Coverage for pygeodesy/fsums.py: 95%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''Class L{Fsum} for precision floating point summation similar to 

5Python's C{math.fsum} enhanced with I{running} summation and as an 

6option, accurate I{TwoProduct} multiplication. 

7 

8Accurate multiplication is based on the C{math.fma} function for 

9Python 3.13 and newer or one of two equivalent C{fma} implementations 

10for Python 3.12 and older. To enable accurate multiplication, set 

11env variable C{PYGEODESY_FSUM_F2PRODUCT} to C{"std"} or any non-empty 

12string or invoke function C{pygeodesy.f2product(True)} or set. With 

13C{"std"} the C{fma} implemention follows the C{math.fma} function, 

14otherwise the C{PyGeodesy 24.09.09} release. 

15 

16Generally, an L{Fsum} instance is considered a C{float} plus a small or 

17zero C{residue} aka C{residual} value, see property L{Fsum.residual}. 

18 

19Set env variable C{PYGEODESY_FSUM_RESIDUAL} to a C{float} string greater 

20than C{"0.0"} as the threshold to throw a L{ResidualError} for a division, 

21power or root operation of an L{Fsum} with a C{residual} I{ratio} exceeding 

22the threshold. See methods L{Fsum.RESIDUAL}, L{Fsum.pow}, L{Fsum.__ipow__} 

23and L{Fsum.__itruediv__}. 

24 

25There are several C{integer} L{Fsum} cases, for example the result from 

26functions C{ceil}, C{floor}, C{Fsum.__floordiv__} and methods L{Fsum.fint}, 

27L{Fsum.fint2} and L{Fsum.is_integer}. Also, L{Fsum} methods L{Fsum.pow}, 

28L{Fsum.__ipow__}, L{Fsum.__pow__} and L{Fsum.__rpow__} return a (very long) 

29C{int} if invoked with optional argument C{mod} set to C{None}. The 

30C{residual} of an C{integer} L{Fsum} is between C{-1.0} and C{+1.0} and 

31will be C{INT0} if that is considered to be I{exact}. 

32 

33Set env variable C{PYGEODESY_FSUM_NONFINITES} to C{"std"} or use function 

34C{pygeodesy.nonfiniterrors(False)} to allow I{non-finite} C{float}s like 

35C{inf}, C{INF}, C{NINF}, C{nan} and C{NAN} and to ignore C{OverflowError} 

36respectively C{ValueError} exceptions. However, in that case I{non-finite} 

37results may differ from Python's C{math.fsum} results. 

38''' 

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

40from __future__ import division as _; del _ # PYCHOK semicolon 

41 

42from pygeodesy.basics import isbool, iscomplex, isint, isscalar, \ 

43 _signOf, itemsorted, signOf, _xiterable, \ 

44 _xiterablen 

45from pygeodesy.constants import INF, INT0, MANT_DIG, NEG0, NINF, _0_0, \ 

46 _1_0, _N_1_0, _isfinite, _pos_self, \ 

47 Float, Int 

48from pygeodesy.errors import _AssertionError, _OverflowError, _TypeError, \ 

49 _ValueError, _xError, _xError2, _xkwds_get, \ 

50 _xkwds, _xkwds_get1, _xkwds_not, _xkwds_pop 

51from pygeodesy.internals import _enquote, _passarg 

52from pygeodesy.interns import NN, _arg_, _COMMASPACE_, _DOT_, _from_, \ 

53 _not_finite_, _SPACE_, _std_, _UNDER_ 

54from pygeodesy.lazily import _ALL_LAZY, _getenv, _sys_version_info2 

55from pygeodesy.named import _name__, _name2__, _Named, _NamedTuple, \ 

56 _NotImplemented 

57from pygeodesy.props import _allPropertiesOf_n, deprecated_method, \ 

58 deprecated_property_RO, Property, \ 

59 Property_RO, property_RO 

60from pygeodesy.streprs import Fmt, fstr, unstr 

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

62 

63from math import fabs, isinf, isnan, \ 

64 ceil as _ceil, floor as _floor # PYCHOK used! .ltp 

65 

66__all__ = _ALL_LAZY.fsums 

67__version__ = '24.09.25' 

68 

69from pygeodesy.interns import ( 

70 _PLUS_ as _add_op_, # in .auxilats.auxAngle 

71 _EQUAL_ as _fset_op_, 

72 _RANGLE_ as _gt_op_, 

73 _LANGLE_ as _lt_op_, 

74 _PERCENT_ as _mod_op_, 

75 _STAR_ as _mul_op_, 

76 _NOTEQUAL_ as _ne_op_, 

77 _DASH_ as _sub_op_, # in .auxilats.auxAngle 

78 _SLASH_ as _truediv_op_ 

79) 

80_eq_op_ = _fset_op_ * 2 # _DEQUAL_ 

81_floordiv_op_ = _truediv_op_ * 2 # _DSLASH_ 

82_divmod_op_ = _floordiv_op_ + _mod_op_ 

83_F2PRODUCT = _getenv('PYGEODESY_FSUM_F2PRODUCT', NN) 

84_ge_op_ = _gt_op_ + _fset_op_ 

85_iadd_op_ = _add_op_ + _fset_op_ # in .auxilats.auxAngle, .fstats 

86_integer_ = 'integer' 

87_isub_op_ = _sub_op_ + _fset_op_ # in .auxilats.auxAngle 

88_le_op_ = _lt_op_ + _fset_op_ 

89_NONFINITES = _getenv('PYGEODESY_FSUM_NONFINITES', NN) == _std_ 

90_non_zero_ = 'non-zero' 

91_pow_op_ = _mul_op_ * 2 # _DSTAR_ 

92_RESIDUAL_0_0 = _getenv('PYGEODESY_FSUM_RESIDUAL', _0_0) 

93_significant_ = 'significant' 

94_2split3s = _passarg 

95_threshold_ = 'threshold' 

96 

97 

98def _2finite(x): # in .fstats 

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

100 ''' 

101 return (float(x) if _isfinite(x) # and isscalar(x) 

102 else _nfError(x)) 

103 

104 

105def _2float(index=None, _isfine=_isfinite, **name_value): # in .fmath, .fstats 

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

107 ''' 

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

109 try: 

110 f = float(v) 

111 return f if _isfine(f) else _nfError(f) 

112 except Exception as X: 

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

114 

115 

116def _X_ps(X): # for _2floats only 

117 return X._ps 

118 

119 

120def _2floats(xs, origin=0, _X=_X_ps, _x=float, _isfine=_isfinite): 

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

122 ''' 

123 try: 

124 i, x = origin, xs 

125 _FsT = _Fsum_Fsum2Tuple_types 

126 for x in _xiterable(xs): 

127 if isinstance(x, _FsT): 

128 for p in _X(x._Fsum): 

129 yield p 

130 else: 

131 f = _x(x) 

132 yield f if _isfine(f) else _nfError(f) 

133 i += 1 

134 except Exception as X: 

135 raise _xsError(X, xs, i, x) 

136 

137 

138try: # MCCABE 17 

139 from math import fma as _fma 

140except ImportError: # Python 3.12- 

141 

142 if _F2PRODUCT == _std_: 

143 _2FACTOR = pow(2, (MANT_DIG + 1) // 2) + 1 

144 

145 def _fma(a, b, c): 

146 # mimick C{math.fma} from Python 3.13+, 

147 # the same accuracy, but ~13x slower 

148 b3s = _2split3(b), 

149 r = fsumf_(c, *_2products(a, b3s)) # two=True 

150 return r if _isfinite(r) else _fmaX(r, a, b, c) 

151 

152 def _2split3(x): 

153 # Split U{Algorithm 3.2 

154 # <https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>} 

155 a = c = x * _2FACTOR 

156 a -= c - x 

157 b = x - a 

158 return x, a, b 

159 

160 def _2split3s(xs): # overwrites 

161 return tuple(map(_2split3, xs)) 

162 

163 else: 

164 def _fma(*a_b_c): # PYCHOK no cover 

165 # mimick C{math.fma} from Python 3.13+, 

166 # the same accuracy, but ~14x slower 

167 (na, da), (nb, db), (nc, dc) = map(_2n_d, a_b_c) 

168 n = na * nb * dc 

169 n += da * db * nc 

170 d = da * db * dc 

171 try: 

172 r = float(n / d) 

173 except OverflowError: # "integer division result too large ..." 

174 r = NINF if (_signOf(n, 0) * _signOf(d, 0)) < 0 else INF 

175 return r if _isfinite(r) else _fmaX(r, *a_b_c) # "overflow in fma" 

176 

177 def _2n_d(x): 

178 try: # int.as_integer_ratio in 3.8+ 

179 return x.as_integer_ratio() 

180 except (AttributeError, OverflowError, TypeError, ValueError): 

181 return (x if isint(x) else float(x)), 1 

182 

183 def _fmaX(r, *a_b_c): # like Python 3.13+ I{Modules/mathmodule.c}: 

184 # raise a ValueError for a NAN result from non-NAN C{a_b_c}s or 

185 # OverflowError for a non-NAN result from all finite C{a_b_c}s. 

186 if isnan(r): 

187 def _is(x): 

188 return not isnan(x) 

189 else: 

190 _is = _isfinite 

191 if all(map(_is, a_b_c)): 

192 raise _nfError(r, unstr(_fma, *a_b_c)) 

193 return r 

194 

195if _2split3s is _passarg: # math._fma or _fma(*a_b_c) 

196 

197 def _2products(x, ys, **unused): 

198 # TwoProductFMA U{Algorithm 3.5 

199 # <https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>} 

200 for y in ys: 

201 f = x * y 

202 yield f 

203 yield _fma(x, y, -f) 

204 

205else: # in _std_ _fma(a, b, c) 

206 

207 def _2products(x, y3s, two=False): # PYCHOK redef 

208 # TwoProduct U{Algorithm 3.3 

209 # <https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>} 

210 # also in Python 3.13+ C{Modules/marhmodule.c} under 

211 # #ifndef UNRELIABLE_FMA ... #else ... #endif 

212 _, a, b = _2split3(x) 

213 for y, c, d in y3s: 

214 y *= x 

215 yield y 

216 if two: # or not a: 

217 yield b * d - (((y - a * c) - b * c) - a * d) 

218 # = b * d + (a * d - ((y - a * c) - b * c)) 

219 # = b * d + (a * d + (b * c - (y - a * c))) 

220 # = b * d + (a * d + (b * c + (a * c - y))) 

221 else: 

222 yield a * c - y 

223 yield b * c 

224 if d: 

225 yield a * d 

226 yield b * d 

227 

228 

229def f2product(*two): 

230 '''Turn accurate I{TwoProduct} multiplication on or off. 

231 

232 @arg two: If C{True}, turn I{TwoProduct} on, if C{False} off or 

233 if C{None} or omitted, keep the current setting. 

234 

235 @return: The previous setting (C{bool}). 

236 

237 @see: I{TwoProduct} multiplication is based on the I{TwoProductFMA} 

238 U{Algorithm 3.5 <https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>} 

239 using function C{math.fma} from Python 3.13 and later or an 

240 equivalent, slower implementation when not available. 

241 ''' 

242 t = Fsum._f2product 

243 if two and two[0] is not None: 

244 Fsum._f2product = bool(two[0]) 

245 return t 

246 

247 

248def _Fsumf_(*xs): # in .auxLat, .ltp, ... 

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

250 ''' 

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

252 

253 

254def _Fsum1f_(*xs): # in .albers 

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

256 ''' 

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

258 

259 

260def _2halfeven(s, r, p): 

261 '''(INTERNAL) Round half-even. 

262 ''' 

263 if (p > 0 and r > 0) or \ 

264 (p < 0 and r < 0): # signs match 

265 r *= 2 

266 t = s + r 

267 if r == (t - s): 

268 s = t 

269 return s 

270 

271 

272def _isFsum(x): # in .fmath 

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

274 ''' 

275 return isinstance(x, Fsum) 

276 

277 

278def _isFsumTuple(x): # in .basics, .constants, .fmath, .fstats 

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

280 ''' 

281 return isinstance(x, _Fsum_Fsum2Tuple_types) 

282 

283 

284def _isOK(unused): 

285 '''(INTERNAL) Helper for C{nonfiniterrors} and C{Fsum.nonfinites}. 

286 ''' 

287 return True 

288 

289 

290def _isOK_or_finite(x, _isfine=_isfinite): 

291 '''(INTERNAL) Is C{x} finite or is I{non-finite} OK?. 

292 ''' 

293 # assert _isfine in (_isOK, _isfinite) 

294 return _isfine(x) 

295 

296 

297def _nfError(x, *args): 

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

299 ''' 

300 E = _NonfiniteError(x) 

301 t = Fmt.PARENSPACED(_not_finite_, x) 

302 if args: # in _fma, _2sum 

303 return E(txt=t, *args) 

304 raise E(t, txt=None) 

305 

306 

307def nonfiniterrors(*raiser): 

308 '''Throw C{OverflowError} and C{ValueError} exceptions for or 

309 handle I{non-finite} C{float}s as C{inf}, C{INF}, C{NINF}, 

310 C{nan} and C{NAN} in summations and multiplications. 

311 

312 @arg raiser: If C{True}, throw exceptions, if C{False} handle 

313 I{non-finites} or if C{None} or omitted, leave 

314 the setting unchanged. 

315 

316 @return: Previous setting (C{bool}). 

