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, itemsorted, \ 

27 signOf, _signOf 

28from pygeodesy.constants import INT0, _isfinite, NEG0, _pos_self, \ 

29 _0_0, _1_0, _N_1_0, Float, Int 

30from pygeodesy.errors import _OverflowError, _TypeError, _ValueError, \ 

31 _xError, _xError2, _xkwds_get 

32from pygeodesy.interns import NN, _arg_, _COMMASPACE_, _DASH_, _DOT_, \ 

33 _enquote, _EQUAL_, _from_, _LANGLE_, _NOTEQUAL_, \ 

34 _not_finite_, _PERCENT_, _PLUS_, _RANGLE_, \ 

35 _SLASH_, _SPACE_, _STAR_, _UNDER_ 

36from pygeodesy.lazily import _ALL_LAZY, _getenv, _sys_version_info2 

37from pygeodesy.named import _Named, _NamedTuple, _NotImplemented 

38from pygeodesy.props import _allPropertiesOf_n, deprecated_property_RO, \ 

39 Property_RO, property_RO 

40from pygeodesy.streprs import Fmt, fstr, unstr 

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

42 

43from math import ceil as _ceil, fabs, floor as _floor # PYCHOK used! .ltp 

44 

45__all__ = _ALL_LAZY.fsums 

46__version__ = '24.05.08' 

47 

48_abs = abs 

49_add_op_ = _PLUS_ # in .auxilats.auxAngle 

50_eq_op_ = _EQUAL_ * 2 # _DEQUAL_ 

51_div_ = 'div' 

52_Float = float # in .fstats 

53_floordiv_op_ = _SLASH_ * 2 # _DSLASH_ 

54_fset_op_ = _EQUAL_ 

55_ge_op_ = _RANGLE_ + _EQUAL_ 

56_gt_op_ = _RANGLE_ 

57_iadd_op_ = _add_op_ + _EQUAL_ # in .auxilats.auxAngle, .fstats 

58_integer_ = 'integer' 

59_isAn = isinstance # in .fstats 

60_le_op_ = _LANGLE_ + _EQUAL_ 

61_len = len 

62_List = list 

63_lt_op_ = _LANGLE_ 

64_mod_ = 'mod' 

65_mod_op_ = _PERCENT_ 

66_mul_op_ = _STAR_ 

67_ne_op_ = _NOTEQUAL_ 

68_non_zero_ = 'non-zero' 

69_pow_op_ = _STAR_ * 2 # _DSTAR_ 

70_significant_ = 'significant' 

71_sub_op_ = _DASH_ # in .auxilats.auxAngle 

72_threshold_ = 'threshold' 

73_truediv_op_ = _SLASH_ 

74_Tuple = tuple # in .fstats 

75_divmod_op_ = _floordiv_op_ + _mod_op_ 

76_isub_op_ = _sub_op_ + _fset_op_ # in .auxilats.auxAngle 

77 

78 

79def _2delta(*ab): 

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

81 ''' 

82 try: 

83 a, b = _2sum(*ab) 

84 except _OverflowError: 

85 a, b = ab 

86 return _Float(a if fabs(a) > fabs(b) else b) 

87 

88 

89def _2error(unused): # in .fstats 

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

91 ''' 

92 raise ValueError(_not_finite_) 

93 

94 

95def _2finite(x): 

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

97 ''' 

98 x = _Float(x) 

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

100 

101 

102def _2float(index=None, **name_value): # in .fmath, .fstats 

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

104 ''' 

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

106 try: 

107 return _2finite(v) 

108 except Exception as X: 

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

110 

111 

112def _X_ps(X): # for _2floats only 

113 return X._ps 

114 

115 

116def _2floats(xs, origin=0, _X=_X_ps, _x=_Float): 

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

118 ''' 

119 try: 

120 i, x = origin, None 

121 _fin = _isfinite 

122 _FsT = _Fsum_Fsum2Tuple_types 

123 _is = _isAn 

124 for x in xs: 

125 if _is(x, _FsT): 

126 for p in _X(x._Fsum): 

127 yield p 

128 else: 

129 f = _x(x) 

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

131 i += 1 

132 except Exception as X: 

133 raise _xError(X, Fmt.INDEX(xs=i), x) 

134 

135 

136def _Fsumf_(*xs): # floats=True, in .auxLat, ... 

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

138 ''' 

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

140 

141 

142def _Fsum1f_(*xs): # floats=True, in .albers, ... 

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

144 ''' 

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

146 

147 

148def _2halfeven(s, r, p): 

149 '''(INTERNAL) Round half-even. 

150 ''' 

151 if (p > 0 and r > 0) or \ 

152 (p < 0 and r < 0): # signs match 

153 r *= 2 

154 t = s + r 

155 if r == (t - s): 

156 s = t 

157 return s 

158 

159 

160def _isFsum(x): # in .fmath 

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

162 ''' 

163 return _isAn(x, Fsum) 

164 

165 

166def _isFsumTuple(x): # in .fmath 

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

168 ''' 

169 return _isAn(x, _Fsum_Fsum2Tuple_types) 

170 

171 

172def _1_Over(x, op, **raiser_RESIDUAL): # vs _1_over 

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

174 ''' 

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

176 

177 

178def _1primed(xs): # in .fmath 

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

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

181 ''' 

182 yield _1_0 

183 for x in xs: 

184 yield x 

185 yield _N_1_0 

186 

187 

188def _psum(ps): # PYCHOK used! 

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

190 ''' 

191 # assert _isAn(ps, _List) 

192 i = _len(ps) - 1 

193 s = _0_0 if i < 0 else ps[i] 

194 _2s = _2sum 

195 while i > 0: 

196 i -= 1 

197 s, r = _2s(s, ps[i]) 

198 if r: # sum(ps) became inexact 

199 if s: 

200 ps[i:] = r, s 

201 if i > 0: 

202 s = _2halfeven(s, r, ps[i-1]) 

203 break # return s 

204 s = r # PYCHOK no cover 

205 ps[i:] = s, 

206 return s 

207 

208 

209def _Psum(ps, **name_RESIDUAL): 

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

211 ''' 

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

213 if ps: 

214 f._ps[:] = ps 

215 f._n = _len(f._ps) 

216 return f 

217 

218 

219def _Psum_(*ps, **name_RESIDUAL): 

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

221 ''' 

222 return _Psum(ps, **name_RESIDUAL) 

223 

224 

225def _2scalar2(other): 

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

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

228 ''' 

229 if _isFsumTuple(other): 

230 s, r = other._fint2 

231 if r: 

232 s, r = other._fprs2 

233 if r: # PYCHOK no cover 

234 s = other # L{Fsum} as-is 

235 else: 

236 r = 0 

237 s = other # C{type} as-is 

238 if isint(s, both=True): 

239 s = int(s) 

240 return s, r 

241 

242 

243def _s_r(s, r): 

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

245 ''' 

246 if r: 

247 if fabs(s) < fabs(r): 

248 s, r = r, (s or INT0) 

249 else: 

250 r = INT0 

251 return s, r 

252 

253 

254def _strcomplex(s, *args): 

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

256 ''' 

257 c = _strcomplex.__name__[4:] 

258 n = _DASH_(_len(args), _arg_) 

259 t = unstr(pow, *args) 

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

261 

262 

263def _stresidual(prefix, residual, R=0, **mod_ratio): 

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

265 ''' 

266 p = _stresidual.__name__[3:] 

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

268 for n, v in itemsorted(mod_ratio): 

269 p = Fmt.PARENSPACED(n, Fmt(v)) 

270 t = _COMMASPACE_(t, p) 

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

272 

273 

274def _2sum(a, b): # by .testFmath 

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

276 ''' 

277 s = a + b 

278 if _isfinite(s): 

279 if fabs(a) < fabs(b): 

280 b, a = a, b 

281 return s, (b - (s - a)) 

282 u = unstr(_2sum, a, b) 

283 t = Fmt.PARENSPACED(_not_finite_, s) 

284 raise _OverflowError(u, txt=t) 

285 

286 

287def _threshold(threshold): 

288 '''(INTERNAL) Get the L{ResidualError}s threshold. 

289 ''' 

290 try: 

291 t = _Float(threshold) or _0_0 

292 return t if _isfinite(t) else _2error(t) # PYCHOK None 

293 except Exception as x: 

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

295 

296 

297class Fsum(_Named): # sync __methods__ with .vector3dBase.Vector3dBase 

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

299 

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

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

302 intermediate, I{running} summuation. 

303 

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

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

306 C{float}, except C{complex}. 

307 

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

309 Python's C{math.fsum}. 

310 

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

312 393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}, 

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

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

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

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

317 ''' 

318 _math_fsum = None 

319 _n = 0 

320# _ps = [] # partial sums 

321# _ps_max = 0 # max(Fsum._ps_max, _len(Fsum._ps)) 

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

323 

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

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

326 

327 @arg xs: No, one or more items to add (each C{scalar} or an L{Fsum} 

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

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

330 the C{B{RESIDUAL}=0.0} threshold for L{ResidualError}s. 

331 

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

333 ''' 

334 if name_RESIDUAL: 

335 

336 def _n_R(name=NN, RESIDUAL=None): 

337 return name, RESIDUAL 

338 

339 n, R = _n_R(**name_RESIDUAL) 

340 if R is not None: 

341 self.RESIDUAL(R) 

342 if n: 

343 self.name = n 

344 

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

346 if xs: 

347 self._facc_1(xs, up=False) 

348 

349 def __abs__(self): 

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

351 ''' 

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

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

354 

355 def __add__(self, other): 

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

357 

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

359 

360 @return: The sum (L{Fsum}). 

