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, _N_1_0, \

12 _1_3rd, _1_5, _1_6th, _2_0, _2_3rd, _3_0, \

13 _isfinite, isnear0, isnear1, _over, remainder

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

15 _xError, _xkwds_get, _xkwds_pop2

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

17 _pow_op_, 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__ = '24.02.29'

29

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

31_0_4142 = 0.414213562373095 # sqrt(_2_0) - _1_0

32

33

34 class Fdot(Fsum):

35 '''Precision dot product.

36 '''

37 def __init__(self, a, *b, **name):

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

39 for i=0..len(a))}.

40

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

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

43 @kwarg name: Optional name (C{str}).

44

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

46

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

48

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

50 '''

51 Fsum.__init__(self, **name)

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

53

54

55 class Fhorner(Fsum):

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

57 '''

58 def __init__(self, x, *cs, **name):

59 '''New L{Fhorner} evaluation of the polynomial

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

61

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

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

64 instances), all positional.

65 @kwarg name: Optional name (C{str}).

66

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

68

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

70

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

72

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

74 '''

75 Fsum.__init__(self, *cs[-1:], **name)

76 if len(cs) > 1:

77 x = _2float(x=x)

78 _a = self._fadd # (other, op)

79 _f = self._finite # (other, op)

80 op = Fhorner.__name__

81 ps = self._ps

82 for c in reversed(cs[:-1]): # multiply-accumulate

83 ps[:] = [_f(p * x, op) for p in ps]

84 _a(c, op)

85 # assert self._ps is ps

86

87

88 class Fhypot(Fsum):

89 '''Precision hypotenuse of summation.

90 '''

91 def __init__(self, *xs, **power_name_RESIDUAL):

92 '''New L{Fhypot} hypotenuse of (the I{power} of) several

93 C{scalar} or C{Fsum} values.

94

95 @arg xs: One or more values to include (each C{scalar}

96 or an C{Fsum} instance).

97 @kwarg power_name_RESIDUAL: Optional exponent and root

98 order C{B{power}=2}, C{B{name}=NN} and

99 C{B{RESIDUAL}=None}, see L{Fsum.__init__}.

100 '''

101 try:

102 p, kwds = _xkwds_pop2(power_name_RESIDUAL, power=2)

103 Fsum.__init__(self, **kwds)

104 if xs:

105 self._facc(_Powers(p, xs), up=False) # PYCHOK yield

106 self._fset(self._fpow(_1_0 / p, _pow_op_), asis=True)

107 except Exception as X:

108 raise self._ErrorX(X, xs, power=p)

109

110

111 class Fpolynomial(Fsum):

112 '''Precision polynomial evaluation.

113 '''

114 def __init__(self, x, *cs, **name):

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

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

117

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

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

120 positional.

121 @kwarg name: Optional name (C{str}).

122

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

124

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

126

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

128

129 @see: Function L{fpolynomial} and method L{Fsum.fadd}.

130 '''

131 Fsum.__init__(self, *cs[:1], **name)

132 n = len(cs) - 1

133 if n > 0:

134 self.fadd(_map_mul(cs[1:], fpowers(x, n), Fpolynomial))

135

136

137 class Fpowers(Fsum):

138 '''Precision summation or powers, optimized for C{power=2}.

139 '''

140 def __init__(self, power, *xs, **name_RESIDUAL):

141 '''New L{Fpowers} sum of (the I{power} of) several C{scalar}

142 or C{Fsum} values.

143

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

145 @arg xs: One or more values to include (each C{scalar}

146 or an C{Fsum} instance).

147 @kwarg power_name_RESIDUAL: Optional exponent and root

148 order C{B{power}=2}, C{B{name}=NN} and

149 C{B{RESIDUAL}=None}, see L{Fsum.__init__}.

150 '''

151 try:

152 Fsum.__init__(self, **name_RESIDUAL)

153 if xs:

154 self._facc(_Powers(power, xs), up=False) # PYCHOK yield

155 except Exception as X:

156 raise self._ErrorX(X, xs, power=power)

157

158

159 class Fn_rt(Fsum):

160 '''Precision n-th root of summation.

161 '''

162 def __init__(self, root, *xs, **name_RESIDUAL):

163 '''New L{Fn_rt} root of the precision sum of several

164 C{scalar} or C{Fsum} values.

165

166 @arg root: The order (C{scalar} or C{Fsum}).

167 @arg xs: Values to include (each C{scalar} or an

168 C{Fsum} instance).

169 @kwarg name_RESIDUAL: See L{Fsum.__init__}.

