Coverage for pygeodesy/fmath.py: 89%

2# -*- coding: utf-8 -*-

4u'''Utilities for precision floating point summation, multiplication,

5C{fused-multiply-add}, polynomials, roots, etc.

6'''

7# make sure int/int division yields float quotient, see .basics

8from __future__ import division as _; del _ # noqa: E702 ;

10from pygeodesy.basics import _copysign, copysign0, isbool, isint, isodd, \

11 isscalar, len2, _xiterable, _zip, typename

12from pygeodesy.constants import EPS0, EPS02, EPS1, NAN, PI, PI_2, PI_4, \

13 _0_0, _0_125, _0_25, _1_3rd, _0_5, _2_3rd, \

14 _1_0, _1_5, _copysign_0_0, isfinite, remainder

15from pygeodesy.errors import _IsnotError, LenError, _TypeError, _ValueError, \

16 _xError, _xkwds, _xkwds_get, _xkwds_pop2, _xsError

17from pygeodesy.fsums import _2float, Fsum, fsum, _isFsum_2Tuple

18# from pygeodesy.internals import typename # from .basics

19from pygeodesy.interns import MISSING, _negative_, _not_scalar_

20from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS

21from pygeodesy.streprs import Fmt, unstr

22from pygeodesy.units import Int_, _isHeight, _isRadius

23# from utily import atan2b, atan2p # _MODS, circular import!

25from math import fabs, sqrt # pow

26import operator as _operator # in .datums, .elliptic, .trf, .utm

28__all__ = _ALL_LAZY.fmath

29__version__ = '26.10.04'

31# 1.4142135623730950_48801688724209 <https://WikiPedia.org/wiki/Square_root_of_2>

32# math.sqrt(2) - 1 == 0.4142135623730950_3 == 3730904090310553 / 9007199254740992

33_0_4142 = 0.4142135623730950488

34_1_6th = 1 / 6

35_h_lt_b_ = 'abs(h) < abs(b)'

38class Fdot(Fsum):

39 '''Precision dot product.

40 '''

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

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

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

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

46 positional.

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

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

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

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

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

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

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

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

60 '''

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

62 Fsum.__init__(self, **kwds)

63 self(s)

65 n = len(b)

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

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

68 self._facc_dot(n, a, b, **kwds)

71class Fdot_(Fdot): # in .elliptic

72 '''Precision dot product.

73 '''

74 def __init__(self, *xys, **start_name_f2product_nonfinites_RESIDUAL):

75 '''New L{Fdot_} precision dot product M{start + sum(xys[i] * xys[i+1] for i in

76 range(0, len(xys), B{2}))}.

78 @arg xys: Pairwise values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}),

79 all positional.

81 @see: Class L{Fdot<Fdot.__init__>} for further details.

82 '''

83 if isodd(len(xys)):

84 raise LenError(Fdot_, xys=len(xys))

85 Fdot.__init__(self, xys[0::2], *xys[1::2], **start_name_f2product_nonfinites_RESIDUAL)

88class Fhorner(Fsum):

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

90 '''

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

92 '''New L{Fhorner} form evaluation of polynomial M{sum(cs[i] * x**i for i=0..n)}

93 with in- or decreasing exponent M{sum(... i=n..0)}, where C{n = len(cs) - 1}.

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

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

97 all positional.

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

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

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

101

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

103

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

105

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

107

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

109 '''

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

111 Fsum.__init__(self, **kwds)

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

113

114

115class Fhypot(Fsum):

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

117 '''

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

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

120 to the power I{root}).

121

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

123 positional.

124 @kwarg root_name_f2product_nonfinites_RESIDUAL_raiser: Optional, exponent

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

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

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

128 L{root<Fsum.root>}.

129 '''

130 def _r_X_kwds(power=None, raiser=True, root=2, **kwds):

131 # DEPRECATED keyword argument C{power=2}, use C{root=2}

132 return (root if power is None else power), raiser, kwds

133

134 r = None # _xkwds_pop2 error

135 try:

136 r, X, kwds = _r_X_kwds(**root_name_f2product_nonfinites_RESIDUAL_raiser)

137 Fsum.__init__(self, **kwds)

138 self(_0_0)

139 if xs:

140 self._facc_power(r, xs, Fhypot, raiser=X)

141 self._fset(self.root(r, raiser=X))

142 except Exception as X:

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

144

145

146class Fpolynomial(Fsum):

147 '''Precision polynomial evaluation.

