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__ = '26.03.31'

31# sqrt(2) - 1 <https://WikiPedia.org/wiki/Square_root_of_2>

32_0_4142 = 0.41421356237309504880 # ~ 3_730_904_090_310_553 / 9_007_199_254_740_992

33_1_6th = 1 / 6

34_h_lt_b_ = 'abs(h) < abs(b)'

37class Fdot(Fsum):

38 '''Precision dot product.

39 '''

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

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

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

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

45 positional.

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

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

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

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

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

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

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

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

59 '''

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

61 Fsum.__init__(self, **kwds)

62 self(s)

64 n = len(b)

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

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

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

70class Fdot_(Fdot): # in .elliptic

71 '''Precision dot product.

72 '''

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

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

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

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

78 all positional.

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

81 '''

82 if isodd(len(xys)):

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

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

87class Fhorner(Fsum):

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

89 '''

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

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

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

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

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

96 all positional.

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

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

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

100

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

102

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

104

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

106

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

108 '''

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

110 Fsum.__init__(self, **kwds)

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

112

113

114class Fhypot(Fsum):

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

116 '''

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

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

119 to the power I{root}).

120

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

122 positional.

123 @kwarg root_name_f2product_nonfinites_RESIDUAL_raiser: Optional, exponent

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

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

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

127 L{root<Fsum.root>}.

128 '''

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

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

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

132

133 r = None # _xkwds_pop2 error

134 try:

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

136 Fsum.__init__(self, **kwds)

137 self(_0_0)

138 if xs:

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

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

141 except Exception as X:

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

143

144

145class Fpolynomial(Fsum):

146 '''Precision polynomial evaluation.

147 '''

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

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

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

151

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

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

154 all positional.

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

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

157

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

159

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

161

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

163

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

165 '''

166 Fsum.__init__(self, **name_f2product_nonfinites_RESIDUAL)

167 n = len(cs) - 1

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

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

170

171

172class Fpowers(Fsum):

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

174 '''

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

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

177

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

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

180 positional.

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

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

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

184 L{fpow<Fsum.fpow>}.

185 '''

186 try:

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

188 Fsum.__init__(self, **kwds)

189 self(_0_0)

190 if xs:

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

192 except Exception as X:

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

194

195

196class Froot(Fsum):

197 '''The root of a precision summation.

198 '''

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

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

201

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

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

204 positional.

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

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

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

208 L{fpow<Fsum.fpow>}.

209 '''

210 try:

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

212 Fsum.__init__(self, **kwds)

213 self(_0_0)

214 if xs:

215 self.fadd(xs)

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

217 except Exception as X:

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

219

220

221class Fcbrt(Froot):

222 '''Cubic root of a precision summation.

223 '''

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

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

226

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

228 '''

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

230

231

232class Fsqrt(Froot):

233 '''Square root of a precision summation.

234 '''

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

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

237

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

239 '''

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

241

242

243def bqrt(x):

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

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

246

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

248

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

250

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

252

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

254

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

256 '''

257 return _root(x, _0_25, bqrt)

258

259

260try:

261 from math import cbrt as _cbrt # Python 3.11+

262except ImportError: # Python 3.10-

263

264 def _cbrt(x):

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

266 '''

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

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

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

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

271

272

273def cbrt(x):

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

275

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

277

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

279

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

281 '''

282 if _isFsum_2Tuple(x):

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

284 if x.signOf() < 0:

285 r = -r

286 else:

287 r = _cbrt(x)

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

289

290

291def cbrt2(x): # PYCHOK attr

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

293

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

295

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

297

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

299 '''

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

301

302

303def euclid(x, y, *xy0):

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

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

306

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

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

309 @arg xy0: Optional reference C{(x0, y0)} (each C{scalar}, an

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

311

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

313 '''

314 x, y = _map0(abs, x, y, *xy0) # NOT fabs!

315 if x < y:

316 x, y = y, x

317 return x + y * _0_4142 # _0_5 before 20.10.02

318

319

320def euclid_(*xs):

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

322 L{euclid}.

323

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

325 all positional.

326

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

328

329 @see: Function L{euclid}.

