1

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

3

4 u'''Class L{Fsum} for precision floating point summation and I{running}

5 summation based on, respectively similar to Python's C{math.fsum}.

6

7 Generally, an L{Fsum} instance is considered a C{float} plus a small or zero

8 C{residual} value, see property L{Fsum.residual}. However, there are several

9 C{integer} L{Fsum} cases, for example the result of C{ceil}, C{floor},

10 C{Fsum.__floordiv__} and methods L{Fsum.fint} and L{Fsum.fint2}.

11

12 Also, L{Fsum} methods L{Fsum.pow}, L{Fsum.__ipow__}, L{Fsum.__pow__} and

13 L{Fsum.__rpow__} return a (very long) C{int} if invoked with optional argument

14 C{mod} set to C{None}. The C{residual} of an C{integer} L{Fsum} may be between

15 C{-1.0} and C{+1.0}, including C{INT0} if considered to be I{exact}.

16

17 Set env variable C{PYGEODESY_FSUM_PARTIALS} to an empty string (or anything

18 other than C{"fsum"}) for backward compatible summation of L{Fsum} partials.

19

20 Set env variable C{PYGEODESY_FSUM_RESIDUAL} to a C{float} string greater

21 than C{"0.0"} as the threshold to throw a L{ResidualError} in division or

22 exponention of an L{Fsum} instance with a I{relative} C{residual} exceeding

23 the threshold, see methods L{Fsum.RESIDUAL}, L{Fsum.pow}, L{Fsum.__ipow__}

24 and L{Fsum.__itruediv__}.

25'''

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

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

28

29 from pygeodesy.basics import iscomplex, isint, isscalar, itemsorted, \

30 signOf, _signOf, _xisscalar

31 from pygeodesy.constants import INT0, _isfinite, isinf, isnan, _pos_self, \

32 _0_0, _1_0, _N_1_0, Float, Int

33 from pygeodesy.errors import _OverflowError, _TypeError, _ValueError, _xError2, \

34 _xkwds_get, _ZeroDivisionError

35 from pygeodesy.interns import NN, _arg_, _COMMASPACE_, _DASH_, _DOT_, _EQUAL_, \

36 _exceeds_, _from_, _iadd_op_, _LANGLE_, _negative_, \

37 _NOTEQUAL_, _not_finite_, _not_scalar_, _PERCENT_, \

38 _PLUS_, _R_, _RANGLE_, _SLASH_, _SPACE_, _STAR_, _UNDER_

39 from pygeodesy.lazily import _ALL_LAZY, _getenv, _sys_version_info2

40 from pygeodesy.named import _Named, _NamedTuple, _NotImplemented, Fmt, unstr

41 from pygeodesy.props import _allPropertiesOf_n, deprecated_property_RO, \

42 Property_RO, property_RO

43# from pygeodesy.streprs import Fmt, unstr # from .named

44# from pygeodesy.units import Float, Int # from .constants

45

46 from math import ceil as _ceil, fabs, floor as _floor # PYCHOK used! .ltp

47

48__all__ = _ALL_LAZY.fsums

49__version__ = '24.03.18'

50

51_add_op_ = _PLUS_ # in .auxilats.auxAngle

52_eq_op_ = _EQUAL_ * 2 # _DEQUAL_

53_COMMASPACE_R_ = _COMMASPACE_ + _R_

54_exceeds_R_ = _SPACE_ + _exceeds_(_R_)

55_floordiv_op_ = _SLASH_ * 2 # _DSLASH_

56_fset_op_ = _EQUAL_

57_ge_op_ = _RANGLE_ + _EQUAL_

58_gt_op_ = _RANGLE_

59_integer_ = 'integer'

60_le_op_ = _LANGLE_ + _EQUAL_

61_lt_op_ = _LANGLE_

62_mod_op_ = _PERCENT_

63_mul_op_ = _STAR_

64_ne_op_ = _NOTEQUAL_

65_non_zero_ = 'non-zero'

66_pow_op_ = _STAR_ * 2 # _DSTAR_, in .fmath

67_sub_op_ = _DASH_ # in .auxilats.auxAngle, .fsums

68_truediv_op_ = _SLASH_

69_divmod_op_ = _floordiv_op_ + _mod_op_

70_isub_op_ = _sub_op_ + _fset_op_ # in .auxilats.auxAngle, .fsums

71

72

73 def _2float(index=None, **name_value): # in .fmath, .fstats

74 '''(INTERNAL) Raise C{TypeError} or C{ValueError} if not scalar or infinite.

75 '''

76 n, v = name_value.popitem() # _xkwds_item2(name_value)

77 try:

78 v = float(v)

79 if _isfinite(v):

80 return v

81 E, t = _ValueError, _not_finite_

82 except Exception as e:

83 E, t = _xError2(e)

84 raise E(Fmt.INDEX(n, index), v, txt=t)

85

86

87 def _2floats(xs, origin=0, sub=False):

88 '''(INTERNAL) Yield each B{C{xs}} as a C{float}.

89 '''

90 try:

91 i, x = origin, None

92 _fin = _isfinite

93 _Fsum = Fsum

94 for x in xs:

95 if isinstance(x, _Fsum):

96 for p in x._ps:

97 yield (-p) if sub else p

98 else:

99 f = float(x)

100 if not _fin(f):

101 raise ValueError(_not_finite_)

102 if f:

103 yield (-f) if sub else f

104 i += 1

105 except Exception as e:

106 E, t = _xError2(e)

107 n = Fmt.SQUARE(xs=i)

108 raise E(n, x, txt=t)

109

110

111 def _Powers(power, xs, origin=1): # in .fmath

112 '''(INTERNAL) Yield each C{xs} as C{float(x**power)}.

113 '''

114 _xisscalar(power=power)

115 try:

116 i, x = origin, None

117 _fin = _isfinite

118 _Fsum = Fsum

119 _pow = pow # XXX math.pow

120 for x in xs:

121 if isinstance(x, _Fsum):

122 P = x.pow(power)

123 for p in P._ps:

124 yield p

125 else:

126 p = _pow(float(x), power)

127 if not _fin(p):

128 raise ValueError(_not_finite_)

129 yield p

130 i += 1

131 except Exception as e:

132 E, t = _xError2(e)

133 n = Fmt.SQUARE(xs=i)

134 raise E(n, x, txt=t)

135

136

137 def _1primed(xs):

138 '''(INTERNAL) 1-Prime the summation of C{xs}

139 arguments I{known} to be C{finite float}.

140 '''

141 yield _1_0

142 for x in xs:

143 if x:

144 yield x

145 yield _N_1_0

146

147

148 def _psum(ps): # PYCHOK used!

149 '''(INTERNAL) Partials summation updating C{ps}, I{overridden below}.

150 '''

151 i = len(ps) - 1 # len(ps) > 2

152 s = ps[i]

153 _2s = _2sum

154 while i > 0:

155 i -= 1

156 s, r = _2s(s, ps[i])

157 if r: # sum(ps) became inexact

158 ps[i:] = [s, r] if s else [r]

159 if i > 0:

160 p = ps[i-1] # round half-even

161 if (p > 0 and r > 0) or \

162 (p < 0 and r < 0): # signs match

163 r *= 2

164 t = s + r

165 if r == (t - s):

166 s = t

167 break

168 ps[i:] = [s]

169 return s

170

171

172 def _2scalar(other, _raiser=None):

173 '''(INTERNAL) Return B{C{other}} as C{int}, C{float} or C{as-is}.

174 '''

175 if isinstance(other, Fsum):

176 s, r = other._fint2

177 if r:

178 s, r = other._fprs2

179 if r: # PYCHOK no cover

180 if _raiser and _raiser(r, s):

181 raise ValueError(_stresidual(_non_zero_, r))

182 s = other # L{Fsum} as-is

183 else:

184 s = other # C{type} as-is

185 if isint(s, both=True):

186 s = int(s)

187 return s

188

189

190 def _strcomplex(s, *args):

191 '''(INTERNAL) C{Complex} 2- or 3-arg C{pow} error C{str}.

192 '''

193 c = iscomplex.__name__[2:]

194 n = _DASH_(len(args), _arg_)

195 t = _SPACE_(c, s, _from_, n, pow.__name__)

196 return unstr(t, *args)

197

198

199 def _stresidual(prefix, residual, **name_values):

200 '''(INTERNAL) Residual error C{str}.

201 '''

202 p = _SPACE_(prefix, Fsum.residual.name)

203 t = Fmt.PARENSPACED(p, Fmt(residual))

204 for n, v in itemsorted(name_values):

205 n = n.replace(_UNDER_, _SPACE_)

206 p = Fmt.PARENSPACED(n, Fmt(v))

207 t = _COMMASPACE_(t, p)

208 return t

209

210

211 def _2sum(a, b): # by .testFmath

212 '''(INTERNAL) Return C{a + b} as 2-tuple (sum, residual).

213 '''

214 s = a + b

215 if not _isfinite(s):

216 u = unstr(_2sum.__name__, a, b)

217 t = Fmt.PARENSPACED(_not_finite_, s)

218 raise _OverflowError(u, txt=t)

219 if fabs(a) < fabs(b):

220 a, b = b, a

221 return s, (b - (s - a))

222

223

224 class Fsum(_Named): # sync __methods__ with .vector3dBase.Vector3dBase

225 '''Precision floating point I{running} summation.

