Coverage for pygeodesy/fmath.py: 90%

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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_pop2, _xsError 

16from pygeodesy.fsums import _2float, Fsum, fsum, _isFsum_2Tuple, _1primed, \ 

17 Fmt, unstr 

18from pygeodesy.interns import MISSING, _negative_, _not_scalar_ 

19from pygeodesy.lazily import _ALL_LAZY, _sys_version_info2 

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

21from pygeodesy.units import Int_, _isHeight, _isRadius, Float_ # PYCHOK for .heights 

22 

23from math import fabs, sqrt # pow 

24import operator as _operator # in .datums, .trf, .utm 

25 

26__all__ = _ALL_LAZY.fmath 

27__version__ = '24.10.04' 

28 

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

30_0_4142 = 0.41421356237309504880 # ... ~ 3730904090310553 / 9007199254740992 

31_2_3rd = _1_3rd * 2 

32_h_lt_b_ = 'abs(h) < abs(b)' 

33 

34 

35class Fdot(Fsum): 

36 '''Precision dot product. 

37 ''' 

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

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

40 

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

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

43 positional. 

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

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

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

47 

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

49 

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

51 

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

53 

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

55 

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

57 ''' 

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

59 Fsum.__init__(self, **kwds) 

60 self(s) 

61 

62 n = len(b) 

63 if len(a) != n: # PYCHOK no cover 

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

65 self._faddot(n, a, b, **kwds) 

66 

67 def _faddot(self, n, xs, ys, **kwds): 

68 if n > 0: 

69 _f = Fsum(**kwds) 

70 r = (_f(x).fmul(y) for x, y in zip(xs, ys)) # PYCHOK attr? 

71 self.fadd(_1primed(r) if n < 4 else r) # PYCHOK attr? 

72 return self 

73 

74 

75class Fhorner(Fsum): 

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

77 ''' 

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

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

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

81 = len(cs) - 1}. 

82 

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

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

85 all positional. 

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

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

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

89 

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

91 

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

93 

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

95 

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

97 ''' 

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

99 Fsum.__init__(self, **kwds) 

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

101 

102 

103class Fhypot(Fsum): 

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

105 ''' 

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

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

108 to the power I{root}). 

109 

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

111 positional. 

112 @kwarg root_name_f2product_nonfinites_RESIDUAL_raiser: Optional, exponent 

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

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

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

116 L{root<Fsum.root>}. 

117 ''' 

118 r = None # _xkwds_pop2 error 

119 try: 

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

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

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

123 Fsum.__init__(self, **kwds) 

124 self(_0_0) 

125 if xs: 

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

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

128 except Exception as X: 

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

130 

131 

132class Fpolynomial(Fdot): 

133 '''Precision polynomial evaluation. 

134 ''' 

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

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

137 i=0..len(cs)-1)}. 

138 

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

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

141 all positional. 

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

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

144 

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

146 

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

148 

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

150 

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

152 ''' 

153 Fsum.__init__(self, *cs[:1], **name_f2product_nonfinites_RESIDUAL) 

154 n = len(cs) - 1 

155 if n > 0: 

156 self._faddot(n, cs[1:], _powers(x, n), **name_f2product_nonfinites_RESIDUAL) 

157 elif n < 0: 

158 self(_0_0) 

159 

160 

161class Fpowers(Fsum): 

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

163 ''' 

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

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

166 

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

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

169 positional. 

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

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

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

173 L{fpow<Fsum.fpow>}. 

174 ''' 

175 try: 

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

177 Fsum.__init__(self, **kwds) 

178 self(_0_0) 

179 if xs: 

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

181 except Exception as X: 

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

183 

184 

185class Froot(Fsum): 

186 '''The root of a precision summation. 

187 ''' 

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

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

190 

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

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

193 positional. 