170 '''

171 try:

172 Fsum.__init__(self, *xs, **name_RESIDUAL)

173 self._fset(self._fpow(_1_0 / root, _pow_op_), asis=True)

174 except Exception as X:

175 raise self._ErrorX(X, xs, root=root)

176

177

178 class Fcbrt(Fn_rt):

179 '''Precision cubic root of summation.

180 '''

181 def __init__(self, *xs, **name_RESIDUAL):

182 '''New L{Fcbrt} cubic root of the precision sum of

183 several C{scalar} or C{Fsum} values.

184

185 @arg xs: Values to include (each C{scalar} or an

186 C{Fsum} instance).

187 @kwarg name_RESIDUAL: See L{Fsum.__init__}.

188 '''

189 Fn_rt.__init__(self, _3_0, *xs, **name_RESIDUAL)

190

191

192 class Fsqrt(Fn_rt):

193 '''Precision square root of summation.

194 '''

195 def __init__(self, *xs, **name_RESIDUAL):

196 '''New L{Fsqrt} square root of the precision sum of

197 several C{scalar} or C{Fsum} values.

198

199 @arg xs: Values to include (each C{scalar} or an

200 C{Fsum} instance).

201 @kwarg name_RESIDUAL: See L{Fsum.__init__}.

202 '''

203 Fn_rt.__init__(self, _2_0, *xs, **name_RESIDUAL)

204

205

206 def bqrt(x):

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

208

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

210

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

212

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

214

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

216 '''

217 return _root(x, _0_25, bqrt)

218

219

220 try:

221 from math import cbrt # Python 3.11+

222

223 def cbrt2(x):

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

225 '''

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

227

228 except ImportError: # Python 3.10-

229

230 def cbrt(x):

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

232

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

234

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

236

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

238 '''

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

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

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

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

243

244 def cbrt2(x): # PYCHOK attr

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

246

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

248

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

250

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

252 '''

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

254

255

256 def euclid(x, y):

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

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

259

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

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

262

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

264

265 @see: Function L{euclid_}.

266 '''

267 x, y = fabs(x), fabs(y)

268 if x < y:

269 x, y = y, x

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

271

272

273 def euclid_(*xs):

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

275 by cascaded L{euclid}.

276

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

278

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

280

281 @see: Function L{euclid}.

282 '''

283 e = _0_0

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

285 # e = euclid(x, e)

286 if e < x:

287 e, x = x, e

288 if x:

289 e += x * _0_4142

290 return e

291

292

293 def facos1(x):

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

295

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

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

298 '''

299 a = fabs(x)

300 if a < EPS0:

301 r = PI_2

302 elif a < EPS1:

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

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

305 if x < 0:

306 r = PI - r

307 else:

308 r = PI if x < 0 else _0_0

309 return r

310

311

312 def fasin1(x): # PYCHOK no cover

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

314

315 @see: L{facos1}.

316 '''

317 return PI_2 - facos1(x)

318

319

320 def fatan(x):

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

322 '''

323 a = fabs(x)

324 if a < _1_0:

325 r = fatan1(a) if a else _0_0

326 elif a > _1_0:

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

328 else:

329 r = PI_4

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

331 r = -r

332 return r

333

334

335 def fatan1(x):

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

337

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

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

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

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

342 IEEE Signal Processing Magazine, 111, May 2006.

343 '''

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

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

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

347 H = Fhorner(x, _0_0, 1.0300982, -0.1784, -0.0663)

348 return float(H)

349

350

351 def fatan2(y, x):

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

353

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

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

356 and L{fatan1}.

357 '''

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

359 if a < b:

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

361 elif b < a:

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

363 elif a: # == b != 0

364 r = PI_4

365 else: # a == b == 0

366 return _0_0

367 if x < 0:

368 r = PI - r

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

370 r = -r

371 return r

372

373

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

375 '''Return the average of two values.

376

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

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

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

380

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

382 '''

383# @raise ValueError: Fraction out of range.

384# '''

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

386# raise _ValueError(fraction=f)

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

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

389

390

391 def fdot(a, *b):

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

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

394

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

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

397

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

399

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

401

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

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

404 '''

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

406

407

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

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

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

411

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

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

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

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

416

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

418

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

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

421

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

423 '''

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

425 return a * b * c

426

427 def _muly(a, b, c, start):

428 yield start

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

430 yield abc

431

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

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

434

435 return fsum(_muly(a, b, c, start) if start else map(_mul3, a, b, c))

436

437

438 def fhorner(x, *cs):

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

440 i=0..len(cs))} using the Horner form.

