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_get, _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__ = '24.05.10' 

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 

43 L{Fsum2Tuple} instance). 

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

45 instance), all positional. 

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

47 threshold, see L{Fsum<Fsum.__init__>}. 

48 

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

50 

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

52 

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

54 

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

56 

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

58 ''' 

59 Fsum.__init__(self, **name_RESIDUAL) 

60 self.fadd(_map_mul(a, b, Fdot)) 

61 

62 

63class Fhorner(Fsum): 

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

65 ''' 

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

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

68 i=0..len(cs)-1)}. 

69 

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

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

72 L{Fsum2Tuple} instance), all positional. 

73 @kwarg name_RESIDUAL: Optional C{B{name}=NN} and C{B{RESIDUAL}=0.0} 

74 threshold, see L{Fsum<Fsum.__init__>}. 

75 

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

77 

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

79 

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

81 

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

83 ''' 

84 Fsum.__init__(self, **name_RESIDUAL) 

85 if cs: 

86 if _isFsumTuple(x): 

87 _mul = self._mul_Fsum 

88 else: 

89 _mul = self._mul_scalar 

90 x = _2float(x=x) 

91 op = Fhorner.__name__ 

92 if len(cs) > 1 and x: 

93 for c in reversed(cs): 

94 self._fset_ps(_mul(x, op)) 

95 self._fadd(c, op, up=False) 

96 self._update() 

97 else: # x == 0 

98 self._fadd(cs[0], op) 

99 else: 

100 self._fset_ps(_0_0) 

101 

102 

103class Fhypot(Fsum): 

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

105 ''' 

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

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

108 (raised to the power I{root}). 

109 

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

111 instance), all positional. 

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

113 order, C{B{name}=NN}, C{B{RESIDUAL}=0.0} threshold and 

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

115 method L{root<Fsum.root>}. 

116 ''' 

117 r = None # _xkwds_pop2 error 

118 try: 

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

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

121 raiser = _Fsum__init__(self, **kwds) 

122 if xs: 

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

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

125 except Exception as X: 

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

127 

128 

129class Fpolynomial(Fsum): 

130 '''Precision polynomial evaluation. 

131 ''' 

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

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

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

135 

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

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

138 L{Fsum2Tuple} instance), all positional. 

139 @kwarg name_RESIDUAL: Optional C{B{name}=NN} and C{B{RESIDUAL}=0.0} 

140 threshold, see L{Fsum<Fsum.__init__>}. 

141 

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

143 

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

145 

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

147 

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

149 ''' 

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

151 n = len(cs) - 1 

152 if n > 0: 

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

154 elif n < 0: 

155 self._fset_ps(_0_0) 

156 

157 

158class Fpowers(Fsum): 

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

160 ''' 

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

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

163 

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

165 @arg xs: One or more bases (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} 

166 instance), all positional. 

167 @kwarg name_RESIDUAL_raiser: Optional C{B{name}=NN}, C{B{RESIDUAL}=0.0} 

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

169 and L{fpow<Fsum.fpow>}. 

170 ''' 

171 try: 

172 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser) 

173 if xs: 

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

175 except Exception as X: 

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

177 

178 

179class Froot(Fsum): 

180 '''The root of a precision summation. 

181 ''' 

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

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

184 

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

186 @arg xs: Items to summate (each a C{scalar} or an L{Fsum} or L{Fsum2Tuple} 

187 instance), all positional. 

188 @kwarg name_RESIDUAL_raiser: Optional C{B{name}=NN}, C{B{RESIDUAL}=0.0} 

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

190 and L{fpow<Fsum.fpow>}. 

191 ''' 

192 try: 

193 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser) 

194 if xs: 

195 self.fadd(xs) 

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

197 except Exception as X: 

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

199 

200 

201class Fcbrt(Froot): 

202 '''Cubic root of a precision summation. 

203 ''' 

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

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

206 

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

208 ''' 

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

210 

211 

212class Fsqrt(Froot): 

213 '''Square root of a precision summation. 

214 ''' 

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

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

217 

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

219 ''' 

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

221 

222 

223def _Fsum__init__(inst, raiser=MISSING, **name_RESIDUAL): 

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

225 ''' 

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

227 inst._fset_ps(_0_0) 

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

229 

230 

231def bqrt(x): 

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

233 preserving C{type(B{x})}. 

