1

2# -*- coding: utf-8 -*-

3

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

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

6 option, accurate I{TwoProduct} multiplication.

7

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

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

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

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

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

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

14 otherwise the C{PyGeodesy 24.09.09} release.

15

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

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

18

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

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

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

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

23 and L{Fsum.__itruediv__}.

24

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

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

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

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

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

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

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

32

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

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

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

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

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

38'''

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

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

41

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

43 _signOf, itemsorted, signOf, _xiterable

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

45 _1_0, _N_1_0, _isfinite, _pos_self, \

46 Float, Int

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

48 _ValueError, _xError, _xError2, _xkwds, \

49 _xkwds_get, _xkwds_get1, _xkwds_not, \

50 _xkwds_pop, _xsError

51 from pygeodesy.internals import _enquote, _getPYGEODESY, _MODS, _passarg

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

53 _not_finite_, _SPACE_, _std_, _UNDER_

54# from pygeodesy.lazily import _ALL_LAZY # from .named

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

56 _NotImplemented, _ALL_LAZY

57 from pygeodesy.props import _allPropertiesOf_n, deprecated_method, \

58 deprecated_property_RO, Property, \

59 Property_RO, property_RO

60 from pygeodesy.streprs import Fmt, fstr, unstr

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

62

63 from math import fabs, isinf, isnan, \

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

65

66__all__ = _ALL_LAZY.fsums

67__version__ = '24.12.02'

68

69 from 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_floordiv_op_ = _truediv_op_ * 2 # _DSLASH_

81_divmod_op_ = _floordiv_op_ + _mod_op_

82_F2PRODUCT = _getPYGEODESY('FSUM_F2PRODUCT')

83_iadd_op_ = _add_op_ + _fset_op_ # in .auxilats.auxAngle, .fstats

84_integer_ = 'integer'

85_isub_op_ = _sub_op_ + _fset_op_ # in .auxilats.auxAngle

86_NONFINITEr = _0_0 # NOT INT0!

87_NONFINITES = _getPYGEODESY('FSUM_NONFINITES')

88_non_zero_ = 'non-zero'

89_pow_op_ = _mul_op_ * 2 # _DSTAR_

90_RESIDUAL_0_0 = _getPYGEODESY('FSUM_RESIDUAL', _0_0)

91_significant_ = 'significant'

92_threshold_ = 'threshold'

93

94

95 def _2finite(x, _isfine=_isfinite): # in .fstats

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

97 '''

98 return (float(x) if _isfine(x) # and isscalar(x)

99 else _nfError(x))

100

101

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

103 '''(INTERNAL) Raise C{TypeError} or C{Overflow-/ValueError} if C{x} not finite.

104 '''

105 n, x = name_x.popitem() # _xkwds_item2(name_x)

106 try:

107 f = float(x)

108 return f if _isfine(f) else _nfError(x)

109 except Exception as X:

110 raise _xError(X, Fmt.INDEX(n, index), x)

111

112

113 try: # MCCABE 26

114 from math import fma as _fma

115

116 def _2products(x, ys, *zs):

117 # yield(x * y for y in ys) + yield(z in zs)

118 # TwoProductFMA U{Algorithm 3.5

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

120 for y in ys:

121 f = x * y

122 yield f

123 if _isfinite(f):

124 yield _fma(x, y, -f)

125 for z in zs:

126 yield z

127

128# _2split3 = \

129 _2split3s = _passarg # in Fsum.is_math_fma

130

131 except ImportError: # PYCHOK DSPACE! Python 3.12-

132

133 if _F2PRODUCT and _F2PRODUCT != _std_:

134 # backward to PyGeodesy 24.09.09, with _fmaX

135

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

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

138 # the same accuracy, but ~14x slower

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

140 n = na * nb * dc

141 n += da * db * nc

142 d = da * db * dc

143 try:

144 n, d = _n_d2(n, d)

145 r = float(n / d)

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

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

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

149

150 def _2n_d(x): # PYCHOK no cover

151 try: # int.as_integer_ratio in 3.8+

152 return x.as_integer_ratio()

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

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

155 else:

156

157 def _fma(a, b, c): # PYCHOK redef

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

159 # the same accuracy, but ~13x slower

160 b3s = _2split3(b), # 1-tuple of 3-tuple

161 r = _fsum(_2products(a, b3s, c))

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

163

164 _2n_d = None # redef

165

166 def _fmaX(r, *a_b_c): # PYCHOK no cover

167 # handle non-finite as Python 3.13+ C-function U{math_fma_impl<https://

168 # GitHub.com/python/cpython/blob/main/Modules/mathmodule.c#L2305>}:

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

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

171 if isnan(r):

172 def _x(x):

173 return not isnan(x)

174 else: # non-NAN non-finite

175 _x = _isfinite

176 if all(map(_x, a_b_c)):

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

178 return r

179

180 def _2products(x, y3s, *zs): # PYCHOK in _fma, ...

181 # yield(x * y3 for y3 in y3s) + yield(z in zs)

182 # TwoProduct U{Algorithm 3.3

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

184 # also in Python 3.13+ C{Modules/mathmodule.c} under

185 # #ifndef UNRELIABLE_FMA ... #else ... #endif

186 _, a, b = _2split3(x)

187 for y, c, d in y3s:

188 y *= x

189 yield y

190 if _isfinite(y):

191 # yield b * d - (((y - a * c) - b * c) - a * d)

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

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

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

195 yield a * c - y

196 yield b * c

197 if d:

198 yield a * d

199 yield b * d

200 for z in zs:

201 yield z

202

203 _2FACTOR = pow(2, (MANT_DIG + 1) // 2) + _1_0 # 134217729 if MANT_DIG == 53

204

205 def _2split3(x):

206 # Split U{Algorithm 3.2

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

208 a = c = x * _2FACTOR

209 a -= c - x

210 b = x - a

211 return x, a, b

212

213 def _2split3s(xs): # in Fsum.is_math_fma

214 return map(_2split3, xs)

215

216

217 def f2product(two=None):

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

219

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

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

222

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

224

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

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

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

228 equivalent, slower implementation when not available.

229 '''

230 t = Fsum._f2product

231 if two is not None:

232 Fsum._f2product = bool(two)

233 return t

234

235

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

237 '''(INTERNAL) An C{Fsum(xs)}, all C{scalar}, an L{Fsum} or L{Fsum2Tuple}.

238 '''

239 return Fsum()._facc_scalarf(xs, up=False)

240

241

242 def _Fsum1f_(*xs): # in .albers

243 '''(INTERNAL) An C{Fsum(xs)}, all C{scalar}, an L{Fsum} or L{Fsum2Tuple}, 1-primed.

244 '''

245 return Fsum()._facc_scalarf(_1primed(xs), origin=-1, up=False)

246

247

248 def _halfeven(s, r, p):

249 '''(INTERNAL) Round half-even.

250 '''

251 if (p > 0 and r > 0) or \

252 (p < 0 and r < 0): # signs match

253 r *= 2

254 t = s + r

255 if r == (t - s):

256 s = t

257 return s

258

259

260 def _isFsum(x): # in .fmath

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

262 '''

263 return isinstance(x, Fsum)

264

265

266 def _isFsum_2Tuple(x): # in .basics, .constants, .fmath, .fstats

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

268 '''

269 return isinstance(x, _Fsum_2Tuple_types)

270

271

272 def _isOK(unused):

273 '''(INTERNAL) Helper for C{Fsum._fsum2} and C{Fsum.nonfinites}.

274 '''

275 return True

276

277

278 def _isOK_or_finite(x, _isfine=_isfinite):

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

280 '''

281 # assert _isfine in (_isOK, _isfinite)

282 return _isfine(x) # C{bool}

283

284

285 try:

286 from math import gcd as _gcd

287

288 def _n_d2(n, d):

289 '''(INTERNAL) Reduce C{n} and C{d} by C{gcd}.

290 '''

291 if n and d:

292 try:

293 c = _gcd(n, d)

294 if c > 1:

295 n, d = (n // c), (d // c)

296 except TypeError: # non-int float

297 pass

298 return n, d

299

300 except ImportError: # 3.4-

301

302 def _n_d2(*n_d): # PYCHOK redef

303 return n_d

304

305

306 def _nfError(x, *args):

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

308 '''

309 E = _NonfiniteError(x)

310 t = Fmt.PARENSPACED(_not_finite_, x)

311 if args: # in _fmaX, _2sum

312 return E(txt=t, *args)

313 raise E(t, txt=None)

314

315

316 def _NonfiniteError(x):

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

318 '''

319 return _OverflowError if isinf(x) else (

320 _ValueError if isnan(x) else _AssertionError)

321

322

323 def nonfiniterrors(raiser=None):

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

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

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

327

328 @kwarg raiser: If C{True}, throw exceptions, if C{False} handle

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

330 the setting unchanged.

331

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

333

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

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

336 '''

337 d = Fsum._isfine

338 if raiser is not None:

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

340 return (False if d is Fsum._nonfinites_isfine_kwds[True] else

341 _xkwds_get1(d, _isfine=_isfinite) is _isfinite) if d else True

342

343

344 def _1primed(xs): # in .fmath

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

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

347 '''

348 yield _1_0

349 for x in xs:

350 yield x

351 yield _N_1_0

352

353

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

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

356 '''

357 # assert isinstance(ps, list)

358 i = len(ps) - 1

359 s = _0_0 if i < 0 else ps[i]

360 while i > 0:

361 i -= 1

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

363 if r: # sum(ps) became inexact

364 if s:

365 ps[i:] = r, s

366 if i > 0:

367 s = _halfeven(s, r, ps[i-1])

368 break # return s

369 s = r # PYCHOK no cover

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

371 i = 0 # collapse ps

372 if ps:

373 s += sum(ps)

374 ps[i:] = s,

375 return s

376

377

378 def _Psum(ps, **name_f2product_nonfinites_RESIDUAL):

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

380 '''

381 F = Fsum(**name_f2product_nonfinites_RESIDUAL)

382 if ps:

383 F._ps[:] = ps

384 F._n = len(F._ps)

385 return F

386

387

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

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

390 '''

391 return _Psum(ps, **name_f2product_nonfinites_RESIDUAL)

392

393

394 def _residue(other):

395 '''(INTERNAL) Return the C{residual} or C{None} for C{scalar}.

396 '''

397 try:

398 r = other.residual

399 except AttributeError:

400 r = None # float, int, other

401 return r

402

403

404 def _s_r2(s, r):

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

406 '''

407 if _isfinite(s):

408 if r:

409 if fabs(s) < fabs(r):

410 s, r = r, (s or INT0)

411 else:

412 r = INT0

413 else:

414 r = _NONFINITEr

415 return s, r

416

417

418 def _strcomplex(s, *args):

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

420 '''

421 c = _strcomplex.__name__[4:]

422 n = _sub_op_(len(args), _arg_)

423 t = unstr(pow, *args)

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

425

426

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

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

429 '''

430 p = _stresidual.__name__[3:]

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

432 for n, v in itemsorted(mod_ratio):

433 p = Fmt.PARENSPACED(n, Fmt(v))

434 t = _COMMASPACE_(t, p)

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

436

437

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

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

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

441 '''

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

443

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

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

446 # 1974, Zeitschrift für Angewandte Mathmatik und Mechanik, vol 51, nr 1, p 39-51

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

448 s = a + b

449 if _isfinite(s):

450 if fabs(a) < fabs(b):

451 r = (b - s) + a

452 else:

453 r = (a - s) + b

454 elif _isfine(s):

455 r = _NONFINITEr

456 else: # non-finite and not OK

457 t = unstr(_2sum, a, b)

458 raise _nfError(s, t)

459 return s, r

460

461

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

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

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

465 '''

466 if kwds:

467 threshold = _xkwds_get1(kwds, RESIDUAL=threshold)

468 try:

469 return _2finite(threshold) # PYCHOK None

470 except Exception as x:

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

472

473

474 def _2tuple2(other):

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

476 C{float} or C{as-is} and C{r} the residual of C{as-is} or 0.

477 '''

478 if _isFsum_2Tuple(other):

479 s, r = other._fint2

480 if r:

481 s, r = other._nfprs2

482 if r: # PYCHOK no cover

483 s = other # L{Fsum} as-is

484 else:

485 r = 0

486 s = other # C{type} as-is

487 if isint(s, both=True):

488 s = int(s)

489 return s, r

490

491

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

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

494

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

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

497 intermediate, I{running} summuation.

