < y: 

315 x, y = y, x 

316 return x + y * _0_4142 # * _0_5 before 20.10.02 

317 

318 

319def euclid_(*xs): 

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

321 L{euclid}. 

322 

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

324 all positional. 

325 

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

327 

328 @see: Function L{euclid}. 

329 ''' 

330 e = _0_0 

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

332 # e = euclid(x, e) 

333 if x: 

334 if e < x: 

335 e, x = x, e 

336 x *= _0_4142 

337# s = e + x 

338# if e < x: # like .fsums._2sum 

339# x -= s # e = (x - s) + e 

340# else: 

341# e -= s # e = (e - s) + x 

342 e += x 

343# e += s 

344 return e 

345 

346 

347def facos1(x): 

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

349 

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

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

352 ''' 

353 a = fabs(x) 

354 if a < EPS0: 

355 r = PI_2 

356 elif a < EPS1: 

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

358 r *= sqrt(_1_0 - a) 

359 if x < 0: 

360 r = PI - r 

361 else: 

362 r = PI if x < 0 else _0_0 

363 return r 

364 

365 

366def fasin1(x): # PYCHOK no cover 

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

368 

369 @see: L{facos1}. 

370 ''' 

371 return PI_2 - facos1(x) 

372 

373 

374def _fast(x, *cs): 

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

376 ''' 

377 h = 0 

378 for c in reversed(cs): 

379 h = _fma(x, h, c) if h else c 

380 return h 

381 

382 

383def fatan(x): 

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

385 ''' 

386 a = fabs(x) 

387 if a < _1_0: 

388 r = fatan1(a) if a else _0_0 

389 elif a > _1_0: 

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

391 else: 

392 r = PI_4 

393 if x < 0: # copysign0(r, x) 

394 r = -r 

395 return r 

396 

397 

398def fatan1(x): 

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

400 

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

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

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

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

405 IEEE Signal Processing Magazine, 111, May 2006. 

406 ''' 

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

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

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

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

411 

412 

413def fatan2(y, x): 

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

415 

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

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

418 and L{fatan1}. 

419 ''' 

420 a, b = fabs(x), fabs(y) 

421 if b > a: 

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

423 elif a > b: 

424 r = fatan1(b / a) if b else _0_0 

425 elif a: # a == b != 0 

426 r = PI_4 

427 else: # a == b == 0 

428 return _0_0 

429 if x < 0: 

430 r = PI - r 

431 if y < 0: # copysign0(r, y) 

432 r = -r 

433 return r 

434 

435 

436def favg(a, b, f=_0_5, nonfinites=True): 

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

438 

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

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

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

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

443 

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

445 ''' 

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

447 return float(F) 

448 

449 

450def fdot(xs, *ys, **start_f2product_nonfinites): 

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

452 

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

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

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

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

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

458 

459 @return: Dot product (C{float}). 

460 

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

462 

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

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

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

466 ''' 

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

468 return float(D) 

469 

470 

471def fdot_(*xys, **start_f2product_nonfinites): 

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

473 

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

475 

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

477 

478 @return: Dot product (C{float}). 

479 ''' 

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

481 return float(D) 

482 

483 

484def fdot3(xs, ys, zs, **start_f2product_nonfinites): 

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

486 

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

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

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

490 

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

492 

493 @return: Dot product (C{float}). 

494 

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

496 ''' 

497 n = len(xs) 

498 if not n == len(ys) == len(zs): 

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

500 

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

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

503 _f = Fsum(**kwds) 

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

505 return float(D) 

506 

507 

508def fhorner(x, *cs, **incx): 

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

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

511 

512 @return: Horner sum (C{float}). 

513 

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

515 ''' 

516 H = Fhorner(x, *cs, **incx) 

517 return float(H) 

518 

519 

520def fidw(xs, ds, beta=2): 

521 '''Interpolate using U{Inverse Distance Weighting 

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

523 

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

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

526 L{Fsum2Tuple}). 

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

528 

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

530 

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

532 

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

534 

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

536 weighted B{C{ds}} below L{EPS}. 