361 

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

363 ''' 

364 f = self._copy_2(self.__add__) 

365 return f._fadd(other, _add_op_) 

366 

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

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

369 ''' 

370 s, r = self._fprs2 

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

372 

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

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

375 

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

377 

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

379 ''' 

380 return self.ceil 

381 

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

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

384 

385 @return: -1, 0 or +1 (C{int}). 

386 

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

388 ''' 

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

390 return _signOf(s, 0) 

391 

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

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

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

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

396 

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

398 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} (C{bool}) to 

399 ignore L{ResidualError}s and C{B{RESIDUAL}=scalar} 

400 to override the L{RESIDUAL<Fsum.RESIDUAL>}. 

401 

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

403 B{C{RESIDUAL}}. 

404 

405 @see: Method L{Fsum.fdiv}. 

406 ''' 

407 f = self._copy_2(self.__divmod__) 

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

409 

410 def __eq__(self, other): 

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

412 ''' 

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

414 

415 def __float__(self): 

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

417 

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

419 ''' 

420 return _Float(self._fprs) 

421 

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

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

424 

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

426 

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

428 ''' 

429 return self.floor 

430 

431 def __floordiv__(self, other): 

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

433 

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

435 

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

437 

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

439 ''' 

440 f = self._copy_2(self.__floordiv__) 

441 return f._floordiv(other, _floordiv_op_) 

442 

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

444 '''Not implemented.''' 

445 return _NotImplemented(self, *other) 

446 

447 def __ge__(self, other): 

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

449 ''' 

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

451 

452 def __gt__(self, other): 

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

454 ''' 

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

456 

457 def __hash__(self): # PYCHOK no cover 

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

459 ''' 

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

461 

462 def __iadd__(self, other): 

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

464 

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

466 

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

468 

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

470 C{scalar} nor L{Fsum}. 

471 

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

473 ''' 

474 return self._fadd(other, _iadd_op_) 

475 

476 def __ifloordiv__(self, other): 

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

478 

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

480 

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

482 

483 @raise ResidualError: Non-zero, significant residual 

484 in B{C{other}}. 

485 

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

487 

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

489 

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

491 

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

493 ''' 

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

495 

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

497 '''Not implemented.''' 

498 return _NotImplemented(self, other) 

499 

500 def __imod__(self, other): 

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

502 

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

504 

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

506 

507 @see: Method L{Fsum.__divmod__}. 

508 ''' 

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

510 

511 def __imul__(self, other): 

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

513 

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

515 

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

517 

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

519 

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

521 

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

523 ''' 

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

525 

526 def __int__(self): 

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

528 

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

530 and L{Fsum.floor}. 

531 ''' 

532 i, _ = self._fint2 

533 return i 

534 

535 def __invert__(self): # PYCHOK no cover 

536 '''Not implemented.''' 

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

538 return _NotImplemented(self) 

539 

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

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

542 

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

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

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

546 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} (C{bool}) to 

547 ignore L{ResidualError}s and C{B{RESIDUAL}=scalar} 

548 to override the L{RESIDUAL<Fsum.RESIDUAL>}. 

549 

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

551 

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

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

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

555 

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

557 

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

559 is non-zero and significant and either 

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

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

562 C{None}. 

563 

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

565 invocation failed. 

566 

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

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

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

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

571 is given as C{0}. 

572 

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

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

575 ''' 

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

577 

578 def __isub__(self, other): 

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

580 

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

582 

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

584 

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

586 

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

588 ''' 

589 return self._fsub(other, _isub_op_) 

590 

591 def __iter__(self): 

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

593 ''' 

594 return iter(self.partials) 

595 

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

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

598 

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

600 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} (C{bool}) to 

601 ignore L{ResidualError}s and C{B{RESIDUAL}=scalar} 

602 to override the L{RESIDUAL<Fsum.RESIDUAL>}. 

603 

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

605 

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

607 

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

609 B{C{RESIDUAL}}. 

610 

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

612 

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

614 

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

616 

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

618 ''' 

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

620 

621 def __le__(self, other): 

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

623 ''' 

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

625 

626 def __len__(self): 

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

628 ''' 

629 return self._n 

630 

631 def __lt__(self, other): 

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

633 ''' 

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

635 

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

637 '''Not implemented.''' 

638 return _NotImplemented(self, other) 

639 

640 def __mod__(self, other): 

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

642 

643 @see: Method L{Fsum.__imod__}. 

644 ''' 

645 f = self._copy_2(self.__mod__) 

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

647 

648 def __mul__(self, other): 

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

650 

651 @see: Method L{Fsum.__imul__}. 

652 ''' 

653 f = self._copy_2(self.__mul__) 

654 return f._fmul(other, _mul_op_) 

655 

656 def __ne__(self, other): 

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

658 ''' 

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

660 

661 def __neg__(self): 

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

663 ''' 

664 f = self._copy_2(self.__neg__) 

665 return f._fset(self._neg) 

666 

667 def __pos__(self): 

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

669 ''' 

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

671 

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

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

674 

675 @see: Method L{Fsum.__ipow__}. 

676 ''' 

677 f = self._copy_2(self.__pow__) 

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

679 

680 def __radd__(self, other): 

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

682 

683 @see: Method L{Fsum.__iadd__}. 

684 ''' 

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

686 return f._fadd(self, _add_op_) 

687 

688 def __rdivmod__(self, other): 

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

690 C{(quotient, remainder)}. 

691 

692 @see: Method L{Fsum.__divmod__}. 

693 ''' 

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

695 return f._fdivmod2(self, _divmod_op_) 

696 

697# def __repr__(self): 

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

699# ''' 

700# return self.toRepr(lenc=True) 

701 

702 def __rfloordiv__(self, other): 

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

704 

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

706 ''' 

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

708 return f._floordiv(self, _floordiv_op_) 

709 

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

711 '''Not implemented.''' 

712 return _NotImplemented(self, other) 

713 

714 def __rmod__(self, other): 

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

716 

717 @see: Method L{Fsum.__imod__}. 

718 ''' 

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

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

721 

722 def __rmul__(self, other): 

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

724 

725 @see: Method L{Fsum.__imul__}. 

726 ''' 

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

728 return f._fmul(self, _mul_op_) 

729 

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

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

732 

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

734 ''' 

735 f = self._copy_2(self.__round__) 

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

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

738 

739 def __rpow__(self, other, *mod): 

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

741 

742 @see: Method L{Fsum.__ipow__}. 

743 ''' 

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

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

746 

747 def __rsub__(self, other): 

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

749 

750 @see: Method L{Fsum.__isub__}. 

751 ''' 

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

753 return f._fsub(self, _sub_op_) 

754 

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

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

757 

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

759 ''' 

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

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

762 

763 def __str__(self): 

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

765 ''' 

766 return self.toStr(lenc=True) 

767 

768 def __sub__(self, other): 

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

770 

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

772 

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

774 

775 @see: Method L{Fsum.__isub__}. 

776 ''' 

777 f = self._copy_2(self.__sub__) 

778 return f._fsub(other, _sub_op_) 

779 

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

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

782 

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

784 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} (C{bool}) to 

785 ignore L{ResidualError}s and C{B{RESIDUAL}=scalar} 

786 to override the L{RESIDUAL<Fsum.RESIDUAL>}. 

787 

788 @return: The quotient (L{Fsum}). 

789 

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

791 B{C{RESIDUAL}}. 

792 

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

794 ''' 

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

796 

797 __trunc__ = __int__ 

798 

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

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

801 __div__ = __truediv__ 

802 __idiv__ = __itruediv__ 

803 __long__ = __int__ 

804 __nonzero__ = __bool__ 

805 __rdiv__ = __rtruediv__ 

806 

807 def as_integer_ratio(self): 

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

809 

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

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

812 non-zero, positive. 

813 

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

815 ''' 

816 n, r = self._fint2 

817 if r: 

818 i, d = _Float(r).as_integer_ratio() 

819 n *= d 

820 n += i 

821 else: # PYCHOK no cover 

822 d = 1 

823 return n, d 

824 

825 @property_RO 

826 def as_iscalar(self): 

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

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

829 ''' 

830 s, r = self._fprs2 

831 return self if r else s 

832 

833 @property_RO 

834 def ceil(self): 

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

836 C{float} in Python 2-). 

837 

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

839 

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

841 L{Fsum.imag} and L{Fsum.real}. 

842 ''' 

843 s, r = self._fprs2 

844 c = _ceil(s) + int(r) - 1 

845 while r > (c - s): # (s + r) > c 

846 c += 1 

847 return c # _ceil(self._n_d) 

848 

849 cmp = __cmp__ 

850 

851 def _cmp_0(self, other, op): 

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

853 ''' 

854 if _isFsumTuple(other): 

855 s = self._ps_1sum(*other._ps) 

856 elif self._scalar(other, op): 

857 s = self._ps_1sum(other) 

858 else: 

859 s = self.signOf() # res=True 

860 return s 

861 

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

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

864 

865 @return: The copy (L{Fsum}). 