498

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

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

501

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

503 determined by function L{nonfiniterrors<fsums.nonfiniterrors>} for the default

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

505 overruling the default. For backward compatibility, I{non-finites} raise

506 exceptions by default.

507

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

509 393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>},

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

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

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

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

514

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

516 multiplication.

517

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

519 C{PYGEODESY_FSUM_NONFINITES} and C{PYGEODESY_FSUM_RESIDUAL}.

520 '''

521 _f2product = _MODS.sys_version_info2 > (3, 12) or bool(_F2PRODUCT)

522 _isfine = {} # == _isfinite, see nonfiniterrors()

523 _n = 0

524# _ps = [] # partial sums

525# _ps_max = 0 # max(Fsum._ps_max, len(Fsum._ps)) # 41

526 _RESIDUAL = _threshold(_RESIDUAL_0_0)

527

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

529 '''New L{Fsum}.

530

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

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

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

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

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

536 L{Fsum}.

537

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

539 L{Fsum.fadd} and L{Fsum.fadd_}.

540 '''

541 if name_f2product_nonfinites_RESIDUAL:

542 self._optionals(**name_f2product_nonfinites_RESIDUAL)

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

544 if xs:

545 self._facc_args(xs, up=False)

546

547 def __abs__(self):

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

549 '''

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

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

552

553 def __add__(self, other):

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

555

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

557

558 @return: The sum (L{Fsum}).

559

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

561 '''

562 f = self._copy_2(self.__add__)

563 return f._fadd(other)

564

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

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

567 '''

568 s, r = self._nfprs2

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

570

571 def __call__(self, other, **up): # in .fmath

572 '''Reset this C{Fsum} to C{other}, default C{B{up}=True}.

573 '''

574 self._ps[:] = 0, # clear for errors

575 self._fset(other, op=_fset_op_, **up)

576 return self

577

578

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

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

581

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

583

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

585 '''

586 return self.ceil

587

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

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

590

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

592

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

594 '''

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

596 return _signOf(s, 0)

597

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

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

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

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

602

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

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

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

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

607

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

609 B{C{RESIDUAL}}.

610

611 @see: Method L{Fsum.fdiv}.

612 '''

613 f = self._copy_2(self.__divmod__)

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

615

616 def __eq__(self, other):

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

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

619 '''

620 return self._cmp_0(other, _fset_op_ + _fset_op_) == 0

621

622 def __float__(self):

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

624

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

626 '''

627 return float(self._fprs)

628

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

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

631

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

633

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

635 '''

636 return self.floor

637

638 def __floordiv__(self, other):

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

640

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

642

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

644

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

646 '''

647 f = self._copy_2(self.__floordiv__)

648 return f._floordiv(other, _floordiv_op_)

649

650 def __ge__(self, other):

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

652 '''

653 return self._cmp_0(other, _gt_op_ + _fset_op_) >= 0

654

655 def __gt__(self, other):

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

657 '''

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

659

660 def __hash__(self): # PYCHOK no cover

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

662 '''

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

664 # 3/library/numbers.html#numbers.Rational>}

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

666

667 def __iadd__(self, other):

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

669

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

671 an iterable of several of the former.

672

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

674

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

676 C{scalar} nor L{Fsum}.

677

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

679 '''

680 try:

681 return self._fadd(other, op=_iadd_op_)

682 except TypeError:

683 pass

684 _xiterable(other)

685 return self._facc(other)

686

687 def __ifloordiv__(self, other):

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

689

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

691

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

693

694 @raise ResidualError: Non-zero, significant residual

695 in B{C{other}}.

696

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

698

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

700

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

702

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

704 '''

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

706

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

708 '''Not implemented.'''

709 return _NotImplemented(self, other)

710

711 def __imod__(self, other):

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

713

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

715

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

717

718 @see: Method L{Fsum.__divmod__}.

719 '''

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

721

722 def __imul__(self, other):

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

724

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

726

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

728

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

730

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

732

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

734 '''

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

736

737 def __int__(self):

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

739

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

741 and L{Fsum.floor}.

742 '''

743 i, _ = self._fint2

744 return i

745

746 def __invert__(self): # PYCHOK no cover

747 '''Not implemented.'''

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

749 return _NotImplemented(self)

750

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

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

753

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

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

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

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

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

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

760

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

762

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

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

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

766

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

768

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

770 is non-zero and significant and either

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

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

773 C{None}.

774

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

776 invocation failed.

777

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

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

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

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

782 is given as C{0}.

783

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

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

786 '''

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

788

789 def __isub__(self, other):

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

791

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

793 an iterable of several of the former.

794

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

796

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

798

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

800 '''

801 try:

802 return self._fsub(other, _isub_op_)

803 except TypeError:

804 pass

805 _xiterable(other)

806 return self._facc_neg(other)

807

808 def __iter__(self):

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

810 '''

811 return iter(self.partials)

812

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

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

815

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

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

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

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

820

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

822

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

824

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

826 B{C{RESIDUAL}}.

827

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

829

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

831

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

833

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

835 '''

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

837

838 def __le__(self, other):

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

840 '''

841 return self._cmp_0(other, _lt_op_ + _fset_op_) <= 0

842

843 def __len__(self):

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

845 '''

846 return self._n

847

848 def __lt__(self, other):

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

850 '''

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

852

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

854 '''Not implemented.'''

855 return _NotImplemented(self, other)

856

857 def __mod__(self, other):

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

859

860 @see: Method L{Fsum.__imod__}.

861 '''

862 f = self._copy_2(self.__mod__)

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

864

865 def __mul__(self, other):

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

867

868 @see: Method L{Fsum.__imul__}.

869 '''

870 f = self._copy_2(self.__mul__)

871 return f._fmul(other, _mul_op_)

872

873 def __ne__(self, other):

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

875 '''

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

877

878 def __neg__(self):

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

880 '''

881 f = self._copy_2(self.__neg__)

882 return f._fset(self._neg)

883

884 def __pos__(self):

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

886 '''

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

888

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

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

891

892 @see: Method L{Fsum.__ipow__}.

893 '''

894 f = self._copy_2(self.__pow__)

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

896

897 def __radd__(self, other):

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

899

900 @see: Method L{Fsum.__iadd__}.

901 '''

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

903 return f._fadd(self)

904

905 def __rdivmod__(self, other):

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

907 C{(quotient, remainder)}.

908

909 @see: Method L{Fsum.__divmod__}.

910 '''

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

912 return f._fdivmod2(self, _divmod_op_)

913

914# turned off, called by _deepcopy and _copy

915# def __reduce__(self): # Python 3.8+

916# ''' Pickle, like std C{fractions.Fraction}, see U{__reduce__

917# <https://docs.Python.org/3/library/pickle.html#object.__reduce__>}

918# '''

919# dict_ = self._Fsum_as().__dict__ # no __setstate__

920# return (self.__class__, self.partials, dict_)

921

922# def __repr__(self):

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

924# '''

925# return self.toRepr(lenc=True)

926

927 def __rfloordiv__(self, other):

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

929

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

931 '''

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

933 return f._floordiv(self, _floordiv_op_)

934

935 def __rmatmul__(self, other): # PYCHOK no coveS

936 '''Not implemented.'''

937 return _NotImplemented(self, other)

938

939 def __rmod__(self, other):

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

941

942 @see: Method L{Fsum.__imod__}.

943 '''

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

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

946

947 def __rmul__(self, other):

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

949

950 @see: Method L{Fsum.__imul__}.

951 '''

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

953 return f._fmul(self, _mul_op_)

954

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

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

957

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

959 '''

960 f = self._copy_2(self.__round__)

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

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

963

964 def __rpow__(self, other, *mod):

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

966

967 @see: Method L{Fsum.__ipow__}.

968 '''

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

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

971

972 def __rsub__(self, other):

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

974

975 @see: Method L{Fsum.__isub__}.

976 '''

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

978 return f._fsub(self, _sub_op_)

979

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

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

982

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

984 '''

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

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

987

988 def __str__(self):

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

990 '''

991 return self.toStr(lenc=True)

992

993 def __sub__(self, other):

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

995

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

997

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

999

1000 @see: Method L{Fsum.__isub__}.

1001 '''

1002 f = self._copy_2(self.__sub__)

1003 return f._fsub(other, _sub_op_)

1004

1005 def __truediv__(self, other, **raiser_RESIDUAL):

1006 '''Return C{B{self} / B{other}} as an L{Fsum}.

1007

1008 @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} divisor.

1009 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore

1010 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}

1011 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

1012

1013 @return: The quotient (L{Fsum}).

1014

1015 @raise ResidualError: Non-zero, significant residual or invalid

1016 B{C{RESIDUAL}}.

1017

1018 @see: Method L{Fsum.__itruediv__}.

1019 '''

1020 return self._truediv(other, _truediv_op_, **raiser_RESIDUAL)

1021

1022 __trunc__ = __int__

1023

1024 if _MODS.sys_version_info2 < (3, 0): # PYCHOK no cover

1025 # <https://docs.Python.org/2/library/operator.html#mapping-operators-to-functions>

1026 __div__ = __truediv__

1027 __idiv__ = __itruediv__

1028 __long__ = __int__

1029 __nonzero__ = __bool__

1030 __rdiv__ = __rtruediv__

1031

1032 def as_integer_ratio(self):

1033 '''Return this instance as the ratio of 2 integers.

1034

1035 @return: 2-Tuple C{(numerator, denominator)} both C{int} with

1036 C{numerator} signed and C{denominator} non-zero and

1037 positive. The C{numerator} is I{non-finite} if this

1038 instance is.

1039

1040 @see: Method L{Fsum.fint2} and C{float.as_integer_ratio} in

1041 Python 2.7+.

1042 '''

1043 n, r = self._fint2

1044 if r:

1045 i, d = float(r).as_integer_ratio()

1046 n, d = _n_d2(n * d + i, d)

1047 else: # PYCHOK no cover

1048 d = 1

1049 return n, d

1050

1051 @property_RO

1052 def as_iscalar(self):

1053 '''Get this instance I{as-is} (L{Fsum} with C{non-zero residual},

1054 C{scalar} or I{non-finite}).

1055 '''

1056 s, r = self._nfprs2

1057 return self if r else s

1058

1059 @property_RO

1060 def ceil(self):

1061 '''Get this instance' C{ceil} value (C{int} in Python 3+, but

1062 C{float} in Python 2-).

1063

1064 @note: This C{ceil} takes the C{residual} into account.

1065

1066 @see: Method L{Fsum.int_float} and properties L{Fsum.floor},

1067 L{Fsum.imag} and L{Fsum.real}.

1068 '''

1069 s, r = self._fprs2

1070 c = _ceil(s) + int(r) - 1

1071 while r > (c - s): # (s + r) > c

1072 c += 1

1073 return c # _ceil(self._n_d)

1074

1075 cmp = __cmp__

1076

1077 def _cmp_0(self, other, op):

1078 '''(INTERNAL) Return C{scalar(self - B{other})} for 0-comparison.

1079 '''

1080 if _isFsum_2Tuple(other):

1081 s = self._ps_1sum(*other._ps)

1082 elif self._scalar(other, op):

1083 s = self._ps_1sum(other)

1084 else:

1085 s = self.signOf() # res=True

1086 return s

1087

1088 def copy(self, deep=False, **name):

1089 '''Copy this instance, C{shallow} or B{C{deep}}.

1090

1091 @kwarg name: Optional, overriding C{B{name}='"copy"} (C{str}).

1092

1093 @return: The copy (L{Fsum}).

1094 '''

1095 n = _name__(name, name__=self.copy)

1096 f = _Named.copy(self, deep=deep, name=n)

1097 if f._ps is self._ps:

1098 f._ps = list(self._ps) # separate list

1099 if not deep:

1100 f._n = 1

1101 # assert f._f2product == self._f2product

1102 # assert f._Fsum is f

1103 # assert f._isfine is self._isfine

1104 # assert f._RESIDUAL is self._RESIDUAL

1105 return f

1106

1107 def _copy_2(self, which, name=NN):

1108 '''(INTERNAL) Copy for I{dyadic} operators.

1109 '''

1110 n = name or which.__name__ # _DUNDER_nameof

1111 # NOT .classof due to .Fdot(a, *b) args, etc.

