Coverage for pygeodesy/fmath.py: 90%

329 statements  

« prev     ^ index     » next       coverage.py v7.6.1, created at 2024-11-12 16:17 -0500

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__ = '24.11.08' 

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 r = None # _xkwds_pop2 error 

111 try: 

112 r, kwds = _xkwds_pop2(root_name_f2product_nonfinites_RESIDUAL_raiser, root=2) 

113 r, kwds = _xkwds_pop2(kwds, power=r) # for backward compatibility 

114 t, kwds = _xkwds_pop2(kwds, raiser=True) 

115 Fsum.__init__(self, **kwds) 

116 self(_0_0) 

117 if xs: 

118 self._facc_power(r, xs, Fhypot, raiser=t) 

119 self._fset(self.root(r, raiser=t)) 

120 except Exception as X: 

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

122 

123 

124class Fpolynomial(Fsum): 

125 '''Precision polynomial evaluation. 

126 ''' 

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

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

129 i=0..len(cs)-1)}. 

130 

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

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

133 all positional. 

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

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

136 

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

138 

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

140 

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

142 

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

144 ''' 

145 Fsum.__init__(self, **name_f2product_nonfinites_RESIDUAL) 

146 n = len(cs) - 1 

147 self(_0_0 if n < 0 else cs[0]) 

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

149 

150 

151class Fpowers(Fsum): 

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

153 ''' 

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

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

156 

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

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

159 positional. 

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

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

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

163 L{fpow<Fsum.fpow>}. 

164 ''' 

165 try: 

166 t, kwds = _xkwds_pop2(name_f2product_nonfinites_RESIDUAL_raiser, raiser=True) 

167 Fsum.__init__(self, **kwds) 

168 self(_0_0) 

169 if xs: 

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

171 except Exception as X: 

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

173 

174 

175class Froot(Fsum): 

176 '''The root of a precision summation. 

177 ''' 

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

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

180 

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

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

183 positional. 

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

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

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

187 L{fpow<Fsum.fpow>}. 

188 ''' 

189 try: 

190 raiser, kwds = _xkwds_pop2(name_f2product_nonfinites_RESIDUAL_raiser, raiser=True) 

191 Fsum.__init__(self, **kwds) 

192 self(_0_0) 

193 if xs: 

194 self.fadd(xs) 

195 self(self.root(root, raiser=raiser)) 

196 except Exception as X: 

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

198 

199 

200class Fcbrt(Froot): 

201 '''Cubic root of a precision summation. 

202 ''' 

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

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

205 

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

207 ''' 

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

209 

210 

211class Fsqrt(Froot): 

212 '''Square root of a precision summation. 

213 ''' 

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

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

216 

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

218 ''' 

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

220 

221 

222def bqrt(x): 

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

224 preserving C{type(B{x})}. 

225 

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

227 

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

229 

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

231 

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

233 

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

235 ''' 

236 return _root(x, _0_25, bqrt) 

237 

238 

239try: 

240 from math import cbrt as _cbrt # Python 3.11+ 

241 

242except ImportError: # Python 3.10- 

243 

244 def _cbrt(x): 

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

246 ''' 

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

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

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

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

251 

252 

253def cbrt(x): 

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

255 

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

257 

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

259 

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

261 ''' 

262 if _isFsum_2Tuple(x): 

263 r = abs(x).fpow(_1_3rd) 

264 if x.signOf() < 0: 

265 r = -r 

266 else: 

267 r = _cbrt(x) 

268 return r # cbrt(-0.0) == -0.0 

269 

270 

271def cbrt2(x): # PYCHOK attr 

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

273 

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

275 

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

277 

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

279 ''' 

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

281 

282 

283def euclid(x, y): 

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

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

286 

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

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

289 

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

291 

292 @see: Function L{euclid_}. 

293 ''' 

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

295 if y > x: 

296 x, y = y, x 

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

298 

299 

300def euclid_(*xs): 

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

302 L{euclid}. 

303 

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

305 all positional. 

306 

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

308 

309 @see: Function L{euclid}. 