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

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

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

197 L{fpow<Fsum.fpow>}. 

198 ''' 

199 try: 

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

201 Fsum.__init__(self, **kwds) 

202 self(_0_0) 

203 if xs: 

204 self.fadd(xs) 

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

206 except Exception as X: 

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

208 

209 

210class Fcbrt(Froot): 

211 '''Cubic root of a precision summation. 

212 ''' 

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

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

215 

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

217 ''' 

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

219 

220 

221class Fsqrt(Froot): 

222 '''Square root of a precision summation. 

223 ''' 

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

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

226 

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

228 ''' 

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

230 

231 

232def bqrt(x): 

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

234 preserving C{type(B{x})}. 

235 

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

237 

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

239 

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

241 

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

243 

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

245 ''' 

246 return _root(x, _0_25, bqrt) 

247 

248 

249try: 

250 from math import cbrt as _cbrt # Python 3.11+ 

251 

252except ImportError: # Python 3.10- 

253 

254 def _cbrt(x): 

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

256 ''' 

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

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

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

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

261 

262 

263def cbrt(x): 

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

265 

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

267 

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

269 

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

271 ''' 

272 if _isFsum_2Tuple(x): 

273 r = abs(x).fpow(_1_3rd) 

274 if x.signOf() < 0: 

275 r = -r 

276 else: 

277 r = _cbrt(x) 

278 return r # cbrt(-0.0) == -0.0 

279 

280 

281def cbrt2(x): # PYCHOK attr 

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

283 

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

285 

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

287 

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

289 ''' 

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

291 

292 

293def euclid(x, y): 

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

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

296 

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

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

299 

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

301 

302 @see: Function L{euclid_}. 

303 ''' 

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

305 if y > x: 

306 x, y = y, x 

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

308 

309 

310def euclid_(*xs): 

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

312 L{euclid}. 

313 

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

315 all positional. 

316 

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

318 

319 @see: Function L{euclid}. 

320 ''' 

321 e = _0_0 

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

323 # e = euclid(x, e) 

324 if e < x: 

325 e, x = x, e 

326 if x: 

327 e += x * _0_4142 

328 return e 

329 

330 

331def facos1(x): 

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

333 

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

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

336 ''' 

337 a = fabs(x) 

338 if a < EPS0: 

339 r = PI_2 

340 elif a < EPS1: 

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

342 r *= sqrt(_1_0 - a) 

343 if x < 0: 

344 r = PI - r 

345 else: 

346 r = PI if x < 0 else _0_0 

347 return r 

348 

349 

350def fasin1(x): # PYCHOK no cover 

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

352 

353 @see: L{facos1}. 

354 ''' 

355 return PI_2 - facos1(x) 

356 

357 

358def _fast(x, *cs): 

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

360 ''' 

361 h = 0 

362 for c in reversed(cs): 

363 h = _fma(x, h, c) if h else c 

364 return h 

365 

366 

367def fatan(x): 

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

369 ''' 

370 a = fabs(x) 

371 if a < _1_0: 

372 r = fatan1(a) if a else _0_0 

373 elif a > _1_0: 

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

375 else: 

376 r = PI_4 

377 if x < 0: # copysign0(r, x) 

378 r = -r 

379 return r 

380 

381 

382def fatan1(x): 

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

384 

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

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

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

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

389 IEEE Signal Processing Magazine, 111, May 2006. 

390 ''' 

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

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

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

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

395 

396 

397def fatan2(y, x): 

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

399 

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

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

402 and L{fatan1}. 

403 ''' 

404 a, b = fabs(x), fabs(y) 

405 if b > a: 

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

407 elif a > b: 

408 r = fatan1(b / a) if b else _0_0 

409 elif a: # a == b != 0 

410 r = PI_4 

411 else: # a == b == 0 

412 return _0_0 

413 if x < 0: 

414 r = PI - r 

415 if y < 0: # copysign0(r, y) 

416 r = -r 

417 return r 

418 

419 

420def favg(a, b, f=_0_5, nonfinites=True): 

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

422 

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

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

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

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

427 

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

429 ''' 

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

431 return float(F) 

432 

433 

434def fdot(a, *b, **start): 

435 '''Return the precision dot product M{sum(a[i] * b[i] for ni=0..len(a))}. 