317 

318 @note: C{inf}, C{INF} and C{NINF} throw an C{OverflowError}, 

319 C{nan} and C{NAN} a C{ValueError}. 

320 ''' 

321 d = Fsum._isfine 

322 if raiser and raiser[0] is not None: 

323 Fsum._isfine = {} if bool(raiser[0]) else Fsum._nonfinites_isfine_kwds[True] 

324 return _xkwds_get1(d, _isfine=_isfinite) is _isfinite 

325 

326 

327def _NonfiniteError(x): 

328 '''(INTERNAL) Return the Error class for C{x}, I{non-finite}. 

329 ''' 

330 return _OverflowError if isinf(x) else ( 

331 _ValueError if isnan(x) else _AssertionError) 

332 

333 

334def _1primed(xs): # in .fmath 

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

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

337 ''' 

338 yield _1_0 

339 for x in xs: 

340 yield x 

341 yield _N_1_0 

342 

343 

344def _psum(ps, **_isfine): # PYCHOK used! 

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

346 ''' 

347 # assert isinstance(ps, list) 

348 i = len(ps) - 1 

349 s = _0_0 if i < 0 else ps[i] 

350 while i > 0: 

351 i -= 1 

352 s, r = _2sum(s, ps[i], **_isfine) 

353 if r: # sum(ps) became inexact 

354 if s: 

355 ps[i:] = r, s 

356 if i > 0: 

357 s = _2halfeven(s, r, ps[i-1]) 

358 break # return s 

359 s = r # PYCHOK no cover 

360 elif not _isfinite(s): # non-finite OK 

361 i = 0 # collapse ps 

362 if ps: 

363 s += _sum(ps) # _fsum(ps) 

364 ps[i:] = s, 

365 return s 

366 

367 

368def _Psum(ps, **name_f2product_nonfinites_RESIDUAL): 

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

370 ''' 

371 F = Fsum(**name_f2product_nonfinites_RESIDUAL) 

372 if ps: 

373 F._ps[:] = ps 

374 F._n = len(F._ps) 

375 return F 

376 

377 

378def _Psum_(*ps, **name_f2product_nonfinites_RESIDUAL): # in .fmath 

379 '''(INTERNAL) Return an C{Fsum} from I{known scalar} C{ps}. 

380 ''' 

381 return _Psum(ps, **name_f2product_nonfinites_RESIDUAL) 

382 

383 

384def _2scalar2(other): 

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

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

387 ''' 

388 if _isFsumTuple(other): 

389 s, r = other._fint2 

390 if r: 

391 s, r = other._fprs2 

392 if r: # PYCHOK no cover 

393 s = other # L{Fsum} as-is 

394 else: 

395 r = 0 

396 s = other # C{type} as-is 

397 if isint(s, both=True): 

398 s = int(s) 

399 return s, r 

400 

401 

402def _s_r(s, r): 

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

404 ''' 

405 if r and _isfinite(s): 

406 if fabs(s) < fabs(r): 

407 s, r = r, (s or INT0) 

408 else: 

409 r = INT0 

410 return s, r 

411 

412 

413def _strcomplex(s, *args): 

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

415 ''' 

416 c = _strcomplex.__name__[4:] 

417 n = _sub_op_(len(args), _arg_) 

418 t = unstr(pow, *args) 

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

420 

421 

422def _stresidual(prefix, residual, R=0, **mod_ratio): 

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

424 ''' 

425 p = _stresidual.__name__[3:] 

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

427 for n, v in itemsorted(mod_ratio): 

428 p = Fmt.PARENSPACED(n, Fmt(v)) 

429 t = _COMMASPACE_(t, p) 

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

431 

432 

433def _2sum(a, b, _isfine=_isfinite): # in .testFmath 

434 '''(INTERNAL) Return C{a + b} as 2-tuple C{(sum, residual)} with finite C{sum}, 

435 otherwise as 2-tuple C{(nonfinite, 0)} iff I{non-finites} are OK. 

436 ''' 

437 # FastTwoSum U{Algorithm 1.1<https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>} 

438 

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

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

441 # 1974, Zeitschrift für Angewandte Mathmatik und Mechanik, vol 51, nr 1, p 39-51 

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

443 s = a + b 

444 if _isfinite(s): 

445 if fabs(a) < fabs(b): 

446 r = (b - s) + a 

447 else: 

448 r = (a - s) + b 

449 elif _isfine(s): 

450 r = 0 

451 else: # non-finite and not OK 

452 t = unstr(_2sum, a, b) 

453 raise _nfError(s, t) 

454 return s, r 

455 

456 

457def _threshold(threshold=_0_0, **kwds): 

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

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

460 ''' 

461 if kwds: 

462 threshold = _xkwds_get1(kwds, RESIDUAL=threshold) 

463 try: 

464 return _2finite(threshold) # PYCHOK None 

465 except Exception as x: 

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

467 

468 

469class Fsum(_Named): # sync __methods__ with .vector3dBase.Vector3dBase, .fstats, ... 

470 '''Precision floating point summation, I{running} summation and accurate multiplication. 

471 

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

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

474 intermediate, I{running} summuation. 

475 

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

477 i.e. any C{type} having method C{__float__}. 

478 

479 @note: Handling of I{non-finites} as C{inf}, C{INF}, C{NINF}, C{nan} and C{NAN} is 

480 determined globally by function L{nonfiniterrors<fsums.nonfiniterrors>} and 

481 by method L{nonfinites<Fsum.nonfinites>} for individual C{Fsum} instances, 

482 overruling the global setting. By default and for backward compatibility, 

483 I{non-finites} raise exceptions. 

484 

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

486 393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}, 

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

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

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

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

491 

492 @see: Method L{f2product<Fsum.f2product>} for details about accurate I{TwoProduct} 

493 multiplication. 

494 

495 @see: Module L{fsums<pygeodesy.fsums>} for env variables C{PYGEODESY_FSUM_F2PRODUCT}, 

496 C{PYGEODESY_FSUM_NONFINITES} and C{PYGEODESY_FSUM_RESIDUAL}. 

497 ''' 

498 _f2product = _sys_version_info2 > (3, 12) or bool(_F2PRODUCT) 

499 _isfine = {} # == _isfinite 

500 _n = 0 

501# _ps = [] # partial sums 

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

503 _RESIDUAL = _threshold(_RESIDUAL_0_0) 

504 

505 def __init__(self, *xs, **name_f2product_nonfinites_RESIDUAL): 

506 '''New L{Fsum}. 

507 

508 @arg xs: No, one or more initial items to accumulate (each C{scalar}, an 

509 L{Fsum} or L{Fsum2Tuple}), all positional. 

510 @kwarg name_f2product_nonfinites_RESIDUAL: Optional C{B{name}=NN} (C{str}) 

511 and settings C{B{f2product}=None} (C{bool}), C{B{nonfinites}=None} 

512 (C{bool}) and C{B{RESIDUAL}=0.0} threshold (C{scalar}) for this 

513 L{Fsum}. 

514 

515 @see: Methods L{Fsum.f2product}, L{Fsum.nonfinites}, L{Fsum.RESIDUAL}, 

516 L{Fsum.fadd} and L{Fsum.fadd_}. 

517 ''' 

518 if name_f2product_nonfinites_RESIDUAL: 

519 self._optionals(**name_f2product_nonfinites_RESIDUAL) 

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

521 if xs: 

522 self._facc_args(xs, up=False) 

523 

524 def __abs__(self): 

525 '''Return C{abs(self)} as an L{Fsum}. 

526 ''' 

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

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

529 

530 def __add__(self, other): 

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

532 

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

534 

535 @return: The sum (L{Fsum}). 

536 

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

538 ''' 

539 f = self._copy_2(self.__add__) 

540 return f._fadd(other, _add_op_) 

541 

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

543 '''Return C{bool(B{self})}, C{True} iff C{residual} is zero. 

544 ''' 

545 s, r = self._fprs2 

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

547 

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

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

550 

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

552 

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

554 ''' 

555 return self.ceil 

556 

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

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

559 

560 @return: -1, 0 or +1 (C{int}). 

561 

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

563 ''' 

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

565 return _signOf(s, 0) 

566 

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

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

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

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

571 

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

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

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

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

576 

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

578 B{C{RESIDUAL}}. 

579 

580 @see: Method L{Fsum.fdiv}. 

581 ''' 

582 f = self._copy_2(self.__divmod__) 

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

584 

585 def __eq__(self, other): 

586 '''Return C{(B{self} == B{other})} as C{bool} where B{C{other}} 

587 is C{scalar}, an other L{Fsum} or L{Fsum2Tuple}. 

588 ''' 

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

590 

591 def __float__(self): 

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

593 

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

595 ''' 

596 return float(self._fprs) 

597 

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

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

600 

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

602 

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

604 ''' 

605 return self.floor 

606 

607 def __floordiv__(self, other): 

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

609 

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

611 

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

613 

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

615 ''' 

616 f = self._copy_2(self.__floordiv__) 

617 return f._floordiv(other, _floordiv_op_) 

618 

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

620 '''Not implemented.''' 

621 return _NotImplemented(self, *other) 

622 

623 def __ge__(self, other): 

624 '''Return C{(B{self} >= B{other})}, see C{__eq__}. 

625 ''' 

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

627 

628 def __gt__(self, other): 

629 '''Return C{(B{self} > B{other})}, see C{__eq__}. 

630 ''' 

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

632 

633 def __hash__(self): # PYCHOK no cover 

634 '''Return C{hash(B{self})} as C{float}. 

635 ''' 

636 # @see: U{Notes for type implementors<https://docs.Python.org/ 

637 # 3/library/numbers.html#numbers.Rational>} 

638 return hash(self.partials) # tuple.__hash__() 

639 

640 def __iadd__(self, other): 

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

642 

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

644 an iterable of several of the former. 

645 

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

647 

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

649 C{scalar} nor L{Fsum}. 

650 

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

652 ''' 

653 try: 

654 return self._fadd(other, _iadd_op_) 

655 except TypeError: 

656 pass 

657 _xiterable(other) 

658 return self._facc(other) 

659 

660 def __ifloordiv__(self, other): 

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

662 

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

664 

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

666 

667 @raise ResidualError: Non-zero, significant residual 

668 in B{C{other}}. 

669 

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

671 

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

673 

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

675 

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

677 ''' 

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

679 

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

681 '''Not implemented.''' 

682 return _NotImplemented(self, other) 

683 

684 def __imod__(self, other): 

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

686 

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

688 

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

690 

691 @see: Method L{Fsum.__divmod__}. 

692 ''' 

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

694 

695 def __imul__(self, other): 

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

697 

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

699 

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

701 

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

703 

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

705 

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

707 ''' 

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

709 

710 def __int__(self): 

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

712 

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

714 and L{Fsum.floor}. 

715 ''' 

716 i, _ = self._fint2 

717 return i 

718 

719 def __invert__(self): # PYCHOK no cover 

720 '''Not implemented.''' 

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

722 return _NotImplemented(self) 

723 

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

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

726 

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

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

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

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

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

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

733 

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

735 

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

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

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

739 

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

741 

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

743 is non-zero and significant and either 

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

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

746 C{None}. 

747 

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

749 invocation failed. 

750 

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

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

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

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

755 is given as C{0}. 

756 

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

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

759 ''' 

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

761 

762 def __isub__(self, other): 

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

764 

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

766 an iterable of several of the former. 

767 

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

769 

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

771 

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

773 ''' 

774 try: 

775 return self._fsub(other, _isub_op_) 

776 except TypeError: 

777 pass 

778 _xiterable(other) 

779 return self._facc_neg(other) 

780 

781 def __iter__(self): 

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

783 ''' 

784 return iter(self.partials) 

785 

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

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

788 

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

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

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

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

793 

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

795 

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

797 

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

799 B{C{RESIDUAL}}. 

800 

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

802 

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

804 

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

806 

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

808 ''' 

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

810 

811 def __le__(self, other): 

812 '''Return C{(B{self} <= B{other})}, see C{__eq__}. 

813 ''' 

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

815 

816 def __len__(self): 

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

818 ''' 

819 return self._n 

820 

821 def __lt__(self, other): 

822 '''Return C{(B{self} < B{other})}, see C{__eq__}. 

823 ''' 

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

825 

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

827 '''Not implemented.''' 

828 return _NotImplemented(self, other) 

829 

830 def __mod__(self, other): 

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

832 

833 @see: Method L{Fsum.__imod__}. 

834 ''' 

835 f = self._copy_2(self.__mod__) 

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

837 

838 def __mul__(self, other): 

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

840 

841 @see: Method L{Fsum.__imul__}. 

842 ''' 

843 f = self._copy_2(self.__mul__) 

844 return f._fmul(other, _mul_op_) 

845 

846 def __ne__(self, other): 

847 '''Return C{(B{self} != B{other})}, see C{__eq__}. 

848 ''' 

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

850 

851 def __neg__(self): 

852 '''Return C{copy(B{self})}, I{negated}. 

853 ''' 

854 f = self._copy_2(self.__neg__) 

855 return f._fset(self._neg) 

856 

857 def __pos__(self): 

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

859 ''' 

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

861 

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

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

864 

865 @see: Method L{Fsum.__ipow__}. 

866 ''' 

867 f = self._copy_2(self.__pow__) 

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

869 

870 def __radd__(self, other): 

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

872 

873 @see: Method L{Fsum.__iadd__}. 