537 

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

539 ''' 

540 n, xs = len2(xs) 

541 if n > 1: 

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

543 if b < 0: 

544 try: # weighted 

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

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

547 x = xs[i] 

548 D = _d(d) 

549 if D < EPS0: 

550 if D < 0: 

551 raise ValueError(_negative_) 

552 x = float(x) 

553 i = n 

554 break 

555 if D.fpow(b): 

556 W += D 

557 X += D.fmul(x) 

558 else: 

559 x = X.fover(W, raiser=False) 

560 i += 1 # len(xs) >= len(ds) 

561 except IndexError: 

562 i += 1 # len(xs) < i < len(ds) 

563 except Exception as X: 

564 _I = Fmt.INDEX 

565 raise _xError(X, _I(xs=i), x, 

566 _I(ds=i), d) 

567 else: # b == 0 

568 x = fsum(xs) / n # fmean(xs) 

569 i = n 

570 elif n: 

571 x = float(xs[0]) 

572 i = n 

573 else: 

574 x = _0_0 

575 i, _ = len2(ds) 

576 if i != n: 

577 raise LenError(fidw, xs=n, ds=i) 

578 return x 

579 

580 

581try: 

582 from math import fma as _fma # in .resections 

583except ImportError: # PYCHOK DSPACE! 

584 

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

586 return x * y + z 

587 

588 

589def fma(x, y, z, **nonfinites): # **raiser 

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

591 or an equivalent implementation. 

592 

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

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

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

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

597 to override default L{nonfiniterrors} 

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

599 

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

601 ''' 

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

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

604 

605 

606def _Fm2(x, nonfinites=None, **raiser): 

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

608 ''' 

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

610 

611 

612def fmean(xs): 

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

614 

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

616 

617 @return: Mean value (C{float}). 

618 

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

620 

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

622 ''' 

623 n, xs = len2(xs) 

624 if n < 1: 

625 raise LenError(fmean, xs=xs) 

626 M = Fsum(*xs, nonfinites=True) 

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

628 

629 

630def fmean_(*xs, **nonfinites): 

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

632 

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

634 ''' 

635 return fmean(xs, **nonfinites) 

636 

637 

638def f2mul_(x, *ys, **nonfinites): # **raiser 

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

640 

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

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

643 positional. 

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

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

646 

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

648 

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

650 ''' 

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

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

653 

654 

655def fpolynomial(x, *cs, **over_f2product_nonfinites): 

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

657 

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

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

660 L{Fpolynomial<Fpolynomial.__init__>}. 

661 

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

663 ''' 

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

665 P = Fpolynomial(x, *cs, **kwds) 

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

667 

668 

669def fpowers(x, n, alts=0): 

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

671 

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

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

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

675 exponent (C{int}). 

676 

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

678 

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

680 

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

682 ''' 

683 if not isint(n): 

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

685 elif n < 1: 

686 raise _ValueError(n=n) 

687 

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

689 ps = tuple(_powers(p, n)) 

690 

691 if alts > 0: # x**2, x**4, ... 

692 # ps[alts-1::2] chokes PyChecker 

693 ps = ps[slice(alts-1, None, 2)] 

694 

695 return ps 

696 

697 

698try: 

699 from math import prod as fprod # Python 3.8 

700except ImportError: 

701 

702 def fprod(xs, start=1): 

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

704 

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

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

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

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

709 

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

711 

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

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

714 ''' 

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

716 

717 

718def frandoms(n, seeded=None): 

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

720 

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

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

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

724 

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

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

727 ''' 

728 from random import gauss, random, seed, shuffle 

729 

730 if seeded is None: 

731 pass 

732 elif seeded and isbool(seeded): 

733 from time import localtime 

734 seed(localtime().tm_yday) 

735 elif isscalar(seeded): 

736 seed(seeded) 

737 

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

739 for _ in range(n): 

740 s = 0 

741 t = list(c) 

742 _a = t.append 

743 for _ in range(n * 8): 

744 v = gauss(0, random())**7 - s 

745 _a(v) 

746 s += v 

747 shuffle(t) 

748 yield t 

749 

750 

751def frange(start, number, step=1): 

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

753 

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

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

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

757 

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

759 

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

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

762 ''' 

