Coverage for pygeodesy/fmath.py: 94%

277 statements  

« prev     ^ index     » next       coverage.py v7.2.2, created at 2024-04-19 10:22 -0400

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, isint, len2 

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

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

12 _1_5, _1_6th, _2_0, _2_3rd, _3_0, \ 

13 _copysign_0_0, _isfinite, _over, remainder 

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

15 _xError, _xkwds_get, _xkwds_pop2 

16from pygeodesy.fsums import _2float, Fsum, _fsum, fsum, fsum1_, _1primed, \ 

17 Fmt, unstr 

18from pygeodesy.interns import MISSING, _few_, _h_, _invokation_, _negative_, \ 

19 _not_scalar_, _SPACE_, _too_ 

20from pygeodesy.lazily import _ALL_LAZY, _sys_version_info2 

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

22from pygeodesy.units import Int_, _isHeight, _isRadius, Float_ # PYCHOK for .heights 

23 

24from math import fabs, sqrt # pow 

25import operator as _operator # in .datums, .trf, .utm 

26 

27__all__ = _ALL_LAZY.fmath 

28__version__ = '24.04.17' 

29 

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

31_0_4142 = 0.41421356237309504880 # ... sqrt(2) - 1 

32_h_lt_b_ = 'abs(h) < abs(b)' 

33 

34 

35def _Fsum__init__(inst, raiser=MISSING, **name_RESIDUAL): 

36 '''(INTERNAL) Init an C{Fsum} instance. 

37 ''' 

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

39 inst._fset_ps(_0_0) 

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

41 

42 

43class Fdot(Fsum): 

44 '''Precision dot product. 

45 ''' 

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

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

48 for i=0..len(a)-1)}. 

49 

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

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

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

53 see L{Fsum<Fsum.__init__>}. 

54 

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

56 

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

58 

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

60 ''' 

61 Fsum.__init__(self, **name_RESIDUAL) 

62 self.fadd(_map_mul(a, b, Fdot)) 

63 

64 

65class Fhorner(Fsum): 

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

67 ''' 

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

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

70 for i=0..len(cs)-1)}. 

71 

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

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

74 instances), all positional. 

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

76 see L{Fsum<Fsum.__init__>}. 

77 

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

79 

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

81 

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

83 

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

85 L{Fsum.fmul}. 

86 ''' 

87 Fsum.__init__(self, **name_RESIDUAL) 

88 if cs: 

89 if isinstance(x, Fsum): 

90 _mul = self._mul_Fsum 

91 else: 

92 _mul = self._mul_scalar 

93 x = _2float(x=x) 

94 op = Fhorner.__name__ 

95 if len(cs) > 1 and x: 

96 for c in reversed(cs): 

97 self._fset_ps(_mul(x, op)) 

98 self._fadd(c, op, up=False) 

99 self._update() 

100 else: # x == 0 

101 self._fadd(cs[0], op) 

102 else: 

103 self._fset_ps(_0_0) 

104 

105 

106class Fhypot(Fsum): 

107 '''Precision summation and hypotenuse, default C{power=2}. 

108 ''' 

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

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

111 

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

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

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

115 class L{Fsum<Fsum.__init__>} and 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 L{Fsum}). 

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

138 all positional. 

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

140 see L{Fsum<Fsum.__init__>}. 

141 

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

143 

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

145 

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

147 

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

149 method L{Fsum.fadd}. 

150 ''' 

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

152 n = len(cs) - 1 

153 if n > 0: 

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

155 elif n < 0: 

156 self._fset_ps(_0_0) 

157 

158 

159class Fpowers(Fsum): 

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

161 ''' 

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

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

164 

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

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

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

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

169 ''' 

170 try: 

171 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser) 

172 if xs: 

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

174 except Exception as X: 

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

176 

177 

178class Froot(Fsum): 

179 '''The root of a precision summation. 

180 ''' 

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

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

183 

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

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

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

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

188 ''' 

189 try: 

190 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser) 

191 if xs: 

192 self. fadd(xs) 

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

194 except Exception as X: 

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

196 

197 

198class Fcbrt(Froot): 

199 '''Cubic root of a precision summation. 

