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__ = '24.05.29' 

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 s = a + b 

277 if _isfinite(s): 

278 if fabs(a) < fabs(b): 

279 b, a = a, b 

280 return s, (b - (s - a)) 

281 u = unstr(_2sum, a, b) 

282 t = Fmt.PARENSPACED(_not_finite_, s) 

283 raise _OverflowError(u, txt=t) 

284 

285 

286def _threshold(threshold=_0_0, **kwds): 

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

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

289 ''' 

290 if kwds: 

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

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

293 if kwds: 

294 raise _UnexpectedError(**kwds) 

295 try: 

296 return _2finite(threshold) # PYCHOK None 

297 except Exception as x: 

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

299 

300 

301class Fsum(_Named): # sync __methods__ with .vector3dBase.Vector3dBase 

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

303 

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

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

306 intermediate, I{running} summuation. 

307 

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

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

310 C{float}, except C{complex}. 

311 

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

313 Python's C{math.fsum}. 

314 

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

316 393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}, 

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

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

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

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

321 ''' 

322 _math_fsum = None 

323 _n = 0 

324# _ps = [] # partial sums 

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

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

327 

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

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

330 

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

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

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

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

335 L{ResidualError}s (C{scalar}). 

336 

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

338 ''' 

339 if name_RESIDUAL: 

340 n, kwds = _name2__(**name_RESIDUAL) 

341 if kwds: 

342 R = Fsum._RESIDUAL 

343 t = _threshold(R, **kwds) 

344 if t != R: 

345 self._RESIDUAL = t 

346 if n: 

347 self.name = n 

348 

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

350 if xs: 

351 self._facc_1(xs, up=False) 

352 

353 def __abs__(self): 

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

355 ''' 

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

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

358 

359 def __add__(self, other): 

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

361 

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

363 

364 @return: The sum (L{Fsum}). 

365 

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

367 ''' 

368 f = self._copy_2(self.__add__) 

369 return f._fadd(other, _add_op_) 

370 

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

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

373 ''' 

374 s, r = self._fprs2 

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

376 

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

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

379 

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

381 

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

383 ''' 

384 return self.ceil 

385 

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

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

388 

389 @return: -1, 0 or +1 (C{int}). 

390 

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

392 ''' 

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

394 return _signOf(s, 0) 

395 

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

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

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

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

400 

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

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

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

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

405 

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

407 B{C{RESIDUAL}}. 

408 

409 @see: Method L{Fsum.fdiv}. 

410 ''' 

411 f = self._copy_2(self.__divmod__) 

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

413 

414 def __eq__(self, other): 

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

416 ''' 

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

418 

419 def __float__(self): 

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

421 

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

423 ''' 

424 return float(self._fprs) 

425 

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

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

428 

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

430 

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

432 ''' 

433 return self.floor 

434 

435 def __floordiv__(self, other): 

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

437 

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

439 

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

441 

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

443 ''' 

444 f = self._copy_2(self.__floordiv__) 

445 return f._floordiv(other, _floordiv_op_) 

446 

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

448 '''Not implemented.''' 

449 return _NotImplemented(self, *other) 

450 

451 def __ge__(self, other): 

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

453 ''' 

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

455 

456 def __gt__(self, other): 

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

458 ''' 

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

460 

461 def __hash__(self): # PYCHOK no cover 

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

463 ''' 

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

465 

466 def __iadd__(self, other): 

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

468 

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

470 an iterable of several of the former. 

471 

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

473 

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

475 C{scalar} nor L{Fsum}. 

476 

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

478 ''' 

479 try: 

480 return self._fadd(other, _iadd_op_) 

481 except TypeError: 

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

483 

484 def __ifloordiv__(self, other): 

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

486 

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

488 

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

490 

491 @raise ResidualError: Non-zero, significant residual 

492 in B{C{other}}. 

493 

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

495 

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

497 

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

499 

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

501 ''' 

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

503 

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

505 '''Not implemented.''' 

506 return _NotImplemented(self, other) 

507 

508 def __imod__(self, other): 

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

510 

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

512 

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

514 

515 @see: Method L{Fsum.__divmod__}. 

516 ''' 

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

518 

519 def __imul__(self, other): 

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

521 

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

523 

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

525 

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

527 

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

529 

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

531 ''' 

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

533 

534 def __int__(self): 

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

536 

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

538 and L{Fsum.floor}. 

539 ''' 

540 i, _ = self._fint2 

541 return i 

542 

543 def __invert__(self): # PYCHOK no cover 

544 '''Not implemented.''' 

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

546 return _NotImplemented(self) 

547 

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

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

550 

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

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

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

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

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

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

557 

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

559 

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

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

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

563 

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

565 

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

567 is non-zero and significant and either 

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

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

570 C{None}. 

571 

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

573 invocation failed. 

574 

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

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

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

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

579 is given as C{0}. 

580 

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

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

583 ''' 

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

585 

586 def __isub__(self, other): 

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

588 

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

590 an iterable of several of the former. 

591 

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

593 

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

595 

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

597 ''' 

598 try: 

599 return self._fsub(other, _isub_op_) 

600 except TypeError: 

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

602 

603 def __iter__(self): 

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

605 ''' 

606 return iter(self.partials) 

607 

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

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

610 

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

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

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

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

615 

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

617 

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

619 

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

621 B{C{RESIDUAL}}. 

622 

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

624 

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

626 

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

628 

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

630 ''' 

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

632 

633 def __le__(self, other): 

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

635 ''' 

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

637 

638 def __len__(self): 

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

640 ''' 

641 return self._n 

642 

643 def __lt__(self, other): 

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

645 ''' 

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

647 

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

649 '''Not implemented.''' 

650 return _NotImplemented(self, other) 

651 

652 def __mod__(self, other): 

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

654 

655 @see: Method L{Fsum.__imod__}. 

656 ''' 

657 f = self._copy_2(self.__mod__) 

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

659 

660 def __mul__(self, other): 

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

662 

663 @see: Method L{Fsum.__imul__}. 

664 ''' 

665 f = self._copy_2(self.__mul__) 

666 return f._fmul(other, _mul_op_) 

667 

668 def __ne__(self, other): 

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

670 ''' 

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

672 

673 def __neg__(self): 

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

675 ''' 

676 f = self._copy_2(self.__neg__) 

677 return f._fset(self._neg) 

678 

679 def __pos__(self): 

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

681 ''' 

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

683 

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

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

686 

687 @see: Method L{Fsum.__ipow__}. 

688 ''' 

689 f = self._copy_2(self.__pow__) 

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

691 

692 def __radd__(self, other): 

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

694 

695 @see: Method L{Fsum.__iadd__}. 

696 ''' 

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

698 return f._fadd(self, _add_op_) 

699 

700 def __rdivmod__(self, other): 

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

702 C{(quotient, remainder)}. 

703 

704 @see: Method L{Fsum.__divmod__}. 

705 ''' 

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

707 return f._fdivmod2(self, _divmod_op_) 

708 

709# def __repr__(self): 

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

711# ''' 

712# return self.toRepr(lenc=True) 

713 

714 def __rfloordiv__(self, other): 

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

716 

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

718 ''' 

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

720 return f._floordiv(self, _floordiv_op_) 

721 

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

723 '''Not implemented.''' 

724 return _NotImplemented(self, other) 

725 

726 def __rmod__(self, other): 

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

728 

729 @see: Method L{Fsum.__imod__}. 

730 ''' 

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

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

733 

734 def __rmul__(self, other): 

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

736 

737 @see: Method L{Fsum.__imul__}. 

738 ''' 

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

740 return f._fmul(self, _mul_op_) 

741 

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

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

744 

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

746 ''' 

747 f = self._copy_2(self.__round__) 

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

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

750 

751 def __rpow__(self, other, *mod): 

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

753 

754 @see: Method L{Fsum.__ipow__}. 

755 ''' 

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

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

758 

759 def __rsub__(self, other): 

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

761 

762 @see: Method L{Fsum.__isub__}. 

763 ''' 

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

765 return f._fsub(self, _sub_op_) 

766 

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

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

769 

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

771 ''' 

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

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

774 

775 def __str__(self): 

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

777 ''' 

778 return self.toStr(lenc=True) 

779 

780 def __sub__(self, other): 

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

782 

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

784 

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

786 

787 @see: Method L{Fsum.__isub__}. 

788 ''' 

789 f = self._copy_2(self.__sub__) 

790 return f._fsub(other, _sub_op_) 

791 

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

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

794 

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

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

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

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

799 

800 @return: The quotient (L{Fsum}). 