874 ''' 

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

876 return f._fadd(self, _add_op_) 

877 

878 def __rdivmod__(self, other): 

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

880 C{(quotient, remainder)}. 

881 

882 @see: Method L{Fsum.__divmod__}. 

883 ''' 

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

885 return f._fdivmod2(self, _divmod_op_) 

886 

887# def __repr__(self): 

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

889# ''' 

890# return self.toRepr(lenc=True) 

891 

892 def __rfloordiv__(self, other): 

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

894 

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

896 ''' 

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

898 return f._floordiv(self, _floordiv_op_) 

899 

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

901 '''Not implemented.''' 

902 return _NotImplemented(self, other) 

903 

904 def __rmod__(self, other): 

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

906 

907 @see: Method L{Fsum.__imod__}. 

908 ''' 

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

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

911 

912 def __rmul__(self, other): 

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

914 

915 @see: Method L{Fsum.__imul__}. 

916 ''' 

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

918 return f._fmul(self, _mul_op_) 

919 

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

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

922 

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

924 ''' 

925 f = self._copy_2(self.__round__) 

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

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

928 

929 def __rpow__(self, other, *mod): 

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

931 

932 @see: Method L{Fsum.__ipow__}. 

933 ''' 

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

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

936 

937 def __rsub__(self, other): 

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

939 

940 @see: Method L{Fsum.__isub__}. 

941 ''' 

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

943 return f._fsub(self, _sub_op_) 

944 

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

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

947 

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

949 ''' 

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

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

952 

953 def __str__(self): 

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

955 ''' 

956 return self.toStr(lenc=True) 

957 

958 def __sub__(self, other): 

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

960 

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

962 

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

964 

965 @see: Method L{Fsum.__isub__}. 

966 ''' 

967 f = self._copy_2(self.__sub__) 

968 return f._fsub(other, _sub_op_) 

969 

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

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

972 

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

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

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

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

977 

978 @return: The quotient (L{Fsum}). 

979 

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

981 B{C{RESIDUAL}}. 

982 

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

984 ''' 

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

986 

987 __trunc__ = __int__ 

988 

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

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

991 __div__ = __truediv__ 

992 __idiv__ = __itruediv__ 

993 __long__ = __int__ 

994 __nonzero__ = __bool__ 

995 __rdiv__ = __rtruediv__ 

996 

997 def as_integer_ratio(self): 

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

999 

1000 @return: 2-Tuple C{(numerator, denominator)} both C{int} with 

1001 C{numerator} signed and C{denominator} non-zero and 

1002 positive. The C{numerator} is I{non-finite} if this 

1003 instance is. 

1004 

1005 @see: Method L{Fsum.fint2} and C{float.as_integer_ratio} in 

1006 Python 2.7+. 

1007 ''' 

1008 n, r = self._fint2 

1009 if r: 

1010 i, d = float(r).as_integer_ratio() 

1011 n *= d 

1012 n += i 

1013 else: # PYCHOK no cover 

1014 d = 1 

1015 return n, d 

1016 

1017 @property_RO 

1018 def as_iscalar(self): 

1019 '''Get this instance I{as-is} (L{Fsum} with C{non-zero residual}, 

1020 C{scalar} or I{non-finite}). 

1021 ''' 

1022 s, r = self._fprs2 

1023 return self if r else s 

1024 

1025 @property_RO 

1026 def ceil(self): 

1027 '''Get this instance' C{ceil} value (C{int} in Python 3+, but 

1028 C{float} in Python 2-). 

1029 

1030 @note: This C{ceil} takes the C{residual} into account. 

1031 

1032 @see: Method L{Fsum.int_float} and properties L{Fsum.floor}, 

1033 L{Fsum.imag} and L{Fsum.real}. 

1034 ''' 

1035 s, r = self._fprs2 

1036 c = _ceil(s) + int(r) - 1 

1037 while r > (c - s): # (s + r) > c 

1038 c += 1 

1039 return c # _ceil(self._n_d) 

1040 

1041 cmp = __cmp__ 

1042 

1043 def _cmp_0(self, other, op): 

1044 '''(INTERNAL) Return C{scalar(self - B{other})} for 0-comparison. 

1045 ''' 

1046 if _isFsumTuple(other): 

1047 s = self._ps_1sum(*other._ps) 

1048 elif self._scalar(other, op): 

1049 s = self._ps_1sum(other) 

1050 else: 

1051 s = self.signOf() # res=True 

1052 return s 

1053 

1054 def copy(self, deep=False, **name): 

1055 '''Copy this instance, C{shallow} or B{C{deep}}. 

1056 

1057 @kwarg name: Optional, overriding C{B{name}='"copy"} (C{str}). 

1058 

1059 @return: The copy (L{Fsum}). 

1060 ''' 

1061 n = _name__(name, name__=self.copy) 

1062 f = _Named.copy(self, deep=deep, name=n) 

1063 if f._ps is self._ps: 

1064 f._ps = list(self._ps) # separate list 

1065 if not deep: 

1066 f._n = 1 

1067 # assert f._f2product == self._f2product 

1068 # assert f._Fsum is f 

1069 return f 

1070 

1071 def _copy_2(self, which, name=NN): 

1072 '''(INTERNAL) Copy for I{dyadic} operators. 

1073 ''' 

1074 n = name or which.__name__ # _dunder_nameof 

1075 # NOT .classof due to .Fdot(a, *b) args, etc. 

1076 f = _Named.copy(self, deep=False, name=n) 

1077 f._ps = list(self._ps) # separate list 

1078 # assert f._n == self._n 

1079 # assert f._f2product == self._f2product 

1080 # assert f._Fsum is f 

1081 return f 

1082 

1083 def _copy_2r(self, other, which): 

1084 '''(INTERNAL) Copy for I{reverse-dyadic} operators. 

1085 ''' 

1086 return other._copy_2(which) if _isFsum(other) else \ 

1087 self._copy_2(which)._fset(other) 

1088 

1089 divmod = __divmod__ 

1090 

1091 def _Error(self, op, other, Error, **txt_cause): 

1092 '''(INTERNAL) Format an B{C{Error}} for C{{self} B{op} B{other}}. 

1093 ''' 

1094 return Error(_SPACE_(self.as_iscalar, op, other), **txt_cause) 

1095 

1096 def _ErrorX(self, X, op, other, *mod): 

1097 '''(INTERNAL) Format the caught exception C{X}. 

1098 ''' 

1099 E, t = _xError2(X) 

1100 if mod: 

1101 t = _COMMASPACE_(Fmt.PARENSPACED(mod=mod[0]), t) 

1102 return self._Error(op, other, E, txt=t, cause=X) 

1103 

1104 def _ErrorXs(self, X, xs, **kwds): # in .fmath 

1105 '''(INTERNAL) Format the caught exception C{X}. 

1106 ''' 

1107 E, t = _xError2(X) 

1108 u = unstr(self.named3, *xs[:3], _ELLIPSIS=len(xs) > 3, **kwds) 

1109 return E(u, txt=t, cause=X) 

1110 

1111 def _facc(self, xs, up=True, **origin_X_x): 

1112 '''(INTERNAL) Accumulate more C{scalars} or L{Fsum}s. 

1113 ''' 

1114 if xs: 

1115 kwds = _xkwds(self._isfine, **origin_X_x) 

1116 _xs = _2floats(xs, **kwds) # PYCHOK yield 

1117 ps = self._ps 

1118 ps[:] = self._ps_acc(list(ps), _xs, up=up) 

1119 return self 

1120 

1121 def _facc_args(self, xs, **up): 

1122 '''(INTERNAL) Accumulate 0, 1 or more C{xs}, all positional 

1123 arguments in the caller of this method. 

1124 ''' 

1125 return self._facc(xs, origin=1, **up) if len(xs) != 1 else \ 

1126 self._fadd(xs[0], _add_op_, **up) 

1127 

1128 def _facc_neg(self, xs, **up_origin): 

1129 '''(INTERNAL) Accumulate more C{xs}, negated. 

1130 ''' 

1131 def _N(X): 

1132 return X._ps_neg 

1133 

1134 def _n(x): 

1135 return -float(x) 

1136 

1137 return self._facc(xs, _X=_N, _x=_n, **up_origin) 

1138 

1139 def _facc_power(self, power, xs, which, **raiser_RESIDUAL): # in .fmath 

1140 '''(INTERNAL) Add each C{xs} as C{float(x**power)}. 

1141 ''' 

1142 def _Pow4(p): 

1143 r = 0 

1144 if _isFsumTuple(p): 

1145 s, r = p._fprs2 

1146 if r: 

1147 m = Fsum._pow 

1148 else: # scalar 

1149 return _Pow4(s) 

1150 elif isint(p, both=True) and int(p) >= 0: 

1151 p = s = int(p) 

1152 m = Fsum._pow_int 

1153 else: 

1154 p = s = _2float(power=p, **self._isfine) 

1155 m = Fsum._pow_scalar 

1156 return m, p, s, r 

1157 

1158 _Pow, p, s, r = _Pow4(power) 

1159 if p: # and xs: 

1160 op = which.__name__ 

1161 _FsT = _Fsum_Fsum2Tuple_types 

1162 _pow = self._pow_2_3 

1163 

1164 def _P(X): 

1165 f = _Pow(X, p, power, op, **raiser_RESIDUAL) 

1166 return f._ps if isinstance(f, _FsT) else (f,) 

1167 

1168 def _p(x): 

1169 x = float(x) 

1170 f = _pow(x, s, power, op, **raiser_RESIDUAL) 

1171 if f and r: 

1172 f *= _pow(x, r, power, op, **raiser_RESIDUAL) 

1173 return f 

1174 

1175 f = self._facc(xs, origin=1, _X=_P, _x=_p) 

1176 else: 

1177 f = self._facc_scalar_(float(len(xs))) # x**0 == 1 

1178 return f 

1179 

1180 def _facc_scalar(self, xs, **up): 

1181 '''(INTERNAL) Accumulate all C{xs}, known to be scalar. 

1182 ''' 

1183 if xs: 

1184 _ = self._ps_acc(self._ps, xs, **up) 

1185 return self 

1186 

1187 def _facc_scalar_(self, *xs, **up): 

1188 '''(INTERNAL) Accumulate all positional C{xs}, known to be scalar. 

1189 ''' 

1190 if xs: 

1191 _ = self._ps_acc(self._ps, xs, **up) 

1192 return self 

1193 

1194# def _facc_up(self, up=True): 

1195# '''(INTERNAL) Update the C{partials}, by removing 

1196# and re-accumulating the final C{partial}. 

1197# ''' 

1198# ps = self._ps 

1199# while len(ps) > 1: 

1200# p = ps.pop() 

1201# if p: 

1202# n = self._n 

1203# _ = self._ps_acc(ps, (p,), up=False) 

1204# self._n = n 

1205# break 

1206# return self._update() if up else self 

1207 

1208 def fadd(self, xs=()): 

1209 '''Add an iterable's items to this instance. 

1210 

1211 @arg xs: Iterable of items to add (each C{scalar} 

1212 or an L{Fsum} or L{Fsum2Tuple} instance). 

1213 

1214 @return: This instance (L{Fsum}). 

1215 

1216 @raise OverflowError: Partial C{2sum} overflow. 

1217 

1218 @raise TypeError: An invalid B{C{xs}} item. 

1219 

1220 @raise ValueError: Invalid or I{non-finite} B{C{xs}} value. 

1221 ''' 

1222 if _isFsumTuple(xs): 

1223 self._facc_scalar(xs._ps) 

1224 elif isscalar(xs): # for backward compatibility # PYCHOK no cover 

1225 x = _2float(x=xs, **self._isfine) 

1226 self._facc_scalar_(x) 

1227 elif xs: # _xiterable(xs) 

1228 self._facc(xs) 

1229 return self 

1230 

1231 def fadd_(self, *xs): 

1232 '''Add all positional items to this instance. 

1233 

1234 @arg xs: Values to add (each C{scalar} or an L{Fsum} 

1235 or L{Fsum2Tuple} instance), all positional. 

1236 

1237 @see: Method L{Fsum.fadd} for further details. 

1238 ''' 

1239 return self._facc_args(xs) 

1240 

1241 def _fadd(self, other, op, **up): # in .fmath.Fhorner 

1242 '''(INTERNAL) Apply C{B{self} += B{other}}. 

1243 ''' 

1244 if _isFsumTuple(other): 

1245 if self._ps: 

1246 self._facc_scalar(other._ps, **up) 

1247 else: 

1248 self._fset(other, op=op, **up) 

1249 elif self._scalar(other, op): 

1250 if self._ps: 

1251 self._facc_scalar_(other, **up) 

1252 else: 

1253 self._fset(other, op=op, **up) 

1254 return self 

1255 

1256 fcopy = copy # for backward compatibility 

1257 fdiv = __itruediv__ 

1258 fdivmod = __divmod__ 

1259 

1260 def _fdivmod2(self, other, op, **raiser_RESIDUAL): 

1261 '''(INTERNAL) Apply C{B{self} %= B{other}} and return a L{DivMod2Tuple}. 

1262 ''' 

1263 # result mostly follows CPython function U{float_divmod 

1264 # <https://GitHub.com/python/cpython/blob/main/Objects/floatobject.c>}, 

1265 # but at least divmod(-3, 2) equals Cpython's result (-2, 1). 

1266 q = self._truediv(other, op, **raiser_RESIDUAL).floor 

1267 if q: # == float // other == floor(float / other) 

1268 self -= self._Fsum_as(q) * other # NOT other * q! 

