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, isint, len2

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

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

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

13 _copysign_0_0, _isfinite, _over, remainder

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

15 _xError, _xkwds_get, _xkwds_pop2

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

17 Fmt, unstr

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

19 _not_scalar_, _SPACE_, _too_

20 from pygeodesy.lazily import _ALL_LAZY, _sys_version_info2

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

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

23

24 from math import fabs, sqrt # pow

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

26

27__all__ = _ALL_LAZY.fmath

28__version__ = '24.04.17'

29

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

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

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

33

34

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

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

37 '''

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

39 inst._fset_ps(_0_0)

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

41

42

43 class Fdot(Fsum):

44 '''Precision dot product.

45 '''

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

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

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

49

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

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

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

53 see L{Fsum<Fsum.__init__>}.

54

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

56

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

58

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

60 '''

61 Fsum.__init__(self, **name_RESIDUAL)

62 self.fadd(_map_mul(a, b, Fdot))

63

64

65 class Fhorner(Fsum):

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

67 '''

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

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

70 for i=0..len(cs)-1)}.

71

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

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

74 instances), all positional.

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

76 see L{Fsum<Fsum.__init__>}.

77

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

79

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

81

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

83

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

85 L{Fsum.fmul}.

86 '''

87 Fsum.__init__(self, **name_RESIDUAL)

88 if cs:

89 if isinstance(x, Fsum):

90 _mul = self._mul_Fsum

91 else:

92 _mul = self._mul_scalar

93 x = _2float(x=x)

94 op = Fhorner.__name__

95 if len(cs) > 1 and x:

96 for c in reversed(cs):

97 self._fset_ps(_mul(x, op))

98 self._fadd(c, op, up=False)

99 self._update()

100 else: # x == 0

101 self._fadd(cs[0], op)

102 else:

103 self._fset_ps(_0_0)

104

105

106 class Fhypot(Fsum):

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

108 '''

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

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

111

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

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

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

115 class L{Fsum<Fsum.__init__>} and method L{root<Fsum.root>}.

116 '''

117 r = None # _xkwds_pop2 error

118 try:

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

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

121 raiser = _Fsum__init__(self, **kwds)

122 if xs:

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

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

125 except Exception as X:

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

127

128

129 class Fpolynomial(Fsum):

130 '''Precision polynomial evaluation.

131 '''

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

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

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

135

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

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

138 all positional.

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

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

141

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

143

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

145

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

147

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

149 method L{Fsum.fadd}.

150 '''

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

152 n = len(cs) - 1

153 if n > 0:

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

155 elif n < 0:

156 self._fset_ps(_0_0)

157

158

159 class Fpowers(Fsum):

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

161 '''

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

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

164

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

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

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

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

169 '''

170 try:

171 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser)

172 if xs:

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

174 except Exception as X:

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

176

177

178 class Froot(Fsum):

179 '''The root of a precision summation.

180 '''

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

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

183

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

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

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

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

188 '''

189 try:

190 raiser = _Fsum__init__(self, **name_RESIDUAL_raiser)

191 if xs:

192 self. fadd(xs)

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

194 except Exception as X:

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

196

197

198 class Fcbrt(Froot):

199 '''Cubic root of a precision summation.

200 '''

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

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

203

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

205 '''

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

207

208

209 class Fsqrt(Froot):

210 '''Square root of a precision summation.

211 '''

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

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

214

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

216 '''

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

218

219

220 def bqrt(x):

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

222

223 @arg x: Value (C{scalar}).

224

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

226

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

228

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

230 '''

231 return _root(x, _0_25, bqrt)

232

233

234 try:

235 from math import cbrt # Python 3.11+

236

237 def cbrt2(x):

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

239 '''

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

241

242 except ImportError: # Python 3.10-

243

244 def cbrt(x):

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

246

247 @arg x: Value (C{scalar}).

248

249 @return: Cubic root (C{float}).

250

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

252 '''

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

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

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

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

257

258 def cbrt2(x): # PYCHOK attr

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

260

261 @arg x: Value (C{scalar}).

262

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

264

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

266 '''

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

268

269

270 def euclid(x, y):

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

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

273

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

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

276

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

278

279 @see: Function L{euclid_}.

280 '''

281 x, y = fabs(x), fabs(y)

282 if x < y:

283 x, y = y, x

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

285

286

287 def euclid_(*xs):

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

289 by cascaded L{euclid}.

290

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

292

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

294

295 @see: Function L{euclid}.

