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, _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# from utily import atan2b, atan2p # _MODS, circular import! 

24 

25from math import fabs, sqrt # pow 

26import operator as _operator # in .datums, .elliptic, .trf, .utm 

27 

28__all__ = _ALL_LAZY.fmath 

29__version__ = '26.03.25' 

30 

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

32_0_4142 = 0.41421356237309504880 # ~ 3_730_904_090_310_553 / 9_007_199_254_740_992 

33_1_6th = 1 / 6 

34_h_lt_b_ = 'abs(h) < abs(b)' 

35 

36 

37class Fdot(Fsum): 

38 '''Precision dot product. 

39 ''' 

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

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

42 

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

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

45 positional. 

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

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

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

49 

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

51 

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

53 

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

55 

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

57 

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

59 ''' 

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

61 Fsum.__init__(self, **kwds) 

62 self(s) 

63 

64 n = len(b) 

65 if len(a) != n: # PYCHOK no cover 

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

67 self._facc_dot(n, a, b, **kwds) 

68 

69 

70class Fdot_(Fdot): # in .elliptic 

71 '''Precision dot product. 

72 ''' 

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

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

75 range(0, len(xys), B{2}))}. 

76 

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

78 all positional. 

79 

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

81 ''' 

82 if isodd(len(xys)): 

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

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

85 

86 

87class Fhorner(Fsum): 

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

89 ''' 

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

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

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

93 

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

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

96 all positional. 

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

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

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

100 

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

102 

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

104 

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

106 

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

108 ''' 

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

110 Fsum.__init__(self, **kwds) 

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

112 

113 

114class Fhypot(Fsum): 

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

116 ''' 

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

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

119 to the power I{root}). 

120 

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

122 positional. 

123 @kwarg root_name_f2product_nonfinites_RESIDUAL_raiser: Optional, exponent 

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

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

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

127 L{root<Fsum.root>}. 

128 ''' 

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

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

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

132 

133 r = None # _xkwds_pop2 error 

134 try: 

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

136 Fsum.__init__(self, **kwds) 

137 self(_0_0) 

138 if xs: 

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

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

141 except Exception as X: 

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

143 

144 

145class Fpolynomial(Fsum): 

146 '''Precision polynomial evaluation. 

147 ''' 

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

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

150 i=0..len(cs)-1)}. 

151 

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

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

154 all positional. 

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

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

157 

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

159 

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

161 

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

163 

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

165 ''' 

166 Fsum.__init__(self, **name_f2product_nonfinites_RESIDUAL) 

167 n = len(cs) - 1 

168 self(_0_0 if n < 0 else cs[0]) 

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

170 

171 

172class Fpowers(Fsum): 

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

174 ''' 

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

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

177 

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

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

180 positional. 

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

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

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

184 L{fpow<Fsum.fpow>}. 

185 ''' 

186 try: 

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

188 Fsum.__init__(self, **kwds) 

189 self(_0_0) 

190 if xs: 

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

192 except Exception as X: 

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

194 

195 

196class Froot(Fsum): 

197 '''The root of a precision summation. 

198 ''' 

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

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

201 

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

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

204 positional. 

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

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

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

208 L{fpow<Fsum.fpow>}. 

209 ''' 

210 try: 

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

212 Fsum.__init__(self, **kwds) 

213 self(_0_0) 

214 if xs: 

215 self.fadd(xs) 

216 self(self.root(root, raiser=X)) 

217 except Exception as X: 

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

219 

220 

221class Fcbrt(Froot): 

222 '''Cubic root of a precision summation. 

223 ''' 

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

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

226 

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

228 ''' 

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

230 

231 

232class Fsqrt(Froot): 

233 '''Square root of a precision summation. 

234 ''' 

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

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

237 

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

239 ''' 

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

241 

242 

243def bqrt(x): 

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

245 preserving C{type(B{x})}. 

246 

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

248 

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

250 

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

252 

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

254 

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

256 ''' 

257 return _root(x, _0_25, bqrt) 

258 

259 

260try: 

261 from math import cbrt as _cbrt # Python 3.11+ 

262except ImportError: # Python 3.10- 

263 

264 def _cbrt(x): 

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

266 ''' 

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

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

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

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

271 

272 

273def cbrt(x): 

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

275 

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

277 

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

279 

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

281 ''' 

282 if _isFsum_2Tuple(x): 

283 r = abs(x).fpow(_1_3rd) 

284 if x.signOf() < 0: 

285 r = -r 

286 else: 

287 r = _cbrt(x) 

288 return r # cbrt(-0.0) == -0.0 

289 

290 

291def cbrt2(x): # PYCHOK attr 

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

293 

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

295 

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

297 

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

299 ''' 

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

301 

302 

303def euclid(x, y, *xy0): 

304 '''I{Appoximate} the norm M{hypot(B{x}, B{y})} by M{max(abs(B{x}), 

305 abs(B{y})) + min(abs(B{x}), abs(B{y})) * 0.4142...}. 

306 

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

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

309 @arg xy0: Optional reference C{(x0, y0)} (each C{scalar}, an 

310 L{Fsum} or L{Fsum2Tuple}). 

311 

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

313 ''' 

314 x, y = _map0(abs, x, y, *xy0) # NOT fabs! 

315 if x < y: 

316 x, y = y, x 

317 return x + y * _0_4142 # _0_5 before 20.10.02 

318 

319 

320def euclid_(*xs): 

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

322 L{euclid}. 

323 

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

325 all positional. 

326 

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

328 

329 @see: Function L{euclid}. 

330 ''' 

331 e = _0_0 

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

333 # e = euclid(x, e) 

334 if x: 

335 if e < x: 

336 e, x = x, e 

337 x *= _0_4142 

338# s = e + x 

339# if e < x: # like .fsums._2sum 

340# x -= s # e = (x - s) + e 

341# else: 

342# e -= s # e = (e - s) + x 

343 e += x 

344# e += s 

345 return e 

346 

347 

348def facos1(x): 

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

350 

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

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

353 ''' 

354 a = fabs(x) 

355 if a < EPS0: 

356 r = PI_2 

357 elif a < EPS1: 

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

359 r *= sqrt(_1_0 - a) 

360 if x < 0: 

361 r = PI - r 

362 else: 

363 r = PI if x < 0 else _0_0 

364 return r 

365 

366 

367def fasin1(x): # PYCHOK no cover 

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

369 

370 @see: L{facos1}. 

371 ''' 

372 return PI_2 - facos1(x) 

373 

374 

375def _fast(x, *cs): 

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

377 ''' 

378 h = 0 

379 for c in reversed(cs): 

380 h = _fma(x, h, c) if h else c 

381 return h 

382 

383 

384def fatan(x): 

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

386 ''' 

387 a = fabs(x) 

388 if a < _1_0: 

389 r = fatan1(a) if a else _0_0 

390 elif a > _1_0: 

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

392 else: 

393 r = PI_4 

394 if x < 0: # copysign0(r, x) 

395 r = -r 

396 return r 

397 

398 

399def fatan1(x): 

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

401 

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

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

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

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

406 IEEE Signal Processing Magazine, 111, May 2006. 

407 ''' 

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

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

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

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

412 

413 

414def fatan2(y, x): 

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

416 

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

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

419 and L{fatan1}. 

420 ''' 

421 a, b = fabs(x), fabs(y) 

422 if b > a: 

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

424 elif a > b: 

425 r = fatan1(b / a) if b else _0_0 

426 elif a: # a == b != 0 

427 r = PI_4 

428 else: # a == b == 0 

429 return _0_0 

430 if x < 0: 

431 r = PI - r 

432 if y < 0: # copysign0(r, y) 

433 r = -r 

434 return r 

435 

436 

437def favg(a, b, f=_0_5, nonfinites=True): 

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

439 

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

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

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

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

444 

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

446 ''' 

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

448 return float(F) 

449 

450 

451def fdot(xs, *ys, **start_f2product_nonfinites): 

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

453 

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

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

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

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

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

459 

460 @return: Dot product (C{float}). 

461 

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

463 

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

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

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

467 ''' 

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

469 return float(D) 

470 

471 

472def fdot_(*xys, **start_f2product_nonfinites): 

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

474 

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

476 

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

478 

479 @return: Dot product (C{float}). 

480 ''' 

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

482 return float(D) 

483 

484 

485def fdot3(xs, ys, zs, **start_f2product_nonfinites): 

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

487 

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

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

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

491 

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

493 

494 @return: Dot product (C{float}). 

495 

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

497 ''' 

498 n = len(xs) 

499 if not n == len(ys) == len(zs): 

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

501 

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

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

504 _f = Fsum(**kwds) 

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

506 return float(D) 

507 

508 

509def fhorner(x, *cs, **incx): 

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

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

512 

513 @return: Horner sum (C{float}). 

