1

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

3

4 u'''Utilities using precision floating point summation.

5'''

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

7 from __future__ import division as _; del _ # PYCHOK semicolon

8

9 from pygeodesy.basics import _copysign, copysign0, isbool, isint, isscalar, \

10 len2, map1, _xiterable

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

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

13 _1_5, _copysign_0_0, isfinite, remainder

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

15 _xError, _xkwds_pop2, _xsError

16 from pygeodesy.fsums import _2float, Fsum, fsum, _isFsum_2Tuple, _1primed, \

17 Fmt, unstr

18 from pygeodesy.interns import MISSING, _negative_, _not_scalar_

19 from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS

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

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

22

23 from math import fabs, sqrt # pow

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

25

26__all__ = _ALL_LAZY.fmath

27__version__ = '24.10.11'

28

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

30_0_4142 = 0.41421356237309504880 # ... ~ 3730904090310553 / 9007199254740992

31_2_3rd = _1_3rd * 2

32_h_lt_b_ = 'abs(h) < abs(b)'

33

34

35 class Fdot(Fsum):

36 '''Precision dot product.

37 '''

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

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

40

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

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

43 positional.

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

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

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

47

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

49

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

51

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

53

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

55

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

57 '''

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

59 Fsum.__init__(self, **kwds)

60 self(s)

61

62 n = len(b)

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

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

65 self._faddot(n, a, b, **kwds)

66

67 def _faddot(self, n, xs, ys, **kwds):

68 if n > 0:

69 _f = Fsum(**kwds)

70 r = (_f(x).fmul(y) for x, y in zip(xs, ys)) # PYCHOK attr?

71 self.fadd(_1primed(r) if n < 4 else r) # PYCHOK attr?

72 return self

73

74

75 class Fhorner(Fsum):

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

77 '''

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

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

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

81 = len(cs) - 1}.

82

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

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

85 all positional.

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

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

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

89

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

91

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

93

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

95

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

97 '''

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

99 Fsum.__init__(self, **kwds)

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

101

102

103 class Fhypot(Fsum):

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

105 '''

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

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

108 to the power I{root}).

109

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

111 positional.

112 @kwarg root_name_f2product_nonfinites_RESIDUAL_raiser: Optional, exponent

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

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

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

116 L{root<Fsum.root>}.

117 '''

118 r = None # _xkwds_pop2 error

119 try:

120 r, kwds = _xkwds_pop2(root_name_f2product_nonfinites_RESIDUAL_raiser, root=2)

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

122 t, kwds = _xkwds_pop2(kwds, raiser=True)

123 Fsum.__init__(self, **kwds)

124 self(_0_0)

125 if xs:

126 self._facc_power(r, xs, Fhypot, raiser=t)

127 self._fset(self.root(r, raiser=t))

128 except Exception as X:

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

130

131

132 class Fpolynomial(Fdot):

133 '''Precision polynomial evaluation.

134 '''

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

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

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

138

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

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

141 all positional.

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

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

144

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

146

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

148

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

150

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

152 '''

153 Fsum.__init__(self, *cs[:1], **name_f2product_nonfinites_RESIDUAL)

154 n = len(cs) - 1

155 if n > 0:

156 self._faddot(n, cs[1:], _powers(x, n), **name_f2product_nonfinites_RESIDUAL)

157 elif n < 0:

158 self(_0_0)

159

160

161 class Fpowers(Fsum):

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

163 '''

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

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

166

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

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

169 positional.

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

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

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

173 L{fpow<Fsum.fpow>}.

174 '''

175 try:

176 t, kwds = _xkwds_pop2(name_f2product_nonfinites_RESIDUAL_raiser, raiser=True)

177 Fsum.__init__(self, **kwds)

178 self(_0_0)

179 if xs:

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

181 except Exception as X:

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

183

184

185 class Froot(Fsum):

186 '''The root of a precision summation.

187 '''

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

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

190

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

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

193 positional.

