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__ = '24.10.22'

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 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): # like Python 3.13+ I{Modules/mathmodule.c}:

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

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

169 if isnan(r):

170 def _x(x):

171 return not isnan(x)

172 else: # non-NAN non-finite

173 _x = _isfinite

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

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

176 return r

177

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

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

180 # TwoProduct U{Algorithm 3.3

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

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

183 # #ifndef UNRELIABLE_FMA ... #else ... #endif

184 _, a, b = _2split3(x)

185 for y, c, d in y3s:

186 y *= x

187 yield y

188 if _isfinite(y):

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

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

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

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

193 yield a * c - y

194 yield b * c

195 if d:

196 yield a * d

197 yield b * d

198 for z in zs:

199 yield z

200

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

202

203 def _2split3(x):

204 # Split U{Algorithm 3.2

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

206 a = c = x * _2FACTOR

207 a -= c - x

208 b = x - a

209 return x, a, b

210

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

212 return map(_2split3, xs)

213

214

215 def f2product(*two):

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

217

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

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

220

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

222

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

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

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

226 equivalent, slower implementation when not available.

227 '''

228 t = Fsum._f2product

229 if two and two[0] is not None:

230 Fsum._f2product = bool(two[0])

231 return t

232

233

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

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

236 '''

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

238

239

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

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

242 '''

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

244

245

246 def _halfeven(s, r, p):

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

248 '''

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

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

251 r *= 2

252 t = s + r

253 if r == (t - s):

254 s = t

255 return s

256

257

258 def _isFsum(x): # in .fmath

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

260 '''

261 return isinstance(x, Fsum)

262

263

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

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

266 '''

267 return isinstance(x, _Fsum_2Tuple_types)

268

269

270 def _isOK(unused):

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

272 '''

273 return True

274

275

276 def _isOK_or_finite(x, _isfine=_isfinite):

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

278 '''

279 # assert _isfine in (_isOK, _isfinite)

280 return _isfine(x) # C{bool}

281

282

283 try:

284 from math import gcd as _gcd

285

286 def _n_d2(n, d):

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

288 '''

289 if n and d:

290 try:

291 c = _gcd(n, d)

292 if c > 1:

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

294 except TypeError: # non-int float

295 pass

296 return n, d

297

298 except ImportError: # 3.4-

299

300 def _n_d2(*n_d): # PYCHOK redef

301 return n_d

302

303

304 def _nfError(x, *args):

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

306 '''

307 E = _NonfiniteError(x)

308 t = Fmt.PARENSPACED(_not_finite_, x)

309 if args: # in _fmaX, _2sum

310 return E(txt=t, *args)

311 raise E(t, txt=None)

312

313

314 def _NonfiniteError(x):

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

316 '''

317 return _OverflowError if isinf(x) else (

318 _ValueError if isnan(x) else _AssertionError)

319

320

321 def nonfiniterrors(*raiser):

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

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

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

325

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

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

328 the setting unchanged.

329

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

331

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

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

334 '''

335 d = Fsum._isfine

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

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

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

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

340

341

342 def _1primed(xs): # in .fmath

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

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

345 '''

346 yield _1_0

347 for x in xs:

348 yield x

349 yield _N_1_0

350

351

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

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

354 '''

355 # assert isinstance(ps, list)

356 i = len(ps) - 1

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

358 while i > 0:

359 i -= 1

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

361 if r: # sum(ps) became inexact

362 if s:

363 ps[i:] = r, s

364 if i > 0:

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

366 break # return s

367 s = r # PYCHOK no cover

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

369 i = 0 # collapse ps

370 if ps:

371 s += sum(ps)

372 ps[i:] = s,

373 return s

374

375

376 def _Psum(ps, **name_f2product_nonfinites_RESIDUAL):

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

378 '''

379 F = Fsum(**name_f2product_nonfinites_RESIDUAL)

380 if ps:

381 F._ps[:] = ps

382 F._n = len(F._ps)

383 return F

384

385

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

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

388 '''