866 ''' 

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

868 if f._ps is self._ps: 

869 f._ps = _List(self._ps) # separate list 

870 if not deep: 

871 f._n = 1 

872 # assert f._Fsum is f 

873 return f 

874 

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

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

877 ''' 

878 n = name or which.__name__ 

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

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

881 f._ps = _List(self._ps) # separate list 

882 # assert f._n == self._n 

883 # assert f._Fsum is f 

884 return f 

885 

886 def _copy_2r(self, other, which): 

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

888 ''' 

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

890 self._copy_2(which)._fset(other) 

891 

892# def _copy_RESIDUAL(self, other): 

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

894# ''' 

895# R = other._RESIDUAL 

896# if R is not Fsum._RESIDUAL: 

897# self._RESIDUAL = R 

898 

899 divmod = __divmod__ 

900 

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

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

903 ''' 

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

905 

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

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

908 ''' 

909 E, t = _xError2(X) 

910 if mod: 

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

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

913 

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

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

916 ''' 

917 E, t = _xError2(X) 

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

919 return E(u, txt=t, cause=X) 

920 

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

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

923 ''' 

924 if xs: 

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

926 ps = self._ps 

927 ps[:] = self._ps_acc(_List(ps), _xs, up=up) 

928 return self 

929 

930 def _facc_1(self, xs, **up): 

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

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

933 ''' 

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

935 self._facc(xs, origin=1, **up) 

936 

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

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

939 ''' 

940 def _N(X): 

941 return X._ps_neg 

942 

943 def _n(x): 

944 return -_Float(x) 

945 

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

947 

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

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

950 ''' 

951 def _Pow4(p): 

952 r = 0 

953 if _isFsumTuple(p): 

954 s, r = p._fprs2 

955 if r: 

956 m = Fsum._pow 

957 else: # scalar 

958 return _Pow4(s) 

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

960 p = s = int(p) 

961 m = Fsum._pow_int 

962 else: 

963 p = s = _2float(power=p) 

964 m = Fsum._pow_scalar 

965 return m, p, s, r 

966 

967 _Pow, p, s, r = _Pow4(power) 

968 if p: # and xs: 

969 op = which.__name__ 

970 _Fs = Fsum 

971 _is = _isAn 

972 _pow = self._pow_2_3 

973 

974 def _P(X): 

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

976 return f._ps if _is(f, _Fs) else (f,) 

977 

978 def _p(x): 

979 x = _Float(x) 

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

981 if f and r: 

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

983 return f 

984 

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

986 else: 

987 f = self._facc_scalar_(_Float(_len(xs))) # x**0 == 1 

988 return f 

989 

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

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

992 ''' 

993 if xs: 

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

995 return self 

996 

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

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

999 ''' 

1000 if xs: 

1001 _ = self._ps_acc(self._ps, xs, **up) 

1002 return self 

1003 

1004# def _facc_up(self, up=True): 

1005# '''(INTERNAL) Update the C{partials}, by removing 

1006# and re-accumulating the final C{partial}. 

1007# ''' 

1008# ps = self._ps 

1009# while _len(ps) > 1: 

1010# p = ps.pop() 

1011# if p: 

1012# n = self._n 

1013# _ = self._ps_acc(ps, (p,), up=False) 

1014# self._n = n 

1015# break 

1016# return self._update() if up else self 

1017 

1018 def fadd(self, xs=()): 

1019 '''Add an iterable's items to this instance. 

1020 

1021 @arg xs: Iterable of items to add (each C{scalar} 

1022 or an L{Fsum} or L{Fsum2Tuple} instance). 

1023 

1024 @return: This instance (L{Fsum}). 

1025 

1026 @raise OverflowError: Partial C{2sum} overflow. 

1027 

1028 @raise TypeError: An invalid B{C{xs}} item. 

1029 

1030 @raise ValueError: Invalid or non-finite B{C{xs}} value. 

1031 ''' 

1032 if _isFsumTuple(xs): 

1033 self._facc_scalar(xs._ps) # _Tuple(xs._ps) 

1034 elif isscalar(xs): # for backward compatibility 

1035 self._facc_scalar_(_2float(x=xs)) # PYCHOK no cover 

1036 elif xs: # assert isiterable(xs) 

1037 self._facc(xs) 

1038 return self 

1039 

1040 def fadd_(self, *xs): 

1041 '''Add all positional items to this instance. 

1042 

1043 @arg xs: Values to add (each C{scalar} or an L{Fsum} 

1044 or L{Fsum2Tuple} instance), all positional. 

1045 

1046 @see: Method L{Fsum.fadd} for further details. 

1047 ''' 

1048 return self._facc_1(xs) 

1049 

1050 def _fadd(self, other, op, **up): # in .fmath.Fhorner 

1051 '''(INTERNAL) Apply C{B{self} += B{other}}. 

1052 ''' 

1053 if not self._ps: # new Fsum(x) 

1054 self._fset(other, as_is=False, **up) 

1055 elif _isFsumTuple(other): 

1056 self._facc_scalar(other._ps, **up) # _Tuple 

1057 elif self._scalar(other, op): 

1058 self._facc_scalar_(other, **up) 

1059 return self 

1060 

1061 fcopy = copy # for backward compatibility 

1062 fdiv = __itruediv__ 

1063 fdivmod = __divmod__ 

1064 

1065 def _fdivmod2(self, other, op, **raiser_RESIDUAL): 

1066 '''(INTERNAL) Apply C{B{self} %= B{other}} and return a L{DivMod2Tuple}. 

1067 ''' 

1068 # result mostly follows CPython function U{float_divmod 

1069 # <https://GitHub.com/python/cpython/blob/main/Objects/floatobject.c>}, 

1070 # but at least divmod(-3, 2) equals Cpython's result (-2, 1). 

1071 q = self._truediv(other, op, **raiser_RESIDUAL).floor 

1072 if q: # == float // other == floor(float / other) 

1073 self -= Fsum(q) * other # NOT other * q! 

1074 

1075 s = signOf(other) # make signOf(self) == signOf(other) 

1076 if s and self.signOf() == -s: # PYCHOK no cover 

1077 self += other 

1078 q -= 1 

1079# t = self.signOf() 

1080# if t and t != s: 

1081# raise self._Error(op, other, _AssertionError, txt=signOf.__name__) 

1082 return DivMod2Tuple(q, self) # q is C{int} in Python 3+, but C{float} in Python 2- 

1083 

1084 def _finite(self, other, op=None): 

1085 '''(INTERNAL) Return B{C{other}} if C{finite}. 

1086 ''' 

1087 if _isfinite(other): 

1088 return other 

1089 raise ValueError(_not_finite_) if op is None else \ 

1090 self._Error(op, other, _ValueError, txt=_not_finite_) 

1091 

1092 def fint(self, name=NN, **raiser_RESIDUAL): 

1093 '''Return this instance' current running sum as C{integer}. 

1094 

1095 @kwarg name: Optional name (C{str}), overriding C{"fint"}. 

1096 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} (C{bool}) to 

1097 ignore L{ResidualError}s and C{B{RESIDUAL}=scalar} 

1098 to override the L{RESIDUAL<Fsum.RESIDUAL>}. 

1099 

1100 @return: The C{integer} sum (L{Fsum}) if this instance C{is_integer} 

1101 with a zero or insignificant I{integer} residual. 

1102 

1103 @raise ResidualError: Non-zero, significant residual or invalid 

1104 B{C{RESIDUAL}}. 

1105 

1106 @see: Methods L{Fsum.fint2}, L{Fsum.int_float} and L{Fsum.is_integer}. 

1107 ''' 

1108 i, r = self._fint2 

1109 if r: 

1110 R = self._raiser(r, i, **raiser_RESIDUAL) 

1111 if R: 

1112 t = _stresidual(_integer_, r, **R) 

1113 raise ResidualError(_integer_, i, txt=t) 

1114 f = self._copy_2(self.fint, name=name) 

1115 return f._fset(i) 

1116 

1117 def fint2(self, **name): 

1118 '''Return this instance' current running sum as C{int} and the 

1119 I{integer} residual. 

1120 

1121 @kwarg name: Optional name (C{str}). 

1122 

1123 @return: An L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} 

1124 an C{int} and I{integer} C{residual} a C{float} or 

1125 C{INT0} if the C{fsum} is considered to be I{exact}. 

1126 ''' 

1127 return Fsum2Tuple(*self._fint2, **name) 

1128 

1129 @Property_RO 

1130 def _fint2(self): # see ._fset 

1131 '''(INTERNAL) Get 2-tuple (C{int}, I{integer} residual). 

1132 ''' 

1133 s, r = self._fprs2 

1134 i = int(s) 

1135 n = _len(self._ps) 

1136 r = self._ps_1sum(i) if r and n > 1 else _Float(s - i) 

1137 return i, (r or INT0) # Fsum2Tuple? 

1138 

1139 @deprecated_property_RO 

1140 def float_int(self): # PYCHOK no cover 

1141 '''DEPRECATED, use method C{Fsum.int_float}.''' 

1142 return self.int_float() # raiser=False 

1143 

1144 @property_RO 

1145 def floor(self): 

1146 '''Get this instance' C{floor} (C{int} in Python 3+, but 

1147 C{float} in Python 2-). 

1148 

1149 @note: This C{floor} takes the C{residual} into account. 

1150 

1151 @see: Method L{Fsum.int_float} and properties L{Fsum.ceil}, 

1152 L{Fsum.imag} and L{Fsum.real}. 

1153 ''' 

1154 s, r = self._fprs2 

1155 f = _floor(s) + _floor(r) + 1 

1156 while (f - s) > r: # f > (s + r) 

1157 f -= 1 

1158 return f # _floor(self._n_d) 

1159 

1160# ffloordiv = __ifloordiv__ # for naming consistency 

1161# floordiv = __floordiv__ # for naming consistency 

1162 

1163 def _floordiv(self, other, op, **raiser_RESIDUAL): # rather _ffloordiv? 

1164 '''Apply C{B{self} //= B{other}}. 

1165 ''' 

1166 q = self._ftruediv(other, op, **raiser_RESIDUAL) # == self 

1167 return self._fset(q.floor) # floor(q) 

1168 

1169 fmul = __imul__ 

1170 

1171 def _fmul(self, other, op): 

1172 '''(INTERNAL) Apply C{B{self} *= B{other}}. 