148 '''

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

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

151 i=0..len(cs)-1)}.

152

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

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

155 all positional.

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

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

158

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

160

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

162

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

164

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

166 '''

167 Fsum.__init__(self, **name_f2product_nonfinites_RESIDUAL)

168 n = len(cs) - 1

169 self(_0_0 if n < 0 else cs[0])

170 self._facc_dot(n, cs[1:], _powers(x, n), **name_f2product_nonfinites_RESIDUAL)

171

172

173class Fpowers(Fsum):

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

175 '''

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

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

178

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

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

181 positional.

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

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

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

185 L{fpow<Fsum.fpow>}.

186 '''

187 try:

188 X, kwds = _xkwds_pop2(name_f2product_nonfinites_RESIDUAL_raiser, raiser=True)

189 Fsum.__init__(self, **kwds)

190 self(_0_0)

191 if xs:

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

193 except Exception as X:

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

195

196

197class Froot(Fsum):

198 '''The root of a precision summation.

199 '''

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

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

202

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

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

205 positional.

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

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

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

209 L{fpow<Fsum.fpow>}.

210 '''

211 try:

212 X, kwds = _xkwds_pop2(name_f2product_nonfinites_RESIDUAL_raiser, raiser=True)

213 Fsum.__init__(self, **kwds)

214 self(_0_0)

215 if xs:

216 self.fadd(xs)

217 self(self.root(root, raiser=X))

218 except Exception as X:

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

220

221

222class Fcbrt(Froot):

223 '''Cubic root of a precision summation.

224 '''

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

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

227

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

229 '''

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

231

232

233class Fsqrt(Froot):

234 '''Square root of a precision summation.

235 '''

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

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

238

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

240 '''

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

242

243

244def bqrt(x):

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

246 preserving C{type(B{x})}.

247

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

249

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

251

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

253

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

255

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

257 '''

258 return _root(x, _0_25, bqrt)

259

260

261try:

262 from math import cbrt as _cbrt # Python 3.11+

263except ImportError: # Python 3.10-

264

265 def _cbrt(x):

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

267 '''

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

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

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

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

272

273

274def cbrt(x):

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

276

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

278

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

280

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

282 '''

283 if _isFsum_2Tuple(x):

284 r = abs(x).fpow(_1_3rd)

285 if x.signOf() < 0:

286 r = -r

287 else:

288 r = _cbrt(x)

289 return r # cbrt(-0.0) == -0.0

290

291

292def cbrt2(x): # PYCHOK attr

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

294

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

296

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

298

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

300 '''

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

302

303

304def euclid(x, y, *x0_y0, **half):

305 '''I{Appoximate} the norm M{hypot(B{x}, B{y})} by M{max(abs(B{x}),

306 abs(B{y})) + min(abs(B{x}), abs(B{y})) * 0.4142...}.

307

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

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

310 @arg x0_y0: Optional reference C{(x0, y0)} (each C{scalar}, an

311 L{Fsum} or L{Fsum2Tuple}).

312 @kwarg half: Use C{B{half}=True} to apply C{0.5} instead of

313 C{sqrt(2) - 1} (C{bool}).

314

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

316 '''

317 x, y = _map0(abs, x, y, *x0_y0) # NOT fabs!

318 if x < y:

319 x, y = y, x

320 y *= _half(**half) if half else _0_4142 # _0_5 before 20.10.02

321 return x + y

322

323

324def euclid_(*xs, **half):

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

326 L{euclid}.

327

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

329 all positional.

330 @kwarg half: Use C{B{half}=True} to apply C{0.5} instead of

331 C{sqrt(2) - 1} (C{bool}).

332

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

334

335 @see: Function L{euclid}.

336 '''

337 e = _0_0

338 f = _half(**half)

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

340 # e = euclid(x, e)

341 if x:

342 if e < x:

343 e, x = x, e

344 x *= f

345# s = e + x

346# if e < x: # like .fsums._2sum

347# x -= s # e = (x - s) + e

348# else:

349# e -= s # e = (e - s) + x

350 e += x

351# e += s

352 return e

353

354

355def facos1(x):

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

357

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

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

360 '''

361 a = fabs(x)

362 if a < EPS0:

363 r = PI_2

364 elif a < EPS1:

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

366 r *= sqrt(_1_0 - a)

367 if x < 0:

368 r = PI - r

369 else:

370 r = PI if x < 0 else _0_0

371 return r

372

373

374def fasin1(x): # PYCHOK no cover

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

376

377 @see: L{facos1}.