1112 f = _Named.copy(self, deep=False, name=n)

1113 f._ps = list(self._ps) # separate list

1114 # assert f._n == self._n

1115 # assert f._f2product == self._f2product

1116 # assert f._Fsum is f

1117 # assert f._isfine is self._isfine

1118 # assert f._RESIDUAL is self._RESIDUAL

1119 return f

1120

1121 def _copy_2r(self, other, which):

1122 '''(INTERNAL) Copy for I{reverse-dyadic} operators.

1123 '''

1124 return other._copy_2(which) if _isFsum(other) else \

1125 self._copy_2(which)._fset(other)

1126

1127 divmod = __divmod__

1128

1129 def _Error(self, op, other, Error, **txt_cause):

1130 '''(INTERNAL) Format an B{C{Error}} for C{{self} B{op} B{other}}.

1131 '''

1132 # self.as_iscalar causes RecursionError for ._fprs2 errors

1133 s = _Psum(self._ps, nonfinites=True, name=self.name)

1134 return Error(_SPACE_(s.as_iscalar, op, other), **txt_cause)

1135

1136 def _ErrorX(self, X, op, other, *mod):

1137 '''(INTERNAL) Format the caught exception C{X}.

1138 '''

1139 E, t = _xError2(X)

1140 if mod:

1141 t = _COMMASPACE_(Fmt.PARENSPACED(mod=mod[0]), t)

1142 return self._Error(op, other, E, txt=t, cause=X)

1143

1144 def _ErrorXs(self, X, xs, **kwds): # in .fmath

1145 '''(INTERNAL) Format the caught exception C{X}.

1146 '''

1147 E, t = _xError2(X)

1148 u = unstr(self.named3, *xs, _ELLIPSIS=4, **kwds)

1149 return E(u, txt=t, cause=X)

1150

1151 def _facc(self, xs, up=True, **_X_x_origin):

1152 '''(INTERNAL) Accumulate more C{scalar}s or L{Fsum}s.

1153 '''

1154 if xs:

1155 kwds = self._isfine

1156 if _X_x_origin:

1157 kwds = _xkwds(_X_x_origin, **kwds)

1158 fs = _xs(xs, **kwds) # PYCHOK yield

1159 ps = self._ps

1160 ps[:] = self._ps_acc(list(ps), fs, up=up)

1161# if len(ps) > 16:

1162# _ = _psum(ps, **self._isfine)

1163 return self

1164

1165 def _facc_args(self, xs, **up):

1166 '''(INTERNAL) Accumulate 0, 1 or more C{xs}, all positional

1167 arguments in the caller of this method.

1168 '''

1169 return self._fadd(xs[0], **up) if len(xs) == 1 else \

1170 self._facc(xs, **up) # origin=1?

1171

1172 def _facc_dot(self, n, xs, ys, **kwds): # in .fmath

1173 '''(INTERNAL) Accumulate C{fdot(B{xs}, *B{ys})}.

1174 '''

1175 if n > 0:

1176 _f = Fsum(**kwds)

1177 self._facc(_f(x).fmul(y) for x, y in zip(xs, ys)) # PYCHOK attr?

1178 return self

1179

1180 def _facc_neg(self, xs, **up_origin):

1181 '''(INTERNAL) Accumulate more C{xs}, negated.

1182 '''

1183 def _N(X):

1184 return X._ps_neg

1185

1186 def _n(x):

1187 return -float(x)

1188

1189 return self._facc(xs, _X=_N, _x=_n, **up_origin)

1190

1191 def _facc_power(self, power, xs, which, **raiser_RESIDUAL): # in .fmath

1192 '''(INTERNAL) Add each C{xs} as C{float(x**power)}.

1193 '''

1194 def _Pow4(p):

1195 r = 0

1196 if _isFsum_2Tuple(p):

1197 s, r = p._fprs2

1198 if r:

1199 m = Fsum._pow

1200 else: # scalar

1201 return _Pow4(s)

1202 elif isint(p, both=True) and int(p) >= 0:

1203 p = s = int(p)

1204 m = Fsum._pow_int

1205 else:

1206 p = s = _2float(power=p, **self._isfine)

1207 m = Fsum._pow_scalar

1208 return m, p, s, r

1209

1210 _Pow, p, s, r = _Pow4(power)

1211 if p: # and xs:

1212 op = which.__name__

1213 _FsT = _Fsum_2Tuple_types

1214 _pow = self._pow_2_3

1215

1216 def _P(X):

1217 f = _Pow(X, p, power, op, **raiser_RESIDUAL)

1218 return f._ps if isinstance(f, _FsT) else (f,)

1219

1220 def _p(x):

1221 x = float(x)

1222 f = _pow(x, s, power, op, **raiser_RESIDUAL)

1223 if f and r:

1224 f *= _pow(x, r, power, op, **raiser_RESIDUAL)

1225 return f

1226

1227 f = self._facc(xs, _X=_P, _x=_p) # origin=1?

1228 else:

1229 f = self._facc_scalar_(float(len(xs))) # x**0 == 1

1230 return f

1231

1232 def _facc_scalar(self, xs, **up):

1233 '''(INTERNAL) Accumulate all C{xs}, each C{scalar}.

1234 '''

1235 if xs:

1236 ps = self._ps

1237 ps[:] = self._ps_acc(list(ps), xs, **up)

1238 return self

1239

1240 def _facc_scalar_(self, *xs, **up):

1241 '''(INTERNAL) Accumulate all positional C{xs}, each C{scalar}.

1242 '''

1243 return self._facc_scalar(xs, **up)

1244

1245 def _facc_scalarf(self, xs, up=True, **origin_which):

1246 '''(INTERNAL) Accumulate all C{xs}, each C{scalar}, an L{Fsum} or

1247 L{Fsum2Tuple}, like function C{_xsum}.

1248 '''

1249 _C = self.__class__

1250 fs = _xs(xs, **_x_isfine(self.nonfinitesOK, _Cdot=_C,

1251 **origin_which)) # PYCHOK yield

1252 return self._facc_scalar(fs, up=up)

1253

1254# def _facc_up(self, up=True):

1255# '''(INTERNAL) Update the C{partials}, by removing

1256# and re-accumulating the final C{partial}.

1257# '''

1258# ps = self._ps

1259# while len(ps) > 1:

1260# p = ps.pop()

1261# if p:

1262# n = self._n

1263# _ = self._ps_acc(ps, (p,), up=False)

1264# self._n = n

1265# break

1266# return self._update() if up else self

1267

1268 def fadd(self, xs=()):

1269 '''Add an iterable's items to this instance.

1270

1271 @arg xs: Iterable of items to add (each C{scalar},

1272 an L{Fsum} or L{Fsum2Tuple}).

1273

1274 @return: This instance (L{Fsum}).

1275

1276 @raise OverflowError: Partial C{2sum} overflow.

1277

1278 @raise TypeError: An invalid B{C{xs}} item.

1279

1280 @raise ValueError: Invalid or I{non-finite} B{C{xs}} value.

1281 '''

1282 if _isFsum_2Tuple(xs):

1283 self._facc_scalar(xs._ps)

1284 elif isscalar(xs): # for backward compatibility # PYCHOK no cover

1285 x = _2float(x=xs, **self._isfine)

1286 self._facc_scalar_(x)

1287 elif xs: # _xiterable(xs)

1288 self._facc(xs)

1289 return self

1290

1291 def fadd_(self, *xs):

1292 '''Add all positional items to this instance.

1293

1294 @arg xs: Values to add (each C{scalar}, an L{Fsum}

1295 or L{Fsum2Tuple}), all positional.

1296

1297 @see: Method L{Fsum.fadd} for further details.

1298 '''

1299 return self._facc_args(xs)

1300

1301 def _fadd(self, other, op=_add_op_, **up):

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

1303 '''

1304 if _isFsum_2Tuple(other):

1305 self._facc_scalar(other._ps, **up)

1306 elif self._scalar(other, op):

1307 self._facc_scalar_(other, **up)

1308 return self

1309

1310 fcopy = copy # for backward compatibility

1311 fdiv = __itruediv__

1312 fdivmod = __divmod__

1313

1314 def _fdivmod2(self, other, op, **raiser_RESIDUAL):

1315 '''(INTERNAL) Apply C{B{self} %= B{other}} and return a L{DivMod2Tuple}.

1316 '''

1317 # result mostly follows CPython function U{float_divmod

1318 # <https://GitHub.com/python/cpython/blob/main/Objects/floatobject.c>},

1319 # but at least divmod(-3, 2) equals Cpython's result (-2, 1).

1320 q = self._truediv(other, op, **raiser_RESIDUAL).floor

1321 if q: # == float // other == floor(float / other)

1322 self -= self._Fsum_as(q) * other # NOT other * q!

1323

1324 s = signOf(other) # make signOf(self) == signOf(other)

1325 if s and self.signOf() == -s: # PYCHOK no cover

1326 self += other

1327 q -= 1

1328# t = self.signOf()

1329# if t and t != s:

1330# raise self._Error(op, other, _AssertionError, txt__=signOf)

1331 return DivMod2Tuple(q, self) # q is C{int} in Python 3+, but C{float} in Python 2-

1332

1333 def _fhorner(self, x, cs, where, incx=True): # in .fmath

1334 '''(INTERNAL) Add an L{Fhorner} evaluation of polynomial

1335 C{sum(cs[i] * B{x}**i for i=0..len(cs)-1) if B{incx}

1336 else sum(... i=len(cs)-1..0)}.

1337 '''

1338 # assert _xiterablen(cs)

1339 try:

1340 n = len(cs)

1341 H = self._Fsum_as(name__=self._fhorner)

1342 _m = H._mul_Fsum if _isFsum_2Tuple(x) else \

1343 H._mul_scalar

1344 if _2finite(x, **self._isfine) and n > 1:

1345 for c in (reversed(cs) if incx else cs):

1346 H._fset(_m(x, _mul_op_), up=False)

1347 H._fadd(c, up=False)

1348 else: # x == 0

1349 H = cs[0] if n else 0

1350 self._fadd(H)

1351 except Exception as X:

1352 t = unstr(where, x, *cs, _ELLIPSIS=4, incx=incx)

1353 raise self._ErrorX(X, _add_op_, t)

1354 return self

1355

1356 def _finite(self, other, op=None):

1357 '''(INTERNAL) Return B{C{other}} if C{finite}.

1358 '''

1359 if _isOK_or_finite(other, **self._isfine):

1360 return other

1361 E = _NonfiniteError(other)

1362 raise self._Error(op, other, E, txt=_not_finite_)

1363

1364 def fint(self, name=NN, **raiser_RESIDUAL):

1365 '''Return this instance' current running sum as C{integer}.

1366

1367 @kwarg name: Optional, overriding C{B{name}="fint"} (C{str}).

1368 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore

1369 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}

1370 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

1371

1372 @return: The C{integer} sum (L{Fsum}) if this instance C{is_integer}

1373 with a zero or insignificant I{integer} residual.

1374

1375 @raise ResidualError: Non-zero, significant residual or invalid

1376 B{C{RESIDUAL}}.

1377

1378 @see: Methods L{Fsum.fint2}, L{Fsum.int_float} and L{Fsum.is_integer}.

1379 '''

1380 i, r = self._fint2

1381 if r:

1382 R = self._raiser(r, i, **raiser_RESIDUAL)

1383 if R:

1384 t = _stresidual(_integer_, r, **R)

1385 raise ResidualError(_integer_, i, txt=t)

1386 return self._Fsum_as(i, name=_name__(name, name__=self.fint))

1387

1388 def fint2(self, **name):

1389 '''Return this instance' current running sum as C{int} and the

1390 I{integer} residual.

1391

1392 @kwarg name: Optional name (C{str}).

1393

1394 @return: An L{Fsum2Tuple}C{(fsum, residual)} with C{fsum}

1395 an C{int} and I{integer} C{residual} a C{float} or

1396 C{INT0} if the C{fsum} is considered to be I{exact}.

1397 The C{fsum} is I{non-finite} if this instance is.

1398 '''

1399 return Fsum2Tuple(*self._fint2, **name)

1400

1401 @Property

1402 def _fint2(self): # see ._fset

1403 '''(INTERNAL) Get 2-tuple (C{int}, I{integer} residual).

1404 '''

1405 s, r = self._nfprs2

1406 if _isfinite(s):

1407 i = int(s)

1408 r = (self._ps_1sum(i) if len(self._ps) > 1 else

1409 float(s - i)) or INT0

1410 else: # INF, NAN, NINF

1411 i = float(s)

1412# r = _NONFINITEr

1413 return i, r # Fsum2Tuple?