436 

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

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

439 positional. 

440 @kwarg start: Optional bias C{B{start}=0} (C{scalar}, an L{Fsum} or 

441 L{Fsum2Tuple}). 

442 

443 @return: Dot product (C{float}). 

444 

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

446 

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

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

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

450 ''' 

451 D = Fdot(a, nonfinites=True, *b, **start) 

452 return float(D) 

453 

454 

455def fdot3(xs, ys, zs, start=0): 

456 '''Return the precision dot product M{start + sum(a[i] * b[i] * c[i] 

457 for i=0..len(a)-1)}. 

458 

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

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

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

462 @kwarg start: Optional bias (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

463 

464 @return: Dot product (C{float}). 

465 

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

467 ''' 

468 n = len(xs) 

469 if not n == len(ys) == len(zs): 

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

471 

472 D = Fdot((), nonfinites=True, start=start) 

473 _f = Fsum(nonfinites=True) # f2product=True 

474 r = (_f(x).f2mul_(y, z) for x, y, z in zip(xs, ys, zs)) 

475 D = D.fadd(_1primed(r) if n < 4 else r) 

476 return float(D) 

477 

478 

479def fhorner(x, *cs, **incx): 

480 '''Horner form evaluation of polynomial M{sum(cs[i] * x**i for 

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

482 where C{n = len(cs) - 1}. 

483 

484 @return: Horner sum (C{float}). 

485 

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

487 ''' 

488 H = Fhorner(x, *cs, **incx) 

489 return float(H) 

490 

491 

492def fidw(xs, ds, beta=2): 

493 '''Interpolate using U{Inverse Distance Weighting 

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

495 

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

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

498 L{Fsum2Tuple}). 

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

500 

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

502 

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

504 

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

506 

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

508 weighted B{C{ds}} below L{EPS}. 

509 

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

511 ''' 

512 n, xs = len2(xs) 

513 if n > 1: 

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

515 if b < 0: 

516 try: # weighted 

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

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

519 x = xs[i] 

520 D = _d(d) 

521 if D < EPS0: 

522 if D < 0: 

523 raise ValueError(_negative_) 

524 x = float(x) 

525 i = n 

526 break 

527 if D.fpow(b): 

528 W += D 

529 X += D.fmul(x) 

530 else: 

531 x = X.fover(W, raiser=False) 

532 i += 1 # len(xs) >= len(ds) 

533 except IndexError: 

534 i += 1 # len(xs) < i < len(ds) 

535 except Exception as X: 

536 _I = Fmt.INDEX 

537 raise _xError(X, _I(xs=i), x, 

538 _I(ds=i), d) 

539 else: # b == 0 

540 x = fsum(xs) / n # fmean(xs) 

541 i = n 

542 elif n: 

543 x = float(xs[0]) 

544 i = n 

545 else: 

546 x = _0_0 

547 i, _ = len2(ds) 

548 if i != n: 

549 raise LenError(fidw, xs=n, ds=i) 

550 return x 

551 

552 

553try: 

554 from math import fma as _fma 

555except ImportError: # PYCHOK DSPACE! 

556 

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

558 return x * y + z 

559 

560 

561def fma(x, y, z, **nonfinites): # **raiser 

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

563 or an equivalent implementation. 

564 

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

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

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

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

569 to override default L{nonfiniterrors} 

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

571 

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

573 ''' 

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

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

576 

577 

578def _Fm2(x, nonfinites=None, **raiser): 

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

580 ''' 

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

582 

583 

584def fmean(xs): 

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

586 

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

588 

589 @return: Mean value (C{float}). 

