Coverage for pygeodesy/fsums.py: 96%

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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_PARTIALS} to string C{"fsum"}) for summation 

18of L{Fsum} partials by Python function C{math.fsum}. 

19 

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

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

22of an L{Fsum} instance with a I{relative} C{residual} exceeding the threshold, 

23see methods L{Fsum.RESIDUAL}, L{Fsum.pow}, L{Fsum.__ipow__} and L{Fsum.__itruediv__}. 

24''' 

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

26from __future__ import division as _; del _ # PYCHOK semicolon 

27 

28from pygeodesy.basics import iscomplex, isint, isscalar, itemsorted, \ 

29 signOf, _signOf 

30from pygeodesy.constants import INT0, _isfinite, NEG0, _pos_self, \ 

31 _0_0, _1_0, _N_1_0, Float, Int 

32from pygeodesy.errors import _OverflowError, _TypeError, _ValueError, _xError, \ 

33 _xError2, _xkwds_get, _ZeroDivisionError 

34from pygeodesy.interns import NN, _arg_, _COMMASPACE_, _DASH_, _DOT_, _EQUAL_, \ 

35 _exceeds_, _from_, _iadd_op_, _LANGLE_, _negative_, \ 

36 _NOTEQUAL_, _not_finite_, _PERCENT_, _PLUS_, _R_, \ 

37 _RANGLE_, _SLASH_, _SPACE_, _STAR_, _UNDER_ 

38from pygeodesy.lazily import _ALL_LAZY, _getenv, _sys_version_info2 

39from pygeodesy.named import _Named, _NamedTuple, _NotImplemented, Fmt, unstr 

40from pygeodesy.props import _allPropertiesOf_n, deprecated_property_RO, \ 

41 Property_RO, property_RO 

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

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

44 

45from math import ceil as _ceil, fabs, floor as _floor # PYCHOK used! .ltp 

46 

47__all__ = _ALL_LAZY.fsums 

48__version__ = '24.04.18' 

49 

50_add_op_ = _PLUS_ # in .auxilats.auxAngle 

51_eq_op_ = _EQUAL_ * 2 # _DEQUAL_ 

52_COMMASPACE_R_ = _COMMASPACE_ + _R_ 

53_div_ = 'div' 

54_exceeds_R_ = _SPACE_ + _exceeds_(_R_) 

55_floordiv_op_ = _SLASH_ * 2 # _DSLASH_ 

56_fset_op_ = _EQUAL_ 

57_ge_op_ = _RANGLE_ + _EQUAL_ 

58_gt_op_ = _RANGLE_ 

59_integer_ = 'integer' 

60_le_op_ = _LANGLE_ + _EQUAL_ 

61_lt_op_ = _LANGLE_ 

62_mod_ = 'mod' 

63_mod_op_ = _PERCENT_ 

64_mul_op_ = _STAR_ 

65_ne_op_ = _NOTEQUAL_ 

66_non_zero_ = 'non-zero' 

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

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

69_truediv_op_ = _SLASH_ 

70_divmod_op_ = _floordiv_op_ + _mod_op_ 

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

72 

73 

74def _2delta(*ab): 

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

76 ''' 

77 try: 

78 a, b = _2sum(*ab) 

79 except _OverflowError: 

80 a, b = ab 

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

82 

83 

84def _2error(unused): 

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

86 ''' 

87 raise ValueError(_not_finite_) 

88 

89 

90def _2float(index=None, **name_value): # in .fmath, .fstats 

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

92 ''' 

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

94 try: 

95 v = float(v) 

96 return v if _isfinite(v) else _2error(v) 

97 except Exception as X: 

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

99 

100 

101def _X_ps(X): # for _2floats only 

102 return X._ps 

103 

104 

105def _2floats(xs, origin=0, _X=_X_ps, _x=float): 

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

107 ''' 

108 try: 

109 i, x = origin, None 

110 _fin = _isfinite 

111 _Fs = Fsum 

112 for x in xs: 

113 if isinstance(x, _Fs): 

114 for p in _X(x): 

115 yield p 

116 else: 

117 f = _x(x) 

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

119 i += 1 

120 except Exception as X: 

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

122 

123 

124def _Fsumf_(*xs): # floats=True, in .auxLat, ... 

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

126 ''' 

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

128 

129 

130def _Fsum1f_(*xs): # floats=True, in .albers, ... 

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

132 ''' 

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

134 

135 

136def _2halfeven(s, r, p): 

137 '''(INTERNAL) Round half-even. 

138 ''' 

139 if (p > 0 and r > 0) or \ 

140 (p < 0 and r < 0): # signs match 

141 r *= 2 

142 t = s + r 

143 if r == (t - s): 

144 s = t 

145 return s 

146 

147 

148def _1primed(xs): # in .fmath 

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

150 items, all I{known} to be C{finite float}. 

151 ''' 

152 yield _1_0 

153 for x in xs: 

154 yield x 

155 yield _N_1_0 

156 

157 

158def _2ps(s, r): 

159 '''(INTERNAL) Return a C{s} and C{r} pair, I{ps-ordered}. 

160 ''' 

161 return (s, r) if fabs(s) < fabs(r) else (r, s) 

162 

163 

164def _psum(ps): # PYCHOK used! 

165 '''(INTERNAL) Partials sum, updating C{ps}, I{overridden below}. 

166 ''' 

167 # assert isinstance(ps, list) 

168 i = len(ps) - 1 

169 s = _0_0 if i < 0 else ps[i] 

170 _2s = _2sum 

171 while i > 0: 

172 i -= 1 

173 s, r = _2s(s, ps[i]) 

174 if r: # sum(ps) became inexact 

175 if s: 

176 ps[i:] = r, s 

177 if i > 0: 

178 s = _2halfeven(s, r, ps[i-1]) 

179 break # return s 

180 s = r # PYCHOK no cover 

181 ps[i:] = s, 

182 return s 

183 

184 

185def _Psum(ps, **name): 

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

187 ''' 

188 F = Fsum(**name) if name else Fsum() 

189 if ps: 

190 F._ps[:] = ps 

191 F._n = len(F._ps) 

192 return F 

193 

194 

195def _Psum_1(p=_1_0, **name): 

196 '''(INTERNAL) Return an C{Fsum} from a single partial C{p}. 

197 ''' 

198 F = Fsum(**name) if name else Fsum() 

199 F._ps[:] = p, 

200 F._n = 1 # len(F._ps) 

201 return F 

202 

203 

204def _2scalar(other, _raiser=None, **mod): 

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

206 ''' 

207 if isinstance(other, Fsum): 

208 s, r = other._fint2 

209 if r: 

210 s, r = other._fprs2 

211 if r: # PYCHOK no cover 

212 if _raiser and _raiser(r, s): 

213 t = _stresidual(_non_zero_, r, **mod) 

214 raise ResidualError(t, txt=None) 

215 s = other # L{Fsum} as-is 

216 else: 

217 s = other # C{type} as-is 

218 if isint(s, both=True): 

219 s = int(s) 

220 return s 

221 

222 

223def _strcomplex(s, *args): 

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

225 ''' 

226 c = _strcomplex.__name__[4:] 

227 n = _DASH_(len(args), _arg_) 

228 t = unstr(pow, *args) 

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

230 

231 

232def _stresidual(prefix, residual, **name_values): 

233 '''(INTERNAL) Residual error as C{str}. 

234 ''' 

235 p = _stresidual.__name__[3:] 

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

237 for n, v in itemsorted(name_values): 

238 n = n.replace(_UNDER_, _SPACE_) 

239 p = Fmt.PARENSPACED(n, Fmt(v)) 

240 t = _COMMASPACE_(t, p) 

241 return _SPACE_(prefix, t) 

242 

243 

244def _2sum(a, b): # by .testFmath 

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

246 ''' 

247 s = a + b 

248 if _isfinite(s): 

249 if fabs(a) < fabs(b): 

250 b, a = a, b 

251 return s, (b - (s - a)) 

252 u = unstr(_2sum, a, b) 

253 t = Fmt.PARENSPACED(_not_finite_, s) 

254 raise _OverflowError(u, txt=t) 

255 

256 

257class Fsum(_Named): # sync __methods__ with .vector3dBase.Vector3dBase 

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

259 

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

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

262 intermediate, I{running} summuation. 

263 

264 @note: Accumulated values may be L{Fsum} or C{scalar} instances, any C{type} having 

265 method C{__float__} to convert the C{scalar} to a single C{float}. 

266 

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

268 Python's C{math.fsum}. 

269 

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

271 393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}, U{Kahan 

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

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

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

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

276 ''' 

277 _math_fsum = None 

278 _n = 0 

279# _ps = [] # partial sums 

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

281 _ratio = None # see method _raiser 

282 _recursive = bool(_getenv('PYGEODESY_FSUM_RECURSIVE', NN)) 

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

284 

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

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

287 

288 @arg xs: No, one or more initial values, all positional (each C{scalar} 

289 or an L{Fsum} instance). 