200 ''' 

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

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

203 

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

205 ''' 

206 Froot.__init__(self, _3_0, *xs, **name_RESIDUAL_raiser) 

207 

208 

209class Fsqrt(Froot): 

210 '''Square root of a precision summation. 

211 ''' 

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

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

214 

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

216 ''' 

217 Froot.__init__(self, _2_0, *xs, **name_RESIDUAL_raiser) 

218 

219 

220def bqrt(x): 

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

222 

223 @arg x: Value (C{scalar}). 

224 

225 @return: I{Quartic} root (C{float}). 

226 

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

228 

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

230 ''' 

231 return _root(x, _0_25, bqrt) 

232 

233 

234try: 

235 from math import cbrt # Python 3.11+ 

236 

237 def cbrt2(x): 

238 '''Compute the cube root I{squared} M{x**(2/3)}. 

239 ''' 

240 return cbrt(x)**2 # cbrt(-0.0*2) == -0.0 

241 

242except ImportError: # Python 3.10- 

243 

244 def cbrt(x): 

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

246 

247 @arg x: Value (C{scalar}). 

248 

249 @return: Cubic root (C{float}). 

250 

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

252 ''' 

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

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

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

256 return _copysign(pow(fabs(x), _1_3rd), x) # cbrt(-0.0) == -0.0 

257 

258 def cbrt2(x): # PYCHOK attr 

259 '''Compute the cube root I{squared} M{x**(2/3)}. 

260 

261 @arg x: Value (C{scalar}). 

262 

263 @return: Cube root I{squared} (C{float}). 

264 

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

266 ''' 

267 return pow(fabs(x), _2_3rd) # XXX pow(fabs(x), _1_3rd)**2 

268 

269 

270def euclid(x, y): 

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

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

273 

274 @arg x: X component (C{scalar}). 

275 @arg y: Y component (C{scalar}). 

276 

277 @return: Appoximate norm (C{float}). 

278 

279 @see: Function L{euclid_}. 

280 ''' 

281 x, y = fabs(x), fabs(y) 

282 if x < y: 

283 x, y = y, x 

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

285 

286 

287def euclid_(*xs): 

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

289 by cascaded L{euclid}. 

290 

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

292 

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

294 

295 @see: Function L{euclid}. 

296 ''' 

297 e = _0_0 

298 for x in sorted(map(fabs, xs)): # XXX not reverse=True 

299 # e = euclid(x, e) 

300 if e < x: 

301 e, x = x, e 

302 if x: 

303 e += x * _0_4142 

304 return e 

305 

306 

307def facos1(x): 

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

309 

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

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

312 ''' 

313 a = fabs(x) 

314 if a < EPS0: 

315 r = PI_2 

316 elif a < EPS1: 

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

318 r = float(H * sqrt(_1_0 - a)) 

319 if x < 0: 

320 r = PI - r 

321 else: 

322 r = PI if x < 0 else _0_0 

323 return r 

324 

325 

326def fasin1(x): # PYCHOK no cover 

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

328 

329 @see: L{facos1}. 

330 ''' 

331 return PI_2 - facos1(x) 

332 

333 

334def fatan(x): 

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

336 ''' 

337 a = fabs(x) 

338 if a < _1_0: 

339 r = fatan1(a) if a else _0_0 

340 elif a > _1_0: 

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

342 else: 

343 r = PI_4 

344 if x < 0: # copysign0(r, x) 

345 r = -r 

346 return r 

347 

348 

349def fatan1(x): 

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

351 

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

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

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

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

356 IEEE Signal Processing Magazine, 111, May 2006. 

357 ''' 

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

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

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

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

362 return float(H) 

363 

364 

365def fatan2(y, x): 

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

367 

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

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

370 and L{fatan1}. 

371 ''' 

372 a, b = fabs(x), fabs(y) 

373 if b > a: 

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

375 elif a > b: 

376 r = fatan1(b / a) if b else _0_0 

377 elif a: # a == b != 0 

378 r = PI_4 

379 else: # a == b == 0 

380 return _0_0 

381 if x < 0: 

382 r = PI - r 

383 if y < 0: # copysign0(r, y) 

384 r = -r 

385 return r 

386 

387 

388def favg(v1, v2, f=_0_5): 

389 '''Return the average of two values. 