590 

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

592 

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

594 ''' 

595 n, xs = len2(xs) 

596 if n < 1: 

597 raise LenError(fmean, xs=xs) 

598 M = Fsum(*xs, nonfinites=True) 

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

600 

601 

602def fmean_(*xs, **nonfinites): 

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

604 

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

606 ''' 

607 return fmean(xs, **nonfinites) 

608 

609 

610def f2mul_(x, *ys, **nonfinites): # **raiser 

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

612 

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

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

615 positional. 

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

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

618 

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

620 

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

622 ''' 

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

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

625 

626 

627def fpolynomial(x, *cs, **over_f2product_nonfinites): 

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

629 

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

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

632 L{Fpolynomial<Fpolynomial.__init__>}. 

633 

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

635 ''' 

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

637 P = Fpolynomial(x, *cs, **kwds) 

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

639 

640 

641def fpowers(x, n, alts=0): 

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

643 

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

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

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

647 exponent (C{int}). 

648 

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

650 

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

652 

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

654 ''' 

655 if not isint(n): 

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

657 elif n < 1: 

658 raise _ValueError(n=n) 

659 

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

661 ps = tuple(_powers(p, n)) 

662 

663 if alts > 0: # x**2, x**4, ... 

664 # ps[alts-1::2] chokes PyChecker 

665 ps = ps[slice(alts-1, None, 2)] 

666 

667 return ps 

668 

669 

670try: 

671 from math import prod as fprod # Python 3.8 

672except ImportError: 

673 

674 def fprod(xs, start=1): 

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

676 

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

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

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

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

681 

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

683 

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

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

686 ''' 

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

688 

689 

690def frandoms(n, seeded=None): 

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

692 

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

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

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

696 

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

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

699 ''' 

700 from random import gauss, random, seed, shuffle 

701 

702 if seeded is None: 

703 pass 

704 elif seeded and isbool(seeded): 

705 from time import localtime 

706 seed(localtime().tm_yday) 

707 elif isscalar(seeded): 

708 seed(seeded) 

709 

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

711 for _ in range(n): 

712 s = 0 

713 t = list(c) 

714 _a = t.append 

715 for _ in range(n * 8): 

716 v = gauss(0, random())**7 - s 

717 _a(v) 

718 s += v 

719 shuffle(t) 

720 yield t 

721 

722 

723def frange(start, number, step=1): 

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

725 

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

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

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

729 

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

731 

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

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

734 ''' 

735 if not isint(number): 

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

737 for i in range(number): 

738 yield start + (step * i) 

739 

740 

741try: 

742 from functools import reduce as freduce 

743except ImportError: 

744 try: 

745 freduce = reduce # PYCHOK expected 

746 except NameError: # Python 3+ 

747 

748 def freduce(f, xs, *start): 

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

750 ''' 

751 if start: 

752 r = v = start[0] 

753 else: 

754 r, v = 0, MISSING 

755 for v in xs: 

756 r = f(r, v) 

757 if v is MISSING: 

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

759 return r 

760 

761 

762def fremainder(x, y): 

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

764 

765 @arg x: Numerator (C{scalar}). 

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

767 

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

769 0.0) or C{NAN} for any non-finite B{C{x}}. 

770 

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

772 

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

774 project/geographiclib/>} and Python 3.7+ 

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

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

777 ''' 

778 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and 

779 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native 

780 # fmod( 0, 360) == 0.0 

781 # fmod( 360, 360) == 0.0 

782 # fmod(-0, 360) == 0.0 

783 # fmod(-0.0, 360) == -0.0 

784 # fmod(-360, 360) == -0.0 

785 # however, using the % operator ... 

786 # 0 % 360 == 0 

787 # 360 % 360 == 0 

788 # 360.0 % 360 == 0.0 

789 # -0 % 360 == 0 

790 # -360 % 360 == 0 == (-360) % 360 

791 # -0.0 % 360 == 0.0 == (-0.0) % 360 

792 # -360.0 % 360 == 0.0 == (-360.0) % 360 

793 

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

795 # == +0.0. This fixes this bug. See also Math::AngNormalize 

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

797 if isfinite(x): 

798 try: 

799 r = remainder(x, y) if x else x 

800 except Exception as e: 

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

802 else: # handle x INF and NINF as NAN 

803 r = NAN 

804 return r 

805 

806 

807if _sys_version_info2 < (3, 8): # PYCHOK no cover 

808 from math import hypot # OK in Python 3.7- 

809 

810 def hypot_(*xs): 

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

812 

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

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

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

816 handled differently. 