226

227 Unlike Python's C{math.fsum}, this class accumulates values and provides intermediate,

228 I{running} precision floating point summation. Accumulation may continue after

229 intermediate, I{running} summuation.

230

231 @note: Accumulated values may be L{Fsum} or C{scalar} instances with C{scalar} meaning

232 type C{float}, C{int} or any C{type} convertible to a single C{float}, having

233 method C{__float__}.

234

235 @note: Handling of exceptions and C{inf}, C{INF}, C{nan} and C{NAN} differs from

236 Python's C{math.fsum}.

237

238 @see: U{Hettinger<https://GitHub.com/ActiveState/code/blob/master/recipes/Python/

239 393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}, U{Kahan

240 <https://WikiPedia.org/wiki/Kahan_summation_algorithm>}, U{Klein

241 <https://Link.Springer.com/article/10.1007/s00607-005-0139-x>}, Python 2.6+

242 file I{Modules/mathmodule.c} and the issue log U{Full precision summation

243 <https://Bugs.Python.org/issue2819>}.

244 '''

245 _math_fsum = None

246 _n = 0

247# _ps = [] # partial sums

248# _px = 0

249 _ratio = None

250 _RESIDUAL = max(float(_getenv('PYGEODESY_FSUM_RESIDUAL', _0_0)), _0_0)

251

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

253 '''New L{Fsum} for precision floating point I{running} summation.

254

255 @arg xs: No, one or more initial values (each C{scalar} or an

256 L{Fsum} instance).

257 @kwarg name_RESIDUAL: Optional C{B{name}=NN} for this L{Fsum}

258 (C{str}) and C{B{RESIDUAL}=None} for the

259 L{ResidualError} threshold.

260

261 @see: Methods L{Fsum.fadd} and L{Fsum.RESIDUAL}.

262 '''

263 if name_RESIDUAL:

264 n = _xkwds_get(name_RESIDUAL, name=NN)

265 if n: # set name ...

266 self.name = n

267 r = _xkwds_get(name_RESIDUAL, RESIDUAL=None)

268 if r is not None:

269 self.RESIDUAL(r) # ... for ResidualError

270# self._n = 0

271 self._ps = [] # [_0_0], see L{Fsum._fprs}

272 if len(xs) > 1:

273 self._facc(_2floats(xs, origin=1), up=False) # PYCHOK yield

274 elif xs: # len(xs) == 1

275 self._ps = [_2float(x=xs[0])]

276 self._n = 1

277

278 def __abs__(self):

279 '''Return this instance' absolute value as an L{Fsum}.

280 '''

281 s = _fsum(self._ps_1()) # == self._cmp_0(0, ...)

282 return self._copy_n(self.__abs__) if s < 0 else \

283 self._copy_2(self.__abs__)

284

285 def __add__(self, other):

286 '''Return the C{Fsum(B{self}, B{other})}.

287

288 @arg other: An L{Fsum} or C{scalar}.

289

290 @return: The sum (L{Fsum}).

291

292 @see: Method L{Fsum.__iadd__}.

293 '''

294 f = self._copy_2(self.__add__)

295 return f._fadd(other, _add_op_)

296

297 def __bool__(self): # PYCHOK not special in Python 2-

298 '''Return C{True} if this instance is I{exactly} non-zero.

299 '''

300 s, r = self._fprs2

301 return bool(s or r) and s != -r # == self != 0

302

303 def __ceil__(self): # PYCHOK not special in Python 2-

304 '''Return this instance' C{math.ceil} as C{int} or C{float}.

305

306 @return: An C{int} in Python 3+, but C{float} in Python 2-.

307

308 @see: Methods L{Fsum.__floor__} and property L{Fsum.ceil}.

309 '''

310 return self.ceil

311

312 def __cmp__(self, other): # Python 2-

313 '''Compare this with an other instance or C{scalar}.

314

315 @return: -1, 0 or +1 (C{int}).

316

317 @raise TypeError: Incompatible B{C{other}} C{type}.

318 '''

319 s = self._cmp_0(other, self.cmp.__name__)

320 return _signOf(s, 0)

321

322 cmp = __cmp__

323

324 def __divmod__(self, other):

325 '''Return C{divmod(B{self}, B{other})} as 2-tuple C{(quotient,

326 remainder)}, an C{int} in Python 3+ or C{float} in Python 2-

327 and an L{Fsum}.

328

329 @arg other: An L{Fsum} or C{scalar} modulus.

330

331 @see: Method L{Fsum.__itruediv__}.

332 '''

333 f = self._copy_2(self.__divmod__)

334 return f._fdivmod2(other, _divmod_op_)

335

336 def __eq__(self, other):

337 '''Compare this with an other instance or C{scalar}.

338 '''

339 return self._cmp_0(other, _eq_op_) == 0

340

341 def __float__(self):

342 '''Return this instance' current precision running sum as C{float}.

343

344 @see: Methods L{Fsum.fsum} and L{Fsum.int_float}.

345 '''

346 return float(self._fprs)

347

348 def __floor__(self): # PYCHOK not special in Python 2-

349 '''Return this instance' C{math.floor} as C{int} or C{float}.

350

351 @return: An C{int} in Python 3+, but C{float} in Python 2-.

352

353 @see: Methods L{Fsum.__ceil__} and property L{Fsum.floor}.

354 '''

355 return self.floor

356

357 def __floordiv__(self, other):

358 '''Return C{B{self} // B{other}} as an L{Fsum}.

359

360 @arg other: An L{Fsum} or C{scalar} divisor.

361

362 @return: The C{floor} quotient (L{Fsum}).

363

364 @see: Methods L{Fsum.__ifloordiv__}.

365 '''

366 f = self._copy_2(self.__floordiv__)

367 return f._floordiv(other, _floordiv_op_)

368

369 def __format__(self, *other): # PYCHOK no cover

370 '''Not implemented.'''

371 return _NotImplemented(self, *other)

372

373 def __ge__(self, other):

374 '''Compare this with an other instance or C{scalar}.

375 '''

376 return self._cmp_0(other, _ge_op_) >= 0

377

378 def __gt__(self, other):

379 '''Compare this with an other instance or C{scalar}.

380 '''

381 return self._cmp_0(other, _gt_op_) > 0

382

383 def __hash__(self): # PYCHOK no cover

384 '''Return this instance' C{hash}.

385 '''

386 return hash(self._ps) # XXX id(self)?

387

388 def __iadd__(self, other):

389 '''Apply C{B{self} += B{other}} to this instance.

390

391 @arg other: An L{Fsum} or C{scalar} instance.

392

393 @return: This instance, updated (L{Fsum}).

394

395 @raise TypeError: Invalid B{C{other}}, not

396 C{scalar} nor L{Fsum}.

397

398 @see: Methods L{Fsum.fadd} and L{Fsum.fadd_}.

399 '''

400 return self._fadd(other, _iadd_op_)

401

402 def __ifloordiv__(self, other):

403 '''Apply C{B{self} //= B{other}} to this instance.

404

405 @arg other: An L{Fsum} or C{scalar} divisor.

406

407 @return: This instance, updated (L{Fsum}).

408

409 @raise ResidualError: Non-zero residual in B{C{other}}.

410

411 @raise TypeError: Invalid B{C{other}} type.

412

413 @raise ValueError: Invalid or non-finite B{C{other}}.

414

415 @raise ZeroDivisionError: Zero B{C{other}}.

416

417 @see: Methods L{Fsum.__itruediv__}.

418 '''

419 return self._floordiv(other, _floordiv_op_ + _fset_op_)

420

421 def __imatmul__(self, other): # PYCHOK no cover

422 '''Not implemented.'''

423 return _NotImplemented(self, other)

424

425 def __imod__(self, other):

426 '''Apply C{B{self} %= B{other}} to this instance.

427

428 @arg other: An L{Fsum} or C{scalar} modulus.

429

430 @return: This instance, updated (L{Fsum}).

431

432 @see: Method L{Fsum.__divmod__}.

433 '''

434 self._fdivmod2(other, _mod_op_ + _fset_op_)

435 return self

436

437 def __imul__(self, other):

438 '''Apply C{B{self} *= B{other}} to this instance.

439

440 @arg other: An L{Fsum} or C{scalar} factor.

441

442 @return: This instance, updated (L{Fsum}).

443

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

445

446 @raise TypeError: Invalid B{C{other}} type.

447

448 @raise ValueError: Invalid or non-finite B{C{other}}.

449 '''

450 return self._fmul(other, _mul_op_ + _fset_op_)

451

452 def __int__(self):

453 '''Return this instance as an C{int}.

454

455 @see: Methods L{Fsum.int_float}, L{Fsum.__ceil__}

456 and L{Fsum.__floor__} and properties

457 L{Fsum.ceil} and L{Fsum.floor}.

458 '''

459 i, _ = self._fint2

460 return i

461

462 def __invert__(self): # PYCHOK no cover

463 '''Not implemented.'''

464 # Luciano Ramalho, "Fluent Python", O'Reilly, 2nd Ed, 2022 p. 567

465 return _NotImplemented(self)

466

467 def __ipow__(self, other, *mod): # PYCHOK 2 vs 3 args

468 '''Apply C{B{self} **= B{other}} to this instance.

469

470 @arg other: The exponent (L{Fsum} or C{scalar}).

471 @arg mod: Optional modulus (C{int} or C{None}) for the

472 3-argument C{pow(B{self}, B{other}, B{mod})}

473 version.

