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, \

34 _xError2, _xkwds_get, _ZeroDivisionError

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

36 _exceeds_, _from_, _iadd_op_, _LANGLE_, \

37 _negative_, _NOTEQUAL_, _not_finite_, \

38 _not_scalar_, _PERCENT_, _PLUS_, _R_, _RANGLE_, \

39 _SLASH_, _SPACE_, _STAR_, _UNDER_

40 from pygeodesy.lazily import _ALL_LAZY, _getenv, _sys_version_info2

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

42 from pygeodesy.props import _allPropertiesOf_n, deprecated_property_RO, \

43 Property_RO, property_RO

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

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

46

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

48

49__all__ = _ALL_LAZY.fsums

50__version__ = '24.02.20'

51

52_add_op_ = _PLUS_ # in .auxilats.auxAngle

53_eq_op_ = _EQUAL_ * 2 # _DEQUAL_

54_COMMASPACE_R_ = _COMMASPACE_ + _R_

55_exceeds_R_ = _SPACE_ + _exceeds_(_R_)

56_floordiv_op_ = _SLASH_ * 2 # _DSLASH_

57_fset_op_ = _EQUAL_

58_ge_op_ = _RANGLE_ + _EQUAL_

59_gt_op_ = _RANGLE_

60_integer_ = 'integer'

61_le_op_ = _LANGLE_ + _EQUAL_

62_lt_op_ = _LANGLE_

63_mod_op_ = _PERCENT_

64_mul_op_ = _STAR_

65_ne_op_ = _NOTEQUAL_

66_non_zero_ = 'non-zero'

67_pow_op_ = _STAR_ * 2 # _DSTAR_, in .fmath

68_sub_op_ = _DASH_ # in .auxilats.auxAngle, .fsums

69_truediv_op_ = _SLASH_

70_divmod_op_ = _floordiv_op_ + _mod_op_

71_isub_op_ = _sub_op_ + _fset_op_ # in .auxilats.auxAngle, .fsums

72

73

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

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

76 '''

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

78 try:

79 v = float(v)

80 if _isfinite(v):

81 return v

82 E, t = _ValueError, _not_finite_

83 except Exception as e:

84 E, t = _xError2(e)

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

86

87

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

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

90 '''

91 try:

92 i, x = origin, None

93 _fin = _isfinite

94 _Fsum = Fsum

95 for x in xs:

96 if isinstance(x, _Fsum):

97 for p in x._ps:

98 yield (-p) if sub else p

99 else:

100 f = float(x)

101 if not _fin(f):

102 raise ValueError(_not_finite_)

103 if f:

104 yield (-f) if sub else f

105 i += 1

106 except Exception as e:

107 E, t = _xError2(e)

108 n = Fmt.SQUARE(xs=i)

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

110

111

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

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

114 '''

115 _xisscalar(power=power)

116 try:

117 i, x = origin, None

118 _fin = _isfinite

119 _Fsum = Fsum

120 _pow = pow # XXX math.pow

121 for x in xs:

122 if isinstance(x, _Fsum):

123 P = x.pow(power)

124 for p in P._ps:

125 yield p

126 else:

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

128 if not _fin(p):

129 raise ValueError(_not_finite_)

130 yield p

131 i += 1

132 except Exception as e:

133 E, t = _xError2(e)

134 n = Fmt.SQUARE(xs=i)

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

136

137

138 def _1primed(xs):

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

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

141 '''

142 yield _1_0

143 for x in xs:

144 if x:

145 yield x

146 yield _N_1_0

147

148

149 def _psum(ps): # PYCHOK used!

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

151 '''

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

153 s = ps[i]

154 _2s = _2sum

155 while i > 0:

156 i -= 1

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

158 if r: # sum(ps) became inexact

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

160 if i > 0:

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

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

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

164 r *= 2

165 t = s + r

166 if r == (t - s):

167 s = t

168 break

169 ps[i:] = [s]

170 return s

171

172

173 def _2scalar(other, _raiser=None):

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

175 '''

176 if isinstance(other, Fsum):

177 s, r = other._fint2

178 if r:

179 s, r = other._fprs2

180 if r: # PYCHOK no cover

181 if _raiser and _raiser(r, s):

182 raise ValueError(_stresidual(_non_zero_, r))

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

184 else:

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

186 if isint(s, both=True):

187 s = int(s)

188 return s

189

190

191 def _strcomplex(s, *args):

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

193 '''

194 c = iscomplex.__name__[2:]

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

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

197 return unstr(t, *args)

198

199

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

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

202 '''

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

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

205 for n, v in itemsorted(name_values):

206 n = n.replace(_UNDER_, _SPACE_)

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

208 t = _COMMASPACE_(t, p)

209 return t

210

211

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

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

214 '''

215 s = a + b

216 if not _isfinite(s):

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

218 t = Fmt.PARENSPACED(_not_finite_, s)

219 raise _OverflowError(u, txt=t)

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

221 a, b = b, a

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

223

224

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

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

227

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

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

230 intermediate, I{running} summuation.

231

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

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

234 method C{__float__}.

235

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

237 Python's C{math.fsum}.

238

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

240 393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}, U{Kahan

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

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

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

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

245 '''

246 _math_fsum = None

247 _n = 0

248# _ps = [] # partial sums

249# _px = 0

250 _ratio = None

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

252

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

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

255

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

257 L{Fsum} instance).

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

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

260 L{ResidualError} threshold.

261

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

263 '''

264 if name_RESIDUAL:

265 n = _xkwds_get(name_RESIDUAL, name=NN)

266 if n: # set name ...

267 self.name = n

268 r = _xkwds_get(name_RESIDUAL, RESIDUAL=None)

269 if r is not None:

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

271# self._n = 0

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

273 if len(xs) > 1:

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

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

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

277 self._n = 1

278

279 def __abs__(self):

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

281 '''

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

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

284 self._copy_2(self.__abs__)

285

286 def __add__(self, other):

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

288

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

290

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

292

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

294 '''

295 f = self._copy_2(self.__add__)

296 return f._fadd(other, _add_op_)

297

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

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

300 '''

301 s, r = self._fprs2

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

303

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

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

306

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

308

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

310 '''

311 return self.ceil

312

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

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

315

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

317

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

319 '''

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

321 return _signOf(s, 0)

322

323 cmp = __cmp__

324

325 def __divmod__(self, other):

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

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

328 and an L{Fsum}.

329

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

331

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

333 '''

334 f = self._copy_2(self.__divmod__)

335 return f._fdivmod2(other, _divmod_op_)

336

337 def __eq__(self, other):

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

339 '''

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

341

342 def __float__(self):

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

344

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

346 '''

347 return float(self._fprs)

348

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

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

351

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

353

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

355 '''

356 return self.floor

357

358 def __floordiv__(self, other):

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

360

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

362

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

364

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

366 '''

367 f = self._copy_2(self.__floordiv__)

368 return f._floordiv(other, _floordiv_op_)

369

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

371 '''Not implemented.'''