389 return _Psum(ps, **name_f2product_nonfinites_RESIDUAL)

390

391

392 def _residue(other):

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

394 '''

395 try:

396 r = other.residual

397 except AttributeError:

398 r = None # float, int, other

399 return r

400

401

402 def _s_r(s, r):

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

404 '''

405 if _isfinite(s):

406 if r:

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

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

409 else:

410 r = INT0

411 else:

412 r = _NONFINITEr

413 return s, r

414

415

416 def _2s_r(other):

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

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

419 '''

420 if _isFsum_2Tuple(other):

421 s, r = other._fint2

422 if r:

423 s, r = other._nfprs2

424 if r: # PYCHOK no cover

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

426 else:

427 r = 0

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

429 if isint(s, both=True):

430 s = int(s)

431 return s, r

432

433

434 def _strcomplex(s, *args):

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

436 '''

437 c = _strcomplex.__name__[4:]

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

439 t = unstr(pow, *args)

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

441

442

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

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

445 '''

446 p = _stresidual.__name__[3:]

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

448 for n, v in itemsorted(mod_ratio):

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

450 t = _COMMASPACE_(t, p)

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

452

453

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

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

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

457 '''

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

459

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

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

462 # 1974, Zeitschrift für Angewandte Mathmatik und Mechanik, vol 51, nr 1, p 39-51

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

464 s = a + b

465 if _isfinite(s):

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

467 r = (b - s) + a

468 else:

469 r = (a - s) + b

470 elif _isfine(s):

471 r = _NONFINITEr

472 else: # non-finite and not OK

473 t = unstr(_2sum, a, b)

474 raise _nfError(s, t)

475 return s, r

476

477

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

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

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

481 '''

482 if kwds:

483 threshold = _xkwds_get1(kwds, RESIDUAL=threshold)

484 try:

485 return _2finite(threshold) # PYCHOK None

486 except Exception as x:

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

488

489

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

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

492

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

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

495 intermediate, I{running} summuation.

496

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

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

499

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

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

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

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

504 exceptions by default.

505

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

507 393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>},

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

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

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

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

512

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

514 multiplication.

515

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

517 C{PYGEODESY_FSUM_NONFINITES} and C{PYGEODESY_FSUM_RESIDUAL}.

518 '''

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

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

521 _n = 0

522# _ps = [] # partial sums

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

524 _RESIDUAL = _threshold(_RESIDUAL_0_0)

525

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

527 '''New L{Fsum}.

528

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

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

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

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

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

534 L{Fsum}.

535

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

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

538 '''

539 if name_f2product_nonfinites_RESIDUAL:

540 self._optionals(**name_f2product_nonfinites_RESIDUAL)

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

542 if xs:

543 self._facc_args(xs, up=False)

544

545 def __abs__(self):

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

547 '''

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

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

550

551 def __add__(self, other):

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

553

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

555

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

557

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

559 '''

560 f = self._copy_2(self.__add__)

561 return f._fadd(other)

562

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

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

565 '''

566 s, r = self._nfprs2

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

568

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

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

571 '''

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

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

574 return self

575

576

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

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

579

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

581

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

583 '''

584 return self.ceil

585

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

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

588

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

590

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

592 '''

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

594 return _signOf(s, 0)

595

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

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

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

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

600

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

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

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

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

605

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

607 B{C{RESIDUAL}}.

608

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

610 '''

611 f = self._copy_2(self.__divmod__)

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

613

614 def __eq__(self, other):

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

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

617 '''

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

619

620 def __float__(self):

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

622

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

624 '''

625 return float(self._fprs)

626

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

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

629

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

631

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

633 '''

634 return self.floor

635

636 def __floordiv__(self, other):

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

638

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

640

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

642

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

644 '''

645 f = self._copy_2(self.__floordiv__)

646 return f._floordiv(other, _floordiv_op_)

647

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

649 '''Not implemented.'''