1173 ''' 

1174 if _isFsumTuple(other): 

1175 if _len(self._ps) != 1: 

1176 f = self._mul_Fsum(other, op) 

1177 elif _len(other._ps) != 1: # and _len(self._ps) == 1 

1178 f = other._mul_scalar(self._ps[0], op) 

1179 else: # _len(other._ps) == _len(self._ps) == 1 

1180 f = self._finite(self._ps[0] * other._ps[0]) 

1181 else: 

1182 s = self._scalar(other, op) 

1183 f = self._mul_scalar(s, op) 

1184 return self._fset(f) # n=_len(self) + 1 

1185 

1186 def fover(self, over, **raiser_RESIDUAL): 

1187 '''Apply C{B{self} /= B{over}} and summate. 

1188 

1189 @arg over: An L{Fsum} or C{scalar} denominator. 

1190 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} (C{bool}) to 

1191 ignore L{ResidualError}s and C{B{RESIDUAL}=scalar} 

1192 to override the L{RESIDUAL<Fsum.RESIDUAL>}. 

1193 

1194 @return: Precision running sum (C{float}). 

1195 

1196 @raise ResidualError: Non-zero, significant residual or invalid 

1197 B{C{RESIDUAL}}. 

1198 

1199 @see: Methods L{Fsum.fsum} and L{Fsum.__itruediv__}. 

1200 ''' 

1201 return _Float(self.fdiv(over, **raiser_RESIDUAL)._fprs) 

1202 

1203 fpow = __ipow__ 

1204 

1205 def _fpow(self, other, op, *mod, **raiser_RESIDUAL): 

1206 '''Apply C{B{self} **= B{other}}, optional B{C{mod}} or C{None}. 

1207 ''' 

1208 if mod: 

1209 if mod[0] is not None: # == 3-arg C{pow} 

1210 f = self._pow_2_3(self, other, other, op, *mod, **raiser_RESIDUAL) 

1211 elif self.is_integer(): 

1212 # return an exact C{int} for C{int}**C{int} 

1213 i, _ = self._fint2 # assert _ == 0 

1214 x, r = _2scalar2(other) # C{int}, C{float} or other 

1215 f = _Psum_(i)._pow_Fsum(other, op, **raiser_RESIDUAL) if r else \ 

1216 self._pow_2_3(i, x, other, op, **raiser_RESIDUAL) 

1217 else: # mod[0] is None, power(self, other) 

1218 f = self._pow(other, other, op, **raiser_RESIDUAL) 

1219 else: # pow(self, other) 

1220 f = self._pow(other, other, op, **raiser_RESIDUAL) 

1221 return self._fset(f, as_is=isint(f)) # n=max(_len(self), 1) 

1222 

1223 @Property_RO 

1224 def _fprs(self): 

1225 '''(INTERNAL) Get and cache this instance' precision 

1226 running sum (C{float} or C{int}), ignoring C{residual}. 

1227 

1228 @note: The precision running C{fsum} after a C{//=} or 

1229 C{//} C{floor} division is C{int} in Python 3+. 

1230 ''' 

1231 s, _ = self._fprs2 

1232 return s # ._fprs2.fsum 

1233 

1234 @Property_RO 

1235 def _fprs2(self): 

1236 '''(INTERNAL) Get and cache this instance' precision 

1237 running sum and residual (L{Fsum2Tuple}). 

1238 ''' 

1239 ps = self._ps 

1240 n = _len(ps) - 2 

1241 if n > 0: # _len(ps) > 2 

1242 s = _psum(ps) 

1243 n = _len(ps) - 2 

1244 if n > 0: 

1245 r = self._ps_1sum(s) 

1246 return Fsum2Tuple(*_s_r(s, r)) 

1247 if n == 0: # _len(ps) == 2 

1248 s, r = _s_r(*_2sum(*ps)) 

1249 ps[:] = (r, s) if r else (s,) 

1250 elif ps: # _len(ps) == 1 

1251 s, r = ps[0], INT0 

1252 else: # _len(ps) == 0 

1253 s, r = _0_0, INT0 

1254 ps[:] = s, 

1255 # assert self._ps is ps 

1256 return Fsum2Tuple(s, r) 

1257 

1258 def fset_(self, *xs): 

1259 '''Replace this instance' value with all positional items. 

1260 

1261 @arg xs: Optional, new values (each C{scalar} or 

1262 an L{Fsum} or L{Fsum2Tuple} instance), 

1263 all positional. 

1264 

1265 @return: This instance, replaced (C{Fsum}). 

1266 

1267 @see: Method L{Fsum.fadd} for further details. 

1268 ''' 

1269 f = xs[0] if len(xs) == 1 else ( 

1270 Fsum(*xs) if xs else _0_0) 

1271 return self._fset(f) 

1272 

1273 def _fset(self, other, as_is=True, n=0, up=True): 

1274 '''(INTERNAL) Overwrite this instance with an other or a C{scalar}. 

1275 ''' 

1276 if other is self: 

1277 pass # from ._fmul, ._ftruediv and ._pow_0_1 

1278 elif _isFsumTuple(other): 

1279 self._ps[:] = other._ps 

1280 self._n = n or other._n 

1281# self._copy_RESIDUAL(other) 

1282 if up: # use or zap the C{Property_RO} values 

1283 Fsum._fint2._update_from(self, other) 

1284 Fsum._fprs ._update_from(self, other) 

1285 Fsum._fprs2._update_from(self, other) 

1286 elif isscalar(other): 

1287 s = other if as_is else _Float(other) 

1288 self._ps[:] = s, 

1289 self._n = n or 1 

1290 if up: 

1291 i = int(s) # see ._fint2 

1292 t = i, ((s - i) or INT0) 

1293 # Property_ROs _fint2, _fprs and _fprs2 can't be a Property: 

1294 # Property's _fset zaps the value just set by the @setter 

1295 self.__dict__.update(_fint2=t, _fprs=s, _fprs2=Fsum2Tuple(s, INT0)) 

1296 else: # PYCHOK no cover 

1297 raise self._Error(_fset_op_, other, _TypeError) 

1298 return self 

1299 

1300 def _fset_ps(self, other, n=0): # in .fmath 

1301 '''(INTERNAL) Set partials from a known C{scalar}, L{Fsum} or L{Fsum2Tuple}. 

1302 ''' 

1303 if _isFsumTuple(other): 

1304 self._ps[:] = other._ps 

1305 self._n = n or other._n 

1306 else: # assert isscalar(other) 

1307 self._ps[:] = other, 

1308 self._n = n or 1 

1309 return self 

1310 

1311 def fsub(self, xs=()): 

1312 '''Subtract an iterable's items from this instance. 

1313 

1314 @see: Method L{Fsum.fadd} for further details. 

1315 ''' 

1316 return self._facc_neg(xs) 

1317 

1318 def fsub_(self, *xs): 

1319 '''Subtract all positional items from this instance. 

1320 

1321 @see: Method L{Fsum.fadd_} for further details. 

1322 ''' 

1323 return self._fsub(xs[0], _sub_op_) if _len(xs) == 1 else \ 

1324 self._facc_neg(xs, origin=1) 

1325 

1326 def _fsub(self, other, op): 

1327 '''(INTERNAL) Apply C{B{self} -= B{other}}. 

1328 ''' 

1329 if _isFsumTuple(other): 

1330 if other is self: # or other._fprs2 == self._fprs2: 

1331 self._fset(_0_0, n=_len(self) * 2) 

1332 elif other._ps: 

1333 self._facc_scalar(other._ps_neg) 

1334 elif self._scalar(other, op): 

1335 self._facc_scalar_(-other) 

1336 return self 

1337 

1338 def fsum(self, xs=()): 

1339 '''Add an iterable's items, summate and return the 

1340 current precision running sum. 

1341 

1342 @arg xs: Iterable of items to add (each item C{scalar} 

1343 or an L{Fsum} or L{Fsum2Tuple} instance). 

1344 

1345 @return: Precision running sum (C{float} or C{int}). 

1346 

1347 @see: Method L{Fsum.fadd}. 

1348 

1349 @note: Accumulation can continue after summation. 

1350 ''' 

1351 return self._facc(xs)._fprs 

1352 

1353 def fsum_(self, *xs): 

1354 '''Add any positional items, summate and return the 

1355 current precision running sum. 

1356 

1357 @arg xs: Items to add (each C{scalar} or an L{Fsum} 

1358 or L{Fsum2Tuple} instance), all positional. 

1359 

1360 @return: Precision running sum (C{float} or C{int}). 

1361 

1362 @see: Methods L{Fsum.fsum}, L{Fsum.Fsum_} and L{Fsum.fsumf_}. 

1363 ''' 

1364 return self._facc_1(xs)._fprs 

1365 

1366 @property_RO 

1367 def _Fsum(self): # like L{Fsum2Tuple._Fsum}, for C{_2floats}, .fstats 

1368 return self # NOT @Property_RO, see .copy and ._copy_2 

1369 

1370 def Fsum_(self, *xs, **name): 

1371 '''Like method L{Fsum.fsum_} but returning a named L{Fsum}. 

1372 

1373 @kwarg name: Optional name (C{str}). 

1374 

1375 @return: Copy of this updated instance (L{Fsum}). 

1376 ''' 

1377 return self._facc_1(xs)._copy_2(self.Fsum_, **name) 

1378 

1379 def Fsum2Tuple_(self, *xs, **name): 

1380 '''Like method L{Fsum.fsum_} but returning a named L{Fsum2Tuple}. 