390 

391 @arg v1: One value (C{scalar}). 

392 @arg v2: Other value (C{scalar}). 

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

394 

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

396 ''' 

397# @raise ValueError: Fraction out of range. 

398# ''' 

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

400# raise _ValueError(fraction=f) 

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

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

403 

404 

405def fdot(a, *b): 

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

407 i=0..len(a))}. 

408 

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

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

411 

412 @return: Dot product (C{float}). 

413 

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

415 

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

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

418 ''' 

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

420 

421 

422def fdot3(a, b, c, start=0): 

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

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

425 

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

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

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

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

430 

431 @return: Dot product (C{float}). 

432 

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

434 and/or C{len(B{c})}. 

435 

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

437 ''' 

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

439 return a * b * c 

440 

441 def _mul3_(a, b, c, start): 

442 yield start 

443 for abc in map(_mul3, a, b, c): 

444 yield abc 

445 

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

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

448 

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

450 

451 

452def fhorner(x, *cs): 

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

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

455 

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

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

458 

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

460 

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

462 

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

464 

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

466 

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

468 ''' 

469 H = Fhorner(x, *cs) 

470 return float(H) 

471 

472 

473def fidw(xs, ds, beta=2): 

474 '''Interpolate using U{Inverse Distance Weighting 

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

476 

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

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

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

480 

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

482 

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

484 

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

486 weighted B{C{ds}} below L{EPS}. 

487 

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

489 ''' 

490 n, xs = len2(xs) 

491 d, ds = len2(ds) 

492 if n != d or n < 1: 

493 raise LenError(fidw, xs=n, ds=d) 

494 

495 d, x = min(zip(ds, xs)) 

496 if d > EPS0 and n > 1: 

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

498 if b < 0: 

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

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

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

502 else: # b == 0 

503 x = fsum(xs) / n # fmean(xs) 

504 elif d < 0: # PYCHOK no cover 

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

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

507 return x 

508 

509 

510def fmean(xs): 

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

512 

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

514 

515 @return: Mean value (C{float}). 

516 

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

518 

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

520 ''' 

521 n, xs = len2(xs) 

522 if n < 1: 

523 raise LenError(fmean, xs=xs) 

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

525 

526 

527def fmean_(*xs): 

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

529 

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

531 ''' 

532 return fmean(xs) 

533 

534 

535def fpolynomial(x, *cs, **over): 

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

537 i=0..len(cs)) [/ over]}. 

538 

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

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

541 positional. 

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

543 (C{scalar}). 

544 

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

546 

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

548 

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

550 

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

552 

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

554 ''' 

555 P = Fpolynomial(x, *cs) 

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

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

558 

559 

560def fpowers(x, n, alts=0): 

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

562 

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

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

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

566 exponent (C{int}). 

567 

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

569 

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

571 

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

573 ''' 

574 if not isint(n): 

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

576 elif n < 1: 

577 raise _ValueError(n=n) 

578 

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

580 ps = tuple(_powers(p, n)) 

581 

582 if alts > 0: # x**2, x**4, ... 

583 # ps[alts-1::2] chokes PyChecker 

584 ps = ps[slice(alts-1, None, 2)] 

585 

586 return ps 

587 

588 

589try: 

590 from math import prod as fprod # Python 3.8 

591except ImportError: 

592 

593 def fprod(xs, start=1): 

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

595 

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

597 tuple, etc. (C{scalar}s). 

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

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

600 

601 @return: The product (C{float}). 

602 

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

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

605 ''' 

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

607 

608 

609def frange(start, number, step=1): 

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

611 

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

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

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

615 

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

617 

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

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

620 ''' 

621 if not isint(number): 

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

623 for i in range(number): 

624 yield start + (step * i) 

625 

626 

627try: 

628 from functools import reduce as freduce 

629except ImportError: 

630 try: 

631 freduce = reduce # PYCHOK expected 

632 except NameError: # Python 3+ 

633 

634 def freduce(f, xs, *start): 

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

636 ''' 

637 if start: 

638 r = v = start[0] 

639 else: 

640 r, v = 0, MISSING 

641 for v in xs: 

642 r = f(r, v) 

643 if v is MISSING: 

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

645 return r 

646 

647 

648def fremainder(x, y): 

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

650 

651 @arg x: Numerator (C{scalar}). 