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

291 C{B{RESIDUAL}=None} for the L{ResidualError} threshold. 

292 

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

294 ''' 

295 if name_RESIDUAL: 

296 n = _xkwds_get(name_RESIDUAL, name=NN) 

297 if n: # set name before ... 

298 self.name = n 

299 r = _xkwds_get(name_RESIDUAL, RESIDUAL=None) 

300 if r is not None: 

301 self.RESIDUAL(r) # ... ResidualError 

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

303 if xs: 

304 self._facc(xs, origin=1, up=False) 

305 

306 def __abs__(self): 

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

308 ''' 

309 s = _fsum(self._ps_1primed()) # == self._cmp_0(0, ...) 

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

311 

312 def __add__(self, other): 

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

314 

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

316 

317 @return: The sum (L{Fsum}). 

318 

319 @see: Method L{Fsum.__iadd__}. 

320 ''' 

321 f = self._copy_2(self.__add__) 

322 return f._fadd(other, _add_op_) 

323 

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

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

326 ''' 

327 s, r = self._fprs2 

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

329 

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

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

332 

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

334 

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

336 ''' 

337 return self.ceil 

338 

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

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

341 

342 @return: -1, 0 or +1 (C{int}). 

343 

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

345 ''' 

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

347 return _signOf(s, 0) 

348 

349 cmp = __cmp__ 

350 

351 def __divmod__(self, other): 

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

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

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

355 

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

357 

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

359 ''' 

360 f = self._copy_2(self.__divmod__) 

361 return f._fdivmod2(other, _divmod_op_) 

362 

363 def __eq__(self, other): 

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

365 ''' 

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

367 

368 def __float__(self): 

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

370 

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

372 ''' 

373 return float(self._fprs) 

374 

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

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

377 

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

379 

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

381 ''' 

382 return self.floor 

383 

384 def __floordiv__(self, other): 

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

386 

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

388 

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

390 

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

392 ''' 

393 f = self._copy_2(self.__floordiv__) 

394 return f._floordiv(other, _floordiv_op_) 

395 

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

397 '''Not implemented.''' 

398 return _NotImplemented(self, *other) 

399 

400 def __ge__(self, other): 

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

402 ''' 

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

404 

405 def __gt__(self, other): 

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

407 ''' 

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

409 

410 def __hash__(self): # PYCHOK no cover 

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

412 ''' 

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

414 

415 def __iadd__(self, other): 

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

417 

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

419 

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

421 

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

423 C{scalar} nor L{Fsum}. 

424 

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

426 ''' 

427 return self._fadd(other, _iadd_op_) 

428 

429 def __ifloordiv__(self, other): 

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

431 

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

433 

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

435 

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

437 

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

439 

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

441 

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

443 

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

445 ''' 

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

447 

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

449 '''Not implemented.''' 

450 return _NotImplemented(self, other) 

451 

452 def __imod__(self, other): 

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

454 

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

456 

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

458 

459 @see: Method L{Fsum.__divmod__}. 

460 ''' 

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

462 

463 def __imul__(self, other): 

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

465 

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

467 

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

469 

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

471 

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

473 

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

475 ''' 

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

477 

478 def __int__(self): 

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

480 

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

482 and L{Fsum.__floor__} and properties 

483 L{Fsum.ceil} and L{Fsum.floor}. 

484 ''' 

485 i, _ = self._fint2 

486 return i 

487 

488 def __invert__(self): # PYCHOK no cover 

489 '''Not implemented.''' 

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

491 return _NotImplemented(self) 

492 

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

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

495 

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

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

498 3-argument C{pow(B{self}, B{other}, B{mod})} 

499 version. 

500 @kwarg raiser: Use C{B{raiser}=False} to ignore L{ResidualError}s 

501 (C{bool}), see also method L{RESIDUAL}. 

502 

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

504 

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

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

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

508 

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

510 

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

512 env var C{PYGEODESY_FSUM_RESIDUAL} 

513 set or this instance has a non-zero 

514 residual and either B{C{mod}} is 

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

516 is a negative or fractional C{scalar}. 

517 

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

519 C{pow} invocation failed. 

520 

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

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

523 is a fractional C{scalar} and this 

524 instance is negative or has a non-zero 

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

526 

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

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

529 ''' 

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

531 

532 def __isub__(self, other): 

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

534 

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

536 

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

538 

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

540 

541 @see: Method L{Fsum.fadd}. 

542 ''' 

543 return self._fsub(other, _isub_op_) 

544 

545 def __iter__(self): 

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

547 ''' 

548 return iter(self.partials) 

549 

550 def __itruediv__(self, other, **raiser): 

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

552 

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

554 @kwarg raiser: Use C{B{raiser}=False} to ignore L{ResidualError}s 

555 (C{bool}), see also method L{RESIDUAL}. 

556 

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

558 

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

560 

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

562 env var C{PYGEODESY_FSUM_RESIDUAL} set. 

563 

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

565 

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

567 

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

569 

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

571 ''' 

572 return self._ftruediv(other, _truediv_op_ + _fset_op_, **raiser) 

573 

574 def __le__(self, other): 

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

576 ''' 

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

578 

579 def __len__(self): 

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

581 ''' 

582 return self._n 

583 

584 def __lt__(self, other): 

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

586 ''' 

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

588 

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

590 '''Not implemented.''' 

591 return _NotImplemented(self, other) 

592 

593 def __mod__(self, other): 

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

595 

596 @see: Method L{Fsum.__imod__}. 

597 ''' 

598 f = self._copy_2(self.__mod__) 

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

600 

601 def __mul__(self, other): 

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

603 

604 @see: Method L{Fsum.__imul__}. 

605 ''' 

606 f = self._copy_2(self.__mul__) 

607 return f._fmul(other, _mul_op_) 

608 

609 def __ne__(self, other): 

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

611 ''' 

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

613 

614 def __neg__(self): 

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

616 ''' 

617 f = self._copy_2(self.__neg__) 

618 return f._fset(self._neg) 

619 

620 def __pos__(self): 

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

622 ''' 

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

624 

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

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

627 

628 @see: Method L{Fsum.__ipow__}. 

629 ''' 

630 f = self._copy_2(self.__pow__) 

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

632 

633 def __radd__(self, other): 

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

635 

636 @see: Method L{Fsum.__iadd__}. 

637 ''' 

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

639 return f._fadd(self, _add_op_) 

640 

641 def __rdivmod__(self, other): 

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

643 remainder)}. 

644 

645 @see: Method L{Fsum.__divmod__}. 

646 ''' 

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

648 return f._fdivmod2(self, _divmod_op_) 

649 

650# def __repr__(self): 

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

652# ''' 

653# return self.toRepr(lenc=True) 

654 

655 def __rfloordiv__(self, other): 

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

657 

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

659 ''' 

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

661 return f._floordiv(self, _floordiv_op_) 

662 

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

664 '''Not implemented.''' 

665 return _NotImplemented(self, other) 

666 

667 def __rmod__(self, other): 

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

669 

670 @see: Method L{Fsum.__imod__}. 

671 ''' 

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

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

674 

675 def __rmul__(self, other): 

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

677 

678 @see: Method L{Fsum.__imul__}. 

679 ''' 

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

681 return f._fmul(self, _mul_op_) 

682 

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

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

685 

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

687 ''' 

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

689 return _Psum_1(round(float(self), *ndigits), # can be C{int} 

690 name=self.__round__.__name__) 

691 

692 def __rpow__(self, other, *mod): 

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

694 

695 @see: Method L{Fsum.__ipow__}. 

696 ''' 

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

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

699 

700 def __rsub__(self, other): 

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

702 

703 @see: Method L{Fsum.__isub__}. 

704 ''' 

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

706 return f._fsub(self, _sub_op_) 

707 

708 def __rtruediv__(self, other, **raiser): 

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

710 

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

712 ''' 

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

714 return f._ftruediv(self, _truediv_op_, **raiser) 

715 

716 def __str__(self): 

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

718 ''' 

719 return self.toStr(lenc=True) 

720 

721 def __sub__(self, other): 

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

723 

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

725 

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

727 

728 @see: Method L{Fsum.__isub__}. 

729 ''' 

730 f = self._copy_2(self.__sub__) 

731 return f._fsub(other, _sub_op_) 

732 

733 def __truediv__(self, other, **raiser): 

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

735 

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

737 @kwarg raiser: Use C{B{raiser}=False} to ignore L{ResidualError}s 

738 (C{bool}), see also method L{RESIDUAL}. 

739 

740 @return: The quotient (L{Fsum}). 

741 

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

743 ''' 

744 f = self._copy_2(self.__truediv__) 

745 return f._ftruediv(other, _truediv_op_, **raiser) 

746 

747 __trunc__ = __int__ 

748 

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

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

751 __div__ = __truediv__ 

752 __idiv__ = __itruediv__ 

753 __long__ = __int__ 

754 __nonzero__ = __bool__ 

755 __rdiv__ = __rtruediv__ 

756 

757 def as_integer_ratio(self): 

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

759 

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

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

762 

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

764 ''' 

765 n, r = self._fint2 

766 if r: 

767 i, d = r.as_integer_ratio() 

768 n *= d 

769 n += i 

770 else: # PYCHOK no cover 

771 d = 1 

772 return n, d 

773 

774 @property_RO 

775 def ceil(self): 

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

777 but C{float} in Python 2-). 