310 ''' 

311 e = _0_0 

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

313 # e = euclid(x, e) 

314 if e < x: 

315 e, x = x, e 

316 if x: 

317 e += x * _0_4142 

318 return e 

319 

320 

321def facos1(x): 

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

323 

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

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

326 ''' 

327 a = fabs(x) 

328 if a < EPS0: 

329 r = PI_2 

330 elif a < EPS1: 

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

332 r *= sqrt(_1_0 - a) 

333 if x < 0: 

334 r = PI - r 

335 else: 

336 r = PI if x < 0 else _0_0 

337 return r 

338 

339 

340def fasin1(x): # PYCHOK no cover 

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

342 

343 @see: L{facos1}. 

344 ''' 

345 return PI_2 - facos1(x) 

346 

347 

348def _fast(x, *cs): 

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

350 ''' 

351 h = 0 

352 for c in reversed(cs): 

353 h = _fma(x, h, c) if h else c 

354 return h 

355 

356 

357def fatan(x): 

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

359 ''' 

360 a = fabs(x) 

361 if a < _1_0: 

362 r = fatan1(a) if a else _0_0 

363 elif a > _1_0: 

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

365 else: 

366 r = PI_4 

367 if x < 0: # copysign0(r, x) 

368 r = -r 

369 return r 

370 

371 

372def fatan1(x): 

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

374 

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

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

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

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

379 IEEE Signal Processing Magazine, 111, May 2006. 

380 ''' 

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

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

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

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

385 

386 

387def fatan2(y, x): 

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

389 

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

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

392 and L{fatan1}. 

393 ''' 

394 a, b = fabs(x), fabs(y) 

395 if b > a: 

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

397 elif a > b: 

398 r = fatan1(b / a) if b else _0_0 

399 elif a: # a == b != 0 

400 r = PI_4 

401 else: # a == b == 0 

402 return _0_0 

403 if x < 0: 

404 r = PI - r 

405 if y < 0: # copysign0(r, y) 

406 r = -r 

407 return r 

408 

409 

410def favg(a, b, f=_0_5, nonfinites=True): 

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

412 

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

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

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

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

417 

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

419 ''' 

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

421 return float(F) 

422 

423 

424def fdot(xs, *ys, **start_f2product_nonfinites): 

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

426 

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

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

429 positional. 

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

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

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

433 

434 @return: Dot product (C{float}). 

435 

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

437 

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

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

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

441 ''' 

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

443 return float(D) 

444 

445 

446def fdot_(*xys, **start_f2product_nonfinites): 

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

448 

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

450 

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

452 

453 @return: Dot product (C{float}). 

454 ''' 

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

456 

457 

458def fdot3(xs, ys, zs, **start_f2product_nonfinites): 

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

460 

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

462 @arg ys: Iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

463 @arg zs: Iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

464 

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

466 

467 @return: Dot product (C{float}). 

468 

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

470 ''' 

471 n = len(xs) 

472 if not n == len(ys) == len(zs): 

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

474 

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

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

477 _f = Fsum(**kwds) 

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

479 return float(D) 

480 

481 

482def fhorner(x, *cs, **incx): 

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

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

485 

486 @return: Horner sum (C{float}). 

487 

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

489 ''' 

490 H = Fhorner(x, *cs, **incx) 

491 return float(H) 

492 

493 

494def fidw(xs, ds, beta=2): 

495 '''Interpolate using U{Inverse Distance Weighting 

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

497 

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

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

500 L{Fsum2Tuple}). 

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

502 

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

504 

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

506 

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

508 

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

510 weighted B{C{ds}} below L{EPS}. 

511 

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

513 ''' 

514 n, xs = len2(xs) 

515 if n > 1: 

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

517 if b < 0: 

518 try: # weighted 

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

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

521 x = xs[i] 

522 D = _d(d) 

523 if D < EPS0: 

524 if D < 0: 

525 raise ValueError(_negative_) 

526 x = float(x) 

527 i = n 

528 break 

529 if D.fpow(b): 

530 W += D 

531 X += D.fmul(x) 

532 else: 

533 x = X.fover(W, raiser=False) 

534 i += 1 # len(xs) >= len(ds) 

535 except IndexError: 

536 i += 1 # len(xs) < i < len(ds) 

537 except Exception as X: 

538 _I = Fmt.INDEX 

539 raise _xError(X, _I(xs=i), x, 

540 _I(ds=i), d) 

541 else: # b == 0 

542 x = fsum(xs) / n # fmean(xs) 

543 i = n 

544 elif n: 

545 x = float(xs[0]) 

546 i = n 

547 else: 

548 x = _0_0 

549 i, _ = len2(ds) 

550 if i != n: 

551 raise LenError(fidw, xs=n, ds=i) 

552 return x 

553 

554 

555try: 

556 from math import fma as _fma 

557except ImportError: # PYCHOK DSPACE! 