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

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

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

197 L{fpow<Fsum.fpow>}.

198 '''

199 try:

200 raiser, kwds = _xkwds_pop2(name_f2product_nonfinites_RESIDUAL_raiser, raiser=True)

201 Fsum.__init__(self, **kwds)

202 self(_0_0)

203 if xs:

204 self.fadd(xs)

205 self(self.root(root, raiser=raiser))

206 except Exception as X:

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

208

209

210 class Fcbrt(Froot):

211 '''Cubic root of a precision summation.

212 '''

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

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

215

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

217 '''

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

219

220

221 class Fsqrt(Froot):

222 '''Square root of a precision summation.

223 '''

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

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

226

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

228 '''

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

230

231

232 def bqrt(x):

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

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

235

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

237

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

239

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

241

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

243

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

245 '''

246 return _root(x, _0_25, bqrt)

247

248

249 try:

250 from math import cbrt as _cbrt # Python 3.11+

251

252 except ImportError: # Python 3.10-

253

254 def _cbrt(x):

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

256 '''

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

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

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

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

261

262

263 def cbrt(x):

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

265

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

267

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

269

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

271 '''

272 if _isFsum_2Tuple(x):

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

274 if x.signOf() < 0:

275 r = -r

276 else:

277 r = _cbrt(x)

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

279

280

281 def cbrt2(x): # PYCHOK attr

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

283

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

285

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

287

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

289 '''

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

291

292

293 def euclid(x, y):

294 '''I{Appoximate} the norm M{sqrt(x**2 + y**2)} by M{max(abs(x),

295 abs(y)) + min(abs(x), abs(y)) * 0.4142...}.

296

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

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

299

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

301

302 @see: Function L{euclid_}.

303 '''

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

305 if y > x:

306 x, y = y, x

307 return x + y * _0_4142 # * _0_5 before 20.10.02

308

309

310 def euclid_(*xs):

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

312 L{euclid}.

313

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

315 all positional.

316

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

318

319 @see: Function L{euclid}.

320 '''

321 e = _0_0

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

323 # e = euclid(x, e)

324 if e < x:

325 e, x = x, e

326 if x:

327 e += x * _0_4142

328 return e

329

330

331 def facos1(x):

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

333

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

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

336 '''

337 a = fabs(x)

338 if a < EPS0:

339 r = PI_2

340 elif a < EPS1:

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

342 r *= sqrt(_1_0 - a)

343 if x < 0:

344 r = PI - r

345 else:

346 r = PI if x < 0 else _0_0

347 return r

348

349

350 def fasin1(x): # PYCHOK no cover

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

352

353 @see: L{facos1}.

354 '''

355 return PI_2 - facos1(x)

356

357

358 def _fast(x, *cs):

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

360 '''

361 h = 0

362 for c in reversed(cs):

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

364 return h

365

366

367 def fatan(x):

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

369 '''

370 a = fabs(x)

371 if a < _1_0:

372 r = fatan1(a) if a else _0_0

373 elif a > _1_0:

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

375 else:

376 r = PI_4

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

378 r = -r

379 return r

380

381

382 def fatan1(x):

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

384

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

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

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

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

389 IEEE Signal Processing Magazine, 111, May 2006.

390 '''

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

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

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

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

395

396

397 def fatan2(y, x):

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

399

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

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

402 and L{fatan1}.

403 '''

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

405 if b > a:

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

407 elif a > b:

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

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

410 r = PI_4

411 else: # a == b == 0

412 return _0_0

413 if x < 0:

414 r = PI - r

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

416 r = -r

417 return r

418

419

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

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

422

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

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

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

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

427

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

429 '''

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

431 return float(F)

432

433

434 def fdot(a, *b, **start):

435 '''Return the precision dot product M{sum(a[i] * b[i] for ni=0..len(a))}.

436

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

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

439 positional.

440 @kwarg start: Optional bias C{B{start}=0} (C{scalar}, an L{Fsum} or

441 L{Fsum2Tuple}).

442

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

444

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

446

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

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

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

450 '''

451 D = Fdot(a, nonfinites=True, *b, **start)

452 return float(D)

453

454

455 def fdot3(xs, ys, zs, start=0):

456 '''Return the precision dot product M{start + sum(a[i] * b[i] * c[i]

457 for i=0..len(a)-1)}.