372 return _NotImplemented(self, *other)

373

374 def __ge__(self, other):

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

376 '''

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

378

379 def __gt__(self, other):

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

381 '''

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

383

384 def __hash__(self): # PYCHOK no cover

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

386 '''

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

388

389 def __iadd__(self, other):

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

391

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

393

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

395

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

397 C{scalar} nor L{Fsum}.

398

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

400 '''

401 return self._fadd(other, _iadd_op_)

402

403 def __ifloordiv__(self, other):

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

405

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

407

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

409

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

411

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

413

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

415

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

417

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

419 '''

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

421

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

423 '''Not implemented.'''

424 return _NotImplemented(self, other)

425

426 def __imod__(self, other):

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

428

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

430

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

432

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

434 '''

435 self._fdivmod2(other, _mod_op_ + _fset_op_)

436 return self

437

438 def __imul__(self, other):

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

440

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

442

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

444

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

446

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

448

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

450 '''

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

452

453 def __int__(self):

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

455

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

457 and L{Fsum.__floor__} and properties

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

459 '''

460 i, _ = self._fint2

461 return i

462

463 def __invert__(self): # PYCHOK no cover

464 '''Not implemented.'''

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

466 return _NotImplemented(self)

467

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

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

470

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

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

473 3-argument C{pow(B{self}, B{other}, B{mod})}

474 version.

475

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

477

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

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

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

481

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

483

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

485 env var C{PYGEODESY_FSUM_RESIDUAL}

486 set or this instance has a non-zero

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

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

489 is a negative or fractional C{scalar}.

490

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

492 C{pow} invocation failed.

493

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

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

496 is a fractional C{scalar} and this

497 instance is negative or has a non-zero

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

499

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

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

502 '''

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

504

505 def __isub__(self, other):

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

507

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

509

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

511

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

513

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

515 '''

516 return self._fsub(other, _isub_op_)

517

518 def __iter__(self):

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

520 '''

521 return iter(self.partials)

522

523 def __itruediv__(self, other):

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

525

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

527

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

529

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

531

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

533 env var C{PYGEODESY_FSUM_RESIDUAL} set.

534

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

536

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

538

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

540

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

542 '''

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

544

545 def __le__(self, other):

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

547 '''

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

549

550 def __len__(self):

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

552 '''

553 return self._n

554

555 def __lt__(self, other):

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

557 '''

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

559

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

561 '''Not implemented.'''

562 return _NotImplemented(self, other)

563

564 def __mod__(self, other):

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

566

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

568 '''

569 f = self._copy_2(self.__mod__)

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

571

572 def __mul__(self, other):

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

574

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

576 '''

577 f = self._copy_2(self.__mul__)

578 return f._fmul(other, _mul_op_)

579

580 def __ne__(self, other):

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

582 '''

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

584

585 def __neg__(self):

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

587 '''

588 return self._copy_n(self.__neg__)

589

590 def __pos__(self):

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

592 '''

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

594

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

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

597

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

599 '''

600 f = self._copy_2(self.__pow__)

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

602

603 def __radd__(self, other):

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

605

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

607 '''

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

609 return f._fadd(self, _add_op_)

610

611 def __rdivmod__(self, other):

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

613 remainder)}.

614

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

616 '''

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

618 return f._fdivmod2(self, _divmod_op_)

619

620# def __repr__(self):

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

622# '''

623# return self.toRepr(lenc=True)

624

625 def __rfloordiv__(self, other):

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

627

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

629 '''

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

631 return f._floordiv(self, _floordiv_op_)

632

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

634 '''Not implemented.'''

635 return _NotImplemented(self, other)

636

637 def __rmod__(self, other):

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

639

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

641 '''

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

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

644

645 def __rmul__(self, other):

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

647

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

649 '''

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

651 return f._fmul(self, _mul_op_)

652

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

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

655

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

657 '''

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

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

660 f._n = 1

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

662 return f

663

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

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

666

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

668 '''

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

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

671

672 def __rsub__(self, other):

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

674

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

676 '''

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

678 return f._fsub(self, _sub_op_)

679

680 def __rtruediv__(self, other):

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

682

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

684 '''

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

686 return f._ftruediv(self, _truediv_op_)

687

688 def __str__(self):

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

690 '''

691 return self.toStr(lenc=True)

692

693 def __sub__(self, other):

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

695

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

697

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

699

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

701 '''

702 f = self._copy_2(self.__sub__)

703 return f._fsub(other, _sub_op_)

704

705 def __truediv__(self, other):

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

707

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

709

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

711

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

713 '''

714 f = self._copy_2(self.__truediv__)

715 return f._ftruediv(other, _truediv_op_)

716

717 __trunc__ = __int__

718

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

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

721 __div__ = __truediv__

722 __idiv__ = __itruediv__

723 __long__ = __int__

724 __nonzero__ = __bool__

725 __rdiv__ = __rtruediv__

726

727 def as_integer_ratio(self):

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

729

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

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

732

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

734 '''

735 n, r = self._fint2

736 if r:

737 i, d = r.as_integer_ratio()

738 n *= d

739 n += i

740 else: # PYCHOK no cover

741 d = 1

742 return n, d

743

744 @property_RO

745 def ceil(self):

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

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

748

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

750

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

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

753 '''

754 s, r = self._fprs2

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

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

757 c += 1

758 return c

759

760 def _cmp_0(self, other, op):

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

762 '''

763 if isscalar(other):

764 if other:

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

766 else:

767 s, r = self._fprs2

768 s = _signOf(s, -r)

769 elif isinstance(other, Fsum):

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

771 else:

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

773 return s

774

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

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

777

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

779 '''

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

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

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

783 return f

784

785 def _copy_0(self, *xs):

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

787 '''

788 # for x in xs:

789 # assert isscalar(x)

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

791 if self.name:

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

793 return f

794

795 def _copy_2(self, which):

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

797 '''

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

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

800 # assert f._n == self._n

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

802 return f

803

804 def _copy_n(self, which):

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

806 '''

807 if self._ps:

808 f = self._Fsum(self._n)

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

810# f._facc_up(up=False)

811 else:

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

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

814 return f

815

816 def _copy_r2(self, other, which):

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

818 '''

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

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

821

822 def _copy_RESIDUAL(self, other):

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

824 '''

825 R = other._RESIDUAL

826 if R is not Fsum._RESIDUAL:

827 self._RESIDUAL = R

828

829 def _copy_up(self, _fprs2=False):

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

831 '''

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

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

834 Fsum._fprs2._update_from(f, self)

835 return f

836

837 def divmod(self, other):

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

839 remainder)}.