558 

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

560 return x * y + z 

561 

562 

563def fma(x, y, z, **nonfinites): # **raiser 

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

565 or an equivalent implementation. 

566 

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

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

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

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

571 to override default L{nonfiniterrors} 

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

573 

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

575 ''' 

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

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

578 

579 

580def _Fm2(x, nonfinites=None, **raiser): 

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

582 ''' 

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

584 

585 

586def fmean(xs): 

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

588 

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

590 

591 @return: Mean value (C{float}). 

592 

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

594 

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

596 ''' 

597 n, xs = len2(xs) 

598 if n < 1: 

599 raise LenError(fmean, xs=xs) 

600 M = Fsum(*xs, nonfinites=True) 

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

602 

603 

604def fmean_(*xs, **nonfinites): 

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

606 

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

608 ''' 

609 return fmean(xs, **nonfinites) 

610 

611 

612def f2mul_(x, *ys, **nonfinites): # **raiser 

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

614 

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

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

617 positional. 

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

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

620 

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

622 

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

624 ''' 

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

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

627 

628 

629def fpolynomial(x, *cs, **over_f2product_nonfinites): 

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

631 

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

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

634 L{Fpolynomial<Fpolynomial.__init__>}. 

635 

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

637 ''' 

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

639 P = Fpolynomial(x, *cs, **kwds) 

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

641 

642 

643def fpowers(x, n, alts=0): 

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

645 

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

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

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

649 exponent (C{int}). 

650 

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

652 

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

654 

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

656 ''' 

657 if not isint(n): 

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

659 elif n < 1: 

660 raise _ValueError(n=n) 

661 

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

663 ps = tuple(_powers(p, n)) 

664 

665 if alts > 0: # x**2, x**4, ... 

666 # ps[alts-1::2] chokes PyChecker 

667 ps = ps[slice(alts-1, None, 2)] 

668 

669 return ps 

670 

671 

672try: 

673 from math import prod as fprod # Python 3.8 

674except ImportError: 

675 

676 def fprod(xs, start=1): 

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

678 

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

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

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

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

683 

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

685 

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

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

688 ''' 

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

690 

691 

692def frandoms(n, seeded=None): 

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

694 

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

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

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

698 

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

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

701 ''' 

702 from random import gauss, random, seed, shuffle 

703 

704 if seeded is None: 

705 pass 

706 elif seeded and isbool(seeded): 

707 from time import localtime 

708 seed(localtime().tm_yday) 

709 elif isscalar(seeded): 

710 seed(seeded) 

711 

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

713 for _ in range(n): 

714 s = 0 

715 t = list(c) 

716 _a = t.append 

717 for _ in range(n * 8): 

718 v = gauss(0, random())**7 - s 

719 _a(v) 

720 s += v 

721 shuffle(t) 

722 yield t 

723 

724 

725def frange(start, number, step=1): 

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

727 

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

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

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

731 

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

733 

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

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

736 ''' 

737 if not isint(number): 

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

739 for i in range(number): 

740 yield start + (step * i) 

741 

742 

743try: 

744 from functools import reduce as freduce 

745except ImportError: 

746 try: 

747 freduce = reduce # PYCHOK expected 

748 except NameError: # Python 3+ 

749 

750 def freduce(f, xs, *start): 

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

752 ''' 

753 if start: 

754 r = v = start[0] 

755 else: 

756 r, v = 0, MISSING 

757 for v in xs: 

758 r = f(r, v) 

759 if v is MISSING: 

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

761 return r 

762 

763 

764def fremainder(x, y): 

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

766 

767 @arg x: Numerator (C{scalar}). 

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

769 

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

771 0.0) or C{NAN} for any non-finite B{C{x}}. 