650 return _NotImplemented(self, *other)

651

652 def __ge__(self, other):

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

654 '''

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

656

657 def __gt__(self, other):

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

659 '''

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

661

662 def __hash__(self): # PYCHOK no cover

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

664 '''

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

666 # 3/library/numbers.html#numbers.Rational>}

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

668

669 def __iadd__(self, other):

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

671

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

673 an iterable of several of the former.

674

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

676

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

678 C{scalar} nor L{Fsum}.

679

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

681 '''

682 try:

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

684 except TypeError:

685 pass

686 _xiterable(other)

687 return self._facc(other)

688

689 def __ifloordiv__(self, other):

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

691

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

693

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

695

696 @raise ResidualError: Non-zero, significant residual

697 in B{C{other}}.

698

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

700

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

702

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

704

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

706 '''

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

708

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

710 '''Not implemented.'''

711 return _NotImplemented(self, other)

712

713 def __imod__(self, other):

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

715

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

717

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

719

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

721 '''

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

723

724 def __imul__(self, other):

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

726

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

728

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

730

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

732

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

734

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

736 '''

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

738

739 def __int__(self):

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

741

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

743 and L{Fsum.floor}.

744 '''

745 i, _ = self._fint2

746 return i

747

748 def __invert__(self): # PYCHOK no cover

749 '''Not implemented.'''

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

751 return _NotImplemented(self)

752

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

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

755

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

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

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

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

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

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

762

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

764

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

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

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

768

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

770

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

772 is non-zero and significant and either

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

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

775 C{None}.

776

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

778 invocation failed.

779

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

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

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

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

784 is given as C{0}.

785

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

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

788 '''

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

790

791 def __isub__(self, other):

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

793

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

795 an iterable of several of the former.

796

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

798

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

800

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

802 '''

803 try:

804 return self._fsub(other, _isub_op_)

805 except TypeError:

806 pass

807 _xiterable(other)

808 return self._facc_neg(other)

809

810 def __iter__(self):

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

812 '''

813 return iter(self.partials)

814

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

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

817

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

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

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

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

822

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

824

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

826

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

828 B{C{RESIDUAL}}.

829

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

831

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

833

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

835

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

837 '''

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

839

840 def __le__(self, other):

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

842 '''

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

844

845 def __len__(self):

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

847 '''

848 return self._n

849

850 def __lt__(self, other):

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

852 '''

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

854

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

856 '''Not implemented.'''

857 return _NotImplemented(self, other)

858

859 def __mod__(self, other):

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

861

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

863 '''

864 f = self._copy_2(self.__mod__)

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

866

867 def __mul__(self, other):

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

869

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

871 '''

872 f = self._copy_2(self.__mul__)

873 return f._fmul(other, _mul_op_)

874

875 def __ne__(self, other):

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

877 '''

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

879

880 def __neg__(self):

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

882 '''

883 f = self._copy_2(self.__neg__)

884 return f._fset(self._neg)

885

886 def __pos__(self):

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

888 '''

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

890

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

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

893

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

895 '''

896 f = self._copy_2(self.__pow__)

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

898

899 def __radd__(self, other):

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

901

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

903 '''

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

905 return f._fadd(self)

906

907 def __rdivmod__(self, other):

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

909 C{(quotient, remainder)}.

910

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

912 '''

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

914 return f._fdivmod2(self, _divmod_op_)

915

916# turned off, called by _deepcopy and _copy

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

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

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

920# '''

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

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

923

924# def __repr__(self):

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

926# '''

927# return self.toRepr(lenc=True)

928

929 def __rfloordiv__(self, other):

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

931

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

933 '''

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

935 return f._floordiv(self, _floordiv_op_)

936

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

938 '''Not implemented.'''