840

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

842

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

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

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

846

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

848 '''

849 f = self._copy_2(self.divmod)

850 return f._fdivmod2(other, _divmod_op_)

851

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

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

854 '''

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

856

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

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

859 '''

860 E, t = _xError2(X)

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

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

863

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

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

866 '''

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

868 for x in xs: # _iter()

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

870 i = 0

871 for p in ps:

872 x, p = _2s(x, p)

873 if p:

874 ps[i] = p

875 i += 1

876 ps[i:] = [x]

877 n += 1

878 # assert self._ps is ps

879 if n:

880 self._n += n

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

882 if up:

883 self._update()

884 return self

885

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

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

888 '''

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

890

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

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

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

894# '''

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

896# p = self._ps.pop()

897# if p:

898# n = self._n

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

900# self._n = n

901# break

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

903

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

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

906 to this instance.

907

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

909 L{Fsum} instances).

910

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

912

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

914

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

916 nor L{Fsum}.

917

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

919 '''

920 if isinstance(xs, Fsum):

921 self._facc(xs._ps)

922 elif isscalar(xs): # for backward compatibility

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

924 elif xs:

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

926 return self

927

928 def fadd_(self, *xs):

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

930 to this instance.

931

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

933 all positional.

934

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

936

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

938

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

940 nor L{Fsum}.

941

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

943 '''

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

945

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

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

948 '''

949 if isinstance(other, Fsum):

950 if other is self:

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

952 elif other._ps:

953 self._facc(other._ps)

954 elif not isscalar(other):

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

956 elif other:

957 self._facc_(other)

958 return self

959

960 fcopy = copy # for backward compatibility

961 fdiv = __itruediv__ # for backward compatibility

962 fdivmod = __divmod__ # for backward compatibility

963

964 def _fdivmod2(self, other, op):

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

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

967 '''

968 # result mostly follows CPython function U{float_divmod

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

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

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

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

973 self -= other * q

974

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

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

977 self += other

978 q -= 1

979

980# t = self.signOf()

981# if t and t != s:

982# from pygeodesy.errors import _AssertionError

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

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

985

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

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

988 '''

989 if _isfinite(other):

990 return other

991 raise ValueError(_not_finite_) if not op else \

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

993

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

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

996

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

998 I{integer} residual is non-zero.

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

1000

1001 @return: The C{integer} (L{Fsum}).

1002

1003 @raise ResidualError: Non-zero I{integer} residual.

1004

1005 @see: Methods L{Fsum.int_float} and L{Fsum.is_integer}.

1006 '''

1007 i, r = self._fint2

1008 if r and raiser:

1009 t = _stresidual(_integer_, r)

1010 raise ResidualError(_integer_, i, txt=t)

1011 n = name or self.fint.__name__

1012 return Fsum(name=n)._fset(i, asis=True)

1013

1014 def fint2(self, **name):

1015 '''Return this instance' current running sum as C{int} and

1016 the I{integer} residual.

1017

1018 @kwarg name: Optional name (C{str}).

1019

1020 @return: An L{Fsum2Tuple}C{(fsum, residual)} with C{fsum}

1021 an C{int} and I{integer} C{residual} a C{float} or

1022 C{INT0} if the C{fsum} is considered to be I{exact}.

1023 '''

1024 return Fsum2Tuple(*self._fint2, **name)

1025

1026 @Property_RO

1027 def _fint2(self): # see ._fset

1028 '''(INTERNAL) Get 2-tuple (C{int}, I{integer} residual).

1029 '''

1030 i = int(self._fprs) # int(self)

1031 r = _fsum(self._ps_1(i)) if len(self._ps) > 1 else (

1032 (self._ps[0] - i) if self._ps else -i)

1033 return i, (r or INT0)

1034

1035 @deprecated_property_RO

1036 def float_int(self): # PYCHOK no cover

1037 '''DEPRECATED, use method C{Fsum.int_float}.'''

1038 return self.int_float() # raiser=False

1039

1040 @property_RO

1041 def floor(self):

1042 '''Get this instance' C{floor} (C{int} in Python 3+, but

1043 C{float} in Python 2-).

1044

1045 @note: The C{floor} takes the C{residual} into account.

1046

1047 @see: Method L{Fsum.int_float} and properties L{Fsum.ceil},

1048 L{Fsum.imag} and L{Fsum.real}.

1049 '''

1050 s, r = self._fprs2

1051 f = _floor(s) + _floor(r) + 1

1052 while r < (f - s): # (s + r) < f

1053 f -= 1

1054 return f

1055

1056# floordiv = __floordiv__ # for naming consistency

1057

1058 def _floordiv(self, other, op): # rather _ffloordiv?

1059 '''Apply C{B{self} //= B{other}}.

1060 '''

1061 q = self._ftruediv(other, op) # == self

1062 return self._fset(q.floor, asis=True) # floor(q)

1063

1064 fmul = __imul__ # for backward compatibility

1065

1066 def _fmul(self, other, op):

1067 '''(INTERNAL) Apply C{B{self} *= B{other}}.

1068 '''

1069 if isscalar(other):

1070 f = self._mul_scalar(other, op)

1071 elif not isinstance(other, Fsum):

1072 raise self._TypeError(op, other) # txt=_invalid_

1073 elif len(self._ps) != 1:

1074 f = self._mul_Fsum(other, op)

1075 elif len(other._ps) != 1: # len(self._ps) == 1

1076 f = other._copy_up()._mul_scalar(self._ps[0], op)

1077 else: # len(other._ps) == len(self._ps) == 1

1078 s = self._finite(self._ps[0] * other._ps[0])

1079 return self._fset(s, asis=True, n=len(self) + 1)

1080 return self._fset(f)

1081

1082 def fover(self, over):

1083 '''Apply C{B{self} /= B{over}} and summate.

1084

1085 @arg over: An L{Fsum} or C{scalar} denominator.

1086

1087 @return: Precision running sum (C{float}).

1088

1089 @see: Methods L{Fsum.fsum} and L{Fsum.__itruediv__}.

1090 '''

1091 return float(self.fdiv(over)._fprs)

1092

1093 fpow = __ipow__ # for backward compatibility

1094

1095 def _fpow(self, other, op, *mod):

1096 '''Apply C{B{self} **= B{other}}, optional B{C{mod}} or C{None}.