458

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

460 @arg ys: Iterable (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

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

462 @kwarg start: Optional bias (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

463

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

465

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

467 '''

468 n = len(xs)

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

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

471

472 D = Fdot((), nonfinites=True, start=start)

473 _f = Fsum(nonfinites=True) # f2product=True

474 r = (_f(x).f2mul_(y, z) for x, y, z in zip(xs, ys, zs))

475 D = D.fadd(_1primed(r) if n < 4 else r)

476 return float(D)

477

478

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

480 '''Horner form evaluation of polynomial M{sum(cs[i] * x**i for

481 i=0..n)} with in- or decreasing exponent M{sum(... i=n..0)},

482 where C{n = len(cs) - 1}.

483

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

485

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

487 '''

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

489 return float(H)

490

491

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

493 '''Interpolate using U{Inverse Distance Weighting

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

495

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

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

498 L{Fsum2Tuple}).

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

500

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

502

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

504

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

506

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

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

509

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

511 '''

512 n, xs = len2(xs)

513 if n > 1:

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

515 if b < 0:

516 try: # weighted

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

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

519 x = xs[i]

520 D = _d(d)

521 if D < EPS0:

522 if D < 0:

523 raise ValueError(_negative_)

524 x = float(x)

525 i = n

526 break

527 if D.fpow(b):

528 W += D

529 X += D.fmul(x)

530 else:

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

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

533 except IndexError:

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

535 except Exception as X:

536 _I = Fmt.INDEX

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

538 _I(ds=i), d)

539 else: # b == 0

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

541 i = n

542 elif n:

543 x = float(xs[0])

544 i = n

545 else:

546 x = _0_0

547 i, _ = len2(ds)

548 if i != n:

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

550 return x

551

552

553 try:

554 from math import fma as _fma

555 except ImportError: # PYCHOK DSPACE!

556

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

558 return x * y + z

559

560

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

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

563 or an equivalent implementation.

564

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

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

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

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

569 to override default L{nonfiniterrors}

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

571

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

573 '''

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

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

576

577

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

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

580 '''

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

582

583

584 def fmean(xs):

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

586

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

588

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

590

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

592

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

594 '''

595 n, xs = len2(xs)

596 if n < 1:

597 raise LenError(fmean, xs=xs)

598 M = Fsum(*xs, nonfinites=True)

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

600

601

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

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

604

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

606 '''

607 return fmean(xs, **nonfinites)

608

609

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

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

612

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

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

615 positional.

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

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

618

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

620

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

622 '''

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

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

625

626

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

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

629

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

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

632 L{Fpolynomial<Fpolynomial.__init__>}.

633

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

635 '''

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

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

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

639

640

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

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

643

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

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

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

647 exponent (C{int}).

648

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

650

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

652

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

654 '''

655 if not isint(n):

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

657 elif n < 1:

658 raise _ValueError(n=n)

659

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

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

662

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

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

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

666

667 return ps

668

669

670 try:

671 from math import prod as fprod # Python 3.8

672 except ImportError:

673

674 def fprod(xs, start=1):

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

676

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

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

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

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

681

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

683

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

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

686 '''

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

688

689

690 def frandoms(n, seeded=None):

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

692

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

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

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

696

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

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

699 '''

700 from random import gauss, random, seed, shuffle

701

702 if seeded is None:

703 pass

704 elif seeded and isbool(seeded):

705 from time import localtime

706 seed(localtime().tm_yday)

707 elif isscalar(seeded):

708 seed(seeded)

709

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

711 for _ in range(n):

712 s = 0

713 t = list(c)

714 _a = t.append

715 for _ in range(n * 8):

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

717 _a(v)

718 s += v

719 shuffle(t)

720 yield t

721

722

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

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

725

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

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

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

729

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

731

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

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

734 '''

735 if not isint(number):

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

737 for i in range(number):

738 yield start + (step * i)

739

740

741 try:

742 from functools import reduce as freduce

743 except ImportError:

744 try:

745 freduce = reduce # PYCHOK expected

746 except NameError: # Python 3+

747

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

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

750 '''

751 if start:

752 r = v = start[0]

753 else:

754 r, v = 0, MISSING

755 for v in xs:

756 r = f(r, v)

757 if v is MISSING:

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

759 return r

760

761

762 def fremainder(x, y):

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

764

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

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

767

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

769 0.0) or C{NAN} for any non-finite B{C{x}}.

