Coverage for pygeodesy/fsums.py: 97%

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1 

2# -*- coding: utf-8 -*- 

3 

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

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

6 

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

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

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

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

11 

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

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

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

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

16 

17Set env variable C{PYGEODESY_FSUM_RESIDUAL} to a C{float} string greater than 

18C{"0.0"} as the threshold to throw a L{ResidualError} for a division, power or 

19root operation of an L{Fsum} instance with a C{residual} I{ratio} exceeding 

20the threshold. See methods L{Fsum.RESIDUAL}, L{Fsum.pow}, L{Fsum.__ipow__} 

21and L{Fsum.__itruediv__}. 

22''' 

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

24from __future__ import division as _; del _ # PYCHOK semicolon 

25 

26from pygeodesy.basics import isbool, iscomplex, isint, isscalar, \ 

27 _signOf, itemsorted, signOf, _xiterable, \ 

28 _xiterablen, _enquote 

29from pygeodesy.constants import INT0, _isfinite, NEG0, _pos_self, \ 

30 _0_0, _1_0, _N_1_0, Float, Int 

31from pygeodesy.errors import _OverflowError, _TypeError, _UnexpectedError, \ 

32 _ValueError, _xError, _xError2, _xkwds_get1, \ 

33 _xkwds_pop2 

34# from pygeodesy.internals import _enquote # from .basics 

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

36 _EQUAL_, _from_, _LANGLE_, _NOTEQUAL_, \ 

37 _not_finite_, _PERCENT_, _PLUS_, \ 

38 _RANGLE_, _SLASH_, _SPACE_, _STAR_, _UNDER_ 

39from pygeodesy.lazily import _ALL_LAZY, _getenv, _sys_version_info2 

40from pygeodesy.named import _name__, _name2__, _Named, _NamedTuple, \ 

41 _NotImplemented 

42from pygeodesy.props import _allPropertiesOf_n, deprecated_property_RO, \ 

43 Property, Property_RO, property_RO 

44from pygeodesy.streprs import Fmt, fstr, unstr 

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

46 

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

48 

49__all__ = _ALL_LAZY.fsums 

50__version__ = '24.06.11' 

51 

52_add_op_ = _PLUS_ # in .auxilats.auxAngle 

53_eq_op_ = _EQUAL_ * 2 # _DEQUAL_ 

54_div_ = 'div' 

55_floordiv_op_ = _SLASH_ * 2 # _DSLASH_ 

56_fset_op_ = _EQUAL_ 

57_ge_op_ = _RANGLE_ + _EQUAL_ 

58_gt_op_ = _RANGLE_ 

59_iadd_op_ = _add_op_ + _EQUAL_ # in .auxilats.auxAngle, .fstats 

60_integer_ = 'integer' 

61_le_op_ = _LANGLE_ + _EQUAL_ 

62_lt_op_ = _LANGLE_ 

63_mod_ = 'mod' 

64_mod_op_ = _PERCENT_ 

65_mul_op_ = _STAR_ 

66_ne_op_ = _NOTEQUAL_ 

67_non_zero_ = 'non-zero' 

68_pow_op_ = _STAR_ * 2 # _DSTAR_ 

69_significant_ = 'significant' 

70_sub_op_ = _DASH_ # in .auxilats.auxAngle 

71_threshold_ = 'threshold' 

72_truediv_op_ = _SLASH_ 

73_divmod_op_ = _floordiv_op_ + _mod_op_ 

74_isub_op_ = _sub_op_ + _fset_op_ # in .auxilats.auxAngle 

75 

76 

77def _2delta(*ab): 

78 '''(INTERNAL) Helper for C{Fsum._fsum2}. 

79 ''' 

80 try: 

81 a, b = _2sum(*ab) 

82 except _OverflowError: 

83 a, b = ab 

84 return float(a if fabs(a) > fabs(b) else b) 

85 

86 

87def _2error(unused): # in .fstats 

88 '''(INTERNAL) Throw a C{not-finite} exception. 

89 ''' 

90 raise ValueError(_not_finite_) 

91 

92 

93def _2finite(x): 

94 '''(INTERNAL) return C{float(x)} if finite. 

95 ''' 

96 x = float(x) 

97 return x if _isfinite(x) else _2error(x) 

98 

99 

100def _2float(index=None, **name_value): # in .fmath, .fstats 

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

102 ''' 

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

104 try: 

105 return _2finite(v) 

106 except Exception as X: 

107 raise _xError(X, Fmt.INDEX(n, index), v) 

108 

109 

110def _X_ps(X): # for _2floats only 

111 return X._ps 

112 

113 

114def _2floats(xs, origin=0, _X=_X_ps, _x=float): 

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

116 ''' 

117 try: 

118 i, x = origin, _X 

119 _fin = _isfinite 

120 _FsT = _Fsum_Fsum2Tuple_types 

121 _isa = isinstance 

122 for x in _xiterable(xs): 

123 if _isa(x, _FsT): 

124 for p in _X(x._Fsum): 

125 yield p 

126 else: 

127 f = _x(x) 

128 yield f if _fin(f) else _2error(f) 

129 i += 1 

130 except Exception as X: 

131 raise _xError(X, xs=xs) if x is _X else \ 

132 _xError(X, Fmt.INDEX(xs=i), x) 

133 

134 

135def _Fsumf_(*xs): # floats=True, in .auxLat, ... 

136 '''(INTERNAL) An C{Fsum} of I{known scalars}. 

137 ''' 

138 return Fsum()._facc_scalar(xs, up=False) 

139 

140 

141def _Fsum1f_(*xs): # floats=True, in .albers, ... 

142 '''(INTERNAL) An C{Fsum} of I{known scalars}, 1-primed. 

143 ''' 

144 return Fsum()._facc_scalar(_1primed(xs), up=False) 

145 

146 

147def _2halfeven(s, r, p): 

148 '''(INTERNAL) Round half-even. 

149 ''' 

150 if (p > 0 and r > 0) or \ 

151 (p < 0 and r < 0): # signs match 

152 r *= 2 

153 t = s + r 

154 if r == (t - s): 

155 s = t 

156 return s 

157 

158 

159def _isFsum(x): # in .fmath 

160 '''(INTERNAL) Is C{x} an C{Fsum} instance? 

161 ''' 

162 return isinstance(x, Fsum) 

163 

164 

165def _isFsumTuple(x): # in .fmath 

166 '''(INTERNAL) Is C{x} an C{Fsum} or C{Fsum2Tuple} instance? 

167 ''' 

168 return isinstance(x, _Fsum_Fsum2Tuple_types) 

169 

170 

171def _1_Over(x, op, **raiser_RESIDUAL): # vs _1_over 

172 '''(INTERNAL) Return C{Fsum(1) / B{x}}. 

173 ''' 

174 return _Psum_(_1_0)._ftruediv(x, op, **raiser_RESIDUAL) 

175 

176 

177def _1primed(xs): # in .fmath 

178 '''(INTERNAL) 1-Primed summation of iterable C{xs} 

179 items, all I{known} to be C{scalar}. 

180 ''' 

181 yield _1_0 

182 for x in xs: 

183 yield x 

184 yield _N_1_0 

185 

186 

187def _psum(ps): # PYCHOK used! 

188 '''(INTERNAL) Partials summation, updating C{ps}. 

189 ''' 

190 # assert isinstance(ps, list) 

191 i = len(ps) - 1 

192 s = _0_0 if i < 0 else ps[i] 

193 _2s = _2sum 

194 while i > 0: 

195 i -= 1 

196 s, r = _2s(s, ps[i]) 

197 if r: # sum(ps) became inexact 

198 if s: 

199 ps[i:] = r, s 

200 if i > 0: 

201 s = _2halfeven(s, r, ps[i-1]) 

202 break # return s 

203 s = r # PYCHOK no cover 

204 ps[i:] = s, 

205 return s 

206 

207 

208def _Psum(ps, **name_RESIDUAL): 

209 '''(INTERNAL) Return an C{Fsum} from I{ordered} partials C{ps}. 

210 ''' 

211 f = Fsum(**name_RESIDUAL) if name_RESIDUAL else Fsum() 

212 if ps: 

213 f._ps[:] = ps 

214 f._n = len(f._ps) 

215 return f 

216 

217 

218def _Psum_(*ps, **name_RESIDUAL): 

219 '''(INTERNAL) Return an C{Fsum} from 1 or 2 known scalar(s) C{ps}. 

220 ''' 

221 return _Psum(ps, **name_RESIDUAL) 

222 

223 

224def _2scalar2(other): 

225 '''(INTERNAL) Return 2-tuple C{(other, r)} with C{other} as C{int}, 

226 C{float} or C{as-is} and C{r} the residual of C{as-is}. 

227 ''' 

228 if _isFsumTuple(other): 

229 s, r = other._fint2 

230 if r: 

231 s, r = other._fprs2 

232 if r: # PYCHOK no cover 

233 s = other # L{Fsum} as-is 

234 else: 

235 r = 0 

236 s = other # C{type} as-is 

237 if isint(s, both=True): 

238 s = int(s) 

239 return s, r 

240 

241 

242def _s_r(s, r): 

243 '''(INTERNAL) Return C{(s, r)}, I{ordered}. 

244 ''' 

245 if r: 

246 if fabs(s) < fabs(r): 

247 s, r = r, (s or INT0) 

248 else: 

249 r = INT0 

250 return s, r 

251 

252 

253def _strcomplex(s, *args): 

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

255 ''' 

256 c = _strcomplex.__name__[4:] 

257 n = _DASH_(len(args), _arg_) 

258 t = unstr(pow, *args) 

259 return _SPACE_(c, s, _from_, n, t) 

260 

261 

262def _stresidual(prefix, residual, R=0, **mod_ratio): 

263 '''(INTERNAL) Residual error txt C{str}. 

264 ''' 

265 p = _stresidual.__name__[3:] 

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

267 for n, v in itemsorted(mod_ratio): 

268 p = Fmt.PARENSPACED(n, Fmt(v)) 

269 t = _COMMASPACE_(t, p) 

270 return _SPACE_(prefix, t, Fmt.exceeds_R(R), _threshold_) 

271 

272 

273def _2sum(a, b): # by .testFmath 

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

275 ''' 

276 # Neumaier, A. U{Rundungsfehleranalyse einiger Verfahren zur Summation endlicher 

277 # Summen<https://OnlineLibrary.Wiley.com/doi/epdf/10.1002/zamm.19740540106>}, 

278 # 1974, Zeitschrift für Angewandte Mathmatik und Mechanik, vol 51, nr 1, p 39-51 

279 # <https://StackOverflow.com/questions/78633770/can-neumaier-summation-be-sped-up> 

280 s = a + b 

281 if _isfinite(s): 

282 if fabs(a) < fabs(b): 

283 r = (b - s) + a 

284 else: 

285 r = (a - s) + b 

286 return s, r 

287 u = unstr(_2sum, a, b) 

288 t = Fmt.PARENSPACED(_not_finite_, s) 

289 raise _OverflowError(u, txt=t) 

290 

291 

292def _threshold(threshold=_0_0, **kwds): 

293 '''(INTERNAL) Get the L{ResidualError}s threshold, 

294 optionally from single kwds C{B{RESIDUAL}=scalar}. 