474

475 @return: This instance, updated (L{Fsum}).

476

477 @note: If B{C{mod}} is given, the result will be an C{integer}

478 L{Fsum} in Python 3+ if this instance C{is_integer} or

479 set to C{as_integer} if B{C{mod}} given as C{None}.

480

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

482

483 @raise ResidualError: Non-zero residual in B{C{other}} and

484 env var C{PYGEODESY_FSUM_RESIDUAL}

485 set or this instance has a non-zero

486 residual and either B{C{mod}} is

487 given and non-C{None} or B{C{other}}

488 is a negative or fractional C{scalar}.

489

490 @raise TypeError: Invalid B{C{other}} type or 3-argument

491 C{pow} invocation failed.

492

493 @raise ValueError: If B{C{other}} is a negative C{scalar}

494 and this instance is C{0} or B{C{other}}

495 is a fractional C{scalar} and this

496 instance is negative or has a non-zero

497 residual or B{C{mod}} is given and C{0}.

498

499 @see: CPython function U{float_pow<https://GitHub.com/

500 python/cpython/blob/main/Objects/floatobject.c>}.

501 '''

502 return self._fpow(other, _pow_op_ + _fset_op_, *mod)

503

504 def __isub__(self, other):

505 '''Apply C{B{self} -= B{other}} to this instance.

506

507 @arg other: An L{Fsum} or C{scalar}.

508

509 @return: This instance, updated (L{Fsum}).

510

511 @raise TypeError: Invalid B{C{other}} type.

512

513 @see: Method L{Fsum.fadd}.

514 '''

515 return self._fsub(other, _isub_op_)

516

517 def __iter__(self):

518 '''Return an C{iter}ator over a C{partials} duplicate.

519 '''

520 return iter(self.partials)

521

522 def __itruediv__(self, other):

523 '''Apply C{B{self} /= B{other}} to this instance.

524

525 @arg other: An L{Fsum} or C{scalar} divisor.

526

527 @return: This instance, updated (L{Fsum}).

528

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

530

531 @raise ResidualError: Non-zero residual in B{C{other}} and

532 env var C{PYGEODESY_FSUM_RESIDUAL} set.

533

534 @raise TypeError: Invalid B{C{other}} type.

535

536 @raise ValueError: Invalid or non-finite B{C{other}}.

537

538 @raise ZeroDivisionError: Zero B{C{other}}.

539

540 @see: Method L{Fsum.__ifloordiv__}.

541 '''

542 return self._ftruediv(other, _truediv_op_ + _fset_op_)

543

544 def __le__(self, other):

545 '''Compare this with an other instance or C{scalar}.

546 '''

547 return self._cmp_0(other, _le_op_) <= 0

548

549 def __len__(self):

550 '''Return the number of values accumulated (C{int}).

551 '''

552 return self._n

553

554 def __lt__(self, other):

555 '''Compare this with an other instance or C{scalar}.

556 '''

557 return self._cmp_0(other, _lt_op_) < 0

558

559 def __matmul__(self, other): # PYCHOK no cover

560 '''Not implemented.'''

561 return _NotImplemented(self, other)

562

563 def __mod__(self, other):

564 '''Return C{B{self} % B{other}} as an L{Fsum}.

565

566 @see: Method L{Fsum.__imod__}.

567 '''

568 f = self._copy_2(self.__mod__)

569 return f._fdivmod2(other, _mod_op_)[1]

570

571 def __mul__(self, other):

572 '''Return C{B{self} * B{other}} as an L{Fsum}.

573

574 @see: Method L{Fsum.__imul__}.

575 '''

576 f = self._copy_2(self.__mul__)

577 return f._fmul(other, _mul_op_)

578

579 def __ne__(self, other):

580 '''Compare this with an other instance or C{scalar}.

581 '''

582 return self._cmp_0(other, _ne_op_) != 0

583

584 def __neg__(self):

585 '''Return I{a copy of} this instance, negated.

586 '''

587 return self._copy_n(self.__neg__)

588

589 def __pos__(self):

590 '''Return this instance I{as-is}, like C{float.__pos__()}.

591 '''

592 return self if _pos_self else self._copy_2(self.__pos__)

593

594 def __pow__(self, other, *mod): # PYCHOK 2 vs 3 args

595 '''Return C{B{self}**B{other}} as an L{Fsum}.

596

597 @see: Method L{Fsum.__ipow__}.

598 '''

599 f = self._copy_2(self.__pow__)

600 return f._fpow(other, _pow_op_, *mod)

601

602 def __radd__(self, other):

603 '''Return C{B{other} + B{self}} as an L{Fsum}.

604

605 @see: Method L{Fsum.__iadd__}.

606 '''

607 f = self._copy_r2(other, self.__radd__)

608 return f._fadd(self, _add_op_)

609

610 def __rdivmod__(self, other):

611 '''Return C{divmod(B{other}, B{self})} as 2-tuple C{(quotient,

612 remainder)}.

613

614 @see: Method L{Fsum.__divmod__}.

615 '''

616 f = self._copy_r2(other, self.__rdivmod__)

617 return f._fdivmod2(self, _divmod_op_)

618

619# def __repr__(self):

620# '''Return the default C{repr(this)}.

621# '''

622# return self.toRepr(lenc=True)

623

624 def __rfloordiv__(self, other):

625 '''Return C{B{other} // B{self}} as an L{Fsum}.

626

627 @see: Method L{Fsum.__ifloordiv__}.

628 '''

629 f = self._copy_r2(other, self.__rfloordiv__)

630 return f._floordiv(self, _floordiv_op_)

631

632 def __rmatmul__(self, other): # PYCHOK no cover

633 '''Not implemented.'''

634 return _NotImplemented(self, other)

635

636 def __rmod__(self, other):

637 '''Return C{B{other} % B{self}} as an L{Fsum}.

638

639 @see: Method L{Fsum.__imod__}.

640 '''

641 f = self._copy_r2(other, self.__rmod__)

642 return f._fdivmod2(self, _mod_op_)[1]

643

644 def __rmul__(self, other):

645 '''Return C{B{other} * B{self}} as an L{Fsum}.

646

647 @see: Method L{Fsum.__imul__}.

648 '''

649 f = self._copy_r2(other, self.__rmul__)

650 return f._fmul(self, _mul_op_)

651

652 def __round__(self, *ndigits): # PYCHOK no cover

653 '''Return C{round(B{self}, *B{ndigits}} as an L{Fsum}.

654

655 @arg ndigits: Optional number of digits (C{int}).

656 '''

657 # <https://docs.Python.org/3.12/reference/datamodel.html?#object.__round__>

658 f = Fsum(name=self.__round__.__name__)

659 f._n = 1

660 f._ps = [round(float(self), *ndigits)] # can be C{int}

661 return f

662

663 def __rpow__(self, other, *mod):

664 '''Return C{B{other}**B{self}} as an L{Fsum}.

665

666 @see: Method L{Fsum.__ipow__}.

667 '''

668 f = self._copy_r2(other, self.__rpow__)

669 return f._fpow(self, _pow_op_, *mod)

670

671 def __rsub__(self, other):

672 '''Return C{B{other} - B{self}} as L{Fsum}.

673

674 @see: Method L{Fsum.__isub__}.

675 '''

676 f = self._copy_r2(other, self.__rsub__)

677 return f._fsub(self, _sub_op_)

678

679 def __rtruediv__(self, other):

680 '''Return C{B{other} / B{self}} as an L{Fsum}.

681

682 @see: Method L{Fsum.__itruediv__}.

683 '''

684 f = self._copy_r2(other, self.__rtruediv__)

685 return f._ftruediv(self, _truediv_op_)

686

687 def __str__(self):

688 '''Return the default C{str(self)}.

689 '''

690 return self.toStr(lenc=True)

691

692 def __sub__(self, other):

693 '''Return C{B{self} - B{other}} as an L{Fsum}.

694

695 @arg other: An L{Fsum} or C{scalar}.

696

697 @return: The difference (L{Fsum}).

698

699 @see: Method L{Fsum.__isub__}.

700 '''

701 f = self._copy_2(self.__sub__)

702 return f._fsub(other, _sub_op_)

703

704 def __truediv__(self, other):

705 '''Return C{B{self} / B{other}} as an L{Fsum}.

706

707 @arg other: An L{Fsum} or C{scalar} divisor.

708

709 @return: The quotient (L{Fsum}).

710

711 @see: Method L{Fsum.__itruediv__}.

712 '''

713 f = self._copy_2(self.__truediv__)

714 return f._ftruediv(other, _truediv_op_)

715

716 __trunc__ = __int__

717

718 if _sys_version_info2 < (3, 0): # PYCHOK no cover

719 # <https://docs.Python.org/2/library/operator.html#mapping-operators-to-functions>

720 __div__ = __truediv__

721 __idiv__ = __itruediv__

722 __long__ = __int__

723 __nonzero__ = __bool__

724 __rdiv__ = __rtruediv__

725

726 def as_integer_ratio(self):

727 '''Return this instance as the ratio of 2 integers.

728

729 @return: 2-Tuple C{(numerator, denominator)} both

730 C{int} and with positive C{denominator}.

731

732 @see: Standard C{float.as_integer_ratio} in Python 3+.