296 '''

297 e = _0_0

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

299 # e = euclid(x, e)

300 if e < x:

301 e, x = x, e

302 if x:

303 e += x * _0_4142

304 return e

305

306

307 def facos1(x):

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

309

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

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

312 '''

313 a = fabs(x)

314 if a < EPS0:

315 r = PI_2

316 elif a < EPS1:

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

318 r = float(H * sqrt(_1_0 - a))

319 if x < 0:

320 r = PI - r

321 else:

322 r = PI if x < 0 else _0_0

323 return r

324

325

326 def fasin1(x): # PYCHOK no cover

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

328

329 @see: L{facos1}.

330 '''

331 return PI_2 - facos1(x)

332

333

334 def fatan(x):

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

336 '''

337 a = fabs(x)

338 if a < _1_0:

339 r = fatan1(a) if a else _0_0

340 elif a > _1_0:

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

342 else:

343 r = PI_4

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

345 r = -r

346 return r

347

348

349 def fatan1(x):

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

351

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

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

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

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

356 IEEE Signal Processing Magazine, 111, May 2006.

357 '''

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

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

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

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

362 return float(H)

363

364

365 def fatan2(y, x):

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

367

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

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

370 and L{fatan1}.

371 '''

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

373 if b > a:

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

375 elif a > b:

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

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

378 r = PI_4

379 else: # a == b == 0

380 return _0_0

381 if x < 0:

382 r = PI - r

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

384 r = -r

385 return r

386

387

388 def favg(v1, v2, f=_0_5):

389 '''Return the average of two values.

390

391 @arg v1: One value (C{scalar}).

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

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

394

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

396 '''

397# @raise ValueError: Fraction out of range.

398# '''

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

400# raise _ValueError(fraction=f)

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

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

403

404

405 def fdot(a, *b):

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

407 i=0..len(a))}.

408

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

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

411

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

413

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

415

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

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

418 '''

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

420

421

422 def fdot3(a, b, c, start=0):

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

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

425

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

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

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

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

430

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

432

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

434 and/or C{len(B{c})}.

435

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

437 '''

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

439 return a * b * c

440

441 def _mul3_(a, b, c, start):

442 yield start

443 for abc in map(_mul3, a, b, c):

444 yield abc

445

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

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

448

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

450

451

452 def fhorner(x, *cs):

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

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

455

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

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

458

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

460

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

462

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

464

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

466

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

468 '''

469 H = Fhorner(x, *cs)

470 return float(H)

471

472

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

474 '''Interpolate using U{Inverse Distance Weighting

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

476

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

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

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

480

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

482

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

484

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

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

487

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

489 '''

490 n, xs = len2(xs)

491 d, ds = len2(ds)

492 if n != d or n < 1:

493 raise LenError(fidw, xs=n, ds=d)

494

495 d, x = min(zip(ds, xs))

496 if d > EPS0 and n > 1:

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

498 if b < 0:

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

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

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

502 else: # b == 0

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

504 elif d < 0: # PYCHOK no cover

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

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

507 return x

508

509

510 def fmean(xs):

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

512

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

514

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

516

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

518

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

520 '''

521 n, xs = len2(xs)

522 if n < 1:

523 raise LenError(fmean, xs=xs)

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

525

526

527 def fmean_(*xs):

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

529

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

531 '''

532 return fmean(xs)

533

534

535 def fpolynomial(x, *cs, **over):

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

537 i=0..len(cs)) [/ over]}.

538

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

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

541 positional.

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

543 (C{scalar}).

544

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

546

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

548

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

550

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

552

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

554 '''

555 P = Fpolynomial(x, *cs)

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

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

558

559

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

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

562

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

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

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

566 exponent (C{int}).

567

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

569

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

571

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

573 '''

574 if not isint(n):

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

576 elif n < 1:

577 raise _ValueError(n=n)

578

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

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

581

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

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

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

585

586 return ps

587

588

589 try:

590 from math import prod as fprod # Python 3.8

591 except ImportError:

592

593 def fprod(xs, start=1):

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

595

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

597 tuple, etc. (C{scalar}s).

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

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

600

601 @return: The product (C{float}).

602

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

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

605 '''

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

607

608

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

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

611

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

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

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

615

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

617

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

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

620 '''

621 if not isint(number):

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

623 for i in range(number):

624 yield start + (step * i)

625

626

627 try:

628 from functools import reduce as freduce

629 except ImportError:

630 try:

631 freduce = reduce # PYCHOK expected

632 except NameError: # Python 3+

633

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

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

636 '''

637 if start:

638 r = v = start[0]

639 else:

640 r, v = 0, MISSING

641 for v in xs:

642 r = f(r, v)

643 if v is MISSING:

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

645 return r

646

647

648 def fremainder(x, y):

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

650

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

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

653

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

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

656

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

658

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

660 project/geographiclib/>} and Python 3.7+

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

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

663 '''

664 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and

665 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native

666 # fmod( 0, 360) == 0.0

667 # fmod( 360, 360) == 0.0

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

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

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

671 # however, using the % operator ...