295 ''' 

296 if kwds: 

297 threshold, kwds = _xkwds_pop2(kwds, RESIDUAL=threshold) 

298# threshold = kwds.pop('RESIDUAL', threshold) 

299 if kwds: 

300 raise _UnexpectedError(**kwds) 

301 try: 

302 return _2finite(threshold) # PYCHOK None 

303 except Exception as x: 

304 raise ResidualError(threshold=threshold, cause=x) 

305 

306 

307class Fsum(_Named): # sync __methods__ with .vector3dBase.Vector3dBase 

308 '''Precision floating point summation and I{running} summation. 

309 

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

311 I{running}, precision floating point summations. Accumulation may continue after any 

312 intermediate, I{running} summuation. 

313 

314 @note: Values may be L{Fsum}, L{Fsum2Tuple}, C{int}, C{float} or C{scalar} instances, 

315 any C{type} having method C{__float__} to convert the C{scalar} to a single 

316 C{float}, except C{complex}. 

317 

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

319 Python's C{math.fsum}. 

320 

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

322 393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}, 

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

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

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

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

327 ''' 

328 _math_fsum = None 

329 _n = 0 

330# _ps = [] # partial sums 

331# _ps_max = 0 # max(Fsum._ps_max, len(Fsum._ps)) 

332 _RESIDUAL = _threshold(_getenv('PYGEODESY_FSUM_RESIDUAL', _0_0)) 

333 

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

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

336 

337 @arg xs: No, one or more initial items to add (each C{scalar} or 

338 an L{Fsum} or L{Fsum2Tuple} instance), all positional. 

339 @kwarg name_RESIDUAL: Optional C{B{name}=NN} (C{str}) for this 

340 L{Fsum} and the C{B{RESIDUAL}=0.0} threshold for 

341 L{ResidualError}s (C{scalar}). 

342 

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

344 ''' 

345 if name_RESIDUAL: 

346 n, kwds = _name2__(**name_RESIDUAL) 

347 if kwds: 

348 R = Fsum._RESIDUAL 

349 t = _threshold(R, **kwds) 

350 if t != R: 

351 self._RESIDUAL = t 

352 if n: 

353 self.name = n 

354 

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

356 if xs: 

357 self._facc_1(xs, up=False) 

358 

359 def __abs__(self): 

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

361 ''' 

362 s = self.signOf() # == self._cmp_0(0) 

363 return (-self) if s < 0 else self._copy_2(self.__abs__) 

364 

365 def __add__(self, other): 

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

367 

368 @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar}. 

369 

370 @return: The sum (L{Fsum}). 

371 

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

373 ''' 

374 f = self._copy_2(self.__add__) 

375 return f._fadd(other, _add_op_) 

376 

377 def __bool__(self): # PYCHOK Python 3+ 

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

379 ''' 

380 s, r = self._fprs2 

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

382 

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

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

385 

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

387 

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

389 ''' 

390 return self.ceil 

391 

392 def __cmp__(self, other): # PYCHOK no cover 

393 '''Compare this with an other instance or C{scalar}, Python 2-. 

394 

395 @return: -1, 0 or +1 (C{int}). 

396 

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

398 ''' 

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

400 return _signOf(s, 0) 

401 

402 def __divmod__(self, other, **raiser_RESIDUAL): 

403 '''Return C{divmod(B{self}, B{other})} as a L{DivMod2Tuple} 

404 with quotient C{div} an C{int} in Python 3+ or C{float} 

405 in Python 2- and remainder C{mod} an L{Fsum} instance. 

406 

407 @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} modulus. 

408 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

409 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

410 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

411 

412 @raise ResidualError: Non-zero, significant residual or invalid 

413 B{C{RESIDUAL}}. 

414 

415 @see: Method L{Fsum.fdiv}. 

416 ''' 

417 f = self._copy_2(self.__divmod__) 

418 return f._fdivmod2(other, _divmod_op_, **raiser_RESIDUAL) 

419 

420 def __eq__(self, other): 

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

422 ''' 

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

424 

425 def __float__(self): 

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

427 

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

429 ''' 

430 return float(self._fprs) 

431 

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

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

434 

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

436 

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

438 ''' 

439 return self.floor 

440 

441 def __floordiv__(self, other): 

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

443 

444 @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} divisor. 

445 

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

447 

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

449 ''' 

450 f = self._copy_2(self.__floordiv__) 

451 return f._floordiv(other, _floordiv_op_) 

452 

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

454 '''Not implemented.''' 

455 return _NotImplemented(self, *other) 

456 

457 def __ge__(self, other): 

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

459 ''' 

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

461 

462 def __gt__(self, other): 

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

464 ''' 

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

466 

467 def __hash__(self): # PYCHOK no cover 

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

469 ''' 

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

471 

472 def __iadd__(self, other): 

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

474 

475 @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} value or 

476 an iterable of several of the former. 

477 

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

479 

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

481 C{scalar} nor L{Fsum}. 

482 

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

484 ''' 

485 try: 

486 return self._fadd(other, _iadd_op_) 

487 except TypeError: 

488 return self._facc_inplace(other, _iadd_op_, self._facc) 

489 

490 def __ifloordiv__(self, other): 

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

492 

493 @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} divisor. 

494 

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

496 

497 @raise ResidualError: Non-zero, significant residual 

498 in B{C{other}}. 

499 

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

501 

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

503 

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

505 

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

507 ''' 

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

509 

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

511 '''Not implemented.''' 

512 return _NotImplemented(self, other) 

513 

514 def __imod__(self, other): 

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

516 

517 @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} modulus. 

518 

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

520 

521 @see: Method L{Fsum.__divmod__}. 

522 ''' 

523 return self._fdivmod2(other, _mod_op_ + _fset_op_).mod 

524 

525 def __imul__(self, other): 

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

527 

528 @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} factor. 

529 

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

531 

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

533 

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

535 

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

537 ''' 

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

539 

540 def __int__(self): 

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

542 

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

544 and L{Fsum.floor}. 

545 ''' 

546 i, _ = self._fint2 

547 return i 

548 

549 def __invert__(self): # PYCHOK no cover 

550 '''Not implemented.''' 

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

552 return _NotImplemented(self) 

553 

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

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

556 

557 @arg other: The exponent (C{scalar}, L{Fsum} or L{Fsum2Tuple}). 

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

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

560 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

561 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

562 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

563 

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

565 

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

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

568 set to C{as_integer} and B{C{mod}} is given and C{None}. 

569 

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

571 

572 @raise ResidualError: Invalid B{C{RESIDUAL}} or the residual 

573 is non-zero and significant and either 

574 B{C{other}} is a fractional or negative 

575 C{scalar} or B{C{mod}} is given and not 

576 C{None}. 

577 

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

579 invocation failed. 

580 

581 @raise ValueError: If B{C{other}} is a negative C{scalar} and this 

582 instance is C{0} or B{C{other}} is a fractional 

583 C{scalar} and this instance is negative or has a 

584 non-zero and significant residual or B{C{mod}} 

585 is given as C{0}. 

586 

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

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

589 ''' 

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

591 

592 def __isub__(self, other): 

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

594 

595 @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} value or 

596 an iterable of several of the former. 

597 

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

599 

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

601 

602 @see: Methods L{Fsum.fsub_} and L{Fsum.fsub}. 

603 ''' 

604 try: 

605 return self._fsub(other, _isub_op_) 

606 except TypeError: 

607 return self._facc_inplace(other, _isub_op_, self._facc_neg) 

608 

609 def __iter__(self): 

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

611 ''' 

612 return iter(self.partials) 

613 

614 def __itruediv__(self, other, **raiser_RESIDUAL): 

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

616 

617 @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} divisor. 

618 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

619 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

620 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

621 

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

623 

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

625 

626 @raise ResidualError: Non-zero, significant residual or invalid 

627 B{C{RESIDUAL}}. 

628 

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

630 

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

632 

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

634 

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

636 ''' 

637 return self._ftruediv(other, _truediv_op_ + _fset_op_, **raiser_RESIDUAL) 

638 

639 def __le__(self, other): 

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

641 ''' 

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

643 

644 def __len__(self): 

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

646 ''' 

647 return self._n 

648 

649 def __lt__(self, other): 

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

651 ''' 

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

653 

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

655 '''Not implemented.''' 

656 return _NotImplemented(self, other) 

657 

658 def __mod__(self, other): 

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

660 

661 @see: Method L{Fsum.__imod__}. 

662 ''' 

663 f = self._copy_2(self.__mod__) 

664 return f._fdivmod2(other, _mod_op_).mod 

665 

666 def __mul__(self, other): 

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

668 

669 @see: Method L{Fsum.__imul__}. 

670 ''' 

671 f = self._copy_2(self.__mul__) 

672 return f._fmul(other, _mul_op_) 

673 

674 def __ne__(self, other): 

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

676 ''' 

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

678 

679 def __neg__(self): 

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

681 ''' 

682 f = self._copy_2(self.__neg__) 

683 return f._fset(self._neg) 

684 

685 def __pos__(self): 

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

687 ''' 

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

689 

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

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

692 

693 @see: Method L{Fsum.__ipow__}. 

694 ''' 

695 f = self._copy_2(self.__pow__) 

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

697 

698 def __radd__(self, other): 

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

700 

701 @see: Method L{Fsum.__iadd__}. 

702 ''' 

703 f = self._copy_2r(other, self.__radd__) 

704 return f._fadd(self, _add_op_) 

705 

706 def __rdivmod__(self, other): 

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

708 C{(quotient, remainder)}. 

709 

710 @see: Method L{Fsum.__divmod__}. 

711 ''' 

712 f = self._copy_2r(other, self.__rdivmod__) 

713 return f._fdivmod2(self, _divmod_op_) 

714 

715# def __repr__(self): 

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

717# ''' 

718# return self.toRepr(lenc=True) 

719 

720 def __rfloordiv__(self, other): 

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

722 

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

724 ''' 

725 f = self._copy_2r(other, self.__rfloordiv__) 

726 return f._floordiv(self, _floordiv_op_) 

727 

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

729 '''Not implemented.''' 

730 return _NotImplemented(self, other) 

731 

732 def __rmod__(self, other): 

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

734 

735 @see: Method L{Fsum.__imod__}. 

736 ''' 

737 f = self._copy_2r(other, self.__rmod__) 

738 return f._fdivmod2(self, _mod_op_).mod 

739 

740 def __rmul__(self, other): 

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

742 

743 @see: Method L{Fsum.__imul__}. 