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

653 

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

655 0.0) or C{NAN} for any non-finite B{C{x}}. 

656 

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

658 

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

660 project/geographiclib/>} and Python 3.7+ 

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

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

663 ''' 

664 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and 

665 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native 

666 # fmod( 0, 360) == 0.0 

667 # fmod( 360, 360) == 0.0 

668 # fmod(-0, 360) == 0.0 

669 # fmod(-0.0, 360) == -0.0 

670 # fmod(-360, 360) == -0.0 

671 # however, using the % operator ... 

672 # 0 % 360 == 0 

673 # 360 % 360 == 0 

674 # 360.0 % 360 == 0.0 

675 # -0 % 360 == 0 

676 # -360 % 360 == 0 == (-360) % 360 

677 # -0.0 % 360 == 0.0 == (-0.0) % 360 

678 # -360.0 % 360 == 0.0 == (-360.0) % 360 

679 

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

681 # == +0.0. This fixes this bug. See also Math::AngNormalize 

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

683 if _isfinite(x): 

684 try: 

685 r = remainder(x, y) if x else x 

686 except Exception as e: 

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

688 else: # handle x INF and NINF as NAN 

689 r = NAN 

690 return r 

691 

692 

693if _sys_version_info2 < (3, 8): # PYCHOK no cover 

694 from math import hypot # OK in Python 3.7- 

695 

696 def hypot_(*xs): 

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

698 

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

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

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

702 handled differently. 

703 

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

705 

706 @return: Norm (C{float}). 

707 

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

709 

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

711 

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

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

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

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

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

717 ''' 

718 h, x2 = _h_x2(xs, hypot_) 

719 return (h * sqrt(x2)) if x2 else _0_0 

720 

721elif _sys_version_info2 < (3, 10): 

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

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

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

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

726 

727 def hypot(x, y): 

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

729 

730 @arg x: X argument (C{scalar}). 

731 @arg y: Y argument (C{scalar}). 

732 

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

734 ''' 

735 if x: 

736 h = sqrt(x**2 + y**2) if y else fabs(x) 

737 elif y: 

738 h = fabs(y) 

739 else: 

740 h = _0_0 

741 return h 

742 

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

744else: 

745 from math import hypot # PYCHOK in Python 3.10+ 

746 hypot_ = hypot 

747 

748 

749def _h_x2(xs, which): 

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

751 ''' 

752 n, xs = len2(xs) 

753 if n > 0: 

754 h = float(max(map(fabs, xs))) 

755 if h < EPS0: 

756 x2 = _0_0 

757 elif n > 1: 

758 _h = (_1_0 / h) if h != _1_0 else _1_0 

759 x2 = _fsum(_1primed((x * _h)**2 for x in xs)) 

760 else: 

761 x2 = _1_0 

762 return h, x2 

763 

764 t = Fmt.PAREN(which.__name__, xs) 

765 raise _ValueError(t, txt=_too_(_few_)) 

766 

767 

768def hypot1(x): 

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

770 

771 @arg x: Argument (C{scalar}). 

772 

773 @return: Norm (C{float}). 

774 ''' 

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

776 

777 

778def hypot2(x, y): 

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

780 

781 @arg x: X argument (C{scalar}). 

782 @arg y: Y argument (C{scalar}). 

783 

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

785 ''' 

786 if fabs(x) < fabs(y): 

787 x, y = y, x 

788 if x: 

789 h2 = x**2 

790 if y: 

791 h2 *= (y / x)**2 + _1_0 

792 else: 

793 h2 = _0_0 

794 return h2 

795 

796 

797def hypot2_(*xs): 

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

799 

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

801 

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

803 

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

805 

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

807 

808 @see: Function L{hypot_}. 