778 

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

780 

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

782 L{Fsum.imag} and L{Fsum.real}. 

783 ''' 

784 s, r = self._fprs2 

785 c = _ceil(s) + int(r) - 1 

786 while r > (c - s): # (s + r) > c 

787 c += 1 

788 return c 

789 

790 def _cmp_0(self, other, op): 

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

792 ''' 

793 if isinstance(other, Fsum): 

794 s = _fsum(self._ps_1primed(*other._ps)) 

795 elif self._scalar(other, op): 

796 s = _fsum(self._ps_1primed(other)) 

797 else: 

798 s, r = self._fprs2 

799 s = _signOf(s, -r) 

800 return s 

801 

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

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

804 

805 @return: The copy (L{Fsum}). 

806 ''' 

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

808 if f._ps is self._ps: 

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

810 if not deep: 

811 f._n = 1 

812 return f 

813 

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

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

816 ''' 

817 n = name or which.__name__ 

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

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

820 # assert f._n == self._n 

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

822 return f 

823 

824 def _copy_2r(self, other, which): 

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

826 ''' 

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

828 Fsum(other, name=which.__name__) 

829 

830# def _copy_RESIDUAL(self, other): 

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

832# ''' 

833# R = other._RESIDUAL 

834# if R is not Fsum._RESIDUAL: 

835# self._RESIDUAL = R 

836 

837 def divmod(self, other, **raiser): 

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

839 remainder)}. 

840 

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

842 @kwarg raiser: Use C{B{raiser}=False} to ignore L{ResidualError}s 

843 (C{bool}), see also method L{RESIDUAL}. 

844 

845 @return: A L{DivMod2Tuple}C{(div, mod)}, with quotient C{div} 

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

847 remainder C{mod} an L{Fsum} instance. 

848 

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

850 ''' 

851 f = self._copy_2(self.divmod) 

852 return f._fdivmod2(other, _divmod_op_, **raiser) 

853 

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

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

856 ''' 

857 return Error(_SPACE_(self.toStr(), op, other), **txt_cause) 

858 

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

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

861 ''' 

862 E, t = _xError2(X) 

863 if mod: 

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

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

866 

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

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

869 ''' 

870 E, t = _xError2(X) 

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

872 return E(n, txt=t, cause=X) 

873 

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

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

876 ''' 

877 if xs: 

878 _x = _2floats(xs, **origin_X_x) # PYCHOK yield 

879 ps = self._ps 

880 ps[:] = self._ps_acc(list(ps), _x, up=up) 

881 return self 

882 

883 def _facc_1(self, xs, **up): 

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

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

886 ''' 

887 # assert islistuple(xs) 

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

889 self._facc(xs, origin=1, **up) 

890 

891 def _facc_neg(self, xs, up=True, **origin): 

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

893 ''' 

894 if xs: 

895 def _neg(x): 

896 return -x 

897 

898 _x = _2floats(xs, **origin) # PYCHOK yield 

899 ps = self._ps 

900 ps[:] = self._ps_acc(list(ps), map(_neg, _x), up=up) 

901 return self 

902 

903 def _facc_power(self, power, xs, which, **raiser): # in .fmath 

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

905 ''' 

906 def _Pow4(p): 

907 r = 0 

908 if isinstance(p, Fsum): 

909 s, r = p._fprs2 

910 if r: 

911 return _Pow4(s) 

912 m = Fsum._pow 

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

914 p = s = int(p) 

915 m = Fsum._pow_int 

916 else: 

917 p = s = _2float(power=p) 

918 m = Fsum._pow_scalar 

919 return m, p, s, r 

920 

921 _Pow, p, s, r = _Pow4(power) 

922 if p: # and xs: 

923 _pow = Fsum._pow_2_3 

924 _Fs = Fsum 

925 _Ps = _Psum_1()._fset_ps_ 

926 op = which.__name__ 

927 

928 def _X(X): 

929 f = _Pow(X, p, power, op, **raiser) 

930 return f._ps if isinstance(f, _Fs) else (f,) 

931 

932 def _x(x): 

933 x = float(x) 

934 X = _Ps(x) 

935 f = _pow(X, x, s, power, op, **raiser) 

936 if r: 

937 f *= _pow(X, x, r, power, op, **raiser) 

938 return f 

939 

940 f = self._facc(xs, origin=1, _X=_X, _x=_x) 

941 else: 

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

943 return f 

944 

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

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

947 ''' 

948 if xs: 

949 self._ps_acc(self._ps, xs, **up) 

950 return self 

951 

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

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

954 ''' 

955 if xs: 

956 self._ps_acc(self._ps, xs, **up) 

957 return self 

958 

959# def _facc_up(self, up=True): 

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

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

962# ''' 

963# while len(self._ps) > 1: 

964# p = self._ps.pop() 

965# if p: 

966# n = self._n 

967# self._facc_scalar_(p, up=False) 

968# self._n = n 

969# break 

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

971 

972 def fadd(self, xs=()): 

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

974 to this instance. 

975 

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

977 L{Fsum} instances). 

978 

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

980 

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

982 

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

984 nor L{Fsum}. 

985 

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

987 ''' 

988 if isinstance(xs, Fsum): 

989 self._facc_scalar(xs._ps) # tuple 

990 elif isscalar(xs): # for backward compatibility 

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

992 elif xs: # assert isiterable(xs) 

993 self._facc(xs) 

994 return self 

995 

996 def fadd_(self, *xs): 

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

998 to this instance. 

999 

1000 @arg xs: Values to add (C{scalar} or L{Fsum} instances), 

1001 all positional. 

1002 

1003 @return: This instance (L{Fsum}). 

1004 

1005 @raise OverflowError: Partial C{2sum} overflow. 

1006 

1007 @raise TypeError: An invalid B{C{xs}} type, not C{scalar} 

1008 nor L{Fsum}. 

1009 

1010 @raise ValueError: Invalid or non-finite B{C{xs}} value. 

1011 ''' 

1012 return self._facc_1(xs) 

1013 

1014 def _fadd(self, other, op, **up): # in .fmath.Fhorner 

1015 '''(INTERNAL) Apply C{B{self} += B{other}}. 

1016 ''' 

1017 if isinstance(other, Fsum): 

1018 self._facc_scalar(other._ps, **up) # tuple 

1019 elif self._scalar(other, op): 

1020 self._facc_scalar_(other, **up) 

1021 return self 

1022 

1023 fcopy = copy # for backward compatibility 

1024 fdiv = __itruediv__ # for backward compatibility 

1025 fdivmod = __divmod__ # for backward compatibility 

1026 

1027 def _fdivmod2(self, other, op, **raiser): 

1028 '''(INTERNAL) Apply C{B{self} %= B{other}} and return a L{DivMod2Tuple}. 

1029 ''' 

1030 # result mostly follows CPython function U{float_divmod 

1031 # <https://GitHub.com/python/cpython/blob/main/Objects/floatobject.c>}, 

1032 # but at least divmod(-3, 2) equals Cpython's result (-2, 1). 

1033 f = self._copy_2(self._fdivmod2) 

1034 q = f._ftruediv(other, op, **raiser).floor 

1035 if q: # == float // other == floor(float / other) 

1036 self -= other * q 

1037 

1038 s = signOf(other) # make signOf(self) == signOf(other) 

1039 if s and self.signOf() == -s: # PYCHOK no cover 

1040 self += other 

1041 q -= 1 

1042# t = self.signOf() 

1043# if t and t != s: 

1044# raise self._Error(op, other, _AssertionError, txt=signOf.__name__) 

1045 return DivMod2Tuple(q, self) # q is C{int} in Python 3+, but C{float} in Python 2- 

1046 

1047 def _finite(self, other, op=None): 

1048 '''(INTERNAL) Return B{C{other}} if C{finite}. 

1049 ''' 

1050 if _isfinite(other): 

1051 return other 

1052 raise ValueError(_not_finite_) if op is None else \ 

1053 self._ValueError(op, other, txt=_not_finite_) 

1054 

1055 def fint(self, raiser=True, **name): 

1056 '''Return this instance' current running sum as C{integer}. 

1057 

1058 @kwarg raiser: Use C{B{raiser}=False} to ignore L{ResidualError}s 

1059 (C{bool}), see also method L{RESIDUAL}. 

1060 @kwarg name: Optional name (C{str}), overriding C{"fint"}. 

1061 

1062 @return: The C{integer} (L{Fsum}). 

1063 

1064 @raise ResidualError: Non-zero I{integer} residual. 

1065 

1066 @see: Methods L{Fsum.int_float} and L{Fsum.is_integer}. 

