Coverage for pygeodesy/fmath.py: 91%

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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, len2 

10from pygeodesy.constants import EPS0, EPS02, EPS1, NAN, PI, PI_2, PI_4, \ 

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

12 _1_5, _copysign_0_0, _isfinite, _over, remainder 

13from pygeodesy.errors import _IsnotError, LenError, _TypeError, _ValueError, \ 

14 _xError, _xkwds_get, _xkwds_pop2 

15from pygeodesy.fsums import _2float, Fsum, fsum, fsum1_, _1primed, Fmt, unstr 

16from pygeodesy.interns import MISSING, _few_, _negative_, _not_scalar_, _too_ 

17from pygeodesy.lazily import _ALL_LAZY, _sys_version_info2 

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

19from pygeodesy.units import Int_, _isHeight, _isRadius, Float_ # PYCHOK for .heights 

20 

21from math import fabs, sqrt # pow 

22import operator as _operator # in .datums, .trf, .utm 

23 

24__all__ = _ALL_LAZY.fmath 

25__version__ = '24.04.24' 

26 

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

28_0_4142 = 0.41421356237309504880 # ... sqrt(2) - 1 

29_2_3rd = _1_3rd * 2 

30_h_lt_b_ = 'abs(h) < abs(b)' 

31 

32 

33class Fdot(Fsum): 

34 '''Precision dot product. 

35 ''' 

36 def __init__(self, a, *b, **name_RESIDUAL): 

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

38 for i=0..len(a)-1)}. 

39 

40 @arg a: Iterable, list, tuple, etc. (C{scalar}s). 

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

42 @kwarg name_RESIDUAL: Optional C{B{name}=NN} and C{B{RESIDUAL}=None}, 

43 see L{Fsum<Fsum.__init__>}. 

44 

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

46 

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

48 

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

50 ''' 

51 Fsum.__init__(self, **name_RESIDUAL) 

52 self.fadd(_map_mul(a, b, Fdot)) 

53 

54 

55class Fhorner(Fsum): 

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

57 ''' 

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

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

60 for i=0..len(cs)-1)}. 

61 

62 @arg x: Polynomial argument (C{scalar} or C{Fsum} instance). 

63 @arg cs: Polynomial coeffients (C{scalar} or C{Fsum} 

64 instances), all positional. 

65 @kwarg name_RESIDUAL: Optional C{B{name}=NN} and C{B{RESIDUAL}=None}, 

66 see L{Fsum<Fsum.__init__>}. 

67 

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

69 

70 @raise TypeError: Non-scalar B{C{x}}. 

71 

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

73 

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

75 L{Fsum.fmul}. 

76 ''' 

77 Fsum.__init__(self, **name_RESIDUAL) 

78 if cs: 

79 if isinstance(x, Fsum): 

80 _mul = self._mul_Fsum 

81 else: 

82 _mul = self._mul_scalar 

83 x = _2float(x=x) 

84 op = Fhorner.__name__ 

85 if len(cs) > 1 and x: 

86 for c in reversed(cs): 

87 self._fset_ps(_mul(x, op)) 

88 self._fadd(c, op, up=False) 

89 self._update() 

90 else: # x == 0 

91 self._fadd(cs[0], op) 

92 else: 

93 self._fset_ps(_0_0) 

94 

95 

96class Fhypot(Fsum): 

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

98 ''' 

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

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

101 

102 @arg xs: One or more components (each a C{scalar} or an C{Fsum} instance). 

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

104 C{B{name}=NN}, C{B{RESIDUAL}=None} and C{B{raiser}=True}, see 

105 class L{Fsum<Fsum.__init__>} and method L{root<Fsum.root>}. 

106 ''' 

107 r = None # _xkwds_pop2 error 

108 try: 

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

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

111 raiser = _Fsum__init__(self, **kwds) 

112 if xs: 

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

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

115 except Exception as X: 

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

117 

118 

119class Fpolynomial(Fsum): 

120 '''Precision polynomial evaluation. 

121 ''' 

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

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

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

125 

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

127 @arg cs: Polynomial coeffients (each a C{scalar} or an L{Fsum} instance), 

128 all positional. 

129 @kwarg name_RESIDUAL: Optional C{B{name}=NN} and C{B{RESIDUAL}=None}, 

130 see L{Fsum<Fsum.__init__>}. 

131 

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

133 

134 @raise TypeError: Non-scalar B{C{x}}. 