763 if not isint(number): 

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

765 for i in range(number): 

766 yield start + (step * i) 

767 

768 

769try: 

770 from functools import reduce as freduce 

771except ImportError: 

772 try: 

773 freduce = reduce # PYCHOK expected 

774 except NameError: # Python 3+ 

775 

776 def freduce(f, xs, *start): 

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

778 ''' 

779 if start: 

780 r = v = start[0] 

781 else: 

782 r, v = 0, MISSING 

783 for v in xs: 

784 r = f(r, v) 

785 if v is MISSING: 

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

787 return r 

788 

789 

790def fremainder(x, y): 

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

792 

793 @arg x: Numerator (C{scalar}). 

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

795 

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

797 0.0) or C{NAN} for any non-finite B{C{x}}. 

798 

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

800 

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

802 project/geographiclib/>} and Python 3.7+ 

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

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

805 ''' 

806 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and 

807 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native 

808 # fmod( 0, 360) == 0.0 

809 # fmod( 360, 360) == 0.0 

810 # fmod(-0, 360) == 0.0 

811 # fmod(-0.0, 360) == -0.0 

812 # fmod(-360, 360) == -0.0 

813 # however, using the % operator ... 

814 # 0 % 360 == 0 

815 # 360 % 360 == 0 

816 # 360.0 % 360 == 0.0 

817 # -0 % 360 == 0 

818 # -360 % 360 == 0 == (-360) % 360 

819 # -0.0 % 360 == 0.0 == (-0.0) % 360 

820 # -360.0 % 360 == 0.0 == (-360.0) % 360 

821 

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

823 # == +0.0. This fixes this bug. See also Math::AngNormalize 

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

825 if isfinite(x): 

826 try: 

827 r = remainder(x, y) if x else x 

828 except Exception as e: 

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

830 else: # handle x INF and NINF as NAN 

831 r = NAN 

832 return r 

833 

834 

835if _MODS.sys_version_info2 < (3, 8): # PYCHOK no cover 

836 from math import hypot # OK in Python 3.7- 

837 

838 def hypot_(*xs): 

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

840 

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

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

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

844 handled differently. 

845 

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

847 

848 @return: Norm (C{float}). 

849 

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

851 

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

853 

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

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

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

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

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

859 ''' 

860 return float(_Hypot(*xs)) 

861 

862elif _MODS.sys_version_info2 < (3, 10): # PYCHOK no cover 

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

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

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

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

867 

868 def hypot(x, y): 

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

870 

871 @arg x: X argument (C{scalar}). 

872 @arg y: Y argument (C{scalar}). 

873 

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

875 ''' 

876 return float(_Hypot(x, y)) 

877 

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

879else: 

880 from math import hypot # PYCHOK in Python 3.10+ 

881 hypot_ = hypot 

882 

883 

884def _Hypot(*xs): 

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

886 ''' 

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

888 

889 

890def hypot1(x): 

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

892 

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

894 

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

896 ''' 

897 h = _1_0 

898 if x: 

899 if _isFsum_2Tuple(x): 

900 h = _Hypot(h, x) 

901 h = float(h) 

902 else: 

903 h = hypot(h, x) 

904 return h 

905 

906 

907def hypot2(x, y): 

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

909 

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

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

912 

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

914 ''' 

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

916 if y > x: 

917 x, y = y, x 

918 h2 = x**2 

919 if h2 and y: 

920 h2 *= (y / x)**2 + _1_0 

921 return float(h2) 

922 

923 

924def hypot2_(*xs): 

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

926 

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

928 L{Fsum2Tuple}), all positional. 

929 

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

931 

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

933 ''' 

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

935 if h2: # and isfinite(h2) 

936 _h = _1_0 / h2 

937 xs = ((x * _h) for x in xs) 

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

939 h2 = H2.fover(_h**2) 

940 return h2 

941 

942 

943def norm2(x, y): 

944 '''Normalize a 2-dimensional vector. 

945 

946 @arg x: X component (C{scalar}). 

947 @arg y: Y component (C{scalar}). 

948 

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

950 

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

952 or zero norm. 

953 ''' 

954 try: