Coverage for pygeodesy/fmath.py: 91%

325 statements  

« prev     ^ index     » next       coverage.py v7.6.1, created at 2025-04-09 11:05 -0400

1 

2# -*- coding: utf-8 -*- 

3 

4u'''Utilities using precision floating point summation. 

5''' 

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

7from __future__ import division as _; del _ # PYCHOK semicolon 

8 

9from pygeodesy.basics import _copysign, copysign0, isbool, isint, isscalar, \ 

10 len2, map1, _xiterable 

11from pygeodesy.constants import EPS0, EPS02, EPS1, NAN, PI, PI_2, PI_4, \ 

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

13 _1_5, _copysign_0_0, isfinite, remainder 

14from pygeodesy.errors import _IsnotError, LenError, _TypeError, _ValueError, \ 

15 _xError, _xkwds, _xkwds_pop2, _xsError 

16from pygeodesy.fsums import _2float, Fsum, fsum, _isFsum_2Tuple, Fmt, unstr 

17from pygeodesy.interns import MISSING, _negative_, _not_scalar_ 

18from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS 

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

20from pygeodesy.units import Int_, _isHeight, _isRadius, Float_ # PYCHOK for .heights 

21 

22from math import fabs, sqrt # pow 

23import operator as _operator # in .datums, .trf, .utm 

24 

25__all__ = _ALL_LAZY.fmath 

26__version__ = '25.01.09' 

27 

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

29_0_4142 = 0.41421356237309504880 # ... ~ 3730904090310553 / 9007199254740992 

30_2_3rd = _1_3rd * 2 

31_h_lt_b_ = 'abs(h) < abs(b)' 

32 

33 

34class Fdot(Fsum): 

35 '''Precision dot product. 

36 ''' 

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

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

39 

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

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

42 positional. 

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

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

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

46 

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

48 

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

50 

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

52 

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

54 

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

56 ''' 

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

58 Fsum.__init__(self, **kwds) 

59 self(s) 

60 

61 n = len(b) 

62 if len(a) != n: # PYCHOK no cover 

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

64 self._facc_dot(n, a, b, **kwds) 

65 

66 

67class Fhorner(Fsum): 

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

69 ''' 

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

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

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

73 = len(cs) - 1}. 

74 

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

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

77 all positional. 

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

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

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

81 

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

83 

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

85 

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

87 

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

89 ''' 

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

91 Fsum.__init__(self, **kwds) 

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

93 

94 

95class Fhypot(Fsum): 

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

97 ''' 

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

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

100 to the power I{root}). 

101 

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

103 positional. 

104 @kwarg root_name_f2product_nonfinites_RESIDUAL_raiser: Optional, exponent 

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

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

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

108 L{root<Fsum.root>}. 

109 ''' 

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

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

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

113 

114 r = None # _xkwds_pop2 error 

115 try: 

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

117 Fsum.__init__(self, **kwds) 

118 self(_0_0) 

119 if xs: 

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

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

122 except Exception as X: 

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

124 

125 

126class Fpolynomial(Fsum): 

127 '''Precision polynomial evaluation. 

128 ''' 

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

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

131 i=0..len(cs)-1)}. 

132 

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

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

135 all positional. 

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

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

138 

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

140 

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

142 

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

144 

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

146 ''' 

147 Fsum.__init__(self, **name_f2product_nonfinites_RESIDUAL) 

148 n = len(cs) - 1 

149 self(_0_0 if n < 0 else cs[0]) 

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

151 

152 

153class Fpowers(Fsum): 

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

155 ''' 

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

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

158 

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

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

161 positional. 

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

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

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

165 L{fpow<Fsum.fpow>}. 

166 ''' 

167 try: 

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

169 Fsum.__init__(self, **kwds) 

170 self(_0_0) 

171 if xs: 

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

173 except Exception as X: 

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

175 

176 

177class Froot(Fsum): 

178 '''The root of a precision summation. 

179 ''' 

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

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

182 

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

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

185 positional. 

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

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

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

189 L{fpow<Fsum.fpow>}. 

190 ''' 

191 try: 

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

193 Fsum.__init__(self, **kwds) 

194 self(_0_0) 

195 if xs: 

196 self.fadd(xs) 

197 self(self.root(root, raiser=X)) 

198 except Exception as X: 

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

200 

201 

202class Fcbrt(Froot): 

203 '''Cubic root of a precision summation. 

204 ''' 

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

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

207 

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

209 ''' 

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

211 

212 

213class Fsqrt(Froot): 

214 '''Square root of a precision summation. 

215 ''' 

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

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

218 

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

220 ''' 

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

222 

223 

224def bqrt(x): 

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

226 preserving C{type(B{x})}. 

227 

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

229 

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

231 

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

233 

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

235 

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

237 ''' 

238 return _root(x, _0_25, bqrt) 

239 

240 

241try: 

242 from math import cbrt as _cbrt # Python 3.11+ 

243 

244except ImportError: # Python 3.10- 

245 

246 def _cbrt(x): 

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

248 ''' 

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

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

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

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

253 

254 

255def cbrt(x): 

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

257 

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

259 

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

261 

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

263 ''' 

264 if _isFsum_2Tuple(x): 

265 r = abs(x).fpow(_1_3rd) 

266 if x.signOf() < 0: 

267 r = -r 

268 else: 

269 r = _cbrt(x) 

270 return r # cbrt(-0.0) == -0.0 

271 

272 

273def cbrt2(x): # PYCHOK attr 

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

275 

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

277 

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

279 

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

281 ''' 

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

283 

284 

285def euclid(x, y): 

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

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

288 

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

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

291 

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

293 

294 @see: Function L{euclid_}. 

295 ''' 

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

297 if y > x: 

298 x, y = y, x 

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

300 

301 

302def euclid_(*xs): 

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

304 L{euclid}. 

305 

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

307 all positional. 

308 

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

310 

311 @see: Function L{euclid}. 

312 ''' 

313 e = _0_0 

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

315 # e = euclid(x, e) 

316 if e < x: 

317 e, x = x, e 

318 if x: 

319 e += x * _0_4142 

320 return e 

321 

322 

323def facos1(x): 

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

325 

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

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

328 ''' 

329 a = fabs(x) 

330 if a < EPS0: 

331 r = PI_2 

332 elif a < EPS1: 

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

334 r *= sqrt(_1_0 - a) 

335 if x < 0: 

336 r = PI - r 

337 else: 

338 r = PI if x < 0 else _0_0 

339 return r 

340 

341 

342def fasin1(x): # PYCHOK no cover 

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

344 

345 @see: L{facos1}. 

346 ''' 

347 return PI_2 - facos1(x) 

348 

349 

350def _fast(x, *cs): 

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

352 ''' 

353 h = 0 

354 for c in reversed(cs): 

355 h = _fma(x, h, c) if h else c 

356 return h 

357 

358 

359def fatan(x): 

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

361 ''' 

362 a = fabs(x) 

363 if a < _1_0: 

364 r = fatan1(a) if a else _0_0 

365 elif a > _1_0: 

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

367 else: 

368 r = PI_4 

369 if x < 0: # copysign0(r, x) 

370 r = -r 

371 return r 

372 

373 

374def fatan1(x): 

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

376 

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

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

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

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

381 IEEE Signal Processing Magazine, 111, May 2006. 

382 ''' 

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

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

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

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

387 

388 

389def fatan2(y, x): 

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

391 

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

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

394 and L{fatan1}. 

395 ''' 

396 a, b = fabs(x), fabs(y) 

397 if b > a: 

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

399 elif a > b: 

400 r = fatan1(b / a) if b else _0_0 

401 elif a: # a == b != 0 

402 r = PI_4 

403 else: # a == b == 0 

404 return _0_0 

405 if x < 0: 

406 r = PI - r 

407 if y < 0: # copysign0(r, y) 

408 r = -r 

409 return r 

410 

411 

412def favg(a, b, f=_0_5, nonfinites=True): 

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

414 

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

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

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

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

419 

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

421 ''' 

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

423 return float(F) 

424 

425 

426def fdot(xs, *ys, **start_f2product_nonfinites): 

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

428 

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

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

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

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

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

434 

435 @return: Dot product (C{float}). 

436 

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

438 

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

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

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

442 ''' 

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

444 return float(D) 

445 

446 

447def fdot_(*xys, **start_f2product_nonfinites): 

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

449 

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

451 

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

453 

454 @return: Dot product (C{float}). 

455 ''' 

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

457 

458 

459def fdot3(xs, ys, zs, **start_f2product_nonfinites): 

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

461 

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

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

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

465 

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

467 

468 @return: Dot product (C{float}). 

469 

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

471 ''' 

472 n = len(xs) 

473 if not n == len(ys) == len(zs): 

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

475 

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

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

478 _f = Fsum(**kwds) 

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

480 return float(D) 

481 

482 

483def fhorner(x, *cs, **incx): 

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

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

486 

487 @return: Horner sum (C{float}). 

488 

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

490 ''' 

491 H = Fhorner(x, *cs, **incx) 

492 return float(H) 

493 

494 

495def fidw(xs, ds, beta=2): 

496 '''Interpolate using U{Inverse Distance Weighting 

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

498 

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

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

501 L{Fsum2Tuple}). 

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

503 

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

505 

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

507 

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

509 

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

511 weighted B{C{ds}} below L{EPS}. 

512 

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

514 ''' 

515 n, xs = len2(xs) 

516 if n > 1: 

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

518 if b < 0: 

519 try: # weighted 

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

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

522 x = xs[i] 

523 D = _d(d) 

524 if D < EPS0: 

525 if D < 0: 

526 raise ValueError(_negative_) 

527 x = float(x) 

528 i = n 

529 break 

530 if D.fpow(b): 

531 W += D 

532 X += D.fmul(x) 

533 else: 

534 x = X.fover(W, raiser=False) 

535 i += 1 # len(xs) >= len(ds) 

536 except IndexError: 

537 i += 1 # len(xs) < i < len(ds) 

538 except Exception as X: 

539 _I = Fmt.INDEX 

540 raise _xError(X, _I(xs=i), x, 

541 _I(ds=i), d) 

542 else: # b == 0 

543 x = fsum(xs) / n # fmean(xs) 

544 i = n 

545 elif n: 

546 x = float(xs[0]) 

547 i = n 

548 else: 

549 x = _0_0 

550 i, _ = len2(ds) 

551 if i != n: 

552 raise LenError(fidw, xs=n, ds=i) 

553 return x 

554 

555 

556try: 

557 from math import fma as _fma 

558except ImportError: # PYCHOK DSPACE! 

559 

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

561 return x * y + z 

562 

563 

564def fma(x, y, z, **nonfinites): # **raiser 

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

566 or an equivalent implementation. 

567 

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

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

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

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

572 to override default L{nonfiniterrors} 

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

574 

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

576 ''' 

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

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

579 

580 

581def _Fm2(x, nonfinites=None, **raiser): 

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

583 ''' 

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

585 

586 

587def fmean(xs): 

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

589 

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

591 

592 @return: Mean value (C{float}). 

593 

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

595 

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

597 ''' 

598 n, xs = len2(xs) 

599 if n < 1: 

600 raise LenError(fmean, xs=xs) 

601 M = Fsum(*xs, nonfinites=True) 

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

603 

604 

605def fmean_(*xs, **nonfinites): 

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

607 

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

609 ''' 

610 return fmean(xs, **nonfinites) 

611 

612 

613def f2mul_(x, *ys, **nonfinites): # **raiser 

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

615 

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

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

618 positional. 

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

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

621 

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

623 

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

625 ''' 

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

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

628 

629 

630def fpolynomial(x, *cs, **over_f2product_nonfinites): 

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

632 

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

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

635 L{Fpolynomial<Fpolynomial.__init__>}. 

636 

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

638 ''' 

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

640 P = Fpolynomial(x, *cs, **kwds) 

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

642 

643 

644def fpowers(x, n, alts=0): 

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

646 

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

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

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

650 exponent (C{int}). 

651 

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

653 

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

655 

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

657 ''' 

658 if not isint(n): 

659 raise _IsnotError(int.__name__, n=n) 

660 elif n < 1: 

661 raise _ValueError(n=n) 

662 

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

664 ps = tuple(_powers(p, n)) 

665 

666 if alts > 0: # x**2, x**4, ... 

667 # ps[alts-1::2] chokes PyChecker 

668 ps = ps[slice(alts-1, None, 2)] 

669 

670 return ps 

671 

672 

673try: 

674 from math import prod as fprod # Python 3.8 

675except ImportError: 

676 

677 def fprod(xs, start=1): 

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

679 

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

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

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

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

684 

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

686 

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

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

689 ''' 

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

691 

692 

693def frandoms(n, seeded=None): 

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

695 

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

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

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

699 

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

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

702 ''' 

703 from random import gauss, random, seed, shuffle 

704 

705 if seeded is None: 

706 pass 

707 elif seeded and isbool(seeded): 

708 from time import localtime 

709 seed(localtime().tm_yday) 

710 elif isscalar(seeded): 

711 seed(seeded) 

712 

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

714 for _ in range(n): 

715 s = 0 

716 t = list(c) 

717 _a = t.append 

718 for _ in range(n * 8): 

719 v = gauss(0, random())**7 - s 

720 _a(v) 

721 s += v 

722 shuffle(t) 

723 yield t 

724 

725 

726def frange(start, number, step=1): 

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

728 

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

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

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

732 

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

734 

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

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

737 ''' 

738 if not isint(number): 

739 raise _IsnotError(int.__name__, number=number) 

740 for i in range(number): 

741 yield start + (step * i) 

742 

743 

744try: 

745 from functools import reduce as freduce 

746except ImportError: 

747 try: 

748 freduce = reduce # PYCHOK expected 

749 except NameError: # Python 3+ 

750 

751 def freduce(f, xs, *start): 

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

753 ''' 

754 if start: 

755 r = v = start[0] 

756 else: 

757 r, v = 0, MISSING 

758 for v in xs: 

759 r = f(r, v) 

760 if v is MISSING: 

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

762 return r 

763 

764 

765def fremainder(x, y): 

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

767 

768 @arg x: Numerator (C{scalar}). 

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

770 

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

772 0.0) or C{NAN} for any non-finite B{C{x}}. 

773 

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

775 

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

777 project/geographiclib/>} and Python 3.7+ 

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

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

780 ''' 

781 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and 

782 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native 

783 # fmod( 0, 360) == 0.0 

784 # fmod( 360, 360) == 0.0 

785 # fmod(-0, 360) == 0.0 

786 # fmod(-0.0, 360) == -0.0 

787 # fmod(-360, 360) == -0.0 

788 # however, using the % operator ... 

789 # 0 % 360 == 0 

790 # 360 % 360 == 0 

791 # 360.0 % 360 == 0.0 

792 # -0 % 360 == 0 

793 # -360 % 360 == 0 == (-360) % 360 

794 # -0.0 % 360 == 0.0 == (-0.0) % 360 

795 # -360.0 % 360 == 0.0 == (-360.0) % 360 

796 

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

798 # == +0.0. This fixes this bug. See also Math::AngNormalize 

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

800 if isfinite(x): 

801 try: 

802 r = remainder(x, y) if x else x 

803 except Exception as e: 

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

805 else: # handle x INF and NINF as NAN 

806 r = NAN 

807 return r 

808 

809 

810if _MODS.sys_version_info2 < (3, 8): # PYCHOK no cover 

811 from math import hypot # OK in Python 3.7- 

812 

813 def hypot_(*xs): 

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

815 

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

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

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

819 handled differently. 

820 

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

822 

823 @return: Norm (C{float}). 

824 

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

826 

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

828 

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

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

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

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

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

834 ''' 

835 return float(_Hypot(*xs)) 

836 

837elif _MODS.sys_version_info2 < (3, 10): # PYCHOK no cover 

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

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

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

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

842 

843 def hypot(x, y): 

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

845 

846 @arg x: X argument (C{scalar}). 

847 @arg y: Y argument (C{scalar}). 

848 

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

850 ''' 

851 return float(_Hypot(x, y)) 

852 

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

854else: 

855 from math import hypot # PYCHOK in Python 3.10+ 

856 hypot_ = hypot 

857 

858 

859def _Hypot(*xs): 

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

861 ''' 

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

863 

864 

865def hypot1(x): 

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

867 

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

869 

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

871 ''' 

872 h = _1_0 

873 if x: 

874 if _isFsum_2Tuple(x): 

875 h = _Hypot(h, x) 

876 h = float(h) 

877 else: 

878 h = hypot(h, x) 

879 return h 

880 

881 

882def hypot2(x, y): 

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

884 

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

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

887 

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

889 ''' 

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

891 if y > x: 

892 x, y = y, x 

893 h2 = x**2 

894 if h2 and y: 

895 h2 *= (y / x)**2 + _1_0 

896 return float(h2) 

897 

898 

899def hypot2_(*xs): 

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

901 

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

903 L{Fsum2Tuple}), all positional. 

904 

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

906 

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

908 ''' 

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

910 if h2: # and isfinite(h2) 

911 _h = _1_0 / h2 

912 xs = ((x * _h) for x in xs) 

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

914 h2 = H2.fover(_h**2) 

915 return h2 

916 

917 

918def norm2(x, y): 

919 '''Normalize a 2-dimensional vector. 

920 

921 @arg x: X component (C{scalar}). 

922 @arg y: Y component (C{scalar}). 

923 

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

925 

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

927 or zero norm. 

928 ''' 

929 try: 

930 h = None 

931 h = hypot(x, y) 

932 if h: 

933 x, y = (x / h), (y / h) 

934 else: 

935 x = _copysign_0_0(x) # pass? 

936 y = _copysign_0_0(y) 

937 except Exception as e: 

938 raise _xError(e, x=x, y=y, h=h) 

939 return x, y 

940 

941 

942def norm_(*xs): 

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

944 

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

946 L{Fsum2Tuple}), all positional. 

947 

948 @return: Yield each component, normalized. 

949 

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

951 or zero norm. 

952 ''' 

953 try: 

954 i = h = None 

955 x = xs 

956 h = hypot_(*xs) 

957 _h = (_1_0 / h) if h else _0_0 

958 for i, x in enumerate(xs): 

959 yield x * _h 

960 except Exception as X: 

961 raise _xsError(X, xs, i, x, h=h) 

962 

963 

964def _powers(x, n): 

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

966 ''' 

967 p = 1 # type(p) == type(x) 

968 for _ in range(n): 

969 p *= x 

970 yield p 

971 

972 

973def _root(x, p, where): 

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

975 ''' 

976 try: 

977 if x > 0: 

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

979 return r.fpow(p).as_iscalar 

980 elif x < 0: 

981 raise ValueError(_negative_) 

982 except Exception as X: 

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

984 return _0_0 if p else _1_0 

985 

986 

987def sqrt0(x, Error=None): 

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

989 preserving C{type(B{x})}. 

990 

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

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

993 

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

995 

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

997 

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

999 returns C{0.0}. 

1000 ''' 

1001 if Error and x < 0: 

1002 raise Error(unstr(sqrt0, x)) 

1003 return _root(x, _0_5, sqrt0) if x > EPS02 else ( 

1004 _0_0 if x < EPS02 else EPS0) 

1005 

1006 

1007def sqrt3(x): 

1008 '''Return the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)}, 

1009 preserving C{type(B{x})}. 

1010 

1011 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1012 

1013 @return: Square root I{cubed} (C{float} or L{Fsum}). 

1014 

1015 @raise TypeeError: Invalid B{C{x}}. 

1016 

1017 @raise ValueError: Negative B{C{x}}. 

1018 

1019 @see: Functions L{cbrt} and L{cbrt2}. 

1020 ''' 

1021 return _root(x, _1_5, sqrt3) 

1022 

1023 

1024def sqrt_a(h, b): 

1025 '''Compute C{I{a}} side of a right-angled triangle from 

1026 C{sqrt(B{h}**2 - B{b}**2)}. 

1027 

1028 @arg h: Hypotenuse or outer annulus radius (C{scalar}). 

1029 @arg b: Triangle side or inner annulus radius (C{scalar}). 

1030 

1031 @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}). 

1032 

1033 @raise TypeError: Non-scalar B{C{h}} or B{C{b}}. 

1034 

1035 @raise ValueError: If C{abs(B{h}) < abs(B{b})}. 

1036 

1037 @see: Inner tangent chord B{I{d}} of an U{annulus 

1038 <https://WikiPedia.org/wiki/Annulus_(mathematics)>} 

1039 and function U{annulus_area<https://People.SC.FSU.edu/ 

1040 ~jburkardt/py_src/geometry/geometry.py>}. 

1041 ''' 

1042 try: 

1043 if not (_isHeight(h) and _isRadius(b)): 

1044 raise TypeError(_not_scalar_) 

1045 c = fabs(h) 

1046 if c > EPS0: 

1047 s = _1_0 - (b / c)**2 

1048 if s < 0: 

1049 raise ValueError(_h_lt_b_) 

1050 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0) 

1051 else: # PYCHOK no cover 

1052 b = fabs(b) 

1053 d = c - b 

1054 if d < 0: 

1055 raise ValueError(_h_lt_b_) 

1056 d *= c + b 

1057 a = sqrt(d) if d else _0_0 

1058 except Exception as x: 

1059 raise _xError(x, h=h, b=b) 

1060 return copysign0(a, h) 

1061 

1062 

1063def zcrt(x): 

1064 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)}, 

1065 preserving C{type(B{x})}. 

1066 

1067 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1068 

1069 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}). 

1070 

1071 @see: Functions L{bqrt} and L{zqrt}. 

1072 

1073 @raise TypeeError: Invalid B{C{x}}. 

1074 

1075 @raise ValueError: Negative B{C{x}}. 

1076 ''' 

1077 return _root(x, _1_6th, zcrt) 

1078 

1079 

1080def zqrt(x): 

1081 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root, 

1082 M{x**(1 / 8)}, preserving C{type(B{x})}. 

1083 

1084 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1085 

1086 @return: I{Zenzi-quartic} root (C{float} or L{Fsum}). 

1087 

1088 @see: Functions L{bqrt} and L{zcrt}. 

1089 

1090 @raise TypeeError: Invalid B{C{x}}. 

1091 

1092 @raise ValueError: Negative B{C{x}}. 

1093 ''' 

1094 return _root(x, _0_125, zqrt) 

1095 

1096# **) MIT License 

1097# 

1098# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved. 

1099# 

1100# Permission is hereby granted, free of charge, to any person obtaining a 

1101# copy of this software and associated documentation files (the "Software"), 

1102# to deal in the Software without restriction, including without limitation 

1103# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1104# and/or sell copies of the Software, and to permit persons to whom the 

1105# Software is furnished to do so, subject to the following conditions: 

1106# 

1107# The above copyright notice and this permission notice shall be included 

1108# in all copies or substantial portions of the Software. 

1109# 

1110# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1111# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1112# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1113# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1114# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1115# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1116# OTHER DEALINGS IN THE SOFTWARE.