939 return _NotImplemented(self, other)

940

941 def __rmod__(self, other):

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

943

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

945 '''

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

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

948

949 def __rmul__(self, other):

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

951

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

953 '''

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

955 return f._fmul(self, _mul_op_)

956

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

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

959

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

961 '''

962 f = self._copy_2(self.__round__)

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

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

965

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

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

968

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

970 '''

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

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

973

974 def __rsub__(self, other):

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

976

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

978 '''

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

980 return f._fsub(self, _sub_op_)

981

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

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

984

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

986 '''

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

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

989

990 def __str__(self):

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

992 '''

993 return self.toStr(lenc=True)

994

995 def __sub__(self, other):

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

997

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

999

1000 @return: The difference (L{Fsum}).

1001

1002 @see: Method L{Fsum.__isub__}.

1003 '''

1004 f = self._copy_2(self.__sub__)

1005 return f._fsub(other, _sub_op_)

1006

1007 def __truediv__(self, other, **raiser_RESIDUAL):

1008 '''Return C{B{self} / B{other}} as an L{Fsum}.

1009

1010 @arg other: An L{Fsum}, L{Fsum2Tuple} or C{scalar} divisor.

1011 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore

1012 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}

1013 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

1014

1015 @return: The quotient (L{Fsum}).

1016

1017 @raise ResidualError: Non-zero, significant residual or invalid

1018 B{C{RESIDUAL}}.

1019

1020 @see: Method L{Fsum.__itruediv__}.

1021 '''

1022 return self._truediv(other, _truediv_op_, **raiser_RESIDUAL)

1023

1024 __trunc__ = __int__

1025

1026 if _MODS.sys_version_info2 < (3, 0): # PYCHOK no cover

1027 # <https://docs.Python.org/2/library/operator.html#mapping-operators-to-functions>

1028 __div__ = __truediv__

1029 __idiv__ = __itruediv__

1030 __long__ = __int__

1031 __nonzero__ = __bool__

1032 __rdiv__ = __rtruediv__

1033

1034 def as_integer_ratio(self):

1035 '''Return this instance as the ratio of 2 integers.

1036

1037 @return: 2-Tuple C{(numerator, denominator)} both C{int} with

1038 C{numerator} signed and C{denominator} non-zero and

1039 positive. The C{numerator} is I{non-finite} if this

1040 instance is.

1041

1042 @see: Method L{Fsum.fint2} and C{float.as_integer_ratio} in

1043 Python 2.7+.

1044 '''

1045 n, r = self._fint2

1046 if r:

1047 i, d = float(r).as_integer_ratio()

1048 n, d = _n_d2(n * d + i, d)

1049 else: # PYCHOK no cover

1050 d = 1

1051 return n, d

1052

1053 @property_RO

1054 def as_iscalar(self):

1055 '''Get this instance I{as-is} (L{Fsum} with C{non-zero residual},

1056 C{scalar} or I{non-finite}).

1057 '''

1058 s, r = self._nfprs2

1059 return self if r else s

1060

1061 @property_RO

1062 def ceil(self):

1063 '''Get this instance' C{ceil} value (C{int} in Python 3+, but

1064 C{float} in Python 2-).

1065

1066 @note: This C{ceil} takes the C{residual} into account.

1067

1068 @see: Method L{Fsum.int_float} and properties L{Fsum.floor},

1069 L{Fsum.imag} and L{Fsum.real}.

1070 '''

1071 s, r = self._fprs2

1072 c = _ceil(s) + int(r) - 1

1073 while r > (c - s): # (s + r) > c

1074 c += 1

1075 return c # _ceil(self._n_d)

1076

1077 cmp = __cmp__

1078

1079 def _cmp_0(self, other, op):

1080 '''(INTERNAL) Return C{scalar(self - B{other})} for 0-comparison.

1081 '''

1082 if _isFsum_2Tuple(other):

1083 s = self._ps_1sum(*other._ps)

1084 elif self._scalar(other, op):

1085 s = self._ps_1sum(other)

1086 else:

1087 s = self.signOf() # res=True

1088 return s

1089

1090 def copy(self, deep=False, **name):

1091 '''Copy this instance, C{shallow} or B{C{deep}}.

1092

1093 @kwarg name: Optional, overriding C{B{name}='"copy"} (C{str}).

1094

1095 @return: The copy (L{Fsum}).