772 

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

774 

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

776 project/geographiclib/>} and Python 3.7+ 

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

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

779 ''' 

780 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and 

781 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native 

782 # fmod( 0, 360) == 0.0 

783 # fmod( 360, 360) == 0.0 

784 # fmod(-0, 360) == 0.0 

785 # fmod(-0.0, 360) == -0.0 

786 # fmod(-360, 360) == -0.0 

787 # however, using the % operator ... 

788 # 0 % 360 == 0 

789 # 360 % 360 == 0 

790 # 360.0 % 360 == 0.0 

791 # -0 % 360 == 0 

792 # -360 % 360 == 0 == (-360) % 360 

793 # -0.0 % 360 == 0.0 == (-0.0) % 360 

794 # -360.0 % 360 == 0.0 == (-360.0) % 360 

795 

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

797 # == +0.0. This fixes this bug. See also Math::AngNormalize 

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

799 if isfinite(x): 

800 try: 

801 r = remainder(x, y) if x else x 

802 except Exception as e: 

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

804 else: # handle x INF and NINF as NAN 

805 r = NAN 

806 return r 

807 

808 

809if _MODS.sys_version_info2 < (3, 8): # PYCHOK no cover 

810 from math import hypot # OK in Python 3.7- 

811 

812 def hypot_(*xs): 

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

814 

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

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

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

818 handled differently. 

819 

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

821 

822 @return: Norm (C{float}). 

823 

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

825 

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

827 

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

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

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

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

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

833 ''' 

834 return float(_Hypot(*xs)) 

835 

836elif _MODS.sys_version_info2 < (3, 10): 

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

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

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

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

841 

842 def hypot(x, y): 

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

844 

845 @arg x: X argument (C{scalar}). 

846 @arg y: Y argument (C{scalar}). 

847 

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

849 ''' 

850 return float(_Hypot(x, y)) 

851 

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

853else: 

854 from math import hypot # PYCHOK in Python 3.10+ 

855 hypot_ = hypot 

856 

857 

858def _Hypot(*xs): 

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

860 ''' 

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

862 

863 

864def hypot1(x): 

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

866 

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

868 

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

870 ''' 

871 h = _1_0 

872 if x: 

873 if _isFsum_2Tuple(x): 

874 h = _Hypot(h, x) 

875 h = float(h) 

876 else: 

877 h = hypot(h, x) 

878 return h 

879 

880 

881def hypot2(x, y): 

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

883 

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

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

886 

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

888 ''' 

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

890 if y > x: 

891 x, y = y, x 

892 h2 = x**2 

893 if h2 and y: 

894 h2 *= (y / x)**2 + _1_0 

895 return float(h2) 

896 

897 

898def hypot2_(*xs): 

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

900 

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

902 L{Fsum2Tuple}), all positional. 

903 

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

905 

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

907 ''' 

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

909 if h2: # and isfinite(h2) 

910 _h = _1_0 / h2 

911 xs = ((x * _h) for x in xs) 

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

913 h2 = H2.fover(_h**2) 

914 return h2 

915 

916 

917def norm2(x, y): 

918 '''Normalize a 2-dimensional vector. 

919 

920 @arg x: X component (C{scalar}). 

921 @arg y: Y component (C{scalar}). 

922 

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

924 

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

926 or zero norm. 

927 ''' 

928 try: 

929 h = None 

930 h = hypot(x, y) 

931 if h: 

932 x, y = (x / h), (y / h) 

933 else: 

934 x = _copysign_0_0(x) # pass? 

935 y = _copysign_0_0(y) 

936 except Exception as e: 

937 raise _xError(e, x=x, y=y, h=h) 

938 return x, y 

939 

940 

941def norm_(*xs): 

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

943 

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

945 L{Fsum2Tuple}), all positional. 

946 

947 @return: Yield each component, normalized. 

948 

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

950 or zero norm. 

951 ''' 

952 try: 

953 i = h = None 

954 x = xs 

955 h = hypot_(*xs) 

956 _h = (_1_0 / h) if h else _0_0 

957 for i, x in enumerate(xs): 

958 yield x * _h 

959 except Exception as X: 

960 raise _xsError(X, xs, i, x, h=h) 

961 

962 

963def _powers(x, n): 

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

965 ''' 

966 p = 1 # type(p) == type(x) 

967 for _ in range(n): 

968 p *= x 

969 yield p 

970 

971 

972def _root(x, p, where): 

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

974 ''' 

975 try: 

976 if x > 0: 

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

978 return r.fpow(p).as_iscalar 

979 elif x < 0: 

980 raise ValueError(_negative_) 

981 except Exception as X: 

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

983 return _0_0 

984 

985 

986def sqrt0(x, Error=None): 

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

988 preserving C{type(B{x})}. 