378 '''

379 return PI_2 - facos1(x)

380

381

382def _fast(x, *cs):

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

384 '''

385 h = 0

386 for c in reversed(cs):

387 h = _fma(x, h, c) if h else c

388 return h

389

390

391def fatan(x):

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

393 '''

394 a = fabs(x)

395 if a < _1_0:

396 r = fatan1(a) if a else _0_0

397 elif a > _1_0:

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

399 else:

400 r = PI_4

401 if x < 0: # copysign0(r, x)

402 r = -r

403 return r

404

405

406def fatan1(x):

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

408

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

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

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

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

413 IEEE Signal Processing Magazine, 111, May 2006.

414 '''

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

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

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

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

419

420

421def fatan2(y, x):

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

423

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

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

426 and L{fatan1}.

427 '''

428 a, b = fabs(x), fabs(y)

429 if b > a:

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

431 elif a > b:

432 r = fatan1(b / a) if b else _0_0

433 elif a: # a == b != 0

434 r = PI_4

435 else: # a == b == 0

436 return _0_0

437 if x < 0:

438 r = PI - r

439 if y < 0: # copysign0(r, y)

440 r = -r

441 return r

442

443

444def favg(a, b, f=_0_5, nonfinites=True):

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

446

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

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

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

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

451

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

453 '''

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

455 return float(F)

456

457

458def fdot(xs, *ys, **start_f2product_nonfinites):

459 '''Return the precision dot product M{start + sum(xs[i] * ys[i] for i in range(len(xs)))}.

460

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

462 @arg ys: Other values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all positional.

463 @kwarg start_f2product_nonfinites: Optional bias C{B{start}=0} (C{scalar}, an

464 L{Fsum} or L{Fsum2Tuple}) and settings C{B{f2product}=None} (C{bool})

465 and C{B{nonfinites=True}} (C{bool}), see class L{Fsum<Fsum.__init__>}.

466

467 @return: Dot product (C{float}).

468

469 @raise LenError: Unequal C{len(B{xs})} and C{len(B{ys})}.

470

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

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

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

474 '''

475 D = Fdot(xs, *ys, **_xkwds(start_f2product_nonfinites, nonfinites=True))

476 return float(D)

477

478

479def fdot_(*xys, **start_f2product_nonfinites):

480 '''Return the (precision) dot product M{start + sum(xys[i] * xys[i+1] for i in range(0, len(xys), B{2}))}.

481

482 @arg xys: Pairwise values (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all positional.

483

484 @see: Function L{fdot} for further details.

485

486 @return: Dot product (C{float}).

487 '''

488 D = Fdot_(*xys, **_xkwds(start_f2product_nonfinites, nonfinites=True))

489 return float(D)

490

491

492def fdot3(xs, ys, zs, **start_f2product_nonfinites):

493 '''Return the (precision) dot product M{start + sum(xs[i] * ys[i] * zs[i] for i in range(len(xs)))}.

494

495 @arg xs: X values iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

496 @arg ys: Y values iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

497 @arg zs: Z values iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

498

499 @see: Function L{fdot} for further details.

500

501 @return: Dot product (C{float}).

502

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

504 '''

505 n = len(xs)

506 if not n == len(ys) == len(zs):

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

508

509 D = Fdot((), **_xkwds(start_f2product_nonfinites, nonfinites=True))

510 kwds = dict(f2product=D.f2product(), nonfinites=D.nonfinites())

511 _f = Fsum(**kwds)

512 D = D._facc(_f(x).f2mul_(y, z, **kwds) for x, y, z in zip(xs, ys, zs))

513 return float(D)

514

515

516if _zip is zip: # Python 3.9-

517 _fdotf = fdot

518else:

519 from pygeodesy.fsums import _fsum # math.fsum

520

521 def _fdotf(xs, *ys): # in .datums, .ecef

522 '''(INTERNAL) Dot product for C{float} tuples of matching C{len}gths.