770

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

772

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

774 project/geographiclib/>} and Python 3.7+

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

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

777 '''

778 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and

779 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native

780 # fmod( 0, 360) == 0.0

781 # fmod( 360, 360) == 0.0

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

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

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

785 # however, using the % operator ...

786 # 0 % 360 == 0

787 # 360 % 360 == 0

788 # 360.0 % 360 == 0.0

789 # -0 % 360 == 0

790 # -360 % 360 == 0 == (-360) % 360

791 # -0.0 % 360 == 0.0 == (-0.0) % 360

792 # -360.0 % 360 == 0.0 == (-360.0) % 360

793

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

795 # == +0.0. This fixes this bug. See also Math::AngNormalize

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

797 if isfinite(x):

798 try:

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

800 except Exception as e:

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

802 else: # handle x INF and NINF as NAN

803 r = NAN

804 return r

805

806

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

808 from math import hypot # OK in Python 3.7-

809

810 def hypot_(*xs):

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

812

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

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

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

816 handled differently.

817

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

819

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

821

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

823

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

825

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

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

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

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

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

831 '''

832 return float(_Hypot(*xs))

833

834 elif _MODS.sys_version_info2 < (3, 10):

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

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

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

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

839

840 def hypot(x, y):

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

842

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

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

845

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

847 '''

848 return float(_Hypot(x, y))

849

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

851 else:

852 from math import hypot # PYCHOK in Python 3.10+

853 hypot_ = hypot

854

855

856 def _Hypot(*xs):

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

858 '''

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

860

861

862 def hypot1(x):

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

864

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

866

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

868 '''

869 h = _1_0

870 if x:

871 if _isFsum_2Tuple(x):

872 h = _Hypot(h, x)

873 h = float(h)

874 else:

875 h = hypot(h, x)

876 return h

877

878

879 def hypot2(x, y):

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

881

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

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

884

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

886 '''

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

888 if y > x:

889 x, y = y, x

890 h2 = x**2

891 if h2 and y:

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

893 return float(h2)

894

895

896 def hypot2_(*xs):

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

898

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

900 L{Fsum2Tuple}), all positional.

901

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

903

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

905 '''

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

907 if h2: # and isfinite(h2)

908 _h = _1_0 / h2

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

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

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

912 return h2

913

914

915 def norm2(x, y):

916 '''Normalize a 2-dimensional vector.

917

918 @arg x: X component (C{scalar}).

919 @arg y: Y component (C{scalar}).

920

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

922

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

924 or zero norm.

925 '''

926 try:

927 h = None

928 h = hypot(x, y)

929 if h:

930 x, y = (x / h), (y / h)

931 else:

932 x = _copysign_0_0(x) # pass?

933 y = _copysign_0_0(y)

934 except Exception as e:

935 raise _xError(e, x=x, y=y, h=h)

936 return x, y

937

938

939 def norm_(*xs):

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

941

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

943 L{Fsum2Tuple}), all positional.

944

945 @return: Yield each component, normalized.

946

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

948 or zero norm.

949 '''

950 try:

951 i = h = None

952 x = xs

953 h = hypot_(*xs)

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

955 for i, x in enumerate(xs):

956 yield x * _h

957 except Exception as X:

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

959

960

961 def _powers(x, n):

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

963 '''

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

965 for _ in range(n):

966 p *= x

967 yield p

968

969

970 def _root(x, p, where):

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

972 '''

973 try:

974 if x > 0:

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

976 return r.fpow(p).as_iscalar

977 elif x < 0:

978 raise ValueError(_negative_)

979 except Exception as X:

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

981 return _0_0

982

983

984 def sqrt0(x, Error=None):

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

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

987

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

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

990

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

992

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

994

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

996 returns C{0.0}.