1269 

1270 s = signOf(other) # make signOf(self) == signOf(other) 

1271 if s and self.signOf() == -s: # PYCHOK no cover 

1272 self += other 

1273 q -= 1 

1274# t = self.signOf() 

1275# if t and t != s: 

1276# raise self._Error(op, other, _AssertionError, txt__=signOf) 

1277 return DivMod2Tuple(q, self) # q is C{int} in Python 3+, but C{float} in Python 2- 

1278 

1279 def _fhorner(self, x, cs, op, incx=True): # in .fmath 

1280 '''(INTERNAL) Add an L{Fhorner} evaluation of polynomial 

1281 C{sum(cs[i] * B{x}**i for i=0..len(cs)-1) if B{incx} 

1282 else sum(... i=len(cs)-1..0)}. 

1283 ''' 

1284 if _xiterablen(cs): 

1285 H = self._Fsum_as(name__=self._fhorner) 

1286 if _isFsumTuple(x): 

1287 _mul = H._mul_Fsum 

1288 else: 

1289 _mul = H._mul_scalar 

1290 x = _2float(x=x, **self._isfine) 

1291 if len(cs) > 1 and x: 

1292 for c in (reversed(cs) if incx else cs): 

1293 H._fset_ps(_mul(x, op)) 

1294 H._fadd(c, op, up=False) 

1295 else: # x == 0 

1296 H = cs[0] if cs else _0_0 

1297 self._fadd(H, op) 

1298 return self 

1299 

1300 def _finite(self, other, op=None): 

1301 '''(INTERNAL) Return B{C{other}} if C{finite}. 

1302 ''' 

1303 if _isOK_or_finite(other, **self._isfine): 

1304 return other 

1305 E = _NonfiniteError(other) 

1306 raise self._Error(op, other, E, txt=_not_finite_) 

1307 

1308 def fint(self, name=NN, **raiser_RESIDUAL): 

1309 '''Return this instance' current running sum as C{integer}. 

1310 

1311 @kwarg name: Optional, overriding C{B{name}="fint"} (C{str}). 

1312 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1313 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1314 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1315 

1316 @return: The C{integer} sum (L{Fsum}) if this instance C{is_integer} 

1317 with a zero or insignificant I{integer} residual. 

1318 

1319 @raise ResidualError: Non-zero, significant residual or invalid 

1320 B{C{RESIDUAL}}. 

1321 

1322 @see: Methods L{Fsum.fint2}, L{Fsum.int_float} and L{Fsum.is_integer}. 

1323 ''' 

1324 i, r = self._fint2 

1325 if r: 

1326 R = self._raiser(r, i, **raiser_RESIDUAL) 

1327 if R: 

1328 t = _stresidual(_integer_, r, **R) 

1329 raise ResidualError(_integer_, i, txt=t) 

1330 return self._Fsum_as(i, name=_name__(name, name__=self.fint)) 

1331 

1332 def fint2(self, **name): 

1333 '''Return this instance' current running sum as C{int} and the 

1334 I{integer} residual. 

1335 

1336 @kwarg name: Optional name (C{str}). 

1337 

1338 @return: An L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} 

1339 an C{int} and I{integer} C{residual} a C{float} or 

1340 C{INT0} if the C{fsum} is considered to be I{exact}. 

1341 The C{fsum} is I{non-finite} if this instance is. 

1342 ''' 

1343 return Fsum2Tuple(*self._fint2, **name) 

1344 

1345 @Property 

1346 def _fint2(self): # see ._fset 

1347 '''(INTERNAL) Get 2-tuple (C{int}, I{integer} residual). 

1348 ''' 

1349 s, _ = self._fprs2 

1350 try: 

1351 i = int(s) 

1352 r = (self._ps_1sum(i) if len(self._ps) > 1 else 

1353 float(s - i)) or INT0 

1354 except (OverflowError, ValueError) as X: 

1355 r = 0 # INF, NAN, NINF 

1356 i = self._fintX(X, sum(self._ps)) 

1357 return i, r # Fsum2Tuple? 

1358 

1359 @_fint2.setter_ # PYCHOK setter_UNDERscore! 

1360 def _fint2(self, s): # in _fset 

1361 '''(INTERNAL) Replace the C{_fint2} value. 

1362 ''' 

1363 try: 

1364 i = int(s) 

1365 r = (s - i) or INT0 

1366 except (OverflowError, ValueError) as X: 

1367 r = 0 # INF, NAN, NINF 

1368 i = self._fintX(X, float(s)) 

1369 return i, r # like _fint2.getter 

1370 

1371 def _fintX(self, X, i): # PYCHOK X 

1372 '''(INTERNAL) Handle I{non-finite} C{int}. 

1373 ''' 

1374 # "cannot convert float infinity to integer" 

1375 return i # ignore such Overflow-/ValueErrors 

1376 # op = int.__name__ 

1377 # return self._nonfiniteX(X, op, i) 

1378 

1379 @deprecated_property_RO 

1380 def float_int(self): # PYCHOK no cover 

1381 '''DEPRECATED, use method C{Fsum.int_float}.''' 

1382 return self.int_float() # raiser=False 

1383 

1384 @property_RO 

1385 def floor(self): 

1386 '''Get this instance' C{floor} (C{int} in Python 3+, but 

1387 C{float} in Python 2-). 

1388 

1389 @note: This C{floor} takes the C{residual} into account. 

1390 

1391 @see: Method L{Fsum.int_float} and properties L{Fsum.ceil}, 

1392 L{Fsum.imag} and L{Fsum.real}. 

1393 ''' 

1394 s, r = self._fprs2 

1395 f = _floor(s) + _floor(r) + 1 

1396 while (f - s) > r: # f > (s + r) 

1397 f -= 1 

1398 return f # _floor(self._n_d) 

1399 

1400# ffloordiv = __ifloordiv__ # for naming consistency? 

1401# floordiv = __floordiv__ # for naming consistency? 

1402 

1403 def _floordiv(self, other, op, **raiser_RESIDUAL): # rather _ffloordiv? 

1404 '''Apply C{B{self} //= B{other}}. 

1405 ''' 

1406 q = self._ftruediv(other, op, **raiser_RESIDUAL) # == self 

1407 return self._fset(q.floor) # floor(q) 

1408 

1409 def fma(self, other1, other2, raiser=False): # in .fmath.fma 

1410 '''Fused-multiply-add C{self *= B{other1}; self += B{other2}}. 

1411 

1412 @arg other1: Multiplier (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1413 @arg other2: Addend (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

1414 @kwarg raiser: If C{True}, throw an exception, otherwise pass 

1415 the I{non-finite} result (C{bool}). 

1416 

1417 @note: Uses C{math.fma} in Python 3.13+, provided C{self}, 

1418 B{C{other1}} and B{C{other2}} are all C{scalar}. 

1419 ''' 

1420 f, r = self._fprs2 

1421 if r == 0 and isscalar(other1, both=True) \ 

1422 and isscalar(other2, both=True): 

1423 try: 

1424 f = _fma(f, other1, other2) 

1425 except (OverflowError, TypeError, ValueError) as X: # from math.fma 

1426 op = self.fma.__name__ # INF, NAN, NINF 

1427 f = self._mul_reduce(op, f, other1) 

1428 f = _sum(self._ps_other(op, f, other2)) 

1429 if raiser: 

1430 f = self._nonfiniteX(X, op, f) 

1431 else: 

1432 f = self._f2mul(self.fma, other1, raiser=raiser) 

1433 f += other2 

1434 return self._fset(f) 

1435 

1436 fmul = __imul__ 

1437 

1438 def _fmul(self, other, op): 

1439 '''(INTERNAL) Apply C{B{self} *= B{other}}. 

1440 ''' 

1441 if _isFsumTuple(other): 

1442 if len(self._ps) != 1: 

1443 f = self._mul_Fsum(other, op) 

1444 elif len(other._ps) != 1: # and len(self._ps) == 1 

1445 f = self._ps_mul(op, *other._ps) 

1446 elif self._f2product: # len(other._ps) == 1 

1447 f = self._mul_scalar(other._ps[0], op) 

1448 else: # len(other._ps) == len(self._ps) == 1 

1449 f = self._finite(self._ps[0] * other._ps[0], op=op) 

1450 else: 

1451 s = self._scalar(other, op) 

1452 f = self._mul_scalar(s, op) 

1453 return self._fset(f) # n=len(self) + 1 

1454 

1455 @deprecated_method 

1456 def f2mul(self, *others, **raiser): 

1457 '''DEPRECATED on 2024.09.13, use method L{f2mul_<Fsum.f2mul_>}.''' 

1458 return self._fset(self.f2mul_(*others, **raiser)) 

1459 

1460 def f2mul_(self, *others, **raiser): # in .fmath.f2mul 

1461 '''Return C{B{self} * B{other} * B{other} ...} for all B{C{others}} using cascaded, 

1462 accurate multiplication like with L{f2product<Fsum.f2product>} set to C{True}. 

1463 

1464 @arg others: Multipliers (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all 

1465 positional. 

1466 @kwarg raiser: Keyword argument C{B{raiser}=False}, if C{True}, throw an exception, 

1467 otherwise pass the I{non-finite} result (C{bool}). 

1468 

1469 @return: The cascaded I{TwoProduct} (L{Fsum} or C{float}). 

1470 

1471 @see: U{Equations 2.3<https://www.TUHH.De/ti3/paper/rump/OzOgRuOi06.pdf>} 

1472 ''' 

1473 return self._f2mul(self.f2mul_, *others, **raiser) 

1474 

1475 def _f2mul(self, where, *others, **raiser): 

1476 '''(INTERNAL) See methods C{fma} and C{f2mul_}. 

1477 ''' 

1478 f = self._copy_2(where) 

1479 if others: 

1480 op = where.__name__ 

1481 ps = f._ps 

1482 if ps: 

1483 try: 

1484 for p in self._ps_other(op, *others): 

1485 pfs = _2products(p, _2split3s(ps)) 

1486 ps[:] = f._ps_acc([], pfs, up=False) 

1487 f._update() 

1488 except (OverflowError, TypeError, ValueError) as X: 

1489 r = self._mul_reduce(op, _sum(ps), *others) # INF, NAN, NINF 

1490 if _xkwds_get1(raiser, raiser=False): 

1491 r = self._nonfiniteX(X, op, r) 

1492 f._fset(r) 

1493 return f 

1494 

1495 def fover(self, over, **raiser_RESIDUAL): 

1496 '''Apply C{B{self} /= B{over}} and summate. 

1497 

1498 @arg over: An L{Fsum} or C{scalar} denominator. 

1499 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1500 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1501 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1502 

1503 @return: Precision running sum (C{float}). 

1504 

1505 @raise ResidualError: Non-zero, significant residual or invalid 

1506 B{C{RESIDUAL}}. 

1507 

1508 @see: Methods L{Fsum.fsum} and L{Fsum.__itruediv__}. 

1509 ''' 

1510 return float(self.fdiv(over, **raiser_RESIDUAL)._fprs) 

1511 

1512 fpow = __ipow__ 

1513 

1514 def _fpow(self, other, op, *mod, **raiser_RESIDUAL): 

1515 '''Apply C{B{self} **= B{other}}, optional B{C{mod}} or C{None}. 

1516 ''' 

1517 if mod: 

1518 if mod[0] is not None: # == 3-arg C{pow} 

1519 f = self._pow_2_3(self, other, other, op, *mod, **raiser_RESIDUAL) 

1520 elif self.is_integer(): 

1521 # return an exact C{int} for C{int}**C{int} 

1522 i, _ = self._fint2 # assert _ == 0 

1523 x, r = _2scalar2(other) # C{int}, C{float} or other 

1524 f = self._Fsum_as(i)._pow_Fsum(other, op, **raiser_RESIDUAL) if r else \ 

1525 self._pow_2_3(i, x, other, op, **raiser_RESIDUAL) 

1526 else: # mod[0] is None, power(self, other) 

1527 f = self._pow(other, other, op, **raiser_RESIDUAL) 

1528 else: # pow(self, other) 

1529 f = self._pow(other, other, op, **raiser_RESIDUAL) 

1530 return self._fset(f) # n=max(len(self), 1) 

1531 

1532 def f2product(self, *two): 

1533 '''Get and set accurate I{TwoProduct} multiplication for this 

1534 L{Fsum}, I{overriding the global setting} from function 

1535 L{f2product<fsums.f2product>}. 

1536 

1537 @arg two: If omitted, leave the override unchanged, if C{True}, 

1538 turn I{TwoProduct} on, if C{False} off, if C{None}e 

1539 remove th override (C{bool} or C{None}). 

1540 

1541 @return: The previous setting (C{bool} or C{None} if not set). 

1542 

1543 @see: Function L{f2product<fsums.f2product>}. 

1544 

1545 @note: Use C{f.f2product() or f2product()} to determine whether 

1546 multiplication is accurate for L{Fsum} C{f}. 

1547 ''' 

1548 if two: # delattrof(self, _f2product=None) 

1549 t = _xkwds_pop(self.__dict__, _f2product=None) 

1550 if two[0] is not None: 

1551 self._f2product = bool(two[0]) 

1552 else: # getattrof(self, _f2product=None) 

1553 t = _xkwds_get(self.__dict__, _f2product=None) 

1554 return t 

1555 

1556 @Property 

1557 def _fprs(self): 

1558 '''(INTERNAL) Get and cache this instance' precision 

1559 running sum (C{float} or C{int}), ignoring C{residual}. 