733 '''

734 n, r = self._fint2

735 if r:

736 i, d = r.as_integer_ratio()

737 n *= d

738 n += i

739 else: # PYCHOK no cover

740 d = 1

741 return n, d

742

743 @property_RO

744 def ceil(self):

745 '''Get this instance' C{ceil} value (C{int} in Python 3+,

746 but C{float} in Python 2-).

747

748 @note: The C{ceil} takes the C{residual} into account.

749

750 @see: Method L{Fsum.int_float} and properties L{Fsum.floor},

751 L{Fsum.imag} and L{Fsum.real}.

752 '''

753 s, r = self._fprs2

754 c = _ceil(s) + int(r) - 1

755 while r > (c - s): # (s + r) > c

756 c += 1

757 return c

758

759 def _cmp_0(self, other, op):

760 '''(INTERNAL) Return C{scalar(self - B{other})} for 0-comparison.

761 '''

762 if isscalar(other):

763 if other:

764 s = _fsum(self._ps_1(other))

765 else:

766 s, r = self._fprs2

767 s = _signOf(s, -r)

768 elif isinstance(other, Fsum):

769 s = _fsum(self._ps_1(*other._ps))

770 else:

771 raise self._TypeError(op, other) # txt=_invalid_

772 return s

773

774 def copy(self, deep=False, name=NN):

775 '''Copy this instance, C{shallow} or B{C{deep}}.

776

777 @return: The copy (L{Fsum}).

778 '''

779 f = _Named.copy(self, deep=deep, name=name)

780 f._n = self._n if deep else 1

781 f._ps = list(self._ps) # separate list

782 return f

783

784 def _copy_0(self, *xs):

785 '''(INTERNAL) Copy with/-out overriding C{partials}.

786 '''

787 # for x in xs:

788 # assert isscalar(x)

789 f = self._Fsum(self._n + len(xs), *xs)

790 if self.name:

791 f._name = self.name # .rename calls _update_attrs

792 return f

793

794 def _copy_2(self, which):

795 '''(INTERNAL) Copy for I{dyadic} operators.

796 '''

797 # NOT .classof due to .Fdot(a, *b) args, etc.

798 f = _Named.copy(self, deep=False, name=which.__name__)

799 # assert f._n == self._n

800 f._ps = list(self._ps) # separate list

801 return f

802

803 def _copy_n(self, which):

804 '''(INTERNAL) Negated copy for I{monadic} C{__abs__} and C{__neg__}.

805 '''

806 if self._ps:

807 f = self._Fsum(self._n)

808 f._ps[:] = self._ps_n()

809# f._facc_up(up=False)

810 else:

811 f = self._Fsum(self._n, _0_0)

812 f._name = which.__name__ # .rename calls _update_attrs

813 return f

814

815 def _copy_r2(self, other, which):

816 '''(INTERNAL) Copy for I{reverse-dyadic} operators.

817 '''

818 return other._copy_2(which) if isinstance(other, Fsum) else \

819 Fsum(other, name=which.__name__) # see ._copy_2

820

821 def _copy_RESIDUAL(self, other):

822 '''(INTERNAL) Copy C{other._RESIDUAL}.

823 '''

824 R = other._RESIDUAL

825 if R is not Fsum._RESIDUAL:

826 self._RESIDUAL = R

827

828 def _copy_up(self, _fprs2=False):

829 '''(INTERNAL) Minimal, anonymous copy.

830 '''

831 f = self._Fsum(self._n, *self._ps)

832 if _fprs2: # only the ._fprs2 2-tuple

833 Fsum._fprs2._update_from(f, self)

834 return f

835

836 def divmod(self, other):

837 '''Return C{divmod(B{self}, B{other})} as 2-tuple C{(quotient,

838 remainder)}.

839

840 @arg other: An L{Fsum} or C{scalar} divisor.

841

842 @return: 2-Tuple C{(quotient, remainder)}, with the C{quotient}

843 an C{int} in Python 3+ or a C{float} in Python 2- and

844 the C{remainder} an L{Fsum} instance.

845

846 @see: Method L{Fsum.__itruediv__}.

847 '''

848 f = self._copy_2(self.divmod)

849 return f._fdivmod2(other, _divmod_op_)

850

851 def _Error(self, op, other, Error, **txt):

852 '''(INTERNAL) Format an B{C{Error}} for C{{self} B{op} B{other}}.

853 '''

854 return Error(_SPACE_(self.toRepr(), op, repr(other)), **txt)

855

856 def _ErrorX(self, X, xs, **kwds): # in .fmath

857 '''(INTERNAL) Format a caught exception.

858 '''

859 E, t = _xError2(X)

860 n = unstr(self.named3, *xs[:3], _ELLIPSIS=len(xs) > 3, **kwds)

861 return E(n, txt=t, cause=X)

862

863 def _facc(self, xs, up=True): # from .elliptic._Defer.Fsum

864 '''(INTERNAL) Accumulate more known C{scalar}s.

865 '''

866 n, ps, _2s = 0, self._ps, _2sum

867 for x in xs: # _iter()

868 # assert isscalar(x) and isfinite(x)

869 i = 0

870 for p in ps:

871 x, p = _2s(x, p)

872 if p:

873 ps[i] = p

874 i += 1

875 ps[i:] = [x]

876 n += 1

877 # assert self._ps is ps

878 if n:

879 self._n += n

880 # Fsum._px = max(Fsum._px, len(ps))

881 if up:

882 self._update()

883 return self

884

885 def _facc_(self, *xs, **up):

886 '''(INTERNAL) Accumulate all positional C{scalar}s.

887 '''

888 return self._facc(xs, **up) if xs else self

889

890# def _facc_up(self, up=True):

891# '''(INTERNAL) Update the C{partials}, by removing

892# and re-accumulating the final C{partial}.

893# '''

894# while len(self._ps) > 1:

895# p = self._ps.pop()

896# if p:

897# n = self._n

898# self._facc_(p, up=False)

899# self._n = n

900# break

901# return self._update() if up else self # ._fpsqz()

902

903 def fadd(self, xs=()):

904 '''Add an iterable of C{scalar} or L{Fsum} instances

905 to this instance.

906

907 @arg xs: Iterable, list, tuple, etc. (C{scalar} or

908 L{Fsum} instances).

909

910 @return: This instance (L{Fsum}).

911

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

913

914 @raise TypeError: An invalid B{C{xs}} type, not C{scalar}

915 nor L{Fsum}.

916

917 @raise ValueError: Invalid or non-finite B{C{xs}} value.

918 '''

919 if isinstance(xs, Fsum):

920 self._facc(xs._ps)

921 elif isscalar(xs): # for backward compatibility

922 self._facc_(_2float(x=xs)) # PYCHOK no cover

923 elif xs:

924 self._facc(_2floats(xs)) # PYCHOK yield

925 return self

926

927 def fadd_(self, *xs):

928 '''Add all positional C{scalar} or L{Fsum} instances

929 to this instance.

930

931 @arg xs: Values to add (C{scalar} or L{Fsum} instances),

932 all positional.

933

934 @return: This instance (L{Fsum}).

935

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

937

938 @raise TypeError: An invalid B{C{xs}} type, not C{scalar}

939 nor L{Fsum}.

940

941 @raise ValueError: Invalid or non-finite B{C{xs}} value.

942 '''

943 return self._facc(_2floats(xs, origin=1)) # PYCHOK yield

944

945 def _fadd(self, other, op): # in .fmath.Fhorner

946 '''(INTERNAL) Apply C{B{self} += B{other}}.

947 '''

948 if isinstance(other, Fsum):

949 if other is self:

950 self._facc_(*other._ps) # == ._facc(tuple(other._ps))

951 elif other._ps:

952 self._facc(other._ps)

953 elif not isscalar(other):

954 raise self._TypeError(op, other) # txt=_invalid_

955 elif other:

956 self._facc_(other)

957 return self

958

959 fcopy = copy # for backward compatibility

960 fdiv = __itruediv__ # for backward compatibility

961 fdivmod = __divmod__ # for backward compatibility

962

963 def _fdivmod2(self, other, op):

964 '''(INTERNAL) C{divmod(B{self}, B{other})} as 2-tuple

965 (C{int} or C{float}, remainder C{self}).

966 '''

967 # result mostly follows CPython function U{float_divmod

968 # <https://GitHub.com/python/cpython/blob/main/Objects/floatobject.c>},

969 # but at least divmod(-3, 2) equals Cpython's result (-2, 1).

970 q = self._copy_up(_fprs2=True)._ftruediv(other, op).floor

971 if q: # == float // other == floor(float / other)

972 self -= other * q

973

974 s = signOf(other) # make signOf(self) == signOf(other)

975 if s and self.signOf() == -s: # PYCHOK no cover

976 self += other

977 q -= 1

978

979# t = self.signOf()

980# if t and t != s:

981# from pygeodesy.errors import _AssertionError

982# raise self._Error(op, other, _AssertionError, txt=signOf.__name__)

983 return q, self # q is C{int} in Python 3+, but C{float} in Python 2-

984

985 def _finite(self, other, op=None):

986 '''(INTERNAL) Return B{C{other}} if C{finite}.

987 '''

988 if _isfinite(other):

989 return other

990 raise ValueError(_not_finite_) if not op else \

991 self._ValueError(op, other, txt=_not_finite_)

992

993 def fint(self, raiser=True, name=NN):

994 '''Return this instance' current running sum as C{integer}.

995

996 @kwarg raiser: If C{True} throw a L{ResidualError} if the

997 I{integer} residual is non-zero.

998 @kwarg name: Optional name (C{str}), overriding C{"fint"}.

