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 _N_1_0, _1_5, _copysign_0_0, _isfinite, remainder 

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

15 _xError, _xkwds_get1, _xkwds_pop2 

16from pygeodesy.fsums import _2float, Fsum, fsum, fsum1_, _isFsumTuple, _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__ = '24.05.29' 

28 

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

30_0_4142 = 0.41421356237309504880 # ... sqrt(2) - 1 

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, **name_RESIDUAL): 

39 '''New L{Fdot} precision dot product M{sum(a[i] * b[i] for 

40 i=0..len(a)-1)}. 

41 

42 @arg a: Iterable of values (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} 

43 instance). 

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

45 all positional. 

46 @kwarg name_RESIDUAL: Optional C{B{name}=NN} (C{str}) and the C{B{RESIDUAL}=0.0} 

47 threshold (C{scalar}) for raising L{ResidualError}s, see class 

48 L{Fsum<Fsum.__init__>}. 

49 

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

51 

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

53 

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

55 

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

57 

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

59 ''' 

60 Fsum.__init__(self, **name_RESIDUAL) 

61 self.fadd(_map_mul(a, b, Fdot)) 

62 

63 

64class Fhorner(Fsum): 

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

66 ''' 

67 def __init__(self, x, *cs, **name_RESIDUAL): 

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

69 i=0..len(cs)-1)}. 

70 

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

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

73 instance), all positional. 

74 @kwarg name_RESIDUAL: Optional C{B{name}=NN} (C{str}) and the C{B{RESIDUAL}=0.0} 

75 threshold (C{scalar}) for raising L{ResidualError}s, see class 

76 L{Fsum<Fsum.__init__>}. 

77 

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

79 

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

81 

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

83 

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

85 ''' 

86 Fsum.__init__(self, **name_RESIDUAL) 

87 if cs: 

88 self._fhorner(x, cs, Fhorner.__name__) 

89 else: 

90 self._fset_ps(_0_0) 

91 

92 

93class Fhypot(Fsum): 

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

95 ''' 

96 def __init__(self, *xs, **root_name_RESIDUAL_raiser): 

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

98 (raised to the power I{root}). 

99 

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

101 all positional. 

102 @kwarg root_name_RESIDUAL_raiser: Optional, exponent and C{B{root}=2} order 

103 (C{scalar}), C{B{name}=NN} (C{str}), the C{B{RESIDUAL}=0.0} 

104 threshold (C{scalar}) and C{B{raiser}=True} (C{bool}) for 

105 raising L{ResidualError}s, see class L{Fsum<Fsum.__init__>} and 

106 method L{root<Fsum.root>}. 

107 ''' 

108 r = None # _xkwds_pop2 error 

109 try: 

110 r, kwds = _xkwds_pop2(root_name_RESIDUAL_raiser, root=2) 

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

112 raiser = _Fsum__init__(self, **kwds) 

113 if xs: 

114 self._facc_power(r, xs, Fhypot, **raiser) 

115 self._fset(self.root(r, **raiser)) 

116 except Exception as X: 

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

118 

119 

120class Fpolynomial(Fsum): 

121 '''Precision polynomial evaluation. 

122 ''' 

123 def __init__(self, x, *cs, **name_RESIDUAL): 

124 '''New L{Fpolynomial} evaluation of the polynomial 

125 M{sum(cs[i] * x**i for i=0..len(cs)-1)}. 

126 

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

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

129 instance), all positional. 

130 @kwarg name_RESIDUAL: Optional C{B{name}=NN} (C{str}) and the C{B{RESIDUAL}=0.0} 

131 threshold (C{scalar}) for raising L{ResidualError}s, see class 

132 L{Fsum<Fsum.__init__>}. 

133 

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

135 

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

137 

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

139 

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

141 ''' 

142 Fsum.__init__(self, *cs[:1], **name_RESIDUAL) 

143 n = len(cs) - 1 

144 if n > 0: 

145 self.fadd(_1map_mul(cs[1:], _powers(x, n))) 

146 elif n < 0: 

147 self._fset_ps(_0_0) 

148 

149 

150class Fpowers(Fsum): 

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

152 ''' 

153 def __init__(self, power, *xs, **name_RESIDUAL_raiser): 

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

155 

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

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

158 all positional. 