135 

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

137 

138 @see: Class L{Fhorner}, function L{fpolynomial} and 

139 method L{Fsum.fadd}. 

140 ''' 

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

142 n = len(cs) - 1 

143 if n > 0: 

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

145 elif n < 0: 

146 self._fset_ps(_0_0) 

147 

148 

149class Fpowers(Fsum): 

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

151 ''' 

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

153 '''New L{Fpowers} sum of (the I{power} of) several values. 

154 

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

156 @arg xs: One or more values (each a C{scalar} or an C{Fsum} instance). 

157 @kwarg name_RESIDUAL_raiser: Optional C{B{name}=NN}, C{B{RESIDUAL}=None} and 

158 C{B{raiser}=True}, see L{Fsum<Fsum.__init__>} and L{fpow<Fsum.fpow>}. 

159 ''' 

160 try: 

161 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser) 

162 if xs: 

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

164 except Exception as X: 

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

166 

167 

168class Froot(Fsum): 

169 '''The root of a precision summation. 

170 ''' 

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

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

173 

174 @arg root: The order (C{scalar} or C{Fsum}), non-zero. 

175 @arg xs: Values to summate (each a C{scalar} or an C{Fsum} instance). 

176 @kwarg name_RESIDUAL_raiser: Optional C{B{name}=NN}, C{B{RESIDUAL}=None} and 

177 C{B{raiser}=True}, see L{Fsum<Fsum.__init__>} and L{fpow<Fsum.fpow>}. 

178 ''' 

179 try: 

180 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser) 

181 if xs: 

182 self.fadd(xs) 

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

184 except Exception as X: 

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

186 

187 

188class Fcbrt(Froot): 

189 '''Cubic root of a precision summation. 

190 ''' 

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

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

193 

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

195 ''' 

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

197 

198 

199class Fsqrt(Froot): 

200 '''Square root of a precision summation. 

201 ''' 

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

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

204 

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

206 ''' 

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

208 

209 

210def _Fsum__init__(inst, raiser=MISSING, **name_RESIDUAL): 

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

212 ''' 

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

214 inst._fset_ps(_0_0) 

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

216 

217 

218def bqrt(x): 

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

220 preserving C{type(B{x})}. 

221 

222 @arg x: Value (C{scalar} or L{Fsum} instance). 

223 

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

225 

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

227 

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

229 ''' 

230 return _root(x, _0_25, bqrt) 

231 

232 

233try: 

234 from math import cbrt as _cbrt # Python 3.11+ 

235 

236except ImportError: # Python 3.10- 

237 

238 def _cbrt(x): 

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

240 ''' 

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

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

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

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

245 

246 

247def cbrt(x): 

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

249 

250 @arg x: Value (C{scalar} or L{Fsum} instance). 

251 

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

253 

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

255 ''' 

256 if isinstance(x, Fsum): 

257 r = (-(-x).pow(_1_3rd)) if x < 0 else x.pow(_1_3rd) 

258 else: 

259 r = _cbrt(x) 

260 return r # cbrt(-0.0) == -0.0 

261 

262 

263def cbrt2(x): # PYCHOK attr 

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

265 

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

267 

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

269 

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

271 ''' 

272 return abs(x).pow(_2_3rd) if isinstance(x, Fsum) else _cbrt(x**2) 

273 

274 

275def euclid(x, y): 

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

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

278 

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

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

281 

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

283 

284 @see: Function L{euclid_}. 

285 ''' 

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

287 if x < y: 

288 x, y = y, x 

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

290 

291 

292def euclid_(*xs): 

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

294 cascaded L{euclid}. 

295 

296 @arg xs: X arguments, positional (C{scalar}s or L{Fsum} instances). 

297 

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

299 

300 @see: Function L{euclid}. 

301 ''' 

302 e = _0_0 

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

304 # e = euclid(x, e) 

305 if e < x: 

306 e, x = x, e 

307 if x: 

308 e += x * _0_4142 

309 return e 

310 

311 

312def facos1(x): 

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

314 

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

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

317 ''' 

318 a = fabs(x) 

319 if a < EPS0: 

320 r = PI_2 

321 elif a < EPS1: 

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

323 r = float(H * sqrt(_1_0 - a)) 

324 if x < 0: 

325 r = PI - r 

326 else: 

327 r = PI if x < 0 else _0_0 

328 return r 

329 

330 

331def fasin1(x): # PYCHOK no cover 

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

333 

334 @see: L{facos1}. 