1097 '''

1098 if mod and mod[0] is not None: # == 3-arg C{pow}

1099 s = self._pow_3(other, mod[0], op)

1100 elif mod and mod[0] is None and self.is_integer():

1101 # return an exact C{int} for C{int}**C{int}

1102 i = self._copy_0(self._fint2[0]) # assert _fint2[1] == 0

1103 x = _2scalar(other) # C{int}, C{float} or other

1104 s = i._pow_2(x, other, op) if isscalar(x) else i._fpow(x, op)

1105 else: # pow(self, other) == pow(self, other, None)

1106 p = None

1107 if isinstance(other, Fsum):

1108 x, r = other._fprs2

1109 if r:

1110 if self._raiser(r, x):

1111 raise self._ResidualError(op, other, r)

1112 p = self._pow_scalar(r, other, op)

1113# p = _2scalar(p) # _raiser = None

1114 elif not isscalar(other):

1115 raise self._TypeError(op, other) # txt=_invalid_

1116 else:

1117 x = self._finite(other, op)

1118 s = self._pow_scalar(x, other, op)

1119 if p is not None:

1120 s *= p

1121 return self._fset(s, asis=isint(s), n=max(len(self), 1))

1122

1123 @Property_RO

1124 def _fprs(self):

1125 '''(INTERNAL) Get and cache this instance' precision

1126 running sum (C{float} or C{int}), ignoring C{residual}.

1127

1128 @note: The precision running C{fsum} after a C{//=} or

1129 C{//} C{floor} division is C{int} in Python 3+.

1130 '''

1131 ps = self._ps

1132 n = len(ps) - 1

1133 if n > 1:

1134 s = _psum(ps)

1135 elif n > 0: # len(ps) == 2

1136 s, p = _2sum(*ps) if ps[1] else ps

1137 ps[:] = ([p, s] if s else [p]) if p else [s]

1138 elif n < 0: # see L{Fsum.__init__}

1139 s = _0_0

1140 ps[:] = [s]

1141 else: # len(ps) == 1

1142 s = ps[0]

1143 # assert self._ps is ps

1144 # assert Fsum._fprs2.name not in self.__dict__

1145 return s

1146

1147 @Property_RO

1148 def _fprs2(self):

1149 '''(INTERNAL) Get and cache this instance' precision

1150 running sum and residual (L{Fsum2Tuple}).

1151 '''

1152 s = self._fprs

1153 r = _fsum(self._ps_1(s)) if len(self._ps) > 1 else INT0

1154 return Fsum2Tuple(s, r) # name=Fsum.fsum2.__name__

1155

1156# def _fpsqz(self):

1157# '''(INTERNAL) Compress, squeeze the C{partials}.

1158# '''

1159# if len(self._ps) > 2:

1160# _ = self._fprs

1161# return self

1162

1163 def _fset(self, other, asis=False, n=1):

1164 '''(INTERNAL) Overwrite this instance with an other or a C{scalar}.

1165 '''

1166 if other is self:

1167 pass # from ._fmul, ._ftruediv and ._pow_scalar

1168 elif isinstance(other, Fsum):

1169 self._n = other._n

1170 self._ps[:] = other._ps

1171 self._copy_RESIDUAL(other)

1172 # use or zap the C{Property_RO} values

1173 Fsum._fint2._update_from(self, other)

1174 Fsum._fprs ._update_from(self, other)

1175 Fsum._fprs2._update_from(self, other)

1176 elif isscalar(other):

1177 s = other if asis else float(other)

1178 i = int(s) # see ._fint2

1179 t = i, ((s - i) or INT0)

1180 self._n = n

1181 self._ps[:] = [s]

1182 # Property_RO _fint2, _fprs and _fprs2 can't be a Property:

1183 # Property's _fset zaps the value just set by the @setter

1184 self.__dict__.update(_fint2=t, _fprs=s, _fprs2=Fsum2Tuple(s, INT0))

1185 else: # PYCHOK no cover

1186 raise self._TypeError(_fset_op_, other) # txt=_invalid_

1187 return self

1188

1189 def fsub(self, xs=()):

1190 '''Subtract an iterable of C{scalar} or L{Fsum} instances

1191 from this instance.

1192

1193 @arg xs: Iterable, list, tuple. etc. (C{scalar}

1194 or L{Fsum} instances).

1195

1196 @return: This instance, updated (L{Fsum}).

1197

1198 @see: Method L{Fsum.fadd}.

1199 '''

1200 return self._facc(_2floats(xs, sub=True)) if xs else self # PYCHOK yield

1201

1202 def fsub_(self, *xs):

1203 '''Subtract all positional C{scalar} or L{Fsum} instances

1204 from this instance.

1205

1206 @arg xs: Values to subtract (C{scalar} or

1207 L{Fsum} instances), all positional.

1208

1209 @return: This instance, updated (L{Fsum}).

1210

1211 @see: Method L{Fsum.fadd}.

1212 '''

1213 return self._facc(_2floats(xs, origin=1, sub=True)) if xs else self # PYCHOK yield

1214

1215 def _fsub(self, other, op):

1216 '''(INTERNAL) Apply C{B{self} -= B{other}}.

1217 '''

1218 if isinstance(other, Fsum):

1219 if other is self: # or other._fprs2 == self._fprs2:

1220 self._fset(_0_0, asis=True, n=len(self) * 2) # self -= self

1221 elif other._ps:

1222 self._facc(other._ps_n())

1223 elif not isscalar(other):

1224 raise self._TypeError(op, other) # txt=_invalid_

1225 elif self._finite(other, op):

1226 self._facc_(-other)

1227 return self

1228

1229 def _Fsum(self, n, *ps):

1230 '''(INTERNAL) New L{Fsum} instance.

1231 '''

1232 f = Fsum()

1233 f._n = n

1234 if ps:

1235 f._ps[:] = ps

1236 f._copy_RESIDUAL(self)

1237 return f

1238

1239 def fsum(self, xs=()):

1240 '''Add more C{scalar} or L{Fsum} instances and summate.

1241

1242 @kwarg xs: Iterable, list, tuple, etc. (C{scalar} or

1243 L{Fsum} instances).

1244

1245 @return: Precision running sum (C{float} or C{int}).

1246

1247 @see: Method L{Fsum.fadd}.

1248

1249 @note: Accumulation can continue after summation.

1250 '''

1251 f = self._facc(_2floats(xs)) if xs else self # PYCHOK yield

1252 return f._fprs

1253

1254 def fsum_(self, *xs):

1255 '''Add all positional C{scalar} or L{Fsum} instances and summate.

1256

1257 @arg xs: Values to add (C{scalar} or L{Fsum} instances),

1258 all positional.

1259

1260 @return: Precision running sum (C{float} or C{int}).

1261

1262 @see: Methods L{Fsum.fsum} and L{Fsum.fsumf_}.