1560 

1561 @note: The precision running C{fsum} after a C{//=} or 

1562 C{//} C{floor} division is C{int} in Python 3+. 

1563 ''' 

1564 s, _ = self._fprs2 

1565 return s # ._fprs2.fsum 

1566 

1567 @_fprs.setter_ # PYCHOK setter_underscore! 

1568 def _fprs(self, s): 

1569 '''(INTERNAL) Replace the C{_fprs} value. 

1570 ''' 

1571 return s 

1572 

1573 @Property 

1574 def _fprs2(self): 

1575 '''(INTERNAL) Get and cache this instance' precision 

1576 running sum and residual (L{Fsum2Tuple}). 

1577 ''' 

1578 ps = self._ps 

1579 try: 

1580 n = len(ps) - 2 

1581 if n > 0: # len(ps) > 2 

1582 s = _psum(ps, **self._isfine) 

1583 n = len(ps) - 2 

1584 if n > 0: 

1585 r = self._ps_1sum(s) 

1586 return Fsum2Tuple(*_s_r(s, r)) 

1587 if n == 0: # len(ps) == 2 

1588 s, r = _s_r(*_2sum(*ps, **self._isfine)) 

1589 ps[:] = (r, s) if r else (s,) 

1590 elif ps: # len(ps) == 1 

1591 s, r = ps[0], INT0 

1592 else: # len(ps) == 0 

1593 s, r = _0_0, INT0 

1594 ps[:] = s, 

1595 except (OverflowError, ValueError) as X: 

1596 op = _sum.__name__ # INF, NAN, NINF 

1597 s = self._nonfiniteX(X, op, _sum(self._ps)) 

1598 r = _0_0 

1599 # assert self._ps is ps 

1600 return Fsum2Tuple(s, r) 

1601 

1602 @_fprs2.setter_ # PYCHOK setter_underscore! 

1603 def _fprs2(self, s_r): 

1604 '''(INTERNAL) Replace the C{_fprs2} value. 

1605 ''' 

1606 return Fsum2Tuple(s_r) 

1607 

1608 def fset_(self, *xs): 

1609 '''Apply C{B{self}.partials = Fsum(*B{xs}).partials}. 

1610 

1611 @arg xs: Optional, new values (each C{scalar} or 

1612 an L{Fsum} or L{Fsum2Tuple} instance), all 

1613 positional. 

1614 

1615 @return: This instance, replaced (C{Fsum}). 

1616 

1617 @see: Method L{Fsum.fadd} for further details. 

1618 ''' 

1619 return self._fset(xs[0], op=_fset_op_) if len(xs) == 1 else \ 

1620 self._fset(_0_0)._facc_args(xs) 

1621 

1622 def _fset(self, other, n=0, up=True, **op): 

1623 '''(INTERNAL) Overwrite this instance with an other or a C{scalar}. 

1624 ''' 

1625 if other is self: 

1626 pass # from ._fmul, ._ftruediv and ._pow_0_1 

1627 elif _isFsumTuple(other): 

1628 self._ps[:] = other._ps 

1629 self._n = n or other._n 

1630 if up: # use or zap the C{Property_RO} values 

1631 Fsum._fint2._update_from(self, other) 

1632 Fsum._fprs ._update_from(self, other) 

1633 Fsum._fprs2._update_from(self, other) 

1634 elif isscalar(other): 

1635 s = float(self._finite(other, **op)) if op else other 

1636 self._ps[:] = s, 

1637 self._n = n or 1 

1638 if up: # Property _fint2, _fprs and _fprs2 all have 

1639 # @.setter_underscore and NOT @.setter because the 

1640 # latter's _fset zaps the value set by @.setter 

1641 self._fint2 = s 

1642 self._fprs = s 

1643 self._fprs2 = s, INT0 

1644 # assert self._fprs is s 

1645 else: 

1646 op = _xkwds_get1(op, op=_fset_op_) 

1647 raise self._Error(op, other, _TypeError) 

1648 return self 

1649 

1650 def _fset_ps(self, other): # in .fmath._Fsum__init__ 

1651 '''(INTERNAL) Set partials from a known C{other}. 

1652 ''' 

1653 return self._fset(other, up=False) 

1654 

1655 def fsub(self, xs=()): 

1656 '''Subtract an iterable's items from this instance. 

1657 

1658 @see: Method L{Fsum.fadd} for further details. 

1659 ''' 

1660 return self._facc_neg(xs) 

1661 

1662 def fsub_(self, *xs): 

1663 '''Subtract all positional items from this instance. 

1664 

1665 @see: Method L{Fsum.fadd_} for further details. 

1666 ''' 

1667 return self._facc_neg(xs, origin=1) if len(xs) != 1 else \ 

1668 self._fsub(xs[0], _sub_op_) 

1669 

1670 def _fsub(self, other, op): 

1671 '''(INTERNAL) Apply C{B{self} -= B{other}}. 

1672 ''' 

1673 if _isFsumTuple(other): 

1674 if other is self: # or other._fprs2 == self._fprs2: 

1675 self._fset(_0_0, n=len(self) * 2) 

1676 elif other._ps: 

1677 self._facc_scalar(other._ps_neg) 

1678 elif self._scalar(other, op): 

1679 self._facc_scalar_(-other) 

1680 return self 

1681 

1682 def fsum(self, xs=()): 

1683 '''Add an iterable's items, summate and return the 

1684 current precision running sum. 

1685 

1686 @arg xs: Iterable of items to add (each item C{scalar} 

1687 or an L{Fsum} or L{Fsum2Tuple} instance). 

1688 

1689 @return: Precision running sum (C{float} or C{int}). 

1690 

1691 @see: Method L{Fsum.fadd}. 

1692 

1693 @note: Accumulation can continue after summation. 

1694 ''' 

1695 return self._facc(xs)._fprs 

1696 

1697 def fsum_(self, *xs): 

1698 '''Add any positional items, summate and return the 

1699 current precision running sum. 

1700 

1701 @arg xs: Items to add (each C{scalar} or an L{Fsum} 

1702 or L{Fsum2Tuple} instance), all positional. 

1703 

1704 @return: Precision running sum (C{float} or C{int}). 

1705 

1706 @see: Methods L{Fsum.fsum}, L{Fsum.Fsum_} and L{Fsum.fsumf_}. 

1707 ''' 

1708 return self._facc_args(xs)._fprs 

1709 

1710 def Fsum_(self, *xs, **name): 

1711 '''Like method L{Fsum.fsum_} but returning a named L{Fsum}. 

1712 

1713 @kwarg name: Optional name (C{str}). 

1714 

1715 @return: Copy of this updated instance (L{Fsum}). 

1716 ''' 

1717 return self._facc_args(xs)._copy_2(self.Fsum_, **name) 

1718 

1719 def Fsum2Tuple_(self, *xs, **name): 

1720 '''Like method L{Fsum.fsum_} but returning a named L{Fsum2Tuple}. 

1721 

1722 @kwarg name: Optional name (C{str}). 

1723 

1724 @return: Precision running sum (L{Fsum2Tuple}). 

1725 ''' 

1726 return Fsum2Tuple(self._facc_args(xs)._fprs2, **name) 

1727 

1728 @property_RO 

1729 def _Fsum(self): # like L{Fsum2Tuple._Fsum}, for C{_2floats}, .fstats 

1730 return self # NOT @Property_RO, see .copy and ._copy_2 

1731 

1732 def _Fsum_as(self, *xs, **name_f2product_nonfinites_RESIDUAL): 

1733 '''(INTERNAL) Return an C{Fsum} with this C{Fsum}'s C{.f2product}, 

1734 C{.nonfinites} and C{.RESIDUAL} setting, optionally 

1735 overridden with C{name_f2product_nonfinites_RESIDUAL} and 

1736 with any C{xs} accumulated. 

1737 ''' 

1738 kwds = _xkwds_not(None, Fsum._RESIDUAL, f2product =self.f2product(), 

1739 nonfinites=self.nonfinites(), 

1740 RESIDUAL =self.RESIDUAL()) 

1741 if name_f2product_nonfinites_RESIDUAL: # overwrites 

1742 kwds.update(name_f2product_nonfinites_RESIDUAL) 

1743 F = Fsum(**kwds) 

1744 # assert all(v == self.__dict__[n] for n, v in F.__dict__.items()) 

1745 return F._fset(xs[0]) if len(xs) == 1 else ( 

1746 F._facc(xs, up=False) if xs else F) 

1747 

1748 def fsum2(self, xs=(), **name): 

1749 '''Add an iterable's items, summate and return the 

1750 current precision running sum I{and} the C{residual}. 

1751 

1752 @arg xs: Iterable of items to add (each item C{scalar} 

1753 or an L{Fsum} or L{Fsum2Tuple} instance). 

1754 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1755 

1756 @return: L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} the 

1757 current precision running sum and C{residual}, the 

1758 (precision) sum of the remaining C{partials}. The 

1759 C{residual is INT0} if the C{fsum} is considered 

1760 to be I{exact}. 

1761 

1762 @see: Methods L{Fsum.fint2}, L{Fsum.fsum} and L{Fsum.fsum2_} 

1763 ''' 

1764 t = self._facc(xs)._fprs2 

1765 return t.dup(name=name) if name else t 

1766 

1767 def fsum2_(self, *xs): 

1768 '''Add any positional items, summate and return the current 

1769 precision running sum and the I{differential}. 

1770 

1771 @arg xs: Values to add (each C{scalar} or an L{Fsum} or 

1772 L{Fsum2Tuple} instance), all positional. 

1773 

1774 @return: 2Tuple C{(fsum, delta)} with the current, precision 

1775 running C{fsum} like method L{Fsum.fsum} and C{delta}, 

1776 the difference with previous running C{fsum}, C{float}. 

1777 

1778 @see: Methods L{Fsum.fsum_} and L{Fsum.fsum}. 

1779 ''' 

1780 return self._fsum2(xs, self._facc_args) 

1781 

1782 def _fsum2(self, xs, _facc, **origin): 

1783 '''(INTERNAL) Helper for L{Fsum.fsum2_} and L{Fsum.fsum2f_}. 

1784 ''' 

1785 p, q = self._fprs2 

1786 if xs: 

1787 s, r = _facc(xs, **origin)._fprs2 

1788 if _isfinite(s): # _fsum(_1primed((s, -p, r, -q)) 

1789 d, r = _2sum(s - p, r - q, _isfine=_isOK) 

1790 r, _ = _s_r(d, r) 

1791 return s, (r if _isfinite(r) else _0_0) 

1792 else: 

1793 return p, _0_0 

1794 

1795 def fsumf_(self, *xs): 

1796 '''Like method L{Fsum.fsum_} iff I{all} C{B{xs}} are I{known to be scalar}. 

1797 ''' 

1798 return self._facc_scalar(xs)._fprs 

1799 

1800 def Fsumf_(self, *xs): 

1801 '''Like method L{Fsum.Fsum_} iff I{all} C{B{xs}} are I{known to be scalar}. 

1802 ''' 

1803 return self._facc_scalar(xs)._copy_2(self.Fsumf_) 

1804 

1805 def fsum2f_(self, *xs): 

1806 '''Like method L{Fsum.fsum2_} iff I{all} C{B{xs}} are I{known to be scalar}. 

1807 ''' 

1808 return self._fsum2(xs, self._facc_scalar, origin=1) 

1809 

1810# ftruediv = __itruediv__ # for naming consistency? 

1811 

1812 def _ftruediv(self, other, op, **raiser_RESIDUAL): 

1813 '''(INTERNAL) Apply C{B{self} /= B{other}}. 

1814 ''' 

1815 n = _1_0 

1816 if _isFsumTuple(other): 

1817 if other is self or self == other: 

1818 return self._fset(n, n=len(self)) 

1819 d, r = other._fprs2 

1820 if r: 

1821 R = self._raiser(r, d, **raiser_RESIDUAL) 

1822 if R: 

1823 raise self._ResidualError(op, other, r, **R) 

1824 d, n = other.as_integer_ratio() 

1825 else: 

1826 d = self._scalar(other, op) 

1827 try: 

1828 s = n / d 

1829 except Exception as X: 

1830 raise self._ErrorX(X, op, other) 

1831 f = self._mul_scalar(s, _mul_op_) # handles 0, INF, NAN 

1832 return self._fset(f) 

1833 

1834 @property_RO 

1835 def imag(self): 

1836 '''Get the C{imaginary} part of this instance (C{0.0}, always). 

1837 

1838 @see: Property L{Fsum.real}. 

1839 ''' 

1840 return _0_0 

1841 

1842 def int_float(self, **raiser_RESIDUAL): 

1843 '''Return this instance' current running sum as C{int} or C{float}. 

1844 

1845 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1846 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1847 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1848 

1849 @return: This C{integer} sum if this instance C{is_integer}, 

1850 otherwise return the C{float} sum if the residual is 

1851 zero or not significant. 

1852 

1853 @raise ResidualError: Non-zero, significant residual or invalid 

1854 B{C{RESIDUAL}}. 

1855 

1856 @see: Methods L{Fsum.fint}, L{Fsum.fint2}, L{Fsum.is_integer}, 

1857 L{Fsum.RESIDUAL} and property L{Fsum.as_iscalar}. 

1858 ''' 

1859 s, r = self._fint2 

1860 if r: 

1861 s, r = self._fprs2 

1862 if r: # PYCHOK no cover 

1863 R = self._raiser(r, s, **raiser_RESIDUAL) 

1864 if R: 

1865 t = _stresidual(_non_zero_, r, **R) 

1866 raise ResidualError(int_float=s, txt=t) 

1867 s = float(s) 

1868 return s 

1869 

1870 def is_exact(self): 

1871 '''Is this instance' running C{fsum} considered to be exact? 