817 

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

819 

820 @return: Norm (C{float}). 

821 

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

823 

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

825 

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

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

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

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

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

831 ''' 

832 return float(_Hypot(*xs)) 

833 

834elif _sys_version_info2 < (3, 10): 

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

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

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

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

839 

840 def hypot(x, y): 

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

842 

843 @arg x: X argument (C{scalar}). 

844 @arg y: Y argument (C{scalar}). 

845 

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

847 ''' 

848 return float(_Hypot(x, y)) 

849 

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

851else: 

852 from math import hypot # PYCHOK in Python 3.10+ 

853 hypot_ = hypot 

854 

855 

856def _Hypot(*xs): 

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

858 ''' 

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

860 

861 

862def hypot1(x): 

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

864 

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

866 

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

868 ''' 

869 h = _1_0 

870 if x: 

871 if _isFsum_2Tuple(x): 

872 h = _Hypot(h, x) 

873 h = float(h) 

874 else: 

875 h = hypot(h, x) 

876 return h 

877 

878 

879def hypot2(x, y): 

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

881 

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

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

884 

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

886 ''' 

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

888 if y > x: 

889 x, y = y, x 

890 h2 = x**2 

891 if h2 and y: 

892 h2 *= (y / x)**2 + _1_0 

893 return float(h2) 

894 

895 

896def hypot2_(*xs): 

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

898 

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

900 L{Fsum2Tuple}), all positional. 

901 

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

903 

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

905 ''' 

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

907 if h2: # and isfinite(h2) 

908 _h = _1_0 / h2 

909 xs = ((x * _h) for x in xs) 

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

911 h2 = H2.fover(_h**2) 

912 return h2 

913 

914 

915def norm2(x, y): 

916 '''Normalize a 2-dimensional vector. 

917 

918 @arg x: X component (C{scalar}). 

919 @arg y: Y component (C{scalar}). 

920 

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

922 

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

924 or zero norm. 

925 ''' 

926 try: 

927 h = None 

928 h = hypot(x, y) 

929 if h: 

930 x, y = (x / h), (y / h) 

931 else: 

932 x = _copysign_0_0(x) # pass? 

933 y = _copysign_0_0(y) 

934 except Exception as e: 

935 raise _xError(e, x=x, y=y, h=h) 

936 return x, y 

937 

938 

939def norm_(*xs): 

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

941 

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

943 L{Fsum2Tuple}), all positional. 

944 

945 @return: Yield each component, normalized. 

946 

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

948 or zero norm. 

949 ''' 

950 try: 

951 i = h = None 

952 x = xs 

953 h = hypot_(*xs) 

954 _h = (_1_0 / h) if h else _0_0 

955 for i, x in enumerate(xs): 

956 yield x * _h 

957 except Exception as X: 

958 raise _xsError(X, xs, i, x, h=h) 

959 

960 

961def _powers(x, n): 

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

963 ''' 

964 p = 1 # type(p) == type(x) 

965 for _ in range(n): 

966 p *= x 

967 yield p 

968 

969 

970def _root(x, p, where): 

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

972 ''' 

973 try: 

974 if x > 0: 

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

976 return r.fpow(p).as_iscalar 

977 elif x < 0: 

978 raise ValueError(_negative_) 

979 except Exception as X: 

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

981 return _0_0 

982 

983 

984def sqrt0(x, Error=None): 

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

986 preserving C{type(B{x})}. 

987 

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

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

990 

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

992 

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

994 

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

996 returns C{0.0}. 