330 '''

331 e = _0_0

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

333 # e = euclid(x, e)

334 if x:

335 if e < x:

336 e, x = x, e

337 x *= _0_4142

338# s = e + x

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

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

341# else:

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

343 e += x

344# e += s

345 return e

346

347

348def facos1(x):

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

350

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

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

353 '''

354 a = fabs(x)

355 if a < EPS0:

356 r = PI_2

357 elif a < EPS1:

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

359 r *= sqrt(_1_0 - a)

360 if x < 0:

361 r = PI - r

362 else:

363 r = PI if x < 0 else _0_0

364 return r

365

366

367def fasin1(x): # PYCHOK no cover

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

369

370 @see: L{facos1}.

371 '''

372 return PI_2 - facos1(x)

373

374

375def _fast(x, *cs):

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

377 '''

378 h = 0

379 for c in reversed(cs):

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

381 return h

382

383

384def fatan(x):

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

386 '''

387 a = fabs(x)

388 if a < _1_0:

389 r = fatan1(a) if a else _0_0

390 elif a > _1_0:

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

392 else:

393 r = PI_4

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

395 r = -r

396 return r

397

398

399def fatan1(x):

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

401

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

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

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

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

406 IEEE Signal Processing Magazine, 111, May 2006.

407 '''

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

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

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

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

412

413

414def fatan2(y, x):

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

416

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

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

419 and L{fatan1}.

420 '''

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

422 if b > a:

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

424 elif a > b:

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

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

427 r = PI_4

428 else: # a == b == 0

429 return _0_0

430 if x < 0:

431 r = PI - r

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

433 r = -r

434 return r

435

436

437def favg(a, b, f=_0_5, nonfinites=True):

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

439

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

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

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

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

444

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

446 '''

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

448 return float(F)

449

450

451def fdot(xs, *ys, **start_f2product_nonfinites):

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

453

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

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

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

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

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

459

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

461

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

463

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

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

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

467 '''

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

469 return float(D)

470

471

472def fdot_(*xys, **start_f2product_nonfinites):

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

474

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

476

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

478

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

480 '''

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

482 return float(D)

483

484

485def fdot3(xs, ys, zs, **start_f2product_nonfinites):

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

487

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

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

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

491

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

493

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

495

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

497 '''

498 n = len(xs)

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

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

501

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

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

504 _f = Fsum(**kwds)

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

506 return float(D)

507

508

509if _zip is zip: # Python 3.9-

510 _fdotf = fdot

511else:

512 from pygeodesy.fsums import _fsum # math.fsum

513

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

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

516 '''

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

518

519

520def fhorner(x, *cs, **incx):

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

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

523

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

525

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

527 '''

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

529 return float(H)

530

531

532def fidw(xs, ds, beta=2):

533 '''Interpolate using U{Inverse Distance Weighting

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

535

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

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

538 L{Fsum2Tuple}).

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

540

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

542

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

544

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

546

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

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

549

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

551 '''

552 n, xs = len2(xs)

553 if n > 1:

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

555 if b < 0:

556 try: # weighted

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

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

559 x = xs[i]

560 D = _d(d)

561 if D < EPS0:

562 if D < 0:

563 raise ValueError(_negative_)

564 x = float(x)

565 i = n

566 break

567 if D.fpow(b):

568 W += D

569 X += D.fmul(x)

570 else:

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

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

573 except IndexError:

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

575 except Exception as X:

576 _I = Fmt.INDEX

577 raise _xError(X, _I(xs=i), x,

578 _I(ds=i), d)

579 else: # b == 0

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

581 i = n

582 elif n:

583 x = float(xs[0])

584 i = n

585 else:

586 x = _0_0

587 i, _ = len2(ds)

588 if i != n:

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

590 return x

591

592

593try:

594 from math import fma as _fma # in .resections

595

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

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

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

599 '''

600 h = _0_0

601 if cs:

602 if _xkwds_get(incx, incx=True):

603 cs = reversed(cs)

604 for c in cs:

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

606 return h

607

608except ImportError: # PYCHOK DSPACE!

609 _fhornerf = fhorner # PYCHOK redef

610

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

612 return x * y + z

613

614

615def fma(x, y, z, **nonfinites): # **raiser

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

617 or an equivalent implementation.