514 

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

516 ''' 

517 H = Fhorner(x, *cs, **incx) 

518 return float(H) 

519 

520 

521def fidw(xs, ds, beta=2): 

522 '''Interpolate using U{Inverse Distance Weighting 

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

524 

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

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

527 L{Fsum2Tuple}). 

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

529 

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

531 

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

533 

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

535 

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

537 weighted B{C{ds}} below L{EPS}. 

538 

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

540 ''' 

541 n, xs = len2(xs) 

542 if n > 1: 

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

544 if b < 0: 

545 try: # weighted 

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

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

548 x = xs[i] 

549 D = _d(d) 

550 if D < EPS0: 

551 if D < 0: 

552 raise ValueError(_negative_) 

553 x = float(x) 

554 i = n 

555 break 

556 if D.fpow(b): 

557 W += D 

558 X += D.fmul(x) 

559 else: 

560 x = X.fover(W, raiser=False) 

561 i += 1 # len(xs) >= len(ds) 

562 except IndexError: 

563 i += 1 # len(xs) < i < len(ds) 

564 except Exception as X: 

565 _I = Fmt.INDEX 

566 raise _xError(X, _I(xs=i), x, 

567 _I(ds=i), d) 

568 else: # b == 0 

569 x = fsum(xs) / n # fmean(xs) 

570 i = n 

571 elif n: 

572 x = float(xs[0]) 

573 i = n 

574 else: 

575 x = _0_0 

576 i, _ = len2(ds) 

577 if i != n: 

578 raise LenError(fidw, xs=n, ds=i) 

579 return x 

580 

581 

582try: 

583 from math import fma as _fma # in .resections 

584except ImportError: # PYCHOK DSPACE! 

585 

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

587 return x * y + z 

588 

589 

590def fma(x, y, z, **nonfinites): # **raiser 

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

592 or an equivalent implementation. 

593 

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

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

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

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

598 to override default L{nonfiniterrors} 

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

600 

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

602 ''' 

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

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

605 

606 

607def _Fm2(x, nonfinites=None, **raiser): 

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

609 ''' 

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

611 

612 

613def fmean(xs, nonfinites=True): 

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

615 

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

617 

618 @return: Mean value (C{float}). 

619 

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

621 

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

623 ''' 

624 n, xs = len2(xs) 

625 if n < 1: 

626 raise LenError(fmean, xs=xs) 

627 M = Fsum(*xs, nonfinites=nonfinites) 

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

629 

630 

631def fmean_(*xs, **nonfinites): 

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

633 

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

635 ''' 

636 return fmean(xs, **nonfinites) 

637 

638 

639def f2mul_(x, *ys, **nonfinites): # **raiser 

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

641 

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

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

644 positional. 

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

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

647 

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

649 

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

651 ''' 

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

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

654 

655 

656def fpolynomial(x, *cs, **over_f2product_nonfinites): 

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

658 

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

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

661 L{Fpolynomial<Fpolynomial.__init__>}. 

662 

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

664 ''' 

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

666 P = Fpolynomial(x, *cs, **kwds) 

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

668 

669 

670def fpowers(x, n, alts=0): 

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

672 

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

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

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

676 exponent (C{int}). 

677 

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

679 

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

681 

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

683 ''' 

684 if not isint(n): 

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

686 elif n < 1: 

687 raise _ValueError(n=n) 

688 

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

690 ps = tuple(_powers(p, n)) 

691 

692 if alts > 0: # x**2, x**4, ... 

693 # ps[alts-1::2] chokes PyChecker 

694 ps = ps[slice(alts-1, None, 2)] 

695 

696 return ps 

697 

698 

699try: 

700 from math import prod as fprod # Python 3.8 

701except ImportError: 

702 

703 def fprod(xs, start=1): 

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

705 

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

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

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

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

710 

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

712 

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

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

715 ''' 

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

717 

718 

719def frandoms(n, seeded=None): 

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

721 

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

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

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

725 

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

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

728 ''' 

729 from random import gauss, random, seed, shuffle 

730 

731 if seeded is None: 

732 pass 

733 elif seeded and isbool(seeded): 

734 from time import localtime 

735 seed(localtime().tm_yday) 

736 elif isscalar(seeded): 

737 seed(seeded) 

738 

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

740 for _ in range(n): 

741 s = 0 

742 t = list(c) 

743 _a = t.append 

744 for _ in range(n * 8): 