1067 ''' 

1068 i, r = self._fint2 

1069 if r and raiser: 

1070 t = _stresidual(_integer_, r) 

1071 raise ResidualError(_integer_, i, txt=t) 

1072 f = self._copy_2(self.fint, **name) 

1073 return f._fset(i) 

1074 

1075 def fint2(self, **name): 

1076 '''Return this instance' current running sum as C{int} and 

1077 the I{integer} residual. 

1078 

1079 @kwarg name: Optional name (C{str}). 

1080 

1081 @return: An L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} 

1082 an C{int} and I{integer} C{residual} a C{float} or 

1083 C{INT0} if the C{fsum} is considered to be I{exact}. 

1084 ''' 

1085 return Fsum2Tuple(*self._fint2, **name) 

1086 

1087 @Property_RO 

1088 def _fint2(self): # see ._fset 

1089 '''(INTERNAL) Get 2-tuple (C{int}, I{integer} residual). 

1090 ''' 

1091 s, r = self._fprs2 

1092 i = int(s) 

1093 r = _fsum(self._ps_1primed(i)) if r else float(s - i) 

1094 return i, (r or INT0) # Fsum2Tuple? 

1095 

1096 @deprecated_property_RO 

1097 def float_int(self): # PYCHOK no cover 

1098 '''DEPRECATED, use method C{Fsum.int_float}.''' 

1099 return self.int_float() # raiser=False 

1100 

1101 @property_RO 

1102 def floor(self): 

1103 '''Get this instance' C{floor} (C{int} in Python 3+, but 

1104 C{float} in Python 2-). 

1105 

1106 @note: The C{floor} takes the C{residual} into account. 

1107 

1108 @see: Method L{Fsum.int_float} and properties L{Fsum.ceil}, 

1109 L{Fsum.imag} and L{Fsum.real}. 

1110 ''' 

1111 s, r = self._fprs2 

1112 f = _floor(s) + _floor(r) + 1 

1113 while (f - s) > r: # f > (s + r) 

1114 f -= 1 

1115 return f 

1116 

1117# floordiv = __floordiv__ # for naming consistency 

1118 

1119 def _floordiv(self, other, op, **raiser): # rather _ffloordiv? 

1120 '''Apply C{B{self} //= B{other}}. 

1121 ''' 

1122 q = self._ftruediv(other, op, **raiser) # == self 

1123 return self._fset(q.floor) # floor(q) 

1124 

1125 fmul = __imul__ # for backward compatibility 

1126 

1127 def _fmul(self, other, op): 

1128 '''(INTERNAL) Apply C{B{self} *= B{other}}. 

1129 ''' 

1130 if isinstance(other, Fsum): 

1131 if len(self._ps) != 1: 

1132 f = self._mul_Fsum(other, op) 

1133 elif len(other._ps) != 1: # and len(self._ps) == 1 

1134 f = other._mul_scalar(self._ps[0], op) 

1135 else: # len(other._ps) == len(self._ps) == 1 

1136 f = self._finite(self._ps[0] * other._ps[0]) 

1137 else: 

1138 s = self._scalar(other, op) 

1139 f = self._mul_scalar(s, op) 

1140 return self._fset(f) # n=len(self) + 1 

1141 

1142 def fover(self, over, **raiser): 

1143 '''Apply C{B{self} /= B{over}} and summate. 

1144 

1145 @arg over: An L{Fsum} or C{scalar} denominator. 

1146 @kwarg raiser: Use C{B{raiser}=False} to ignore L{ResidualError}s 

1147 (C{bool}), see also method L{RESIDUAL}. 

1148 

1149 @return: Precision running sum (C{float}). 

1150 

1151 @see: Methods L{Fsum.fsum} and L{Fsum.__itruediv__}. 

1152 ''' 

1153 return float(self.fdiv(over, **raiser)._fprs) 

1154 

1155 fpow = __ipow__ # for backward compatibility 

1156 

1157 def _fpow(self, other, op, *mod, **raiser): 

1158 '''Apply C{B{self} **= B{other}}, optional B{C{mod}} or C{None}. 

1159 ''' 

1160 if mod: 

1161 if mod[0] is not None: # == 3-arg C{pow} 

1162 f = self._pow_2_3(self, other, other, op, *mod, **raiser) 

1163 elif self.is_integer(): 

1164 # return an exact C{int} for C{int}**C{int} 

1165 i, _ = self._fint2 # assert _ == 0 

1166 x = _2scalar(other) # C{int}, C{float} or other 

1167 f = self._pow_2_3(i, x, other, op, **raiser) if isscalar(x) else \ 

1168 _Psum_1(i)._pow( x, other, op, **raiser) # x is Fsum 

1169 else: # mod[0] is None, power(self, other) 

1170 f = self._pow(other, other, op, **raiser) 

1171 else: # pow(self, other) 

1172 f = self._pow(other, other, op, **raiser) 

1173 return self._fset(f, asis=isint(f)) # n=max(len(self), 1) 

1174 

1175 @Property_RO 

1176 def _fprs(self): 

1177 '''(INTERNAL) Get and cache this instance' precision 

1178 running sum (C{float} or C{int}), ignoring C{residual}. 

1179 

1180 @note: The precision running C{fsum} after a C{//=} or 

1181 C{//} C{floor} division is C{int} in Python 3+. 

1182 ''' 

1183 return self._fprs2.fsum 

1184 

1185 @Property_RO 

1186 def _fprs2(self): 

1187 '''(INTERNAL) Get and cache this instance' precision 

1188 running sum and residual (L{Fsum2Tuple}). 

1189 ''' 

1190 ps = self._ps 

1191 n = len(ps) - 2 

1192 if n > 0: # len(ps) > 2 

1193 s = _psum(ps) 

1194 n = len(ps) - 2 

1195 if n > 0: 

1196 r = _fsum(self._ps_1primed(s)) or INT0 

1197 return Fsum2Tuple(s, r) 

1198 if n == 0: # len(ps) == 2 

1199 ps[:] = _2ps(*_2sum(*ps)) 

1200 r, s = (INT0, ps[0]) if len(ps) != 2 else ps 

1201 elif ps: # len(ps) == 1 

1202 s, r = ps[0], INT0 

1203 else: # len(ps) == 0 

1204 s, r = _0_0, INT0 

1205 ps[:] = s, 

1206 # assert self._ps is ps 

1207 return Fsum2Tuple(s, r) 

1208 

1209# def _fpsqz(self): 

1210# '''(INTERNAL) Compress, squeeze the C{partials}. 

1211# ''' 

1212# if len(self._ps) > 2: 

1213# _ = self._fprs2 

1214# return self 

1215 

1216 def fset_(self, *xs): 

1217 '''Replace this instance' value with C{xs}. 

1218 

1219 @arg xs: Optional, new values (C{scalar} or L{Fsum} 

1220 instances), all positional. 

1221 

1222 @return: This instance (C{Fsum}). 

1223 

1224 @see: Method L{Fsum.fadd} for further details. 

1225 ''' 

1226 self._ps[:] = 0, 

1227 self._n = 0 

1228 return self.fadd(xs) if xs else self._update() 

1229 

1230 def _fset(self, other, asis=True, n=0): 

1231 '''(INTERNAL) Overwrite this instance with an other or a C{scalar}. 

1232 ''' 

1233 if other is self: 

1234 pass # from ._fmul, ._ftruediv and ._pow_0_1 

1235 elif isinstance(other, Fsum): 

1236 self._ps[:] = other._ps 

1237 self._n = n or other._n 

1238# self._copy_RESIDUAL(other) 

1239 # use or zap the C{Property_RO} values 

1240 Fsum._fint2._update_from(self, other) 

1241 Fsum._fprs ._update_from(self, other) 

1242 Fsum._fprs2._update_from(self, other) 

1243 elif isscalar(other): 

1244 s = other if asis else float(other) 

1245 i = int(s) # see ._fint2 

1246 t = i, ((s - i) or INT0) 

1247 self._ps[:] = s, 

1248 self._n = n or 1 

1249 # Property_ROs _fint2, _fprs and _fprs2 can't be a Property: 

1250 # Property's _fset zaps the value just set by the @setter 

1251 self.__dict__.update(_fint2=t, _fprs=s, _fprs2=Fsum2Tuple(s, INT0)) 

1252 else: # PYCHOK no cover 

1253 raise self._TypeError(_fset_op_, other) # AssertionError 

1254 return self 

1255 

1256 def _fset_ps(self, other, n=0): # in .fmath 

1257 '''(INTERNAL) Set partials from a known C{Fsum} or C{scalar}. 

1258 ''' 

1259 if isinstance(other, Fsum): 

1260 self._ps[:] = other._ps 

1261 self._n = n or other._n 

1262 else: # assert isscalar(other) 

1263 self._ps[:] = other, 

1264 self._n = n or 1 

1265 return self 

1266 

1267 def _fset_ps_(self, *xs): 

1268 '''(INTERNAL) Set partials to all known scalar C{xs}. 

1269 ''' 

1270 self._ps[:] = xs 

1271 self.n = len(xs) 

1272 return self 

1273 

1274 def fsub(self, xs=()): 

1275 '''Subtract an iterable of C{scalar} or L{Fsum} instances from 

1276 this instance. 