801 

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

803 B{C{RESIDUAL}}. 

804 

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

806 ''' 

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

808 

809 __trunc__ = __int__ 

810 

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

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

813 __div__ = __truediv__ 

814 __idiv__ = __itruediv__ 

815 __long__ = __int__ 

816 __nonzero__ = __bool__ 

817 __rdiv__ = __rtruediv__ 

818 

819 def as_integer_ratio(self): 

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

821 

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

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

824 non-zero, positive. 

825 

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

827 ''' 

828 n, r = self._fint2 

829 if r: 

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

831 n *= d 

832 n += i 

833 else: # PYCHOK no cover 

834 d = 1 

835 return n, d 

836 

837 @property_RO 

838 def as_iscalar(self): 

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

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

841 ''' 

842 s, r = self._fprs2 

843 return self if r else s 

844 

845 @property_RO 

846 def ceil(self): 

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

848 C{float} in Python 2-). 

849 

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

851 

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

853 L{Fsum.imag} and L{Fsum.real}. 

854 ''' 

855 s, r = self._fprs2 

856 c = _ceil(s) + int(r) - 1 

857 while r > (c - s): # (s + r) > c 

858 c += 1 

859 return c # _ceil(self._n_d) 

860 

861 cmp = __cmp__ 

862 

863 def _cmp_0(self, other, op): 

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

865 ''' 

866 if _isFsumTuple(other): 

867 s = self._ps_1sum(*other._ps) 

868 elif self._scalar(other, op): 

869 s = self._ps_1sum(other) 

870 else: 

871 s = self.signOf() # res=True 

872 return s 

873 

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

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

876 

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

878 

879 @return: The copy (L{Fsum}). 

880 ''' 

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

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

883 if f._ps is self._ps: 

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

885 if not deep: 

886 f._n = 1 

887 # assert f._Fsum is f 

888 return f 

889 

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

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

892 ''' 

893 n = name or which.__name__ # _dunder_nameof 

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

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

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

897 # assert f._n == self._n 

898 # assert f._Fsum is f 

899 return f 

900 

901 def _copy_2r(self, other, which): 

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

903 ''' 

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

905 self._copy_2(which)._fset(other) 

906 

907# def _copy_RESIDUAL(self, other): 

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

909# ''' 

910# R = other._RESIDUAL 

911# if R is not Fsum._RESIDUAL: 

912# self._RESIDUAL = R 

913 

914 divmod = __divmod__ 

915 

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

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

918 ''' 

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

920 

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

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

923 ''' 

924 E, t = _xError2(X) 

925 if mod: 

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

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

928 

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

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

931 ''' 

932 E, t = _xError2(X) 

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

934 return E(u, txt=t, cause=X) 

935 

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

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

938 ''' 

939 if xs: 

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

941 ps = self._ps 

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

943 return self 

944 

945 def _facc_1(self, xs, **up): 

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

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

948 ''' 

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

950 self._facc(xs, origin=1, **up) 

951 

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

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

954 ''' 

955 try: 

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

957 except Exception as X: 

958 raise self._ErrorX(X, op, other) 

959 

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

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

962 ''' 

963 def _N(X): 

964 return X._ps_neg 

965 

966 def _n(x): 

967 return -float(x) 

968 

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

970 

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

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

973 ''' 

974 def _Pow4(p): 

975 r = 0 

976 if _isFsumTuple(p): 

977 s, r = p._fprs2 

978 if r: 

979 m = Fsum._pow 

980 else: # scalar 

981 return _Pow4(s) 

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

983 p = s = int(p) 

984 m = Fsum._pow_int 

985 else: 

986 p = s = _2float(power=p) 

987 m = Fsum._pow_scalar 

988 return m, p, s, r 

989 

990 _Pow, p, s, r = _Pow4(power) 

991 if p: # and xs: 

992 op = which.__name__ 

993 _flt = float 

994 _Fs = Fsum 

995 _isa = isinstance 

996 _pow = self._pow_2_3 

997 

998 def _P(X): 

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

1000 return f._ps if _isa(f, _Fs) else (f,) 

1001 

1002 def _p(x): 

1003 x = _flt(x) 

1004 f = _pow(x, s, power, op, **raiser_RESIDUAL) 

1005 if f and r: 

1006 f *= _pow(x, r, power, op, **raiser_RESIDUAL) 

1007 return f 

1008 

1009 f = self._facc(xs, origin=1, _X=_P, _x=_p) 

1010 else: 

1011 f = self._facc_scalar_(float(len(xs))) # x**0 == 1 

1012 return f 

1013 

1014 def _facc_scalar(self, xs, **up): 

1015 '''(INTERNAL) Accumulate all C{xs}, known to be scalar. 

1016 ''' 

1017 if xs: 

1018 _ = self._ps_acc(self._ps, xs, **up) 

1019 return self 

1020 

1021 def _facc_scalar_(self, *xs, **up): 

1022 '''(INTERNAL) Accumulate all positional C{xs}, known to be scalar. 

1023 ''' 

1024 if xs: 

1025 _ = self._ps_acc(self._ps, xs, **up) 

1026 return self 

1027 

1028# def _facc_up(self, up=True): 

1029# '''(INTERNAL) Update the C{partials}, by removing 

1030# and re-accumulating the final C{partial}. 

1031# ''' 

1032# ps = self._ps 

1033# while len(ps) > 1: 

1034# p = ps.pop() 

1035# if p: 

1036# n = self._n 

1037# _ = self._ps_acc(ps, (p,), up=False) 

1038# self._n = n 

1039# break 

1040# return self._update() if up else self 

1041 

1042 def fadd(self, xs=()): 

1043 '''Add an iterable's items to this instance. 

1044 

1045 @arg xs: Iterable of items to add (each C{scalar} 

1046 or an L{Fsum} or L{Fsum2Tuple} instance). 

1047 

1048 @return: This instance (L{Fsum}). 

1049 

1050 @raise OverflowError: Partial C{2sum} overflow. 

1051 

1052 @raise TypeError: An invalid B{C{xs}} item. 

1053 

1054 @raise ValueError: Invalid or non-finite B{C{xs}} value. 

1055 ''' 

1056 if _isFsumTuple(xs): 

1057 self._facc_scalar(xs._ps) 

1058 elif isscalar(xs): # for backward compatibility 

1059 self._facc_scalar_(_2float(x=xs)) # PYCHOK no cover 

1060 elif xs: # _xiterable(xs) 

1061 self._facc(xs) 

1062 return self 

1063 

1064 def fadd_(self, *xs): 

1065 '''Add all positional items to this instance. 

1066 

1067 @arg xs: Values to add (each C{scalar} or an L{Fsum} 

1068 or L{Fsum2Tuple} instance), all positional. 

1069 

1070 @see: Method L{Fsum.fadd} for further details. 

1071 ''' 

1072 return self._facc_1(xs) 

1073 

1074 def _fadd(self, other, op, **up): # in .fmath.Fhorner 

1075 '''(INTERNAL) Apply C{B{self} += B{other}}. 

1076 ''' 

1077 if not self._ps: # new Fsum(x) 

1078 self._fset(other, op=op, **up) 

1079 elif _isFsumTuple(other): 

1080 self._facc_scalar(other._ps, **up) 

1081 elif self._scalar(other, op): 

1082 self._facc_scalar_(other, **up) 

1083 return self 

1084 

1085 fcopy = copy # for backward compatibility 

1086 fdiv = __itruediv__ 

1087 fdivmod = __divmod__ 

1088 

1089 def _fdivmod2(self, other, op, **raiser_RESIDUAL): 

1090 '''(INTERNAL) Apply C{B{self} %= B{other}} and return a L{DivMod2Tuple}. 

1091 ''' 

1092 # result mostly follows CPython function U{float_divmod 

1093 # <https://GitHub.com/python/cpython/blob/main/Objects/floatobject.c>}, 

1094 # but at least divmod(-3, 2) equals Cpython's result (-2, 1). 

1095 q = self._truediv(other, op, **raiser_RESIDUAL).floor 

1096 if q: # == float // other == floor(float / other) 

1097 self -= Fsum(q) * other # NOT other * q! 