989 

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

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

992 

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

994 

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

996 

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

998 returns C{0.0}. 

999 ''' 

1000 if Error and x < 0: 

1001 raise Error(unstr(sqrt0, x)) 

1002 return _root(x, _0_5, sqrt0) if x > EPS02 else ( 

1003 _0_0 if x < EPS02 else EPS0) 

1004 

1005 

1006def sqrt3(x): 

1007 '''Return the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)}, 

1008 preserving C{type(B{x})}. 

1009 

1010 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1011 

1012 @return: Square root I{cubed} (C{float} or L{Fsum}). 

1013 

1014 @raise TypeeError: Invalid B{C{x}}. 

1015 

1016 @raise ValueError: Negative B{C{x}}. 

1017 

1018 @see: Functions L{cbrt} and L{cbrt2}. 

1019 ''' 

1020 return _root(x, _1_5, sqrt3) 

1021 

1022 

1023def sqrt_a(h, b): 

1024 '''Compute C{I{a}} side of a right-angled triangle from 

1025 C{sqrt(B{h}**2 - B{b}**2)}. 

1026 

1027 @arg h: Hypotenuse or outer annulus radius (C{scalar}). 

1028 @arg b: Triangle side or inner annulus radius (C{scalar}). 

1029 

1030 @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}). 

1031 

1032 @raise TypeError: Non-scalar B{C{h}} or B{C{b}}. 

1033 

1034 @raise ValueError: If C{abs(B{h}) < abs(B{b})}. 

1035 

1036 @see: Inner tangent chord B{I{d}} of an U{annulus 

1037 <https://WikiPedia.org/wiki/Annulus_(mathematics)>} 

1038 and function U{annulus_area<https://People.SC.FSU.edu/ 

1039 ~jburkardt/py_src/geometry/geometry.py>}. 

1040 ''' 

1041 try: 

1042 if not (_isHeight(h) and _isRadius(b)): 

1043 raise TypeError(_not_scalar_) 

1044 c = fabs(h) 

1045 if c > EPS0: 

1046 s = _1_0 - (b / c)**2 

1047 if s < 0: 

1048 raise ValueError(_h_lt_b_) 

1049 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0) 

1050 else: # PYCHOK no cover 

1051 b = fabs(b) 

1052 d = c - b 

1053 if d < 0: 

1054 raise ValueError(_h_lt_b_) 

1055 d *= c + b 

1056 a = sqrt(d) if d else _0_0 

1057 except Exception as x: 

1058 raise _xError(x, h=h, b=b) 

1059 return copysign0(a, h) 

1060 

1061 

1062def zcrt(x): 

1063 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)}, 

1064 preserving C{type(B{x})}. 

1065 

1066 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1067 

1068 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}). 

1069 

1070 @see: Functions L{bqrt} and L{zqrt}. 

1071 

1072 @raise TypeeError: Invalid B{C{x}}. 

1073 

1074 @raise ValueError: Negative B{C{x}}. 

1075 ''' 

1076 return _root(x, _1_6th, zcrt) 

1077 

1078 

1079def zqrt(x): 

1080 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root, 

1081 M{x**(1 / 8)}, preserving C{type(B{x})}. 

1082 

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

1084 

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

1086 

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

1088 

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

1090 

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

1092 ''' 

1093 return _root(x, _0_125, zqrt) 

1094 

1095# **) MIT License 

1096# 

1097# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1098# 

1099# Permission is hereby granted, free of charge, to any person obtaining a 

1100# copy of this software and associated documentation files (the "Software"), 

1101# to deal in the Software without restriction, including without limitation 

1102# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1103# and/or sell copies of the Software, and to permit persons to whom the 

1104# Software is furnished to do so, subject to the following conditions: 

1105# 

1106# The above copyright notice and this permission notice shall be included 

1107# in all copies or substantial portions of the Software. 

1108# 

1109# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1110# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1111# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1112# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1113# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1114# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1115# OTHER DEALINGS IN THE SOFTWARE.