1872 (C{bool}), C{True} only if the C{residual is }L{INT0}. 

1873 ''' 

1874 return self.residual is INT0 

1875 

1876 def is_finite(self): # in .constants 

1877 '''Is this instance C{finite}? (C{bool}). 

1878 

1879 @see: Function L{isfinite<pygeodesy.isfinite>}. 

1880 ''' 

1881 return _isfinite(_sum(self._ps)) # _sum(self) 

1882 

1883 def is_integer(self): 

1884 '''Is this instance' running sum C{integer}? (C{bool}). 

1885 

1886 @see: Methods L{Fsum.fint}, L{Fsum.fint2} and L{Fsum.is_scalar}. 

1887 ''' 

1888 s, r = self._fint2 

1889 return False if r else (_isfinite(s) and isint(s)) 

1890 

1891 def is_math_fma(self): 

1892 '''Is accurate L{f2product} multiplication based on Python's C{math.fma}? 

1893 

1894 @return: C{True} if accurate multiplication uses C{math.fma}, C{False} 

1895 an C{fma} implementation as C{math.fma} or C{None}, a previous 

1896 C{PyGeodesy} implementation. 

1897 ''' 

1898 return (_fma.__module__ is fabs.__module__ or None) if _2split3s is _passarg else False 

1899 

1900 def is_math_fsum(self): 

1901 '''Are the summation functions L{fsum}, L{fsum_}, L{fsumf_}, L{fsum1}, 

1902 L{fsum1_} and L{fsum1f_} based on Python's C{math.fsum}? 

1903 

1904 @return: C{True} if summation functions use C{math.fsum}, C{False} 

1905 otherwise. 

1906 ''' 

1907 return _sum is _fsum # _fsum.__module__ is fabs.__module__ 

1908 

1909 def is_scalar(self, **raiser_RESIDUAL): 

1910 '''Is this instance' running sum C{scalar} without residual or with 

1911 a residual I{ratio} not exceeding the RESIDUAL threshold? 

1912 

1913 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

1914 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

1915 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

1916 

1917 @return: C{True} if this instance' non-zero residual C{ratio} exceeds 

1918 the L{RESIDUAL<Fsum.RESIDUAL>} threshold (C{bool}). 

1919 

1920 @raise ResidualError: Non-zero, significant residual or invalid 

1921 B{C{RESIDUAL}}. 

1922 

1923 @see: Method L{Fsum.RESIDUAL}, L{Fsum.is_integer} and property 

1924 L{Fsum.as_iscalar}. 

1925 ''' 

1926 s, r = self._fprs2 

1927 return False if r and self._raiser(r, s, **raiser_RESIDUAL) else True 

1928 

1929 def _mul_Fsum(self, other, op=_mul_op_): # in .fmath.Fhorner 

1930 '''(INTERNAL) Return C{B{self} * B{other}} as L{Fsum} or C{0}. 

1931 ''' 

1932 # assert _isFsumTuple(other) 

1933 if self._ps and other._ps: 

1934 f = self._ps_mul(op, *other._ps) # NO .as_iscalar! 

1935 else: 

1936 f = _0_0 

1937 return f 

1938 

1939 def _mul_reduce(self, op, start, *others): 

1940 '''(INTERNAL) Like fmath.freduce(_operator.mul, ...) 

1941 for I{non-finite} C{start} and/or C{others}. 

1942 ''' 

1943 for p in self._ps_other(op, *others): 

1944 start *= p 

1945 return start 

1946 

1947 def _mul_scalar(self, factor, op): # in .fmath.Fhorner 

1948 '''(INTERNAL) Return C{B{self} * scalar B{factor}} as L{Fsum}, C{0.0} or C{self}. 

1949 ''' 

1950 # assert isscalar(factor) 

1951 if self._ps and self._finite(factor, op): 

1952 f = self if factor == _1_0 else ( 

1953 self._neg if factor == _N_1_0 else 

1954 self._ps_mul(op, factor).as_iscalar) 

1955 else: 

1956 f = _0_0 

1957 return f 

1958 

1959# @property_RO 

1960# def _n_d(self): 

1961# n, d = self.as_integer_ratio() 

1962# return n / d 

1963 

1964 @property_RO 

1965 def _neg(self): 

1966 '''(INTERNAL) Return C{Fsum(-self)} or scalar C{NEG0}. 

1967 ''' 

1968 return _Psum(self._ps_neg) if self._ps else NEG0 

1969 

1970 def nonfinites(self, *OK): 

1971 '''Handle I{non-finite} C{float}s as C{inf}, C{INF}, C{NINF}, C{nan} 

1972 and C{NAN} for this L{Fsum} or throw C{OverflowError} respectively 

1973 C{ValueError} exceptions, I{overriding the global setting} from 

1974 function L{nonfiniterrors<fsums.nonfiniterrors>}. 

1975 

1976 @arg OK: If omitted, leave the override unchanged, if C{True}, 

1977 I{non-finites} are C{OK}, if C{False} throw exceptions 

1978 or if C{None} remove the override (C{bool} or C{None}). 

1979 

1980 @return: The previous setting (C{bool} or C{None} if not set). 

1981 

1982 @see: Function L{nonfiniterrors<fsums.nonfiniterrors>}. 

1983 

1984 @note: Use C{f.nonfinites() or not nonfiniterrors()} to determine 

1985 whether L{Fsum} C{f} handles I{non-finites}. 

1986 ''' 

1987 _ks = Fsum._nonfinites_isfine_kwds 

1988 if OK: # delattrof(self, _isfine=None) 

1989 k = _xkwds_pop(self.__dict__, _isfine=None) 

1990 if OK[0] is not None: 

1991 self._isfine = _ks[bool(OK[0])] 

1992 else: # getattrof(self, _isfine=None) 

1993 k = _xkwds_get(self.__dict__, _isfine=None) 

1994 # dict(map(reversed, _ks.items())).get(k, None) 

1995 # raises a TypeError: unhashable type: 'dict' 

1996 return True if k is _ks[True] else ( 

1997 False if k is _ks[False] else None) 

1998 

1999 _nonfinites_isfine_kwds = {True: dict(_isfine=_isOK), 

2000 False: dict(_isfine=_isfinite)} 

2001 

2002 def _nonfiniteX(self, X, op, f): 

2003 '''(INTERNAL) Handle a I{non-finite} exception. 

2004 ''' 

2005 if not _isOK_or_finite(f, **self._isfine): 

2006 raise self._ErrorX(X, op, f) 

2007 return f 

2008 

2009 def _optionals(self, f2product=None, nonfinites=None, **name_RESIDUAL): 

2010 '''(INTERNAL) Re/set options from keyword arguments. 

2011 ''' 

2012 if f2product is not None: 

2013 self.f2product(f2product) 

2014 if nonfinites is not None: 

2015 self.nonfinites(nonfinites) 

2016 if name_RESIDUAL: # MUST be last 

2017 n, kwds = _name2__(**name_RESIDUAL) 

2018 if kwds: 

2019 R = Fsum._RESIDUAL 

2020 t = _threshold(R, **kwds) 

2021 if t != R: 

2022 self._RESIDUAL = t 

2023 if n: 

2024 self.name = n # self.rename(n) 

2025 

2026 def _1_Over(self, x, op, **raiser_RESIDUAL): # vs _1_over 

2027 '''(INTERNAL) Return C{Fsum(1) / B{x}}. 

2028 ''' 

2029 return self._Fsum_as(_1_0)._ftruediv(x, op, **raiser_RESIDUAL) 

2030 

2031 @property_RO 

2032 def partials(self): 

2033 '''Get this instance' current, partial sums (C{tuple} of C{float}s). 

2034 ''' 

2035 return tuple(self._ps) 

2036 

2037 def pow(self, x, *mod, **raiser_RESIDUAL): 

2038 '''Return C{B{self}**B{x}} as L{Fsum}. 

2039 

2040 @arg x: The exponent (C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

2041 @arg mod: Optional modulus (C{int} or C{None}) for the 3-argument 

2042 C{pow(B{self}, B{other}, B{mod})} version. 

2043 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore 

2044 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar} 

2045 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

2046 

2047 @return: The C{pow(self, B{x})} or C{pow(self, B{x}, *B{mod})} 

2048 result (L{Fsum}). 

2049 

2050 @raise ResidualError: Non-zero, significant residual or invalid 

2051 B{C{RESIDUAL}}. 

2052 

2053 @note: If B{C{mod}} is given and C{None}, the result will be an 

2054 C{integer} L{Fsum} provided this instance C{is_integer} 

2055 or set to C{integer} by an L{Fsum.fint} call. 

2056 

2057 @see: Methods L{Fsum.__ipow__}, L{Fsum.fint}, L{Fsum.is_integer} 

2058 and L{Fsum.root}. 

2059 ''' 

2060 f = self._copy_2(self.pow) 

2061 return f._fpow(x, _pow_op_, *mod, **raiser_RESIDUAL) # f = pow(f, x, *mod) 

2062 

2063 def _pow(self, other, unused, op, **raiser_RESIDUAL): 

2064 '''Return C{B{self} ** B{other}}. 

2065 ''' 

2066 if _isFsumTuple(other): 

2067 f = self._pow_Fsum(other, op, **raiser_RESIDUAL) 

2068 elif self._scalar(other, op): 

2069 x = self._finite(other, op) 

2070 f = self._pow_scalar(x, other, op, **raiser_RESIDUAL) 

2071 else: 

2072 f = self._pow_0_1(0, other) 

2073 return f 

2074 

2075 def _pow_0_1(self, x, other): 

2076 '''(INTERNAL) Return B{C{self}**1} or C{B{self}**0 == 1.0}. 

2077 ''' 

2078 return self if x else (1 if isint(other) and self.is_integer() else _1_0) 

2079 

2080 def _pow_2_3(self, b, x, other, op, *mod, **raiser_RESIDUAL): 

2081 '''(INTERNAL) 2-arg C{pow(B{b}, scalar B{x})} and 3-arg C{pow(B{b}, 

2082 B{x}, int B{mod} or C{None})}, embellishing errors. 

2083 ''' 

2084 

2085 if mod: # b, x, mod all C{int}, unless C{mod} is C{None} 

2086 m = mod[0] 

2087 # assert _isFsumTuple(b) 

2088 

2089 def _s(s, r): 

2090 R = self._raiser(r, s, **raiser_RESIDUAL) 

2091 if R: 

2092 raise self._ResidualError(op, other, r, mod=m, **R) 

2093 return s 

2094 

2095 b = _s(*(b._fprs2 if m is None else b._fint2)) 

2096 x = _s(*_2scalar2(x)) 

2097 

2098 try: 

2099 # 0**INF == 0.0, 1**INF == 1.0, -1**2.3 == -(1**2.3) 

2100 s = pow(b, x, *mod) 

2101 if iscomplex(s): 

2102 # neg**frac == complex in Python 3+, but ValueError in 2- 

2103 raise ValueError(_strcomplex(s, b, x, *mod)) 

2104 return self._finite(s) 

2105 except Exception as X: 

2106 raise self._ErrorX(X, op, other, *mod) 

2107 

2108 def _pow_Fsum(self, other, op, **raiser_RESIDUAL): 

2109 '''(INTERNAL) Return C{B{self} **= B{other}} for C{_isFsumTuple(other)}. 

2110 ''' 

2111 # assert _isFsumTuple(other) 

2112 x, r = other._fprs2 

2113 f = self._pow_scalar(x, other, op, **raiser_RESIDUAL) 

2114 if f and r: 

2115 f *= self._pow_scalar(r, other, op, **raiser_RESIDUAL) 

2116 return f 

2117 

2118 def _pow_int(self, x, other, op, **raiser_RESIDUAL): 

2119 '''(INTERNAL) Return C{B{self} **= B{x}} for C{int B{x} >= 0}. 

2120 ''' 

2121 # assert isint(x) and x >= 0 

2122 ps = self._ps 

2123 if len(ps) > 1: 

2124 _mul_Fsum = Fsum._mul_Fsum 

2125 if x > 4: 

2126 p = self 

2127 f = self if (x & 1) else self._Fsum_as(_1_0) 

2128 m = x >> 1 # // 2 

2129 while m: 

2130 p = _mul_Fsum(p, p, op) # p **= 2 

2131 if (m & 1): 

2132 f = _mul_Fsum(f, p, op) # f *= p 

2133 m >>= 1 # //= 2 

2134 elif x > 1: # self**2, 3, or 4 

2135 f = _mul_Fsum(self, self, op) 

2136 if x > 2: # self**3 or 4 

2137 p = self if x < 4 else f 

2138 f = _mul_Fsum(f, p, op) 

2139 else: # self**1 or self**0 == 1 or _1_0 

2140 f = self._pow_0_1(x, other) 

2141 elif ps: # self._ps[0]**x 

2142 f = self._pow_2_3(ps[0], x, other, op, **raiser_RESIDUAL) 

2143 else: # PYCHOK no cover 

2144 # 0**pos_int == 0, but 0**0 == 1 

2145 f = 0 if x else 1 

2146 return f 

2147 

2148 def _pow_scalar(self, x, other, op, **raiser_RESIDUAL): 

2149 '''(INTERNAL) Return C{self**B{x}} for C{scalar B{x}}. 