1263 '''

1264 f = self._facc(_2floats(xs, origin=1)) if xs else self # PYCHOK yield

1265 return f._fprs

1266

1267 def fsum2(self, xs=(), **name):

1268 '''Add more C{scalar} or L{Fsum} instances and return the

1269 current precision running sum and the C{residual}.

1270

1271 @kwarg xs: Iterable, list, tuple, etc. (C{scalar} or

1272 L{Fsum} instances).

1273 @kwarg name: Optional name (C{str}).

1274

1275 @return: L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} the

1276 current precision running sum and C{residual}, the

1277 (precision) sum of the remaining C{partials}. The

1278 C{residual is INT0} if the C{fsum} is considered

1279 to be I{exact}.

1280

1281 @see: Methods L{Fsum.fint2}, L{Fsum.fsum} and L{Fsum.fsum2_}

1282 '''

1283 f = self._facc(_2floats(xs)) if xs else self # PYCHOK yield

1284 t = f._fprs2

1285 if name:

1286 t = t.dup(name=_xkwds_get(name, name=NN))

1287 return t

1288

1289 def fsum2_(self, *xs):

1290 '''Add any positional C{scalar} or L{Fsum} instances and return

1291 the precision running sum and the C{differential}.

1292

1293 @arg xs: Values to add (C{scalar} or L{Fsum} instances),

1294 all positional.

1295

1296 @return: 2-Tuple C{(fsum, delta)} with the current precision

1297 running C{fsum} and C{delta}, the difference with

1298 the previous running C{fsum} (C{float}s).

1299

1300 @see: Methods L{Fsum.fsum_} and L{Fsum.fsum}.

1301 '''

1302 p, r = self._fprs2

1303 if xs:

1304 s, t = self._facc(_2floats(xs, origin=1))._fprs2 # PYCHOK yield

1305 return s, _fsum((s, -p, r, -t)) # ((s - p) + (r - t))

1306 else: # PYCHOK no cover

1307 return p, _0_0

1308

1309 def fsumf_(self, *xs):

1310 '''Like method L{Fsum.fsum_} but only for known C{float B{xs}}.

1311 '''

1312 f = self._facc(xs) if xs else self # PYCHOK yield

1313 return f._fprs

1314

1315# ftruediv = __itruediv__ # for naming consistency

1316

1317 def _ftruediv(self, other, op):

1318 '''(INTERNAL) Apply C{B{self} /= B{other}}.

1319 '''

1320 n = _1_0

1321 if isinstance(other, Fsum):

1322 if other is self or other._fprs2 == self._fprs2:

1323 return self._fset(_1_0, asis=True, n=len(self))

1324 d, r = other._fprs2

1325 if r:

1326 if not d: # PYCHOK no cover

1327 d = r

1328 elif self._raiser(r, d):

1329 raise self._ResidualError(op, other, r)

1330 else:

1331 d, n = other.as_integer_ratio()

1332 elif isscalar(other):

1333 d = other

1334 else: # PYCHOK no cover

1335 raise self._TypeError(op, other) # txt=_invalid_

1336 try:

1337 s = 0 if isinf(d) else (

1338 d if isnan(d) else self._finite(n / d))

1339 except Exception as x:

1340 E, t = _xError2(x)

1341 raise self._Error(op, other, E, txt=t)

1342 f = self._mul_scalar(s, _mul_op_) # handles 0, NAN, etc.

1343 return self._fset(f)

1344

1345 @property_RO

1346 def imag(self):

1347 '''Get the C{imaginary} part of this instance (C{0.0}, always).

1348

1349 @see: Properties L{Fsum.ceil}, L{Fsum.floor} and L{Fsum.real}.

1350 '''

1351 return _0_0

1352

1353 def int_float(self, raiser=False):

1354 '''Return this instance' current running sum as C{int} or C{float}.

1355

1356 @kwarg raiser: If C{True} throw a L{ResidualError} if the

1357 residual is non-zero.

1358

1359 @return: This C{integer} sum if this instance C{is_integer},

1360 otherwise return the C{float} sum if the residual

1361 is zero or if C{B{raiser}=False}.

1362

1363 @raise ResidualError: Non-zero residual and C{B{raiser}=True}.

1364

1365 @see: Methods L{Fsum.fint} and L{Fsum.fint2}.

1366 '''

1367 s, r = self._fint2

1368 if r:

1369 s, r = self._fprs2

1370 if r and raiser: # PYCHOK no cover

1371 t = _stresidual(_non_zero_, r)

1372 raise ResidualError(int_float=s, txt=t)

1373 s = float(s) # redundant

1374 return s

1375

1376 def is_exact(self):

1377 '''Is this instance' current running C{fsum} considered to

1378 be exact? (C{bool}).

1379 '''

1380 return self.residual is INT0

1381

1382 def is_integer(self):

1383 '''Is this instance' current running sum C{integer}? (C{bool}).

1384

1385 @see: Methods L{Fsum.fint} and L{Fsum.fint2}.

1386 '''

1387 _, r = self._fint2

1388 return False if r else True

1389

1390 def is_math_fsum(self):

1391 '''Return whether functions L{fsum}, L{fsum_}, L{fsum1}

1392 and L{fsum1_} plus partials summation are based on

1393 Python's C{math.fsum} or not.

1394

1395 @return: C{2} if all functions and partials summation

1396 are based on C{math.fsum}, C{True} if only

1397 the functions are based on C{math.fsum} (and

1398 partials summation is not) or C{False} if

1399 none are.

1400 '''

1401 f = Fsum._math_fsum

1402 return 2 if _psum is f else bool(f)

1403

1404 def _mul_Fsum(self, other, op=_mul_op_):

1405 '''(INTERNAL) Return C{B{self} * Fsum B{other}} as L{Fsum}.

1406 '''

1407 # assert isinstance(other, Fsum)

1408 return self._copy_0()._facc(self._ps_x(op, *other._ps), up=False)

1409

1410 def _mul_scalar(self, factor, op):

1411 '''(INTERNAL) Return C{B{self} * scalar B{factor}} as L{Fsum} or C{0}.

1412 '''

1413 # assert isscalar(factor)

1414 if self._finite(factor, op) and self._ps:

1415 if factor == _1_0:

1416 return self

1417 f = self._copy_0()._facc(self._ps_x(op, factor), up=False)

1418 else:

1419 f = self._copy_0(_0_0)

1420 return f

1421

1422 @property_RO

1423 def partials(self):

1424 '''Get this instance' current partial sums (C{tuple} of C{float}s and/or C{int}s).

1425 '''

1426 return tuple(self._ps)

1427

1428 def pow(self, x, *mod):

1429 '''Return C{B{self}**B{x}} as L{Fsum}.

1430

1431 @arg x: The exponent (L{Fsum} or C{scalar}).