1381 

1382 @kwarg name: Optional name (C{str}). 

1383 

1384 @return: Precision running sum (L{Fsum2Tuple}). 

1385 ''' 

1386 return Fsum2Tuple(self._facc_1(xs)._fprs2, **name) 

1387 

1388 def fsum2(self, xs=(), name=NN): 

1389 '''Add an iterable's items, summate and return the 

1390 current precision running sum I{and} the C{residual}. 

1391 

1392 @arg xs: Iterable of items to add (each item C{scalar} 

1393 or an L{Fsum} or L{Fsum2Tuple} instance). 

1394 @kwarg name: Optional name (C{str}). 

1395 

1396 @return: L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} the 

1397 current precision running sum and C{residual}, the 

1398 (precision) sum of the remaining C{partials}. The 

1399 C{residual is INT0} if the C{fsum} is considered 

1400 to be I{exact}. 

1401 

1402 @see: Methods L{Fsum.fint2}, L{Fsum.fsum} and L{Fsum.fsum2_} 

1403 ''' 

1404 t = self._facc(xs)._fprs2 

1405 return t.dup(name=name) if name else t 

1406 

1407 def fsum2_(self, *xs): 

1408 '''Add any positional items, summate and return the current 

1409 precision running sum and the I{differential}. 

1410 

1411 @arg xs: Values to add (each C{scalar} or an L{Fsum} or 

1412 L{Fsum2Tuple} instance), all positional. 

1413 

1414 @return: 2Tuple C{(fsum, delta)} with the current, precision 

1415 running C{fsum} like method L{Fsum.fsum} and C{delta}, 

1416 the difference with previous running C{fsum}, C{float}. 

1417 

1418 @see: Methods L{Fsum.fsum_} and L{Fsum.fsum}. 

1419 ''' 

1420 return self._fsum2(xs, self._facc_1) 

1421 

1422 def _fsum2(self, xs, _f, **origin): 

1423 '''(INTERNAL) Helper for L{Fsum.fsum2_} and L{Fsum.fsum2f_}. 

1424 ''' 

1425 p, q = self._fprs2 

1426 if xs: 

1427 s, r = _f(xs, **origin)._fprs2 

1428 return s, _2delta(s - p, r - q) # _fsum(_1primed((s, -p, r, -q)) 

1429 else: 

1430 return p, _0_0 

1431 

1432 def fsumf_(self, *xs): 

1433 '''Like method L{Fsum.fsum_} iff I{all} C{B{xs}} are I{known to be scalar}. 

1434 ''' 

1435 return self._facc_scalar(xs)._fprs 

1436 

1437 def Fsumf_(self, *xs): 

1438 '''Like method L{Fsum.Fsum_} iff I{all} C{B{xs}} are I{known to be scalar}. 

1439 ''' 

1440 return self._facc_scalar(xs)._copy_2(self.Fsumf_) 

1441 

1442 def fsum2f_(self, *xs): 

1443 '''Like method L{Fsum.fsum2_} iff I{all} C{B{xs}} are I{known to be scalar}. 

1444 ''' 

1445 return self._fsum2(xs, self._facc_scalar, origin=1) 

1446 

1447# ftruediv = __itruediv__ # for naming consistency? 

1448 

1449 def _ftruediv(self, other, op, **raiser_RESIDUAL): 

1450 '''(INTERNAL) Apply C{B{self} /= B{other}}. 

1451 ''' 

1452 n = _1_0 

1453 if _isFsumTuple(other): 

1454 if other is self or self == other: 

1455 return self._fset(n) # n=_len(self) 

1456 d, r = other._fprs2 

1457 if r: 

1458 R = self._raiser(r, d, **raiser_RESIDUAL) 

1459 if R: 

1460 raise self._ResidualError(op, other, r, **R) 

1461 d, n = other.as_integer_ratio() 

1462 else: 

1463 d = self._scalar(other, op) 

1464 try: 

1465 s = n / d 

1466 except Exception as X: 

1467 raise self._ErrorX(X, op, other) 

1468 f = self._mul_scalar(s, _mul_op_) # handles 0, INF, NAN 

1469 return self._fset(f) # as_is=False 

1470 

1471 @property_RO 

1472 def imag(self): 

1473 '''Get the C{imaginary} part of this instance (C{0.0}, always). 

1474 

1475 @see: Property L{Fsum.real}. 

1476 ''' 

1477 return _0_0 

1478 

1479 def int_float(self, **raiser_RESIDUAL): 

1480 '''Return this instance' current running sum as C{int} or C{float}. 

1481 

1482 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} (C{bool}) to 

1483 ignore L{ResidualError}s and C{B{RESIDUAL}=scalar} 

1484 to override the L{RESIDUAL<Fsum.RESIDUAL>}. 

1485 

1486 @return: This C{integer} sum if this instance C{is_integer}, 

1487 otherwise return the C{float} sum if the residual is 

1488 zero or not significant. 

1489 

1490 @raise ResidualError: Non-zero, significant residual or invalid 

1491 B{C{RESIDUAL}}. 

1492 

1493 @see: Methods L{Fsum.fint}, L{Fsum.fint2}, L{Fsum.RESIDUAL} and 

1494 property L{Fsum.as_iscalar}. 

1495 ''' 

1496 s, r = self._fint2 

1497 if r: 

1498 s, r = self._fprs2 

1499 if r: # PYCHOK no cover 

1500 R = self._raiser(r, s, **raiser_RESIDUAL) 

1501 if R: 

1502 t = _stresidual(_non_zero_, r, **R) 

1503 raise ResidualError(int_float=s, txt=t) 

1504 s = _Float(s) # redundant 

1505 return s 

1506 

1507 def is_exact(self): 

1508 '''Is this instance' running C{fsum} considered to be exact? 

1509 (C{bool}), C{True} only if the C{residual is }L{INT0}. 

1510 ''' 

1511 return self.residual is INT0 

1512 

1513 def is_integer(self): 

1514 '''Is this instance' running sum C{integer}? (C{bool}). 

1515 

1516 @see: Methods L{Fsum.fint}, L{Fsum.fint2} and L{Fsum.is_scalar}. 

1517 ''' 

1518 _, r = self._fint2 

1519 return False if r else True 

1520 

1521 def is_math_fsum(self): 

1522 '''Return whether functions L{fsum}, L{fsum_}, L{fsum1} and 

1523 L{fsum1_} plus partials summation are based on Python's 

1524 C{math.fsum} or not. 

1525 

1526 @return: C{2} if all functions and partials summation 

1527 are based on C{math.fsum}, C{True} if only 

1528 the functions are based on C{math.fsum} (and 

1529 partials summation is not) or C{False} if 

1530 none are. 

1531 ''' 

1532 f = Fsum._math_fsum 

1533 return 2 if _psum is f else bool(f) 

1534 

1535 def is_scalar(self, **raiser_RESIDUAL): 

1536 '''Is this instance' running sum C{scalar} without residual or with 

1537 a residual I{ratio} not exceeding the RESIDUAL threshold? 

1538 

1539 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} (C{bool}) to ignore 

1540 L{ResidualError}s and C{B{RESIDUAL}=scalar} to override 

1541 the L{RESIDUAL<Fsum.RESIDUAL>} threshold. 

1542 

1543 @return: C{True} if this instance' non-zero residual C{ratio} exceeds 

1544 the L{RESIDUAL<Fsum.RESIDUAL>} threshold (C{bool}). 

1545 

1546 @raise ResidualError: Non-zero, significant residual or invalid 

1547 B{C{RESIDUAL}}. 

1548 

1549 @see: Method L{Fsum.RESIDUAL}, L{Fsum.is_integer} and property 

1550 L{Fsum.as_iscalar}. 

1551 ''' 

1552 s, r = self._fprs2 

1553 return False if r and self._raiser(r, s, **raiser_RESIDUAL) else True 

1554 

1555 def _mul_Fsum(self, other, op=_mul_op_): # in .fmath.Fhorner 

1556 '''(INTERNAL) Return C{B{self} * B{other}} as L{Fsum} or C{0}. 

1557 ''' 

1558 # assert _isFsumTuple(other) 

1559 if self._ps and other._ps: 

1560 f = self._ps_mul(op, *other._ps) # NO .as_iscalar! 

1561 else: 

1562 f = _0_0 

1563 return f 

1564 

1565 def _mul_scalar(self, factor, op): # in .fmath.Fhorner 

1566 '''(INTERNAL) Return C{B{self} * scalar B{factor}} as L{Fsum}, C{0.0} or C{self}. 

1567 ''' 

1568 # assert isscalar(factor) 

1569 if self._ps and self._finite(factor, op): 

1570 f = self if factor == _1_0 else ( 

1571 self._neg if factor == _N_1_0 else 

1572 self._ps_mul(op, factor).as_iscalar) 

1573 else: 

1574 f = _0_0 

1575 return f 

1576 

1577# @property_RO 

1578# def _n_d(self): 

1579# n, d = self.as_integer_ratio() 

1580# return n / d 

1581 

1582 @property_RO 

1583 def _neg(self): 

1584 '''(INTERNAL) Return C{Fsum(-self)} or scalar C{NEG0}. 

1585 ''' 

1586 return _Psum(self._ps_neg) if self._ps else NEG0 

1587 

1588 @property_RO 

1589 def partials(self): 

1590 '''Get this instance' current, partial sums (C{tuple} of C{float}s). 

1591 ''' 

1592 return _Tuple(self._ps) 

1593 

1594 def pow(self, x, *mod, **raiser_RESIDUAL): 

1595 '''Return C{B{self}**B{x}} as L{Fsum}. 