2150 ''' 

2151 s, r = self._fprs2 

2152 if r: 

2153 # assert s != 0 

2154 if isint(x, both=True): # self**int 

2155 x = int(x) 

2156 y = abs(x) 

2157 if y > 1: 

2158 f = self._pow_int(y, other, op, **raiser_RESIDUAL) 

2159 if x > 0: # i.e. > 1 

2160 return f # Fsum or scalar 

2161 # assert x < 0 # i.e. < -1 

2162 if _isFsum(f): 

2163 s, r = f._fprs2 

2164 if r: 

2165 return self._1_Over(f, op, **raiser_RESIDUAL) 

2166 else: # scalar 

2167 s = f 

2168 # use s**(-1) to get the CPython 

2169 # float_pow error iff s is zero 

2170 x = -1 

2171 elif x < 0: # self**(-1) 

2172 return self._1_Over(self, op, **raiser_RESIDUAL) # 1 / self 

2173 else: # self**1 or self**0 

2174 return self._pow_0_1(x, other) # self, 1 or 1.0 

2175 else: # self**fractional 

2176 R = self._raiser(r, s, **raiser_RESIDUAL) 

2177 if R: 

2178 raise self._ResidualError(op, other, r, **R) 

2179 n, d = self.as_integer_ratio() 

2180 if abs(n) > abs(d): 

2181 n, d, x = d, n, (-x) 

2182 s = n / d 

2183 # assert isscalar(s) and isscalar(x) 

2184 return self._pow_2_3(s, x, other, op, **raiser_RESIDUAL) 

2185 

2186 def _ps_acc(self, ps, xs, up=True, **unused): 

2187 '''(INTERNAL) Accumulate C{xs} known scalars into list C{ps}. 

2188 ''' 

2189 n = 0 

2190 _2s = _2sum 

2191 _fi = self._isfine 

2192 for x in (tuple(xs) if xs is ps else xs): 

2193 # assert isscalar(x) and _isOK_or_finite(x, **self._isfine) 

2194 if x: 

2195 i = 0 

2196 for p in ps: 

2197 x, p = _2s(x, p, **_fi) 

2198 if p: 

2199 ps[i] = p 

2200 i += 1 

2201 ps[i:] = (x,) if x else () 

2202 n += 1 

2203 if n: 

2204 self._n += n 

2205# if _fi: # collapse ps if non-finite 

2206# x = _sum(ps) 

2207# if not _isfinite(x): 

2208# ps[:] = x, 

2209 # Fsum._ps_max = max(Fsum._ps_max, len(ps)) 

2210 if up: 

2211 self._update() 

2212 return ps 

2213 

2214 def _ps_mul(self, op, *factors): 

2215 '''(INTERNAL) Multiply this instance' C{partials} with 

2216 each scalar C{factor} and accumulate into an C{Fsum}. 

2217 ''' 

2218 def _psfs(ps, fs, _isfine=_isfinite): 

2219 if len(ps) < len(fs): 

2220 ps, fs = fs, ps 

2221 if self._f2product: 

2222 ps = _2split3s(ps) 

2223 _fps = _2products 

2224 else: 

2225 def _fps(f, ps): 

2226 return (f * p for p in ps) 

2227 

2228 for f in fs: 

2229 for p in _fps(f, ps): 

2230 yield p if _isfine(p) else self._finite(p, op) 

2231 

2232 F = self._Fsum_as(name=op) # assert F._ps is not self._ps 

2233 _s = _psfs(self._ps, factors, **self._isfine) 

2234 return F._facc_scalar(_s, up=False) 

2235 

2236 @property_RO 

2237 def _ps_neg(self): 

2238 '''(INTERNAL) Yield the partials, I{negated}. 

2239 ''' 

2240 for p in self._ps: 

2241 yield -p 

2242 

2243 def _ps_other(self, op, *others): 

2244 '''(INTERNAL) Yield all C{other}s as C{scalar}. 

2245 ''' 

2246 for other in others: 

2247 if _isFsumTuple(other): 

2248 for p in other._ps: 

2249 yield p 

2250 else: 

2251 yield self._scalar(other, op) 

2252 

2253 def _ps_1sum(self, *less): 

2254 '''(INTERNAL) Return the partials sum, 1-primed C{less} some scalars. 

2255 ''' 

2256 def _1psls(ps, ls): 

2257 yield _1_0 

2258 for p in ps: 

2259 yield p 

2260 for p in ls: 

2261 yield -p 

2262 yield _N_1_0 

2263 

2264 return _fsum(_1psls(self._ps, less)) 

2265 

2266 def _raiser(self, r, s, raiser=True, **RESIDUAL): 

2267 '''(INTERNAL) Does ratio C{r / s} exceed the RESIDUAL threshold 

2268 I{and} is residual C{r} I{non-zero} or I{significant} (for a 

2269 negative respectively positive C{RESIDUAL} threshold)? 

2270 ''' 

2271 if r and raiser: 

2272 t = self._RESIDUAL 

2273 if RESIDUAL: 

2274 t = _threshold(t, **RESIDUAL) 

2275 if t < 0 or (s + r) != s: 

2276 q = (r / s) if s else s # == 0. 

2277 if fabs(q) > fabs(t): 

2278 return dict(ratio=q, R=t) 

2279 return {} 

2280 

2281 rdiv = __rtruediv__ 

2282 

2283 @property_RO 

2284 def real(self): 

2285 '''Get the C{real} part of this instance (C{float}). 

2286 

2287 @see: Methods L{Fsum.__float__} and L{Fsum.fsum} 

2288 and properties L{Fsum.ceil}, L{Fsum.floor}, 

2289 L{Fsum.imag} and L{Fsum.residual}. 

2290 ''' 

2291 return float(self) 

2292 

2293 @property_RO 

2294 def residual(self): 

2295 '''Get this instance' residual or residue (C{float} or C{int}): 

2296 the C{sum(partials)} less the precision running sum C{fsum}. 

2297 

2298 @note: The C{residual is INT0} iff the precision running 

2299 C{fsum} is considered to be I{exact}. 

2300 

2301 @see: Methods L{Fsum.fsum}, L{Fsum.fsum2} and L{Fsum.is_exact}. 

2302 ''' 

2303 return self._fprs2.residual 

2304 

2305 def RESIDUAL(self, *threshold): 

2306 '''Get and set this instance' I{ratio} for raising L{ResidualError}s, 

2307 overriding the default from env variable C{PYGEODESY_FSUM_RESIDUAL}. 

2308 

2309 @arg threshold: If C{scalar}, the I{ratio} to exceed for raising 

2310 L{ResidualError}s in division and exponention, if 

2311 C{None}, restore the default set with env variable 

2312 C{PYGEODESY_FSUM_RESIDUAL} or if omitted, keep the 

2313 current setting. 

2314 

2315 @return: The previous C{RESIDUAL} setting (C{float}), default C{0.0}. 

2316 

2317 @raise ResidualError: Invalid B{C{threshold}}. 

2318 

2319 @note: L{ResidualError}s may be thrown if (1) the non-zero I{ratio} 

2320 C{residual / fsum} exceeds the given B{C{threshold}} and (2) 

2321 the C{residual} is non-zero and (3) is I{significant} vs the 

2322 C{fsum}, i.e. C{(fsum + residual) != fsum} and (4) optional 

2323 keyword argument C{raiser=False} is missing. Specify a 

2324 negative B{C{threshold}} for only non-zero C{residual} 

2325 testing without the I{significant} case. 

2326 ''' 

2327 r = self._RESIDUAL 

2328 if threshold: 

2329 t = threshold[0] 

2330 self._RESIDUAL = Fsum._RESIDUAL if t is None else ( # for ... 

2331 (_0_0 if t else _1_0) if isbool(t) else 

2332 _threshold(t)) # ... backward compatibility 

2333 return r 

2334 

2335 def _ResidualError(self, op, other, residual, **mod_R): 

2336 '''(INTERNAL) Non-zero B{C{residual}} etc. 

2337 ''' 

2338 def _p(mod=None, R=0, **unused): # ratio=0 

2339 return (_non_zero_ if R < 0 else _significant_) \ 

2340 if mod is None else _integer_ 

2341 

2342 t = _stresidual(_p(**mod_R), residual, **mod_R) 

2343 return self._Error(op, other, ResidualError, txt=t) 

2344 

2345 def root(self, root, **raiser_RESIDUAL): 

2346 '''Return C{B{self}**(1 / B{root})} as L{Fsum}. 

2347 

2348 @arg root: The order (C{scalar}, L{Fsum} or L{Fsum2Tuple}), non-zero. 

2349 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore any 

2350 L{ResidualError}s (C{bool}) or C{B{RESIDUAL}=scalar} 

2351 to override the current L{RESIDUAL<Fsum.RESIDUAL>}. 

2352 

2353 @return: The C{self ** (1 / B{root})} result (L{Fsum}). 

2354 

2355 @raise ResidualError: Non-zero, significant residual or invalid 

2356 B{C{RESIDUAL}}. 

2357 

2358 @see: Method L{Fsum.pow}. 

2359 ''' 

2360 x = self._1_Over(root, _truediv_op_, **raiser_RESIDUAL) 

2361 f = self._copy_2(self.root) 

2362 return f._fpow(x, f.name, **raiser_RESIDUAL) # == pow(f, x) 

2363 

2364 def _scalar(self, other, op, **txt): 

2365 '''(INTERNAL) Return scalar C{other}. 

2366 ''' 

2367 if isscalar(other): 

2368 return other 

2369 raise self._Error(op, other, _TypeError, **txt) # _invalid_ 

2370 

2371 def signOf(self, res=True): 

2372 '''Determine the sign of this instance. 

2373 

2374 @kwarg res: If C{True}, consider, otherwise ignore 

2375 the residual (C{bool}). 

2376 

2377 @return: The sign (C{int}, -1, 0 or +1). 

2378 ''' 

2379 s, r = self._fprs2 

2380 r = (-r) if res else 0 

2381 return _signOf(s, r) 

2382 

2383 def toRepr(self, **lenc_prec_sep_fmt): # PYCHOK signature 

2384 '''Return this C{Fsum} instance as representation. 

2385 

2386 @kwarg lenc_prec_sep_fmt: Optional keyword arguments 

2387 for method L{Fsum.toStr}. 

2388 

2389 @return: This instance (C{repr}). 

2390 ''' 

2391 return Fmt.repr_at(self, self.toStr(**lenc_prec_sep_fmt)) 

2392 

2393 def toStr(self, lenc=True, **prec_sep_fmt): # PYCHOK signature 

2394 '''Return this C{Fsum} instance as string. 

2395 

2396 @kwarg lenc: If C{True}, include the current C{[len]} of this 

2397 L{Fsum} enclosed in I{[brackets]} (C{bool}). 

2398 @kwarg prec_sep_fmt: Optional keyword arguments for method 

2399 L{Fsum2Tuple.toStr}. 

2400 

2401 @return: This instance (C{str}). 

2402 ''' 

2403 p = self.classname 

2404 if lenc: 

2405 p = Fmt.SQUARE(p, len(self)) 

2406 n = _enquote(self.name, white=_UNDER_) 

2407 t = self._fprs2.toStr(**prec_sep_fmt) 

2408 return NN(p, _SPACE_, n, t) 

2409 

2410 def _truediv(self, other, op, **raiser_RESIDUAL): 

2411 '''(INTERNAL) Return C{B{self} / B{other}} as an L{Fsum}. 

2412 ''' 

2413 f = self._copy_2(self.__truediv__) 

2414 return f._ftruediv(other, op, **raiser_RESIDUAL) 

2415 

2416 def _update(self, updated=True): # see ._fset 

2417 '''(INTERNAL) Zap all cached C{Property_RO} values. 

2418 ''' 

2419 if updated: 

2420 _pop = self.__dict__.pop 

2421 for p in _ROs: 

2422 _ = _pop(p, None) 

2423# Fsum._fint2._update(self) 

2424# Fsum._fprs ._update(self) 

2425# Fsum._fprs2._update(self) 

2426 return self # for .fset_ 

2427 

2428_ROs = _allPropertiesOf_n(3, Fsum, Property_RO) # PYCHOK see Fsum._update 

2429 

2430if _NONFINITES: # PYCHOK no cover 

2431 _ = nonfiniterrors(False) 

2432 

2433 

2434def _Float_Int(arg, **name_Error): 

2435 '''(INTERNAL) L{DivMod2Tuple}, L{Fsum2Tuple} Unit. 

2436 ''' 

2437 U = Int if isint(arg) else Float 

2438 return U(arg, **name_Error) 

2439 

2440 

2441class DivMod2Tuple(_NamedTuple): 

2442 '''2-Tuple C{(div, mod)} with the quotient C{div} and remainder 

2443 C{mod} results of a C{divmod} operation. 

2444 

2445 @note: Quotient C{div} an C{int} in Python 3+ but a C{float} 

2446 in Python 2-. Remainder C{mod} an L{Fsum} instance. 

2447 ''' 

2448 _Names_ = ('div', 'mod') 

2449 _Units_ = (_Float_Int, Fsum) 

2450 

2451 

2452class Fsum2Tuple(_NamedTuple): # in .fstats 

2453 '''2-Tuple C{(fsum, residual)} with the precision running C{fsum} 

2454 and the C{residual}, the sum of the remaining partials. Each 

2455 item is C{float} or C{int}. 