159 @kwarg name_RESIDUAL_raiser: Optional C{B{name}=NN} (C{str}), the C{B{RESIDUAL}=0.0} 

160 threshold (C{scalar}) and C{B{raiser}=True} (C{bool}) for raising 

161 L{ResidualError}s, see class L{Fsum<Fsum.__init__>} and method 

162 L{fpow<Fsum.fpow>}. 

163 ''' 

164 try: 

165 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser) 

166 if xs: 

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

168 except Exception as X: 

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

170 

171 

172class Froot(Fsum): 

173 '''The root of a precision summation. 

174 ''' 

175 def __init__(self, root, *xs, **name_RESIDUAL_raiser): 

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

177 

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

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

180 all positional. 

181 @kwarg name_RESIDUAL_raiser: Optional C{B{name}=NN} (C{str}), the C{B{RESIDUAL}=0.0} 

182 threshold (C{scalar}) and C{B{raiser}=True} (C{bool}) for raising 

183 L{ResidualError}s, see class L{Fsum<Fsum.__init__>} and method 

184 L{fpow<Fsum.fpow>}. 

185 ''' 

186 try: 

187 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser) 

188 if xs: 

189 self.fadd(xs) 

190 self._fset(self.root(root, **raiser)) 

191 except Exception as X: 

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

193 

194 

195class Fcbrt(Froot): 

196 '''Cubic root of a precision summation. 

197 ''' 

198 def __init__(self, *xs, **name_RESIDUAL_raiser): 

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

200 

201 @see: Class L{Froot} for further details. 

202 ''' 

203 Froot.__init__(self, 3, *xs, **name_RESIDUAL_raiser) 

204 

205 

206class Fsqrt(Froot): 

207 '''Square root of a precision summation. 

208 ''' 

209 def __init__(self, *xs, **name_RESIDUAL_raiser): 

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

211 

212 @see: Class L{Froot} for further details. 

213 ''' 

214 Froot.__init__(self, 2, *xs, **name_RESIDUAL_raiser) 

215 

216 

217def _Fsum__init__(inst, raiser=MISSING, **name_RESIDUAL): 

218 '''(INTERNAL) Init an C{F...} instance above. 

219 ''' 

220 Fsum.__init__(inst, **name_RESIDUAL) # PYCHOK self 

221 inst._fset_ps(_0_0) 

222 return {} if raiser is MISSING else dict(raiser=raiser) 

223 

224 

225def bqrt(x): 

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

227 preserving C{type(B{x})}. 

228 

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

230 

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

232 

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

234 

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

236 

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

238 ''' 

239 return _root(x, _0_25, bqrt) 

240 

241 

242try: 

243 from math import cbrt as _cbrt # Python 3.11+ 

244 

245except ImportError: # Python 3.10- 

246 

247 def _cbrt(x): 

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

249 ''' 

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

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

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

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

254 

255 

256def cbrt(x): 

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

258 

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

260 

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

262 

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

264 ''' 

265 if _isFsumTuple(x): 

266 r = abs(x).fpow(_1_3rd) 

267 if x.signOf() < 0: 

268 r = -r 

269 else: 

270 r = _cbrt(x) 

271 return r # cbrt(-0.0) == -0.0 

272 

273 

274def cbrt2(x): # PYCHOK attr 

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

276 

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

278 

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

280 

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

282 ''' 

283 return abs(x).fpow(_2_3rd) if _isFsumTuple(x) else _cbrt(x**2) 

284 

285 

286def euclid(x, y): 

287 '''I{Appoximate} the norm M{sqrt(x**2 + y**2)} by 

288 M{max(abs(x), abs(y)) + min(abs(x), abs(y)) * 0.4142...}. 

289 

290 @arg x: X component (C{scalar} or L{Fsum} instance). 