672 # 0 % 360 == 0

673 # 360 % 360 == 0

674 # 360.0 % 360 == 0.0

675 # -0 % 360 == 0

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

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

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

679

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

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

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

683 if _isfinite(x):

684 try:

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

686 except Exception as e:

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

688 else: # handle x INF and NINF as NAN

689 r = NAN

690 return r

691

692

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

694 from math import hypot # OK in Python 3.7-

695

696 def hypot_(*xs):

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

698

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

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

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

702 handled differently.

703

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

705

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

707

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

709

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

711

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

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

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

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

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

717 '''

718 h, x2 = _h_x2(xs, hypot_)

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

720

721 elif _sys_version_info2 < (3, 10):

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

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

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

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

726

727 def hypot(x, y):

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

729

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

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

732

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

734 '''

735 if x:

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

737 elif y:

738 h = fabs(y)

739 else:

740 h = _0_0

741 return h

742

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

744 else:

745 from math import hypot # PYCHOK in Python 3.10+

746 hypot_ = hypot

747

748

749 def _h_x2(xs, which):

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

751 '''

752 n, xs = len2(xs)

753 if n > 0:

754 h = float(max(map(fabs, xs)))

755 if h < EPS0:

756 x2 = _0_0

757 elif n > 1:

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

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

760 else:

761 x2 = _1_0

762 return h, x2

763

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

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

766

767

768 def hypot1(x):

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

770

771 @arg x: Argument (C{scalar}).

772

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

774 '''

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

776

777

778 def hypot2(x, y):

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

780

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

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

783

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

785 '''

786 if fabs(x) < fabs(y):

787 x, y = y, x

788 if x:

789 h2 = x**2

790 if y:

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

792 else:

793 h2 = _0_0

794 return h2

795

796

797 def hypot2_(*xs):

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

799

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

801

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

803

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

805

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

807

808 @see: Function L{hypot_}.

809 '''

810 h, x2 = _h_x2(xs, hypot2_)

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

812

813

814 def _map_mul(a, b, where):

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

816 '''

817 n = len(b)

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

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

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

821

822

823 def _1map_mul(a, b):

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

825 '''

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

827

828

829 def norm2(x, y):

830 '''Normalize a 2-dimensional vector.

831

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

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

834

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

836

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

838 or zero norm.

839 '''

840 try:

841 h = hypot(x, y)

842 if h:

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

844 else:

845 x = _copysign_0_0(x) # pass?

846 y = _copysign_0_0(y)

847 except Exception as e:

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

849 return x, y

850

851

852 def norm_(*xs):

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

854

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

856

857 @return: Yield each component, normalized.

858

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

860 or zero norm.

861 '''

862 try:

863 h = hypot_(*xs)

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

865 for i, x in enumerate(xs):

866 yield x * _h

867 except Exception as e:

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

869

870

871 def _powers(x, n):

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

873 '''

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

875 for _ in range(n):

876 p *= x

877 yield p

878

879

880 def _root(x, p, where):

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

882 '''

883 if x < 0:

884 t = _SPACE_(_invokation_, where.__name__)

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

886 return pow(x, p) if x else _0_0

887

888

889 def sqrt0(x, Error=None):

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

891

892 @arg x: Value (C{scalar}).

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

894

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

896

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

898 returns C{0.0}.

899 '''

900 if Error and x < 0:

901 raise Error(Fmt.PAREN(sqrt=x))

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

903

904

905 def sqrt3(x):

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

907

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

909

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

911

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

913

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

915 '''

916 return _root(x, _1_5, sqrt3)

917

918

919 def sqrt_a(h, b):

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

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

922

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

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

925

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

927

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

929

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

931

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

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

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

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

936 '''

937 try:

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

939 raise TypeError(_not_scalar_)

940 c = fabs(h)

941 if c > EPS0:

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

943 if s < 0:

944 raise ValueError(_h_lt_b_)

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

946 else: # PYCHOK no cover

947 b = fabs(b)

948 d = c - b

949 if d < 0:

950 raise ValueError(_h_lt_b_)

951 d *= c + b

952 a = sqrt(d) if d else _0_0

953 except Exception as x:

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

955 return copysign0(a, h)

956

957

958 def zcrt(x):

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

960

961 @arg x: Value (C{scalar}).

962

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

964

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

966

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

968 '''

969 return _root(x, _1_6th, zcrt)

970

971

972 def zqrt(x):

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

974

975 @arg x: Value (C{scalar}).

976

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

978

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

980

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

982 '''

983 return _root(x, _0_125, zqrt)

984

985# **) MIT License

986#

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

988#

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

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

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

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

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

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

995#

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

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

998#

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

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

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

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

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

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

1005# OTHER DEALINGS IN THE SOFTWARE.