1596 

1597 @arg x: The exponent (C{scalar} or L{Fsum}). 

1598 @arg mod: Optional modulus (C{int} or C{None}) for the 3-argument 

1599 C{pow(B{self}, B{other}, B{mod})} version. 

1600 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} (C{bool}) to 

1601 ignore L{ResidualError}s and C{B{RESIDUAL}=scalar} 

1602 to override the L{RESIDUAL<Fsum.RESIDUAL>}. 

1603 

1604 @return: The C{pow(self, B{x})} or C{pow(self, B{x}, *B{mod})} 

1605 result (L{Fsum}). 

1606 

1607 @raise ResidualError: Non-zero, significant residual or invalid 

1608 B{C{RESIDUAL}}. 

1609 

1610 @note: If B{C{mod}} is given as C{None}, the result will be an 

1611 C{integer} L{Fsum} provided this instance C{is_integer} 

1612 or set to C{integer} by an L{Fsum.fint} call. 

1613 

1614 @see: Methods L{Fsum.__ipow__}, L{Fsum.fint}, L{Fsum.is_integer} 

1615 and L{Fsum.root}. 

1616 ''' 

1617 f = self._copy_2(self.pow) 

1618 return f._fpow(x, _pow_op_, *mod, **raiser_RESIDUAL) # f = pow(f, x, *mod) 

1619 

1620 def _pow(self, other, unused, op, **raiser_RESIDUAL): 

1621 '''Return C{B{self} ** B{other}}. 

1622 ''' 

1623 if _isFsumTuple(other): 

1624 f = self._pow_Fsum(other, op, **raiser_RESIDUAL) 

1625 elif self._scalar(other, op): 

1626 x = self._finite(other, op) 

1627 f = self._pow_scalar(x, other, op, **raiser_RESIDUAL) 

1628 else: 

1629 f = self._pow_0_1(0, other) 

1630 return f 

1631 

1632 def _pow_0_1(self, x, other): 

1633 '''(INTERNAL) Return B{C{self}**1} or C{B{self}**0 == 1.0}. 

1634 ''' 

1635 return self if x else (1 if isint(other) and self.is_integer() else _1_0) 

1636 

1637 def _pow_2_3(self, b, x, other, op, *mod, **raiser_RESIDUAL): 

1638 '''(INTERNAL) 2-arg C{pow(B{b}, scalar B{x})} and 3-arg C{pow(B{b}, 

1639 B{x}, int B{mod} or C{None})}, embellishing errors. 

1640 ''' 

1641 

1642 if mod: # b, x, mod all C{int}, unless C{mod} is C{None} 

1643 m = mod[0] 

1644 # assert _isFsumTuple(b) 

1645 

1646 def _s(s, r): 

1647 R = self._raiser(r, s, **raiser_RESIDUAL) 

1648 if R: 

1649 raise self._ResidualError(op, other, r, mod=m, **R) 

1650 return s 

1651 

1652 b = _s(*(b._fprs2 if m is None else b._fint2)) 

1653 x = _s(*_2scalar2(x)) 

1654 

1655 try: 

1656 # 0**INF == 0.0, 1**INF == 1.0, -1**2.3 == -(1**2.3) 

1657 s = pow(b, x, *mod) 

1658 if iscomplex(s): 

1659 # neg**frac == complex in Python 3+, but ValueError in 2- 

1660 raise ValueError(_strcomplex(s, b, x, *mod)) 

1661 return self._finite(s) 

1662 except Exception as X: 

1663 raise self._ErrorX(X, op, other, *mod) 

1664 

1665 def _pow_Fsum(self, other, op, **raiser_RESIDUAL): 

1666 '''(INTERNAL) Return C{B{self} **= B{other}} for C{_isFsumTuple(other)}. 

1667 ''' 

1668 # assert _isFsumTuple(other) 

1669 x, r = other._fprs2 

1670 f = self._pow_scalar(x, other, op, **raiser_RESIDUAL) 

1671 if f and r: 

1672 f *= self._pow_scalar(r, other, op, **raiser_RESIDUAL) 

1673 return f 

1674 

1675 def _pow_int(self, x, other, op, **raiser_RESIDUAL): 

1676 '''(INTERNAL) Return C{B{self} **= B{x}} for C{int B{x} >= 0}. 

1677 ''' 

1678 # assert isint(x) and x >= 0 

1679 ps = self._ps 

1680 if _len(ps) > 1: 

1681 _mul_Fsum = Fsum._mul_Fsum 

1682 if x > 4: 

1683 p = self 

1684 f = self if (x & 1) else _Psum_(_1_0) 

1685 m = x >> 1 # // 2 

1686 while m: 

1687 p = _mul_Fsum(p, p, op) # p **= 2 

1688 if (m & 1): 

1689 f = _mul_Fsum(f, p, op) # f *= p 

1690 m >>= 1 # //= 2 

1691 elif x > 1: # self**2, 3, or 4 

1692 f = _mul_Fsum(self, self, op) 

1693 if x > 2: # self**3 or 4 

1694 p = self if x < 4 else f 

1695 f = _mul_Fsum(f, p, op) 

1696 else: # self**1 or self**0 == 1 or _1_0 

1697 f = self._pow_0_1(x, other) 

1698 elif ps: # self._ps[0]**x 

1699 f = self._pow_2_3(ps[0], x, other, op, **raiser_RESIDUAL) 

1700 else: # PYCHOK no cover 

1701 # 0**pos_int == 0, but 0**0 == 1 

1702 f = 0 if x else 1 

1703 return f 

1704 

1705 def _pow_scalar(self, x, other, op, **raiser_RESIDUAL): 

1706 '''(INTERNAL) Return C{self**B{x}} for C{scalar B{x}}. 

1707 ''' 

1708 s, r = self._fprs2 

1709 if r: 

1710 # assert s != 0 

1711 if isint(x, both=True): # self**int 

1712 x = int(x) 

1713 y = _abs(x) 

1714 if y > 1: 

1715 f = self._pow_int(y, other, op, **raiser_RESIDUAL) 

1716 if x > 0: # i.e. > 1 

1717 return f # Fsum or scalar 

1718 # assert x < 0 # i.e. < -1 

1719 if _isFsum(f): 

1720 s, r = f._fprs2 

1721 if r: 

1722 return _1_Over(f, op, **raiser_RESIDUAL) 

1723 else: # scalar 

1724 s = f 

1725 # use s**(-1) to get the CPython 

1726 # float_pow error iff s is zero 

1727 x = -1 

1728 elif x < 0: # self**(-1) 

1729 return _1_Over(self, op, **raiser_RESIDUAL) # 1 / self 

1730 else: # self**1 or self**0 

1731 return self._pow_0_1(x, other) # self, 1 or 1.0 

1732 else: # self**fractional 

1733 R = self._raiser(r, s, **raiser_RESIDUAL) 

1734 if R: 

1735 raise self._ResidualError(op, other, r, **R) 

1736 n, d = self.as_integer_ratio() 

1737 if _abs(n) > _abs(d): 

1738 n, d, x = d, n, (-x) 

1739 s = n / d 

1740 # assert isscalar(s) and isscalar(x) 

1741 return self._pow_2_3(s, x, other, op, **raiser_RESIDUAL) 

1742 

1743 def _ps_acc(self, ps, xs, up=True, **unused): 

1744 '''(INTERNAL) Accumulate C{xs} scalars into list C{ps}. 

1745 ''' 

1746 n = 0 

1747 _2s = _2sum 

1748 for x in (_Tuple(xs) if xs is ps else xs): 

1749 # assert isscalar(x) and _isfinite(x) 

1750 if x: 

1751 i = 0 

1752 for p in ps: 

1753 x, p = _2s(x, p) 

1754 if p: 

1755 ps[i] = p 

1756 i += 1 

1757 ps[i:] = (x,) if x else () 

1758 n += 1 

1759 if n: 

1760 self._n += n 

1761 # Fsum._ps_max = max(Fsum._ps_max, _len(ps)) 

1762 if up: 

1763 self._update() 

1764 return ps 

1765 

1766 def _ps_mul(self, op, *factors): 

1767 '''(INTERNAL) Multiply this instance' C{partials} with 

1768 each scalar C{factor} and accumulate into an C{Fsum}. 

1769 ''' 

1770 def _pfs(ps, fs): 

1771 if _len(ps) < _len(fs): 

1772 ps, fs = fs, ps 

1773 _fin = _isfinite 

1774 for f in fs: 

1775 for p in ps: 

1776 p *= f 

1777 yield p if _fin(p) else self._finite(p, op) 

1778 

1779 return Fsum()._facc_scalar(_pfs(self._ps, factors), up=False) 

1780 

1781 @property_RO 

1782 def _ps_neg(self): 

1783 '''(INTERNAL) Yield the partials, I{negated}. 

1784 ''' 

1785 for p in self._ps: 

1786 yield -p 

1787 

1788 def _ps_1sum(self, *less): 

1789 '''(INTERNAL) Return the partials sum, 1-primed C{less} some scalars. 

1790 ''' 

1791 def _1pls(ps, ls): 

1792 yield _1_0 

1793 for p in ps: 

1794 yield p 

1795 for p in ls: 

1796 yield -p 

1797 yield _N_1_0 

1798 

1799 return _fsum(_1pls(self._ps, less)) 

1800 

1801 def _raiser(self, r, s, raiser=True, **RESIDUAL): 

1802 '''(INTERNAL) Does ratio C{r / s} exceed the RESIDUAL threshold 

1803 I{and} is residual C{r} I{non-zero} or I{significant} (for a 

1804 negative respectively positive C{RESIDUAL} threshold)? 