1098 

1099 s = signOf(other) # make signOf(self) == signOf(other) 

1100 if s and self.signOf() == -s: # PYCHOK no cover 

1101 self += other 

1102 q -= 1 

1103# t = self.signOf() 

1104# if t and t != s: 

1105# raise self._Error(op, other, _AssertionError, txt__=signOf) 

1106 return DivMod2Tuple(q, self) # q is C{int} in Python 3+, but C{float} in Python 2- 

1107 

1108 def _fhorner(self, x, cs, op): # in .fmath 

1109 '''(INTERNAL) Add an L{Fhorner} evaluation of polynomial 

1110 M{sum(cs[i] * x**i for i=0..len(cs)-1)}. 

1111 ''' 

1112 if _xiterablen(cs): 

1113 H = Fsum(name__=self._fhorner) 

1114 if _isFsumTuple(x): 

1115 _mul = H._mul_Fsum 

1116 else: 

1117 _mul = H._mul_scalar 

1118 x = _2float(x=x) 

1119 if len(cs) > 1 and x: 

1120 for c in reversed(cs): 

1121 H._fset_ps(_mul(x, op)) 

1122 H._fadd(c, op, up=False) 

1123 else: # x == 0 

1124 H = cs[0] 

1125 self._fadd(H, op) 

1126 return self 

1127 

1128 def _finite(self, other, op=None): 

1129 '''(INTERNAL) Return B{C{other}} if C{finite}. 

1130 ''' 

1131 if _isfinite(other): 

1132 return other 

1133 raise ValueError(_not_finite_) if op is None else \ 

1134 self._Error(op, other, _ValueError, txt=_not_finite_) 

1135 

1136 def fint(self, name=NN, **raiser_RESIDUAL): 

1137 '''Return this instance' current running sum as C{integer}. 

1138 

1139 @kwarg name: Optional, overriding C{B{name}="fint"} (C{str}). 

1140 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1141 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1142 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1143 

1144 @return: The C{integer} sum (L{Fsum}) if this instance C{is_integer} 

1145 with a zero or insignificant I{integer} residual. 

1146 

1147 @raise ResidualError: Non-zero, significant residual or invalid 

1148 B{C{RESIDUAL}}. 

1149 

1150 @see: Methods L{Fsum.fint2}, L{Fsum.int_float} and L{Fsum.is_integer}. 

1151 ''' 

1152 i, r = self._fint2 

1153 if r: 

1154 R = self._raiser(r, i, **raiser_RESIDUAL) 

1155 if R: 

1156 t = _stresidual(_integer_, r, **R) 

1157 raise ResidualError(_integer_, i, txt=t) 

1158 return _Psum_(i, name=_name__(name, name__=self.fint)) 

1159 

1160 def fint2(self, **name): 

1161 '''Return this instance' current running sum as C{int} and the 

1162 I{integer} residual. 

1163 

1164 @kwarg name: Optional name (C{str}). 

1165 

1166 @return: An L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} 

1167 an C{int} and I{integer} C{residual} a C{float} or 

1168 C{INT0} if the C{fsum} is considered to be I{exact}. 

1169 ''' 

1170 return Fsum2Tuple(*self._fint2, **name) 

1171 

1172 @Property 

1173 def _fint2(self): # see ._fset 

1174 '''(INTERNAL) Get 2-tuple (C{int}, I{integer} residual). 

1175 ''' 

1176 s, r = self._fprs2 

1177 i = int(s) 

1178 n = len(self._ps) 

1179 r = self._ps_1sum(i) if r and n > 1 else float(s - i) 

1180 return i, (r or INT0) # Fsum2Tuple? 

1181 

1182 @_fint2.setter_ # PYCHOK setter_underscore! 

1183 def _fint2(self, s): 

1184 '''(INTERNAL) Replace the C{_fint2} value. 

1185 ''' 

1186 i = int(s) 

1187 return i, ((s - i) or INT0) 

1188 

1189 @deprecated_property_RO 

1190 def float_int(self): # PYCHOK no cover 

1191 '''DEPRECATED, use method C{Fsum.int_float}.''' 

1192 return self.int_float() # raiser=False 

1193 

1194 @property_RO 

1195 def floor(self): 

1196 '''Get this instance' C{floor} (C{int} in Python 3+, but 

1197 C{float} in Python 2-). 

1198 

1199 @note: This C{floor} takes the C{residual} into account. 

1200 

1201 @see: Method L{Fsum.int_float} and properties L{Fsum.ceil}, 

1202 L{Fsum.imag} and L{Fsum.real}. 

1203 ''' 

1204 s, r = self._fprs2 

1205 f = _floor(s) + _floor(r) + 1 

1206 while (f - s) > r: # f > (s + r) 

1207 f -= 1 

1208 return f # _floor(self._n_d) 

1209 

1210# ffloordiv = __ifloordiv__ # for naming consistency 

1211# floordiv = __floordiv__ # for naming consistency 

1212 

1213 def _floordiv(self, other, op, **raiser_RESIDUAL): # rather _ffloordiv? 

1214 '''Apply C{B{self} //= B{other}}. 

1215 ''' 

1216 q = self._ftruediv(other, op, **raiser_RESIDUAL) # == self 

1217 return self._fset(q.floor) # floor(q) 

1218 

1219 fmul = __imul__ 

1220 

1221 def _fmul(self, other, op): 

1222 '''(INTERNAL) Apply C{B{self} *= B{other}}. 

1223 ''' 

1224 if _isFsumTuple(other): 

1225 if len(self._ps) != 1: 

1226 f = self._mul_Fsum(other, op) 

1227 elif len(other._ps) != 1: # and len(self._ps) == 1 

1228 f = other._mul_scalar(self._ps[0], op) 

1229 else: # len(other._ps) == len(self._ps) == 1 

1230 f = self._finite(self._ps[0] * other._ps[0]) 

1231 else: 

1232 s = self._scalar(other, op) 

1233 f = self._mul_scalar(s, op) 

1234 return self._fset(f) # n=len(self) + 1 

1235 

1236 def fover(self, over, **raiser_RESIDUAL): 

1237 '''Apply C{B{self} /= B{over}} and summate. 

1238 

1239 @arg over: An L{Fsum} or C{scalar} denominator. 

1240 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1241 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1242 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1243 

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

1245 

1246 @raise ResidualError: Non-zero, significant residual or invalid 

1247 B{C{RESIDUAL}}. 

1248 

1249 @see: Methods L{Fsum.fsum} and L{Fsum.__itruediv__}. 

1250 ''' 

1251 return float(self.fdiv(over, **raiser_RESIDUAL)._fprs) 

1252 

1253 fpow = __ipow__ 

1254 

1255 def _fpow(self, other, op, *mod, **raiser_RESIDUAL): 

1256 '''Apply C{B{self} **= B{other}}, optional B{C{mod}} or C{None}. 

1257 ''' 

1258 if mod: 

1259 if mod[0] is not None: # == 3-arg C{pow} 

1260 f = self._pow_2_3(self, other, other, op, *mod, **raiser_RESIDUAL) 

1261 elif self.is_integer(): 

1262 # return an exact C{int} for C{int}**C{int} 

1263 i, _ = self._fint2 # assert _ == 0 

1264 x, r = _2scalar2(other) # C{int}, C{float} or other 

1265 f = _Psum_(i)._pow_Fsum(other, op, **raiser_RESIDUAL) if r else \ 

1266 self._pow_2_3(i, x, other, op, **raiser_RESIDUAL) 

1267 else: # mod[0] is None, power(self, other) 

1268 f = self._pow(other, other, op, **raiser_RESIDUAL) 

1269 else: # pow(self, other) 

1270 f = self._pow(other, other, op, **raiser_RESIDUAL) 

1271 return self._fset(f) # n=max(len(self), 1) 

1272 

1273 @Property 

1274 def _fprs(self): 

1275 '''(INTERNAL) Get and cache this instance' precision 

1276 running sum (C{float} or C{int}), ignoring C{residual}. 

1277 

1278 @note: The precision running C{fsum} after a C{//=} or 

1279 C{//} C{floor} division is C{int} in Python 3+. 

1280 ''' 

1281 s, _ = self._fprs2 

1282 return s # ._fprs2.fsum 

1283 

1284 @_fprs.setter_ # PYCHOK setter_underscore! 

1285 def _fprs(self, s): 

1286 '''(INTERNAL) Replace the C{_fprs} value. 

1287 ''' 

1288 return s 

1289 

1290 @Property 

1291 def _fprs2(self): 

1292 '''(INTERNAL) Get and cache this instance' precision 

1293 running sum and residual (L{Fsum2Tuple}). 

1294 ''' 

1295 ps = self._ps 

1296 n = len(ps) - 2 

1297 if n > 0: # len(ps) > 2 

1298 s = _psum(ps) 

1299 n = len(ps) - 2 

1300 if n > 0: 

1301 r = self._ps_1sum(s) 

1302 return Fsum2Tuple(*_s_r(s, r)) 

1303 if n == 0: # len(ps) == 2 

1304 s, r = _s_r(*_2sum(*ps)) 

1305 ps[:] = (r, s) if r else (s,) 

1306 elif ps: # len(ps) == 1 

1307 s, r = ps[0], INT0 

1308 else: # len(ps) == 0 

1309 s, r = _0_0, INT0 

1310 ps[:] = s, 

1311 # assert self._ps is ps 

1312 return Fsum2Tuple(s, r) 

1313 

1314 @_fprs2.setter_ # PYCHOK setter_underscore! 