1432 @arg mod: Optional modulus (C{int} or C{None}) for the 3-argument

1433 C{pow(B{self}, B{other}, B{mod})} version.

1434

1435 @return: The C{pow(self, B{x})} or C{pow(self, B{x}, *B{mod})}

1436 result (L{Fsum}).

1437

1438 @note: If B{C{mod}} is given as C{None}, the result will be an

1439 C{integer} L{Fsum} provided this instance C{is_integer}

1440 or set C{integer} with L{Fsum.fint}.

1441

1442 @see: Methods L{Fsum.__ipow__}, L{Fsum.fint} and L{Fsum.is_integer}.

1443 '''

1444 f = self._copy_2(self.pow)

1445 if f and isint(x) and x >= 0 and not mod:

1446 f._pow_int(x, x, _pow_op_) # f **= x

1447 else:

1448 f._fpow(x, _pow_op_, *mod) # f = pow(f, x, *mod)

1449 return f

1450

1451 def _pow_0_1(self, x, other):

1452 '''(INTERNAL) Return B{C{self}**1} or C{B{self}**0 == 1.0}.

1453 '''

1454 return self if x else (1 if self.is_integer() and isint(other) else _1_0)

1455

1456 def _pow_2(self, x, other, op):

1457 '''(INTERNAL) 2-arg C{pow(B{self}, scalar B{x})} embellishing errors.

1458 '''

1459 # assert len(self._ps) == 1 and isscalar(x)

1460 b = self._ps[0] # assert isscalar(b)

1461 try: # type(s) == type(x) if x in (_1_0, 1)

1462 s = pow(b, x) # -1**2.3 == -(1**2.3)

1463 if not iscomplex(s):

1464 return self._finite(s) # 0**INF == 0.0, 1**INF==1.0

1465 # neg**frac == complex in Python 3+, but ValueError in 2-

1466 E, t = _ValueError, _strcomplex(s, b, x) # PYCHOK no cover

1467 except Exception as x:

1468 E, t = _xError2(x)

1469 raise self._Error(op, other, E, txt=t)

1470

1471 def _pow_3(self, other, mod, op):

1472 '''(INTERNAL) 3-arg C{pow(B{self}, B{other}, int B{mod} or C{None})}.

1473 '''

1474 b, r = self._fprs2 if mod is None else self._fint2

1475 if r and self._raiser(r, b):

1476 t = _non_zero_ if mod is None else _integer_

1477 E, t = ResidualError, _stresidual(t, r, mod=mod)

1478 else:

1479 try: # b, other, mod all C{int}, unless C{mod} is C{None}

1480 x = _2scalar(other, _raiser=self._raiser)

1481 s = pow(b, x, mod)

1482 if not iscomplex(s):

1483 return self._finite(s)

1484 # neg**frac == complex in Python 3+, but ValueError in 2-

1485 E, t = _ValueError, _strcomplex(s, b, x, mod) # PYCHOK no cover

1486 except Exception as x:

1487 E, t = _xError2(x)

1488 t = _COMMASPACE_(Fmt.PARENSPACED(mod=mod), t)

1489 raise self._Error(op, other, E, txt=t)

1490

1491 def _pow_int(self, x, other, op):

1492 '''(INTERNAL) Return C{B{self} **= B{x}} for C{int B{x} >= 0}.

1493 '''

1494 # assert isint(x) and x >= 0

1495 if len(self._ps) > 1:

1496 if x > 2:

1497 p = self._copy_up()

1498 m = 1 # single-bit mask

1499 if x & m:

1500 x -= m # x ^= m

1501 f = p._copy_up()

1502 else:

1503 f = self._copy_0(_1_0)

1504 while x:

1505 p = p._mul_Fsum(p, op) # p **= 2

1506 m += m # m <<= 1

1507 if x & m:

1508 x -= m # x ^= m

1509 f = f._mul_Fsum(p, op) # f *= p

1510 elif x > 1: # self**2

1511 f = self._mul_Fsum(self, op)

1512 else: # self**1 or self**0

1513 f = self._pow_0_1(x, other)

1514 elif self._ps: # self._ps[0]**x

1515 f = self._pow_2(x, other, op)

1516 else: # PYCHOK no cover

1517 # 0**pos_int == 0, but 0**0 == 1

1518 f = 0 if x else 1 # like ._fprs

1519 return self._fset(f, asis=isint(f), n=len(self))

1520

1521 def _pow_scalar(self, x, other, op):

1522 '''(INTERNAL) Return C{self**B{x}} for C{scalar B{x}}.

1523 '''

1524 s, r = self._fprs2

1525 if isint(x, both=True):

1526 x = int(x) # Fsum**int

1527 y = abs(x)

1528 if y > 1:

1529 if r:

1530 f = self._copy_up()._pow_int(y, other, op)

1531 if x > 0: # > 1

1532 return f

1533 # assert x < 0 # < -1

1534 s, r = f._fprs2

1535 if r:

1536 return self._copy_0(_1_0)._ftruediv(f, op)

1537 # use **= -1 for the CPython float_pow

1538 # error if s is zero, and not s = 1 / s

1539 x = -1

1540# elif y > 1: # self**2 or self**-2

1541# f = self._mul_Fsum(self, op)

1542# if x < 0:

1543# f = f._copy_0(_1_0)._ftruediv(f, op)

1544# return f

1545 elif x < 0: # self**-1 == 1 / self

1546 if r:

1547 return self._copy_0(_1_0)._ftruediv(self, op)

1548 else: # self**1 or self**0

1549 return self._pow_0_1(x, other) # self or 0.0

1550 elif not isscalar(x): # assert ...

1551 raise self._TypeError(op, other, txt=_not_scalar_)

1552 elif r and self._raiser(r, s): # non-zero residual**fractional

1553 # raise self._ResidualError(op, other, r, fractional_power=x)

1554 t = _stresidual(_non_zero_, r, fractional_power=x)

1555 raise self._Error(op, other, ResidualError, txt=t)

1556 # assert isscalar(s) and isscalar(x)

1557 return self._copy_0(s)._pow_2(x, other, op)

1558

1559 def _ps_1(self, *less):

1560 '''(INTERNAL) Yield partials, 1-primed and subtract any C{less}.

1561 '''

1562 yield _1_0

1563 for p in self._ps:

1564 if p:

1565 yield p

1566 for p in less:

1567 if p:

1568 yield -p

1569 yield _N_1_0

1570

1571 def _ps_n(self):

1572 '''(INTERNAL) Yield partials, negated.

1573 '''

1574 for p in self._ps:

1575 if p:

1576 yield -p

1577

1578 def _ps_x(self, op, *factors): # see .fmath.Fhorner

1579 '''(INTERNAL) Yield all C{partials} times each B{C{factor}},

1580 in total, up to C{len(partials) * len(factors)} items.