335 ''' 

336 return PI_2 - facos1(x) 

337 

338 

339def fatan(x): 

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

341 ''' 

342 a = fabs(x) 

343 if a < _1_0: 

344 r = fatan1(a) if a else _0_0 

345 elif a > _1_0: 

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

347 else: 

348 r = PI_4 

349 if x < 0: # copysign0(r, x) 

350 r = -r 

351 return r 

352 

353 

354def fatan1(x): 

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

356 

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

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

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

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

361 IEEE Signal Processing Magazine, 111, May 2006. 

362 ''' 

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

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

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

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

367 return float(H) 

368 

369 

370def fatan2(y, x): 

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

372 

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

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

375 and L{fatan1}. 

376 ''' 

377 a, b = fabs(x), fabs(y) 

378 if b > a: 

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

380 elif a > b: 

381 r = fatan1(b / a) if b else _0_0 

382 elif a: # a == b != 0 

383 r = PI_4 

384 else: # a == b == 0 

385 return _0_0 

386 if x < 0: 

387 r = PI - r 

388 if y < 0: # copysign0(r, y) 

389 r = -r 

390 return r 

391 

392 

393def favg(v1, v2, f=_0_5): 

394 '''Return the average of two values. 

395 

396 @arg v1: One value (C{scalar} or L{Fsum} instance). 

397 @arg v2: Other value (C{scalar} or L{Fsum} instance). 

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

399 

400 @return: M{v1 + f * (v2 - v1)} (C{float}). 

401 ''' 

402# @raise ValueError: Fraction out of range. 

403# ''' 

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

405# raise _ValueError(fraction=f) 

406 # v1 + f * (v2 - v1) == v1 * (1 - f) + v2 * f 

407 return fsum1_(v1, -f * v1, f * v2) 

408 

409 

410def fdot(a, *b): 

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

412 i=0..len(a))}. 

413 

414 @arg a: Iterable, list, tuple, etc. (C{scalar}s). 

415 @arg b: All positional arguments (C{scalar}s). 

416 

417 @return: Dot product (C{float}). 

418 

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

420 

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

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

423 ''' 

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

425 

426 

427def fdot3(a, b, c, start=0): 

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

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

430 

431 @arg a: Iterable, list, tuple, etc. (C{scalar}s). 

432 @arg b: Iterable, list, tuple, etc. (C{scalar}s). 

433 @arg c: Iterable, list, tuple, etc. (C{scalar}s). 

434 @kwarg start: Optional bias (C{scalar}). 

435 

436 @return: Dot product (C{float}). 

437 

438 @raise LenError: Unequal C{len(B{a})}, C{len(B{b})} 

439 and/or C{len(B{c})}. 

440 

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

442 ''' 

443 def _mul3(a, b, c): # map function 

444 return a * b * c 

445 

446 def _mul3_(a, b, c, start): 

447 yield start 

448 for abc in map(_mul3, a, b, c): 

449 yield abc 

450 

451 if not len(a) == len(b) == len(c): 

452 raise LenError(fdot3, a=len(a), b=len(b), c=len(c)) 

453 

454 return fsum(_mul3_(a, b, c, start) if start else map(_mul3, a, b, c)) 

455 

456 

457def fhorner(x, *cs): 

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

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

460 

461 @arg x: Polynomial argument (C{scalar}). 

462 @arg cs: Polynomial coeffients (C{scalar}s). 

463 

464 @return: Horner value (C{float}). 

465 

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

467 

468 @raise TypeError: Non-scalar B{C{x}}. 

469 

470 @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. 

471 

472 @see: Function L{fpolynomial} and class L{Fhorner}. 

473 ''' 

474 H = Fhorner(x, *cs) 

475 return float(H) 

476 

477 

478def fidw(xs, ds, beta=2): 

479 '''Interpolate using U{Inverse Distance Weighting 

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

481 

482 @arg xs: Known values (C{scalar}s). 

483 @arg ds: Non-negative distances (C{scalar}s). 

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

485 

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

487 

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

489 

490 @raise ValueError: Invalid B{C{beta}}, negative B{C{ds}} value, 

491 weighted B{C{ds}} below L{EPS}. 