999

1000 @return: The C{integer} (L{Fsum}).

1001

1002 @raise ResidualError: Non-zero I{integer} residual.

1003

1004 @see: Methods L{Fsum.int_float} and L{Fsum.is_integer}.

1005 '''

1006 i, r = self._fint2

1007 if r and raiser:

1008 t = _stresidual(_integer_, r)

1009 raise ResidualError(_integer_, i, txt=t)

1010 n = name or self.fint.__name__

1011 return Fsum(name=n)._fset(i, asis=True)

1012

1013 def fint2(self, **name):

1014 '''Return this instance' current running sum as C{int} and

1015 the I{integer} residual.

1016

1017 @kwarg name: Optional name (C{str}).

1018

1019 @return: An L{Fsum2Tuple}C{(fsum, residual)} with C{fsum}

1020 an C{int} and I{integer} C{residual} a C{float} or

1021 C{INT0} if the C{fsum} is considered to be I{exact}.

1022 '''

1023 return Fsum2Tuple(*self._fint2, **name)

1024

1025 @Property_RO

1026 def _fint2(self): # see ._fset

1027 '''(INTERNAL) Get 2-tuple (C{int}, I{integer} residual).

1028 '''

1029 i = int(self._fprs) # int(self)

1030 r = _fsum(self._ps_1(i)) if len(self._ps) > 1 else (

1031 (self._ps[0] - i) if self._ps else -i)

1032 return i, (r or INT0)

1033

1034 @deprecated_property_RO

1035 def float_int(self): # PYCHOK no cover

1036 '''DEPRECATED, use method C{Fsum.int_float}.'''

1037 return self.int_float() # raiser=False

1038

1039 @property_RO

1040 def floor(self):

1041 '''Get this instance' C{floor} (C{int} in Python 3+, but

1042 C{float} in Python 2-).

1043

1044 @note: The C{floor} takes the C{residual} into account.

1045

1046 @see: Method L{Fsum.int_float} and properties L{Fsum.ceil},

1047 L{Fsum.imag} and L{Fsum.real}.

1048 '''

1049 s, r = self._fprs2

1050 f = _floor(s) + _floor(r) + 1

1051 while r < (f - s): # (s + r) < f

1052 f -= 1

1053 return f

1054

1055# floordiv = __floordiv__ # for naming consistency

1056

1057 def _floordiv(self, other, op): # rather _ffloordiv?

1058 '''Apply C{B{self} //= B{other}}.

1059 '''

1060 q = self._ftruediv(other, op) # == self

1061 return self._fset(q.floor, asis=True) # floor(q)

1062

1063 fmul = __imul__ # for backward compatibility

1064

1065 def _fmul(self, other, op):

1066 '''(INTERNAL) Apply C{B{self} *= B{other}}.

1067 '''

1068 if isscalar(other):

1069 f = self._mul_scalar(other, op)

1070 elif not isinstance(other, Fsum):

1071 raise self._TypeError(op, other) # txt=_invalid_

1072 elif len(self._ps) != 1:

1073 f = self._mul_Fsum(other, op)

1074 elif len(other._ps) != 1: # len(self._ps) == 1

1075 f = other._copy_up()._mul_scalar(self._ps[0], op)

1076 else: # len(other._ps) == len(self._ps) == 1

1077 s = self._finite(self._ps[0] * other._ps[0])

1078 return self._fset(s, asis=True, n=len(self) + 1)

1079 return self._fset(f)

1080

1081 def fover(self, over):

1082 '''Apply C{B{self} /= B{over}} and summate.

1083

1084 @arg over: An L{Fsum} or C{scalar} denominator.

1085

1086 @return: Precision running sum (C{float}).

1087

1088 @see: Methods L{Fsum.fsum} and L{Fsum.__itruediv__}.

1089 '''

1090 return float(self.fdiv(over)._fprs)

1091

1092 fpow = __ipow__ # for backward compatibility

1093

1094 def _fpow(self, other, op, *mod):

1095 '''Apply C{B{self} **= B{other}}, optional B{C{mod}} or C{None}.

1096 '''

1097 if mod and mod[0] is not None: # == 3-arg C{pow}

1098 s = self._pow_3(other, mod[0], op)

1099 elif mod and mod[0] is None and self.is_integer():

1100 # return an exact C{int} for C{int}**C{int}

1101 i = self._copy_0(self._fint2[0]) # assert _fint2[1] == 0

1102 x = _2scalar(other) # C{int}, C{float} or other

1103 s = i._pow_2(x, other, op) if isscalar(x) else i._fpow(x, op)

1104 else: # pow(self, other) == pow(self, other, None)

1105 p = None

1106 if isinstance(other, Fsum):

1107 x, r = other._fprs2

1108 if r:

1109 if self._raiser(r, x):

1110 raise self._ResidualError(op, other, r)

1111 p = self._pow_scalar(r, other, op)

1112# p = _2scalar(p) # _raiser = None

1113 elif not isscalar(other):

1114 raise self._TypeError(op, other) # txt=_invalid_

1115 else:

1116 x = self._finite(other, op)

1117 s = self._pow_scalar(x, other, op)

1118 if p is not None:

1119 s *= p

1120 return self._fset(s, asis=isint(s), n=max(len(self), 1))

1121

1122 @Property_RO

1123 def _fprs(self):

1124 '''(INTERNAL) Get and cache this instance' precision

1125 running sum (C{float} or C{int}), ignoring C{residual}.

1126

1127 @note: The precision running C{fsum} after a C{//=} or

1128 C{//} C{floor} division is C{int} in Python 3+.

1129 '''

1130 ps = self._ps

1131 n = len(ps) - 1

1132 if n > 1:

1133 s = _psum(ps)

1134 elif n > 0: # len(ps) == 2

1135 s, p = _2sum(*ps) if ps[1] else ps

1136 ps[:] = ([p, s] if s else [p]) if p else [s]

1137 elif n < 0: # see L{Fsum.__init__}

1138 s = _0_0

1139 ps[:] = [s]

1140 else: # len(ps) == 1

1141 s = ps[0]

1142 # assert self._ps is ps

1143 # assert Fsum._fprs2.name not in self.__dict__

1144 return s

1145

1146 @Property_RO

1147 def _fprs2(self):

1148 '''(INTERNAL) Get and cache this instance' precision

1149 running sum and residual (L{Fsum2Tuple}).

1150 '''

1151 s = self._fprs

1152 r = _fsum(self._ps_1(s)) if len(self._ps) > 1 else INT0

1153 return Fsum2Tuple(s, r) # name=Fsum.fsum2.__name__

1154

1155# def _fpsqz(self):

1156# '''(INTERNAL) Compress, squeeze the C{partials}.

1157# '''

1158# if len(self._ps) > 2:

1159# _ = self._fprs

1160# return self

1161

1162 def _fset(self, other, asis=False, n=1):

1163 '''(INTERNAL) Overwrite this instance with an other or a C{scalar}.

1164 '''

1165 if other is self:

1166 pass # from ._fmul, ._ftruediv and ._pow_scalar

1167 elif isinstance(other, Fsum):

1168 self._n = other._n

1169 self._ps[:] = other._ps

1170 self._copy_RESIDUAL(other)

1171 # use or zap the C{Property_RO} values

1172 Fsum._fint2._update_from(self, other)

1173 Fsum._fprs ._update_from(self, other)

1174 Fsum._fprs2._update_from(self, other)

1175 elif isscalar(other):

1176 s = other if asis else float(other)

1177 i = int(s) # see ._fint2

1178 t = i, ((s - i) or INT0)

1179 self._n = n

1180 self._ps[:] = [s]

1181 # Property_RO _fint2, _fprs and _fprs2 can't be a Property:

1182 # Property's _fset zaps the value just set by the @setter

1183 self.__dict__.update(_fint2=t, _fprs=s, _fprs2=Fsum2Tuple(s, INT0))

1184 else: # PYCHOK no cover

1185 raise self._TypeError(_fset_op_, other) # txt=_invalid_

1186 return self

1187

1188 def fsub(self, xs=()):

1189 '''Subtract an iterable of C{scalar} or L{Fsum} instances

1190 from this instance.

1191

1192 @arg xs: Iterable, list, tuple. etc. (C{scalar}

1193 or L{Fsum} instances).

1194

1195 @return: This instance, updated (L{Fsum}).

1196

1197 @see: Method L{Fsum.fadd}.

1198 '''

1199 return self._facc(_2floats(xs, sub=True)) if xs else self # PYCHOK yield

1200

1201 def fsub_(self, *xs):

1202 '''Subtract all positional C{scalar} or L{Fsum} instances

1203 from this instance.

1204

1205 @arg xs: Values to subtract (C{scalar} or

1206 L{Fsum} instances), all positional.

1207

1208 @return: This instance, updated (L{Fsum}).

1209

1210 @see: Method L{Fsum.fadd}.

1211 '''

1212 return self._facc(_2floats(xs, origin=1, sub=True)) if xs else self # PYCHOK yield

1213

1214 def _fsub(self, other, op):

1215 '''(INTERNAL) Apply C{B{self} -= B{other}}.