1277 

1278 @arg xs: Iterable, list, tuple. etc. (C{scalar} or L{Fsum} 

1279 instances). 

1280 

1281 @return: This instance, updated (L{Fsum}). 

1282 

1283 @see: Method L{Fsum.fadd}. 

1284 ''' 

1285 return self._facc_neg(xs) 

1286 

1287 def fsub_(self, *xs): 

1288 '''Subtract all positional C{scalar} or L{Fsum} instances from 

1289 this instance. 

1290 

1291 @arg xs: Values to subtract (C{scalar} or L{Fsum} instances), 

1292 all positional. 

1293 

1294 @return: This instance, updated (L{Fsum}). 

1295 

1296 @see: Method L{Fsum.fadd}. 

1297 ''' 

1298 return self._fsub(xs[0], _sub_op_) if len(xs) == 1 else \ 

1299 self._facc_neg(xs, origin=1) 

1300 

1301 def _fsub(self, other, op): 

1302 '''(INTERNAL) Apply C{B{self} -= B{other}}. 

1303 ''' 

1304 if isinstance(other, Fsum): 

1305 if other is self: # or other._fprs2 == self._fprs2: 

1306 self._fset(_0_0) # n=len(self) * 2, self -= self 

1307 elif other._ps: 

1308 self._facc_scalar(other._ps_neg) 

1309 elif self._scalar(other, op): 

1310 self._facc_scalar_(-self._finite(other, op)) 

1311 return self 

1312 

1313 def fsum(self, xs=()): 

1314 '''Add more C{scalar} or L{Fsum} instances and summate. 

1315 

1316 @kwarg xs: Iterable, list, tuple, etc. (C{scalar} or 

1317 L{Fsum} instances). 

1318 

1319 @return: Precision running sum (C{float} or C{int}). 

1320 

1321 @see: Method L{Fsum.fadd}. 

1322 

1323 @note: Accumulation can continue after summation. 

1324 ''' 

1325 return self._facc(xs)._fprs 

1326 

1327 def fsum_(self, *xs): 

1328 '''Add all positional C{scalar} or L{Fsum} instances and summate. 

1329 

1330 @arg xs: Values to add (C{scalar} or L{Fsum} instances), all 

1331 positional. 

1332 

1333 @return: Precision running sum (C{float} or C{int}). 

1334 

1335 @see: Methods L{Fsum.fsum}, L{Fsum.Fsum_} and L{Fsum.fsumf_}. 

1336 ''' 

1337 return self._facc_1(xs)._fprs 

1338 

1339 def Fsum_(self, *xs): 

1340 '''Like method L{Fsum.fsum_} but returning an L{Fsum}. 

1341 

1342 @return: Current, precision running sum (L{Fsum}). 

1343 ''' 

1344 return self._facc_1(xs)._copy_2(self.Fsum_) 

1345 

1346 def fsum2(self, xs=(), name=NN): 

1347 '''Add more C{scalar} or L{Fsum} instances and return the 

1348 current precision running sum and the C{residual}. 

1349 

1350 @kwarg xs: Iterable, list, tuple, etc. (C{scalar} or L{Fsum} 

1351 instances). 

1352 @kwarg name: Optional name (C{str}). 

1353 

1354 @return: L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} the 

1355 current precision running sum and C{residual}, the 

1356 (precision) sum of the remaining C{partials}. The 

1357 C{residual is INT0} if the C{fsum} is considered 

1358 to be I{exact}. 

1359 

1360 @see: Methods L{Fsum.fint2}, L{Fsum.fsum} and L{Fsum.fsum2_} 

1361 ''' 

1362 t = self._facc(xs)._fprs2 

1363 return t.dup(name=name) if name else t 

1364 

1365 def fsum2_(self, *xs): 

1366 '''Add any positional C{scalar} or L{Fsum} instances and return 

1367 the precision running sum and the C{differential}. 

1368 

1369 @arg xs: Values to add (C{scalar} or L{Fsum} instances), all 

1370 positional. 

1371 

1372 @return: 2Tuple C{(fsum, delta)} with the current, precision 

1373 running C{fsum} like method L{Fsum.fsum} and C{delta}, 

1374 the difference with previous running C{fsum}, C{float}. 

1375 

1376 @see: Methods L{Fsum.fsum_} and L{Fsum.fsum}. 

1377 ''' 

1378 return self._fsum2(xs, self._facc_1) 

1379 

1380 def _fsum2(self, xs, _f, **origin): 

1381 '''(INTERNAL) Helper for L{Fsum.fsum2_} and L{Fsum.fsum2f_}. 

1382 ''' 

1383 p, q = self._fprs2 

1384 if xs: 

1385 s, r = _f(xs, **origin)._fprs2 

1386 return s, _2delta(s - p, r - q) # _fsum(_1primed((s, -p, r, -q)) 

1387 else: 

1388 return p, _0_0 

1389 

1390 def fsumf_(self, *xs): 

1391 '''Like method L{Fsum.fsum_} but only for C{B{xs}}, I{known to be scalar}. 

1392 ''' 

1393 return self._facc_scalar(xs)._fprs 

1394 

1395 def Fsumf_(self, *xs): 

1396 '''Like method L{Fsum.Fsum_} but only for C{B{xs}}, I{known to be scalar}. 

1397 ''' 

1398 return self._facc_scalar(xs)._copy_2(self.Fsumf_) 

1399 

1400 def fsum2f_(self, *xs): 

1401 '''Like method L{Fsum.fsum2_} but only for C{B{xs}}, I{known to be scalar}. 

1402 ''' 

1403 return self._fsum2(xs, self._facc_scalar, origin=1) 

1404 

1405# ftruediv = __itruediv__ # for naming consistency? 

1406 

1407 def _ftruediv(self, other, op, **raiser): 

1408 '''(INTERNAL) Apply C{B{self} /= B{other}}. 

1409 ''' 

1410 n = _1_0 

1411 if isinstance(other, Fsum): 

1412 if other is self or other == self: 

1413 return self._fset(n) # n=len(self) 

1414 d, r = other._fprs2 

1415 if r: 

1416 if d: 

1417 if self._raiser(r, d, **raiser): 

1418 raise self._ResidualError(op, other, r) 

1419 d, n = other.as_integer_ratio() 

1420 else: # PYCHOK no cover 

1421 d = r 

1422 else: 

1423 d = self._scalar(other, op) 

1424 try: 

1425 s = n / d 

1426 except Exception as X: 

1427 raise self._ErrorX(X, op, other) 

1428 f = self._mul_scalar(s, _mul_op_) # handles 0, INF, NAN 

1429 return self._fset(f) # asis=False 

1430 

1431 @property_RO 

1432 def imag(self): 

1433 '''Get the C{imaginary} part of this instance (C{0.0}, always). 

1434 

1435 @see: Properties L{Fsum.ceil}, L{Fsum.floor} and L{Fsum.real}. 

1436 ''' 

1437 return _0_0 

1438 

1439 def int_float(self, raiser=False): 

1440 '''Return this instance' current running sum as C{int} or C{float}. 

1441 

1442 @kwarg raiser: If C{True} throw a L{ResidualError} if the 

1443 residual is non-zero (C{bool}). 

1444 

1445 @return: This C{integer} sum if this instance C{is_integer}, 

1446 otherwise return the C{float} sum if the residual 

1447 is zero or if C{B{raiser}=False}. 

1448 

1449 @raise ResidualError: Non-zero residual and C{B{raiser}=True}. 

1450 

1451 @see: Methods L{Fsum.fint} and L{Fsum.fint2}. 

1452 ''' 

1453 s, r = self._fint2 

1454 if r: 

1455 s, r = self._fprs2 

1456 if r and raiser: # PYCHOK no cover 

1457 t = _stresidual(_non_zero_, r) 

1458 raise ResidualError(int_float=s, txt=t) 

1459 s = float(s) # redundant 

1460 return s 

1461 

1462 def is_exact(self): 

1463 '''Is this instance' running C{fsum} considered to be exact? (C{bool}). 

1464 ''' 

1465 return self.residual is INT0 

1466 

1467 def is_integer(self): 

1468 '''Is this instance' running sum C{integer}? (C{bool}). 

1469 

1470 @see: Methods L{Fsum.fint} and L{Fsum.fint2}. 

1471 ''' 

1472 _, r = self._fint2 

1473 return False if r else True 

1474 

1475 def is_math_fsum(self): 

1476 '''Return whether functions L{fsum}, L{fsum_}, L{fsum1} and 

1477 L{fsum1_} plus partials summation are based on Python's 

1478 C{math.fsum} or not. 

1479 

1480 @return: C{2} if all functions and partials summation 

1481 are based on C{math.fsum}, C{True} if only 

1482 the functions are based on C{math.fsum} (and 

1483 partials summation is not) or C{False} if 

1484 none are. 