441

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

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

444

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

446

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

448

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

450

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

452

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

454 '''

455 H = Fhorner(x, *cs)

456 return float(H)

457

458

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

460 '''Interpolate using U{Inverse Distance Weighting

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

462

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

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

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

466

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

468

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

470

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

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

473

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

475 '''

476 n, xs = len2(xs)

477 d, ds = len2(ds)

478 if n != d or n < 1:

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

480

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

482 if d > EPS0 and n > 1:

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

484 if b < 0:

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

486 t = fsum(_map_mul1(xs, ws)) # fdot(xs, *ws)

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

488 else: # b == 0

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

490 elif d < 0: # PYCHOK no cover

491 n = Fmt.SQUARE(ds=ds.index(d))

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

493 return x

494

495

496 def fmean(xs):

497 '''Compute the accurate mean M{sum(xs[i] for

498 i=0..len(xs)) / len(xs)}.

499

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

501

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

503

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

505

506 @raise ValueError: No B{C{xs}} values.

507 '''

508 n, xs = len2(xs)

509 if n > 0:

510 return fsum(xs) / n # if n > 1 else _2float(index=0, xs=xs[0])

511 raise _ValueError(xs=xs)

512

513

514 def fmean_(*xs):

515 '''Compute the accurate mean M{sum(xs[i] for

516 i=0..len(xs)) / len(xs)}.

517

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

519

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

521

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

523

524 @raise ValueError: No B{C{xs}} values.

525 '''

526 return fmean(xs)

527

528

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

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

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

532

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

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

535 positional.

536 @kwarg over: Optional, final divisor (C{scalar}

537

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

539

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

541

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

543

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

545

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

547 '''

548 P = Fpolynomial(x, *cs)

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

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

551

552

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

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

555

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

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

558 @kwarg alts: Only alternating powers, starting with

559 this exponent (C{int}).

560

561 @return: Powers of B{C{x}} (C{float}s or C{int}s).

562

563 @raise TypeError: Non-scalar B{C{x}} or B{C{n}} not C{int}.

564

565 @raise ValueError: Non-finite B{C{x}} or non-positive B{C{n}}.

566 '''

567 if not isint(n):

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

569 elif n < 1:

570 raise _ValueError(n=n)

571

572 p = t = x if isint(x) else _2float(x=x)

573 ps = [p]

574 _a = ps.append

575 for _ in range(1, n):

576 p *= t

577 _a(p)

578

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

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

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

582

583 return ps

584

585

586 try:

587 from math import prod as fprod # Python 3.8

588 except ImportError:

589

590 def fprod(xs, start=1):

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

592

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

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

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

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

597

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

599

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

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

602 '''

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

604

605

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

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

608

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

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

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

612

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

614

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

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

617 '''

618 if not isint(number):

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

620 for i in range(number):

621 yield start + i * step

622

623

624 try:

625 from functools import reduce as freduce

626 except ImportError:

627 try:

628 freduce = reduce # PYCHOK expected

629 except NameError: # Python 3+

630

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

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

633 '''

634 if start:

635 r = v = start[0]

636 else:

637 r, v = 0, MISSING

638 for v in xs:

639 r = f(r, v)

640 if v is MISSING:

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

642 return r

643

644

645 def fremainder(x, y):

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

647

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

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

650

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

652 0.0) or C{NAN} for any non-finite B{C{x}}.

653

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

655

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

657 project/geographiclib/>} and Python 3.7+

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

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

660 '''

661 # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and

662 # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native

663 # fmod( 0, 360) == 0.0

664 # fmod( 360, 360) == 0.0

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

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

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

668 # however, using the % operator ...

669 # 0 % 360 == 0

670 # 360 % 360 == 0

671 # 360.0 % 360 == 0.0

672 # -0 % 360 == 0

673 # -360 % 360 == 0 == (-360) % 360

674 # -0.0 % 360 == 0.0 == (-0.0) % 360

675 # -360.0 % 360 == 0.0 == (-360.0) % 360

676

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

678 # == +0.0. This fixes this bug. See also Math::AngNormalize

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

680 if _isfinite(x):

681 try:

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

683 except Exception as e:

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

685 else: # handle x INF and NINF as NAN

686 r = NAN

687 return r

688

689

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

691 from math import hypot # OK in Python 3.7-

692

693 def hypot_(*xs):

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

695

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

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

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

699 handled differently.