523 '''

524 return _fsum(x * y for x, y in _zip(xs, ys)) # strict=True

525

526

527def fhorner(x, *cs, **incx):

528 '''Horner form evaluation of polynomial M{sum(cs[i] * x**i for i=0..n)} as

529 in- or decreasing exponents M{sum(... i=n..0)}, where C{n = len(cs) - 1}.

530

531 @return: Horner sum (C{float}).

532

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

534 '''

535 H = Fhorner(x, *cs, **incx)

536 return float(H)

537

538

539def fidw(xs, ds, beta=2):

540 '''Interpolate using U{Inverse Distance Weighting

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

542

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

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

545 L{Fsum2Tuple}).

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

547

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

549

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

551

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

553

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

555 weighted B{C{ds}} below L{EPS}.

556

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

558 '''

559 n, xs = len2(xs)

560 if n > 1:

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

562 if b < 0:

563 try: # weighted

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

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

566 x = xs[i]

567 D = _d(d)

568 if D < EPS0:

569 if D < 0:

570 raise ValueError(_negative_)

571 x = float(x)

572 i = n

573 break

574 if D.fpow(b):

575 W += D

576 X += D.fmul(x)

577 else:

578 x = X.fover(W, raiser=False)

579 i += 1 # len(xs) >= len(ds)

580 except IndexError:

581 i += 1 # len(xs) < i < len(ds)

582 except Exception as X:

583 _I = Fmt.INDEX

584 raise _xError(X, _I(xs=i), x, # PYLINT E1120

585 _I(ds=i), d) # PYLINT E1120

586 else: # b == 0

587 x = fsum(xs) / n # fmean(xs)

588 i = n

589 elif n:

590 x = float(xs[0])

591 i = n

592 else:

593 x = _0_0

594 i, _ = len2(ds)

595 if i != n:

596 raise LenError(fidw, xs=n, ds=i)

597 return x

598

599

600try:

601 from math import fma as _fma # in .resections

602

603 def _fhornerf(x, *cs, **incx):

604 '''(INTERNAL) Horner form for C{float} coefficients C{cs} and C{x}

605 and C{B{incx}=True} in- or decreasing with exponents=.

606 '''

607 h = _0_0

608 if cs:

609 if _xkwds_get(incx, incx=True):

610 cs = reversed(cs)

611 for c in cs:

612 h = _fma(x, h, c) if h else c

613 return h

614

615except ImportError: # PYCHOK DSPACE!

616 _fhornerf = fhorner # PYCHOK redef

617

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

619 return x * y + z

620

621

622def fma(x, y, z, **nonfinites): # **raiser

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

624 or an equivalent implementation.

625

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

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

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

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

630 to override default L{nonfiniterrors}

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

632

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

634 '''

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

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

637

638

639def _Fm2(x, nonfinites=None, **raiser):

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

641 '''

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

643

644

645def fmean(xs, nonfinites=True):

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

647

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

649

650 @return: Mean value (C{float}).

651

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

653

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

655 '''

656 n, xs = len2(xs)

657 if n < 1:

658 raise LenError(fmean, xs=xs)

659 M = Fsum(*xs, nonfinites=nonfinites)

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

661

662

663def fmean_(*xs, **nonfinites):

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

665

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

667 '''

668 return fmean(xs, **nonfinites)

669

670

671def f2mul_(x, *ys, **nonfinites): # **raiser

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

673

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

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

676 positional.

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

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

679

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

681

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

683 '''

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

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

686

687

688def fpolynomial(x, *cs, **over_f2product_nonfinites):

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

690

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

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

693 L{Fpolynomial<Fpolynomial.__init__>}.

694

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

696 '''

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

698 P = Fpolynomial(x, *cs, **kwds)

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

700

701

702def fpowers(x, n, alts=0):

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

704

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

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

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

708 exponent (C{int}).

709

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

711

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

713

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

715 '''

716 if not isint(n):

717 raise _IsnotError(typename(int), n=n)

718 elif n < 1:

719 raise _ValueError(n=n)

720

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

722 ps = tuple(_powers(p, n))

723

724 if alts > 0: # x**2, x**4, ...