618

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

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

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

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

623 to override default L{nonfiniterrors}

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

625

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

627 '''

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

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

630

631

632def _Fm2(x, nonfinites=None, **raiser):

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

634 '''

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

636

637

638def fmean(xs, nonfinites=True):

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

640

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

642

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

644

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

646

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

648 '''

649 n, xs = len2(xs)

650 if n < 1:

651 raise LenError(fmean, xs=xs)

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

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

654

655

656def fmean_(*xs, **nonfinites):

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

658

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

660 '''

661 return fmean(xs, **nonfinites)

662

663

664def f2mul_(x, *ys, **nonfinites): # **raiser

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

666

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

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

669 positional.

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

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

672

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

674

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

676 '''

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

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

679

680

681def fpolynomial(x, *cs, **over_f2product_nonfinites):

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

683

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

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

686 L{Fpolynomial<Fpolynomial.__init__>}.

687

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

689 '''

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

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

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

693

694

695def fpowers(x, n, alts=0):

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

697

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

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

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

701 exponent (C{int}).

702

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

704

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

706

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

708 '''

709 if not isint(n):

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

711 elif n < 1:

712 raise _ValueError(n=n)

713

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

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

716

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

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

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

720

721 return ps

722

723

724try:

725 from math import prod as fprod # Python 3.8

726except ImportError:

727

728 def fprod(xs, start=1):

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

730

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

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

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

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

735

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

737

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

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

740 '''

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

742

743

744def frandoms(n, seeded=None):

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

746

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

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

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

750

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

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

753 '''

754 from random import gauss, random, seed, shuffle

755

756 if seeded is None:

757 pass

758 elif seeded and isbool(seeded):

759 from time import localtime

760 seed(localtime().tm_yday)

761 elif isscalar(seeded):

762 seed(seeded)

763

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

765 for _ in range(n):

766 s = 0

767 t = list(c)

768 _a = t.append

769 for _ in range(n * 8):

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

771 _a(v)

772 s += v

773 shuffle(t)

774 yield t

775

776

777def frange(start, number, step=1):

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

779

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

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

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

783

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

785

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

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

788 '''

789 if not isint(number):

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

791 for i in range(number):

792 yield start + (step * i)

793

794

795try:

796 from functools import reduce as freduce

797except ImportError:

798 try:

799 freduce = reduce # PYCHOK expected

800 except NameError: # Python 3+

801

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

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

804 '''

805 if start:

806 r = v = start[0]

807 else:

808 r, v = 0, MISSING

809 for v in xs:

810 r = f(r, v)

811 if v is MISSING:

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

813 return r

814

815

816def fremainder(x, y):

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

818

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

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

821

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

823 0.0) or C{NAN} for any non-finite B{C{x}}.

824

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

826

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

828 project/geographiclib/>} and Python 3.7+

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

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

831 '''

832 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and

833 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native

834 # fmod( 0, 360) == 0.0

835 # fmod( 360, 360) == 0.0

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

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

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

839 # however, using the % operator ...

840 # 0 % 360 == 0

841 # 360 % 360 == 0

842 # 360.0 % 360 == 0.0

843 # -0 % 360 == 0

844 # -360 % 360 == 0 == (-360) % 360

845 # -0.0 % 360 == 0.0 == (-0.0) % 360

846 # -360.0 % 360 == 0.0 == (-360.0) % 360

847

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

849 # == +0.0. This fixes this bug. See also Math::AngNormalize

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

851 if isfinite(x):

852 try:

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

854 except Exception as e:

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

856 else: # handle x INF and NINF as NAN

857 r = NAN

858 return r

859

860

861if _MODS.sys_version_info2 < (3, 8): # PYCHOK no cover

862 from math import hypot # OK in Python 3.7-

863

864 def hypot_(*xs):

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

866

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

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

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

870 handled differently.

871

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

873

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

875

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

877

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

879

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

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

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

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

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

885 '''

886 return float(_Hypot(*xs))

887

888elif _MODS.sys_version_info2 < (3, 10): # PYCHOK no cover

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

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

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

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

893

894 def hypot(x, y):

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

896

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

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

899

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

901 '''

902 return float(_Hypot(x, y))

903

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

905else:

906 from math import hypot # PYCHOK in Python 3.10+

907 hypot_ = hypot

908

909

910def _Hypot(*xs):

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

912 '''

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

914

915

916def hypot1(x):

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

918

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

920

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

922 '''

923 h = _1_0

924 if x:

925 if _isFsum_2Tuple(x):

926 h = _Hypot(h, x)

927 h = float(h)

928 else:

929 h = hypot(h, x)

930 return h

931

932

933def hypot2(x, y, *xy0):

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

935

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

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

938 @arg xy0: Optional reference C{(x0, y0)} (each C{scalar},

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

940

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

942 '''

943 x, y = _map0(abs, x, y, *xy0) # NOT fabs!