291 @arg y: Y component (C{scalar} or L{Fsum} instance). 

292 

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

294 

295 @see: Function L{euclid_}. 

296 ''' 

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

298 if y > x: 

299 x, y = y, x 

300 return x + y * _0_4142 # XXX * _0_5 before 20.10.02 

301 

302 

303def euclid_(*xs): 

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

305 cascaded L{euclid}. 

306 

307 @arg xs: X arguments (each C{scalar} or an L{Fsum} 

308 instance), all positional. 

309 

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

311 

312 @see: Function L{euclid}. 

313 ''' 

314 e = _0_0 

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

316 # e = euclid(x, e) 

317 if e < x: 

318 e, x = x, e 

319 if x: 

320 e += x * _0_4142 

321 return e 

322 

323 

324def facos1(x): 

325 '''Fast approximation of L{pygeodesy.acos1}C{(B{x})}. 

326 

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

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

329 ''' 

330 a = fabs(x) 

331 if a < EPS0: 

332 r = PI_2 

333 elif a < EPS1: 

334 H = Fhorner(-a, 1.5707288, 0.2121144, 0.0742610, 0.0187293) 

335 H *= Fsqrt(_1_0, -a) 

336 r = float(H) 

337 if x < 0: 

338 r = PI - r 

339 else: 

340 r = PI if x < 0 else _0_0 

341 return r 

342 

343 

344def fasin1(x): # PYCHOK no cover 

345 '''Fast approximation of L{pygeodesy.asin1}C{(B{x})}. 

346 

347 @see: L{facos1}. 

348 ''' 

349 return PI_2 - facos1(x) 

350 

351 

352def fatan(x): 

353 '''Fast approximation of C{atan(B{x})}. 

354 ''' 

355 a = fabs(x) 

356 if a < _1_0: 

357 r = fatan1(a) if a else _0_0 

358 elif a > _1_0: 

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

360 else: 

361 r = PI_4 

362 if x < 0: # copysign0(r, x) 

363 r = -r 

364 return r 

365 

366 

367def fatan1(x): 

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

369 

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

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

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

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

374 IEEE Signal Processing Magazine, 111, May 2006. 

375 ''' 

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

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

378 # x * (1.0300981633974482 + x * (-0.1784 - x * 0.0663)) 

379 H = Fhorner(x, _0_0, 1.0300981634, -0.1784, -0.0663) 

380 return float(H) 

381 

382 

383def fatan2(y, x): 

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

385 

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

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

388 and L{fatan1}. 

389 ''' 

390 a, b = fabs(x), fabs(y) 

391 if b > a: 

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

393 elif a > b: 

394 r = fatan1(b / a) if b else _0_0 

395 elif a: # a == b != 0 

396 r = PI_4 

397 else: # a == b == 0 

398 return _0_0 

399 if x < 0: 

400 r = PI - r 

401 if y < 0: # copysign0(r, y) 

402 r = -r 

403 return r 

404 

405 

406def favg(a, b, f=_0_5): 

407 '''Return the precision average of two values. 

408 

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

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

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

412 

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

414 ''' 

415# @raise ValueError: Fraction out of range. 

416# ''' 

417# if not 0 <= f <= 1: # XXX restrict fraction? 

418# raise _ValueError(fraction=f) 

419 # a + f * (b - a) == a * (1 - f) + b * f 

420 return fsum1_(a, a * (-f), b * f) 

421 

422 

423def fdot(a, *b): 

424 '''Return the precision dot product M{sum(a[i] * b[i] for 

425 i=0..len(a))}. 

426 

427 @arg a: Iterable of values (each C{scalar}). 

428 @arg b: Other values (each C{scalar}), all positional. 

429 

430 @return: Dot product (C{float}). 