1485 ''' 

1486 f = Fsum._math_fsum 

1487 return 2 if _psum is f else bool(f) 

1488 

1489 def _mul_Fsum(self, other, op=_mul_op_): # in .fmath.Fhorner 

1490 '''(INTERNAL) Return C{B{self} * Fsum B{other}} as L{Fsum} or C{0}. 

1491 ''' 

1492 # assert isinstance(other, Fsum) 

1493 if self._ps and other._ps: 

1494 f = self._ps_mul(op, *other._ps) # NO ._2scalar 

1495 else: 

1496 f = _0_0 

1497 return f 

1498 

1499 def _mul_scalar(self, factor, op): # in .fmath.Fhorner 

1500 '''(INTERNAL) Return C{B{self} * scalar B{factor}} as L{Fsum}, C{0} or C{self}. 

1501 ''' 

1502 # assert isscalar(factor) 

1503 if self._ps and self._finite(factor, op): 

1504 f = self if factor == _1_0 else ( 

1505 self._neg if factor == _N_1_0 else 

1506 self._ps_mul(op, factor)._2scalar) 

1507 else: 

1508 f = _0_0 

1509 return f 

1510 

1511 @property_RO 

1512 def _neg(self): 

1513 '''(INTERNAL) Return C{Fsum(-self)} or scalar C{NEG0}. 

1514 ''' 

1515 return _Psum(self._ps_neg) if self._ps else NEG0 

1516 

1517 @property_RO 

1518 def partials(self): 

1519 '''Get this instance' current, partial sums (C{tuple} of C{float}s). 

1520 ''' 

1521 return tuple(self._ps) 

1522 

1523 def pow(self, x, *mod, **raiser): 

1524 '''Return C{B{self}**B{x}} as L{Fsum}. 

1525 

1526 @arg x: The exponent (L{Fsum} or C{scalar}). 

1527 @arg mod: Optional modulus (C{int} or C{None}) for the 3-argument 

1528 C{pow(B{self}, B{other}, B{mod})} version. 

1529 @kwarg raiser: Use C{B{raiser}=False} to ignore L{ResidualError}s 

1530 (C{bool}), see also method L{RESIDUAL}. 

1531 

1532 @return: The C{pow(self, B{x})} or C{pow(self, B{x}, *B{mod})} 

1533 result (L{Fsum}). 

1534 

1535 @note: If B{C{mod}} is given as C{None}, the result will be an 

1536 C{integer} L{Fsum} provided this instance C{is_integer} 

1537 or set to C{integer} by an L{Fsum.fint} call. 

1538 

1539 @see: Methods L{Fsum.__ipow__}, L{Fsum.fint}, L{Fsum.is_integer} 

1540 and L{Fsum.root}. 

1541 ''' 

1542 f = self._copy_2(self.pow) 

1543 return f._fpow(x, _pow_op_, *mod, **raiser) # f = pow(f, x, *mod) 

1544 

1545 def _pow(self, other, unused, op, **raiser): 

1546 '''Return C{B{self} ** B{other}}. 

1547 ''' 

1548 if isinstance(other, Fsum): 

1549 x, r = other._fprs2 

1550 if r and self._raiser(r, x, **raiser): 

1551 raise self._ResidualError(op, other, r) 

1552 f = self._pow_scalar(x, other, op, **raiser) 

1553 if r: 

1554 f *= self._pow_scalar(r, other, op, **raiser) 

1555 elif self._scalar(other, op): 

1556 x = self._finite(other, op) 

1557 f = self._pow_scalar(x, other, op, **raiser) 

1558 else: 

1559 f = self._pow_0_1(0, other) 

1560 return f 

1561 

1562 def _pow_0_1(self, x, other): 

1563 '''(INTERNAL) Return B{C{self}**1} or C{B{self}**0 == 1.0}. 

1564 ''' 

1565 return self if x else (1 if isint(other) and self.is_integer() else _1_0) 

1566 

1567 def _pow_2_3(self, b, x, other, op, *mod, **raiser): 

1568 '''(INTERNAL) 2-arg C{pow(B{b}, scalar B{x})} and 3-arg C{pow(B{b}, 

1569 B{x}, int B{mod} or C{None})}, embellishing errors. 

1570 ''' 

1571 try: 

1572 if mod: # b, x, mod all C{int}, unless C{mod} is C{None} 

1573 m = mod[0] 

1574 b, r = b._fprs2 if m is None else b._fint2 

1575 if r and self._raiser(r, b, **raiser): 

1576 t = _non_zero_ if m is None else _integer_ 

1577 raise ResidualError(_stresidual(t, r, mod=m), txt=None) 

1578 x = _2scalar(x, _raiser=self._raiser, mod=m) 

1579 # 0**INF == 0.0, 1**INF == 1.0, -1**2.3 == -(1**2.3) 

1580 s = pow(b, x, *mod) 

1581 if iscomplex(s): 

1582 # neg**frac == complex in Python 3+, but ValueError in 2- 

1583 raise ValueError(_strcomplex(s, b, x, *mod)) 

1584 return self._finite(s) 

1585 except Exception as X: 

1586 raise self._ErrorX(X, op, other, *mod) 

1587 

1588 def _pow_int(self, x, other, op, **raiser): 

1589 '''(INTERNAL) Return C{B{self} **= B{x}} for C{int B{x} >= 0}. 

1590 ''' 

1591 # assert isint(x) and x >= 0 

1592 ps = self._ps 

1593 if len(ps) > 1: 

1594 _mul_Fsum = Fsum._mul_Fsum 

1595 if x > 4: 

1596 p = self 

1597 f = self if (x & 1) else _Psum_1() 

1598 m = x >> 1 # // 2 

1599 while m: 

1600 p = _mul_Fsum(p, p, op) # p **= 2 

1601 if (m & 1): 

1602 f = _mul_Fsum(f, p, op) # f *= p 

1603 m >>= 1 # //= 2 

1604 elif x > 1: # self**2, 3 or 4 

1605 f = _mul_Fsum(self, self, op) 

1606 if x > 2: # self**3 or 4 

1607 p = self if x < 4 else f 

1608 f = _mul_Fsum(f, p, op)._2scalar 

1609 else: # self**1 or self**0 == 1 or _1_0 

1610 f = self._pow_0_1(x, other) 

1611 elif ps: # self._ps[0]**x 

1612 f = self._pow_2_3(ps[0], x, other, op, **raiser) 

1613 else: # PYCHOK no cover 

1614 # 0**pos_int == 0, but 0**0 == 1 

1615 f = 0 if x else 1 

1616 return f 

1617 

1618 def _pow_scalar(self, x, other, op, **raiser): 

1619 '''(INTERNAL) Return C{self**B{x}} for C{scalar B{x}}. 

1620 ''' 

1621 s, r = self._fprs2 

1622 if isint(x, both=True): 

1623 x = int(x) # Fsum**int 

1624 y = abs(x) 

1625 if y > 1: 

1626 if r: 

1627 f = self._pow_int(y, other, op, **raiser) 

1628 if x > 0: # > 1 

1629 return f 

1630 # assert x < 0 # < -1 

1631 s, r = f._fprs2 if isinstance(f, Fsum) else (f, 0) 

1632 if r: 

1633 return _Psum_1()._ftruediv(f, op, **raiser) 

1634 # use **= -1 for the CPython float_pow 

1635 # error if s is zero, and not s = 1 / s 

1636 x = -1 

1637 elif x < 0: # == -1: self**(-1) == 1 / self 

1638 if r: 

1639 return _Psum_1()._ftruediv(self, op, **raiser) 

1640 else: # self**1 or self**0 

1641 return self._pow_0_1(x, other) # self, 1 or 1.0 

1642 elif r and self._raiser(r, s, **raiser): # non-zero residual**fractional 

1643 # raise self._ResidualError(op, other, r, fractional_power=x) 

1644 t = _stresidual(_non_zero_, r, fractional_power=x) 

1645 raise self._Error(op, other, ResidualError, txt=t) 

1646 # assert isscalar(s) and isscalar(x) 

1647 return self._pow_2_3(s, x, other, op, **raiser) 

1648 

1649 def _ps_acc(self, ps, xs, up=True, **unused): 

1650 '''(INTERNAL) Accumulate all scalar C{xs} into C{ps}. 

1651 ''' 

1652 n = 0 

1653 _2s = _2sum 

1654 for x in (tuple(xs) if xs is ps else xs): 

1655 # assert isscalar(x) and _isfinite(x) 

1656 if x: 

1657 i = 0 

1658 for p in ps: 

1659 x, p = _2s(x, p) 

1660 if p: 

1661 ps[i] = p 

1662 i += 1 

1663 ps[i:] = (x,) if x else () 

1664 n += 1 

1665 if n: 

1666 self._n += n 

1667 # Fsum._ps_max = max(Fsum._ps_max, len(ps)) 

1668 if up: 

1669 self._update() 

1670 return ps 

1671 

1672 def _ps_mul(self, op, *factors): 

1673 '''(INTERNAL) Multiply this instance' C{partials} with 

1674 each of the scalar B{C{factors}} and accumulate. 