1581 '''

1582 ps = self._ps

1583 if len(ps) < len(factors):

1584 ps, factors = factors, ps

1585 _f = _isfinite

1586 for f in factors:

1587 for p in ps:

1588 p *= f

1589 if _f(p):

1590 yield p

1591 else: # PYCHOK no cover

1592 self._finite(p, op) # throw ValueError

1593

1594 @property_RO

1595 def real(self):

1596 '''Get the C{real} part of this instance (C{float}).

1597

1598 @see: Methods L{Fsum.__float__} and L{Fsum.fsum}

1599 and properties L{Fsum.ceil}, L{Fsum.floor},

1600 L{Fsum.imag} and L{Fsum.residual}.

1601 '''

1602 return float(self._fprs)

1603

1604 @property_RO

1605 def residual(self):

1606 '''Get this instance' residual (C{float} or C{int}), the

1607 C{sum(partials)} less the precision running sum C{fsum}.

1608

1609 @note: If the C{residual is INT0}, the precision running

1610 C{fsum} is considered to be I{exact}.

1611

1612 @see: Methods L{Fsum.fsum}, L{Fsum.fsum2} and L{Fsum.is_exact}.

1613 '''

1614 return self._fprs2.residual

1615

1616 def _raiser(self, r, s):

1617 '''(INTERNAL) Does the ratio C{r / s} exceed threshold?

1618 '''

1619 self._ratio = t = fabs((r / s) if s else r)

1620 return t > self._RESIDUAL

1621

1622 def RESIDUAL(self, *threshold):

1623 '''Get and set this instance' I{ratio} for raising L{ResidualError}s,

1624 overriding the default from env variable C{PYGEODESY_FSUM_RESIDUAL}.

1625

1626 @arg threshold: If C{scalar}, the I{ratio} to exceed for raising

1627 L{ResidualError}s in division and exponention, if

1628 C{None} restore the default set with env variable

1629 C{PYGEODESY_FSUM_RESIDUAL} or if omitted, keep the

1630 current setting.

1631

1632 @return: The previous C{RESIDUAL} setting (C{float}).

1633

1634 @raise ValueError: Negative B{C{threshold}}.

1635

1636 @note: A L{ResidualError} is thrown if the non-zero I{ratio}

1637 C{residual} / C{fsum} exceeds the B{C{threshold}}.

1638 '''

1639 r = self._RESIDUAL

1640 if threshold:

1641 t = threshold[0]

1642 t = Fsum._RESIDUAL if t is None else (

1643 float(t) if isscalar(t) else ( # for backward ...

1644 _0_0 if bool(t) else _1_0)) # ... compatibility

1645 if t < 0:

1646 u = self._unstr(self.RESIDUAL, *threshold)

1647 raise _ValueError(u, RESIDUAL=t, txt=_negative_)

1648 self._RESIDUAL = t

1649 return r

1650

1651 def _ResidualError(self, op, other, residual):

1652 '''(INTERNAL) Non-zero B{C{residual}} etc.

1653 '''

1654 t = _stresidual(_non_zero_, residual, ratio=self._ratio,

1655 RESIDUAL=self._RESIDUAL)

1656 t = t.replace(_COMMASPACE_R_, _exceeds_R_)

1657 return self._Error(op, other, ResidualError, txt=t)

1658

1659 def signOf(self, res=True):

1660 '''Determine the sign of this instance.

1661

1662 @kwarg res: If C{True} consider, otherwise

1663 ignore the residual (C{bool}).

1664

1665 @return: The sign (C{int}, -1, 0 or +1).

1666 '''

1667 s, r = self._fprs2 if res else (self._fprs, 0)

1668 return _signOf(s, -r)

1669

1670 def toRepr(self, **prec_sep_fmt_lenc): # PYCHOK signature

1671 '''Return this C{Fsum} instance as representation.

1672

1673 @kwarg prec_sep_fmt_lenc: Optional keyword arguments for

1674 method L{Fsum2Tuple.toRepr} plus C{B{lenc}=True}

1675 (C{bool}) to in-/exclude the current C{[len]}

1676 of this L{Fsum} enclosed in I{[brackets]}.

1677

1678 @return: This instance (C{repr}).

1679 '''

1680 return self._toT(self._fprs2.toRepr, **prec_sep_fmt_lenc)

1681

1682 def toStr(self, **prec_sep_fmt_lenc): # PYCHOK signature

1683 '''Return this C{Fsum} instance as string.

1684

1685 @kwarg prec_sep_fmt_lenc: Optional keyword arguments for

1686 method L{Fsum2Tuple.toStr} plus C{B{lenc}=True}

1687 (C{bool}) to in-/exclude the current C{[len]}

1688 of this L{Fsum} enclosed in I{[brackets]}.

1689

1690 @return: This instance (C{str}).

1691 '''

1692 return self._toT(self._fprs2.toStr, **prec_sep_fmt_lenc)

1693

1694 def _toT(self, toT, fmt=Fmt.g, lenc=True, **kwds):

1695 '''(INTERNAL) Helper for C{toRepr} and C{toStr}.

1696 '''

1697 n = self.named3

1698 if lenc:

1699 n = Fmt.SQUARE(n, len(self))

1700 return _SPACE_(n, toT(fmt=fmt, **kwds))

1701

1702 def _TypeError(self, op, other, **txt): # PYCHOK no cover

1703 '''(INTERNAL) Return a C{TypeError}.

1704 '''

1705 return self._Error(op, other, _TypeError, **txt)

1706

1707 def _update(self): # see ._fset

1708 '''(INTERNAL) Zap all cached C{Property_RO} values.

1709 '''

1710 Fsum._fint2._update(self)

1711 Fsum._fprs ._update(self)

1712 Fsum._fprs2._update(self)

1713 return self

1714

1715 def _ValueError(self, op, other, **txt): # PYCHOK no cover

1716 '''(INTERNAL) Return a C{ValueError}.

1717 '''

1718 return self._Error(op, other, _ValueError, **txt)

1719

1720 def _ZeroDivisionError(self, op, other, **txt): # PYCHOK no cover

1721 '''(INTERNAL) Return a C{ZeroDivisionError}.

1722 '''

1723 return self._Error(op, other, _ZeroDivisionError, **txt)

1724

1725_allPropertiesOf_n(3, Fsum, Property_RO) # PYCHOK assert, see Fsum._fset, -._update

1726

1727

1728 def _Float_Int(arg, **name_Error):

1729 '''(INTERNAL) Unit of L{Fsum2Tuple} items.