1315 def _fprs2(self, s_r): 

1316 '''(INTERNAL) Replace the C{_fprs2} value. 

1317 ''' 

1318 return Fsum2Tuple(s_r) 

1319 

1320 def fset_(self, *xs): 

1321 '''Replace this instance' value with all positional items. 

1322 

1323 @arg xs: Optional, new values (each C{scalar} or 

1324 an L{Fsum} or L{Fsum2Tuple} instance), 

1325 all positional. 

1326 

1327 @return: This instance, replaced (C{Fsum}). 

1328 

1329 @see: Method L{Fsum.fadd} for further details. 

1330 ''' 

1331 f = xs[0] if len(xs) == 1 else ( 

1332 Fsum(*xs) if xs else _0_0) 

1333 return self._fset(f) 

1334 

1335 def _fset(self, other, n=0, up=True, **op): 

1336 '''(INTERNAL) Overwrite this instance with an other or a C{scalar}. 

1337 ''' 

1338 if other is self: 

1339 pass # from ._fmul, ._ftruediv and ._pow_0_1 

1340 elif _isFsumTuple(other): 

1341 self._ps[:] = other._ps 

1342 self._n = n or other._n 

1343# self._copy_RESIDUAL(other) 

1344 if up: # use or zap the C{Property_RO} values 

1345 Fsum._fint2._update_from(self, other) 

1346 Fsum._fprs ._update_from(self, other) 

1347 Fsum._fprs2._update_from(self, other) 

1348 elif isscalar(other): 

1349 s = float(self._finite(other, **op)) if op else other 

1350 self._ps[:] = s, 

1351 self._n = n or 1 

1352 if up: # Property _fint2, _fprs and _fprs2 all have 

1353 # @.setter_underscore and NOT @.setter because the 

1354 # latter's _fset zaps the value set by @.setter 

1355 self._fint2 = s 

1356 self._fprs = s 

1357 self._fprs2 = s, INT0 

1358 # assert self._fprs is s 

1359 else: # PYCHOK no cover 

1360 op = _xkwds_get1(op, op=_fset_op_) 

1361 raise self._Error(op, other, _TypeError) 

1362 return self 

1363 

1364 def _fset_ps(self, other): # in .fmath 

1365 '''(INTERNAL) Set partials from a known C{scalar}, L{Fsum} or L{Fsum2Tuple}. 

1366 ''' 

1367 return self._fset(other, up=False) 

1368 

1369 def fsub(self, xs=()): 

1370 '''Subtract an iterable's items from this instance. 

1371 

1372 @see: Method L{Fsum.fadd} for further details. 

1373 ''' 

1374 return self._facc_neg(xs) 

1375 

1376 def fsub_(self, *xs): 

1377 '''Subtract all positional items from this instance. 

1378 

1379 @see: Method L{Fsum.fadd_} for further details. 

1380 ''' 

1381 return self._fsub(xs[0], _sub_op_) if len(xs) == 1 else \ 

1382 self._facc_neg(xs, origin=1) 

1383 

1384 def _fsub(self, other, op): 

1385 '''(INTERNAL) Apply C{B{self} -= B{other}}. 

1386 ''' 

1387 if _isFsumTuple(other): 

1388 if other is self: # or other._fprs2 == self._fprs2: 

1389 self._fset(_0_0, n=len(self) * 2) 

1390 elif other._ps: 

1391 self._facc_scalar(other._ps_neg) 

1392 elif self._scalar(other, op): 

1393 self._facc_scalar_(-other) 

1394 return self 

1395 

1396 def fsum(self, xs=()): 

1397 '''Add an iterable's items, summate and return the 

1398 current precision running sum. 

1399 

1400 @arg xs: Iterable of items to add (each item C{scalar} 

1401 or an L{Fsum} or L{Fsum2Tuple} instance). 

1402 

1403 @return: Precision running sum (C{float} or C{int}). 

1404 

1405 @see: Method L{Fsum.fadd}. 

1406 

1407 @note: Accumulation can continue after summation. 

1408 ''' 

1409 return self._facc(xs)._fprs 

1410 

1411 def fsum_(self, *xs): 

1412 '''Add any positional items, summate and return the 

1413 current precision running sum. 

1414 

1415 @arg xs: Items to add (each C{scalar} or an L{Fsum} 

1416 or L{Fsum2Tuple} instance), all positional. 

1417 

1418 @return: Precision running sum (C{float} or C{int}). 

1419 

1420 @see: Methods L{Fsum.fsum}, L{Fsum.Fsum_} and L{Fsum.fsumf_}. 

1421 ''' 

1422 return self._facc_1(xs)._fprs 

1423 

1424 @property_RO 

1425 def _Fsum(self): # like L{Fsum2Tuple._Fsum}, for C{_2floats}, .fstats 

1426 return self # NOT @Property_RO, see .copy and ._copy_2 

1427 

1428 def Fsum_(self, *xs, **name): 

1429 '''Like method L{Fsum.fsum_} but returning a named L{Fsum}. 

1430 

1431 @kwarg name: Optional name (C{str}). 

1432 

1433 @return: Copy of this updated instance (L{Fsum}). 

1434 ''' 

1435 return self._facc_1(xs)._copy_2(self.Fsum_, **name) 

1436 

1437 def Fsum2Tuple_(self, *xs, **name): 

1438 '''Like method L{Fsum.fsum_} but returning a named L{Fsum2Tuple}. 

1439 

1440 @kwarg name: Optional name (C{str}). 

1441 

1442 @return: Precision running sum (L{Fsum2Tuple}). 

1443 ''' 

1444 return Fsum2Tuple(self._facc_1(xs)._fprs2, **name) 

1445 

1446 def fsum2(self, xs=(), **name): 

1447 '''Add an iterable's items, summate and return the 

1448 current precision running sum I{and} the C{residual}. 

1449 

1450 @arg xs: Iterable of items to add (each item C{scalar} 

1451 or an L{Fsum} or L{Fsum2Tuple} instance). 

1452 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1453 

1454 @return: L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} the 

1455 current precision running sum and C{residual}, the 

1456 (precision) sum of the remaining C{partials}. The 

1457 C{residual is INT0} if the C{fsum} is considered 

1458 to be I{exact}. 

1459 

1460 @see: Methods L{Fsum.fint2}, L{Fsum.fsum} and L{Fsum.fsum2_} 

1461 ''' 

1462 t = self._facc(xs)._fprs2 

1463 return t.dup(name=name) if name else t 

1464 

1465 def fsum2_(self, *xs): 

1466 '''Add any positional items, summate and return the current 

1467 precision running sum and the I{differential}. 

1468 

1469 @arg xs: Values to add (each C{scalar} or an L{Fsum} or 

1470 L{Fsum2Tuple} instance), all positional. 

1471 

1472 @return: 2Tuple C{(fsum, delta)} with the current, precision 

1473 running C{fsum} like method L{Fsum.fsum} and C{delta}, 

1474 the difference with previous running C{fsum}, C{float}. 

1475 

1476 @see: Methods L{Fsum.fsum_} and L{Fsum.fsum}. 

1477 ''' 

1478 return self._fsum2(xs, self._facc_1) 

1479 

1480 def _fsum2(self, xs, _facc, **origin): 

1481 '''(INTERNAL) Helper for L{Fsum.fsum2_} and L{Fsum.fsum2f_}. 

1482 ''' 

1483 p, q = self._fprs2 

1484 if xs: 

1485 s, r = _facc(xs, **origin)._fprs2 

1486 return s, _2delta(s - p, r - q) # _fsum(_1primed((s, -p, r, -q)) 

1487 else: 

1488 return p, _0_0 

1489 

1490 def fsumf_(self, *xs): 

1491 '''Like method L{Fsum.fsum_} iff I{all} C{B{xs}} are I{known to be scalar}. 

1492 ''' 

1493 return self._facc_scalar(xs)._fprs 

1494 

1495 def Fsumf_(self, *xs): 

1496 '''Like method L{Fsum.Fsum_} iff I{all} C{B{xs}} are I{known to be scalar}. 

1497 ''' 

1498 return self._facc_scalar(xs)._copy_2(self.Fsumf_) 

1499 

1500 def fsum2f_(self, *xs): 

1501 '''Like method L{Fsum.fsum2_} iff I{all} C{B{xs}} are I{known to be scalar}. 