1805 ''' 

1806 if r and raiser: 

1807 t = self._RESIDUAL 

1808 if RESIDUAL: 

1809 t = _threshold(_xkwds_get(RESIDUAL, RESIDUAL=t)) 

1810 if t < 0 or (s + r) != s: 

1811 q = (r / s) if s else s # == 0. 

1812 if fabs(q) > fabs(t): 

1813 return dict(ratio=q, R=t) 

1814 return {} 

1815 

1816 rdiv = __rtruediv__ 

1817 

1818 @property_RO 

1819 def real(self): 

1820 '''Get the C{real} part of this instance (C{float}). 

1821 

1822 @see: Methods L{Fsum.__float__} and L{Fsum.fsum} 

1823 and properties L{Fsum.ceil}, L{Fsum.floor}, 

1824 L{Fsum.imag} and L{Fsum.residual}. 

1825 ''' 

1826 return _Float(self._fprs) 

1827 

1828 @property_RO 

1829 def residual(self): 

1830 '''Get this instance' residual (C{float} or C{int}): the 

1831 C{sum(partials)} less the precision running sum C{fsum}. 

1832 

1833 @note: The C{residual is INT0} iff the precision running 

1834 C{fsum} is considered to be I{exact}. 

1835 

1836 @see: Methods L{Fsum.fsum}, L{Fsum.fsum2} and L{Fsum.is_exact}. 

1837 ''' 

1838 return self._fprs2.residual 

1839 

1840 def RESIDUAL(self, *threshold): 

1841 '''Get and set this instance' I{ratio} for raising L{ResidualError}s, 

1842 overriding the default from env variable C{PYGEODESY_FSUM_RESIDUAL}. 

1843 

1844 @arg threshold: If C{scalar}, the I{ratio} to exceed for raising 

1845 L{ResidualError}s in division and exponention, if 

1846 C{None} restore the default set with env variable 

1847 C{PYGEODESY_FSUM_RESIDUAL} or if omitted, keep the 

1848 current setting. 

1849 

1850 @return: The previous C{RESIDUAL} setting (C{float}), default C{0.0}. 

1851 

1852 @raise ResidualError: Invalid B{C{threshold}}. 

1853 

1854 @note: L{ResidualError}s may be thrown if the non-zero I{ratio} 

1855 C{residual / fsum} exceeds the given B{C{threshold}} and 

1856 if the C{residual} is non-zero and I{significant} vs the 

1857 C{fsum}, i.e. C{(fsum + residual) != fsum} and if optional 

1858 keyword argument C{raiser=False} is missing. Specify a 

1859 negative B{C{threshold}} for only non-zero C{residual} 

1860 testing without I{significant}. 

1861 ''' 

1862 r = self._RESIDUAL 

1863 if threshold: 

1864 t = threshold[0] 

1865 self._RESIDUAL = Fsum._RESIDUAL if t is None else ( # for ... 

1866 (_0_0 if t else _1_0) if isbool(t) else 

1867 _threshold(t)) # ... backward compatibility 

1868 return r 

1869 

1870 def _ResidualError(self, op, other, residual, **mod_R): 

1871 '''(INTERNAL) Non-zero B{C{residual}} etc. 

1872 ''' 

1873 def _p(mod=None, R=0, **unused): # ratio=0 

1874 return (_non_zero_ if R < 0 else _significant_) \ 

1875 if mod is None else _integer_ 

1876 

1877 t = _stresidual(_p(**mod_R), residual, **mod_R) 

1878 return self._Error(op, other, ResidualError, txt=t) 

1879 

1880 def root(self, root, **raiser_RESIDUAL): 

1881 '''Return C{B{self}**(1 / B{root})} as L{Fsum}. 

1882 

1883 @arg root: The order (C{scalar} or L{Fsum}), non-zero. 

1884 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} (C{bool}) to 

1885 ignore L{ResidualError}s and C{B{RESIDUAL}=scalar} 

1886 to override the L{RESIDUAL<Fsum.RESIDUAL>}. 

1887 

1888 @return: The C{self ** (1 / B{root})} result (L{Fsum}). 

1889 

1890 @raise ResidualError: Non-zero, significant residual or invalid 

1891 B{C{RESIDUAL}}. 

1892 

1893 @see: Method L{Fsum.pow}. 

1894 ''' 

1895 x = _1_Over(root, _truediv_op_, **raiser_RESIDUAL) 

1896 f = self._copy_2(self.root) 

1897 return f._fpow(x, f.name, **raiser_RESIDUAL) # == pow(f, x) 

1898 

1899 def _scalar(self, other, op, **txt): 

1900 '''(INTERNAL) Return scalar C{other}. 

1901 ''' 

1902 if isscalar(other): 

1903 return other 

1904 raise self._Error(op, other, _TypeError, **txt) # _invalid_ 

1905 

1906 def signOf(self, res=True): 

1907 '''Determine the sign of this instance. 

1908 

1909 @kwarg res: If C{True} consider, otherwise 

1910 ignore the residual (C{bool}). 

1911 

1912 @return: The sign (C{int}, -1, 0 or +1). 

1913 ''' 

1914 s, r = self._fprs2 

1915 r = (-r) if res else 0 

1916 return _signOf(s, r) 

1917 

1918 def toRepr(self, **lenc_prec_sep_fmt): # PYCHOK signature 

1919 '''Return this C{Fsum} instance as representation. 

1920 

1921 @kwarg lenc_prec_sep_fmt: Optional keyword arguments 

1922 for method L{Fsum.toStr}. 

1923 

1924 @return: This instance (C{repr}). 

1925 ''' 

1926 return Fmt.repr_at(self, self.toStr(**lenc_prec_sep_fmt)) 

1927 

1928 def toStr(self, lenc=True, **prec_sep_fmt): # PYCHOK signature 

1929 '''Return this C{Fsum} instance as string. 

1930 

1931 @kwarg lenc: If C{True} include the current C{[len]} of this 

1932 L{Fsum} enclosed in I{[brackets]} (C{bool}). 

1933 @kwarg prec_sep_fmt: Optional keyword arguments for method 

1934 L{Fsum2Tuple.toStr}. 

1935 

1936 @return: This instance (C{str}). 

1937 ''' 

1938 p = self.classname 

1939 if lenc: 

1940 p = Fmt.SQUARE(p, _len(self)) 

1941 n = _enquote(self.name, white=_UNDER_) 

1942 t = self._fprs2.toStr(**prec_sep_fmt) 

1943 return NN(p, _SPACE_, n, t) 

1944 

1945 def _truediv(self, other, op, **raiser_RESIDUAL): 

1946 '''(INTERNAL) Return C{B{self} / B{other}} as an L{Fsum}. 

1947 ''' 

1948 f = self._copy_2(self.__truediv__) 

1949 return f._ftruediv(other, op, **raiser_RESIDUAL) 

1950 

1951 def _update(self, updated=True): # see ._fset 

1952 '''(INTERNAL) Zap all cached C{Property_RO} values. 

1953 ''' 

1954 if updated: 

1955 _pop = self.__dict__.pop 

1956 for p in _ROs: 

1957 _ = _pop(p, None) 

1958# Fsum._fint2._update(self) 

1959# Fsum._fprs ._update(self) 

1960# Fsum._fprs2._update(self) 

1961 return self # for .fset_ 

1962 

1963_ROs = _allPropertiesOf_n(3, Fsum, Property_RO) # PYCHOK see Fsum._update 

1964 

1965 

1966def _Float_Int(arg, **name_Error): 

1967 '''(INTERNAL) Unit of L{Fsum2Tuple} items. 

1968 ''' 

1969 U = Int if isint(arg) else Float 

1970 return U(arg, **name_Error) 

1971 

1972 

1973class DivMod2Tuple(_NamedTuple): 

1974 '''2-Tuple C{(div, mod)} with the quotient C{div} and remainder 

1975 C{mod} results of a C{divmod} operation. 

1976 

1977 @note: Quotient C{div} an C{int} in Python 3+ but a C{float} 

1978 in Python 2-. Remainder C{mod} an L{Fsum} instance. 

1979 ''' 

1980 _Names_ = (_div_, _mod_) 

1981 _Units_ = (_Float_Int, Fsum) 

1982 

1983 

1984class Fsum2Tuple(_NamedTuple): # in .fstats 

1985 '''2-Tuple C{(fsum, residual)} with the precision running C{fsum} 

1986 and the C{residual}, the sum of the remaining partials. Each 

1987 item is C{float} or C{int}. 

1988 

1989 @note: If the C{residual is INT0}, the C{fsum} is considered 

1990 to be I{exact}, see method L{Fsum2Tuple.is_exact}. 

1991 ''' 

1992 _Names_ = ( Fsum.fsum.__name__, Fsum.residual.name) 

1993 _Units_ = (_Float_Int, _Float_Int) 

1994 

1995 def __abs__(self): # in .fmath 

1996 return self._Fsum.__abs__() 

1997 

1998 def __bool__(self): # PYCHOK Python 3+ 

1999 return bool(self._Fsum) 

2000 

2001 def __eq__(self, other): 

2002 return self._other_op(other, self.__eq__) 

2003 

2004 def __float__(self): 

2005 return self._Fsum.__float__() 

2006 

2007 def __ge__(self, other): 

2008 return self._other_op(other, self.__ge__) 

2009 

2010 def __gt__(self, other): 

2011 return self._other_op(other, self.__gt__) 

2012 

2013 def __le__(self, other): 

2014 return self._other_op(other, self.__le__) 

2015 

2016 def __lt__(self, other): 

2017 return self._other_op(other, self.__lt__) 

2018 

2019 def __int__(self): 

2020 return self._Fsum.__int__() 

2021 

2022 def __ne__(self, other): 

2023 return self._other_op(other, self.__ne__) 

2024 

2025 def __neg__(self): 

2026 return self._Fsum.__neg__() 

2027 

2028 __nonzero__ = __bool__ # Python 2- 

2029 

2030 def __pos__(self): 

2031 return self._Fsum.__pos__() 

2032 

2033 def as_integer_ratio(self): 

2034 '''Return this instance as the ratio of 2 integers. 