431 

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

433 

434 @see: Class L{Fdot} and U{Algorithm 5.10 B{DotK} 

435 <https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>}. 

436 ''' 

437 return fsum(_map_mul(a, b, fdot)) 

438 

439 

440def fdot3(xs, ys, zs, start=0): 

441 '''Return the precision dot product M{start + 

442 sum(a[i] * b[i] * c[i] for i=0..len(a)-1)}. 

443 

444 @arg xs: Iterable (each C{scalar} or an L{Fsum} or 

445 L{Fsum2Tuple} instance). 

446 @arg ys: Iterable (each C{scalar} or an L{Fsum} or 

447 L{Fsum2Tuple} instance). 

448 @arg zs: Iterable (each C{scalar} or an L{Fsum} or 

449 L{Fsum2Tuple} instance). 

450 @kwarg start: Optional bias (C{scalar} or an L{Fsum} 

451 or L{Fsum2Tuple}). 

452 

453 @return: Dot product (C{float}). 

454 

455 @raise LenError: Unequal C{len(B{xs})}, C{len(B{ys})} 

456 and/or C{len(B{zs})}. 

457 

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

459 ''' 

460 def _mul3(xs, ys, zs, s, p): 

461 if s: 

462 yield s 

463 if p: 

464 yield _1_0 

465 _F = Fsum 

466 for x, y, z in zip(xs, ys, zs): 

467 yield (_F(x) * y) * z 

468 if p: 

469 yield _N_1_0 

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 return fsum(_mul3(xs, ys, zs, start, n < 4)) 

476 

477 

478def fhorner(x, *cs): 

479 '''Evaluate the polynomial M{sum(cs[i] * x**i for 

480 i=0..len(cs)-1)} using the Horner form. 

481 

482 @return: Horner sum (C{float}). 

483 

484 @see: Class L{Fhorner} for further details. 

485 ''' 

486 H = Fhorner(x, *cs) 

487 return float(H) 

488 

489 

490def fidw(xs, ds, beta=2): 

491 '''Interpolate using U{Inverse Distance Weighting 

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

493 

494 @arg xs: Known values (each C{scalar} or an L{Fsum} or 

495 L{Fsum2Tuple} instance). 

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

497 or L{Fsum2Tuple} instance). 

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

499 

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

501 

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

503 

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

505 

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

507 weighted B{C{ds}} below L{EPS}. 

508 

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

510 ''' 

511 n, xs = len2(xs) 

512 if n > 1: 

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

514 if b < 0: 

515 try: # weighted 

516 _F = Fsum 

517 W = _F() 

518 X = _F() 

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

520 x = xs[i] 

521 D = _F(d) 

522 if D < EPS0: 

523 if D < 0: 

524 raise ValueError(_negative_) 

525 x = float(x) 

526 i = n 

527 break 

528 if D.fpow(b): 

529 W += D 

530 X += D.fmul(x) 

531 else: 

532 x = X.fover(W, raiser=False) 

533 i += 1 # len(xs) >= len(ds) 

534 except IndexError: 

535 i += 1 # len(xs) < i < len(ds) 

536 except Exception as X: 

537 _I = Fmt.INDEX 

538 raise _xError(X, _I(xs=i), x, _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 

553def fmean(xs): 

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

555 

556 @arg xs: Values (C{scalar} or L{Fsum} instances). 

557 

558 @return: Mean value (C{float}). 

559 

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

561 

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

563 ''' 

564 n, xs = len2(xs) 

565 if n < 1: 

566 raise LenError(fmean, xs=xs) 

567 return Fsum(*xs).fover(n) if n > 1 else _2float(index=0, xs=xs[0]) 

568 

569 

570def fmean_(*xs): 

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

572 

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

574 ''' 

575 return fmean(xs) 

576 

577 

578def fpolynomial(x, *cs, **over): 

579 '''Evaluate the polynomial M{sum(cs[i] * x**i for 

580 i=0..len(cs)) [/ over]}. 