1502 ''' 

1503 return self._fsum2(xs, self._facc_scalar, origin=1) 

1504 

1505# ftruediv = __itruediv__ # for naming consistency? 

1506 

1507 def _ftruediv(self, other, op, **raiser_RESIDUAL): 

1508 '''(INTERNAL) Apply C{B{self} /= B{other}}. 

1509 ''' 

1510 n = _1_0 

1511 if _isFsumTuple(other): 

1512 if other is self or self == other: 

1513 return self._fset(n, n=len(self)) 

1514 d, r = other._fprs2 

1515 if r: 

1516 R = self._raiser(r, d, **raiser_RESIDUAL) 

1517 if R: 

1518 raise self._ResidualError(op, other, r, **R) 

1519 d, n = other.as_integer_ratio() 

1520 else: 

1521 d = self._scalar(other, op) 

1522 try: 

1523 s = n / d 

1524 except Exception as X: 

1525 raise self._ErrorX(X, op, other) 

1526 f = self._mul_scalar(s, _mul_op_) # handles 0, INF, NAN 

1527 return self._fset(f) 

1528 

1529 @property_RO 

1530 def imag(self): 

1531 '''Get the C{imaginary} part of this instance (C{0.0}, always). 

1532 

1533 @see: Property L{Fsum.real}. 

1534 ''' 

1535 return _0_0 

1536 

1537 def int_float(self, **raiser_RESIDUAL): 

1538 '''Return this instance' current running sum as C{int} or C{float}. 

1539 

1540 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1541 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1542 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1543 

1544 @return: This C{integer} sum if this instance C{is_integer}, 

1545 otherwise return the C{float} sum if the residual is 

1546 zero or not significant. 

1547 

1548 @raise ResidualError: Non-zero, significant residual or invalid 

1549 B{C{RESIDUAL}}. 

1550 

1551 @see: Methods L{Fsum.fint}, L{Fsum.fint2}, L{Fsum.RESIDUAL} and 

1552 property L{Fsum.as_iscalar}. 

1553 ''' 

1554 s, r = self._fint2 

1555 if r: 

1556 s, r = self._fprs2 

1557 if r: # PYCHOK no cover 

1558 R = self._raiser(r, s, **raiser_RESIDUAL) 

1559 if R: 

1560 t = _stresidual(_non_zero_, r, **R) 

1561 raise ResidualError(int_float=s, txt=t) 

1562 s = float(s) 

1563 return s 

1564 

1565 def is_exact(self): 

1566 '''Is this instance' running C{fsum} considered to be exact? 

1567 (C{bool}), C{True} only if the C{residual is }L{INT0}. 

1568 ''' 

1569 return self.residual is INT0 

1570 

1571 def is_integer(self): 

1572 '''Is this instance' running sum C{integer}? (C{bool}). 

1573 

1574 @see: Methods L{Fsum.fint}, L{Fsum.fint2} and L{Fsum.is_scalar}. 

1575 ''' 

1576 _, r = self._fint2 

1577 return False if r else True 

1578 

1579 def is_math_fsum(self): 

1580 '''Return whether functions L{fsum}, L{fsum_}, L{fsum1} and 

1581 L{fsum1_} plus partials summation are based on Python's 

1582 C{math.fsum} or not. 

1583 

1584 @return: C{2} if all functions and partials summation 

1585 are based on C{math.fsum}, C{True} if only 

1586 the functions are based on C{math.fsum} (and 

1587 partials summation is not) or C{False} if 

1588 none are. 

1589 ''' 

1590 f = Fsum._math_fsum 

1591 return 2 if _psum is f else bool(f) 

1592 

1593 def is_scalar(self, **raiser_RESIDUAL): 

1594 '''Is this instance' running sum C{scalar} without residual or with 

1595 a residual I{ratio} not exceeding the RESIDUAL threshold? 

1596 

1597 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1598 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1599 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1600 

1601 @return: C{True} if this instance' non-zero residual C{ratio} exceeds 

1602 the L{RESIDUAL<Fsum.RESIDUAL>} threshold (C{bool}). 

1603 

1604 @raise ResidualError: Non-zero, significant residual or invalid 

1605 B{C{RESIDUAL}}. 

1606 

1607 @see: Method L{Fsum.RESIDUAL}, L{Fsum.is_integer} and property 

1608 L{Fsum.as_iscalar}. 

1609 ''' 

1610 s, r = self._fprs2 

1611 return False if r and self._raiser(r, s, **raiser_RESIDUAL) else True 

1612 

1613 def _mul_Fsum(self, other, op=_mul_op_): # in .fmath.Fhorner 

1614 '''(INTERNAL) Return C{B{self} * B{other}} as L{Fsum} or C{0}. 

1615 ''' 

1616 # assert _isFsumTuple(other) 

1617 if self._ps and other._ps: 

1618 f = self._ps_mul(op, *other._ps) # NO .as_iscalar! 

1619 else: 

1620 f = _0_0 

1621 return f 

1622 

1623 def _mul_scalar(self, factor, op): # in .fmath.Fhorner 

1624 '''(INTERNAL) Return C{B{self} * scalar B{factor}} as L{Fsum}, C{0.0} or C{self}. 

1625 ''' 

1626 # assert isscalar(factor) 

1627 if self._ps and self._finite(factor, op): 

1628 f = self if factor == _1_0 else ( 

1629 self._neg if factor == _N_1_0 else 

1630 self._ps_mul(op, factor).as_iscalar) 

1631 else: 

1632 f = _0_0 

1633 return f 

1634 

1635# @property_RO 

1636# def _n_d(self): 

1637# n, d = self.as_integer_ratio() 

1638# return n / d 

1639 

1640 @property_RO 

1641 def _neg(self): 

1642 '''(INTERNAL) Return C{Fsum(-self)} or scalar C{NEG0}. 

1643 ''' 

1644 return _Psum(self._ps_neg) if self._ps else NEG0 

1645 

1646 @property_RO 

1647 def partials(self): 

1648 '''Get this instance' current, partial sums (C{tuple} of C{float}s). 

1649 ''' 

1650 return tuple(self._ps) 

1651 

1652 def pow(self, x, *mod, **raiser_RESIDUAL): 

1653 '''Return C{B{self}**B{x}} as L{Fsum}. 

1654 

1655 @arg x: The exponent (C{scalar} or L{Fsum}). 

1656 @arg mod: Optional modulus (C{int} or C{None}) for the 3-argument 

1657 C{pow(B{self}, B{other}, B{mod})} version. 

1658 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1659 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1660 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1661 

1662 @return: The C{pow(self, B{x})} or C{pow(self, B{x}, *B{mod})} 

1663 result (L{Fsum}). 

1664 

1665 @raise ResidualError: Non-zero, significant residual or invalid 

1666 B{C{RESIDUAL}}. 

1667 

1668 @note: If B{C{mod}} is given as C{None}, the result will be an 

1669 C{integer} L{Fsum} provided this instance C{is_integer} 

1670 or set to C{integer} by an L{Fsum.fint} call. 

1671 

1672 @see: Methods L{Fsum.__ipow__}, L{Fsum.fint}, L{Fsum.is_integer} 

1673 and L{Fsum.root}. 

1674 ''' 

1675 f = self._copy_2(self.pow) 

1676 return f._fpow(x, _pow_op_, *mod, **raiser_RESIDUAL) # f = pow(f, x, *mod) 

1677 

1678 def _pow(self, other, unused, op, **raiser_RESIDUAL): 

1679 '''Return C{B{self} ** B{other}}. 

1680 ''' 

1681 if _isFsumTuple(other): 

1682 f = self._pow_Fsum(other, op, **raiser_RESIDUAL) 

1683 elif self._scalar(other, op): 

1684 x = self._finite(other, op) 

1685 f = self._pow_scalar(x, other, op, **raiser_RESIDUAL) 

1686 else: 

1687 f = self._pow_0_1(0, other) 

1688 return f 

1689 

1690 def _pow_0_1(self, x, other): 

1691 '''(INTERNAL) Return B{C{self}**1} or C{B{self}**0 == 1.0}. 