809 ''' 

810 h, x2 = _h_x2(xs, hypot2_) 

811 return (h**2 * x2) if x2 else _0_0 

812 

813 

814def _map_mul(a, b, where): 

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

816 ''' 

817 n = len(b) 

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

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

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

821 

822 

823def _1map_mul(a, b): 

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

825 ''' 

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

827 

828 

829def norm2(x, y): 

830 '''Normalize a 2-dimensional vector. 

831 

832 @arg x: X component (C{scalar}). 

833 @arg y: Y component (C{scalar}). 

834 

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

836 

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

838 or zero norm. 

839 ''' 

840 try: 

841 h = hypot(x, y) 

842 if h: 

843 x, y = (x / h), (y / h) 

844 else: 

845 x = _copysign_0_0(x) # pass? 

846 y = _copysign_0_0(y) 

847 except Exception as e: 

848 raise _xError(e, x=x, y=y, h=h) 

849 return x, y 

850 

851 

852def norm_(*xs): 

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

854 

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

856 

857 @return: Yield each component, normalized. 

858 

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

860 or zero norm. 

861 ''' 

862 try: 

863 h = hypot_(*xs) 

864 _h = (_1_0 / h) if h else _0_0 

865 for i, x in enumerate(xs): 

866 yield x * _h 

867 except Exception as e: 

868 raise _xError(e, Fmt.SQUARE(xs=i), x, _h_, h) 

869 

870 

871def _powers(x, n): 

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

873 ''' 

874 p = 1 # type(p) == type(x) 

875 for _ in range(n): 

876 p *= x 

877 yield p 

878 

879 

880def _root(x, p, where): 

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

882 ''' 

883 if x < 0: 

884 t = _SPACE_(_invokation_, where.__name__) 

885 raise _ValueError(unstr(t, x), txt=_negative_) 

886 return pow(x, p) if x else _0_0 

887 

888 

889def sqrt0(x, Error=None): 

890 '''Return the square root iff C{B{x} >} L{EPS02}. 

891 

892 @arg x: Value (C{scalar}). 

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

894 

895 @return: Square root (C{float}) or C{0.0}. 

896 

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

898 returns C{0.0}. 

899 ''' 

900 if Error and x < 0: 

901 raise Error(Fmt.PAREN(sqrt=x)) 

902 return sqrt(x) if x > EPS02 else (_0_0 if x < EPS02 else EPS0) 

903 

904 

905def sqrt3(x): 

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

907 

908 @arg x: Value (C{scalar}). 

909 

910 @return: Square root I{cubed} (C{float}). 

911 

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

913 

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

915 ''' 

916 return _root(x, _1_5, sqrt3) 

917 

918 

919def sqrt_a(h, b): 

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

921 C{sqrt(B{h}**2 - B{b}**2)}. 

922 

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

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

925 

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

927 

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

929 

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

931 

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

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

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

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

936 ''' 

937 try: 

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

939 raise TypeError(_not_scalar_) 

940 c = fabs(h) 

941 if c > EPS0: 

942 s = _1_0 - (b / c)**2 

943 if s < 0: 

944 raise ValueError(_h_lt_b_) 

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

946 else: # PYCHOK no cover 

947 b = fabs(b) 

948 d = c - b 

949 if d < 0: 

950 raise ValueError(_h_lt_b_) 

951 d *= c + b 

952 a = sqrt(d) if d else _0_0 

953 except Exception as x: 

954 raise _xError(x, h=h, b=b) 

955 return copysign0(a, h) 

956 

957 

958def zcrt(x): 

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

960 

961 @arg x: Value (C{scalar}). 

962 

963 @return: I{Zenzi-cubic} root (C{float}). 

964 

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

966 

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

968 ''' 

969 return _root(x, _1_6th, zcrt) 

970 

971 

972def zqrt(x): 

973 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root, M{x**(1 / 8)}. 

974 

975 @arg x: Value (C{scalar}). 

976 

977 @return: I{Zenzi-quartic} root (C{float}). 

978 

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

980 

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

982 ''' 

983 return _root(x, _0_125, zqrt) 

984 

985# **) MIT License 

986# 

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

988# 

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

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

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

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

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

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

995# 

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

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

998# 

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

1000# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1001# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1002# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1003# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1004# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1005# OTHER DEALINGS IN THE SOFTWARE.