1414

1415 @_fint2.setter_ # PYCHOK setter_UNDERscore!

1416 def _fint2(self, s): # in _fset

1417 '''(INTERNAL) Replace the C{_fint2} value.

1418 '''

1419 if _isfinite(s):

1420 i = int(s)

1421 r = (s - i) or INT0

1422 else: # INF, NAN, NINF

1423 i = float(s)

1424 r = _NONFINITEr

1425 return i, r # like _fint2.getter

1426

1427 @deprecated_property_RO

1428 def float_int(self): # PYCHOK no cover

1429 '''DEPRECATED, use method C{Fsum.int_float}.'''

1430 return self.int_float() # raiser=False

1431

1432 @property_RO

1433 def floor(self):

1434 '''Get this instance' C{floor} (C{int} in Python 3+, but

1435 C{float} in Python 2-).

1436

1437 @note: This C{floor} takes the C{residual} into account.

1438

1439 @see: Method L{Fsum.int_float} and properties L{Fsum.ceil},

1440 L{Fsum.imag} and L{Fsum.real}.

1441 '''

1442 s, r = self._fprs2

1443 f = _floor(s) + _floor(r) + 1

1444 while (f - s) > r: # f > (s + r)

1445 f -= 1

1446 return f # _floor(self._n_d)

1447

1448# ffloordiv = __ifloordiv__ # for naming consistency?

1449# floordiv = __floordiv__ # for naming consistency?

1450

1451 def _floordiv(self, other, op, **raiser_RESIDUAL): # rather _ffloordiv?

1452 '''Apply C{B{self} //= B{other}}.

1453 '''

1454 q = self._ftruediv(other, op, **raiser_RESIDUAL) # == self

1455 return self._fset(q.floor) # floor(q)

1456

1457 def fma(self, other1, other2, **nonfinites): # in .fmath.fma

1458 '''Fused-multiply-add C{self *= B{other1}; self += B{other2}}.

1459

1460 @arg other1: Multiplier (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1461 @arg other2: Addend (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1462 @kwarg nonfinites: Use C{B{nonfinites}=True} or C{False}, to

1463 override L{nonfinites<Fsum.nonfinites>} and

1464 L{nonfiniterrors} default (C{bool}).

1465 '''

1466 op = self.fma.__name__

1467 _fs = self._ps_other

1468 try:

1469 s, r = self._fprs2

1470 if r:

1471 f = self._f2mul(self.fma, (other1,), **nonfinites)

1472 f += other2

1473 elif _residue(other1) or _residue(other2):

1474 fs = _2split3s(_fs(op, other1))

1475 fs = _2products(s, fs, *_fs(op, other2))

1476 f = _Psum(self._ps_acc([], fs, up=False), name=op)

1477 else:

1478 f = _fma(s, other1, other2)

1479 f = _2finite(f, **self._isfine)

1480 except TypeError as X:

1481 raise self._ErrorX(X, op, (other1, other2))

1482 except (OverflowError, ValueError) as X: # from math.fma

1483 f = self._mul_reduce(s, other1) # INF, NAN, NINF

1484 f += sum(_fs(op, other2))

1485 f = self._nonfiniteX(X, op, f, **nonfinites)

1486 return self._fset(f)

1487

1488 fmul = __imul__

1489

1490 def _fmul(self, other, op):

1491 '''(INTERNAL) Apply C{B{self} *= B{other}}.

1492 '''

1493 if _isFsum_2Tuple(other):

1494 if len(self._ps) != 1:

1495 f = self._mul_Fsum(other, op)

1496 elif len(other._ps) != 1: # and len(self._ps) == 1

1497 f = self._ps_mul(op, *other._ps) if other._ps else _0_0

1498 elif self._f2product: # len(other._ps) == 1

1499 f = self._mul_scalar(other._ps[0], op)

1500 else: # len(other._ps) == len(self._ps) == 1

1501 f = self._finite(self._ps[0] * other._ps[0], op=op)

1502 else:

1503 s = self._scalar(other, op)

1504 f = self._mul_scalar(s, op)

1505 return self._fset(f) # n=len(self) + 1

1506

1507 @deprecated_method

1508 def f2mul(self, *others, **raiser):

1509 '''DEPRECATED on 2024.09.13, use method L{f2mul_<Fsum.f2mul_>}.'''

1510 return self._fset(self.f2mul_(others, **raiser))

1511

1512 def f2mul_(self, *others, **f2product_nonfinites): # in .fmath.f2mul

1513 '''Return C{B{self} * B{other} * B{other} ...} for all B{C{others}} using cascaded,

1514 accurate multiplication like with L{f2product<Fsum.f2product>}C{(B{True})}.

1515

1516 @arg others: Multipliers (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all

1517 positional.

1518 @kwarg f2product_nonfinites: Use C{B{f2product=False}} to override the default

1519 C{True} and C{B{nonfinites}=True} or C{False}, to override

1520 settings L{nonfinites<Fsum.nonfinites>} and L{nonfiniterrors}.

1521

1522 @return: The cascaded I{TwoProduct} (L{Fsum} or C{float}).

1523

1524 @see: U{Equations 2.3<https://www.TUHH.De/ti3/paper/rump/OzOgRuOi06.pdf>}

1525 '''

1526 return self._f2mul(self.f2mul_, others, **f2product_nonfinites)

1527

1528 def _f2mul(self, where, others, f2product=True, **nonfinites_raiser):

1529 '''(INTERNAL) See methods C{fma} and C{f2mul_}.

1530 '''

1531 f = _Psum(self._ps, f2product=f2product, name=where.__name__)

1532 if others and f:

1533 if f.f2product():

1534 def _pfs(f, ps):

1535 return _2products(f, _2split3s(ps))

1536 else:

1537 def _pfs(f, ps): # PYCHOK redef

1538 return (f * p for p in ps)

1539

1540 op, ps = where.__name__, f._ps

1541 try: # as if self.f2product(True)

1542 for other in others: # to pinpoint errors

1543 for p in self._ps_other(op, other):

1544 ps[:] = f._ps_acc([], _pfs(p, ps), update=False)

1545 f._update()

1546 except TypeError as X:

1547 raise self._ErrorX(X, op, other)

1548 except (OverflowError, ValueError) as X:

1549 r = self._mul_reduce(sum(ps), other) # INF, NAN, NINF

1550 r = self._nonfiniteX(X, op, r, **nonfinites_raiser)

1551 f._fset(r)

1552 return f

1553

1554 def fover(self, over, **raiser_RESIDUAL):

1555 '''Apply C{B{self} /= B{over}} and summate.

1556

1557 @arg over: An L{Fsum} or C{scalar} denominator.

1558 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore

1559 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}

1560 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

1561

1562 @return: Precision running sum (C{float}).

1563

1564 @raise ResidualError: Non-zero, significant residual or invalid

1565 B{C{RESIDUAL}}.

1566

1567 @see: Methods L{Fsum.fsum} and L{Fsum.__itruediv__}.

1568 '''

1569 return float(self.fdiv(over, **raiser_RESIDUAL)._fprs)

1570

1571 fpow = __ipow__

1572

1573 def _fpow(self, other, op, *mod, **raiser_RESIDUAL):

1574 '''Apply C{B{self} **= B{other}}, optional B{C{mod}} or C{None}.

1575 '''

1576 if mod:

1577 if mod[0] is not None: # == 3-arg C{pow}

1578 f = self._pow_2_3(self, other, other, op, *mod, **raiser_RESIDUAL)

1579 elif self.is_integer():

1580 # return an exact C{int} for C{int}**C{int}

1581 i, _ = self._fint2 # assert _ == 0

1582 x, r = _2tuple2(other) # C{int}, C{float} or other

1583 f = self._Fsum_as(i)._pow_Fsum(other, op, **raiser_RESIDUAL) if r else \

1584 self._pow_2_3(i, x, other, op, **raiser_RESIDUAL)

1585 else: # mod[0] is None, power(self, other)

1586 f = self._pow(other, other, op, **raiser_RESIDUAL)

1587 else: # pow(self, other)

1588 f = self._pow(other, other, op, **raiser_RESIDUAL)

1589 return self._fset(f) # n=max(len(self), 1)

1590

1591 def f2product(self, *two):

1592 '''Get and set accurate I{TwoProduct} multiplication for this

1593 L{Fsum}, overriding the L{f2product} default.

1594

1595 @arg two: If omitted, leave the override unchanged, if C{True},

1596 turn I{TwoProduct} on, if C{False} off, if C{None}e

1597 remove th override (C{bool} or C{None}).

1598

1599 @return: The previous setting (C{bool} or C{None} if not set).

1600

1601 @see: Function L{f2product<fsums.f2product>}.

1602

1603 @note: Use C{f.f2product() or f2product()} to determine whether

1604 multiplication is accurate for L{Fsum} C{f}.

1605 '''

1606 if two: # delattrof(self, _f2product=None)

1607 t = _xkwds_pop(self.__dict__, _f2product=None)

1608 if two[0] is not None:

1609 self._f2product = bool(two[0])

1610 else: # getattrof(self, _f2product=None)

1611 t = _xkwds_get(self.__dict__, _f2product=None)

1612 return t

1613

1614 @Property

1615 def _fprs(self):

1616 '''(INTERNAL) Get and cache this instance' precision

1617 running sum (C{float} or C{int}), ignoring C{residual}.

1618

1619 @note: The precision running C{fsum} after a C{//=} or

1620 C{//} C{floor} division is C{int} in Python 3+.

1621 '''

1622 s, _ = self._fprs2

1623 return s # ._fprs2.fsum

1624

1625 @_fprs.setter_ # PYCHOK setter_UNDERscore!

1626 def _fprs(self, s):

1627 '''(INTERNAL) Replace the C{_fprs} value.

1628 '''

1629 return s

1630

1631 @Property

1632 def _fprs2(self):

1633 '''(INTERNAL) Get and cache this instance' precision

1634 running sum and residual (L{Fsum2Tuple}).

1635 '''

1636 ps = self._ps

1637 n = len(ps)

1638 try:

1639 if n > 2:

1640 s = _psum(ps, **self._isfine)

1641 if not _isfinite(s):

1642 ps[:] = s, # collapse ps

1643 return Fsum2Tuple(s, _NONFINITEr)

1644 n = len(ps)

1645# Fsum._ps_max = max(Fsum._ps_max, n)

1646 if n > 2:

1647 r = self._ps_1sum(s)

1648 return Fsum2Tuple(*_s_r2(s, r))

1649 if n > 1: # len(ps) == 2

1650 s, r = _s_r2(*_2sum(*ps, **self._isfine))

1651 ps[:] = (r, s) if r else (s,)

1652 elif ps: # len(ps) == 1

1653 s = ps[0]

1654 r = INT0 if _isfinite(s) else _NONFINITEr

1655 else: # len(ps) == 0

1656 s = _0_0

1657 r = INT0 if _isfinite(s) else _NONFINITEr

1658 ps[:] = s,

1659 except (OverflowError, ValueError) as X:

1660 op = _fset_op_ # INF, NAN, NINF

1661 ps[:] = sum(ps), # collapse ps

1662 s = self._nonfiniteX(X, op, ps[0])

1663 r = _NONFINITEr

1664 # assert self._ps is ps

1665 return Fsum2Tuple(s, r)

1666

1667 @_fprs2.setter_ # PYCHOK setter_UNDERscore!

1668 def _fprs2(self, s_r):

1669 '''(INTERNAL) Replace the C{_fprs2} value.

1670 '''

1671 return Fsum2Tuple(s_r)

1672

1673 def fset_(self, *xs):

1674 '''Apply C{B{self}.partials = Fsum(*B{xs}).partials}.

1675

1676 @arg xs: Optional, new values (each C{scalar} or an L{Fsum}

1677 or L{Fsum2Tuple} instance), all positional.

1678

1679 @return: This instance, replaced (C{Fsum}).

1680

1681 @see: Method L{Fsum.fadd} for further details.

1682 '''

1683 f = (xs[0] if xs else _0_0) if len(xs) < 2 else \

1684 Fsum(*xs, nonfinites=self.nonfinites()) # self._Fsum_as(*xs)

1685 return self._fset(f, op=_fset_op_)

1686

1687 def _fset(self, other, n=0, up=True, **op):

1688 '''(INTERNAL) Overwrite this instance with an other or a C{scalar}.

1689 '''

1690 if other is self:

1691 pass # from ._fmul, ._ftruediv and ._pow_0_1

1692 elif _isFsum_2Tuple(other):

1693 if op: # and not self.nonfinitesOK:

1694 self._finite(other._fprs, **op)

1695 self._ps[:] = other._ps

1696 self._n = n or other._n

1697 if up: # use or zap the C{Property_RO} values

1698 Fsum._fint2._update_from(self, other)

1699 Fsum._fprs ._update_from(self, other)

1700 Fsum._fprs2._update_from(self, other)

1701 elif isscalar(other):

1702 s = float(self._finite(other, **op)) if op else other

1703 self._ps[:] = s,

1704 self._n = n or 1

1705 if up: # Property _fint2, _fprs and _fprs2 all have

1706 # @.setter_underscore and NOT @.setter because the

1707 # latter's _fset zaps the value set by @.setter

1708 self._fint2 = s

1709 self._fprs = s

1710 self._fprs2 = s, INT0

1711 # assert self._fprs is s

1712 else:

1713 op = _xkwds_get1(op, op=_fset_op_)

1714 raise self._Error(op, other, _TypeError)

1715 return self

1716

1717 def fsub(self, xs=()):

1718 '''Subtract an iterable's items from this instance.

1719

1720 @see: Method L{Fsum.fadd} for further details.

1721 '''

1722 return self._facc_neg(xs)

1723

1724 def fsub_(self, *xs):

1725 '''Subtract all positional items from this instance.

1726

1727 @see: Method L{Fsum.fadd_} for further details.

1728 '''

1729 return self._fsub(xs[0], _sub_op_) if len(xs) == 1 else \

1730 self._facc_neg(xs) # origin=1?

1731

1732 def _fsub(self, other, op):

1733 '''(INTERNAL) Apply C{B{self} -= B{other}}.

1734 '''

1735 if _isFsum_2Tuple(other):

1736 if other is self: # or other._fprs2 == self._fprs2:

1737 self._fset(_0_0, n=len(self) * 2)

1738 elif other._ps:

1739 self._facc_scalar(other._ps_neg)

1740 elif self._scalar(other, op):

1741 self._facc_scalar_(-other)

1742 return self

1743

1744 def fsum(self, xs=()):

1745 '''Add an iterable's items, summate and return the current

1746 precision running sum.

1747

1748 @arg xs: Iterable of items to add (each item C{scalar},

1749 an L{Fsum} or L{Fsum2Tuple}).

1750

1751 @return: Precision running sum (C{float} or C{int}).

1752

1753 @see: Method L{Fsum.fadd}.

1754

1755 @note: Accumulation can continue after summation.

1756 '''

1757 return self._facc(xs)._fprs

1758

1759 def fsum_(self, *xs):

1760 '''Add any positional items, summate and return the current

1761 precision running sum.

1762

1763 @arg xs: Items to add (each C{scalar}, an L{Fsum} or

1764 L{Fsum2Tuple}), all positional.

1765

1766 @return: Precision running sum (C{float} or C{int}).

1767

1768 @see: Methods L{Fsum.fsum}, L{Fsum.Fsum_} and L{Fsum.fsumf_}.

1769 '''

1770 return self._facc_args(xs)._fprs

1771

1772 def Fsum_(self, *xs, **name):

1773 '''Like method L{Fsum.fsum_} but returning a named L{Fsum}.

1774

1775 @kwarg name: Optional name (C{str}).

1776

1777 @return: Copy of this updated instance (L{Fsum}).

1778 '''

1779 return self._facc_args(xs)._copy_2(self.Fsum_, **name)

1780

1781 def Fsum2Tuple_(self, *xs, **name):

1782 '''Like method L{Fsum.fsum_} but returning a named L{Fsum2Tuple}.

1783

1784 @kwarg name: Optional name (C{str}).

1785

1786 @return: Precision running sum (L{Fsum2Tuple}).

1787 '''

1788 return Fsum2Tuple(self._facc_args(xs)._nfprs2, **name)

1789

1790 @property_RO

1791 def _Fsum(self): # like L{Fsum2Tuple._Fsum}, in .fstats

1792 return self # NOT @Property_RO, see .copy and ._copy_2

1793

1794 def _Fsum_as(self, *xs, **name_f2product_nonfinites_RESIDUAL):

1795 '''(INTERNAL) Return an C{Fsum} with this C{Fsum}'s C{.f2product},

1796 C{.nonfinites} and C{.RESIDUAL} setting, optionally

1797 overridden with C{name_f2product_nonfinites_RESIDUAL} and

1798 with any C{xs} accumulated.

1799 '''

1800 kwds = _xkwds_not(None, Fsum._RESIDUAL, f2product =self.f2product(),

1801 nonfinites=self.nonfinites(),

1802 RESIDUAL =self.RESIDUAL())

1803 if name_f2product_nonfinites_RESIDUAL: # overwrites

1804 kwds.update(name_f2product_nonfinites_RESIDUAL)

1805 f = Fsum(**kwds)

1806 # assert all(v == self.__dict__[n] for n, v in f.__dict__.items())

1807 return f._fset(xs[0], op=_fset_op_) if len(xs) == 1 else (

1808 f._facc(xs, up=False) if xs else f)

1809

1810 def fsum2(self, xs=(), **name):

1811 '''Add an iterable's items, summate and return the

1812 current precision running sum I{and} the C{residual}.

1813

1814 @arg xs: Iterable of items to add (each item C{scalar},

1815 an L{Fsum} or L{Fsum2Tuple}).

1816 @kwarg name: Optional C{B{name}=NN} (C{str}).

1817

1818 @return: L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} the

1819 current precision running sum and C{residual}, the

1820 (precision) sum of the remaining C{partials}. The

1821 C{residual is INT0} if the C{fsum} is considered

1822 to be I{exact}.

1823

1824 @see: Methods L{Fsum.fint2}, L{Fsum.fsum} and L{Fsum.fsum2_}

1825 '''

1826 t = self._facc(xs)._fprs2

1827 return t.dup(name=name) if name else t

1828

1829 def fsum2_(self, *xs):

1830 '''Add any positional items, summate and return the current

1831 precision running sum and the I{differential}.

1832

1833 @arg xs: Values to add (each C{scalar}, an L{Fsum} or

1834 L{Fsum2Tuple}), all positional.

1835

1836 @return: 2Tuple C{(fsum, delta)} with the current, precision

1837 running C{fsum} like method L{Fsum.fsum} and C{delta},

1838 the difference with previous running C{fsum}, C{float}.

1839

1840 @see: Methods L{Fsum.fsum_} and L{Fsum.fsum}.

1841 '''

1842 return self._fsum2(xs, self._facc_args)

1843

1844 def _fsum2(self, xs, _facc, **facc_kwds):

1845 '''(INTERNAL) Helper for L{Fsum.fsum2_} and L{Fsum.fsum2f_}.

1846 '''

1847 p, q = self._fprs2

1848 if xs:

1849 s, r = _facc(xs, **facc_kwds)._fprs2

1850 if _isfinite(s): # _fsum(_1primed((s, -p, r, -q))

1851 d, r = _2sum(s - p, r - q, _isfine=_isOK)

1852 r, _ = _s_r2(d, r)

1853 return s, (r if _isfinite(r) else _NONFINITEr)

1854 else:

1855 return p, _0_0

1856

1857 def fsumf_(self, *xs):

1858 '''Like method L{Fsum.fsum_} iff I{all} C{B{xs}}, each I{known to be}

1859 C{scalar}, an L{Fsum} or L{Fsum2Tuple}.

1860 '''

1861 return self._facc_scalarf(xs, which=self.fsumf_)._fprs # origin=1?

1862

1863 def Fsumf_(self, *xs):

1864 '''Like method L{Fsum.Fsum_} iff I{all} C{B{xs}}, each I{known to be}

1865 C{scalar}, an L{Fsum} or L{Fsum2Tuple}.

1866 '''

1867 return self._facc_scalarf(xs, which=self.Fsumf_)._copy_2(self.Fsumf_) # origin=1?

1868

1869 def fsum2f_(self, *xs):

1870 '''Like method L{Fsum.fsum2_} iff I{all} C{B{xs}}, each I{known to be}

1871 C{scalar}, an L{Fsum} or L{Fsum2Tuple}.

1872 '''

1873 return self._fsum2(xs, self._facc_scalarf, which=self.fsum2f_) # origin=1?

1874

1875# ftruediv = __itruediv__ # for naming consistency?

1876

1877 def _ftruediv(self, other, op, **raiser_RESIDUAL):

1878 '''(INTERNAL) Apply C{B{self} /= B{other}}.

1879 '''

1880 n = _1_0

1881 if _isFsum_2Tuple(other):

1882 if other is self or self == other:

1883 return self._fset(n, n=len(self))

1884 d, r = other._fprs2

1885 if r:

1886 R = self._raiser(r, d, **raiser_RESIDUAL)

1887 if R:

1888 raise self._ResidualError(op, other, r, **R)

1889 d, n = other.as_integer_ratio()

1890 else:

1891 d = self._scalar(other, op)

1892 try:

1893 s = n / d

1894 except Exception as X:

1895 raise self._ErrorX(X, op, other)

1896 f = self._mul_scalar(s, _mul_op_) # handles 0, INF, NAN

1897 return self._fset(f)

1898

1899 @property_RO

1900 def imag(self):

1901 '''Get the C{imaginary} part of this instance (C{0.0}, always).

1902

1903 @see: Property L{Fsum.real}.

1904 '''

1905 return _0_0

1906

1907 def int_float(self, **raiser_RESIDUAL):

1908 '''Return this instance' current running sum as C{int} or C{float}.

1909

1910 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore

1911 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}

1912 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

1913

1914 @return: This C{int} sum if this instance C{is_integer} and

1915 I{finite}, otherwise the C{float} sum if the residual

1916 is zero or not significant.

1917

1918 @raise ResidualError: Non-zero, significant residual or invalid

1919 B{C{RESIDUAL}}.

1920

1921 @see: Methods L{Fsum.fint}, L{Fsum.fint2}, L{Fsum.is_integer},

1922 L{Fsum.RESIDUAL} and property L{Fsum.as_iscalar}.

1923 '''

1924 s, r = self._fint2

1925 if r:

1926 s, r = self._fprs2

1927 if r: # PYCHOK no cover

1928 R = self._raiser(r, s, **raiser_RESIDUAL)

1929 if R:

1930 t = _stresidual(_non_zero_, r, **R)

1931 raise ResidualError(int_float=s, txt=t)

1932 s = float(s)

1933 return s

1934

1935 def is_exact(self):

1936 '''Is this instance' running C{fsum} considered to be exact?

1937 (C{bool}), C{True} only if the C{residual is }L{INT0}.

1938 '''

1939 return self.residual is INT0

1940

1941 def is_finite(self): # in .constants

1942 '''Is this instance C{finite}? (C{bool}).

1943

1944 @see: Function L{isfinite<pygeodesy.isfinite>}.

1945 '''

1946 return _isfinite(sum(self._ps)) # == sum(self)

1947

1948 def is_integer(self):

1949 '''Is this instance' running sum C{integer}? (C{bool}).

1950

1951 @see: Methods L{Fsum.fint}, L{Fsum.fint2} and L{Fsum.is_scalar}.

1952 '''

1953 s, r = self._fint2

1954 return False if r else (_isfinite(s) and isint(s))

1955

1956 def is_math_fma(self):

1957 '''Is accurate L{f2product} multiplication based on Python's C{math.fma}?

1958

1959 @return: C{True} if accurate multiplication uses C{math.fma}, C{False}

1960 an C{fma} implementation as C{math.fma} or C{None}, a previous

1961 C{PyGeodesy} implementation.

1962 '''

1963 return (_2split3s is _passarg) or (False if _2n_d is None else None)

1964

1965 def is_math_fsum(self):

1966 '''Are the summation functions L{fsum}, L{fsum_}, L{fsumf_}, L{fsum1},

1967 L{fsum1_} and L{fsum1f_} based on Python's C{math.fsum}?

1968

1969 @return: C{True} if summation functions use C{math.fsum}, C{False}

1970 otherwise.

1971 '''

1972 return _sum is _fsum # _fsum.__module__ is fabs.__module__

1973

1974 def is_scalar(self, **raiser_RESIDUAL):

1975 '''Is this instance' running sum C{scalar} with C{0} residual or with

1976 a residual I{ratio} not exceeding the RESIDUAL threshold?