1692 ''' 

1693 return self if x else (1 if isint(other) and self.is_integer() else _1_0) 

1694 

1695 def _pow_2_3(self, b, x, other, op, *mod, **raiser_RESIDUAL): 

1696 '''(INTERNAL) 2-arg C{pow(B{b}, scalar B{x})} and 3-arg C{pow(B{b}, 

1697 B{x}, int B{mod} or C{None})}, embellishing errors. 

1698 ''' 

1699 

1700 if mod: # b, x, mod all C{int}, unless C{mod} is C{None} 

1701 m = mod[0] 

1702 # assert _isFsumTuple(b) 

1703 

1704 def _s(s, r): 

1705 R = self._raiser(r, s, **raiser_RESIDUAL) 

1706 if R: 

1707 raise self._ResidualError(op, other, r, mod=m, **R) 

1708 return s 

1709 

1710 b = _s(*(b._fprs2 if m is None else b._fint2)) 

1711 x = _s(*_2scalar2(x)) 

1712 

1713 try: 

1714 # 0**INF == 0.0, 1**INF == 1.0, -1**2.3 == -(1**2.3) 

1715 s = pow(b, x, *mod) 

1716 if iscomplex(s): 

1717 # neg**frac == complex in Python 3+, but ValueError in 2- 

1718 raise ValueError(_strcomplex(s, b, x, *mod)) 

1719 return self._finite(s) 

1720 except Exception as X: 

1721 raise self._ErrorX(X, op, other, *mod) 

1722 

1723 def _pow_Fsum(self, other, op, **raiser_RESIDUAL): 

1724 '''(INTERNAL) Return C{B{self} **= B{other}} for C{_isFsumTuple(other)}. 

1725 ''' 

1726 # assert _isFsumTuple(other) 

1727 x, r = other._fprs2 

1728 f = self._pow_scalar(x, other, op, **raiser_RESIDUAL) 

1729 if f and r: 

1730 f *= self._pow_scalar(r, other, op, **raiser_RESIDUAL) 

1731 return f 

1732 

1733 def _pow_int(self, x, other, op, **raiser_RESIDUAL): 

1734 '''(INTERNAL) Return C{B{self} **= B{x}} for C{int B{x} >= 0}. 

1735 ''' 

1736 # assert isint(x) and x >= 0 

1737 ps = self._ps 

1738 if len(ps) > 1: 

1739 _mul_Fsum = Fsum._mul_Fsum 

1740 if x > 4: 

1741 p = self 

1742 f = self if (x & 1) else _Psum_(_1_0) 

1743 m = x >> 1 # // 2 

1744 while m: 

1745 p = _mul_Fsum(p, p, op) # p **= 2 

1746 if (m & 1): 

1747 f = _mul_Fsum(f, p, op) # f *= p 

1748 m >>= 1 # //= 2 

1749 elif x > 1: # self**2, 3, or 4 

1750 f = _mul_Fsum(self, self, op) 

1751 if x > 2: # self**3 or 4 

1752 p = self if x < 4 else f 

1753 f = _mul_Fsum(f, p, op) 

1754 else: # self**1 or self**0 == 1 or _1_0 

1755 f = self._pow_0_1(x, other) 

1756 elif ps: # self._ps[0]**x 

1757 f = self._pow_2_3(ps[0], x, other, op, **raiser_RESIDUAL) 

1758 else: # PYCHOK no cover 

1759 # 0**pos_int == 0, but 0**0 == 1 

1760 f = 0 if x else 1 

1761 return f 

1762 

1763 def _pow_scalar(self, x, other, op, **raiser_RESIDUAL): 

1764 '''(INTERNAL) Return C{self**B{x}} for C{scalar B{x}}. 

1765 ''' 

1766 s, r = self._fprs2 

1767 if r: 

1768 # assert s != 0 

1769 if isint(x, both=True): # self**int 

1770 x = int(x) 

1771 y = abs(x) 

1772 if y > 1: 

1773 f = self._pow_int(y, other, op, **raiser_RESIDUAL) 

1774 if x > 0: # i.e. > 1 

1775 return f # Fsum or scalar 

1776 # assert x < 0 # i.e. < -1 

1777 if _isFsum(f): 

1778 s, r = f._fprs2 

1779 if r: 

1780 return _1_Over(f, op, **raiser_RESIDUAL) 

1781 else: # scalar 

1782 s = f 

1783 # use s**(-1) to get the CPython 

1784 # float_pow error iff s is zero 

1785 x = -1 

1786 elif x < 0: # self**(-1) 

1787 return _1_Over(self, op, **raiser_RESIDUAL) # 1 / self 

1788 else: # self**1 or self**0 

1789 return self._pow_0_1(x, other) # self, 1 or 1.0 

1790 else: # self**fractional 

1791 R = self._raiser(r, s, **raiser_RESIDUAL) 

1792 if R: 

1793 raise self._ResidualError(op, other, r, **R) 

1794 n, d = self.as_integer_ratio() 

1795 if abs(n) > abs(d): 

1796 n, d, x = d, n, (-x) 

1797 s = n / d 

1798 # assert isscalar(s) and isscalar(x) 

1799 return self._pow_2_3(s, x, other, op, **raiser_RESIDUAL) 

1800 

1801 def _ps_acc(self, ps, xs, up=True, **unused): 

1802 '''(INTERNAL) Accumulate C{xs} known scalars into list C{ps}. 

1803 ''' 

1804 n = 0 

1805 _2s = _2sum 

1806 for x in (tuple(xs) if xs is ps else xs): 

1807 # assert isscalar(x) and _isfinite(x) 

1808 if x: 

1809 i = 0 

1810 for p in ps: 

1811 x, p = _2s(x, p) 

1812 if p: 

1813 ps[i] = p 

1814 i += 1 

1815 ps[i:] = (x,) if x else () 

1816 n += 1 

1817 if n: 

1818 self._n += n 

1819 # Fsum._ps_max = max(Fsum._ps_max, len(ps)) 

1820 if up: 

1821 self._update() 

1822 return ps 

1823 

1824 def _ps_mul(self, op, *factors): 

1825 '''(INTERNAL) Multiply this instance' C{partials} with 

1826 each scalar C{factor} and accumulate into an C{Fsum}. 

1827 ''' 

1828 def _pfs(ps, fs): 

1829 if len(ps) < len(fs): 

1830 ps, fs = fs, ps 

1831 _fin = _isfinite 

1832 for f in fs: 

1833 for p in ps: 

1834 p *= f 

1835 yield p if _fin(p) else self._finite(p, op) 

1836 

1837 return Fsum()._facc_scalar(_pfs(self._ps, factors), up=False) 

1838 

1839 @property_RO 

1840 def _ps_neg(self): 

1841 '''(INTERNAL) Yield the partials, I{negated}. 

1842 ''' 

1843 for p in self._ps: 

1844 yield -p 

1845 

1846 def _ps_1sum(self, *less): 

1847 '''(INTERNAL) Return the partials sum, 1-primed C{less} some scalars. 

1848 ''' 

1849 def _1pls(ps, ls): 

1850 yield _1_0 

1851 for p in ps: 

1852 yield p 

1853 for p in ls: 

1854 yield -p 

1855 yield _N_1_0 

1856 

1857 return _fsum(_1pls(self._ps, less)) 

1858 

1859 def _raiser(self, r, s, raiser=True, **RESIDUAL): 

1860 '''(INTERNAL) Does ratio C{r / s} exceed the RESIDUAL threshold 

1861 I{and} is residual C{r} I{non-zero} or I{significant} (for a 

1862 negative respectively positive C{RESIDUAL} threshold)? 

1863 ''' 

1864 if r and raiser: 

1865 t = self._RESIDUAL 

1866 if RESIDUAL: 

1867 t = _threshold(t, **RESIDUAL) 

1868 if t < 0 or (s + r) != s: 

1869 q = (r / s) if s else s # == 0. 

1870 if fabs(q) > fabs(t): 

1871 return dict(ratio=q, R=t) 

1872 return {} 

1873 

1874 rdiv = __rtruediv__ 

1875 

1876 @property_RO 

1877 def real(self): 

1878 '''Get the C{real} part of this instance (C{float}). 

1879 

1880 @see: Methods L{Fsum.__float__} and L{Fsum.fsum} 

1881 and properties L{Fsum.ceil}, L{Fsum.floor}, 

1882 L{Fsum.imag} and L{Fsum.residual}. 

1883 ''' 

1884 return float(self._fprs) 

1885 

1886 @property_RO 

1887 def residual(self): 

1888 '''Get this instance' residual (C{float} or C{int}): the 

1889 C{sum(partials)} less the precision running sum C{fsum}. 

1890 

1891 @note: The C{residual is INT0} iff the precision running 

1892 C{fsum} is considered to be I{exact}. 

1893 

1894 @see: Methods L{Fsum.fsum}, L{Fsum.fsum2} and L{Fsum.is_exact}. 

1895 ''' 

1896 return self._fprs2.residual 

1897 

1898 def RESIDUAL(self, *threshold): 

1899 '''Get and set this instance' I{ratio} for raising L{ResidualError}s, 

1900 overriding the default from env variable C{PYGEODESY_FSUM_RESIDUAL}. 