581 

582 @kwarg over: Optional final, I{non-zero} divisor (C{scalar}). 

583 

584 @return: Polynomial value (C{float}). 

585 

586 @see: Class L{Fpolynomial} for further details. 

587 ''' 

588 P = Fpolynomial(x, *cs) 

589 d = _xkwds_get1(over, over=0) if over else 0 

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

591 

592 

593def fpowers(x, n, alts=0): 

594 '''Return a series of powers M{[x**i for i=1..n]}. 

595 

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

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

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

599 exponent (C{int}). 

600 

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

602 

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

604 

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

606 ''' 

607 if not isint(n): 

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

609 elif n < 1: 

610 raise _ValueError(n=n) 

611 

612 p = x if isint(x) or _isFsumTuple(x) else _2float(x=x) 

613 ps = tuple(_powers(p, n)) 

614 

615 if alts > 0: # x**2, x**4, ... 

616 # ps[alts-1::2] chokes PyChecker 

617 ps = ps[slice(alts-1, None, 2)] 

618 

619 return ps 

620 

621 

622try: 

623 from math import prod as fprod # Python 3.8 

624except ImportError: 

625 

626 def fprod(xs, start=1): 

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

628 

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

630 C{scalar} or an L{Fsum}). 

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

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

633 

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

635 

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

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

638 ''' 

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

640 

641 

642def frandoms(n, seeded=None): 

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

644 

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

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

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

648 

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

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

651 ''' 

652 from random import gauss, random, seed, shuffle 

653 

654 if seeded is None: 

655 pass 

656 elif seeded and isbool(seeded): 

657 from time import localtime 

658 seed(localtime().tm_yday) 

659 elif isscalar(seeded): 

660 seed(seeded) 

661 

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

663 for _ in range(n): 

664 s = 0 

665 t = list(c) 

666 _a = t.append 

667 for _ in range(n * 8): 

668 v = gauss(0, random())**7 - s 

669 _a(v) 

670 s += v 

671 shuffle(t) 

672 yield t 

673 

674 

675def frange(start, number, step=1): 

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

677 

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

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

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

681 

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

683 

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

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

686 ''' 

687 if not isint(number): 

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

689 for i in range(number): 

690 yield start + (step * i) 

691 

692 

693try: 

694 from functools import reduce as freduce 

695except ImportError: 

696 try: 

697 freduce = reduce # PYCHOK expected 

698 except NameError: # Python 3+ 

699 

700 def freduce(f, xs, *start): 

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

702 ''' 

703 if start: 

704 r = v = start[0] 

705 else: 

706 r, v = 0, MISSING 

707 for v in xs: 

708 r = f(r, v) 

709 if v is MISSING: 

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

711 return r 

712 

713 

714def fremainder(x, y): 

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

716 

717 @arg x: Numerator (C{scalar}). 

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

719 

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

721 0.0) or C{NAN} for any non-finite B{C{x}}. 

722 

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

724 

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

726 project/geographiclib/>} and Python 3.7+ 

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

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

729 ''' 

730 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and 

731 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native 

732 # fmod( 0, 360) == 0.0 

733 # fmod( 360, 360) == 0.0 

734 # fmod(-0, 360) == 0.0 

735 # fmod(-0.0, 360) == -0.0 

736 # fmod(-360, 360) == -0.0 

737 # however, using the % operator ... 

738 # 0 % 360 == 0 

739 # 360 % 360 == 0 

740 # 360.0 % 360 == 0.0 

741 # -0 % 360 == 0 

742 # -360 % 360 == 0 == (-360) % 360 

743 # -0.0 % 360 == 0.0 == (-0.0) % 360 

744 # -360.0 % 360 == 0.0 == (-360.0) % 360 

745 

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

747 # == +0.0. This fixes this bug. See also Math::AngNormalize 

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

749 if _isfinite(x): 

750 try: 

751 r = remainder(x, y) if x else x 

752 except Exception as e: 

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

754 else: # handle x INF and NINF as NAN 

755 r = NAN 

756 return r 

757 

758 

759if _sys_version_info2 < (3, 8): # PYCHOK no cover 

760 from math import hypot # OK in Python 3.7- 

761 

762 def hypot_(*xs): 

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

764 

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

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

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

768 handled differently. 