234 

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

236 

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

238 

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

240 

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

242 

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

244 ''' 

245 return _root(x, _0_25, bqrt) 

246 

247 

248try: 

249 from math import cbrt as _cbrt # Python 3.11+ 

250 

251except ImportError: # Python 3.10- 

252 

253 def _cbrt(x): 

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

255 ''' 

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

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

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

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

260 

261 

262def cbrt(x): 

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

264 

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

266 

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

268 

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

270 ''' 

271 if _isFsumTuple(x): 

272 r = abs(x).fpow(_1_3rd) 

273 if x.signOf() < 0: 

274 r = -r 

275 else: 

276 r = _cbrt(x) 

277 return r # cbrt(-0.0) == -0.0 

278 

279 

280def cbrt2(x): # PYCHOK attr 

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

282 

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

284 

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

286 

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

288 ''' 

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

290 

291 

292def euclid(x, y): 

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

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

295 

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

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

298 

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

300 

301 @see: Function L{euclid_}. 

302 ''' 

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

304 if y > x: 

305 x, y = y, x 

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

307 

308 

309def euclid_(*xs): 

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

311 cascaded L{euclid}. 

312 

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

314 instance), all positional. 

315 

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

317 

318 @see: Function L{euclid}. 

319 ''' 

320 e = _0_0 

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

322 # e = euclid(x, e) 

323 if e < x: 

324 e, x = x, e 

325 if x: 

326 e += x * _0_4142 

327 return e 

328 

329 

330def facos1(x): 

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

332 

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

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

335 ''' 

336 a = fabs(x) 

337 if a < EPS0: 

338 r = PI_2 

339 elif a < EPS1: 

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

341 r = float(H * sqrt(_1_0 - a)) 

342 if x < 0: 

343 r = PI - r 

344 else: 

345 r = PI if x < 0 else _0_0 

346 return r 

347 

348 

349def fasin1(x): # PYCHOK no cover 

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

351 

352 @see: L{facos1}. 

353 ''' 

354 return PI_2 - facos1(x) 

355 

356 

357def fatan(x): 

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

359 ''' 

360 a = fabs(x) 

361 if a < _1_0: 

362 r = fatan1(a) if a else _0_0 

363 elif a > _1_0: 

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

365 else: 

366 r = PI_4 

367 if x < 0: # copysign0(r, x) 

368 r = -r 

369 return r 

370 

371 

372def fatan1(x): 

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

374 

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

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

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

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

379 IEEE Signal Processing Magazine, 111, May 2006. 

380 ''' 

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

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

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

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

385 return float(H) 

386 

387 

388def fatan2(y, x): 

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

390 

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

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

393 and L{fatan1}. 

394 ''' 

395 a, b = fabs(x), fabs(y) 

396 if b > a: 

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

398 elif a > b: 

399 r = fatan1(b / a) if b else _0_0 

400 elif a: # a == b != 0 

401 r = PI_4 

402 else: # a == b == 0 

403 return _0_0 

404 if x < 0: 

405 r = PI - r 

406 if y < 0: # copysign0(r, y) 

407 r = -r 

408 return r 

409 

410 

411def favg(a, b, f=_0_5): 

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

413 

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

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

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

417 

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

419 ''' 

420# @raise ValueError: Fraction out of range. 

421# ''' 

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

423# raise _ValueError(fraction=f) 

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

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

426 

427 

428def fdot(a, *b): 

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

430 i=0..len(a))}. 

431 

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

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

434 

435 @return: Dot product (C{float}). 

436 

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

438 

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

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

441 ''' 

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

443 

444 

445def fdot3(xs, ys, zs, start=0): 

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

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

448 

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

450 L{Fsum2Tuple} instance). 

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

452 L{Fsum2Tuple} instance). 

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

454 L{Fsum2Tuple} instance). 

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

456 or L{Fsum2Tuple}). 

457 

458 @return: Dot product (C{float}). 

459 

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

461 and/or C{len(B{zs})}. 