725 # ps[alts-1::2] chokes PyChecker

726 ps = ps[slice(alts-1, None, 2)]

727

728 return ps

729

730

731try:

732 from math import prod as fprod # PYLINT W0611 Python 3.8

733except ImportError:

734

735 def fprod(xs, start=1):

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

737

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

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

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

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

742

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

744

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

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

747 '''

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

749

750

751def frandoms(n, seeded=None):

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

753

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

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

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

757

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

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

760 '''

761 from random import gauss, random, seed, shuffle

762

763 if seeded is None:

764 pass

765 elif seeded and isbool(seeded):

766 from time import localtime

767 seed(localtime().tm_yday)

768 elif isscalar(seeded):

769 seed(seeded)

770

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

772 for _ in range(n):

773 s = 0

774 t = list(c)

775 _a = t.append

776 for _ in range(n * 8):

777 v = gauss(0, random())**7 - s

778 _a(v)

779 s += v

780 shuffle(t)

781 yield t

782

783

784def frange(start, number, step=1):

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

786

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

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

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

790

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

792

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

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

795 '''

796 if not isint(number):

797 raise _IsnotError(typename(int), number=number)

798 for i in range(number):

799 yield start + (step * i)

800

801

802try:

803 from functools import reduce as freduce

804except ImportError:

805 try:

806 freduce = reduce # PYCHOK expected

807 except NameError: # Python 3+

808

809 def freduce(f, xs, *start):

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

811 '''

812 if start:

813 r = v = start[0]

814 else:

815 r, v = 0, MISSING

816 for v in xs:

817 r = f(r, v)

818 if v is MISSING:

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

820 return r

821

822

823def fremainder(x, y):

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

825

826 @arg x: Numerator (C{scalar}).

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

828

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

830 0.0) or C{NAN} for any non-finite B{C{x}}.

831

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

833

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

835 project/geographiclib/>} and Python 3.7+

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

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

838 '''

839 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and

840 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native

841 # fmod( 0, 360) == 0.0

842 # fmod( 360, 360) == 0.0

843 # fmod(-0, 360) == 0.0

844 # fmod(-0.0, 360) == -0.0

845 # fmod(-360, 360) == -0.0

846 # however, using the % operator ...

847 # 0 % 360 == 0

848 # 360 % 360 == 0

849 # 360.0 % 360 == 0.0

850 # -0 % 360 == 0

851 # -360 % 360 == 0 == (-360) % 360

852 # -0.0 % 360 == 0.0 == (-0.0) % 360

853 # -360.0 % 360 == 0.0 == (-360.0) % 360

854

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

856 # == +0.0. This fixes this bug. See also Math::AngNormalize

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

858 if isfinite(x):

859 try:

860 r = remainder(x, y) if x else x

861 except Exception as e:

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

863 else: # handle x INF and NINF as NAN

864 r = NAN

865 return r

866

867

868def _half(half=False):

869 return _0_5 if half else _0_4142

870

871

872if _MODS.sys_version_info2 < (3, 8): # PYCHOK no cover

873 from math import hypot # OK in Python 3.7-

874

875 def hypot_(*xs):

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

877

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

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

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

881 handled differently.

882

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

884

885 @return: Norm (C{float}).

886

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

888

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

890

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

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

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

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

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

896 '''

897 return float(_Hypot(*xs))

898

899elif _MODS.sys_version_info2 < (3, 10): # PYCHOK no cover

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

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

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

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

904

905 def hypot(x, y):

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

907

908 @arg x: X argument (C{scalar}).

909 @arg y: Y argument (C{scalar}).

910

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

912 '''

913 return float(_Hypot(x, y))

914

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

916else:

917 from math import hypot # PYCHOK in Python 3.10+

918 hypot_ = hypot

919

920

921def _Hypot(*xs):

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

923 '''

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

925

926

927def hypot1(x):

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

929

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

931

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

933 '''

934 h = _1_0

935 if x:

936 if _isFsum_2Tuple(x):

937 h = _Hypot(h, x)

938 h = float(h)

939 else:

940 h = hypot(h, x)

941 return h

942

943

944def hypot2(x, y, *x0_y0):

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

946

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

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

949 @arg x0_y0: Optional reference C{(x0, y0)} (each C{scalar},

950 an L{Fsum} or L{Fsum2Tuple}).

951

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

953 '''

954 x, y = _map0(abs, x, y, *x0_y0) # NOT fabs!