1216 '''

1217 if isinstance(other, Fsum):

1218 if other is self: # or other._fprs2 == self._fprs2:

1219 self._fset(_0_0, asis=True, n=len(self) * 2) # self -= self

1220 elif other._ps:

1221 self._facc(other._ps_n())

1222 elif not isscalar(other):

1223 raise self._TypeError(op, other) # txt=_invalid_

1224 elif self._finite(other, op):

1225 self._facc_(-other)

1226 return self

1227

1228 def _Fsum(self, n, *ps):

1229 '''(INTERNAL) New L{Fsum} instance.

1230 '''

1231 f = Fsum()

1232 f._n = n

1233 if ps:

1234 f._ps[:] = ps

1235 f._copy_RESIDUAL(self)

1236 return f

1237

1238 def fsum(self, xs=()):

1239 '''Add more C{scalar} or L{Fsum} instances and summate.

1240

1241 @kwarg xs: Iterable, list, tuple, etc. (C{scalar} or

1242 L{Fsum} instances).

1243

1244 @return: Precision running sum (C{float} or C{int}).

1245

1246 @see: Method L{Fsum.fadd}.

1247

1248 @note: Accumulation can continue after summation.

1249 '''

1250 f = self._facc(_2floats(xs)) if xs else self # PYCHOK yield

1251 return f._fprs

1252

1253 def fsum_(self, *xs):

1254 '''Add all positional C{scalar} or L{Fsum} instances and summate.

1255

1256 @arg xs: Values to add (C{scalar} or L{Fsum} instances),

1257 all positional.

1258

1259 @return: Precision running sum (C{float} or C{int}).

1260

1261 @see: Methods L{Fsum.fsum} and L{Fsum.fsumf_}.

1262 '''

1263 f = self._facc(_2floats(xs, origin=1)) if xs else self # PYCHOK yield

1264 return f._fprs

1265

1266 def fsum2(self, xs=(), **name):

1267 '''Add more C{scalar} or L{Fsum} instances and return the

1268 current precision running sum and the C{residual}.

1269

1270 @kwarg xs: Iterable, list, tuple, etc. (C{scalar} or

1271 L{Fsum} instances).

1272 @kwarg name: Optional name (C{str}).

1273

1274 @return: L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} the

1275 current precision running sum and C{residual}, the

1276 (precision) sum of the remaining C{partials}. The

1277 C{residual is INT0} if the C{fsum} is considered

1278 to be I{exact}.

1279

1280 @see: Methods L{Fsum.fint2}, L{Fsum.fsum} and L{Fsum.fsum2_}

1281 '''

1282 f = self._facc(_2floats(xs)) if xs else self # PYCHOK yield

1283 t = f._fprs2

1284 if name:

1285 t = t.dup(name=_xkwds_get(name, name=NN))

1286 return t

1287

1288 def fsum2_(self, *xs):

1289 '''Add any positional C{scalar} or L{Fsum} instances and return

1290 the precision running sum and the C{differential}.

1291

1292 @arg xs: Values to add (C{scalar} or L{Fsum} instances),

1293 all positional.

1294

1295 @return: 2-Tuple C{(fsum, delta)} with the current precision

1296 running C{fsum} and C{delta}, the difference with

1297 the previous running C{fsum} (C{float}s).

1298

1299 @see: Methods L{Fsum.fsum_} and L{Fsum.fsum}.

1300 '''

1301 p, r = self._fprs2

1302 if xs:

1303 s, t = self._facc(_2floats(xs, origin=1))._fprs2 # PYCHOK yield

1304 return s, _fsum((s, -p, r, -t)) # ((s - p) + (r - t))

1305 else: # PYCHOK no cover

1306 return p, _0_0

1307

1308 def fsumf_(self, *xs):

1309 '''Like method L{Fsum.fsum_} but only for known C{float B{xs}}.

1310 '''

1311 f = self._facc(xs) if xs else self # PYCHOK yield

1312 return f._fprs

1313

1314# ftruediv = __itruediv__ # for naming consistency

1315

1316 def _ftruediv(self, other, op):

1317 '''(INTERNAL) Apply C{B{self} /= B{other}}.

1318 '''

1319 n = _1_0

1320 if isinstance(other, Fsum):

1321 if other is self or other._fprs2 == self._fprs2:

1322 return self._fset(_1_0, asis=True, n=len(self))

1323 d, r = other._fprs2

1324 if r:

1325 if not d: # PYCHOK no cover

1326 d = r

1327 elif self._raiser(r, d):

1328 raise self._ResidualError(op, other, r)

1329 else:

1330 d, n = other.as_integer_ratio()

1331 elif isscalar(other):

1332 d = other

1333 else: # PYCHOK no cover

1334 raise self._TypeError(op, other) # txt=_invalid_

1335 try:

1336 s = 0 if isinf(d) else (

1337 d if isnan(d) else self._finite(n / d))

1338 except Exception as x:

1339 E, t = _xError2(x)

1340 raise self._Error(op, other, E, txt=t)

1341 f = self._mul_scalar(s, _mul_op_) # handles 0, NAN, etc.

1342 return self._fset(f)

1343

1344 @property_RO

1345 def imag(self):

1346 '''Get the C{imaginary} part of this instance (C{0.0}, always).

1347

1348 @see: Properties L{Fsum.ceil}, L{Fsum.floor} and L{Fsum.real}.

1349 '''

1350 return _0_0

1351

1352 def int_float(self, raiser=False):

1353 '''Return this instance' current running sum as C{int} or C{float}.

1354

1355 @kwarg raiser: If C{True} throw a L{ResidualError} if the

1356 residual is non-zero.

1357

1358 @return: This C{integer} sum if this instance C{is_integer},

1359 otherwise return the C{float} sum if the residual

1360 is zero or if C{B{raiser}=False}.

1361

1362 @raise ResidualError: Non-zero residual and C{B{raiser}=True}.

1363

1364 @see: Methods L{Fsum.fint} and L{Fsum.fint2}.

1365 '''

1366 s, r = self._fint2

1367 if r:

1368 s, r = self._fprs2

1369 if r and raiser: # PYCHOK no cover

1370 t = _stresidual(_non_zero_, r)

1371 raise ResidualError(int_float=s, txt=t)

1372 s = float(s) # redundant

1373 return s

1374

1375 def is_exact(self):

1376 '''Is this instance' current running C{fsum} considered to

1377 be exact? (C{bool}).

1378 '''

1379 return self.residual is INT0

1380

1381 def is_integer(self):

1382 '''Is this instance' current running sum C{integer}? (C{bool}).

1383

1384 @see: Methods L{Fsum.fint} and L{Fsum.fint2}.

1385 '''

1386 _, r = self._fint2

1387 return False if r else True

1388

1389 def is_math_fsum(self):

1390 '''Return whether functions L{fsum}, L{fsum_}, L{fsum1}

1391 and L{fsum1_} plus partials summation are based on

1392 Python's C{math.fsum} or not.

1393

1394 @return: C{2} if all functions and partials summation

1395 are based on C{math.fsum}, C{True} if only

1396 the functions are based on C{math.fsum} (and

1397 partials summation is not) or C{False} if

1398 none are.

1399 '''

1400 f = Fsum._math_fsum

1401 return 2 if _psum is f else bool(f)

1402

1403 def _mul_Fsum(self, other, op=_mul_op_):

1404 '''(INTERNAL) Return C{B{self} * Fsum B{other}} as L{Fsum}.

1405 '''

1406 # assert isinstance(other, Fsum)

1407 return self._copy_0()._facc(self._ps_x(op, *other._ps), up=False)

1408

1409 def _mul_scalar(self, factor, op):

1410 '''(INTERNAL) Return C{B{self} * scalar B{factor}} as L{Fsum} or C{0}.

1411 '''

1412 # assert isscalar(factor)

1413 if self._finite(factor, op) and self._ps:

1414 if factor == _1_0:

1415 return self

1416 f = self._copy_0()._facc(self._ps_x(op, factor), up=False)

1417 else:

1418 f = self._copy_0(_0_0)

1419 return f

1420

1421 @property_RO

1422 def partials(self):

1423 '''Get this instance' current partial sums (C{tuple} of C{float}s and/or C{int}s).

1424 '''

1425 return tuple(self._ps)

1426

1427 def pow(self, x, *mod):

1428 '''Return C{B{self}**B{x}} as L{Fsum}.

1429

1430 @arg x: The exponent (L{Fsum} or C{scalar}).

1431 @arg mod: Optional modulus (C{int} or C{None}) for the 3-argument

1432 C{pow(B{self}, B{other}, B{mod})} version.

1433

1434 @return: The C{pow(self, B{x})} or C{pow(self, B{x}, *B{mod})}

1435 result (L{Fsum}).

1436

1437 @note: If B{C{mod}} is given as C{None}, the result will be an

1438 C{integer} L{Fsum} provided this instance C{is_integer}

1439 or set C{integer} with L{Fsum.fint}.

1440

1441 @see: Methods L{Fsum.__ipow__}, L{Fsum.fint} and L{Fsum.is_integer}.

1442 '''

1443 f = self._copy_2(self.pow)

1444 if f and isint(x) and x >= 0 and not mod:

1445 f._pow_int(x, x, _pow_op_) # f **= x

1446 else:

1447 f._fpow(x, _pow_op_, *mod) # f = pow(f, x, *mod)

1448 return f

1449

1450 def _pow_0_1(self, x, other):

1451 '''(INTERNAL) Return B{C{self}**1} or C{B{self}**0 == 1.0}.