1675 ''' 

1676 def _pfs(ps, fs): 

1677 if len(ps) < len(fs): 

1678 ps, fs = fs, ps 

1679 _fin = _isfinite 

1680 for f in fs: 

1681 for p in ps: 

1682 p *= f 

1683 yield p if _fin(p) else self._finite(p, op) 

1684 

1685 return _Psum(self._ps_acc([], _pfs(self._ps, factors))) 

1686 

1687 @property_RO 

1688 def _ps_neg(self): 

1689 '''(INTERNAL) Yield the partials, I{negated}. 

1690 ''' 

1691 for p in self._ps: 

1692 yield -p 

1693 

1694 def _ps_1primed(self, *less): 

1695 '''(INTERNAL) Yield partials, 1-primed and subtract any C{less} scalars. 

1696 ''' 

1697 yield _1_0 

1698 for p in self._ps: 

1699 yield p 

1700 for p in less: 

1701 yield -p 

1702 yield _N_1_0 

1703 

1704 def _raiser(self, r, s, raiser=True): 

1705 '''(INTERNAL) Does ratio C{r / s} exceed the RESIDUAL threshold? 

1706 ''' 

1707 self._ratio = t = fabs(r / s) if s else 0 # _0_0 

1708 return raiser and (t > self._RESIDUAL) 

1709 

1710 @property_RO 

1711 def real(self): 

1712 '''Get the C{real} part of this instance (C{float}). 

1713 

1714 @see: Methods L{Fsum.__float__} and L{Fsum.fsum} 

1715 and properties L{Fsum.ceil}, L{Fsum.floor}, 

1716 L{Fsum.imag} and L{Fsum.residual}. 

1717 ''' 

1718 return float(self._fprs) 

1719 

1720 @property_RO 

1721 def residual(self): 

1722 '''Get this instance' residual (C{float} or C{int}), the 

1723 C{sum(partials)} less the precision running sum C{fsum}. 

1724 

1725 @note: If the C{residual is INT0}, the precision running 

1726 C{fsum} is considered to be I{exact}. 

1727 

1728 @see: Methods L{Fsum.fsum}, L{Fsum.fsum2} and L{Fsum.is_exact}. 

1729 ''' 

1730 return self._fprs2.residual 

1731 

1732 def RESIDUAL(self, *threshold): 

1733 '''Get and set this instance' I{ratio} for raising L{ResidualError}s, 

1734 overriding the default from env variable C{PYGEODESY_FSUM_RESIDUAL}. 

1735 

1736 @arg threshold: If C{scalar}, the I{ratio} to exceed for raising 

1737 L{ResidualError}s in division and exponention, if 

1738 C{None} restore the default set with env variable 

1739 C{PYGEODESY_FSUM_RESIDUAL} or if omitted, keep the 

1740 current setting. 

1741 

1742 @return: The previous C{RESIDUAL} setting (C{float}), default C{0.}. 

1743 

1744 @raise ValueError: Negative B{C{threshold}}. 

1745 

1746 @note: L{ResidualError}s will be thrown if the non-zero I{ratio} 

1747 C{residual / fsum} exceeds the B{C{threshold}}. 

1748 ''' 

1749 r = self._RESIDUAL 

1750 if threshold: 

1751 t = threshold[0] 

1752 t = Fsum._RESIDUAL if t is None else ( 

1753 float(t) if isscalar(t) else ( # for backward ... 

1754 _0_0 if bool(t) else _1_0)) # ... compatibility 

1755 if t < 0: 

1756 u = _DOT_(self, unstr(self.RESIDUAL, *threshold)) 

1757 raise _ValueError(u, RESIDUAL=t, txt=_negative_) 

1758 self._RESIDUAL = t 

1759 return r 

1760 

1761 def _ResidualError(self, op, other, residual): 

1762 '''(INTERNAL) Non-zero B{C{residual}} etc. 

1763 ''' 

1764 t = _stresidual(_non_zero_, residual, ratio=self._ratio, 

1765 RESIDUAL=self._RESIDUAL) 

1766 t = t.replace(_COMMASPACE_R_, _exceeds_R_) 

1767 return self._Error(op, other, ResidualError, txt=t) 

1768 

1769 def root(self, root, **raiser): 

1770 '''Return C{B{self}**(1 / B{root})} as L{Fsum}. 

1771 

1772 @arg root: The order (C{scalar} or C{Fsum}), non-zero. 

1773 @kwarg raiser: Use C{B{raiser}=False} to ignore L{ResidualError}s 

1774 (C{bool}), see also method L{RESIDUAL}. 

1775 

1776 @return: The C{self ** (1 / B{root})} result (L{Fsum}). 

1777 

1778 @see: Method L{Fsum.pow}. 

1779 ''' 

1780 _root_ = self.root.__name__ 

1781 if isinstance(root, Fsum): 

1782 x = root.__rtruediv__(_1_0, **raiser) 

1783 else: 

1784 try: 

1785 x = _1_0 / _2float(root=root) 

1786 except Exception as X: 

1787 E, t = _xError2(X) 

1788 n = _SPACE_(_1_0, _truediv_op_, _root_) 

1789 raise E(n, root, txt=t, cause=X) 

1790 f = self._copy_2(self.root) 

1791 return f._fpow(x, _root_, **raiser) # == pow(f, x) 

1792 

1793 def _scalar(self, other, op, **txt): 

1794 '''(INTERNAL) Return scalar C{other}. 

1795 ''' 

1796 if isscalar(other): 

1797 return other 

1798 raise self._TypeError(op, other, **txt) # _invalid_ 

1799 

1800 @property_RO 

1801 def _2scalar(self): 

1802 '''(INTERNAL) Get this instance as C{scalar} or C{as-is}. 

1803 ''' 

1804 s, r = self._fprs2 

1805 return self if r else s 

1806 

1807 def signOf(self, res=True): 

1808 '''Determine the sign of this instance. 

1809 

1810 @kwarg res: If C{True} consider, otherwise 

1811 ignore the residual (C{bool}). 

1812 

1813 @return: The sign (C{int}, -1, 0 or +1). 

1814 ''' 

1815 s, r = self._fprs2 

1816 return _signOf(s, (-r) if res else 0) 

1817 

1818 def toRepr(self, **prec_sep_fmt_lenc): # PYCHOK signature 

1819 '''Return this C{Fsum} instance as representation. 

1820 

1821 @kwarg prec_sep_fmt_lenc: Optional keyword arguments for 

1822 method L{Fsum2Tuple.toRepr} plus C{B{lenc}=True} 

1823 (C{bool}) to in-/exclude the current C{[len]} 

1824 of this L{Fsum} enclosed in I{[brackets]}. 

1825 

1826 @return: This instance (C{repr}). 

1827 ''' 

1828 return self._toT(self._fprs2.toRepr, **prec_sep_fmt_lenc) 

1829 

1830 def toStr(self, **prec_sep_fmt_lenc): # PYCHOK signature 

1831 '''Return this C{Fsum} instance as string. 

1832 

1833 @kwarg prec_sep_fmt_lenc: Optional keyword arguments for 

1834 method L{Fsum2Tuple.toStr} plus C{B{lenc}=True} 

1835 (C{bool}) to in-/exclude the current C{[len]} 

1836 of this L{Fsum} enclosed in I{[brackets]}. 

1837 

1838 @return: This instance (C{str}). 

1839 ''' 

1840 return self._toT(self._fprs2.toStr, **prec_sep_fmt_lenc) 

1841 

1842 def _toT(self, toT, fmt=Fmt.g, lenc=True, **kwds): 

1843 '''(INTERNAL) Helper for C{toRepr} and C{toStr}. 

1844 ''' 

1845 n = self.named3 

1846 if lenc: 

1847 n = Fmt.SQUARE(n, len(self)) 

1848 return _SPACE_(n, toT(fmt=fmt, **kwds)) 

1849 

1850 def _TypeError(self, op, other, **txt): # PYCHOK no cover 

1851 '''(INTERNAL) Return a C{TypeError}. 

1852 ''' 

1853 return self._Error(op, other, _TypeError, **txt) 

1854 

1855 def _update(self, updated=True): # see ._fset 

1856 '''(INTERNAL) Zap all cached C{Property_RO} values. 

1857 ''' 

1858 if updated: 

1859 _pop = self.__dict__.pop 

1860 for p in _ROs: 

1861 _ = _pop(p, None) 

1862# Fsum._fint2._update(self) 

1863# Fsum._fprs ._update(self) 

1864# Fsum._fprs2._update(self) 

1865 return self # for .fset_ 

1866 

1867 def _ValueError(self, op, other, **txt): # PYCHOK no cover 

1868 '''(INTERNAL) Return a C{ValueError}. 

1869 ''' 

1870 return self._Error(op, other, _ValueError, **txt) 

1871 

1872 def _ZeroDivisionError(self, op, other, **txt): # PYCHOK no cover 

1873 '''(INTERNAL) Return a C{ZeroDivisionError}. 