997 '''

998 if Error and x < 0:

999 raise Error(unstr(sqrt0, x))

1000 return _root(x, _0_5, sqrt0) if x > EPS02 else (

1001 _0_0 if x < EPS02 else EPS0)

1002

1003

1004 def sqrt3(x):

1005 '''Return the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)},

1006 preserving C{type(B{x})}.

1007

1008 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1009

1010 @return: Square root I{cubed} (C{float} or L{Fsum}).

1011

1012 @raise TypeeError: Invalid B{C{x}}.

1013

1014 @raise ValueError: Negative B{C{x}}.

1015

1016 @see: Functions L{cbrt} and L{cbrt2}.

1017 '''

1018 return _root(x, _1_5, sqrt3)

1019

1020

1021 def sqrt_a(h, b):

1022 '''Compute C{I{a}} side of a right-angled triangle from

1023 C{sqrt(B{h}**2 - B{b}**2)}.

1024

1025 @arg h: Hypotenuse or outer annulus radius (C{scalar}).

1026 @arg b: Triangle side or inner annulus radius (C{scalar}).

1027

1028 @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}).

1029

1030 @raise TypeError: Non-scalar B{C{h}} or B{C{b}}.

1031

1032 @raise ValueError: If C{abs(B{h}) < abs(B{b})}.

1033

1034 @see: Inner tangent chord B{I{d}} of an U{annulus

1035 <https://WikiPedia.org/wiki/Annulus_(mathematics)>}

1036 and function U{annulus_area<https://People.SC.FSU.edu/

1037 ~jburkardt/py_src/geometry/geometry.py>}.

1038 '''

1039 try:

1040 if not (_isHeight(h) and _isRadius(b)):

1041 raise TypeError(_not_scalar_)

1042 c = fabs(h)

1043 if c > EPS0:

1044 s = _1_0 - (b / c)**2

1045 if s < 0:

1046 raise ValueError(_h_lt_b_)

1047 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0)

1048 else: # PYCHOK no cover

1049 b = fabs(b)

1050 d = c - b

1051 if d < 0:

1052 raise ValueError(_h_lt_b_)

1053 d *= c + b

1054 a = sqrt(d) if d else _0_0

1055 except Exception as x:

1056 raise _xError(x, h=h, b=b)

1057 return copysign0(a, h)

1058

1059

1060 def zcrt(x):

1061 '''Return the 6-th, I{zenzi-cubic} root, M{x**(1 / 6)},

1062 preserving C{type(B{x})}.

1063

1064 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1065

1066 @return: I{Zenzi-cubic} root (C{float} or L{Fsum}).

1067

1068 @see: Functions L{bqrt} and L{zqrt}.

1069

1070 @raise TypeeError: Invalid B{C{x}}.

1071

1072 @raise ValueError: Negative B{C{x}}.

1073 '''

1074 return _root(x, _1_6th, zcrt)

1075

1076

1077 def zqrt(x):

1078 '''Return the 8-th, I{zenzi-quartic} or I{squared-quartic} root,

1079 M{x**(1 / 8)}, preserving C{type(B{x})}.

1080

1081 @arg x: Value (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1082

1083 @return: I{Zenzi-quartic} root (C{float} or L{Fsum}).

1084

1085 @see: Functions L{bqrt} and L{zcrt}.

1086

1087 @raise TypeeError: Invalid B{C{x}}.

1088

1089 @raise ValueError: Negative B{C{x}}.

1090 '''

1091 return _root(x, _0_125, zqrt)

1092

1093# **) MIT License

1094#

1095# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.

1096#

1097# Permission is hereby granted, free of charge, to any person obtaining a

1098# copy of this software and associated documentation files (the "Software"),

1099# to deal in the Software without restriction, including without limitation

1100# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1101# and/or sell copies of the Software, and to permit persons to whom the

1102# Software is furnished to do so, subject to the following conditions:

1103#

1104# The above copyright notice and this permission notice shall be included

1105# in all copies or substantial portions of the Software.

1106#

1107# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1108# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1109# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1110# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1111# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1112# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1113# OTHER DEALINGS IN THE SOFTWARE.