1901 

1902 @arg threshold: If C{scalar}, the I{ratio} to exceed for raising 

1903 L{ResidualError}s in division and exponention, if 

1904 C{None} restore the default set with env variable 

1905 C{PYGEODESY_FSUM_RESIDUAL} or if omitted, keep the 

1906 current setting. 

1907 

1908 @return: The previous C{RESIDUAL} setting (C{float}), default C{0.0}. 

1909 

1910 @raise ResidualError: Invalid B{C{threshold}}. 

1911 

1912 @note: L{ResidualError}s may be thrown if the non-zero I{ratio} 

1913 C{residual / fsum} exceeds the given B{C{threshold}} and 

1914 if the C{residual} is non-zero and I{significant} vs the 

1915 C{fsum}, i.e. C{(fsum + residual) != fsum} and if optional 

1916 keyword argument C{raiser=False} is missing. Specify a 

1917 negative B{C{threshold}} for only non-zero C{residual} 

1918 testing without I{significant}. 

1919 ''' 

1920 r = self._RESIDUAL 

1921 if threshold: 

1922 t = threshold[0] 

1923 self._RESIDUAL = Fsum._RESIDUAL if t is None else ( # for ... 

1924 (_0_0 if t else _1_0) if isbool(t) else 

1925 _threshold(t)) # ... backward compatibility 

1926 return r 

1927 

1928 def _ResidualError(self, op, other, residual, **mod_R): 

1929 '''(INTERNAL) Non-zero B{C{residual}} etc. 

1930 ''' 

1931 def _p(mod=None, R=0, **unused): # ratio=0 

1932 return (_non_zero_ if R < 0 else _significant_) \ 

1933 if mod is None else _integer_ 

1934 

1935 t = _stresidual(_p(**mod_R), residual, **mod_R) 

1936 return self._Error(op, other, ResidualError, txt=t) 

1937 

1938 def root(self, root, **raiser_RESIDUAL): 

1939 '''Return C{B{self}**(1 / B{root})} as L{Fsum}. 

1940 

1941 @arg root: The order (C{scalar} or L{Fsum}), non-zero. 

1942 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1943 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1944 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1945 

1946 @return: The C{self ** (1 / B{root})} result (L{Fsum}). 

1947 

1948 @raise ResidualError: Non-zero, significant residual or invalid 

1949 B{C{RESIDUAL}}. 

1950 

1951 @see: Method L{Fsum.pow}. 

1952 ''' 

1953 x = _1_Over(root, _truediv_op_, **raiser_RESIDUAL) 

1954 f = self._copy_2(self.root) 

1955 return f._fpow(x, f.name, **raiser_RESIDUAL) # == pow(f, x) 

1956 

1957 def _scalar(self, other, op, **txt): 

1958 '''(INTERNAL) Return scalar C{other}. 

1959 ''' 

1960 if isscalar(other): 

1961 return other 

1962 raise self._Error(op, other, _TypeError, **txt) # _invalid_ 

1963 

1964 def signOf(self, res=True): 

1965 '''Determine the sign of this instance. 

1966 

1967 @kwarg res: If C{True} consider, otherwise 

1968 ignore the residual (C{bool}). 

1969 

1970 @return: The sign (C{int}, -1, 0 or +1). 

1971 ''' 

1972 s, r = self._fprs2 

1973 r = (-r) if res else 0 

1974 return _signOf(s, r) 

1975 

1976 def toRepr(self, **lenc_prec_sep_fmt): # PYCHOK signature 

1977 '''Return this C{Fsum} instance as representation. 

1978 

1979 @kwarg lenc_prec_sep_fmt: Optional keyword arguments 

1980 for method L{Fsum.toStr}. 

1981 

1982 @return: This instance (C{repr}). 

1983 ''' 

1984 return Fmt.repr_at(self, self.toStr(**lenc_prec_sep_fmt)) 

1985 

1986 def toStr(self, lenc=True, **prec_sep_fmt): # PYCHOK signature 

1987 '''Return this C{Fsum} instance as string. 

1988 

1989 @kwarg lenc: If C{True} include the current C{[len]} of this 

1990 L{Fsum} enclosed in I{[brackets]} (C{bool}). 

1991 @kwarg prec_sep_fmt: Optional keyword arguments for method 

1992 L{Fsum2Tuple.toStr}. 

1993 

1994 @return: This instance (C{str}). 

1995 ''' 

1996 p = self.classname 

1997 if lenc: 

1998 p = Fmt.SQUARE(p, len(self)) 

1999 n = _enquote(self.name, white=_UNDER_) 

2000 t = self._fprs2.toStr(**prec_sep_fmt) 

2001 return NN(p, _SPACE_, n, t) 

2002 

2003 def _truediv(self, other, op, **raiser_RESIDUAL): 

2004 '''(INTERNAL) Return C{B{self} / B{other}} as an L{Fsum}. 

2005 ''' 

2006 f = self._copy_2(self.__truediv__) 

2007 return f._ftruediv(other, op, **raiser_RESIDUAL) 

2008 

2009 def _update(self, updated=True): # see ._fset 

2010 '''(INTERNAL) Zap all cached C{Property_RO} values. 

2011 ''' 

2012 if updated: 

2013 _pop = self.__dict__.pop 

2014 for p in _ROs: 

2015 _ = _pop(p, None) 

2016# Fsum._fint2._update(self) 

2017# Fsum._fprs ._update(self) 

2018# Fsum._fprs2._update(self) 

2019 return self # for .fset_ 

2020 

2021_ROs = _allPropertiesOf_n(3, Fsum, Property_RO) # PYCHOK see Fsum._update 

2022 

2023 

2024def _Float_Int(arg, **name_Error): 

2025 '''(INTERNAL) Unit of L{Fsum2Tuple} items. 

2026 ''' 

2027 U = Int if isint(arg) else Float 

2028 return U(arg, **name_Error) 

2029 

2030 

2031class DivMod2Tuple(_NamedTuple): 

2032 '''2-Tuple C{(div, mod)} with the quotient C{div} and remainder 

2033 C{mod} results of a C{divmod} operation. 

2034 

2035 @note: Quotient C{div} an C{int} in Python 3+ but a C{float} 

2036 in Python 2-. Remainder C{mod} an L{Fsum} instance. 

2037 ''' 

2038 _Names_ = (_div_, _mod_) 

2039 _Units_ = (_Float_Int, Fsum) 

2040 

2041 

2042class Fsum2Tuple(_NamedTuple): # in .fstats 

2043 '''2-Tuple C{(fsum, residual)} with the precision running C{fsum} 

2044 and the C{residual}, the sum of the remaining partials. Each 

2045 item is C{float} or C{int}. 

2046 

2047 @note: If the C{residual is INT0}, the C{fsum} is considered 

2048 to be I{exact}, see method L{Fsum2Tuple.is_exact}. 

2049 ''' 

2050 _Names_ = ( Fsum.fsum.__name__, Fsum.residual.name) 

2051 _Units_ = (_Float_Int, _Float_Int) 

2052 

2053 def __abs__(self): # in .fmath 

2054 return self._Fsum.__abs__() 

2055 

2056 def __bool__(self): # PYCHOK Python 3+ 

2057 return bool(self._Fsum) 

2058 

2059 def __eq__(self, other): 

2060 return self._other_op(other, self.__eq__) 

2061 

2062 def __float__(self): 

2063 return self._Fsum.__float__() 

2064 

2065 def __ge__(self, other): 

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

2067 

2068 def __gt__(self, other): 

2069 return self._other_op(other, self.__gt__) 

2070 

2071 def __le__(self, other): 

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

2073 

2074 def __lt__(self, other): 

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

2076 

2077 def __int__(self): 

2078 return self._Fsum.__int__() 

2079 

2080 def __ne__(self, other): 

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

2082 

2083 def __neg__(self): 

2084 return self._Fsum.__neg__() 

2085 

2086 __nonzero__ = __bool__ # Python 2- 

2087 

2088 def __pos__(self): 

2089 return self._Fsum.__pos__() 

2090 

2091 def as_integer_ratio(self): 

2092 '''Return this instance as the ratio of 2 integers. 

2093 

2094 @see: Method L{Fsum.as_integer_ratio} for further details. 