744 ''' 

745 f = self._copy_2r(other, self.__rmul__) 

746 return f._fmul(self, _mul_op_) 

747 

748 def __round__(self, *ndigits): # PYCHOK Python 3+ 

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

750 

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

752 ''' 

753 f = self._copy_2(self.__round__) 

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

755 return f._fset(round(float(self), *ndigits)) # can be C{int} 

756 

757 def __rpow__(self, other, *mod): 

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

759 

760 @see: Method L{Fsum.__ipow__}. 

761 ''' 

762 f = self._copy_2r(other, self.__rpow__) 

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

764 

765 def __rsub__(self, other): 

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

767 

768 @see: Method L{Fsum.__isub__}. 

769 ''' 

770 f = self._copy_2r(other, self.__rsub__) 

771 return f._fsub(self, _sub_op_) 

772 

773 def __rtruediv__(self, other, **raiser_RESIDUAL): 

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

775 

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

777 ''' 

778 f = self._copy_2r(other, self.__rtruediv__) 

779 return f._ftruediv(self, _truediv_op_, **raiser_RESIDUAL) 

780 

781 def __str__(self): 

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

783 ''' 

784 return self.toStr(lenc=True) 

785 

786 def __sub__(self, other): 

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

788 

789 @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar}. 

790 

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

792 

793 @see: Method L{Fsum.__isub__}. 

794 ''' 

795 f = self._copy_2(self.__sub__) 

796 return f._fsub(other, _sub_op_) 

797 

798 def __truediv__(self, other, **raiser_RESIDUAL): 

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

800 

801 @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} divisor. 

802 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

803 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

804 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

805 

806 @return: The quotient (L{Fsum}). 

807 

808 @raise ResidualError: Non-zero, significant residual or invalid 

809 B{C{RESIDUAL}}. 

810 

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

812 ''' 

813 return self._truediv(other, _truediv_op_, **raiser_RESIDUAL) 

814 

815 __trunc__ = __int__ 

816 

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

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

819 __div__ = __truediv__ 

820 __idiv__ = __itruediv__ 

821 __long__ = __int__ 

822 __nonzero__ = __bool__ 

823 __rdiv__ = __rtruediv__ 

824 

825 def as_integer_ratio(self): 

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

827 

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

829 with C{numerator} signed and C{denominator} 

830 non-zero, positive. 

831 

832 @see: Standard C{float.as_integer_ratio} in Python 2.7+. 

833 ''' 

834 n, r = self._fint2 

835 if r: 

836 i, d = float(r).as_integer_ratio() 

837 n *= d 

838 n += i 

839 else: # PYCHOK no cover 

840 d = 1 

841 return n, d 

842 

843 @property_RO 

844 def as_iscalar(self): 

845 '''Get this instance I{as-is} (L{Fsum} or C{scalar}), the 

846 latter only if the C{residual} equals C{zero}. 

847 ''' 

848 s, r = self._fprs2 

849 return self if r else s 

850 

851 @property_RO 

852 def ceil(self): 

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

854 C{float} in Python 2-). 

855 

856 @note: This C{ceil} takes the C{residual} into account. 

857 

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

859 L{Fsum.imag} and L{Fsum.real}. 

860 ''' 

861 s, r = self._fprs2 

862 c = _ceil(s) + int(r) - 1 

863 while r > (c - s): # (s + r) > c 

864 c += 1 

865 return c # _ceil(self._n_d) 

866 

867 cmp = __cmp__ 

868 

869 def _cmp_0(self, other, op): 

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

871 ''' 

872 if _isFsumTuple(other): 

873 s = self._ps_1sum(*other._ps) 

874 elif self._scalar(other, op): 

875 s = self._ps_1sum(other) 

876 else: 

877 s = self.signOf() # res=True 

878 return s 

879 

880 def copy(self, deep=False, **name): 

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

882 

883 @kwarg name: Optional, overriding C{B{name}='"copy"} (C{str}). 

884 

885 @return: The copy (L{Fsum}). 

886 ''' 

887 n = _name__(name, name__=self.copy) 

888 f = _Named.copy(self, deep=deep, name=n) 

889 if f._ps is self._ps: 

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

891 if not deep: 

892 f._n = 1 

893 # assert f._Fsum is f 

894 return f 

895 

896 def _copy_2(self, which, name=NN): 

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

898 ''' 

899 n = name or which.__name__ # _dunder_nameof 

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

901 f = _Named.copy(self, deep=False, name=n) 

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

903 # assert f._n == self._n 

904 # assert f._Fsum is f 

905 return f 

906 

907 def _copy_2r(self, other, which): 

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

909 ''' 

910 return other._copy_2(which) if _isFsum(other) else \ 

911 self._copy_2(which)._fset(other) 

912 

913# def _copy_RESIDUAL(self, other): 

914# '''(INTERNAL) Copy C{other._RESIDUAL}. 

915# ''' 

916# R = other._RESIDUAL 

917# if R is not Fsum._RESIDUAL: 

918# self._RESIDUAL = R 

919 

920 divmod = __divmod__ 

921 

922 def _Error(self, op, other, Error, **txt_cause): 

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

924 ''' 

925 return Error(_SPACE_(self.as_iscalar, op, other), **txt_cause) 

926 

927 def _ErrorX(self, X, op, other, *mod): 

928 '''(INTERNAL) Format the caught exception C{X}. 

929 ''' 

930 E, t = _xError2(X) 

931 if mod: 

932 t = _COMMASPACE_(Fmt.PARENSPACED(mod=mod[0]), t) 

933 return self._Error(op, other, E, txt=t, cause=X) 

934 

935 def _ErrorXs(self, X, xs, **kwds): # in .fmath 

936 '''(INTERNAL) Format the caught exception C{X}. 

937 ''' 

938 E, t = _xError2(X) 

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

940 return E(u, txt=t, cause=X) 

941 

942 def _facc(self, xs, up=True, **origin_X_x): 

943 '''(INTERNAL) Accumulate more C{scalars} or L{Fsum}s. 

944 ''' 

945 if xs: 

946 _xs = _2floats(xs, **origin_X_x) # PYCHOK yield 

947 ps = self._ps 

948 ps[:] = self._ps_acc(list(ps), _xs, up=up) 

949 return self 

950 

951 def _facc_1(self, xs, **up): 

952 '''(INTERNAL) Accumulate 0, 1 or more C{scalars} or L{Fsum}s, 

953 all positional C{xs} in the caller of this method. 

954 ''' 

955 return self._fadd(xs[0], _add_op_, **up) if len(xs) == 1 else \ 

956 self._facc(xs, origin=1, **up) 

957 

958 def _facc_inplace(self, other, op, _facc): 

959 '''(INTERNAL) Accumulate from an iterable. 

960 ''' 

961 try: 

962 return _facc(other, origin=1) if _xiterable(other) else self 

963 except Exception as X: 

964 raise self._ErrorX(X, op, other) 

965 

966 def _facc_neg(self, xs, **up_origin): 

967 '''(INTERNAL) Accumulate more C{scalars} or L{Fsum}s, negated. 

968 ''' 

969 def _N(X): 

970 return X._ps_neg 

971 

972 def _n(x): 

973 return -float(x) 

974 

975 return self._facc(xs, _X=_N, _x=_n, **up_origin) 

976 

977 def _facc_power(self, power, xs, which, **raiser_RESIDUAL): # in .fmath 

978 '''(INTERNAL) Add each C{xs} as C{float(x**power)}. 

979 ''' 

980 def _Pow4(p): 

981 r = 0 

982 if _isFsumTuple(p): 

983 s, r = p._fprs2 

984 if r: 

985 m = Fsum._pow 

986 else: # scalar 

987 return _Pow4(s) 

988 elif isint(p, both=True) and int(p) >= 0: 

989 p = s = int(p) 

990 m = Fsum._pow_int 

991 else: 

992 p = s = _2float(power=p) 

993 m = Fsum._pow_scalar 

994 return m, p, s, r 

995 

996 _Pow, p, s, r = _Pow4(power) 

997 if p: # and xs: 

998 op = which.__name__ 

999 _flt = float 

1000 _Fs = Fsum 

1001 _isa = isinstance 

1002 _pow = self._pow_2_3 

1003 

1004 def _P(X): 

1005 f = _Pow(X, p, power, op, **raiser_RESIDUAL) 

1006 return f._ps if _isa(f, _Fs) else (f,) 

1007 

1008 def _p(x): 

1009 x = _flt(x) 

1010 f = _pow(x, s, power, op, **raiser_RESIDUAL) 

1011 if f and r: 

1012 f *= _pow(x, r, power, op, **raiser_RESIDUAL) 

1013 return f 

1014 

1015 f = self._facc(xs, origin=1, _X=_P, _x=_p) 

1016 else: 

1017 f = self._facc_scalar_(float(len(xs))) # x**0 == 1 

1018 return f 

1019 

1020 def _facc_scalar(self, xs, **up): 

1021 '''(INTERNAL) Accumulate all C{xs}, known to be scalar. 

1022 ''' 

1023 if xs: 

1024 _ = self._ps_acc(self._ps, xs, **up) 

1025 return self 

1026 

1027 def _facc_scalar_(self, *xs, **up): 

1028 '''(INTERNAL) Accumulate all positional C{xs}, known to be scalar. 

1029 ''' 

1030 if xs: 

1031 _ = self._ps_acc(self._ps, xs, **up) 

1032 return self 

1033 

1034# def _facc_up(self, up=True): 

1035# '''(INTERNAL) Update the C{partials}, by removing 

1036# and re-accumulating the final C{partial}. 

1037# ''' 

1038# ps = self._ps 

1039# while len(ps) > 1: 

1040# p = ps.pop() 

1041# if p: 

1042# n = self._n 

1043# _ = self._ps_acc(ps, (p,), up=False) 

1044# self._n = n 

1045# break 

1046# return self._update() if up else self 

1047 

1048 def fadd(self, xs=()): 

1049 '''Add an iterable's items to this instance. 

1050 

1051 @arg xs: Iterable of items to add (each C{scalar} 

1052 or an L{Fsum} or L{Fsum2Tuple} instance). 

1053 

1054 @return: This instance (L{Fsum}). 

1055 

1056 @raise OverflowError: Partial C{2sum} overflow. 

1057 

1058 @raise TypeError: An invalid B{C{xs}} item. 

1059 

1060 @raise ValueError: Invalid or non-finite B{C{xs}} value. 

1061 ''' 

1062 if _isFsumTuple(xs): 

1063 self._facc_scalar(xs._ps) 

1064 elif isscalar(xs): # for backward compatibility 

1065 self._facc_scalar_(_2float(x=xs)) # PYCHOK no cover 

1066 elif xs: # _xiterable(xs) 

1067 self._facc(xs) 

1068 return self 

1069 

1070 def fadd_(self, *xs): 

1071 '''Add all positional items to this instance. 