492 

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

494 ''' 

495 n, xs = len2(xs) 

496 d, ds = len2(ds) 

497 if n != d or n < 1: 

498 raise LenError(fidw, xs=n, ds=d) 

499 

500 d, x = min(zip(ds, xs)) 

501 if d > EPS0 and n > 1: 

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

503 if b < 0: 

504 ws = tuple(float(d)**b for d in ds) 

505 t = fsum(_1map_mul(xs, ws)) # Fdot(xs, *ws) 

506 x = _over(t, fsum(ws, floats=True)) 

507 else: # b == 0 

508 x = fsum(xs) / n # fmean(xs) 

509 elif d < 0: # PYCHOK no cover 

510 n = Fmt.SQUARE(distance=ds.index(d)) 

511 raise _ValueError(n, d, txt=_negative_) 

512 return x 

513 

514 

515def fmean(xs): 

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

517 

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

519 

520 @return: Mean value (C{float}). 

521 

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

523 

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

525 ''' 

526 n, xs = len2(xs) 

527 if n < 1: 

528 raise LenError(fmean, xs=xs) 

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

530 

531 

532def fmean_(*xs): 

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

534 

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

536 ''' 

537 return fmean(xs) 

538 

539 

540def fpolynomial(x, *cs, **over): 

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

542 i=0..len(cs)) [/ over]}. 

543 

544 @arg x: Polynomial argument (C{scalar}). 

545 @arg cs: Polynomial coeffients (C{scalar}s), all 

546 positional. 

547 @kwarg over: Optional final, I{non-zero} divisor 

548 (C{scalar}). 

549 

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

551 

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

553 

554 @raise TypeError: Non-scalar B{C{x}}. 

555 

556 @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. 

557 

558 @see: Function L{fhorner} and class L{Fpolynomial}. 

559 ''' 

560 P = Fpolynomial(x, *cs) 

561 d = _xkwds_get(over, over=0) if over else 0 

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

563 

564 

565def fpowers(x, n, alts=0): 

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

567 

568 @arg x: Value (C{scalar} or L{Fsum}). 

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

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

571 exponent (C{int}). 

572 

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

574 

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

576 

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

578 ''' 

579 if not isint(n): 

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

581 elif n < 1: 

582 raise _ValueError(n=n) 

583 

584 p = x if isint(x) or isinstance(x, Fsum) else _2float(x=x) 

585 ps = tuple(_powers(p, n)) 

586 

587 if alts > 0: # x**2, x**4, ... 

588 # ps[alts-1::2] chokes PyChecker 

589 ps = ps[slice(alts-1, None, 2)] 

590 

591 return ps 

592 

593 

594try: 

595 from math import prod as fprod # Python 3.8 

596except ImportError: 

597 

598 def fprod(xs, start=1): 

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

600 

601 @arg xs: Terms to be multiplied, an iterable, list, 

602 tuple, etc. (C{scalar}s). 

603 @kwarg start: Initial term, also the value returned 

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

605 

606 @return: The product (C{float}). 

607 

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

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

610 ''' 

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

612 

613 

614def frandoms(n, seeded=None): 

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

616 

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

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

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

620 

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

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

623 ''' 

624 from random import gauss, random, seed, shuffle 

625 

626 if seeded is None: 

627 pass 

628 elif seeded and isbool(seeded): 

629 from time import localtime 

630 seed(localtime().tm_yday) 

631 elif isscalar(seeded): 

632 seed(seeded) 

633 

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

635 for _ in range(n): 

636 s = 0 

637 t = list(c) 

638 _a = t.append 

639 for _ in range(n * 8): 

640 v = gauss(0, random())**7 - s 

641 _a(v) 

642 s += v 

643 shuffle(t) 

644 yield t 

645 

646 

647def frange(start, number, step=1): 

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

649 

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

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

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

653 

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

655 

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

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

658 ''' 

659 if not isint(number): 

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

661 for i in range(number): 

662 yield start + (step * i) 

663 

664 

665try: 

666 from functools import reduce as freduce 

667except ImportError: 

668 try: 

669 freduce = reduce # PYCHOK expected 

670 except NameError: # Python 3+ 

671 

672 def freduce(f, xs, *start): 

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

674 ''' 

675 if start: 

676 r = v = start[0] 

677 else: 

678 r, v = 0, MISSING 

679 for v in xs: 

680 r = f(r, v) 

681 if v is MISSING: 

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

683 return r 

684 

685 

686def fremainder(x, y): 

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

688 

689 @arg x: Numerator (C{scalar}). 

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

691 

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

693 0.0) or C{NAN} for any non-finite B{C{x}}. 