2035 

2036 @see: Method L{Fsum.as_integer_ratio} for further details. 

2037 ''' 

2038 return self._Fsum.as_integer_ratio() 

2039 

2040 @property_RO 

2041 def _fint2(self): 

2042 return self._Fsum._fint2 

2043 

2044 @property_RO 

2045 def _fprs2(self): 

2046 return self._Fsum._fprs2 

2047 

2048 @Property_RO 

2049 def _Fsum(self): # this C{Fsum2Tuple} as L{Fsum}, in .fstats 

2050 s, r = _s_r(*self) 

2051 ps = (r, s) if r else (s,) 

2052 return _Psum(ps, name=self.name) 

2053 

2054 def Fsum_(self, *xs, **name_RESIDUAL): 

2055 '''Return this C{Fsum2Tuple} as an L{Fsum} plus some C{xs}. 

2056 ''' 

2057 f = _Psum(self._Fsum._ps, **name_RESIDUAL) 

2058 return f._facc_1(xs, up=False) if xs else f 

2059 

2060 def is_exact(self): 

2061 '''Is this L{Fsum2Tuple} considered to be exact? (C{bool}). 

2062 ''' 

2063 return self._Fsum.is_exact() 

2064 

2065 def is_integer(self): 

2066 '''Is this L{Fsum2Tuple} C{integer}? (C{bool}). 

2067 ''' 

2068 return self._Fsum.is_integer() 

2069 

2070 def _mul_scalar(self, other, op): # for Fsum._fmul 

2071 return self._Fsum._mul_scalar(other, op) 

2072 

2073 @property_RO 

2074 def _n(self): 

2075 return self._Fsum._n 

2076 

2077 def _other_op(self, other, which): 

2078 C, s = (_Tuple, self) if _isAn(other, _Tuple) else (Fsum, self._Fsum) 

2079 return getattr(C, which.__name__)(s, other) 

2080 

2081 @property_RO 

2082 def _ps(self): 

2083 return self._Fsum._ps 

2084 

2085 @property_RO 

2086 def _ps_neg(self): 

2087 return self._Fsum._ps_neg 

2088 

2089 def signOf(self, **res): 

2090 '''Like method L{Fsum.signOf}. 

2091 ''' 

2092 return self._Fsum.signOf(**res) 

2093 

2094 def toStr(self, fmt=Fmt.g, **prec_sep): # PYCHOK signature 

2095 '''Return this L{Fsum2Tuple} as string (C{str}). 

2096 

2097 @kwarg fmt: Optional C{float} format (C{letter}). 

2098 @kwarg prec_sep: Optional keyword arguments for function 

2099 L{fstr<streprs.fstr>}. 

2100 ''' 

2101 return Fmt.PAREN(fstr(self, fmt=fmt, strepr=str, force=False, **prec_sep)) 

2102 

2103_Fsum_Fsum2Tuple_types = Fsum, Fsum2Tuple # PYCHOK lines 

2104 

2105 

2106class ResidualError(_ValueError): 

2107 '''Error raised for a division, power or root operation of 

2108 an L{Fsum} instance with a C{residual} I{ratio} exceeding 

2109 the L{RESIDUAL<Fsum.RESIDUAL>} threshold. 

2110 

2111 @see: Module L{pygeodesy.fsums} and method L{Fsum.RESIDUAL}. 

2112 ''' 

2113 pass 

2114 

2115 

2116try: 

2117 from math import fsum as _fsum # precision IEEE-754 sum, Python 2.6+ 

2118 

2119 # make sure _fsum works as expected (XXX check 

2120 # float.__getformat__('float')[:4] == 'IEEE'?) 

2121 if _fsum((1, 1e101, 1, -1e101)) != 2: # PYCHOK no cover 

2122 del _fsum # nope, remove _fsum ... 

2123 raise ImportError # ... use _fsum below 

2124 

2125 Fsum._math_fsum = _sum = _fsum # PYCHOK exported 

2126except ImportError: 

2127 _sum = sum # Fsum(NAN) exception fall-back, in .elliptic 

2128 

2129 def _fsum(xs): 

2130 '''(INTERNAL) Precision summation, Python 2.5-. 

2131 ''' 

2132 F = Fsum() 

2133 F.name = _fsum.__name__ 

2134 return F._facc(xs, up=False)._fprs2.fsum 

2135 

2136 

2137def fsum(xs, floats=False): 

2138 '''Precision floating point summation based on/like Python's C{math.fsum}. 

2139 

2140 @arg xs: Iterable of items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} 

2141 instance). 

2142 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to 

2143 be scalar} (C{bool}). 

2144 

2145 @return: Precision C{fsum} (C{float}). 

2146 

2147 @raise OverflowError: Partial C{2sum} overflow. 

2148 

2149 @raise TypeError: Non-scalar B{C{xs}} item. 

2150 

2151 @raise ValueError: Invalid or non-finite B{C{xs}} item. 

2152 

2153 @note: Exception and I{non-finite} handling may differ if not based 

2154 on Python's C{math.fsum}. 

2155 

2156 @see: Class L{Fsum} and methods L{Fsum.fsum} and L{Fsum.fadd}. 

2157 ''' 

2158 return _fsum(xs if floats is True else _2floats(xs)) if xs else _0_0 # PYCHOK yield 

2159 

2160 

2161def fsum_(*xs, **floats): 

2162 '''Precision floating point summation of all positional items. 

2163 

2164 @arg xs: Items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} instance), 

2165 all positional. 

2166 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to 

2167 be scalar} (C{bool}). 

2168 

2169 @see: Function L{fsum<fsums.fsum>} for further details. 

2170 ''' 

2171 return _fsum(xs if _xkwds_get(floats, floats=False) is True else 

2172 _2floats(xs, origin=1)) if xs else _0_0 # PYCHOK yield 

2173 

2174 

2175def fsumf_(*xs): 

2176 '''Precision floating point summation iff I{all} C{B{xs}} items are I{known to be scalar}. 

2177 

2178 @see: Function L{fsum_<fsums.fsum_>} for further details. 

2179 ''' 

2180 return _fsum(xs) if xs else _0_0 

2181 

2182 

2183def fsum1(xs, floats=False): 

2184 '''Precision floating point summation, 1-primed. 

2185 

2186 @arg xs: Iterable of items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} 

2187 instance). 

2188 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to 

2189 be scalar} (C{bool}). 

2190 

2191 @see: Function L{fsum<fsums.fsum>} for further details. 

2192 ''' 

2193 return _fsum(_1primed(xs if floats is True else _2floats(xs))) if xs else _0_0 # PYCHOK yield 

2194 

2195 

2196def fsum1_(*xs, **floats): 

2197 '''Precision floating point summation, 1-primed of all positional items. 

2198 

2199 @arg xs: Items to add (each C{scalar} or an L{Fsum} or L{Fsum2Tuple} instance), 

2200 all positional. 

2201 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} items are I{known to 

2202 be scalar} (C{bool}). 

2203 

2204 @see: Function L{fsum_<fsums.fsum_>} for further details. 

2205 ''' 

2206 return _fsum(_1primed(xs if _xkwds_get(floats, floats=False) is True else 

2207 _2floats(xs, origin=1))) if xs else _0_0 # PYCHOK yield 

2208 

2209 

2210def fsum1f_(*xs): 

2211 '''Precision floating point summation iff I{all} C{B{xs}} items are I{known to be scalar}. 

2212 

2213 @see: Function L{fsum_<fsums.fsum_>} for further details. 

2214 ''' 

2215 return _fsum(_1primed(xs)) if xs else _0_0 

2216 

2217 

2218if __name__ == '__main__': 

2219 

2220 # usage: [env _psum=fsum] python3 -m pygeodesy.fsums 

2221 

2222 if _getenv(_psum.__name__, NN) == _fsum.__name__: 

2223 _psum = _fsum 

2224 

2225 def _test(n): 

2226 # copied from Hettinger, see L{Fsum} reference 

2227 from pygeodesy import frandoms, printf 

2228 

2229 printf(_fsum.__name__, end=_COMMASPACE_) 

2230 printf(_psum.__name__, end=_COMMASPACE_) 

2231 

2232 F = Fsum() 

2233 if F.is_math_fsum(): 

2234 for t in frandoms(n, seeded=True): 

2235 assert _Float(F.fset_(*t)) == _fsum(t) 

2236 printf(_DOT_, end=NN) 

2237 printf(NN) 

2238 

2239 _test(128) 

2240 

2241# **) MIT License 

2242# 

2243# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

2244# 

2245# Permission is hereby granted, free of charge, to any person obtaining a 

2246# copy of this software and associated documentation files (the "Software"), 

2247# to deal in the Software without restriction, including without limitation 

2248# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

2249# and/or sell copies of the Software, and to permit persons to whom the 

2250# Software is furnished to do so, subject to the following conditions: 

2251# 

2252# The above copyright notice and this permission notice shall be included 

2253# in all copies or substantial portions of the Software. 

2254# 

2255# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

2256# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

2257# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

2258# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

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2260# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

2261# OTHER DEALINGS IN THE SOFTWARE.