1072 

1073 @arg xs: Values to add (each C{scalar} or an L{Fsum} 

1074 or L{Fsum2Tuple} instance), all positional. 

1075 

1076 @see: Method L{Fsum.fadd} for further details. 

1077 ''' 

1078 return self._facc_1(xs) 

1079 

1080 def _fadd(self, other, op, **up): # in .fmath.Fhorner 

1081 '''(INTERNAL) Apply C{B{self} += B{other}}. 

1082 ''' 

1083 if not self._ps: # new Fsum(x) 

1084 self._fset(other, op=op, **up) 

1085 elif _isFsumTuple(other): 

1086 self._facc_scalar(other._ps, **up) 

1087 elif self._scalar(other, op): 

1088 self._facc_scalar_(other, **up) 

1089 return self 

1090 

1091 fcopy = copy # for backward compatibility 

1092 fdiv = __itruediv__ 

1093 fdivmod = __divmod__ 

1094 

1095 def _fdivmod2(self, other, op, **raiser_RESIDUAL): 

1096 '''(INTERNAL) Apply C{B{self} %= B{other}} and return a L{DivMod2Tuple}. 

1097 ''' 

1098 # result mostly follows CPython function U{float_divmod 

1099 # <https://GitHub.com/python/cpython/blob/main/Objects/floatobject.c>}, 

1100 # but at least divmod(-3, 2) equals Cpython's result (-2, 1). 

1101 q = self._truediv(other, op, **raiser_RESIDUAL).floor 

1102 if q: # == float // other == floor(float / other) 

1103 self -= Fsum(q) * other # NOT other * q! 

1104 

1105 s = signOf(other) # make signOf(self) == signOf(other) 

1106 if s and self.signOf() == -s: # PYCHOK no cover 

1107 self += other 

1108 q -= 1 

1109# t = self.signOf() 

1110# if t and t != s: 

1111# raise self._Error(op, other, _AssertionError, txt__=signOf) 

1112 return DivMod2Tuple(q, self) # q is C{int} in Python 3+, but C{float} in Python 2- 

1113 

1114 def _fhorner(self, x, cs, op): # in .fmath 

1115 '''(INTERNAL) Add an L{Fhorner} evaluation of polynomial 

1116 M{sum(cs[i] * x**i for i=0..len(cs)-1)}. 

1117 ''' 

1118 if _xiterablen(cs): 

1119 H = Fsum(name__=self._fhorner) 

1120 if _isFsumTuple(x): 

1121 _mul = H._mul_Fsum 

1122 else: 

1123 _mul = H._mul_scalar 

1124 x = _2float(x=x) 

1125 if len(cs) > 1 and x: 

1126 for c in reversed(cs): 

1127 H._fset_ps(_mul(x, op)) 

1128 H._fadd(c, op, up=False) 

1129 else: # x == 0 

1130 H = cs[0] 

1131 self._fadd(H, op) 

1132 return self 

1133 

1134 def _finite(self, other, op=None): 

1135 '''(INTERNAL) Return B{C{other}} if C{finite}. 

1136 ''' 

1137 if _isfinite(other): 

1138 return other 

1139 raise ValueError(_not_finite_) if op is None else \ 

1140 self._Error(op, other, _ValueError, txt=_not_finite_) 

1141 

1142 def fint(self, name=NN, **raiser_RESIDUAL): 

1143 '''Return this instance' current running sum as C{integer}. 

1144 

1145 @kwarg name: Optional, overriding C{B{name}="fint"} (C{str}). 

1146 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1147 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1148 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1149 

1150 @return: The C{integer} sum (L{Fsum}) if this instance C{is_integer} 

1151 with a zero or insignificant I{integer} residual. 

1152 

1153 @raise ResidualError: Non-zero, significant residual or invalid 

1154 B{C{RESIDUAL}}. 

1155 

1156 @see: Methods L{Fsum.fint2}, L{Fsum.int_float} and L{Fsum.is_integer}. 

1157 ''' 

1158 i, r = self._fint2 

1159 if r: 

1160 R = self._raiser(r, i, **raiser_RESIDUAL) 

1161 if R: 

1162 t = _stresidual(_integer_, r, **R) 

1163 raise ResidualError(_integer_, i, txt=t) 

1164 return _Psum_(i, name=_name__(name, name__=self.fint)) 

1165 

1166 def fint2(self, **name): 

1167 '''Return this instance' current running sum as C{int} and the 

1168 I{integer} residual. 

1169 

1170 @kwarg name: Optional name (C{str}). 

1171 

1172 @return: An L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} 

1173 an C{int} and I{integer} C{residual} a C{float} or 

1174 C{INT0} if the C{fsum} is considered to be I{exact}. 

1175 ''' 

1176 return Fsum2Tuple(*self._fint2, **name) 

1177 

1178 @Property 

1179 def _fint2(self): # see ._fset 

1180 '''(INTERNAL) Get 2-tuple (C{int}, I{integer} residual). 

1181 ''' 

1182 s, r = self._fprs2 

1183 i = int(s) 

1184 n = len(self._ps) 

1185 r = self._ps_1sum(i) if r and n > 1 else float(s - i) 

1186 return i, (r or INT0) # Fsum2Tuple? 

1187 

1188 @_fint2.setter_ # PYCHOK setter_underscore! 

1189 def _fint2(self, s): 

1190 '''(INTERNAL) Replace the C{_fint2} value. 

1191 ''' 

1192 i = int(s) 

1193 return i, ((s - i) or INT0) 

1194 

1195 @deprecated_property_RO 

1196 def float_int(self): # PYCHOK no cover 

1197 '''DEPRECATED, use method C{Fsum.int_float}.''' 

1198 return self.int_float() # raiser=False 

1199 

1200 @property_RO 

1201 def floor(self): 

1202 '''Get this instance' C{floor} (C{int} in Python 3+, but 

1203 C{float} in Python 2-). 

1204 

1205 @note: This C{floor} takes the C{residual} into account. 

1206 

1207 @see: Method L{Fsum.int_float} and properties L{Fsum.ceil}, 

1208 L{Fsum.imag} and L{Fsum.real}. 

1209 ''' 

1210 s, r = self._fprs2 

1211 f = _floor(s) + _floor(r) + 1 

1212 while (f - s) > r: # f > (s + r) 

1213 f -= 1 

1214 return f # _floor(self._n_d) 

1215 

1216# ffloordiv = __ifloordiv__ # for naming consistency 

1217# floordiv = __floordiv__ # for naming consistency 

1218 

1219 def _floordiv(self, other, op, **raiser_RESIDUAL): # rather _ffloordiv? 

1220 '''Apply C{B{self} //= B{other}}. 

1221 ''' 

1222 q = self._ftruediv(other, op, **raiser_RESIDUAL) # == self 

1223 return self._fset(q.floor) # floor(q) 

1224 

1225 fmul = __imul__ 

1226 

1227 def _fmul(self, other, op): 

1228 '''(INTERNAL) Apply C{B{self} *= B{other}}. 

1229 ''' 

1230 if _isFsumTuple(other): 

1231 if len(self._ps) != 1: 

1232 f = self._mul_Fsum(other, op) 

1233 elif len(other._ps) != 1: # and len(self._ps) == 1 

1234 f = other._mul_scalar(self._ps[0], op) 

1235 else: # len(other._ps) == len(self._ps) == 1 

1236 f = self._finite(self._ps[0] * other._ps[0]) 

1237 else: 

1238 s = self._scalar(other, op) 

1239 f = self._mul_scalar(s, op) 

1240 return self._fset(f) # n=len(self) + 1 

1241 

1242 def fover(self, over, **raiser_RESIDUAL): 

1243 '''Apply C{B{self} /= B{over}} and summate. 

1244 

1245 @arg over: An L{Fsum} or C{scalar} denominator. 

1246 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1247 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1248 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1249 

1250 @return: Precision running sum (C{float}). 

1251 

1252 @raise ResidualError: Non-zero, significant residual or invalid 

1253 B{C{RESIDUAL}}. 

1254 

1255 @see: Methods L{Fsum.fsum} and L{Fsum.__itruediv__}. 

1256 ''' 

1257 return float(self.fdiv(over, **raiser_RESIDUAL)._fprs) 

1258 

1259 fpow = __ipow__ 

1260 

1261 def _fpow(self, other, op, *mod, **raiser_RESIDUAL): 

1262 '''Apply C{B{self} **= B{other}}, optional B{C{mod}} or C{None}. 

1263 ''' 

1264 if mod: 

1265 if mod[0] is not None: # == 3-arg C{pow} 

1266 f = self._pow_2_3(self, other, other, op, *mod, **raiser_RESIDUAL) 

1267 elif self.is_integer(): 

1268 # return an exact C{int} for C{int}**C{int} 

1269 i, _ = self._fint2 # assert _ == 0 

1270 x, r = _2scalar2(other) # C{int}, C{float} or other 

1271 f = _Psum_(i)._pow_Fsum(other, op, **raiser_RESIDUAL) if r else \ 

1272 self._pow_2_3(i, x, other, op, **raiser_RESIDUAL) 

1273 else: # mod[0] is None, power(self, other) 

1274 f = self._pow(other, other, op, **raiser_RESIDUAL) 

1275 else: # pow(self, other) 

1276 f = self._pow(other, other, op, **raiser_RESIDUAL) 

1277 return self._fset(f) # n=max(len(self), 1) 

1278 

1279 @Property 

1280 def _fprs(self): 

1281 '''(INTERNAL) Get and cache this instance' precision 

1282 running sum (C{float} or C{int}), ignoring C{residual}. 

1283 

1284 @note: The precision running C{fsum} after a C{//=} or 

1285 C{//} C{floor} division is C{int} in Python 3+. 

1286 ''' 

1287 s, _ = self._fprs2 

1288 return s # ._fprs2.fsum 

1289 

1290 @_fprs.setter_ # PYCHOK setter_underscore! 

1291 def _fprs(self, s): 

1292 '''(INTERNAL) Replace the C{_fprs} value. 

1293 ''' 

1294 return s 

1295 

1296 @Property 

1297 def _fprs2(self): 

1298 '''(INTERNAL) Get and cache this instance' precision 

1299 running sum and residual (L{Fsum2Tuple}). 