1452 '''

1453 return self if x else (1 if self.is_integer() and isint(other) else _1_0)

1454

1455 def _pow_2(self, x, other, op):

1456 '''(INTERNAL) 2-arg C{pow(B{self}, scalar B{x})} embellishing errors.

1457 '''

1458 # assert len(self._ps) == 1 and isscalar(x)

1459 b = self._ps[0] # assert isscalar(b)

1460 try: # type(s) == type(x) if x in (_1_0, 1)

1461 s = pow(b, x) # -1**2.3 == -(1**2.3)

1462 if not iscomplex(s):

1463 return self._finite(s) # 0**INF == 0.0, 1**INF==1.0

1464 # neg**frac == complex in Python 3+, but ValueError in 2-

1465 E, t = _ValueError, _strcomplex(s, b, x) # PYCHOK no cover

1466 except Exception as x:

1467 E, t = _xError2(x)

1468 raise self._Error(op, other, E, txt=t)

1469

1470 def _pow_3(self, other, mod, op):

1471 '''(INTERNAL) 3-arg C{pow(B{self}, B{other}, int B{mod} or C{None})}.

1472 '''

1473 b, r = self._fprs2 if mod is None else self._fint2

1474 if r and self._raiser(r, b):

1475 t = _non_zero_ if mod is None else _integer_

1476 E, t = ResidualError, _stresidual(t, r, mod=mod)

1477 else:

1478 try: # b, other, mod all C{int}, unless C{mod} is C{None}

1479 x = _2scalar(other, _raiser=self._raiser)

1480 s = pow(b, x, mod)

1481 if not iscomplex(s):

1482 return self._finite(s)

1483 # neg**frac == complex in Python 3+, but ValueError in 2-

1484 E, t = _ValueError, _strcomplex(s, b, x, mod) # PYCHOK no cover

1485 except Exception as x:

1486 E, t = _xError2(x)

1487 t = _COMMASPACE_(Fmt.PARENSPACED(mod=mod), t)

1488 raise self._Error(op, other, E, txt=t)

1489

1490 def _pow_int(self, x, other, op):

1491 '''(INTERNAL) Return C{B{self} **= B{x}} for C{int B{x} >= 0}.

1492 '''

1493 # assert isint(x) and x >= 0

1494 if len(self._ps) > 1:

1495 if x > 2:

1496 p = self._copy_up()

1497 m = 1 # single-bit mask

1498 if x & m:

1499 x -= m # x ^= m

1500 f = p._copy_up()

1501 else:

1502 f = self._copy_0(_1_0)

1503 while x:

1504 p = p._mul_Fsum(p, op) # p **= 2

1505 m += m # m <<= 1

1506 if x & m:

1507 x -= m # x ^= m

1508 f = f._mul_Fsum(p, op) # f *= p

1509 elif x > 1: # self**2

1510 f = self._mul_Fsum(self, op)

1511 else: # self**1 or self**0

1512 f = self._pow_0_1(x, other)

1513 elif self._ps: # self._ps[0]**x

1514 f = self._pow_2(x, other, op)

1515 else: # PYCHOK no cover

1516 # 0**pos_int == 0, but 0**0 == 1

1517 f = 0 if x else 1 # like ._fprs

1518 return self._fset(f, asis=isint(f), n=len(self))

1519

1520 def _pow_scalar(self, x, other, op):

1521 '''(INTERNAL) Return C{self**B{x}} for C{scalar B{x}}.

1522 '''

1523 s, r = self._fprs2

1524 if isint(x, both=True):

1525 x = int(x) # Fsum**int

1526 y = abs(x)

1527 if y > 1:

1528 if r:

1529 f = self._copy_up()._pow_int(y, other, op)

1530 if x > 0: # > 1

1531 return f

1532 # assert x < 0 # < -1

1533 s, r = f._fprs2

1534 if r:

1535 return self._copy_0(_1_0)._ftruediv(f, op)

1536 # use **= -1 for the CPython float_pow

1537 # error if s is zero, and not s = 1 / s

1538 x = -1

1539# elif y > 1: # self**2 or self**-2

1540# f = self._mul_Fsum(self, op)

1541# if x < 0:

1542# f = f._copy_0(_1_0)._ftruediv(f, op)

1543# return f

1544 elif x < 0: # self**-1 == 1 / self

1545 if r:

1546 return self._copy_0(_1_0)._ftruediv(self, op)

1547 else: # self**1 or self**0

1548 return self._pow_0_1(x, other) # self or 0.0

1549 elif not isscalar(x): # assert ...

1550 raise self._TypeError(op, other, txt=_not_scalar_)

1551 elif r and self._raiser(r, s): # non-zero residual**fractional

1552 # raise self._ResidualError(op, other, r, fractional_power=x)

1553 t = _stresidual(_non_zero_, r, fractional_power=x)

1554 raise self._Error(op, other, ResidualError, txt=t)

1555 # assert isscalar(s) and isscalar(x)

1556 return self._copy_0(s)._pow_2(x, other, op)

1557

1558 def _ps_1(self, *less):

1559 '''(INTERNAL) Yield partials, 1-primed and subtract any C{less}.

1560 '''

1561 yield _1_0

1562 for p in self._ps:

1563 if p:

1564 yield p

1565 for p in less:

1566 if p:

1567 yield -p

1568 yield _N_1_0

1569

1570 def _ps_n(self):

1571 '''(INTERNAL) Yield partials, negated.

1572 '''

1573 for p in self._ps:

1574 if p:

1575 yield -p

1576

1577 def _ps_x(self, op, *factors): # see .fmath.Fhorner

1578 '''(INTERNAL) Yield all C{partials} times each B{C{factor}},

1579 in total, up to C{len(partials) * len(factors)} items.

1580 '''

1581 ps = self._ps

1582 if len(ps) < len(factors):

1583 ps, factors = factors, ps

1584 _f = _isfinite

1585 for f in factors:

1586 for p in ps:

1587 p *= f

1588 if _f(p):

1589 yield p

1590 else: # PYCHOK no cover

1591 self._finite(p, op) # throw ValueError

1592

1593 @property_RO

1594 def real(self):

1595 '''Get the C{real} part of this instance (C{float}).

1596

1597 @see: Methods L{Fsum.__float__} and L{Fsum.fsum}

1598 and properties L{Fsum.ceil}, L{Fsum.floor},

1599 L{Fsum.imag} and L{Fsum.residual}.

1600 '''

1601 return float(self._fprs)

1602

1603 @property_RO

1604 def residual(self):

1605 '''Get this instance' residual (C{float} or C{int}), the

1606 C{sum(partials)} less the precision running sum C{fsum}.

1607

1608 @note: If the C{residual is INT0}, the precision running

1609 C{fsum} is considered to be I{exact}.

1610

1611 @see: Methods L{Fsum.fsum}, L{Fsum.fsum2} and L{Fsum.is_exact}.

1612 '''

1613 return self._fprs2.residual

1614

1615 def _raiser(self, r, s):

1616 '''(INTERNAL) Does the ratio C{r / s} exceed threshold?

1617 '''

1618 self._ratio = t = fabs((r / s) if s else r)

1619 return t > self._RESIDUAL

1620

1621 def RESIDUAL(self, *threshold):

1622 '''Get and set this instance' I{ratio} for raising L{ResidualError}s,

1623 overriding the default from env variable C{PYGEODESY_FSUM_RESIDUAL}.

1624

1625 @arg threshold: If C{scalar}, the I{ratio} to exceed for raising

1626 L{ResidualError}s in division and exponention, if

1627 C{None} restore the default set with env variable

1628 C{PYGEODESY_FSUM_RESIDUAL} or if omitted, keep the

1629 current setting.

1630

1631 @return: The previous C{RESIDUAL} setting (C{float}).

1632

1633 @raise ValueError: Negative B{C{threshold}}.

1634

1635 @note: A L{ResidualError} is thrown if the non-zero I{ratio}

1636 C{residual} / C{fsum} exceeds the B{C{threshold}}.

1637 '''

1638 r = self._RESIDUAL

1639 if threshold:

1640 t = threshold[0]

1641 t = Fsum._RESIDUAL if t is None else (

1642 float(t) if isscalar(t) else ( # for backward ...

1643 _0_0 if bool(t) else _1_0)) # ... compatibility

1644 if t < 0:

1645 u = _DOT_(self, unstr(self.RESIDUAL, *threshold))

1646 raise _ValueError(u, RESIDUAL=t, txt=_negative_)

1647 self._RESIDUAL = t

1648 return r

1649

1650 def _ResidualError(self, op, other, residual):

1651 '''(INTERNAL) Non-zero B{C{residual}} etc.

1652 '''

1653 t = _stresidual(_non_zero_, residual, ratio=self._ratio,

1654 RESIDUAL=self._RESIDUAL)

1655 t = t.replace(_COMMASPACE_R_, _exceeds_R_)

1656 return self._Error(op, other, ResidualError, txt=t)

1657

1658 def signOf(self, res=True):

1659 '''Determine the sign of this instance.

1660

1661 @kwarg res: If C{True} consider, otherwise

1662 ignore the residual (C{bool}).

1663

1664 @return: The sign (C{int}, -1, 0 or +1).

1665 '''

1666 s, r = self._fprs2 if res else (self._fprs, 0)

1667 return _signOf(s, -r)

1668

1669 def toRepr(self, **prec_sep_fmt_lenc): # PYCHOK signature

1670 '''Return this C{Fsum} instance as representation.