944 if x < y:

945 x, y = y, x

946 h2 = x**2

947 if h2 and y:

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

949 return float(h2)

950

951

952def hypot2_(*xs):

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

954

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

956 L{Fsum2Tuple}), all positional.

957

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

959

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

961 '''

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

963 if h2: # and isfinite(h2)

964 _h = _1_0 / h2

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

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

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

968 return h2

969

970

971def _map0(_f, x, y, x0=_0_0, y0=_0_0, *unused):

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

973

974

975def norm2(x, y, *xy0):

976 '''Normalize a 2-dimensional vector.

977

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

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

980 @arg xy0: Optional reference C{(x0, y0)} (each C{scalar}, an

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

982

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

984

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

986 '''

987 try:

988 h = None

989 x, y = _map0(float, x, y, *xy0)

990 h = hypot(x, y)

991 if h:

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

993 else:

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

995 _copysign_0_0(y))

996 except Exception as X:

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

998 return t

999

1000

1001def norm_(*xs):

1002 '''Normalize the components of an n-dimensional vector.

1003

1004 @arg xs: Components (each C{scalar}, an L{Fsum} or

1005 L{Fsum2Tuple}), all positional.

1006

1007 @return: Yield each component, normalized.

1008

1009 @raise ValueError: Invalid or insufficent B{C{xs}}

1010 or zero norm.

1011 '''

1012 try:

1013 i = h = None

1014 x = xs

1015 h = hypot_(*xs)

1016 _h = (_1_0 / h) if h else _0_0

1017 for i, x in enumerate(xs):

1018 yield x * _h

1019 except Exception as X:

1020 raise _xsError(X, xs, i, x, h=h)

1021

1022

1023def polar2(x, y, *xy0):

1024 '''Return 2-tuple C{(length, angle)} with C{angle} in radians M{[0..+PI2)}.

1025

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

1027 @arg y: Y (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1028 @arg xy0: Optional reference C{(x0, y0)} (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1029

1030 @note: Use C{polar(B{y}, B{x}, *B{yx0})} to get the angle as C{bearing} from North.

1031 '''

1032 x, y = _map0(float, x, y, *xy0)

1033 return hypot(x, y), _MODS.utily.atan2p(y, x)

1034

1035

1036def polar2d(x, y, *xy0):

1037 '''Return 2-tuple C{(length, angle)} with C{angle} in degrees M{[0..+360)}.

1038

1039 @arg x: X (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1040 @arg y: Y (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1041 @arg xy0: Optional reference C{(x0, y0)} (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1042

1043 @note: Use C{polar2d(B{y}, B{x}, *B{yx0})} to get the angle as C{bearing} from North.

1044 '''

1045 x, y = _map0(float, x, y, *xy0)

1046 return hypot(x, y), _MODS.utily.atan2b(y, x)

1047

1048

1049def _powers(x, n):

1050 '''(INTERNAL) Yield C{x**i for i=1..n}.

1051 '''

1052 p = 1 # type(p) == type(x)

1053 for _ in range(n):

1054 p *= x

1055 yield p

1056

1057

1058def _root(x, p, where):

1059 '''(INTERNAL) Raise C{x} to power C{0 <= p < 1}.

1060 '''

1061 try:

1062 if x > 0:

1063 r = Fsum(f2product=True, nonfinites=True)(x)

1064 return r.fpow(p).as_iscalar

1065 elif x < 0:

1066 raise ValueError(_negative_)

1067 except Exception as X:

1068 raise _xError(X, unstr(where, x))

1069 return _0_0 if p else _1_0 # x == 0

1070

1071

1072def sqrt0(x, Error=None):

1073 '''Return the square root C{sqrt(B{x})} iff C{B{x} > }L{EPS02},

1074 preserving C{type(B{x})}.