1300 ''' 

1301 ps = self._ps 

1302 n = len(ps) - 2 

1303 if n > 0: # len(ps) > 2 

1304 s = _psum(ps) 

1305 n = len(ps) - 2 

1306 if n > 0: 

1307 r = self._ps_1sum(s) 

1308 return Fsum2Tuple(*_s_r(s, r)) 

1309 if n == 0: # len(ps) == 2 

1310 s, r = _s_r(*_2sum(*ps)) 

1311 ps[:] = (r, s) if r else (s,) 

1312 elif ps: # len(ps) == 1 

1313 s, r = ps[0], INT0 

1314 else: # len(ps) == 0 

1315 s, r = _0_0, INT0 

1316 ps[:] = s, 

1317 # assert self._ps is ps 

1318 return Fsum2Tuple(s, r) 

1319 

1320 @_fprs2.setter_ # PYCHOK setter_underscore! 

1321 def _fprs2(self, s_r): 

1322 '''(INTERNAL) Replace the C{_fprs2} value. 

1323 ''' 

1324 return Fsum2Tuple(s_r) 

1325 

1326 def fset_(self, *xs): 

1327 '''Replace this instance' value with all positional items. 

1328 

1329 @arg xs: Optional, new values (each C{scalar} or 

1330 an L{Fsum} or L{Fsum2Tuple} instance), 

1331 all positional. 

1332 

1333 @return: This instance, replaced (C{Fsum}). 

1334 

1335 @see: Method L{Fsum.fadd} for further details. 

1336 ''' 

1337 f = xs[0] if len(xs) == 1 else ( 

1338 Fsum(*xs) if xs else _0_0) 

1339 return self._fset(f) 

1340 

1341 def _fset(self, other, n=0, up=True, **op): 

1342 '''(INTERNAL) Overwrite this instance with an other or a C{scalar}. 

1343 ''' 

1344 if other is self: 

1345 pass # from ._fmul, ._ftruediv and ._pow_0_1 

1346 elif _isFsumTuple(other): 

1347 self._ps[:] = other._ps 

1348 self._n = n or other._n 

1349# self._copy_RESIDUAL(other) 

1350 if up: # use or zap the C{Property_RO} values 

1351 Fsum._fint2._update_from(self, other) 

1352 Fsum._fprs ._update_from(self, other) 

1353 Fsum._fprs2._update_from(self, other) 

1354 elif isscalar(other): 

1355 s = float(self._finite(other, **op)) if op else other 

1356 self._ps[:] = s, 

1357 self._n = n or 1 

1358 if up: # Property _fint2, _fprs and _fprs2 all have 

1359 # @.setter_underscore and NOT @.setter because the 

1360 # latter's _fset zaps the value set by @.setter 

1361 self._fint2 = s 

1362 self._fprs = s 

1363 self._fprs2 = s, INT0 

1364 # assert self._fprs is s 

1365 else: # PYCHOK no cover 

1366 op = _xkwds_get1(op, op=_fset_op_) 

1367 raise self._Error(op, other, _TypeError) 

1368 return self 

1369 

1370 def _fset_ps(self, other): # in .fmath 

1371 '''(INTERNAL) Set partials from a known C{scalar}, L{Fsum} or L{Fsum2Tuple}. 

1372 ''' 

1373 return self._fset(other, up=False) 

1374 

1375 def fsub(self, xs=()): 

1376 '''Subtract an iterable's items from this instance. 

1377 

1378 @see: Method L{Fsum.fadd} for further details. 

1379 ''' 

1380 return self._facc_neg(xs) 

1381 

1382 def fsub_(self, *xs): 

1383 '''Subtract all positional items from this instance. 

1384 

1385 @see: Method L{Fsum.fadd_} for further details. 

1386 ''' 

1387 return self._fsub(xs[0], _sub_op_) if len(xs) == 1 else \ 

1388 self._facc_neg(xs, origin=1) 

1389 

1390 def _fsub(self, other, op): 

1391 '''(INTERNAL) Apply C{B{self} -= B{other}}. 

1392 ''' 

1393 if _isFsumTuple(other): 

1394 if other is self: # or other._fprs2 == self._fprs2: 

1395 self._fset(_0_0, n=len(self) * 2) 

1396 elif other._ps: 

1397 self._facc_scalar(other._ps_neg) 

1398 elif self._scalar(other, op): 

1399 self._facc_scalar_(-other) 

1400 return self 

1401 

1402 def fsum(self, xs=()): 

1403 '''Add an iterable's items, summate and return the 

1404 current precision running sum. 

1405 

1406 @arg xs: Iterable of items to add (each item C{scalar} 

1407 or an L{Fsum} or L{Fsum2Tuple} instance). 

1408 

1409 @return: Precision running sum (C{float} or C{int}). 

1410 

1411 @see: Method L{Fsum.fadd}. 

1412 

1413 @note: Accumulation can continue after summation. 

1414 ''' 

1415 return self._facc(xs)._fprs 

1416 

1417 def fsum_(self, *xs): 

1418 '''Add any positional items, summate and return the 

1419 current precision running sum. 

1420 

1421 @arg xs: Items to add (each C{scalar} or an L{Fsum} 

1422 or L{Fsum2Tuple} instance), all positional. 

1423 

1424 @return: Precision running sum (C{float} or C{int}). 

1425 

1426 @see: Methods L{Fsum.fsum}, L{Fsum.Fsum_} and L{Fsum.fsumf_}. 

1427 ''' 

1428 return self._facc_1(xs)._fprs 

1429 

1430 @property_RO 

1431 def _Fsum(self): # like L{Fsum2Tuple._Fsum}, for C{_2floats}, .fstats 

1432 return self # NOT @Property_RO, see .copy and ._copy_2 

1433 

1434 def Fsum_(self, *xs, **name): 

1435 '''Like method L{Fsum.fsum_} but returning a named L{Fsum}. 

1436 

1437 @kwarg name: Optional name (C{str}). 

1438 

1439 @return: Copy of this updated instance (L{Fsum}). 

1440 ''' 

1441 return self._facc_1(xs)._copy_2(self.Fsum_, **name) 

1442 

1443 def Fsum2Tuple_(self, *xs, **name): 

1444 '''Like method L{Fsum.fsum_} but returning a named L{Fsum2Tuple}. 

1445 

1446 @kwarg name: Optional name (C{str}). 

1447 

1448 @return: Precision running sum (L{Fsum2Tuple}). 

1449 ''' 

1450 return Fsum2Tuple(self._facc_1(xs)._fprs2, **name) 

1451 

1452 def fsum2(self, xs=(), **name): 

1453 '''Add an iterable's items, summate and return the 

1454 current precision running sum I{and} the C{residual}. 

1455 

1456 @arg xs: Iterable of items to add (each item C{scalar} 

1457 or an L{Fsum} or L{Fsum2Tuple} instance). 

1458 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1459 

1460 @return: L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} the 

1461 current precision running sum and C{residual}, the 

1462 (precision) sum of the remaining C{partials}. The 

1463 C{residual is INT0} if the C{fsum} is considered 

1464 to be I{exact}. 

1465 

1466 @see: Methods L{Fsum.fint2}, L{Fsum.fsum} and L{Fsum.fsum2_} 

1467 ''' 

1468 t = self._facc(xs)._fprs2 

1469 return t.dup(name=name) if name else t 

1470 

1471 def fsum2_(self, *xs): 

1472 '''Add any positional items, summate and return the current 

1473 precision running sum and the I{differential}. 

1474 

1475 @arg xs: Values to add (each C{scalar} or an L{Fsum} or 

1476 L{Fsum2Tuple} instance), all positional. 

1477 

1478 @return: 2Tuple C{(fsum, delta)} with the current, precision 

1479 running C{fsum} like method L{Fsum.fsum} and C{delta}, 

1480 the difference with previous running C{fsum}, C{float}. 

1481 

1482 @see: Methods L{Fsum.fsum_} and L{Fsum.fsum}. 

1483 ''' 

1484 return self._fsum2(xs, self._facc_1) 

1485 

1486 def _fsum2(self, xs, _facc, **origin): 

1487 '''(INTERNAL) Helper for L{Fsum.fsum2_} and L{Fsum.fsum2f_}. 

1488 ''' 

1489 p, q = self._fprs2 

1490 if xs: 

1491 s, r = _facc(xs, **origin)._fprs2 

1492 return s, _2delta(s - p, r - q) # _fsum(_1primed((s, -p, r, -q)) 

1493 else: 

1494 return p, _0_0 

1495 

1496 def fsumf_(self, *xs): 

1497 '''Like method L{Fsum.fsum_} iff I{all} C{B{xs}} are I{known to be scalar}. 

1498 ''' 

1499 return self._facc_scalar(xs)._fprs 

1500 

1501 def Fsumf_(self, *xs): 

1502 '''Like method L{Fsum.Fsum_} iff I{all} C{B{xs}} are I{known to be scalar}. 

1503 ''' 

1504 return self._facc_scalar(xs)._copy_2(self.Fsumf_) 

1505 

1506 def fsum2f_(self, *xs): 

1507 '''Like method L{Fsum.fsum2_} iff I{all} C{B{xs}} are I{known to be scalar}. 

1508 ''' 

1509 return self._fsum2(xs, self._facc_scalar, origin=1) 

1510 

1511# ftruediv = __itruediv__ # for naming consistency? 

1512 

1513 def _ftruediv(self, other, op, **raiser_RESIDUAL): 

1514 '''(INTERNAL) Apply C{B{self} /= B{other}}. 

1515 ''' 

1516 n = _1_0 

1517 if _isFsumTuple(other): 

1518 if other is self or self == other: 

1519 return self._fset(n, n=len(self)) 

1520 d, r = other._fprs2 

1521 if r: 

1522 R = self._raiser(r, d, **raiser_RESIDUAL) 

1523 if R: 

1524 raise self._ResidualError(op, other, r, **R) 

1525 d, n = other.as_integer_ratio() 

1526 else: 

1527 d = self._scalar(other, op) 

1528 try: 

1529 s = n / d 

1530 except Exception as X: 

1531 raise self._ErrorX(X, op, other) 

1532 f = self._mul_scalar(s, _mul_op_) # handles 0, INF, NAN 

1533 return self._fset(f) 

1534 

1535 @property_RO 

1536 def imag(self): 

1537 '''Get the C{imaginary} part of this instance (C{0.0}, always). 

1538 

1539 @see: Property L{Fsum.real}. 

1540 ''' 

1541 return _0_0 

1542 

1543 def int_float(self, **raiser_RESIDUAL): 

1544 '''Return this instance' current running sum as C{int} or C{float}. 

1545 

1546 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1547 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1548 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1549 

1550 @return: This C{integer} sum if this instance C{is_integer}, 

1551 otherwise return the C{float} sum if the residual is 

1552 zero or not significant. 

1553 

1554 @raise ResidualError: Non-zero, significant residual or invalid 

1555 B{C{RESIDUAL}}. 

1556 

1557 @see: Methods L{Fsum.fint}, L{Fsum.fint2}, L{Fsum.RESIDUAL} and 

1558 property L{Fsum.as_iscalar}. 