955 if x < y:

956 x, y = y, x

957 h2 = x**2

958 if h2 and y:

959 h2 *= (y / x)**2 + _1_0

960 return float(h2)

961

962

963def hypot2_(*xs):

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

965

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

967 L{Fsum2Tuple}), all positional.

968

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

970

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

972 '''

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

974 if h2: # and isfinite(h2)

975 _h = _1_0 / h2

976 xs = ((x * _h) for x in xs)

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

978 h2 = H2.fover(_h**2)

979 return h2

980

981

982def _map0(_f, x, y, x0=_0_0, y0=_0_0, *unused): # PYLINT W1113

983 return _f(x - x0), _f(y - y0)

984

985

986def norm2(x, y, *x0_y0):

987 '''Normalize a 2-dimensional vector.

988

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

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

991 @arg x0_y0: Optional reference C{(x0, y0)} (each C{scalar},

992 an L{Fsum} or L{Fsum2Tuple}).

993

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

995

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

997 '''

998 try:

999 h = None

1000 x, y = _map0(float, x, y, *x0_y0)

1001 h = hypot(x, y)

1002 if h:

1003 t = (x / h), (y / h)

1004 else:

1005 t = (_copysign_0_0(x), # pass?

1006 _copysign_0_0(y))

1007 except Exception as X:

1008 raise _xError(X, x=x, y=y, h=h)

1009 return t

1010

1011

1012def norm_(*xs):

1013 '''Normalize the components of an n-dimensional vector.

1014

1015 @arg xs: Components (each C{scalar}, an L{Fsum} or

1016 L{Fsum2Tuple}), all positional.

1017

1018 @return: Yield each component, normalized.

1019

1020 @raise ValueError: Invalid or insufficent B{C{xs}}

1021 or zero norm.

1022 '''

1023 try:

1024 i = h = None

1025 x = xs

1026 h = hypot_(*xs)

1027 _h = (_1_0 / h) if h else _0_0

1028 for i, x in enumerate(xs):

1029 yield x * _h

1030 except Exception as X:

1031 raise _xsError(X, xs, i, x, h=h) # PYLINT E0601

1032

1033

1034def polar2(x, y, *x0_y0):

1035 '''Return 2-tuple C{(length, angle)} with C{angle} in radians M{[0..+PI2)}.

1036

1037 @arg x: X (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1038 @arg y: Y (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1039 @arg x0_y0: Optional reference C{(x0, y0)} (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1040

1041 @note: Use C{polar(B{y}, B{x}, *B{yx0})} to get the angle as C{bearing} from North.

1042 '''

1043 x, y = _map0(float, x, y, *x0_y0)

1044 return hypot(x, y), _MODS.utily.atan2p(y, x)

1045

1046

1047def polar2d(x, y, *x0_y0):

1048 '''Return 2-tuple C{(length, angle)} with C{angle} in degrees M{[0..+360)}.

1049

1050 @arg x: X (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1051 @arg y: Y (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1052 @arg x0_y0: Optional reference C{(x0, y0)} (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1053

1054 @note: Use C{polar2d(B{y}, B{x}, *B{yx0})} to get the angle as C{bearing} from North.

1055 '''

1056 x, y = _map0(float, x, y, *x0_y0)

1057 return hypot(x, y), _MODS.utily.atan2b(y, x)

1058

1059

1060def _powers(x, n):

1061 '''(INTERNAL) Yield C{x**i for i=1..n}.

1062 '''

1063 p = 1 # type(p) == type(x)

1064 for _ in range(n):

1065 p *= x

1066 yield p

1067

1068

1069def _root(x, p, where):

1070 '''(INTERNAL) Raise C{x} to power C{0 <= p < 1}.

1071 '''

1072 try:

1073 if x > 0:

1074 r = Fsum(f2product=True, nonfinites=True)(x)

1075 return r.fpow(p).as_iscalar

1076 elif x < 0:

1077 raise ValueError(_negative_)

1078 except Exception as X:

1079 raise _xError(X, unstr(where, x))

1080 return _0_0 if p else _1_0 # x == 0

1081

1082

1083def sqrt0(x, Error=None):

1084 '''Return the square root C{sqrt(B{x})} iff C{B{x} > }L{EPS02},

1085 preserving C{type(B{x})}.