2095 ''' 

2096 return self._Fsum.as_integer_ratio() 

2097 

2098 @property_RO 

2099 def _fint2(self): 

2100 return self._Fsum._fint2 

2101 

2102 @property_RO 

2103 def _fprs2(self): 

2104 return self._Fsum._fprs2 

2105 

2106 @Property_RO 

2107 def _Fsum(self): # this C{Fsum2Tuple} as L{Fsum}, in .fstats 

2108 s, r = _s_r(*self) 

2109 ps = (r, s) if r else (s,) 

2110 return _Psum(ps, name=self.name) 

2111 

2112 def Fsum_(self, *xs, **name_RESIDUAL): 

2113 '''Return this C{Fsum2Tuple} as an L{Fsum} plus some C{xs}. 

2114 ''' 

2115 f = _Psum(self._Fsum._ps, **name_RESIDUAL) 

2116 return f._facc_1(xs, up=False) if xs else f 

2117 

2118 def is_exact(self): 

2119 '''Is this L{Fsum2Tuple} considered to be exact? (C{bool}). 

2120 ''' 

2121 return self._Fsum.is_exact() 

2122 

2123 def is_integer(self): 

2124 '''Is this L{Fsum2Tuple} C{integer}? (C{bool}). 

2125 ''' 

2126 return self._Fsum.is_integer() 

2127 

2128 def _mul_scalar(self, other, op): # for Fsum._fmul 

2129 return self._Fsum._mul_scalar(other, op) 

2130 

2131 @property_RO 

2132 def _n(self): 

2133 return self._Fsum._n 

2134 

2135 def _other_op(self, other, which): 

2136 C, s = (tuple, self) if isinstance(other, tuple) else (Fsum, self._Fsum) 

2137 return getattr(C, which.__name__)(s, other) 

2138 

2139 @property_RO 

2140 def _ps(self): 

2141 return self._Fsum._ps 

2142 

2143 @property_RO 

2144 def _ps_neg(self): 

2145 return self._Fsum._ps_neg 

2146 

2147 def signOf(self, **res): 

2148 '''Like method L{Fsum.signOf}. 

2149 ''' 

2150 return self._Fsum.signOf(**res) 

2151 

2152 def toStr(self, fmt=Fmt.g, **prec_sep): # PYCHOK signature 

2153 '''Return this L{Fsum2Tuple} as string (C{str}). 

2154 

2155 @kwarg fmt: Optional C{float} format (C{letter}). 

2156 @kwarg prec_sep: Optional keyword arguments for function 

2157 L{fstr<streprs.fstr>}. 

2158 ''' 

2159 return Fmt.PAREN(fstr(self, fmt=fmt, strepr=str, force=False, **prec_sep)) 

2160 

2161_Fsum_Fsum2Tuple_types = Fsum, Fsum2Tuple # PYCHOK lines 

2162 

2163 

2164class ResidualError(_ValueError): 

2165 '''Error raised for a division, power or root operation of 

2166 an L{Fsum} instance with a C{residual} I{ratio} exceeding 

2167 the L{RESIDUAL<Fsum.RESIDUAL>} threshold. 

2168 

2169 @see: Module L{pygeodesy.fsums} and method L{Fsum.RESIDUAL}. 

2170 ''' 

2171 pass 

2172 

2173 

2174try: 

2175 from math import fsum as _fsum # precision IEEE-754 sum, Python 2.6+ 

2176 

2177 # make sure _fsum works as expected (XXX check 

2178 # float.__getformat__('float')[:4] == 'IEEE'?) 

2179 if _fsum((1, 1e101, 1, -1e101)) != 2: # PYCHOK no cover 

2180 del _fsum # nope, remove _fsum ... 

2181 raise ImportError # ... use _fsum below 

2182 

2183 Fsum._math_fsum = _sum = _fsum # PYCHOK exported 

2184except ImportError: 

2185 _sum = sum # Fsum(NAN) exception fall-back, in .elliptic 

2186 

2187 def _fsum(xs): 

2188 '''(INTERNAL) Precision summation, Python 2.5-. 

2189 ''' 

2190 F = Fsum() 

2191 F.name = _fsum.__name__ 

2192 return F._facc(xs, up=False)._fprs2.fsum 

2193 

2194 

2195def fsum(xs, floats=False): 

2196 '''Precision floating point summation based on/like Python's C{math.fsum}. 

2197 

2198 @arg xs: Iterable of items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} 

2199 instance). 

2200 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to 

2201 be scalar} (C{bool}). 

2202 

2203 @return: Precision C{fsum} (C{float}). 

2204 

2205 @raise OverflowError: Partial C{2sum} overflow. 

2206 

2207 @raise TypeError: Non-scalar B{C{xs}} item. 

2208 

2209 @raise ValueError: Invalid or non-finite B{C{xs}} item. 

2210 

2211 @note: Exception and I{non-finite} handling may differ if not based 

2212 on Python's C{math.fsum}. 

2213 

2214 @see: Class L{Fsum} and methods L{Fsum.fsum} and L{Fsum.fadd}. 

2215 ''' 

2216 return _fsum(xs if floats is True else _2floats(xs)) if xs else _0_0 # PYCHOK yield 

2217 

2218 

2219def fsum_(*xs, **floats): 

2220 '''Precision floating point summation of all positional items. 

2221 

2222 @arg xs: Items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} instance), 

2223 all positional. 

2224 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to 

2225 be scalar} (C{bool}). 

2226 

2227 @see: Function L{fsum<fsums.fsum>} for further details. 

2228 ''' 

2229 return _fsum(xs if _xkwds_get1(floats, floats=False) is True else 

2230 _2floats(xs, origin=1)) if xs else _0_0 # PYCHOK yield 

2231 

2232 

2233def fsumf_(*xs): 

2234 '''Precision floating point summation iff I{all} C{B{xs}} items are I{known to be scalar}. 

2235 

2236 @see: Function L{fsum_<fsums.fsum_>} for further details. 

2237 ''' 

2238 return _fsum(xs) if xs else _0_0 

2239 

2240 

2241def fsum1(xs, floats=False): 

2242 '''Precision floating point summation, 1-primed. 

2243 

2244 @arg xs: Iterable of items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} 

2245 instance). 

2246 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to 

2247 be scalar} (C{bool}). 

2248 

2249 @see: Function L{fsum<fsums.fsum>} for further details. 

2250 ''' 

2251 return _fsum(_1primed(xs if floats is True else _2floats(xs))) if xs else _0_0 # PYCHOK yield 

2252 

2253 

2254def fsum1_(*xs, **floats): 

2255 '''Precision floating point summation, 1-primed of all positional items. 

2256 

2257 @arg xs: Items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} instance), 

2258 all positional. 

2259 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to 

2260 be scalar} (C{bool}). 

2261 

2262 @see: Function L{fsum_<fsums.fsum_>} for further details. 

2263 ''' 

2264 return _fsum(_1primed(xs if _xkwds_get1(floats, floats=False) is True else 

2265 _2floats(xs, origin=1))) if xs else _0_0 # PYCHOK yield 

2266 

2267 

2268def fsum1f_(*xs): 

2269 '''Precision floating point summation iff I{all} C{B{xs}} items are I{known to be scalar}. 

2270 

2271 @see: Function L{fsum_<fsums.fsum_>} for further details. 

2272 ''' 

2273 return _fsum(_1primed(xs)) if xs else _0_0 

2274 

2275 

2276if __name__ == '__main__': 

2277 

2278 # usage: [env _psum=fsum] python3 -m pygeodesy.fsums 

2279 

2280 if _getenv(_psum.__name__, NN) == _fsum.__name__: 

2281 _psum = _fsum 

2282 

2283 def _test(n): 

2284 # copied from Hettinger, see L{Fsum} reference 

2285 from pygeodesy import frandoms, printf 

2286 

2287 printf(_fsum.__name__, end=_COMMASPACE_) 

2288 printf(_psum.__name__, end=_COMMASPACE_) 

2289 

2290 F = Fsum() 

2291 if F.is_math_fsum(): 

2292 for t in frandoms(n, seeded=True): 

2293 assert float(F.fset_(*t)) == _fsum(t) 

2294 printf(_DOT_, end=NN) 

2295 printf(NN) 

2296 

2297 _test(128) 

2298 

2299# **) MIT License 

2300# 

2301# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

2302# 

2303# Permission is hereby granted, free of charge, to any person obtaining a 

2304# copy of this software and associated documentation files (the "Software"), 

2305# to deal in the Software without restriction, including without limitation 

2306# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

2307# and/or sell copies of the Software, and to permit persons to whom the 

2308# Software is furnished to do so, subject to the following conditions: 

2309# 

2310# The above copyright notice and this permission notice shall be included 

2311# in all copies or substantial portions of the Software. 

2312# 

2313# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

2314# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

2315# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

2316# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

2317# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

2318# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

2319# OTHER DEALINGS IN THE SOFTWARE.