1559 ''' 

1560 s, r = self._fint2 

1561 if r: 

1562 s, r = self._fprs2 

1563 if r: # PYCHOK no cover 

1564 R = self._raiser(r, s, **raiser_RESIDUAL) 

1565 if R: 

1566 t = _stresidual(_non_zero_, r, **R) 

1567 raise ResidualError(int_float=s, txt=t) 

1568 s = float(s) 

1569 return s 

1570 

1571 def is_exact(self): 

1572 '''Is this instance' running C{fsum} considered to be exact? 

1573 (C{bool}), C{True} only if the C{residual is }L{INT0}. 

1574 ''' 

1575 return self.residual is INT0 

1576 

1577 def is_integer(self): 

1578 '''Is this instance' running sum C{integer}? (C{bool}). 

1579 

1580 @see: Methods L{Fsum.fint}, L{Fsum.fint2} and L{Fsum.is_scalar}. 

1581 ''' 

1582 _, r = self._fint2 

1583 return False if r else True 

1584 

1585 def is_math_fsum(self): 

1586 '''Return whether functions L{fsum}, L{fsum_}, L{fsum1} and 

1587 L{fsum1_} plus partials summation are based on Python's 

1588 C{math.fsum} or not. 

1589 

1590 @return: C{2} if all functions and partials summation 

1591 are based on C{math.fsum}, C{True} if only 

1592 the functions are based on C{math.fsum} (and 

1593 partials summation is not) or C{False} if 

1594 none are. 

1595 ''' 

1596 f = Fsum._math_fsum 

1597 return 2 if _psum is f else bool(f) 

1598 

1599 def is_scalar(self, **raiser_RESIDUAL): 

1600 '''Is this instance' running sum C{scalar} without residual or with 

1601 a residual I{ratio} not exceeding the RESIDUAL threshold? 

1602 

1603 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1604 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1605 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1606 

1607 @return: C{True} if this instance' non-zero residual C{ratio} exceeds 

1608 the L{RESIDUAL<Fsum.RESIDUAL>} threshold (C{bool}). 

1609 

1610 @raise ResidualError: Non-zero, significant residual or invalid 

1611 B{C{RESIDUAL}}. 

1612 

1613 @see: Method L{Fsum.RESIDUAL}, L{Fsum.is_integer} and property 

1614 L{Fsum.as_iscalar}. 

1615 ''' 

1616 s, r = self._fprs2 

1617 return False if r and self._raiser(r, s, **raiser_RESIDUAL) else True 

1618 

1619 def _mul_Fsum(self, other, op=_mul_op_): # in .fmath.Fhorner 

1620 '''(INTERNAL) Return C{B{self} * B{other}} as L{Fsum} or C{0}. 

1621 ''' 

1622 # assert _isFsumTuple(other) 

1623 if self._ps and other._ps: 

1624 f = self._ps_mul(op, *other._ps) # NO .as_iscalar! 

1625 else: 

1626 f = _0_0 

1627 return f 

1628 

1629 def _mul_scalar(self, factor, op): # in .fmath.Fhorner 

1630 '''(INTERNAL) Return C{B{self} * scalar B{factor}} as L{Fsum}, C{0.0} or C{self}. 

1631 ''' 

1632 # assert isscalar(factor) 

1633 if self._ps and self._finite(factor, op): 

1634 f = self if factor == _1_0 else ( 

1635 self._neg if factor == _N_1_0 else 

1636 self._ps_mul(op, factor).as_iscalar) 

1637 else: 

1638 f = _0_0 

1639 return f 

1640 

1641# @property_RO 

1642# def _n_d(self): 

1643# n, d = self.as_integer_ratio() 

1644# return n / d 

1645 

1646 @property_RO 

1647 def _neg(self): 

1648 '''(INTERNAL) Return C{Fsum(-self)} or scalar C{NEG0}. 

1649 ''' 

1650 return _Psum(self._ps_neg) if self._ps else NEG0 

1651 

1652 @property_RO 

1653 def partials(self): 

1654 '''Get this instance' current, partial sums (C{tuple} of C{float}s). 

1655 ''' 

1656 return tuple(self._ps) 

1657 

1658 def pow(self, x, *mod, **raiser_RESIDUAL): 

1659 '''Return C{B{self}**B{x}} as L{Fsum}. 

1660 

1661 @arg x: The exponent (C{scalar} or L{Fsum}). 

1662 @arg mod: Optional modulus (C{int} or C{None}) for the 3-argument 

1663 C{pow(B{self}, B{other}, B{mod})} version. 

1664 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1665 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1666 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1667 

1668 @return: The C{pow(self, B{x})} or C{pow(self, B{x}, *B{mod})} 

1669 result (L{Fsum}). 

1670 

1671 @raise ResidualError: Non-zero, significant residual or invalid 

1672 B{C{RESIDUAL}}. 

1673 

1674 @note: If B{C{mod}} is given and C{None}, the result will be an 

1675 C{integer} L{Fsum} provided this instance C{is_integer} 

1676 or set to C{integer} by an L{Fsum.fint} call. 

1677 

1678 @see: Methods L{Fsum.__ipow__}, L{Fsum.fint}, L{Fsum.is_integer} 

1679 and L{Fsum.root}. 

1680 ''' 

1681 f = self._copy_2(self.pow) 

1682 return f._fpow(x, _pow_op_, *mod, **raiser_RESIDUAL) # f = pow(f, x, *mod) 

1683 

1684 def _pow(self, other, unused, op, **raiser_RESIDUAL): 

1685 '''Return C{B{self} ** B{other}}. 

1686 ''' 

1687 if _isFsumTuple(other): 

1688 f = self._pow_Fsum(other, op, **raiser_RESIDUAL) 

1689 elif self._scalar(other, op): 

1690 x = self._finite(other, op) 

1691 f = self._pow_scalar(x, other, op, **raiser_RESIDUAL) 

1692 else: 

1693 f = self._pow_0_1(0, other) 

1694 return f 

1695 

1696 def _pow_0_1(self, x, other): 

1697 '''(INTERNAL) Return B{C{self}**1} or C{B{self}**0 == 1.0}. 

1698 ''' 

1699 return self if x else (1 if isint(other) and self.is_integer() else _1_0) 

1700 

1701 def _pow_2_3(self, b, x, other, op, *mod, **raiser_RESIDUAL): 

1702 '''(INTERNAL) 2-arg C{pow(B{b}, scalar B{x})} and 3-arg C{pow(B{b}, 

1703 B{x}, int B{mod} or C{None})}, embellishing errors. 

1704 ''' 

1705 

1706 if mod: # b, x, mod all C{int}, unless C{mod} is C{None} 

1707 m = mod[0] 

1708 # assert _isFsumTuple(b) 

1709 

1710 def _s(s, r): 

1711 R = self._raiser(r, s, **raiser_RESIDUAL) 

1712 if R: 

1713 raise self._ResidualError(op, other, r, mod=m, **R) 

1714 return s 

1715 

1716 b = _s(*(b._fprs2 if m is None else b._fint2)) 

1717 x = _s(*_2scalar2(x)) 

1718 

1719 try: 

1720 # 0**INF == 0.0, 1**INF == 1.0, -1**2.3 == -(1**2.3) 

1721 s = pow(b, x, *mod) 

1722 if iscomplex(s): 

1723 # neg**frac == complex in Python 3+, but ValueError in 2- 

1724 raise ValueError(_strcomplex(s, b, x, *mod)) 

1725 return self._finite(s) 

1726 except Exception as X: 

1727 raise self._ErrorX(X, op, other, *mod) 

1728 

1729 def _pow_Fsum(self, other, op, **raiser_RESIDUAL): 

1730 '''(INTERNAL) Return C{B{self} **= B{other}} for C{_isFsumTuple(other)}. 

1731 ''' 

1732 # assert _isFsumTuple(other) 

1733 x, r = other._fprs2 

1734 f = self._pow_scalar(x, other, op, **raiser_RESIDUAL) 

1735 if f and r: 

1736 f *= self._pow_scalar(r, other, op, **raiser_RESIDUAL) 

1737 return f 

1738 

1739 def _pow_int(self, x, other, op, **raiser_RESIDUAL): 

1740 '''(INTERNAL) Return C{B{self} **= B{x}} for C{int B{x} >= 0}. 

1741 ''' 

1742 # assert isint(x) and x >= 0 

1743 ps = self._ps 

1744 if len(ps) > 1: 

1745 _mul_Fsum = Fsum._mul_Fsum 

1746 if x > 4: 

1747 p = self 

1748 f = self if (x & 1) else _Psum_(_1_0) 

1749 m = x >> 1 # // 2 

1750 while m: 

1751 p = _mul_Fsum(p, p, op) # p **= 2 

1752 if (m & 1): 

1753 f = _mul_Fsum(f, p, op) # f *= p 

1754 m >>= 1 # //= 2 

1755 elif x > 1: # self**2, 3, or 4 

1756 f = _mul_Fsum(self, self, op) 

1757 if x > 2: # self**3 or 4 

1758 p = self if x < 4 else f 

1759 f = _mul_Fsum(f, p, op) 

1760 else: # self**1 or self**0 == 1 or _1_0 

1761 f = self._pow_0_1(x, other) 

1762 elif ps: # self._ps[0]**x 

1763 f = self._pow_2_3(ps[0], x, other, op, **raiser_RESIDUAL) 

1764 else: # PYCHOK no cover 

1765 # 0**pos_int == 0, but 0**0 == 1 

1766 f = 0 if x else 1 

1767 return f 

1768 

1769 def _pow_scalar(self, x, other, op, **raiser_RESIDUAL): 

1770 '''(INTERNAL) Return C{self**B{x}} for C{scalar B{x}}. 

1771 ''' 

1772 s, r = self._fprs2 

1773 if r: 

1774 # assert s != 0 

1775 if isint(x, both=True): # self**int 

1776 x = int(x) 

1777 y = abs(x) 

1778 if y > 1: 

1779 f = self._pow_int(y, other, op, **raiser_RESIDUAL) 

1780 if x > 0: # i.e. > 1 

1781 return f # Fsum or scalar 

1782 # assert x < 0 # i.e. < -1 

1783 if _isFsum(f): 

1784 s, r = f._fprs2 

1785 if r: 

1786 return _1_Over(f, op, **raiser_RESIDUAL) 

1787 else: # scalar 

1788 s = f 

1789 # use s**(-1) to get the CPython 

1790 # float_pow error iff s is zero 

1791 x = -1 

1792 elif x < 0: # self**(-1) 

1793 return _1_Over(self, op, **raiser_RESIDUAL) # 1 / self 

1794 else: # self**1 or self**0 

1795 return self._pow_0_1(x, other) # self, 1 or 1.0 

1796 else: # self**fractional 

1797 R = self._raiser(r, s, **raiser_RESIDUAL) 

1798 if R: 

1799 raise self._ResidualError(op, other, r, **R) 

1800 n, d = self.as_integer_ratio() 

1801 if abs(n) > abs(d): 

1802 n, d, x = d, n, (-x) 

1803 s = n / d 

1804 # assert isscalar(s) and isscalar(x) 

1805 return self._pow_2_3(s, x, other, op, **raiser_RESIDUAL) 

1806 

1807 def _ps_acc(self, ps, xs, up=True, **unused): 

1808 '''(INTERNAL) Accumulate C{xs} known scalars into list C{ps}. 