769 

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

771 

772 @return: Norm (C{float}). 

773 

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

775 

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

777 

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

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

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

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

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

783 ''' 

784 return float(Fhypot(*xs, raiser=False)) 

785 

786elif _sys_version_info2 < (3, 10): 

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

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

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

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

791 

792 def hypot(x, y): 

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

794 

795 @arg x: X argument (C{scalar}). 

796 @arg y: Y argument (C{scalar}). 

797 

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

799 ''' 

800 return float(Fhypot(x, y, raiser=False)) 

801 

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

803else: 

804 from math import hypot # PYCHOK in Python 3.10+ 

805 hypot_ = hypot 

806 

807 

808def hypot1(x): 

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

810 

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

812 

813 @return: Norm (C{float}). 

814 ''' 

815 if _isFsumTuple(x): 

816 h = float(Fhypot(_1_0, x)) if x else _1_0 

817 else: 

818 h = hypot(_1_0, x) if x else _1_0 

819 return h 

820 

821 

822def hypot2(x, y): 

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

824 

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

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

827 

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

829 ''' 

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

831 if y > x: 

832 x, y = y, x 

833 if x: 

834 h2 = x**2 

835 if y: 

836 h2 *= (y / x)**2 + _1_0 

837 h2 = float(h2) 

838 else: 

839 h2 = _0_0 

840 return h2 

841 

842 

843def hypot2_(*xs): 

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

845 

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

847 L{Fsum2Tuple} instance), all positional. 

848 

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

850 

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

852 ''' 

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

854 if h2: 

855 _h = _1_0 / h2 

856 h2 = Fpowers(2, *((x * _h) for x in xs)) 

857 h2 = h2.fover(_h**2) 

858 return h2 

859 

860 

861def _map_mul(xs, ys, where): 

862 '''(INTERNAL) Yield each B{C{x * y}}. 

863 ''' 

864 n = len(ys) 

865 if len(xs) != n: # PYCHOK no cover 

866 raise LenError(where, xs=len(xs), ys=n) 

867 return _1map_mul(xs, ys) if n < 4 else map( 

868 _operator.mul, map(Fsum, xs), ys) 

869 

870 

871def _1map_mul(xs, ys): 

872 '''(INTERNAL) Yield each B{C{x * y}}, 1-primed. 

873 ''' 

874 return _1primed(map(_operator.mul, map(Fsum, xs), ys)) 

875 

876 

877def norm2(x, y): 

878 '''Normalize a 2-dimensional vector. 

879 

880 @arg x: X component (C{scalar}). 

881 @arg y: Y component (C{scalar}). 

882 

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

884 

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

886 or zero norm. 

887 ''' 

888 try: 

889 h = hypot(x, y) 

890 if h: 

891 x, y = (x / h), (y / h) 

892 else: 

893 x = _copysign_0_0(x) # pass? 

894 y = _copysign_0_0(y) 

895 except Exception as e: 

896 raise _xError(e, x=x, y=y, h=h) 

897 return x, y 

898 

899 

900def norm_(*xs): 

901 '''Normalize all n-dimensional vector components. 

902 

903 @arg xs: Components (C{scalar}s), all positional. 

904 

905 @return: Yield each component, normalized. 

906 

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

908 or zero norm. 