1730 '''

1731 U = Int if isint(arg) else Float

1732 return U(arg, **name_Error)

1733

1734

1735 class Fsum2Tuple(_NamedTuple):

1736 '''2-Tuple C{(fsum, residual)} with the precision running C{fsum}

1737 and the C{residual}, the sum of the remaining partials. Each

1738 item is either C{float} or C{int}.

1739

1740 @note: If the C{residual is INT0}, the C{fsum} is considered

1741 to be I{exact}, see method L{Fsum2Tuple.is_exact}.

1742 '''

1743 _Names_ = ( Fsum.fsum.__name__, Fsum.residual.name)

1744 _Units_ = (_Float_Int, _Float_Int)

1745

1746 @Property_RO

1747 def Fsum(self):

1748 '''Get this L{Fsum2Tuple} as an L{Fsum}.

1749 '''

1750 f = Fsum(name=self.name)

1751 return f._copy_0(*(s for s in reversed(self) if s))

1752

1753 def is_exact(self):

1754 '''Is this L{Fsum2Tuple} considered to be exact? (C{bool}).

1755 '''

1756 return self.Fsum.is_exact()

1757

1758 def is_integer(self):

1759 '''Is this L{Fsum2Tuple} C{integer}? (C{bool}).

1760 '''

1761 return self.Fsum.is_integer()

1762

1763

1764 class ResidualError(_ValueError):

1765 '''Error raised for an operation involving a L{pygeodesy.sums.Fsum}

1766 instance with a non-zero C{residual}, I{integer} or otherwise.

1767

1768 @see: Module L{pygeodesy.fsums} and method L{Fsum.RESIDUAL}.

1769 '''

1770 pass

1771

1772

1773 try:

1774 from math import fsum as _fsum # precision IEEE-754 sum, Python 2.6+

1775

1776 # make sure _fsum works as expected (XXX check

1777 # float.__getformat__('float')[:4] == 'IEEE'?)

1778 if _fsum((1, 1e101, 1, -1e101)) != 2: # PYCHOK no cover

1779 del _fsum # nope, remove _fsum ...

1780 raise ImportError # ... use _fsum below

1781

1782 Fsum._math_fsum = _sum = _fsum # PYCHOK exported

1783

1784 if _getenv('PYGEODESY_FSUM_PARTIALS', _fsum.__name__) == _fsum.__name__:

1785 _psum = _fsum # PYCHOK redef

1786

1787 except ImportError:

1788 _sum = sum # Fsum(NAN) exception fall-back

1789

1790 def _fsum(xs):

1791 '''(INTERNAL) Precision summation, Python 2.5-.

1792 '''

1793 return Fsum(name=_fsum.__name__)._facc(xs, up=False)._fprs

1794

1795

1796 def fsum(xs, floats=False):

1797 '''Precision floating point summation based on or like Python's C{math.fsum}.

1798

1799 @arg xs: Iterable, list, tuple, etc. of values (C{scalar} or

1800 L{Fsum} instances).

1801 @kwarg floats: Optionally, use C{B{floats}=True} iff I{all}

1802 B{C{xs}} are known to be C{float}.

1803

1804 @return: Precision C{fsum} (C{float}).

1805

1806 @raise OverflowError: Partial C{2sum} overflow.

1807

1808 @raise TypeError: Non-scalar B{C{xs}} value.

1809

1810 @raise ValueError: Invalid or non-finite B{C{xs}} value.

1811

1812 @note: Exceptions and I{non-finite} handling may differ if not

1813 based on Python's C{math.fsum}.

1814

1815 @see: Class L{Fsum} and methods L{Fsum.fsum} and L{Fsum.fadd}.

1816 '''

1817 return _fsum(xs if floats else _2floats(xs)) if xs else _0_0 # PYCHOK yield

1818

1819

1820 def fsum_(*xs, **floats):

1821 '''Precision floating point summation of all positional arguments.

1822

1823 @arg xs: Values to be added (C{scalar} or L{Fsum} instances),

1824 all positional.

1825 @kwarg floats: Optionally, use C{B{floats}=True} iff I{all}

1826 B{C{xs}} are known to be C{float}.

1827

1828 @return: Precision C{fsum} (C{float}).

1829

1830 @see: Function C{fsum}.

1831 '''

1832 return _fsum(xs if _xkwds_get(floats, floats=False) else

1833 _2floats(xs, origin=1)) if xs else _0_0 # PYCHOK yield

1834

1835

1836 def fsumf_(*xs):

1837 '''Precision floating point summation L{fsum_}C{(*xs, floats=True)}.

1838 '''

1839 return _fsum(xs) if xs else _0_0

1840

1841

1842 def fsum1(xs, floats=False):

1843 '''Precision floating point summation of a few arguments, 1-primed.

1844

1845 @arg xs: Iterable, list, tuple, etc. of values (C{scalar} or

1846 L{Fsum} instances).

1847 @kwarg floats: Optionally, use C{B{floats}=True} iff I{all}

1848 B{C{xs}} are known to be C{float}.

1849

1850 @return: Precision C{fsum} (C{float}).

1851

1852 @see: Function C{fsum}.

1853 '''

1854 return _fsum(_1primed(xs if floats else _2floats(xs))) if xs else _0_0 # PYCHOK yield

1855

1856

1857 def fsum1_(*xs, **floats):

1858 '''Precision floating point summation of a few arguments, 1-primed.

1859

1860 @arg xs: Values to be added (C{scalar} or L{Fsum} instances),

1861 all positional.

1862 @kwarg floats: Optionally, use C{B{floats}=True} iff I{all}

1863 B{C{xs}} are known to be C{float}.

1864

1865 @return: Precision C{fsum} (C{float}).

1866

1867 @see: Function C{fsum}

1868 '''

1869 return _fsum(_1primed(xs if _xkwds_get(floats, floats=False) else

1870 _2floats(xs, origin=1))) if xs else _0_0 # PYCHOK yield

1871

1872

1873 def fsum1f_(*xs):

1874 '''Precision floating point summation L{fsum1_}C{(*xs, floats=True)}.

1875 '''

1876 return _fsum(_1primed(xs)) if xs else _0_0

1877

1878

1879# **) MIT License

1880#

1881# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.

1882#

1883# Permission is hereby granted, free of charge, to any person obtaining a

1884# copy of this software and associated documentation files (the "Software"),

1885# to deal in the Software without restriction, including without limitation

1886# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1887# and/or sell copies of the Software, and to permit persons to whom the

1888# Software is furnished to do so, subject to the following conditions:

1889#

1890# The above copyright notice and this permission notice shall be included

1891# in all copies or substantial portions of the Software.

1892#

1893# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1894# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1895# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1896# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1897# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1898# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1899# OTHER DEALINGS IN THE SOFTWARE.