2456 

2457 @note: If the C{residual is INT0}, the C{fsum} is considered 

2458 to be I{exact}, see method L{Fsum2Tuple.is_exact}. 

2459 ''' 

2460 _Names_ = ( Fsum.fsum.__name__, Fsum.residual.name) 

2461 _Units_ = (_Float_Int, _Float_Int) 

2462 

2463 def __abs__(self): # in .fmath 

2464 return self._Fsum.__abs__() 

2465 

2466 def __bool__(self): # PYCHOK Python 3+ 

2467 return bool(self._Fsum) 

2468 

2469 def __eq__(self, other): 

2470 return self._other_op(other, self.__eq__) 

2471 

2472 def __float__(self): 

2473 return self._Fsum.__float__() 

2474 

2475 def __ge__(self, other): 

2476 return self._other_op(other, self.__ge__) 

2477 

2478 def __gt__(self, other): 

2479 return self._other_op(other, self.__gt__) 

2480 

2481 def __le__(self, other): 

2482 return self._other_op(other, self.__le__) 

2483 

2484 def __lt__(self, other): 

2485 return self._other_op(other, self.__lt__) 

2486 

2487 def __int__(self): 

2488 return self._Fsum.__int__() 

2489 

2490 def __ne__(self, other): 

2491 return self._other_op(other, self.__ne__) 

2492 

2493 def __neg__(self): 

2494 return self._Fsum.__neg__() 

2495 

2496 __nonzero__ = __bool__ # Python 2- 

2497 

2498 def __pos__(self): 

2499 return self._Fsum.__pos__() 

2500 

2501 def as_integer_ratio(self): 

2502 '''Return this instance as the ratio of 2 integers. 

2503 

2504 @see: Method L{Fsum.as_integer_ratio} for further details. 

2505 ''' 

2506 return self._Fsum.as_integer_ratio() 

2507 

2508 @property_RO 

2509 def _fint2(self): 

2510 return self._Fsum._fint2 

2511 

2512 @property_RO 

2513 def _fprs2(self): 

2514 return self._Fsum._fprs2 

2515 

2516 @Property_RO 

2517 def _Fsum(self): # this C{Fsum2Tuple} as L{Fsum}, in .fstats 

2518 s, r = _s_r(*self) 

2519 ps = (r, s) if r else (s,) 

2520 return _Psum(ps, name=self.name) 

2521 

2522 def Fsum_(self, *xs, **name_f2product_nonfinites_RESIDUAL): 

2523 '''Return this C{Fsum2Tuple} as an L{Fsum} plus some C{xs}. 

2524 ''' 

2525 return Fsum(self, *xs, **name_f2product_nonfinites_RESIDUAL) 

2526 

2527 def is_exact(self): 

2528 '''Is this L{Fsum2Tuple} considered to be exact? (C{bool}). 

2529 ''' 

2530 return self._Fsum.is_exact() 

2531 

2532 def is_finite(self): # in .constants 

2533 '''Is this L{Fsum2Tuple} C{finite}? (C{bool}). 

2534 

2535 @see: Function L{isfinite<pygeodesy.isfinite>}. 

2536 ''' 

2537 return self._Fsum.is_finite() 

2538 

2539 def is_integer(self): 

2540 '''Is this L{Fsum2Tuple} C{integer}? (C{bool}). 

2541 ''' 

2542 return self._Fsum.is_integer() 

2543 

2544 def _mul_scalar(self, other, op): # for Fsum._fmul 

2545 return self._Fsum._mul_scalar(other, op) 

2546 

2547 @property_RO 

2548 def _n(self): 

2549 return self._Fsum._n 

2550 

2551 def _other_op(self, other, which): 

2552 C, s = (tuple, self) if isinstance(other, tuple) else (Fsum, self._Fsum) 

2553 return getattr(C, which.__name__)(s, other) 

2554 

2555 @property_RO 

2556 def _ps(self): 

2557 return self._Fsum._ps 

2558 

2559 @property_RO 

2560 def _ps_neg(self): 

2561 return self._Fsum._ps_neg 

2562 

2563 def signOf(self, **res): 

2564 '''Like method L{Fsum.signOf}. 

2565 ''' 

2566 return self._Fsum.signOf(**res) 

2567 

2568 def toStr(self, fmt=Fmt.g, **prec_sep): # PYCHOK signature 

2569 '''Return this L{Fsum2Tuple} as string (C{str}). 

2570 

2571 @kwarg fmt: Optional C{float} format (C{letter}). 

2572 @kwarg prec_sep: Optional keyword arguments for function 

2573 L{fstr<streprs.fstr>}. 

2574 ''' 

2575 return Fmt.PAREN(fstr(self, fmt=fmt, strepr=str, force=False, **prec_sep)) 

2576 

2577_Fsum_Fsum2Tuple_types = Fsum, Fsum2Tuple # PYCHOK lines 

2578 

2579 

2580class ResidualError(_ValueError): 

2581 '''Error raised for a division, power or root operation of 

2582 an L{Fsum} instance with a C{residual} I{ratio} exceeding 

2583 the L{RESIDUAL<Fsum.RESIDUAL>} threshold. 

2584 

2585 @see: Module L{pygeodesy.fsums} and method L{Fsum.RESIDUAL}. 

2586 ''' 

2587 pass 

2588 

2589 

2590try: 

2591 from math import fsum as _fsum # precision IEEE-754 sum, Python 2.6+ 

2592 

2593 # make sure _fsum works as expected (XXX check 

2594 # float.__getformat__('float')[:4] == 'IEEE'?) 

2595 if _fsum((1, 1e101, 1, -1e101)) != 2: # PYCHOK no cover 

2596 del _fsum # nope, remove _fsum ... 

2597 raise ImportError() # ... use _fsum below 

2598 

2599 _sum = _fsum # in .elliptic 

2600except ImportError: 

2601 _sum = sum # in .elliptic 

2602 

2603 def _fsum(xs): 

2604 '''(INTERNAL) Precision summation, Python 2.5-. 

2605 ''' 

2606 F = Fsum(name=_fsum.name, nonfinites=True) 

2607 return float(F._facc(xs, up=False)) 

2608 

2609 

2610def fsum(xs, nonfinites=None, **floats): 

2611 '''Precision floating point summation from Python's C{math.fsum}. 

2612 

2613 @arg xs: Iterable of items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

2614 @kwarg nonfinites: Use C{B{nonfinites}=True} if I{non-finites} are C{OK}, if 

2615 C{False} I{non-finites} raise an Overflow-/ValueError or if 

2616 C{None}, apply C{B{nonfinites}=not }L{nonfiniterrors()} 

2617 (C{bool} or C{None}). 

2618 @kwarg floats: DEPRECATED keyword argument C{B{floats}=False} (C{bool}), use 

2619 keyword argument C{B{nonfinites}=False} instead. 

2620 

2621 @return: Precision C{fsum} (C{float}). 

2622 

2623 @raise OverflowError: Infinite B{C{xs}} item or intermediate C{math.fsum} overflow. 

2624 

2625 @raise TypeError: Invalid B{C{xs}} item. 

2626 

2627 @raise ValueError: Invalid or C{NAN} B{C{xs}} item. 

2628 

2629 @see: Function L{nonfiniterrors}, class L{Fsum} and methods L{Fsum.nonfinites}, 

2630 L{Fsum.fsum}, L{Fsum.fadd} and L{Fsum.fadd_}. 

2631 ''' 

2632 return _xsum(fsum, xs, nonfinites=nonfinites, **floats) if xs else _0_0 

2633 

2634 

2635def fsum_(*xs, **nonfinites): 

2636 '''Precision floating point summation of all positional items. 

2637 

2638 @arg xs: Items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all positional. 

2639 @kwarg nonfinites: Use C{B{nonfinites}=True} if I{non-finites} are C{OK} (C{bool}). 

2640 

2641 @see: Function L{fsum<fsums.fsum>} for further details. 

2642 ''' 

2643 return _xsum(fsum_, xs, origin=1, **nonfinites) if xs else _0_0 

2644 

2645 

2646def fsumf_(*xs): 

2647 '''Precision floating point summation of all positional items with I{non-finites} C{OK}. 

2648 

2649 @arg xs: Items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), 

2650 all positional. 

2651 

2652 @see: Function L{fsum_<fsums.fsum_>} for further details. 

2653 ''' 

2654 return _xsum(fsumf_, xs, nonfinites=True, origin=1) if xs else _0_0 

2655 

2656 

2657def fsum1(xs, **nonfinites): 

2658 '''Precision floating point summation, 1-primed. 

2659 

2660 @arg xs: Iterable of items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}). 

2661 @kwarg nonfinites: Use C{B{nonfinites}=True} if I{non-finites} are C{OK} (C{bool}). 

2662 

2663 @see: Function L{fsum<fsums.fsum>} for further details. 

2664 ''' 

2665 return _xsum(fsum1, xs, primed=1, **nonfinites) if xs else _0_0 

2666 

2667 

2668def fsum1_(*xs, **nonfinites): 

2669 '''Precision floating point summation of all positional items, 1-primed. 

2670 

2671 @arg xs: Items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all positional. 

2672 @kwarg nonfinites: Use C{B{nonfinites}=True} if I{non-finites} are C{OK} (C{bool}). 

2673 

2674 @see: Function L{fsum_<fsums.fsum_>} for further details. 

2675 ''' 

2676 return _xsum(fsum1_, xs, origin=1, primed=1, **nonfinites) if xs else _0_0 

2677 

2678 

2679def fsum1f_(*xs): 

2680 '''Precision floating point summation of all positional items, 1-primed and 

2681 with I{non-finites} C{OK}. 

2682 

2683 @see: Function L{fsum_<fsums.fsum_>} for further details. 

2684 ''' 

2685 return _xsum(fsum1f_, xs, nonfinites=True, primed=1) if xs else _0_0 

2686 

2687 

2688def _xs(xs, _x, i_x): 

2689 '''(INTERNAL) Yield all C{xs} as C{scalar}. 

2690 ''' 

2691 for i, x in enumerate(xs): 

2692 i_x[:] = i, x 

2693 if _isFsumTuple(x): 

2694 for p in map(_x, x._ps): 

2695 yield p 

2696 else: 

2697 yield _x(x) 

2698 

2699 

2700def _xsError(X, xs, i, x, *n): # in _2floats, ._fstats 

2701 '''(INTERNAL) Error for C{xs} or C{x}, item C{xs[i]}. 

2702 ''' 

2703 return ((_xError(X, n[0], xs) if n else 

2704 _xError(X, xs=xs)) if x is xs else 

2705 _xError(X, Fmt.INDEX(xs=i), x)) 

2706 

2707 

2708def _xsum(which, xs, nonfinites=None, origin=0, primed=0, **floats): 

2709 '''(INTERNAL) Precision summation of C{xs} with conditions. 

2710 ''' 

2711 i_x = [0, xs] 

2712 try: 

2713 if nonfinites is None: 

2714 nonfinites = not nonfiniterrors() 

2715 elif floats: 

2716 nonfinites = _xkwds_get1(floats, floats=nonfinites) 

2717 fs = _xs(xs, (_passarg if nonfinites else _2finite), i_x) 

2718 return _fsum(_1primed(fs) if primed else fs) 

2719 except (OverflowError, TypeError, ValueError) as X: 

2720 i, x = i_x 

2721 i += origin - (1 if primed else 0) 

2722 t = _xsError(X, xs, i, x) 

2723 t = _COMMASPACE_(unstr(which), t) 

2724 raise _xError(X, t, txt=None) 

2725 

2726 

2727# delete all decorators, etc. 

2728del _allPropertiesOf_n, deprecated_method, deprecated_property_RO, \ 

2729 Property, Property_RO, property_RO, _ALL_LAZY, _F2PRODUCT, \ 

2730 MANT_DIG, _NONFINITES, _RESIDUAL_0_0, _getenv, _std_ 

2731 

2732if __name__ == '__main__': 

2733 

2734 # usage: python3 -m pygeodesy.fsums 

2735 

2736 def _test(n): 

2737 # copied from Hettinger, see L{Fsum} reference 

2738 from pygeodesy import frandoms, printf 

2739 

2740 printf(_fsum.__name__, end=_COMMASPACE_) 

2741 printf(_psum.__name__, end=_COMMASPACE_) 

2742 

2743 F = Fsum() 

2744 if F.is_math_fsum(): 

2745 for t in frandoms(n, seeded=True): 

2746 assert float(F.fset_(*t)) == _fsum(t) 

2747 printf(_DOT_, end=NN) 

2748 printf(NN) 

2749 

2750 _test(128) 

2751 

2752# **) MIT License 

2753# 

2754# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

2755# 

2756# Permission is hereby granted, free of charge, to any person obtaining a 

2757# copy of this software and associated documentation files (the "Software"), 

2758# to deal in the Software without restriction, including without limitation 

2759# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

2760# and/or sell copies of the Software, and to permit persons to whom the 

2761# Software is furnished to do so, subject to the following conditions: 

2762# 

2763# The above copyright notice and this permission notice shall be included 

2764# in all copies or substantial portions of the Software. 

2765# 

2766# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

2767# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

2768# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

2769# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

2770# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

2771# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

2772# OTHER DEALINGS IN THE SOFTWARE.