909 ''' 

910 try: 

911 i = x = h = None 

912 h = hypot_(*xs) 

913 _h = (_1_0 / h) if h else _0_0 

914 for i, x in enumerate(xs): 

915 yield x * _h 

916 except Exception as X: 

917 raise _xError(X, Fmt.SQUARE(xs=i), x, h=h) 

918 

919 

920def _powers(x, n): 

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

922 ''' 

923 p = 1 # type(p) == type(x) 

924 for _ in range(n): 

925 p *= x 

926 yield p 

927 

928 

929def _root(x, p, where): 

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

931 ''' 

932 try: 

933 if x > 0: 

934 return Fsum(x).fpow(p).as_iscalar 

935 elif x < 0: 

936 raise ValueError(_negative_) 

937 except Exception as X: 

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

939 return _0_0 

940 

941 

942def sqrt0(x, Error=None): 

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

944 preserving C{type(B{x})}. 

945 

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

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

948 

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

950 

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

952 

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

954 returns C{0.0}. 

955 ''' 

956 if Error and x < 0: 

957 raise Error(unstr(sqrt0, x)) 

958 return _root(x, _0_5, sqrt0) if x > EPS02 else (_0_0 if x < EPS02 else EPS0) 

959 

960 

961def sqrt3(x): 

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

963 preserving C{type(B{x})}. 

964 

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

966 

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

968 

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

970 

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

972 

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

974 ''' 

975 return _root(x, _1_5, sqrt3) 

976 

977 

978def sqrt_a(h, b): 

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

980 C{sqrt(B{h}**2 - B{b}**2)}. 

981 

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

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

984 

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

986 

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

988 

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

990 

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

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

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

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

995 ''' 

996 try: 

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

998 raise TypeError(_not_scalar_) 

999 c = fabs(h) 

1000 if c > EPS0: 

1001 s = _1_0 - (b / c)**2 

1002 if s < 0: 

1003 raise ValueError(_h_lt_b_) 

1004 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0) 

1005 else: # PYCHOK no cover 

1006 b = fabs(b) 

1007 d = c - b 

1008 if d < 0: 

1009 raise ValueError(_h_lt_b_) 

1010 d *= c + b 

1011 a = sqrt(d) if d else _0_0 

1012 except Exception as x: 

1013 raise _xError(x, h=h, b=b) 

1014 return copysign0(a, h) 

1015 

1016 

1017def zcrt(x): 

1018 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)}, 

1019 preserving C{type(B{x})}. 

1020 

1021 @arg x: Value (C{scalar} or an L{Fsum} or L{Fsum2Tuple}). 

1022 

1023 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}). 

1024 

1025 @see: Functions L{bqrt} and L{zqrt}. 

1026 

1027 @raise TypeeError: Invalid B{C{x}}. 

1028 

1029 @raise ValueError: Negative B{C{x}}. 

1030 ''' 

1031 return _root(x, _1_6th, zcrt) 

1032 

1033 

1034def zqrt(x): 

1035 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root, 

1036 M{x**(1 / 8)}, preserving C{type(B{x})}. 

1037 

1038 @arg x: Value (C{scalar} or an L{Fsum} or L{Fsum2Tuple}). 

1039 

1040 @return: I{Zenzi-quartic} root (C{float} or L{Fsum}). 

1041 

1042 @see: Functions L{bqrt} and L{zcrt}. 

1043 

1044 @raise TypeeError: Invalid B{C{x}}. 

1045 

1046 @raise ValueError: Negative B{C{x}}. 

1047 ''' 

1048 return _root(x, _0_125, zqrt) 

1049 

1050# **) MIT License 

1051# 

1052# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1053# 

1054# Permission is hereby granted, free of charge, to any person obtaining a 

1055# copy of this software and associated documentation files (the "Software"), 

1056# to deal in the Software without restriction, including without limitation 

1057# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1058# and/or sell copies of the Software, and to permit persons to whom the 

1059# Software is furnished to do so, subject to the following conditions: 

1060# 

1061# The above copyright notice and this permission notice shall be included 

1062# in all copies or substantial portions of the Software. 

1063# 

1064# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1065# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1066# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1067# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1068# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1069# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1070# OTHER DEALINGS IN THE SOFTWARE.