1809 ''' 

1810 n = 0 

1811 _2s = _2sum 

1812 for x in (tuple(xs) if xs is ps else xs): 

1813 # assert isscalar(x) and _isfinite(x) 

1814 if x: 

1815 i = 0 

1816 for p in ps: 

1817 x, p = _2s(x, p) 

1818 if p: 

1819 ps[i] = p 

1820 i += 1 

1821 ps[i:] = (x,) if x else () 

1822 n += 1 

1823 if n: 

1824 self._n += n 

1825 # Fsum._ps_max = max(Fsum._ps_max, len(ps)) 

1826 if up: 

1827 self._update() 

1828 return ps 

1829 

1830 def _ps_mul(self, op, *factors): 

1831 '''(INTERNAL) Multiply this instance' C{partials} with 

1832 each scalar C{factor} and accumulate into an C{Fsum}. 

1833 ''' 

1834 def _pfs(ps, fs): 

1835 if len(ps) < len(fs): 

1836 ps, fs = fs, ps 

1837 _fin = _isfinite 

1838 for f in fs: 

1839 for p in ps: 

1840 p *= f 

1841 yield p if _fin(p) else self._finite(p, op) 

1842 

1843 return Fsum()._facc_scalar(_pfs(self._ps, factors), up=False) 

1844 

1845 @property_RO 

1846 def _ps_neg(self): 

1847 '''(INTERNAL) Yield the partials, I{negated}. 

1848 ''' 

1849 for p in self._ps: 

1850 yield -p 

1851 

1852 def _ps_1sum(self, *less): 

1853 '''(INTERNAL) Return the partials sum, 1-primed C{less} some scalars. 

1854 ''' 

1855 def _1pls(ps, ls): 

1856 yield _1_0 

1857 for p in ps: 

1858 yield p 

1859 for p in ls: 

1860 yield -p 

1861 yield _N_1_0 

1862 

1863 return _fsum(_1pls(self._ps, less)) 

1864 

1865 def _raiser(self, r, s, raiser=True, **RESIDUAL): 

1866 '''(INTERNAL) Does ratio C{r / s} exceed the RESIDUAL threshold 

1867 I{and} is residual C{r} I{non-zero} or I{significant} (for a 

1868 negative respectively positive C{RESIDUAL} threshold)? 

1869 ''' 

1870 if r and raiser: 

1871 t = self._RESIDUAL 

1872 if RESIDUAL: 

1873 t = _threshold(t, **RESIDUAL) 

1874 if t < 0 or (s + r) != s: 

1875 q = (r / s) if s else s # == 0. 

1876 if fabs(q) > fabs(t): 

1877 return dict(ratio=q, R=t) 

1878 return {} 

1879 

1880 rdiv = __rtruediv__ 

1881 

1882 @property_RO 

1883 def real(self): 

1884 '''Get the C{real} part of this instance (C{float}). 

1885 

1886 @see: Methods L{Fsum.__float__} and L{Fsum.fsum} 

1887 and properties L{Fsum.ceil}, L{Fsum.floor}, 

1888 L{Fsum.imag} and L{Fsum.residual}. 

1889 ''' 

1890 return float(self._fprs) 

1891 

1892 @property_RO 

1893 def residual(self): 

1894 '''Get this instance' residual (C{float} or C{int}): the 

1895 C{sum(partials)} less the precision running sum C{fsum}. 

1896 

1897 @note: The C{residual is INT0} iff the precision running 

1898 C{fsum} is considered to be I{exact}. 

1899 

1900 @see: Methods L{Fsum.fsum}, L{Fsum.fsum2} and L{Fsum.is_exact}. 

1901 ''' 

1902 return self._fprs2.residual 

1903 

1904 def RESIDUAL(self, *threshold): 

1905 '''Get and set this instance' I{ratio} for raising L{ResidualError}s, 

1906 overriding the default from env variable C{PYGEODESY_FSUM_RESIDUAL}. 

1907 

1908 @arg threshold: If C{scalar}, the I{ratio} to exceed for raising 

1909 L{ResidualError}s in division and exponention, if 

1910 C{None} restore the default set with env variable 

1911 C{PYGEODESY_FSUM_RESIDUAL} or if omitted, keep the 

1912 current setting. 

1913 

1914 @return: The previous C{RESIDUAL} setting (C{float}), default C{0.0}. 

1915 

1916 @raise ResidualError: Invalid B{C{threshold}}. 

1917 

1918 @note: L{ResidualError}s may be thrown if (1) the non-zero I{ratio} 

1919 C{residual / fsum} exceeds the given B{C{threshold}} and (2) 

1920 the C{residual} is non-zero and (3) I{significant} vs the 

1921 C{fsum}, i.e. C{(fsum + residual) != fsum} and (4) optional 

1922 keyword argument C{raiser=False} is missing. Specify a 

1923 negative B{C{threshold}} for only non-zero C{residual} 

1924 testing without I{significant}. 

1925 ''' 

1926 r = self._RESIDUAL 

1927 if threshold: 

1928 t = threshold[0] 

1929 self._RESIDUAL = Fsum._RESIDUAL if t is None else ( # for ... 

1930 (_0_0 if t else _1_0) if isbool(t) else 

1931 _threshold(t)) # ... backward compatibility 

1932 return r 

1933 

1934 def _ResidualError(self, op, other, residual, **mod_R): 

1935 '''(INTERNAL) Non-zero B{C{residual}} etc. 

1936 ''' 

1937 def _p(mod=None, R=0, **unused): # ratio=0 

1938 return (_non_zero_ if R < 0 else _significant_) \ 

1939 if mod is None else _integer_ 

1940 

1941 t = _stresidual(_p(**mod_R), residual, **mod_R) 

1942 return self._Error(op, other, ResidualError, txt=t) 

1943 

1944 def root(self, root, **raiser_RESIDUAL): 

1945 '''Return C{B{self}**(1 / B{root})} as L{Fsum}. 

1946 

1947 @arg root: The order (C{scalar} or L{Fsum}), non-zero. 

1948 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1949 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1950 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1951 

1952 @return: The C{self ** (1 / B{root})} result (L{Fsum}). 

1953 

1954 @raise ResidualError: Non-zero, significant residual or invalid 

1955 B{C{RESIDUAL}}. 

1956 

1957 @see: Method L{Fsum.pow}. 

1958 ''' 

1959 x = _1_Over(root, _truediv_op_, **raiser_RESIDUAL) 

1960 f = self._copy_2(self.root) 

1961 return f._fpow(x, f.name, **raiser_RESIDUAL) # == pow(f, x) 

1962 

1963 def _scalar(self, other, op, **txt): 

1964 '''(INTERNAL) Return scalar C{other}. 

1965 ''' 

1966 if isscalar(other): 

1967 return other 

1968 raise self._Error(op, other, _TypeError, **txt) # _invalid_ 

1969 

1970 def signOf(self, res=True): 

1971 '''Determine the sign of this instance. 

1972 

1973 @kwarg res: If C{True} consider, otherwise 

1974 ignore the residual (C{bool}). 

1975 

1976 @return: The sign (C{int}, -1, 0 or +1). 

1977 ''' 

1978 s, r = self._fprs2 

1979 r = (-r) if res else 0 

1980 return _signOf(s, r) 

1981 

1982 def toRepr(self, **lenc_prec_sep_fmt): # PYCHOK signature 

1983 '''Return this C{Fsum} instance as representation. 

1984 

1985 @kwarg lenc_prec_sep_fmt: Optional keyword arguments 

1986 for method L{Fsum.toStr}. 

1987 

1988 @return: This instance (C{repr}). 

1989 ''' 

1990 return Fmt.repr_at(self, self.toStr(**lenc_prec_sep_fmt)) 

1991 

1992 def toStr(self, lenc=True, **prec_sep_fmt): # PYCHOK signature 

1993 '''Return this C{Fsum} instance as string. 

1994 

1995 @kwarg lenc: If C{True} include the current C{[len]} of this 

1996 L{Fsum} enclosed in I{[brackets]} (C{bool}). 

1997 @kwarg prec_sep_fmt: Optional keyword arguments for method 

1998 L{Fsum2Tuple.toStr}. 

1999 

2000 @return: This instance (C{str}). 

2001 ''' 

2002 p = self.classname 

2003 if lenc: 

2004 p = Fmt.SQUARE(p, len(self)) 

2005 n = _enquote(self.name, white=_UNDER_) 

2006 t = self._fprs2.toStr(**prec_sep_fmt) 

2007 return NN(p, _SPACE_, n, t) 

2008 

2009 def _truediv(self, other, op, **raiser_RESIDUAL): 

2010 '''(INTERNAL) Return C{B{self} / B{other}} as an L{Fsum}. 

2011 ''' 

2012 f = self._copy_2(self.__truediv__) 

2013 return f._ftruediv(other, op, **raiser_RESIDUAL) 

2014 

2015 def _update(self, updated=True): # see ._fset 

2016 '''(INTERNAL) Zap all cached C{Property_RO} values. 

2017 ''' 

2018 if updated: 

2019 _pop = self.__dict__.pop 

2020 for p in _ROs: 

2021 _ = _pop(p, None) 

2022# Fsum._fint2._update(self) 

2023# Fsum._fprs ._update(self) 

2024# Fsum._fprs2._update(self) 

2025 return self # for .fset_ 

2026 

2027_ROs = _allPropertiesOf_n(3, Fsum, Property_RO) # PYCHOK see Fsum._update 

2028 

2029 

2030def _Float_Int(arg, **name_Error): 

2031 '''(INTERNAL) Unit of L{Fsum2Tuple} items. 

2032 ''' 

2033 U = Int if isint(arg) else Float 

2034 return U(arg, **name_Error) 

2035 

2036 

2037class DivMod2Tuple(_NamedTuple): 

2038 '''2-Tuple C{(div, mod)} with the quotient C{div} and remainder 

2039 C{mod} results of a C{divmod} operation. 

2040 

2041 @note: Quotient C{div} an C{int} in Python 3+ but a C{float} 

2042 in Python 2-. Remainder C{mod} an L{Fsum} instance. 

2043 ''' 

2044 _Names_ = (_div_, _mod_) 

2045 _Units_ = (_Float_Int, Fsum) 

2046 

2047 

2048class Fsum2Tuple(_NamedTuple): # in .fstats 

2049 '''2-Tuple C{(fsum, residual)} with the precision running C{fsum} 

2050 and the C{residual}, the sum of the remaining partials. Each 

2051 item is C{float} or C{int}. 