1874 ''' 

1875 return self._Error(op, other, _ZeroDivisionError, **txt) 

1876 

1877_ROs = _allPropertiesOf_n(3, Fsum, Property_RO) # PYCHOK assert, see Fsum._fset, -._update 

1878 

1879 

1880def _Float_Int(arg, **name_Error): 

1881 '''(INTERNAL) Unit of L{Fsum2Tuple} items. 

1882 ''' 

1883 U = Int if isint(arg) else Float 

1884 return U(arg, **name_Error) 

1885 

1886 

1887class DivMod2Tuple(_NamedTuple): 

1888 '''2-Tuple C{(div, mod)} with the quotient C{div} and remainder 

1889 C{mod} results of a C{divmod} operation. 

1890 

1891 @note: Quotient C{div} an C{int} in Python 3+ or a C{float} in 

1892 Python 2-. Remainder C{mod} an L{Fsum} instance. 

1893 ''' 

1894 _Names_ = (_div_, _mod_) 

1895 _Units_ = (_Float_Int, Fsum) 

1896 

1897 

1898class Fsum2Tuple(_NamedTuple): 

1899 '''2-Tuple C{(fsum, residual)} with the precision running C{fsum} 

1900 and the C{residual}, the sum of the remaining partials. Each 

1901 item is either C{float} or C{int}. 

1902 

1903 @note: If the C{residual is INT0}, the C{fsum} is considered 

1904 to be I{exact}, see method L{Fsum2Tuple.is_exact}. 

1905 ''' 

1906 _Names_ = ( Fsum.fsum.__name__, Fsum.residual.name) 

1907 _Units_ = (_Float_Int, _Float_Int) 

1908 

1909 @Property_RO 

1910 def _Fsum(self): 

1911 '''(INTERNAL) Get this L{Fsum2Tuple} as an L{Fsum}. 

1912 ''' 

1913 s, r = map(float, self) 

1914 return _Psum(_2ps(s, r), name=self.name) 

1915 

1916 def is_exact(self): 

1917 '''Is this L{Fsum2Tuple} considered to be exact? (C{bool}). 

1918 ''' 

1919 return self._Fsum.is_exact() 

1920 

1921 def is_integer(self): 

1922 '''Is this L{Fsum2Tuple} C{integer}? (C{bool}). 

1923 ''' 

1924 return self._Fsum.is_integer() 

1925 

1926 

1927class ResidualError(_ValueError): 

1928 '''Error raised for an operation involving a L{pygeodesy.sums.Fsum} 

1929 instance with a non-zero C{residual}, I{integer} or otherwise. 

1930 

1931 @see: Module L{pygeodesy.fsums} and method L{Fsum.RESIDUAL}. 

1932 ''' 

1933 pass 

1934 

1935 

1936try: 

1937 from math import fsum as _fsum # precision IEEE-754 sum, Python 2.6+ 

1938 

1939 # make sure _fsum works as expected (XXX check 

1940 # float.__getformat__('float')[:4] == 'IEEE'?) 

1941 if _fsum((1, 1e101, 1, -1e101)) != 2: # PYCHOK no cover 

1942 del _fsum # nope, remove _fsum ... 

1943 raise ImportError # ... use _fsum below 

1944 

1945 Fsum._math_fsum = _sum = _fsum # PYCHOK exported 

1946 

1947 if _getenv('PYGEODESY_FSUM_PARTIALS', NN) == _fsum.__name__: 

1948 _psum = _fsum # PYCHOK re-def 

1949 

1950except ImportError: 

1951 _sum = sum # Fsum(NAN) exception fall-back, in .elliptic 

1952 

1953 def _fsum(xs): 

1954 '''(INTERNAL) Precision summation, Python 2.5-. 

1955 ''' 

1956 f = Fsum() 

1957 f.name = _fsum.__name__ 

1958 return f.fsum(xs) 

1959 

1960 

1961def fsum(xs, floats=False): 

1962 '''Precision floating point summation based on or like Python's C{math.fsum}. 

1963 

1964 @arg xs: Iterable, list, tuple, etc. of values (C{scalar} or L{Fsum} 

1965 instances). 

1966 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} are known 

1967 to be C{float} scalars (C{bool}). 

1968 

1969 @return: Precision C{fsum} (C{float}). 

1970 

1971 @raise OverflowError: Partial C{2sum} overflow. 

1972 

1973 @raise TypeError: Non-scalar B{C{xs}} value. 

1974 

1975 @raise ValueError: Invalid or non-finite B{C{xs}} value. 

1976 

1977 @note: Exception and I{non-finite} handling may differ if not based 

1978 on Python's C{math.fsum}. 

1979 

1980 @see: Class L{Fsum} and methods L{Fsum.fsum} and L{Fsum.fadd}. 

1981 ''' 

1982 return _fsum(xs if floats else _2floats(xs)) if xs else _0_0 # PYCHOK yield 

1983 

1984 

1985def fsum_(*xs, **floats): 

1986 '''Precision floating point summation of all positional arguments. 

1987 

1988 @arg xs: Values to be added (C{scalar} or L{Fsum} instances), all 

1989 positional. 

1990 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} are I{known 

1991 to be scalar} (C{bool}). 

1992 

1993 @return: Precision C{fsum} (C{float}). 

1994 

1995 @see: Function C{fsum}. 

1996 ''' 

1997 return _fsum(xs if _xkwds_get(floats, floats=False) else 

1998 _2floats(xs, origin=1)) if xs else _0_0 # PYCHOK yield 

1999 

2000 

2001def fsumf_(*xs): 

2002 '''Precision floating point summation, L{fsum_}C{(*B{xs}, floats=True)}, 

2003 but only for C{B{xs}} I{known to be scalar}. 

2004 ''' 

2005 return _fsum(xs) if xs else _0_0 

2006 

2007 

2008def fsum1(xs, floats=False): 

2009 '''Precision floating point summation, 1-primed. 

2010 

2011 @arg xs: Iterable, list, tuple, etc. of values (C{scalar} or L{Fsum} 

2012 instances). 

2013 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} are known 

2014 to be C{float}. 

2015 

2016 @return: Precision C{fsum} (C{float}). 

2017 

2018 @see: Function C{fsum}. 

2019 ''' 

2020 return _fsum(_1primed(xs if floats else _2floats(xs))) if xs else _0_0 # PYCHOK yield 

2021 

2022 

2023def fsum1_(*xs, **floats): 

2024 '''Precision floating point summation, 1-primed. 

2025 

2026 @arg xs: Values to be added (C{scalar} or L{Fsum} instances), all 

2027 positional. 

2028 @kwarg floats: Use C{B{floats}=True} iff I{all} B{C{xs}} are I{known 

2029 to be scalar} (C{bool}). 

2030 

2031 @return: Precision C{fsum} (C{float}). 

2032 

2033 @see: Function C{fsum} 

2034 ''' 

2035 return _fsum(_1primed(xs if _xkwds_get(floats, floats=False) else 

2036 _2floats(xs, origin=1))) if xs else _0_0 # PYCHOK yield 

2037 

2038 

2039def fsum1f_(*xs): 

2040 '''Precision floating point summation, L{fsum1_}C{(*B{xs}, floats=True)}, 

2041 but only for C{B{xs}} I{known to be scalar}. 

2042 ''' 

2043 return _fsum(_1primed(xs)) if xs else _0_0 

2044 

2045 

2046if __name__ == '__main__': 

2047 

2048 # usage: [env PYGEODESY_FSUM_PARTIALS=fsum] python3 -m pygeodesy.fsums 

2049 

2050 def _test(n): 

2051 # copied from Hettinger, see L{Fsum} reference 

2052 from pygeodesy import printf 

2053 from random import gauss, random, shuffle 

2054 

2055 printf(_fsum.__name__, end=_COMMASPACE_) 

2056 printf(_psum.__name__, end=_COMMASPACE_) 

2057 

2058 F = Fsum() 

2059 if F.is_math_fsum(): 

2060 c = (7, 1e100, -7, -1e100, -9e-20, 8e-20) * 10 

2061 for _ in range(n): 

2062 t = list(c) 

2063 s = 0 

2064 for _ in range(n * 8): 

2065 v = gauss(0, random())**7 - s 

2066 t.append(v) 

2067 s += v 

2068 shuffle(t) 

2069 assert float(F.fset_(*t)) == _fsum(t) 

2070 printf(_DOT_, end=NN) 

2071 printf(NN) 

2072 

2073 _test(128) 

2074 

2075# **) MIT License 

2076# 

2077# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

2078# 

2079# Permission is hereby granted, free of charge, to any person obtaining a 

2080# copy of this software and associated documentation files (the "Software"), 

2081# to deal in the Software without restriction, including without limitation 

2082# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

2083# and/or sell copies of the Software, and to permit persons to whom the 

2084# Software is furnished to do so, subject to the following conditions: 

2085# 

2086# The above copyright notice and this permission notice shall be included 

2087# in all copies or substantial portions of the Software. 

2088# 

2089# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

2090# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

2091# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

2092# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

2093# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

2094# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

2095# OTHER DEALINGS IN THE SOFTWARE.