694 

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

696 

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

698 project/geographiclib/>} and Python 3.7+ 

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

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

701 ''' 

702 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and 

703 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native 

704 # fmod( 0, 360) == 0.0 

705 # fmod( 360, 360) == 0.0 

706 # fmod(-0, 360) == 0.0 

707 # fmod(-0.0, 360) == -0.0 

708 # fmod(-360, 360) == -0.0 

709 # however, using the % operator ... 

710 # 0 % 360 == 0 

711 # 360 % 360 == 0 

712 # 360.0 % 360 == 0.0 

713 # -0 % 360 == 0 

714 # -360 % 360 == 0 == (-360) % 360 

715 # -0.0 % 360 == 0.0 == (-0.0) % 360 

716 # -360.0 % 360 == 0.0 == (-360.0) % 360 

717 

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

719 # == +0.0. This fixes this bug. See also Math::AngNormalize 

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

721 if _isfinite(x): 

722 try: 

723 r = remainder(x, y) if x else x 

724 except Exception as e: 

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

726 else: # handle x INF and NINF as NAN 

727 r = NAN 

728 return r 

729 

730 

731if _sys_version_info2 < (3, 8): # PYCHOK no cover 

732 from math import hypot # OK in Python 3.7- 

733 

734 def hypot_(*xs): 

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

736 

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

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

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

740 handled differently. 

741 

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

743 

744 @return: Norm (C{float}). 

745 

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

747 

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

749 

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

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

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

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

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

755 ''' 

756 _, R = _h_xs2(xs, True, hypot_) 

757 return float(R) 

758 

759elif _sys_version_info2 < (3, 10): 

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

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

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

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

764 

765 def hypot(x, y): 

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

767 

768 @arg x: X argument (C{scalar}). 

769 @arg y: Y argument (C{scalar}). 

770 

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

772 ''' 

773 return (float(Fhypot(x, y, raiser=False)) if y else 

774 fabs(x)) if x else fabs(y) 

775 

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

777else: 

778 from math import hypot # PYCHOK in Python 3.10+ 

779 hypot_ = hypot 

780 

781 

782def _h_xs2(xs, pot_, which): 

783 '''(INTERNAL) Helper for L{hypot_} and L{hypot2_}. 

784 ''' 

785 n, xs = len2(xs) 

786 if n > 0: 

787 h = float(max(map(abs, xs))) # NOT fabs! 

788 if h < EPS0: 

789 R = _0_0 

790 elif n > 1: 

791 if pot_: 

792 if h != _1_0: 

793 xs = ((x / h) for x in xs) 

794 R = Fhypot(*xs, raiser=False) * h 

795 else: 

796 R = Fpowers(2, *xs) 

797 else: 

798 R = h if pot_ else (h**2) 

799 return h, R 

800 

801 raise _ValueError(unstr(which, xs), txt=_too_(_few_)) 

802 

803 

804def hypot1(x): 

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

806 

807 @arg x: Argument (C{scalar} or L{Fsum} instance). 

808 

809 @return: Norm (C{float}). 

810 ''' 

811 return hypot(_1_0, x) if x else _1_0 

812 

813 

814def hypot2(x, y): 

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

816 

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

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

819 

820 @return: C{B{x}**2 + B{y}**2} (C{float} or L{Fsum}). 

821 ''' 

822 if abs(x) < abs(y): # NOT fabs! 

823 x, y = y, x 

824 if x: 

825 h2 = x**2 

826 if y: 

827 h2 *= (y / x)**2 + _1_0 

828 else: 

829 h2 = _0_0 

830 return h2 

831 

832 

833def hypot2_(*xs): 

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

835 

836 @arg xs: X arguments (C{scalar}s or L{Fsum} instances), 

837 all positional. 

838 

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

840 

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

842 

843 @raise ValueError: Invalid or no B{C{xs}} value. 

844 

845 @see: Function L{hypot_}. 

846 ''' 

847 _, R = _h_xs2(xs, False, hypot2_) 

848 return float(R) 

849 

850 

851def _map_mul(a, b, where): 

852 '''(INTERNAL) Yield each B{C{a * b}}. 

853 ''' 

854 n = len(b) 

855 if len(a) != n: # PYCHOK no cover 

856 raise LenError(where, a=len(a), b=n) 

857 return _1map_mul(a, b) if n < 4 else map(_operator.mul, a, b) 

858 

859 

860def _1map_mul(a, b): 

861 '''(INTERNAL) Yield each B{C{a * b}}, 1-primed. 