1977

1978 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore

1979 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}

1980 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

1981

1982 @return: C{True} if this instance' residual is C{0} or C{insignificant},

1983 i.e. its residual C{ratio} doesn't exceed the L{RESIDUAL

1984 <Fsum.RESIDUAL>} threshold (C{bool}).

1985

1986 @raise ResidualError: Non-zero, significant residual or invalid

1987 B{C{RESIDUAL}}.

1988

1989 @see: Methods L{Fsum.RESIDUAL} and L{Fsum.is_integer} and property

1990 L{Fsum.as_iscalar}.

1991 '''

1992 s, r = self._fprs2

1993 return False if r and self._raiser(r, s, **raiser_RESIDUAL) else True

1994

1995 def _mul_Fsum(self, other, op):

1996 '''(INTERNAL) Return C{B{self} * B{other}} as L{Fsum} or C{0}.

1997 '''

1998 # assert _isFsum_2Tuple(other)

1999 if self._ps and other._ps:

2000 try:

2001 f = self._ps_mul(op, *other._ps) # NO .as_iscalar!

2002 except Exception as X:

2003 raise self._ErrorX(X, op, other)

2004 else:

2005 f = _0_0

2006 return f

2007

2008 def _mul_reduce(self, *others):

2009 '''(INTERNAL) Like fmath.fprod for I{non-finite} C{other}s.

2010 '''

2011 r = _1_0

2012 for f in others:

2013 r *= sum(f._ps) if _isFsum_2Tuple(f) else float(f)

2014 return r

2015

2016 def _mul_scalar(self, factor, op):

2017 '''(INTERNAL) Return C{B{self} * scalar B{factor}} as L{Fsum}, C{0.0} or C{self}.

2018 '''

2019 # assert isscalar(factor)

2020 if self._ps and self._finite(factor, op=op):

2021 f = self if factor == _1_0 else (

2022 self._neg if factor == _N_1_0 else

2023 self._ps_mul(op, factor).as_iscalar)

2024 else:

2025 f = _0_0

2026 return f

2027

2028# @property_RO

2029# def _n_d(self):

2030# n, d = self.as_integer_ratio()

2031# return n / d

2032

2033 @property_RO

2034 def _neg(self):

2035 '''(INTERNAL) Return C{Fsum(-self)} or scalar C{NEG0}.

2036 '''

2037 return _Psum(self._ps_neg) if self._ps else NEG0

2038

2039 @property_RO

2040 def _nfprs2(self):

2041 '''(INTERNAL) Handle I{non-finite} C{_fprs2}.

2042 '''

2043 try: # to handle nonfiniterrors, etc.

2044 t = self._fprs2

2045 except (OverflowError, ValueError):

2046 t = Fsum2Tuple(sum(self._ps), _NONFINITEr)

2047 return t

2048

2049 def nonfinites(self, *OK):

2050 '''Handle I{non-finite} C{float}s as C{inf}, C{INF}, C{NINF}, C{nan}

2051 and C{NAN} for this L{Fsum} or throw C{OverflowError} respectively

2052 C{ValueError} exceptions, overriding the L{nonfiniterrors} default.

2053

2054 @arg OK: If omitted, leave the override unchanged, if C{True},

2055 I{non-finites} are C{OK}, if C{False} throw exceptions

2056 or if C{None} remove the override (C{bool} or C{None}).

2057

2058 @return: The previous setting (C{bool} or C{None} if not set).

2059

2060 @see: Function L{nonfiniterrors<fsums.nonfiniterrors>}.

2061

2062 @note: Use property L{nonfinitesOK<Fsum.nonfinitesOK>} to determine

2063 whether I{non-finites} are C{OK} for this L{Fsum} and by the

2064 L{nonfiniterrors} default.

2065 '''

2066 _ks = Fsum._nonfinites_isfine_kwds

2067 if OK: # delattrof(self, _isfine=None)

2068 k = _xkwds_pop(self.__dict__, _isfine=None)

2069 if OK[0] is not None:

2070 self._isfine = _ks[bool(OK[0])]

2071 self._update()

2072 else: # getattrof(self, _isfine=None)

2073 k = _xkwds_get(self.__dict__, _isfine=None)

2074 # dict(map(reversed, _ks.items())).get(k, None)

2075 # raises a TypeError: unhashable type: 'dict'

2076 return True if k is _ks[True] else (

2077 False if k is _ks[False] else None)

2078

2079 _nonfinites_isfine_kwds = {True: dict(_isfine=_isOK),

2080 False: dict(_isfine=_isfinite)}

2081

2082 @property_RO

2083 def nonfinitesOK(self):

2084 '''Are I{non-finites} C{OK} for this L{Fsum} or by default? (C{bool}).

2085 '''

2086# nf = self.nonfinites()

2087# if nf is None:

2088# nf = not nonfiniterrors()

2089 return _isOK_or_finite(INF, **self._isfine)

2090

2091 def _nonfiniteX(self, X, op, f, nonfinites=None, raiser=None):

2092 '''(INTERNAL) Handle a I{non-finite} exception.

2093 '''

2094 if nonfinites is None:

2095 nonfinites = _isOK_or_finite(f, **self._isfine) if raiser is None else (not raiser)

2096 if not nonfinites:

2097 raise self._ErrorX(X, op, f)

2098 return f

2099

2100 def _optionals(self, f2product=None, nonfinites=None, **name_RESIDUAL):

2101 '''(INTERNAL) Re/set options from keyword arguments.

2102 '''

2103 if f2product is not None:

2104 self.f2product(f2product)

2105 if nonfinites is not None:

2106 self.nonfinites(nonfinites)

2107 if name_RESIDUAL: # MUST be last

2108 n, kwds = _name2__(**name_RESIDUAL)

2109 if kwds:

2110 R = Fsum._RESIDUAL

2111 t = _threshold(R, **kwds)

2112 if t != R:

2113 self._RESIDUAL = t

2114 if n:

2115 self.name = n # self.rename(n)

2116

2117 def _1_Over(self, x, op, **raiser_RESIDUAL): # vs _1_over

2118 '''(INTERNAL) Return C{Fsum(1) / B{x}}.

2119 '''

2120 return self._Fsum_as(_1_0)._ftruediv(x, op, **raiser_RESIDUAL)

2121

2122 @property_RO

2123 def partials(self):

2124 '''Get this instance' current, partial sums (C{tuple} of C{float}s).

2125 '''

2126 return tuple(self._ps)

2127

2128 def pow(self, x, *mod, **raiser_RESIDUAL):

2129 '''Return C{B{self}**B{x}} as L{Fsum}.

2130

2131 @arg x: The exponent (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

2132 @arg mod: Optional modulus (C{int} or C{None}) for the 3-argument

2133 C{pow(B{self}, B{other}, B{mod})} version.

2134 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore

2135 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}

2136 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

2137

2138 @return: The C{pow(self, B{x})} or C{pow(self, B{x}, *B{mod})}

2139 result (L{Fsum}).

2140

2141 @raise ResidualError: Non-zero, significant residual or invalid

2142 B{C{RESIDUAL}}.

2143

2144 @note: If B{C{mod}} is given and C{None}, the result will be an

2145 C{integer} L{Fsum} provided this instance C{is_integer}

2146 or set to C{integer} by an L{Fsum.fint} call.

2147

2148 @see: Methods L{Fsum.__ipow__}, L{Fsum.fint}, L{Fsum.is_integer}

2149 and L{Fsum.root}.

2150 '''

2151 f = self._copy_2(self.pow)

2152 return f._fpow(x, _pow_op_, *mod, **raiser_RESIDUAL) # f = pow(f, x, *mod)

2153

2154 def _pow(self, other, unused, op, **raiser_RESIDUAL):

2155 '''Return C{B{self} ** B{other}}.

2156 '''

2157 if _isFsum_2Tuple(other):

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

2159 elif self._scalar(other, op):

2160 x = self._finite(other, op=op)

2161 f = self._pow_scalar(x, other, op, **raiser_RESIDUAL)

2162 else:

2163 f = self._pow_0_1(0, other)

2164 return f

2165

2166 def _pow_0_1(self, x, other):

2167 '''(INTERNAL) Return B{C{self}**1} or C{B{self}**0 == 1.0}.

2168 '''

2169 return self if x else (1 if isint(other) and self.is_integer() else _1_0)

2170

2171 def _pow_2_3(self, b, x, other, op, *mod, **raiser_RESIDUAL):

2172 '''(INTERNAL) 2-arg C{pow(B{b}, scalar B{x})} and 3-arg C{pow(B{b},

2173 B{x}, int B{mod} or C{None})}, embellishing errors.

2174 '''

2175

2176 if mod: # b, x, mod all C{int}, unless C{mod} is C{None}

2177 m = mod[0]

2178 # assert _isFsum_2Tuple(b)

2179

2180 def _s(s, r):

2181 R = self._raiser(r, s, **raiser_RESIDUAL)

2182 if R:

2183 raise self._ResidualError(op, other, r, mod=m, **R)

2184 return s

2185

2186 b = _s(*(b._fprs2 if m is None else b._fint2))

2187 x = _s(*_2tuple2(x))

2188

2189 try:

2190 # 0**INF == 0.0, 1**INF == 1.0, -1**2.3 == -(1**2.3)

2191 s = pow(b, x, *mod)

2192 if iscomplex(s):

2193 # neg**frac == complex in Python 3+, but ValueError in 2-

2194 raise ValueError(_strcomplex(s, b, x, *mod))

2195 _ = _2finite(s, **self._isfine) # ignore float

2196 return s

2197 except Exception as X:

2198 raise self._ErrorX(X, op, other, *mod)

2199

2200 def _pow_Fsum(self, other, op, **raiser_RESIDUAL):

2201 '''(INTERNAL) Return C{B{self} **= B{other}} for C{_isFsum_2Tuple(other)}.

2202 '''

2203 # assert _isFsum_2Tuple(other)

2204 x, r = other._fprs2

2205 f = self._pow_scalar(x, other, op, **raiser_RESIDUAL)

2206 if f and r:

2207 f *= self._pow_scalar(r, other, op, **raiser_RESIDUAL)

2208 return f

2209

2210 def _pow_int(self, x, other, op, **raiser_RESIDUAL):

2211 '''(INTERNAL) Return C{B{self} **= B{x}} for C{int B{x} >= 0}.

2212 '''

2213 # assert isint(x) and x >= 0

2214 ps = self._ps

2215 if len(ps) > 1:

2216 _mul_Fsum = Fsum._mul_Fsum

2217 if x > 4:

2218 p = self

2219 f = self if (x & 1) else self._Fsum_as(_1_0)

2220 m = x >> 1 # // 2

2221 while m:

2222 p = _mul_Fsum(p, p, op) # p **= 2

2223 if (m & 1):

2224 f = _mul_Fsum(f, p, op) # f *= p

2225 m >>= 1 # //= 2

2226 elif x > 1: # self**2, 3, or 4

2227 f = _mul_Fsum(self, self, op)

2228 if x > 2: # self**3 or 4

2229 p = self if x < 4 else f

2230 f = _mul_Fsum(f, p, op)

2231 else: # self**1 or self**0 == 1 or _1_0

2232 f = self._pow_0_1(x, other)

2233 elif ps: # self._ps[0]**x

2234 f = self._pow_2_3(ps[0], x, other, op, **raiser_RESIDUAL)

2235 else: # PYCHOK no cover

2236 # 0**pos_int == 0, but 0**0 == 1

2237 f = 0 if x else 1

2238 return f

2239

2240 def _pow_scalar(self, x, other, op, **raiser_RESIDUAL):

2241 '''(INTERNAL) Return C{self**B{x}} for C{scalar B{x}}.