2052 

2053 @note: If the C{residual is INT0}, the C{fsum} is considered 

2054 to be I{exact}, see method L{Fsum2Tuple.is_exact}. 

2055 ''' 

2056 _Names_ = ( Fsum.fsum.__name__, Fsum.residual.name) 

2057 _Units_ = (_Float_Int, _Float_Int) 

2058 

2059 def __abs__(self): # in .fmath 

2060 return self._Fsum.__abs__() 

2061 

2062 def __bool__(self): # PYCHOK Python 3+ 

2063 return bool(self._Fsum) 

2064 

2065 def __eq__(self, other): 

2066 return self._other_op(other, self.__eq__) 

2067 

2068 def __float__(self): 

2069 return self._Fsum.__float__() 

2070 

2071 def __ge__(self, other): 

2072 return self._other_op(other, self.__ge__) 

2073 

2074 def __gt__(self, other): 

2075 return self._other_op(other, self.__gt__) 

2076 

2077 def __le__(self, other): 

2078 return self._other_op(other, self.__le__) 

2079 

2080 def __lt__(self, other): 

2081 return self._other_op(other, self.__lt__) 

2082 

2083 def __int__(self): 

2084 return self._Fsum.__int__() 

2085 

2086 def __ne__(self, other): 

2087 return self._other_op(other, self.__ne__) 

2088 

2089 def __neg__(self): 

2090 return self._Fsum.__neg__() 

2091 

2092 __nonzero__ = __bool__ # Python 2- 

2093 

2094 def __pos__(self): 

2095 return self._Fsum.__pos__() 

2096 

2097 def as_integer_ratio(self): 

2098 '''Return this instance as the ratio of 2 integers. 

2099 

2100 @see: Method L{Fsum.as_integer_ratio} for further details. 

2101 ''' 

2102 return self._Fsum.as_integer_ratio() 

2103 

2104 @property_RO 

2105 def _fint2(self): 

2106 return self._Fsum._fint2 

2107 

2108 @property_RO 

2109 def _fprs2(self): 

2110 return self._Fsum._fprs2 

2111 

2112 @Property_RO 

2113 def _Fsum(self): # this C{Fsum2Tuple} as L{Fsum}, in .fstats 

2114 s, r = _s_r(*self) 

2115 ps = (r, s) if r else (s,) 

2116 return _Psum(ps, name=self.name) 

2117 

2118 def Fsum_(self, *xs, **name_RESIDUAL): 

2119 '''Return this C{Fsum2Tuple} as an L{Fsum} plus some C{xs}. 

2120 ''' 

2121 f = _Psum(self._Fsum._ps, **name_RESIDUAL) 

2122 return f._facc_1(xs, up=False) if xs else f 

2123 

2124 def is_exact(self): 

2125 '''Is this L{Fsum2Tuple} considered to be exact? (C{bool}). 

2126 ''' 

2127 return self._Fsum.is_exact() 

2128 

2129 def is_integer(self): 

2130 '''Is this L{Fsum2Tuple} C{integer}? (C{bool}). 

2131 ''' 

2132 return self._Fsum.is_integer() 

2133 

2134 def _mul_scalar(self, other, op): # for Fsum._fmul 

2135 return self._Fsum._mul_scalar(other, op) 

2136 

2137 @property_RO 

2138 def _n(self): 

2139 return self._Fsum._n 

2140 

2141 def _other_op(self, other, which): 

2142 C, s = (tuple, self) if isinstance(other, tuple) else (Fsum, self._Fsum) 

2143 return getattr(C, which.__name__)(s, other) 

2144 

2145 @property_RO 

2146 def _ps(self): 

2147 return self._Fsum._ps 

2148 

2149 @property_RO 

2150 def _ps_neg(self): 

2151 return self._Fsum._ps_neg 

2152 

2153 def signOf(self, **res): 

2154 '''Like method L{Fsum.signOf}. 

2155 ''' 

2156 return self._Fsum.signOf(**res) 

2157 

2158 def toStr(self, fmt=Fmt.g, **prec_sep): # PYCHOK signature 

2159 '''Return this L{Fsum2Tuple} as string (C{str}). 

2160 

2161 @kwarg fmt: Optional C{float} format (C{letter}). 

2162 @kwarg prec_sep: Optional keyword arguments for function 

2163 L{fstr<streprs.fstr>}. 

2164 ''' 

2165 return Fmt.PAREN(fstr(self, fmt=fmt, strepr=str, force=False, **prec_sep)) 

2166 

2167_Fsum_Fsum2Tuple_types = Fsum, Fsum2Tuple # PYCHOK lines 

2168 

2169 

2170class ResidualError(_ValueError): 

2171 '''Error raised for a division, power or root operation of 

2172 an L{Fsum} instance with a C{residual} I{ratio} exceeding 

2173 the L{RESIDUAL<Fsum.RESIDUAL>} threshold. 

2174 

2175 @see: Module L{pygeodesy.fsums} and method L{Fsum.RESIDUAL}. 

2176 ''' 

2177 pass 

2178 

2179 

2180try: 

2181 from math import fsum as _fsum # precision IEEE-754 sum, Python 2.6+ 

2182 

2183 # make sure _fsum works as expected (XXX check 

2184 # float.__getformat__('float')[:4] == 'IEEE'?) 

2185 if _fsum((1, 1e101, 1, -1e101)) != 2: # PYCHOK no cover 

2186 del _fsum # nope, remove _fsum ... 

2187 raise ImportError() # ... use _fsum below 

2188 

2189 Fsum._math_fsum = _sum = _fsum # PYCHOK exported 

2190except ImportError: 

2191 _sum = sum # Fsum(NAN) exception fall-back, in .elliptic 

2192 

2193 def _fsum(xs): 

2194 '''(INTERNAL) Precision summation, Python 2.5-. 

2195 ''' 

2196 F = Fsum() 

2197 F.name = _fsum.__name__ 

2198 return F._facc(xs, up=False)._fprs2.fsum 

2199 

2200 

2201def fsum(xs, floats=False): 

2202 '''Precision floating point summation based on/like Python's C{math.fsum}. 

2203 

2204 @arg xs: Iterable of items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} 

2205 instance). 

2206 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to 

2207 be scalar} (C{bool}). 

2208 

2209 @return: Precision C{fsum} (C{float}). 

2210 

2211 @raise OverflowError: Partial C{2sum} overflow. 

2212 

2213 @raise TypeError: Non-scalar B{C{xs}} item. 

2214 

2215 @raise ValueError: Invalid or non-finite B{C{xs}} item. 

2216 

2217 @note: Exception and I{non-finite} handling may differ if not based 

2218 on Python's C{math.fsum}. 

2219 

2220 @see: Class L{Fsum} and methods L{Fsum.fsum} and L{Fsum.fadd}. 

2221 ''' 

2222 return _fsum(xs if floats is True else _2floats(xs)) if xs else _0_0 # PYCHOK yield 

2223 

2224 

2225def fsum_(*xs, **floats): 

2226 '''Precision floating point summation of all positional items. 

2227 

2228 @arg xs: Items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} instance), 

2229 all positional. 

2230 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to 

2231 be scalar} (C{bool}). 

2232 

2233 @see: Function L{fsum<fsums.fsum>} for further details. 

2234 ''' 

2235 return _fsum(xs if _xkwds_get1(floats, floats=False) is True else 

2236 _2floats(xs, origin=1)) if xs else _0_0 # PYCHOK yield 

2237 

2238 

2239def fsumf_(*xs): 

2240 '''Precision floating point summation iff I{all} C{B{xs}} items are I{known to be scalar}. 

2241 

2242 @see: Function L{fsum_<fsums.fsum_>} for further details. 

2243 ''' 

2244 return _fsum(xs) if xs else _0_0 

2245 

2246 

2247def fsum1(xs, floats=False): 

2248 '''Precision floating point summation, 1-primed. 

2249 

2250 @arg xs: Iterable of items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} 

2251 instance). 

2252 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to 

2253 be scalar} (C{bool}). 

2254 

2255 @see: Function L{fsum<fsums.fsum>} for further details. 

2256 ''' 

2257 return _fsum(_1primed(xs if floats is True else _2floats(xs))) if xs else _0_0 # PYCHOK yield 

2258 

2259 

2260def fsum1_(*xs, **floats): 

2261 '''Precision floating point summation, 1-primed of all positional items. 

2262 

2263 @arg xs: Items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} instance), 

2264 all positional. 

2265 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to 

2266 be scalar} (C{bool}). 

2267 

2268 @see: Function L{fsum_<fsums.fsum_>} for further details. 

2269 ''' 

2270 return _fsum(_1primed(xs if _xkwds_get1(floats, floats=False) is True else 

2271 _2floats(xs, origin=1))) if xs else _0_0 # PYCHOK yield 

2272 

2273 

2274def fsum1f_(*xs): 

2275 '''Precision floating point summation iff I{all} C{B{xs}} items are I{known to be scalar}. 

2276 

2277 @see: Function L{fsum_<fsums.fsum_>} for further details. 

2278 ''' 

2279 return _fsum(_1primed(xs)) if xs else _0_0 

2280 

2281 

2282if __name__ == '__main__': 

2283 

2284 # usage: [env _psum=fsum] python3 -m pygeodesy.fsums 

2285 

2286 if _getenv(_psum.__name__, NN) == _fsum.__name__: 

2287 _psum = _fsum 

2288 

2289 def _test(n): 

2290 # copied from Hettinger, see L{Fsum} reference 

2291 from pygeodesy import frandoms, printf 

2292 

2293 printf(_fsum.__name__, end=_COMMASPACE_) 

2294 printf(_psum.__name__, end=_COMMASPACE_) 

2295 

2296 F = Fsum() 

2297 if F.is_math_fsum(): 

2298 for t in frandoms(n, seeded=True): 

2299 assert float(F.fset_(*t)) == _fsum(t) 

2300 printf(_DOT_, end=NN) 

2301 printf(NN) 

2302 

2303 _test(128) 

2304 

2305# **) MIT License 

2306# 

2307# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

2308# 

2309# Permission is hereby granted, free of charge, to any person obtaining a 

2310# copy of this software and associated documentation files (the "Software"), 

2311# to deal in the Software without restriction, including without limitation 

2312# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

2313# and/or sell copies of the Software, and to permit persons to whom the 

2314# Software is furnished to do so, subject to the following conditions: 

2315# 

2316# The above copyright notice and this permission notice shall be included 

2317# in all copies or substantial portions of the Software. 

2318# 

2319# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

2320# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

2321# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

2322# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

2323# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

2324# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

2325# OTHER DEALINGS IN THE SOFTWARE.