997 ''' 

998 if Error and x < 0: 

999 raise Error(unstr(sqrt0, x)) 

1000 return _root(x, _0_5, sqrt0) if x > EPS02 else ( 

1001 _0_0 if x < EPS02 else EPS0) 

1002 

1003 

1004def sqrt3(x): 

1005 '''Return the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)}, 

1006 preserving C{type(B{x})}. 

1007 

1008 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1009 

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

1011 

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

1013 

1014 @raise ValueError: Negative B{C{x}}. 

1015 

1016 @see: Functions L{cbrt} and L{cbrt2}. 

1017 ''' 

1018 return _root(x, _1_5, sqrt3) 

1019 

1020 

1021def sqrt_a(h, b): 

1022 '''Compute C{I{a}} side of a right-angled triangle from 

1023 C{sqrt(B{h}**2 - B{b}**2)}. 

1024 

1025 @arg h: Hypotenuse or outer annulus radius (C{scalar}). 

1026 @arg b: Triangle side or inner annulus radius (C{scalar}). 

1027 

1028 @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}). 

1029 

1030 @raise TypeError: Non-scalar B{C{h}} or B{C{b}}. 

1031 

1032 @raise ValueError: If C{abs(B{h}) < abs(B{b})}. 

1033 

1034 @see: Inner tangent chord B{I{d}} of an U{annulus 

1035 <https://WikiPedia.org/wiki/Annulus_(mathematics)>} 

1036 and function U{annulus_area<https://People.SC.FSU.edu/ 

1037 ~jburkardt/py_src/geometry/geometry.py>}. 

1038 ''' 

1039 try: 

1040 if not (_isHeight(h) and _isRadius(b)): 

1041 raise TypeError(_not_scalar_) 

1042 c = fabs(h) 

1043 if c > EPS0: 

1044 s = _1_0 - (b / c)**2 

1045 if s < 0: 

1046 raise ValueError(_h_lt_b_) 

1047 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0) 

1048 else: # PYCHOK no cover 

1049 b = fabs(b) 

1050 d = c - b 

1051 if d < 0: 

1052 raise ValueError(_h_lt_b_) 

1053 d *= c + b 

1054 a = sqrt(d) if d else _0_0 

1055 except Exception as x: 

1056 raise _xError(x, h=h, b=b) 

1057 return copysign0(a, h) 

1058 

1059 

1060def zcrt(x): 

1061 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)}, 

1062 preserving C{type(B{x})}. 

1063 

1064 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1065 

1066 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}). 

1067 

1068 @see: Functions L{bqrt} and L{zqrt}. 

1069 

1070 @raise TypeeError: Invalid B{C{x}}. 

1071 

1072 @raise ValueError: Negative B{C{x}}. 

1073 ''' 

1074 return _root(x, _1_6th, zcrt) 

1075 

1076 

1077def zqrt(x): 

1078 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root, 

1079 M{x**(1 / 8)}, preserving C{type(B{x})}. 

1080 

1081 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1082 

1083 @return: I{Zenzi-quartic} root (C{float} or L{Fsum}). 

1084 

1085 @see: Functions L{bqrt} and L{zcrt}. 

1086 

1087 @raise TypeeError: Invalid B{C{x}}. 

1088 

1089 @raise ValueError: Negative B{C{x}}. 

1090 ''' 

1091 return _root(x, _0_125, zqrt) 

1092 

1093# **) MIT License 

1094# 

1095# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1096# 

1097# Permission is hereby granted, free of charge, to any person obtaining a 

1098# copy of this software and associated documentation files (the "Software"), 

1099# to deal in the Software without restriction, including without limitation 

1100# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1101# and/or sell copies of the Software, and to permit persons to whom the 

1102# Software is furnished to do so, subject to the following conditions: 

1103# 

1104# The above copyright notice and this permission notice shall be included 

1105# in all copies or substantial portions of the Software. 

1106# 

1107# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1108# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1109# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1110# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1111# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1112# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1113# OTHER DEALINGS IN THE SOFTWARE.