462 

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

464 ''' 

465 def _mul3(xs, ys, zs, s, p): 

466 if s: 

467 yield s 

468 if p: 

469 yield _1_0 

470 _F = Fsum 

471 for x, y, z in zip(xs, ys, zs): 

472 yield (_F(x) * y) * z 

473 if p: 

474 yield _N_1_0 

475 

476 n = len(xs) 

477 if not n == len(ys) == len(zs): 

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

479 

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

481 

482 

483def fhorner(x, *cs): 

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

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

486 

487 @return: Horner sum (C{float}). 

488 

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

490 ''' 

491 H = Fhorner(x, *cs) 

492 return float(H) 

493 

494 

495def fidw(xs, ds, beta=2): 

496 '''Interpolate using U{Inverse Distance Weighting 

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

498 

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

500 L{Fsum2Tuple} instance). 

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

502 or L{Fsum2Tuple} instance). 

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

504 

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

506 

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

508 

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

510 

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

512 weighted B{C{ds}} below L{EPS}. 

513 

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

515 ''' 

516 n, xs = len2(xs) 

517 if n > 1: 

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

519 if b < 0: 

520 try: # weighted 

521 _F = Fsum 

522 W = _F() 

523 X = _F() 

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

525 x = xs[i] 

526 D = _F(d) 

527 if D < EPS0: 

528 if D < 0: 

529 raise ValueError(_negative_) 

530 x = float(x) 

531 i = n 

532 break 

533 if D.fpow(b): 

534 W += D 

535 X += D.fmul(x) 

536 else: 

537 x = X.fover(W, raiser=False) 

538 i += 1 # len(xs) >= len(ds) 

539 except IndexError: 

540 i += 1 # len(xs) < i < len(ds) 

541 except Exception as X: 

542 _I = Fmt.INDEX 

543 raise _xError(X, _I(xs=i), x, _I(ds=i), d) 

544 else: # b == 0 

545 x = fsum(xs) / n # fmean(xs) 

546 i = n 

547 elif n: 

548 x = float(xs[0]) 

549 i = n 

550 else: 

551 x = _0_0 

552 i, _ = len2(ds) 

553 if i != n: 

554 raise LenError(fidw, xs=n, ds=i) 

555 return x 

556 

557 

558def fmean(xs): 

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

560 

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

562 

563 @return: Mean value (C{float}). 

564 

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

566 

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

568 ''' 

569 n, xs = len2(xs) 

570 if n < 1: 

571 raise LenError(fmean, xs=xs) 

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

573 

574 

575def fmean_(*xs): 

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

577 

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

579 ''' 

580 return fmean(xs) 

581 

582 

583def fpolynomial(x, *cs, **over): 

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

585 i=0..len(cs)) [/ over]}. 

586 

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

588 

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

590 

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

592 ''' 

593 P = Fpolynomial(x, *cs) 

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

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

596 

597 

598def fpowers(x, n, alts=0): 

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

600 

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

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

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

604 exponent (C{int}). 

605 

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

607 

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

609 

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

611 ''' 

612 if not isint(n): 

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

614 elif n < 1: 

615 raise _ValueError(n=n) 

616 

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

618 ps = tuple(_powers(p, n)) 

619 

620 if alts > 0: # x**2, x**4, ... 

621 # ps[alts-1::2] chokes PyChecker 

622 ps = ps[slice(alts-1, None, 2)] 

623 

624 return ps 

625 

626 

627try: 

628 from math import prod as fprod # Python 3.8 

629except ImportError: 

630 

631 def fprod(xs, start=1): 

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

633 

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

635 C{scalar} or an L{Fsum}). 

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

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

638 

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

640 

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

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

643 ''' 

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

645 

646 

647def frandoms(n, seeded=None): 

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

649 

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

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

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

653 

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

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

656 ''' 

657 from random import gauss, random, seed, shuffle 

658 

659 if seeded is None: 

660 pass 

661 elif seeded and isbool(seeded): 

662 from time import localtime 

663 seed(localtime().tm_yday) 

664 elif isscalar(seeded): 

665 seed(seeded) 

666 

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

668 for _ in range(n): 

669 s = 0 

670 t = list(c) 

671 _a = t.append 

672 for _ in range(n * 8): 

673 v = gauss(0, random())**7 - s 

674 _a(v) 

675 s += v 

676 shuffle(t) 

677 yield t 

678 

679 

680def frange(start, number, step=1): 

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

682 

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

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

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

686 

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

688 

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

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

691 ''' 

692 if not isint(number): 