1086

1087 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1088 @kwarg Error: Error to raise for negative B{C{x}}.

1089

1090 @return: Square root (C{float} or L{Fsum}) or C{0.0}.

1091

1092 @raise TypeeError: Invalid B{C{x}}.

1093

1094 @note: Any C{B{x} < }L{EPS02} I{including} C{B{x} < 0}

1095 returns C{0.0}.

1096 '''

1097 if Error and x < 0:

1098 raise Error(unstr(sqrt0, x))

1099 return _root(x, _0_5, sqrt0) if x > EPS02 else (

1100 _0_0 if x < EPS02 else EPS0)

1101

1102

1103def sqrt3(x):

1104 '''Return the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)},

1105 preserving C{type(B{x})}.

1106

1107 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1108

1109 @return: Square root I{cubed} (C{float} or L{Fsum}).

1110

1111 @raise TypeeError: Invalid B{C{x}}.

1112

1113 @raise ValueError: Negative B{C{x}}.

1114

1115 @see: Functions L{cbrt} and L{cbrt2}.

1116 '''

1117 return _root(x, _1_5, sqrt3)

1118

1119

1120def sqrt_a(h, b):

1121 '''Compute the C{I{a}} side of a right-angled triangle from

1122 C{sqrt(B{h}**2 - B{b}**2)}.

1123

1124 @arg h: Hypotenuse or outer annulus radius (C{scalar}).

1125 @arg b: Triangle side or inner annulus radius (C{scalar}).

1126

1127 @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}).

1128

1129 @raise TypeError: Non-scalar B{C{h}} or B{C{b}}.

1130

1131 @raise ValueError: If C{abs(B{h}) < abs(B{b})}.

1132

1133 @see: Inner tangent chord B{I{d}} of an U{annulus

1134 <https://WikiPedia.org/wiki/Annulus_(mathematics)>}

1135 and function U{annulus_area<https://People.SC.FSU.edu/

1136 ~jburkardt/py_src/geometry/geometry.py>}.

1137 '''

1138 try:

1139 if not (_isHeight(h) and _isRadius(b)):

1140 raise TypeError(_not_scalar_)

1141 c = fabs(h)

1142 if c > EPS0:

1143 s = _1_0 - (b / c)**2

1144 if s < 0:

1145 raise ValueError(_h_lt_b_)

1146 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0)

1147 else: # PYCHOK no cover

1148 b = fabs(b)

1149 d = c - b

1150 if d < 0:

1151 raise ValueError(_h_lt_b_)

1152 d *= c + b

1153 a = sqrt(d) if d else _0_0

1154 except Exception as x:

1155 raise _xError(x, h=h, b=b)

1156 return copysign0(a, h)

1157

1158

1159def zcrt(x):

1160 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)},

1161 preserving C{type(B{x})}.

1162

1163 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1164

1165 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}).

1166

1167 @see: Functions L{bqrt} and L{zqrt}.

1168

1169 @raise TypeeError: Invalid B{C{x}}.

1170

1171 @raise ValueError: Negative B{C{x}}.

1172 '''

1173 return _root(x, _1_6th, zcrt)

1174

1175

1176def zqrt(x):

1177 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root,

1178 M{x**(1 / 8)}, preserving C{type(B{x})}.

1179

1180 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1181

1182 @return: I{Zenzi-quartic} root (C{float} or L{Fsum}).

1183

1184 @see: Functions L{bqrt} and L{zcrt}.

1185

1186 @raise TypeeError: Invalid B{C{x}}.

1187

1188 @raise ValueError: Negative B{C{x}}.

1189 '''

1190 return _root(x, _0_125, zqrt)

1191

1192# **) MIT License

1193#

1195#

1196# Permission is hereby granted, free of charge, to any person obtaining a

1197# copy of this software and associated documentation files (the "Software"),

1198# to deal in the Software without restriction, including without limitation

1199# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1200# and/or sell copies of the Software, and to permit persons to whom the

1201# Software is furnished to do so, subject to the following conditions:

1202#

1203# The above copyright notice and this permission notice shall be included

1204# in all copies or substantial portions of the Software.

1205#

1206# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1207# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1208# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1209# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1210# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1211# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1212# OTHER DEALINGS IN THE SOFTWARE.

Coverage for pygeodesy / fmath.py: 89%

358 statements