2242 '''

2243 s, r = self._fprs2

2244 if r:

2245 # assert s != 0

2246 if isint(x, both=True): # self**int

2247 x = int(x)

2248 y = abs(x)

2249 if y > 1:

2250 f = self._pow_int(y, other, op, **raiser_RESIDUAL)

2251 if x > 0: # i.e. > 1

2252 return f # Fsum or scalar

2253 # assert x < 0 # i.e. < -1

2254 if _isFsum(f):

2255 s, r = f._fprs2

2256 if r:

2257 return self._1_Over(f, op, **raiser_RESIDUAL)

2258 else: # scalar

2259 s = f

2260 # use s**(-1) to get the CPython

2261 # float_pow error iff s is zero

2262 x = -1

2263 elif x < 0: # self**(-1)

2264 return self._1_Over(self, op, **raiser_RESIDUAL) # 1 / self

2265 else: # self**1 or self**0

2266 return self._pow_0_1(x, other) # self, 1 or 1.0

2267 else: # self**fractional

2268 R = self._raiser(r, s, **raiser_RESIDUAL)

2269 if R:

2270 raise self._ResidualError(op, other, r, **R)

2271 n, d = self.as_integer_ratio()

2272 if abs(n) > abs(d):

2273 n, d, x = d, n, (-x)

2274 s = n / d

2275 # assert isscalar(s) and isscalar(x)

2276 return self._pow_2_3(s, x, other, op, **raiser_RESIDUAL)

2277

2278 def _ps_acc(self, ps, xs, up=True, **unused):

2279 '''(INTERNAL) Accumulate C{xs} known scalars into list C{ps}.

2280 '''

2281 n = 0

2282 _2s = _2sum

2283 _fi = self._isfine

2284 for x in (tuple(xs) if xs is ps else xs):

2285 # assert isscalar(x) and _isOK_or_finite(x, **self._isfine)

2286 if x:

2287 i = 0

2288 for p in ps:

2289 x, p = _2s(x, p, **_fi)

2290 if p:

2291 ps[i] = p

2292 i += 1

2293 ps[i:] = (x,) if x else ()

2294 n += 1

2295 if n:

2296 self._n += n

2297 # Fsum._ps_max = max(Fsum._ps_max, len(ps))

2298 if up:

2299 self._update()

2300# x = sum(ps)

2301# if not _isOK_or_finite(x, **fi):

2302# ps[:] = x, # collapse ps

2303 return ps

2304

2305 def _ps_mul(self, op, *factors):

2306 '''(INTERNAL) Multiply this instance' C{partials} with

2307 each scalar C{factor} and accumulate into an C{Fsum}.

2308 '''

2309 def _psfs(ps, fs, _isfine=_isfinite):

2310 if len(ps) < len(fs):

2311 ps, fs = fs, ps

2312 if self._f2product:

2313 fs, p = _2split3s(fs), fs

2314 if len(ps) > 1 and fs is not p:

2315 fs = tuple(fs) # several ps

2316 _pfs = _2products

2317 else:

2318 def _pfs(p, fs):

2319 return (p * f for f in fs)

2320

2321 for p in ps:

2322 for x in _pfs(p, fs):

2323 yield x if _isfine(x) else _nfError(x)

2324

2325 xs = _psfs(self._ps, factors, **self._isfine)

2326 f = _Psum(self._ps_acc([], xs, up=False), name=op)

2327 return f

2328

2329 @property_RO

2330 def _ps_neg(self):

2331 '''(INTERNAL) Yield the partials, I{negated}.

2332 '''

2333 for p in self._ps:

2334 yield -p

2335

2336 def _ps_other(self, op, other):

2337 '''(INTERNAL) Yield C{other} as C{scalar}s.

2338 '''

2339 if _isFsum_2Tuple(other):

2340 for p in other._ps:

2341 yield p

2342 else:

2343 yield self._scalar(other, op)

2344

2345 def _ps_1sum(self, *less):

2346 '''(INTERNAL) Return the partials sum, 1-primed C{less} some scalars.

2347 '''

2348 def _1psls(ps, ls):

2349 yield _1_0

2350 for p in ps:

2351 yield p

2352 for p in ls:

2353 yield -p

2354 yield _N_1_0

2355

2356 return _fsum(_1psls(self._ps, less))

2357

2358 def _raiser(self, r, s, raiser=True, **RESIDUAL):

2359 '''(INTERNAL) Does ratio C{r / s} exceed the RESIDUAL threshold

2360 I{and} is residual C{r} I{non-zero} or I{significant} (for a

2361 negative respectively positive C{RESIDUAL} threshold)?

2362 '''

2363 if r and raiser:

2364 t = self._RESIDUAL

2365 if RESIDUAL:

2366 t = _threshold(t, **RESIDUAL)

2367 if t < 0 or (s + r) != s:

2368 q = (r / s) if s else s # == 0.

2369 if fabs(q) > fabs(t):

2370 return dict(ratio=q, R=t)

2371 return {}

2372

2373 rdiv = __rtruediv__

2374

2375 @property_RO

2376 def real(self):

2377 '''Get the C{real} part of this instance (C{float}).

2378

2379 @see: Methods L{Fsum.__float__} and L{Fsum.fsum}

2380 and properties L{Fsum.ceil}, L{Fsum.floor},

2381 L{Fsum.imag} and L{Fsum.residual}.

2382 '''

2383 return float(self)

2384

2385 @property_RO

2386 def residual(self):

2387 '''Get this instance' residual or residue (C{float} or C{int}):

2388 the C{sum(partials)} less the precision running sum C{fsum}.

2389

2390 @note: The C{residual is INT0} iff the precision running

2391 C{fsum} is considered to be I{exact}.

2392

2393 @see: Methods L{Fsum.fsum}, L{Fsum.fsum2} and L{Fsum.is_exact}.

2394 '''

2395 return self._fprs2.residual

2396

2397 def RESIDUAL(self, *threshold):

2398 '''Get and set this instance' I{ratio} for raising L{ResidualError}s,

2399 overriding the default from env variable C{PYGEODESY_FSUM_RESIDUAL}.

2400

2401 @arg threshold: If C{scalar}, the I{ratio} to exceed for raising

2402 L{ResidualError}s in division and exponention, if

2403 C{None}, restore the default set with env variable

2404 C{PYGEODESY_FSUM_RESIDUAL} or if omitted, keep the

2405 current setting.

2406

2407 @return: The previous C{RESIDUAL} setting (C{float}), default C{0.0}.

2408

2409 @raise ResidualError: Invalid B{C{threshold}}.

2410

2411 @note: L{ResidualError}s may be thrown if (1) the non-zero I{ratio}

2412 C{residual / fsum} exceeds the given B{C{threshold}} and (2)

2413 the C{residual} is non-zero and (3) is I{significant} vs the

2414 C{fsum}, i.e. C{(fsum + residual) != fsum} and (4) optional

2415 keyword argument C{raiser=False} is missing. Specify a

2416 negative B{C{threshold}} for only non-zero C{residual}

2417 testing without the I{significant} case.

2418 '''

2419 r = self._RESIDUAL

2420 if threshold:

2421 t = threshold[0]

2422 self._RESIDUAL = Fsum._RESIDUAL if t is None else ( # for ...

2423 (_0_0 if t else _1_0) if isbool(t) else

2424 _threshold(t)) # ... backward compatibility

2425 return r

2426

2427 def _ResidualError(self, op, other, residual, **mod_R):

2428 '''(INTERNAL) Non-zero B{C{residual}} etc.

2429 '''

2430 def _p(mod=None, R=0, **unused): # ratio=0

2431 return (_non_zero_ if R < 0 else _significant_) \

2432 if mod is None else _integer_

2433

2434 t = _stresidual(_p(**mod_R), residual, **mod_R)

2435 return self._Error(op, other, ResidualError, txt=t)

2436

2437 def root(self, root, **raiser_RESIDUAL):

2438 '''Return C{B{self}**(1 / B{root})} as L{Fsum}.

2439

2440 @arg root: Non-zero order (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

2441 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore any

2442 L{ResidualError}s (C{bool}) or C{B{RESIDUAL}=scalar}

2443 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

2444

2445 @return: The C{self ** (1 / B{root})} result (L{Fsum}).

2446

2447 @raise ResidualError: Non-zero, significant residual or invalid

2448 B{C{RESIDUAL}}.

2449

2450 @see: Method L{Fsum.pow}.

2451 '''

2452 x = self._1_Over(root, _truediv_op_, **raiser_RESIDUAL)

2453 f = self._copy_2(self.root)

2454 return f._fpow(x, f.name, **raiser_RESIDUAL) # == pow(f, x)

2455

2456 def _scalar(self, other, op, **txt):

2457 '''(INTERNAL) Return scalar C{other} or throw a C{TypeError}.

2458 '''

2459 if isscalar(other):

2460 return other

2461 raise self._Error(op, other, _TypeError, **txt) # _invalid_

2462

2463 def signOf(self, res=True):

2464 '''Determine the sign of this instance.

2465

2466 @kwarg res: If C{True}, consider the residual,

2467 otherwise ignore the latter (C{bool}).

2468

2469 @return: The sign (C{int}, -1, 0 or +1).

2470 '''

2471 s, r = self._nfprs2

2472 r = (-r) if res else 0

2473 return _signOf(s, r)

2474

2475 def toRepr(self, **lenc_prec_sep_fmt): # PYCHOK signature

2476 '''Return this C{Fsum} instance as representation.

2477

2478 @kwarg lenc_prec_sep_fmt: Optional keyword arguments

2479 for method L{Fsum.toStr}.

2480

2481 @return: This instance (C{repr}).

2482 '''

2483 return Fmt.repr_at(self, self.toStr(**lenc_prec_sep_fmt))

2484

2485 def toStr(self, lenc=True, **prec_sep_fmt): # PYCHOK signature

2486 '''Return this C{Fsum} instance as string.

2487

2488 @kwarg lenc: If C{True}, include the current C{[len]} of this

2489 L{Fsum} enclosed in I{[brackets]} (C{bool}).

2490 @kwarg prec_sep_fmt: Optional keyword arguments for method

2491 L{Fsum2Tuple.toStr}.

2492

2493 @return: This instance (C{str}).

2494 '''

2495 p = self.classname

2496 if lenc:

2497 p = Fmt.SQUARE(p, len(self))

2498 n = _enquote(self.name, white=_UNDER_)

2499 t = self._nfprs2.toStr(**prec_sep_fmt)

2500 return NN(p, _SPACE_, n, t)

2501

2502 def _truediv(self, other, op, **raiser_RESIDUAL):

2503 '''(INTERNAL) Return C{B{self} / B{other}} as an L{Fsum}.

2504 '''

2505 f = self._copy_2(self.__truediv__)

2506 return f._ftruediv(other, op, **raiser_RESIDUAL)

2507

2508 def _update(self, updated=True): # see ._fset

2509 '''(INTERNAL) Zap all cached C{Property_RO} values.

2510 '''

2511 if updated:

2512 _pop = self.__dict__.pop

2513 for p in _ROs:

2514 _ = _pop(p, None)

2515# Fsum._fint2._update(self)

2516# Fsum._fprs ._update(self)

2517# Fsum._fprs2._update(self)

2518 return self # for .fset_

2519

2520_ROs = _allPropertiesOf_n(3, Fsum, Property_RO) # PYCHOK see Fsum._update

2521

2522 if _NONFINITES == _std_: # PYCHOK no cover

2523 _ = nonfiniterrors(False)

2524

2525

2526 def _Float_Int(arg, **name_Error):

2527 '''(INTERNAL) L{DivMod2Tuple}, L{Fsum2Tuple} Unit.

2528 '''

2529 U = Int if isint(arg) else Float

2530 return U(arg, **name_Error)

2531

2532

2533 class DivMod2Tuple(_NamedTuple):

2534 '''2-Tuple C{(div, mod)} with the quotient C{div} and remainder

2535 C{mod} results of a C{divmod} operation.

2536

2537 @note: Quotient C{div} an C{int} in Python 3+ but a C{float}

2538 in Python 2-. Remainder C{mod} an L{Fsum} instance.

2539 '''

2540 _Names_ = ('div', 'mod')

2541 _Units_ = (_Float_Int, Fsum)

2542

2543

2544 class Fsum2Tuple(_NamedTuple): # in .fstats

2545 '''2-Tuple C{(fsum, residual)} with the precision running C{fsum}

2546 and the C{residual}, the sum of the remaining partials. Each

2547 item is C{float} or C{int}.

2548

2549 @note: If the C{residual is INT0}, the C{fsum} is considered

2550 to be I{exact}, see method L{Fsum2Tuple.is_exact}.