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

694 for i in range(number): 

695 yield start + (step * i) 

696 

697 

698try: 

699 from functools import reduce as freduce 

700except ImportError: 

701 try: 

702 freduce = reduce # PYCHOK expected 

703 except NameError: # Python 3+ 

704 

705 def freduce(f, xs, *start): 

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

707 ''' 

708 if start: 

709 r = v = start[0] 

710 else: 

711 r, v = 0, MISSING 

712 for v in xs: 

713 r = f(r, v) 

714 if v is MISSING: 

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

716 return r 

717 

718 

719def fremainder(x, y): 

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

721 

722 @arg x: Numerator (C{scalar}). 

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

724 

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

726 0.0) or C{NAN} for any non-finite B{C{x}}. 

727 

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

729 

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

731 project/geographiclib/>} and Python 3.7+ 

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

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

734 ''' 

735 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and 

736 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native 

737 # fmod( 0, 360) == 0.0 

738 # fmod( 360, 360) == 0.0 

739 # fmod(-0, 360) == 0.0 

740 # fmod(-0.0, 360) == -0.0 

741 # fmod(-360, 360) == -0.0 

742 # however, using the % operator ... 

743 # 0 % 360 == 0 

744 # 360 % 360 == 0 

745 # 360.0 % 360 == 0.0 

746 # -0 % 360 == 0 

747 # -360 % 360 == 0 == (-360) % 360 

748 # -0.0 % 360 == 0.0 == (-0.0) % 360 

749 # -360.0 % 360 == 0.0 == (-360.0) % 360 

750 

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

752 # == +0.0. This fixes this bug. See also Math::AngNormalize 

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

754 if _isfinite(x): 

755 try: 

756 r = remainder(x, y) if x else x 

757 except Exception as e: 

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

759 else: # handle x INF and NINF as NAN 

760 r = NAN 

761 return r 

762 

763 

764if _sys_version_info2 < (3, 8): # PYCHOK no cover 

765 from math import hypot # OK in Python 3.7- 

766 

767 def hypot_(*xs): 

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

769 

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

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

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

773 handled differently. 

774 

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

776 

777 @return: Norm (C{float}). 

778 

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

780 

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

782 

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

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

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

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

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

788 ''' 

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

790 

791elif _sys_version_info2 < (3, 10): 

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

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

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

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

796 

797 def hypot(x, y): 

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

799 

800 @arg x: X argument (C{scalar}). 

801 @arg y: Y argument (C{scalar}). 

802 

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

804 ''' 

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

806 

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

808else: 

809 from math import hypot # PYCHOK in Python 3.10+ 

810 hypot_ = hypot 

811 

812 

813def hypot1(x): 

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

815 

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

817 

818 @return: Norm (C{float}). 

819 ''' 

820 if _isFsumTuple(x): 

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

822 else: 

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

824 return h 

825 

826 

827def hypot2(x, y): 

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

829 

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

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

832 

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

834 ''' 

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

836 if y > x: 

837 x, y = y, x 

838 if x: 

839 h2 = x**2 

840 if y: 

841 h2 *= (y / x)**2 + _1_0 

842 h2 = float(h2) 

843 else: 

844 h2 = _0_0 

845 return h2 

846 

847 

848def hypot2_(*xs): 

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

850 

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

852 L{Fsum2Tuple} instance), all positional. 

853 

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

855 

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

857 ''' 

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

859 if h2: 

860 _h = _1_0 / h2 

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

862 h2 = h2.fover(_h**2) 

863 return h2 

864 

865 

866def _map_mul(xs, ys, where): 

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

868 ''' 

869 n = len(ys) 

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

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

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

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

874 

875 

876def _1map_mul(xs, ys): 

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

878 ''' 

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

880 

881 

882def norm2(x, y): 

883 '''Normalize a 2-dimensional vector. 

884 

885 @arg x: X component (C{scalar}). 

886 @arg y: Y component (C{scalar}). 

887 

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

889 

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

891 or zero norm. 

892 ''' 

893 try: 

894 h = hypot(x, y) 

895 if h: 

896 x, y = (x / h), (y / h) 

897 else: 

898 x = _copysign_0_0(x) # pass? 