700

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

702

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

704

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

706

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

708

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

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

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

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

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

714 '''

715 h, x2 = _h_x2(xs)

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

717

718 elif _sys_version_info2 < (3, 10):

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

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

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

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

723

724 def hypot(x, y):

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

726

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

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

729

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

731 '''

732 if x:

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

734 elif y:

735 h = fabs(y)

736 else:

737 h = _0_0

738 return h

739

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

741 else:

742 from math import hypot # PYCHOK in Python 3.10+

743 hypot_ = hypot

744

745

746 def _h_x2(xs):

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

748 '''

749 if xs:

750 n, xs = len2(xs)

751 if n > 0:

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

753 if h < EPS0:

754 x2 = _0_0

755 else: # math.fsum, see C{_hypot21_} below

756 x2 = _fsum(_x2_h2(_1_0, xs, h, _N_1_0))

757 return h, x2

758

759 raise _ValueError(xs=xs, txt=_too_(_few_))

760

761

762 def hypot1(x):

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

764

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

766

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

768 '''

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

770

771

772 def hypot2(x, y):

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

774

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

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

777

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

779 '''

780 if x:

781 if y:

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

783 x, y = y, x

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

785 else:

786 h2 = x**2

787 elif y:

788 h2 = y**2

789 else:

790 h2 = _0_0

791 return h2

792

793

794 def hypot2_(*xs):

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

796

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

798

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

800

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

802

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

804

805 @see: Function L{hypot_}.

806 '''

807 h, x2 = _h_x2(xs)

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

809

810

811 def _map_mul(a, b, where):

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

813 '''

814 n = len(b)

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

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

817 return map(_operator.mul, a, b) if n > 3 else _map_mul1(a, b)

818

819

820 def _map_mul1(a, b):

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

822 '''

823 yield _1_0

824 for ab in map(_operator.mul, a, b):

825 if ab:

826 yield ab

827 yield _N_1_0

828

829

830 def norm2(x, y):

831 '''Normalize a 2-dimensional vector.

832

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

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

835

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

837

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

839 or zero norm.

840 '''

841 h = hypot(x, y)

842 if not h:

843 x = y = _0_0 # pass?

844 elif not isnear1(h):

845 try:

846 x, y = x / h, y / h

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 h = hypot_(*xs)

863 if h:

864 try:

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 else:

870 for _ in xs:

871 yield _0_0

872

873

874 def _root(x, p, where):

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

876 '''

877 if x < 0:

878 t = _SPACE_(_invokation_, where.__name__)

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

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

881

882

883 def sqrt0(x, Error=None):

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

885

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

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

888

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

890

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

892 returns C{0.0}.

893 '''

894 if Error and x < 0:

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

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

897

898

899 def sqrt3(x):

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

901

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

903

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

905

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

907

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

909 '''

910 return _root(x, _1_5, sqrt3)

911

912

913 def sqrt_a(h, b):

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

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

916

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

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

919

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

921

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

923

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

925

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

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

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

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

930 '''

931 try:

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

933 raise TypeError(_not_scalar_)

934 elif isnear0(h): # PYCHOK no cover

935 c, b = fabs(h), fabs(b)

936 d = c - b

937 if d < 0:

938 raise ValueError('abs(h) < abs(b)')

939 a = copysign0(sqrt((c + b) * d), h) if d > 0 else _0_0

940 else:

941 c = float(h)

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

943 if s < 0:

944 raise ValueError('abs(h) < abs(b)')

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

946 except Exception as x:

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

948 return a

949

950

951 def _x2_h2(s, xs, h, e):

952 '''(INTERNAL) Yield M{(x / h)**2 for x in xs}.

953 '''

954 yield s

955 if h in (_0_0, _1_0):

956 for x in xs:

957 if x:

958 yield x**2

959 else:

960 for x in xs:

961 if x:

962 yield (x / h)**2

963 yield e

964

965

966 def zcrt(x):

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

968

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

970

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

972

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

974

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

976 '''

977 return _root(x, _1_6th, zcrt)

978

979

980 def zqrt(x):

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

982

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

984

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

986

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

988

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

990 '''

991 return _root(x, _0_125, zqrt)

992

993# **) MIT License

994#

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

996#

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

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

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

1000# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1001# and/or sell copies of the Software, and to permit persons to whom the

1002# Software is furnished to do so, subject to the following conditions:

1003#

1004# The above copyright notice and this permission notice shall be included

1005# in all copies or substantial portions of the Software.

1006#

1007# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1008# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1009# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1010# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1011# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1012# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1013# OTHER DEALINGS IN THE SOFTWARE.