1075

1076 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1077 @kwarg Error: Error to raise for negative B{C{x}}.

1078

1079 @return: Square root (C{float} or L{Fsum}) or C{0.0}.

1080

1081 @raise TypeeError: Invalid B{C{x}}.

1082

1083 @note: Any C{B{x} < }L{EPS02} I{including} C{B{x} < 0}

1084 returns C{0.0}.

1085 '''

1086 if Error and x < 0:

1087 raise Error(unstr(sqrt0, x))

1088 return _root(x, _0_5, sqrt0) if x > EPS02 else (

1089 _0_0 if x < EPS02 else EPS0)

1090

1091

1092def sqrt3(x):

1093 '''Return the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)},

1094 preserving C{type(B{x})}.

1095

1096 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1097

1098 @return: Square root I{cubed} (C{float} or L{Fsum}).

1099

1100 @raise TypeeError: Invalid B{C{x}}.

1101

1102 @raise ValueError: Negative B{C{x}}.

1103

1104 @see: Functions L{cbrt} and L{cbrt2}.

1105 '''

1106 return _root(x, _1_5, sqrt3)

1107

1108

1109def sqrt_a(h, b):

1110 '''Compute the C{I{a}} side of a right-angled triangle from

1111 C{sqrt(B{h}**2 - B{b}**2)}.

1112

1113 @arg h: Hypotenuse or outer annulus radius (C{scalar}).

1114 @arg b: Triangle side or inner annulus radius (C{scalar}).

1115

1116 @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}).

1117

1118 @raise TypeError: Non-scalar B{C{h}} or B{C{b}}.

1119

1120 @raise ValueError: If C{abs(B{h}) < abs(B{b})}.

1121

1122 @see: Inner tangent chord B{I{d}} of an U{annulus

1123 <https://WikiPedia.org/wiki/Annulus_(mathematics)>}

1124 and function U{annulus_area<https://People.SC.FSU.edu/

1125 ~jburkardt/py_src/geometry/geometry.py>}.

1126 '''

1127 try:

1128 if not (_isHeight(h) and _isRadius(b)):

1129 raise TypeError(_not_scalar_)

1130 c = fabs(h)

1131 if c > EPS0:

1132 s = _1_0 - (b / c)**2

1133 if s < 0:

1134 raise ValueError(_h_lt_b_)

1135 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0)

1136 else: # PYCHOK no cover

1137 b = fabs(b)

1138 d = c - b

1139 if d < 0:

1140 raise ValueError(_h_lt_b_)

1141 d *= c + b

1142 a = sqrt(d) if d else _0_0

1143 except Exception as x:

1144 raise _xError(x, h=h, b=b)

1145 return copysign0(a, h)

1146

1147

1148def zcrt(x):

1149 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)},

1150 preserving C{type(B{x})}.

1151

1152 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1153

1154 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}).

1155

1156 @see: Functions L{bqrt} and L{zqrt}.

1157

1158 @raise TypeeError: Invalid B{C{x}}.

1159

1160 @raise ValueError: Negative B{C{x}}.

1161 '''

1162 return _root(x, _1_6th, zcrt)

1163

1164

1165def zqrt(x):

1166 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root,

1167 M{x**(1 / 8)}, preserving C{type(B{x})}.

1168

1169 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1170

1171 @return: I{Zenzi-quartic} root (C{float} or L{Fsum}).

1172

1173 @see: Functions L{bqrt} and L{zcrt}.

1174

1175 @raise TypeeError: Invalid B{C{x}}.

1176

1177 @raise ValueError: Negative B{C{x}}.

1178 '''

1179 return _root(x, _0_125, zqrt)

1180

1181# **) MIT License

1182#

1184#

1185# Permission is hereby granted, free of charge, to any person obtaining a

1186# copy of this software and associated documentation files (the "Software"),

1187# to deal in the Software without restriction, including without limitation

1188# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1189# and/or sell copies of the Software, and to permit persons to whom the

1190# Software is furnished to do so, subject to the following conditions:

1191#

1192# The above copyright notice and this permission notice shall be included

1193# in all copies or substantial portions of the Software.

1194#

1195# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1196# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1197# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

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Coverage for pygeodesy / fmath.py: 89%

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