899 y = _copysign_0_0(y) 

900 except Exception as e: 

901 raise _xError(e, x=x, y=y, h=h) 

902 return x, y 

903 

904 

905def norm_(*xs): 

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

907 

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

909 

910 @return: Yield each component, normalized. 

911 

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

913 or zero norm. 

914 ''' 

915 try: 

916 i = x = h = None 

917 h = hypot_(*xs) 

918 _h = (_1_0 / h) if h else _0_0 

919 for i, x in enumerate(xs): 

920 yield x * _h 

921 except Exception as X: 

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

923 

924 

925def _powers(x, n): 

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

927 ''' 

928 p = 1 # type(p) == type(x) 

929 for _ in range(n): 

930 p *= x 

931 yield p 

932 

933 

934def _root(x, p, where): 

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

936 ''' 

937 try: 

938 if x > 0: 

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

940 elif x < 0: 

941 raise ValueError(_negative_) 

942 except Exception as X: 

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

944 return _0_0 

945 

946 

947def sqrt0(x, Error=None): 

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

949 preserving C{type(B{x})}. 

950 

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

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

953 

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

955 

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

957 

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

959 returns C{0.0}. 

960 ''' 

961 if Error and x < 0: 

962 raise Error(unstr(sqrt0, x)) 

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

964 

965 

966def sqrt3(x): 

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

968 preserving C{type(B{x})}. 

969 

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

971 

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

973 

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

975 

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

977 

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

979 ''' 

980 return _root(x, _1_5, sqrt3) 

981 

982 

983def sqrt_a(h, b): 

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

985 C{sqrt(B{h}**2 - B{b}**2)}. 

986 

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

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

989 

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

991 

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

993 

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

995 

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

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

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

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

1000 ''' 

1001 try: 

1002 if not (_isHeight(h) and _isRadius(b)): 

1003 raise TypeError(_not_scalar_) 

1004 c = fabs(h) 

1005 if c > EPS0: 

1006 s = _1_0 - (b / c)**2 

1007 if s < 0: 

1008 raise ValueError(_h_lt_b_) 

1009 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0) 

1010 else: # PYCHOK no cover 

1011 b = fabs(b) 

1012 d = c - b 

1013 if d < 0: 

1014 raise ValueError(_h_lt_b_) 

1015 d *= c + b 

1016 a = sqrt(d) if d else _0_0 

1017 except Exception as x: 

1018 raise _xError(x, h=h, b=b) 

1019 return copysign0(a, h) 

1020 

1021 

1022def zcrt(x): 

1023 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)}, 

1024 preserving C{type(B{x})}. 

1025 

1026 @arg x: Value (C{scalar} or an L{Fsum} or L{Fsum2Tuple}). 

1027 

1028 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}). 

1029 

1030 @see: Functions L{bqrt} and L{zqrt}. 

1031 

1032 @raise TypeeError: Invalid B{C{x}}. 

1033 

1034 @raise ValueError: Negative B{C{x}}. 

1035 ''' 

1036 return _root(x, _1_6th, zcrt) 

1037 

1038 

1039def zqrt(x): 

1040 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root, 

1041 M{x**(1 / 8)}, preserving C{type(B{x})}. 

1042 

1043 @arg x: Value (C{scalar} or an L{Fsum} or L{Fsum2Tuple}). 

1044 

1045 @return: I{Zenzi-quartic} root (C{float} or L{Fsum}). 

1046 

1047 @see: Functions L{bqrt} and L{zcrt}. 

1048 

1049 @raise TypeeError: Invalid B{C{x}}. 

1050 

1051 @raise ValueError: Negative B{C{x}}. 

1052 ''' 

1053 return _root(x, _0_125, zqrt) 

1054 

1055# **) MIT License 

1056# 

1057# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1058# 

1059# Permission is hereby granted, free of charge, to any person obtaining a 

1060# copy of this software and associated documentation files (the "Software"), 

1061# to deal in the Software without restriction, including without limitation 

1062# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1063# and/or sell copies of the Software, and to permit persons to whom the 

1064# Software is furnished to do so, subject to the following conditions: 

1065# 

1066# The above copyright notice and this permission notice shall be included 

1067# in all copies or substantial portions of the Software. 

1068# 

1069# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1070# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1071# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1072# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1073# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1074# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1075# OTHER DEALINGS IN THE SOFTWARE.