2551 '''

2552 _Names_ = ( Fsum.fsum.__name__, Fsum.residual.name)

2553 _Units_ = (_Float_Int, _Float_Int)

2554

2555 def __abs__(self): # in .fmath

2556 return self._Fsum.__abs__()

2557

2558 def __bool__(self): # PYCHOK Python 3+

2559 return bool(self._Fsum)

2560

2561 def __eq__(self, other):

2562 return self._other_op(other, self.__eq__)

2563

2564 def __float__(self):

2565 return self._Fsum.__float__()

2566

2567 def __ge__(self, other):

2568 return self._other_op(other, self.__ge__)

2569

2570 def __gt__(self, other):

2571 return self._other_op(other, self.__gt__)

2572

2573 def __le__(self, other):

2574 return self._other_op(other, self.__le__)

2575

2576 def __lt__(self, other):

2577 return self._other_op(other, self.__lt__)

2578

2579 def __int__(self):

2580 return self._Fsum.__int__()

2581

2582 def __ne__(self, other):

2583 return self._other_op(other, self.__ne__)

2584

2585 def __neg__(self):

2586 return self._Fsum.__neg__()

2587

2588 __nonzero__ = __bool__ # Python 2-

2589

2590 def __pos__(self):

2591 return self._Fsum.__pos__()

2592

2593 def as_integer_ratio(self):

2594 '''Return this instance as the ratio of 2 integers.

2595

2596 @see: Method L{Fsum.as_integer_ratio} for further details.

2597 '''

2598 return self._Fsum.as_integer_ratio()

2599

2600 @property_RO

2601 def _fint2(self):

2602 return self._Fsum._fint2

2603

2604 @property_RO

2605 def _fprs2(self):

2606 return self._Fsum._fprs2

2607

2608 @Property_RO

2609 def _Fsum(self): # this C{Fsum2Tuple} as L{Fsum}, in .fstats

2610 s, r = _s_r2(*self)

2611 ps = (r, s) if r else (s,)

2612 return _Psum(ps, name=self.name)

2613

2614 def Fsum_(self, *xs, **name_f2product_nonfinites_RESIDUAL):

2615 '''Return this C{Fsum2Tuple} as an L{Fsum} plus some C{xs}.

2616 '''

2617 return Fsum(self, *xs, **name_f2product_nonfinites_RESIDUAL)

2618

2619 def is_exact(self):

2620 '''Is this L{Fsum2Tuple} considered to be exact? (C{bool}).

2621 '''

2622 return self._Fsum.is_exact()

2623

2624 def is_finite(self): # in .constants

2625 '''Is this L{Fsum2Tuple} C{finite}? (C{bool}).

2626

2627 @see: Function L{isfinite<pygeodesy.isfinite>}.

2628 '''

2629 return self._Fsum.is_finite()

2630

2631 def is_integer(self):

2632 '''Is this L{Fsum2Tuple} C{integer}? (C{bool}).

2633 '''

2634 return self._Fsum.is_integer()

2635

2636 def _mul_scalar(self, other, op): # for Fsum._fmul

2637 return self._Fsum._mul_scalar(other, op)

2638

2639 @property_RO

2640 def _n(self):

2641 return self._Fsum._n

2642

2643 def _other_op(self, other, which):

2644 C, s = (tuple, self) if isinstance(other, tuple) else (Fsum, self._Fsum)

2645 return getattr(C, which.__name__)(s, other)

2646

2647 @property_RO

2648 def _ps(self):

2649 return self._Fsum._ps

2650

2651 @property_RO

2652 def _ps_neg(self):

2653 return self._Fsum._ps_neg

2654

2655 def signOf(self, **res):

2656 '''Like method L{Fsum.signOf}.

2657 '''

2658 return self._Fsum.signOf(**res)

2659

2660 def toStr(self, fmt=Fmt.g, **prec_sep): # PYCHOK signature

2661 '''Return this L{Fsum2Tuple} as string (C{str}).

2662

2663 @kwarg fmt: Optional C{float} format (C{letter}).

2664 @kwarg prec_sep: Optional keyword arguments for function

2665 L{fstr<streprs.fstr>}.

2666 '''

2667 return Fmt.PAREN(fstr(self, fmt=fmt, strepr=str, force=False, **prec_sep))

2668

2669_Fsum_2Tuple_types = Fsum, Fsum2Tuple # PYCHOK lines

2670

2671

2672 class ResidualError(_ValueError):

2673 '''Error raised for a division, power or root operation of

2674 an L{Fsum} instance with a C{residual} I{ratio} exceeding

2675 the L{RESIDUAL<Fsum.RESIDUAL>} threshold.

2676

2677 @see: Module L{pygeodesy.fsums} and method L{Fsum.RESIDUAL}.

2678 '''

2679 pass

2680

2681

2682 try:

2683 from math import fsum as _fsum # precision IEEE-754 sum, Python 2.6+

2684

2685 # make sure _fsum works as expected (XXX check

2686 # float.__getformat__('float')[:4] == 'IEEE'?)

2687 if _fsum((1, 1e101, 1, -1e101)) != 2: # PYCHOK no cover

2688 del _fsum # nope, remove _fsum ...

2689 raise ImportError() # ... use _fsum below

2690

2691 _sum = _fsum # in .elliptic

2692 except ImportError:

2693 _sum = sum # in .elliptic

2694

2695 def _fsum(xs):

2696 '''(INTERNAL) Precision summation, Python 2.5-.

2697 '''

2698 F = Fsum(name=_fsum.name, f2product=False, nonfinites=True)

2699 return float(F._facc(xs, up=False))

2700

2701

2702 def fsum(xs, nonfinites=None, **floats):

2703 '''Precision floating point summation from Python's C{math.fsum}.

2704

2705 @arg xs: Iterable of items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

2706 @kwarg nonfinites: Use C{B{nonfinites}=True} if I{non-finites} are C{OK}, if

2707 C{False} I{non-finites} raise an Overflow-/ValueError or if

2708 C{None}, L{nonfiniterrors} applies (C{bool} or C{None}).

2709 @kwarg floats: DEPRECATED keyword argument C{B{floats}=False} (C{bool}), use

2710 keyword argument C{B{nonfinites}=False} instead.

2711

2712 @return: Precision C{fsum} (C{float}).

2713

2714 @raise OverflowError: Infinite B{C{xs}} item or intermediate C{math.fsum} overflow.

2715

2716 @raise TypeError: Invalid B{C{xs}} item.

2717

2718 @raise ValueError: Invalid or C{NAN} B{C{xs}} item.

2719

2720 @see: Function L{nonfiniterrors}, class L{Fsum} and methods L{Fsum.nonfinites},

2721 L{Fsum.fsum}, L{Fsum.fadd} and L{Fsum.fadd_}.

2722 '''

2723 return _xsum(fsum, xs, nonfinites=nonfinites, **floats) if xs else _0_0

2724

2725

2726 def fsum_(*xs, **nonfinites):

2727 '''Precision floating point summation of all positional items.

2728

2729 @arg xs: Items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all positional.

2730 @kwarg nonfinites: Use C{B{nonfinites}=True} if I{non-finites} are C{OK} (C{bool}).

2731

2732 @see: Function L{fsum<fsums.fsum>} for further details.

2733 '''

2734 return _xsum(fsum_, xs, **nonfinites) if xs else _0_0 # origin=1?

2735

2736

2737 def fsumf_(*xs):

2738 '''Precision floating point summation of all positional items with I{non-finites} C{OK}.

2739

2740 @arg xs: Items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}),

2741 all positional.

2742

2743 @see: Function L{fsum_<fsums.fsum_>} for further details.

2744 '''

2745 return _xsum(fsumf_, xs, nonfinites=True) if xs else _0_0 # origin=1?

2746

2747

2748 def fsum1(xs, **nonfinites):

2749 '''Precision floating point summation, 1-primed.

2750

2751 @arg xs: Iterable of items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

2752 @kwarg nonfinites: Use C{B{nonfinites}=True} if I{non-finites} are C{OK} (C{bool}).

2753

2754 @see: Function L{fsum<fsums.fsum>} for further details.

2755 '''

2756 return _xsum(fsum1, xs, primed=1, **nonfinites) if xs else _0_0

2757

2758

2759 def fsum1_(*xs, **nonfinites):

2760 '''Precision floating point summation of all positional items, 1-primed.

2761

2762 @arg xs: Items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all positional.

2763 @kwarg nonfinites: Use C{B{nonfinites}=True} if I{non-finites} are C{OK} (C{bool}).

2764

2765 @see: Function L{fsum_<fsums.fsum_>} for further details.

2766 '''

2767 return _xsum(fsum1_, xs, primed=1, **nonfinites) if xs else _0_0 # origin=1?

2768

2769

2770 def fsum1f_(*xs):

2771 '''Precision floating point summation of all positional items, 1-primed and

2772 with I{non-finites} C{OK}.

2773

2774 @see: Function L{fsum_<fsums.fsum_>} for further details.

2775 '''

2776 return _xsum(fsum1f_, xs, nonfinites=True, primed=1) if xs else _0_0

2777

2778

2779 def _x_isfine(nfOK, **kwds): # get the C{_x} and C{_isfine} handlers.

2780 _x_kwds = dict(_x= (_passarg if nfOK else _2finite),

2781 _isfine=(_isOK if nfOK else _isfinite)) # PYCHOK kwds

2782 _x_kwds.update(kwds)

2783 return _x_kwds

2784

2785

2786 def _X_ps(X): # default C{_X} handler

2787 return X._ps # lambda X: X._ps

2788

2789

2790 def _xs(xs, _X=_X_ps, _x=float, _isfine=_isfinite, # defaults for Fsum._facc

2791 origin=0, which=None, **_Cdot):

2792 '''(INTERNAL) Yield each C{xs} item as 1 or more C{float}s.

2793 '''

2794 i, x = 0, xs

2795 try:

2796 for i, x in enumerate(_xiterable(xs)):

2797 if _isFsum_2Tuple(x):

2798 for p in _X(x):

2799 yield p if _isfine(p) else _nfError(p)

2800 else:

2801 f = _x(x)

2802 yield f if _isfine(f) else _nfError(f)

2803

2804 except (OverflowError, TypeError, ValueError) as X:

2805 t = _xsError(X, xs, i + origin, x)

2806 if which: # prefix invokation

2807 w = unstr(which, *xs, _ELLIPSIS=4, **_Cdot)

2808 t = _COMMASPACE_(w, t)

2809 raise _xError(X, t, txt=None)

2810

2811

2812 def _xsum(which, xs, nonfinites=None, primed=0, **floats): # origin=0

2813 '''(INTERNAL) Precision summation of C{xs} with conditions.

2814 '''

2815 if floats: # for backward compatibility

2816 nonfinites = _xkwds_get1(floats, floats=nonfinites)

2817 elif nonfinites is None:

2818 nonfinites = not nonfiniterrors()

2819 fs = _xs(xs, **_x_isfine(nonfinites, which=which)) # PYCHOK yield

2820 return _fsum(_1primed(fs) if primed else fs)

2821

2822

2823# delete all decorators, etc.

2824 del _allPropertiesOf_n, deprecated_method, deprecated_property_RO, \

2825 Property, Property_RO, property_RO, _ALL_LAZY, _F2PRODUCT, \

2826 MANT_DIG, _NONFINITES, _RESIDUAL_0_0, _getPYGEODESY, _std_

2827

2828 if __name__ == '__main__':

2829

2830 # usage: python3 -m pygeodesy.fsums

2831

2832 def _test(n):

2833 # copied from Hettinger, see L{Fsum} reference

2834 from pygeodesy import frandoms, printf

2835

2836 printf(_fsum.__name__, end=_COMMASPACE_)

2837 printf(_psum.__name__, end=_COMMASPACE_)

2838

2839 F = Fsum()

2840 if F.is_math_fsum():

2841 for t in frandoms(n, seeded=True):

2842 assert float(F.fset_(*t)) == _fsum(t)

2843 printf(_DOT_, end=NN)

2844 printf(NN)

2845

2846 _test(128)

2847

2848# **) MIT License

2849#

2850# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.

2851#

2852# Permission is hereby granted, free of charge, to any person obtaining a

2853# copy of this software and associated documentation files (the "Software"),

2854# to deal in the Software without restriction, including without limitation

2855# the rights to use, copy, modify, merge, publish, distribute, sublicense,

2856# and/or sell copies of the Software, and to permit persons to whom the

2857# Software is furnished to do so, subject to the following conditions:

2858#

2859# The above copyright notice and this permission notice shall be included

2860# in all copies or substantial portions of the Software.

2861#

2862# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

2863# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

2864# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

2865# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

2866# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

2867# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

2868# OTHER DEALINGS IN THE SOFTWARE.