1671

1672 @kwarg prec_sep_fmt_lenc: Optional keyword arguments for

1673 method L{Fsum2Tuple.toRepr} plus C{B{lenc}=True}

1674 (C{bool}) to in-/exclude the current C{[len]}

1675 of this L{Fsum} enclosed in I{[brackets]}.

1676

1677 @return: This instance (C{repr}).

1678 '''

1679 return self._toT(self._fprs2.toRepr, **prec_sep_fmt_lenc)

1680

1681 def toStr(self, **prec_sep_fmt_lenc): # PYCHOK signature

1682 '''Return this C{Fsum} instance as string.

1683

1684 @kwarg prec_sep_fmt_lenc: Optional keyword arguments for

1685 method L{Fsum2Tuple.toStr} plus C{B{lenc}=True}

1686 (C{bool}) to in-/exclude the current C{[len]}

1687 of this L{Fsum} enclosed in I{[brackets]}.

1688

1689 @return: This instance (C{str}).

1690 '''

1691 return self._toT(self._fprs2.toStr, **prec_sep_fmt_lenc)

1692

1693 def _toT(self, toT, fmt=Fmt.g, lenc=True, **kwds):

1694 '''(INTERNAL) Helper for C{toRepr} and C{toStr}.

1695 '''

1696 n = self.named3

1697 if lenc:

1698 n = Fmt.SQUARE(n, len(self))

1699 return _SPACE_(n, toT(fmt=fmt, **kwds))

1700

1701 def _TypeError(self, op, other, **txt): # PYCHOK no cover

1702 '''(INTERNAL) Return a C{TypeError}.

1703 '''

1704 return self._Error(op, other, _TypeError, **txt)

1705

1706 def _update(self): # see ._fset

1707 '''(INTERNAL) Zap all cached C{Property_RO} values.

1708 '''

1709 Fsum._fint2._update(self)

1710 Fsum._fprs ._update(self)

1711 Fsum._fprs2._update(self)

1712 return self

1713

1714 def _ValueError(self, op, other, **txt): # PYCHOK no cover

1715 '''(INTERNAL) Return a C{ValueError}.

1716 '''

1717 return self._Error(op, other, _ValueError, **txt)

1718

1719 def _ZeroDivisionError(self, op, other, **txt): # PYCHOK no cover

1720 '''(INTERNAL) Return a C{ZeroDivisionError}.

1721 '''

1722 return self._Error(op, other, _ZeroDivisionError, **txt)

1723

1724_allPropertiesOf_n(3, Fsum, Property_RO) # PYCHOK assert, see Fsum._fset, -._update

1725

1726

1727 def _Float_Int(arg, **name_Error):

1728 '''(INTERNAL) Unit of L{Fsum2Tuple} items.

1729 '''

1730 U = Int if isint(arg) else Float

1731 return U(arg, **name_Error)

1732

1733

1734 class Fsum2Tuple(_NamedTuple):

1735 '''2-Tuple C{(fsum, residual)} with the precision running C{fsum}

1736 and the C{residual}, the sum of the remaining partials. Each

1737 item is either C{float} or C{int}.

1738

1739 @note: If the C{residual is INT0}, the C{fsum} is considered

1740 to be I{exact}, see method L{Fsum2Tuple.is_exact}.

1741 '''

1742 _Names_ = ( Fsum.fsum.__name__, Fsum.residual.name)

1743 _Units_ = (_Float_Int, _Float_Int)

1744

1745 @Property_RO

1746 def Fsum(self):

1747 '''Get this L{Fsum2Tuple} as an L{Fsum}.

1748 '''

1749 f = Fsum(name=self.name)

1750 return f._copy_0(*(s for s in reversed(self) if s))

1751

1752 def is_exact(self):

1753 '''Is this L{Fsum2Tuple} considered to be exact? (C{bool}).

1754 '''

1755 return self.Fsum.is_exact()

1756

1757 def is_integer(self):

1758 '''Is this L{Fsum2Tuple} C{integer}? (C{bool}).

1759 '''

1760 return self.Fsum.is_integer()

1761

1762

1763 class ResidualError(_ValueError):

1764 '''Error raised for an operation involving a L{pygeodesy.sums.Fsum}

1765 instance with a non-zero C{residual}, I{integer} or otherwise.

1766

1767 @see: Module L{pygeodesy.fsums} and method L{Fsum.RESIDUAL}.

1768 '''

1769 pass

1770

1771

1772 try:

1773 from math import fsum as _fsum # precision IEEE-754 sum, Python 2.6+

1774

1775 # make sure _fsum works as expected (XXX check

1776 # float.__getformat__('float')[:4] == 'IEEE'?)

1777 if _fsum((1, 1e101, 1, -1e101)) != 2: # PYCHOK no cover

1778 del _fsum # nope, remove _fsum ...

1779 raise ImportError # ... use _fsum below

1780

1781 Fsum._math_fsum = _sum = _fsum # PYCHOK exported

1782

1783 if _getenv('PYGEODESY_FSUM_PARTIALS', _fsum.__name__) == _fsum.__name__:

1784 _psum = _fsum # PYCHOK redef

1785

1786 except ImportError:

1787 _sum = sum # Fsum(NAN) exception fall-back

1788

1789 def _fsum(xs):

1790 '''(INTERNAL) Precision summation, Python 2.5-.

1791 '''

1792 return Fsum(name=_fsum.__name__)._facc(xs, up=False)._fprs

1793

1794

1795 def fsum(xs, floats=False):

1796 '''Precision floating point summation based on or like Python's C{math.fsum}.

1797

1798 @arg xs: Iterable, list, tuple, etc. of values (C{scalar} or

1799 L{Fsum} instances).

1800 @kwarg floats: Optionally, use C{B{floats}=True} iff I{all}

1801 B{C{xs}} are known to be C{float}.

1802

1803 @return: Precision C{fsum} (C{float}).

1804

1805 @raise OverflowError: Partial C{2sum} overflow.

1806

1807 @raise TypeError: Non-scalar B{C{xs}} value.

1808

1809 @raise ValueError: Invalid or non-finite B{C{xs}} value.

1810

1811 @note: Exceptions and I{non-finite} handling may differ if not

1812 based on Python's C{math.fsum}.

1813

1814 @see: Class L{Fsum} and methods L{Fsum.fsum} and L{Fsum.fadd}.

1815 '''

1816 return _fsum(xs if floats else _2floats(xs)) if xs else _0_0 # PYCHOK yield

1817

1818

1819 def fsum_(*xs, **floats):

1820 '''Precision floating point summation of all positional arguments.

1821

1822 @arg xs: Values to be added (C{scalar} or L{Fsum} instances),

1823 all positional.

1824 @kwarg floats: Optionally, use C{B{floats}=True} iff I{all}

1825 B{C{xs}} are known to be C{float}.

1826

1827 @return: Precision C{fsum} (C{float}).

1828

1829 @see: Function C{fsum}.

1830 '''

1831 return _fsum(xs if _xkwds_get(floats, floats=False) else

1832 _2floats(xs, origin=1)) if xs else _0_0 # PYCHOK yield

1833

1834

1835 def fsumf_(*xs):

1836 '''Precision floating point summation L{fsum_}C{(*xs, floats=True)}.

1837 '''

1838 return _fsum(xs) if xs else _0_0

1839

1840

1841 def fsum1(xs, floats=False):

1842 '''Precision floating point summation of a few arguments, 1-primed.

1843

1844 @arg xs: Iterable, list, tuple, etc. of values (C{scalar} or

1845 L{Fsum} instances).

1846 @kwarg floats: Optionally, use C{B{floats}=True} iff I{all}

1847 B{C{xs}} are known to be C{float}.

1848

1849 @return: Precision C{fsum} (C{float}).

1850

1851 @see: Function C{fsum}.

1852 '''

1853 return _fsum(_1primed(xs if floats else _2floats(xs))) if xs else _0_0 # PYCHOK yield

1854

1855

1856 def fsum1_(*xs, **floats):

1857 '''Precision floating point summation of a few arguments, 1-primed.

1858

1859 @arg xs: Values to be added (C{scalar} or L{Fsum} instances),

1860 all positional.

1861 @kwarg floats: Optionally, use C{B{floats}=True} iff I{all}

1862 B{C{xs}} are known to be C{float}.

1863

1864 @return: Precision C{fsum} (C{float}).

1865

1866 @see: Function C{fsum}

1867 '''

1868 return _fsum(_1primed(xs if _xkwds_get(floats, floats=False) else

1869 _2floats(xs, origin=1))) if xs else _0_0 # PYCHOK yield

1870

1871

1872 def fsum1f_(*xs):

1873 '''Precision floating point summation L{fsum1_}C{(*xs, floats=True)}.

1874 '''

1875 return _fsum(_1primed(xs)) if xs else _0_0

1876

1877

1878# **) MIT License

1879#

1880# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.

1881#

1882# Permission is hereby granted, free of charge, to any person obtaining a

1883# copy of this software and associated documentation files (the "Software"),

1884# to deal in the Software without restriction, including without limitation

1885# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1886# and/or sell copies of the Software, and to permit persons to whom the

1887# Software is furnished to do so, subject to the following conditions:

1888#

1889# The above copyright notice and this permission notice shall be included

1890# in all copies or substantial portions of the Software.

1891#

1892# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1893# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1894# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1895# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1896# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1897# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1898# OTHER DEALINGS IN THE SOFTWARE.