745 v = gauss(0, random())**7 - s 

746 _a(v) 

747 s += v 

748 shuffle(t) 

749 yield t 

750 

751 

752def frange(start, number, step=1): 

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

754 

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

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

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

758 

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

760 

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

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

763 ''' 

764 if not isint(number): 

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

766 for i in range(number): 

767 yield start + (step * i) 

768 

769 

770try: 

771 from functools import reduce as freduce 

772except ImportError: 

773 try: 

774 freduce = reduce # PYCHOK expected 

775 except NameError: # Python 3+ 

776 

777 def freduce(f, xs, *start): 

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

779 ''' 

780 if start: 

781 r = v = start[0] 

782 else: 

783 r, v = 0, MISSING 

784 for v in xs: 

785 r = f(r, v) 

786 if v is MISSING: 

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

788 return r 

789 

790 

791def fremainder(x, y): 

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

793 

794 @arg x: Numerator (C{scalar}). 

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

796 

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

798 0.0) or C{NAN} for any non-finite B{C{x}}. 

799 

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

801 

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

803 project/geographiclib/>} and Python 3.7+ 

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

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

806 ''' 

807 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and 

808 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native 

809 # fmod( 0, 360) == 0.0 

810 # fmod( 360, 360) == 0.0 

811 # fmod(-0, 360) == 0.0 

812 # fmod(-0.0, 360) == -0.0 

813 # fmod(-360, 360) == -0.0 

814 # however, using the % operator ... 

815 # 0 % 360 == 0 

816 # 360 % 360 == 0 

817 # 360.0 % 360 == 0.0 

818 # -0 % 360 == 0 

819 # -360 % 360 == 0 == (-360) % 360 

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

821 # -360.0 % 360 == 0.0 == (-360.0) % 360 

822 

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

824 # == +0.0. This fixes this bug. See also Math::AngNormalize 

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

826 if isfinite(x): 

827 try: 

828 r = remainder(x, y) if x else x 

829 except Exception as e: 

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

831 else: # handle x INF and NINF as NAN 

832 r = NAN 

833 return r 

834 

835 

836if _MODS.sys_version_info2 < (3, 8): # PYCHOK no cover 

837 from math import hypot # OK in Python 3.7- 

838 

839 def hypot_(*xs): 

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

841 

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

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

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

845 handled differently. 

846 

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

848 

849 @return: Norm (C{float}). 

850 

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

852 

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

854 

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

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

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

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

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

860 ''' 

861 return float(_Hypot(*xs)) 

862 

863elif _MODS.sys_version_info2 < (3, 10): # PYCHOK no cover 

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

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

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

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

868 

869 def hypot(x, y): 

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

871 

872 @arg x: X argument (C{scalar}). 

873 @arg y: Y argument (C{scalar}). 

874 

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

876 ''' 

877 return float(_Hypot(x, y)) 

878 

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

880else: 

881 from math import hypot # PYCHOK in Python 3.10+ 

882 hypot_ = hypot 

883 

884 

885def _Hypot(*xs): 

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

887 ''' 

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

889 

890 

891def hypot1(x): 

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

893 

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

895 

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

897 ''' 

898 h = _1_0 

899 if x: 

900 if _isFsum_2Tuple(x): 

901 h = _Hypot(h, x) 

902 h = float(h) 

903 else: 

904 h = hypot(h, x) 

905 return h 

906 

907 

908def hypot2(x, y, *xy0): 

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

910 

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

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

913 @arg xy0: Optional reference C{(x0, y0)} (each C{scalar}, 

914 an L{Fsum} or L{Fsum2Tuple}). 

915 

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

917 ''' 

918 x, y = _map0(abs, x, y, *xy0) # NOT fabs! 

919 if y > x: 

920 x, y = y, x 

921 h2 = x**2 

922 if h2 and y: 

923 h2 *= (y / x)**2 + _1_0 

924 return float(h2) 

925 

926 

927def hypot2_(*xs): 

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

929 

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

931 L{Fsum2Tuple}), all positional. 

932 

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

934 

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

936 ''' 

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

938 if h2: # and isfinite(h2) 

939 _h = _1_0 / h2 

940 xs = ((x * _h) for x in xs) 

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

942 h2 = H2.fover(_h**2) 

943 return h2 

944 

945 

946def _map0(_f, x, y, x0=_0_0, y0=_0_0, *unused): 

947 return _f(x - x0), _f(y - y0)