862 ''' 

863 return _1primed(map(_operator.mul, a, b)) 

864 

865 

866def norm2(x, y): 

867 '''Normalize a 2-dimensional vector. 

868 

869 @arg x: X component (C{scalar}). 

870 @arg y: Y component (C{scalar}). 

871 

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

873 

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

875 or zero norm. 

876 ''' 

877 try: 

878 h = hypot(x, y) 

879 if h: 

880 x, y = (x / h), (y / h) 

881 else: 

882 x = _copysign_0_0(x) # pass? 

883 y = _copysign_0_0(y) 

884 except Exception as e: 

885 raise _xError(e, x=x, y=y, h=h) 

886 return x, y 

887 

888 

889def norm_(*xs): 

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

891 

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

893 

894 @return: Yield each component, normalized. 

895 

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

897 or zero norm. 

898 ''' 

899 try: 

900 i = x = h = None 

901 h = hypot_(*xs) 

902 _h = (_1_0 / h) if h else _0_0 

903 for i, x in enumerate(xs): 

904 yield x * _h 

905 except Exception as X: 

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

907 

908 

909def _powers(x, n): 

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

911 ''' 

912 p = 1 # type(p) == type(x) 

913 for _ in range(n): 

914 p *= x 

915 yield p 

916 

917 

918def _root(x, p, where): 

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

920 ''' 

921 try: 

922 if x > 0: 

923 return pow(x, p) 

924 elif x < 0: 

925 raise ValueError(_negative_) 

926 except Exception as X: 

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

928 return _0_0 

929 

930 

931def sqrt0(x, Error=None): 

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

933 preserving C{type(B{x})}. 

934 

935 @arg x: Value (C{scalar} or L{Fsum} instance). 

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

937 

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

939 

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

941 returns C{0.0}. 

942 ''' 

943 if Error and x < 0: 

944 raise Error(unstr(sqrt0, x)) 

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

946 

947 

948def sqrt3(x): 

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

950 preserving C{type(B{x})}. 

951 

952 @arg x: Value (C{scalar} or L{Fsum} instance). 

953 

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

955 

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

957 

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

959 ''' 

960 return _root(x, _1_5, sqrt3) 

961 

962 

963def sqrt_a(h, b): 

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

965 C{sqrt(B{h}**2 - B{b}**2)}. 

966 

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

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

969 

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

971 

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

973 

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

975 

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

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

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

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

980 ''' 

981 try: 

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

983 raise TypeError(_not_scalar_) 

984 c = fabs(h) 

985 if c > EPS0: 

986 s = _1_0 - (b / c)**2 

987 if s < 0: 

988 raise ValueError(_h_lt_b_) 

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

990 else: # PYCHOK no cover 

991 b = fabs(b) 

992 d = c - b 

993 if d < 0: 

994 raise ValueError(_h_lt_b_) 

995 d *= c + b 

996 a = sqrt(d) if d else _0_0 

997 except Exception as x: 

998 raise _xError(x, h=h, b=b) 

999 return copysign0(a, h) 

1000 

1001 

1002def zcrt(x): 

1003 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)}, 

1004 preserving C{type(B{x})}. 

1005 

1006 @arg x: Value (C{scalar} or L{Fsum} instance). 

1007 

1008 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}). 

1009 

1010 @see: Functions L{bqrt} and L{zqrt}. 

1011 

1012 @raise ValueError: Negative B{C{x}}. 

1013 ''' 

1014 return _root(x, _1_6th, zcrt) 

1015 

1016 

1017def zqrt(x): 

1018 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root, 

1019 M{x**(1 / 8)}, preserving C{type(B{x})}. 

1020 

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

1022 

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

1024 

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

1026 

1027 @raise ValueError: Negative B{C{x}}. 

1028 ''' 

1029 return _root(x, _0_125, zqrt) 

1030 

1031# **) MIT License 

1032# 

1033# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1034# 

1035# Permission is hereby granted, free of charge, to any person obtaining a 

1036# copy of this software and associated documentation files (the "Software"), 

1037# to deal in the Software without restriction, including without limitation 

1038# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1039# and/or sell copies of the Software, and to permit persons to whom the 

1040# Software is furnished to do so, subject to the following conditions: 

1041# 

1042# The above copyright notice and this permission notice shall be included 

1043# in all copies or substantial portions of the Software. 

1044# 

1045# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1046# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1047# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1048# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1049# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1050# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1051# OTHER DEALINGS IN THE SOFTWARE.