1096 '''

1097 n = _name__(name, name__=self.copy)

1098 f = _Named.copy(self, deep=deep, name=n)

1099 if f._ps is self._ps:

1100 f._ps = list(self._ps) # separate list

1101 if not deep:

1102 f._n = 1

1103 # assert f._f2product == self._f2product

1104 # assert f._Fsum is f

1105 # assert f._isfine is self._isfine

1106 # assert f._RESIDUAL is self._RESIDUAL

1107 return f

1108

1109 def _copy_2(self, which, name=NN):

1110 '''(INTERNAL) Copy for I{dyadic} operators.

1111 '''

1112 n = name or which.__name__ # _DUNDER_nameof

1113 # NOT .classof due to .Fdot(a, *b) args, etc.

1114 f = _Named.copy(self, deep=False, name=n)

1115 f._ps = list(self._ps) # separate list

1116 # assert f._n == self._n

1117 # assert f._f2product == self._f2product

1118 # assert f._Fsum is f

1119 # assert f._isfine is self._isfine

1120 # assert f._RESIDUAL is self._RESIDUAL

1121 return f

1122

1123 def _copy_2r(self, other, which):

1124 '''(INTERNAL) Copy for I{reverse-dyadic} operators.

1125 '''

1126 return other._copy_2(which) if _isFsum(other) else \

1127 self._copy_2(which)._fset(other)

1128

1129 divmod = __divmod__

1130

1131 def _Error(self, op, other, Error, **txt_cause):

1132 '''(INTERNAL) Format an B{C{Error}} for C{{self} B{op} B{other}}.

1133 '''

1134 # self.as_iscalar causes RecursionError for ._fprs2 errors

1135 s = _Psum(self._ps, nonfinites=True, name=self.name)

1136 return Error(_SPACE_(s.as_iscalar, op, other), **txt_cause)

1137

1138 def _ErrorX(self, X, op, other, *mod):

1139 '''(INTERNAL) Format the caught exception C{X}.

1140 '''

1141 E, t = _xError2(X)

1142 if mod:

1143 t = _COMMASPACE_(Fmt.PARENSPACED(mod=mod[0]), t)

1144 return self._Error(op, other, E, txt=t, cause=X)

1145

1146 def _ErrorXs(self, X, xs, **kwds): # in .fmath

1147 '''(INTERNAL) Format the caught exception C{X}.

1148 '''

1149 E, t = _xError2(X)

1150 u = unstr(self.named3, *xs, _ELLIPSIS=4, **kwds)

1151 return E(u, txt=t, cause=X)

1152

1153 def _facc(self, xs, up=True, **_X_x_origin):

1154 '''(INTERNAL) Accumulate more C{scalar}s or L{Fsum}s.

1155 '''

1156 if xs:

1157 kwds = self._isfine

1158 if _X_x_origin:

1159 kwds = _xkwds(_X_x_origin, **kwds)

1160 fs = _xs(xs, **kwds) # PYCHOK yield

1161 ps = self._ps

1162 ps[:] = self._ps_acc(list(ps), fs, up=up)

1163# if len(ps) > 16:

1164# _ = _psum(ps, **self._isfine)

1165 return self

1166

1167 def _facc_args(self, xs, **up):

1168 '''(INTERNAL) Accumulate 0, 1 or more C{xs}, all positional

1169 arguments in the caller of this method.

1170 '''

1171 return self._fadd(xs[0], **up) if len(xs) == 1 else \

1172 self._facc(xs, **up) # origin=1?

1173

1174 def _facc_neg(self, xs, **up_origin):

1175 '''(INTERNAL) Accumulate more C{xs}, negated.

1176 '''

1177 def _N(X):

1178 return X._ps_neg

1179

1180 def _n(x):

1181 return -float(x)

1182

1183 return self._facc(xs, _X=_N, _x=_n, **up_origin)

1184

1185 def _facc_power(self, power, xs, which, **raiser_RESIDUAL): # in .fmath

1186 '''(INTERNAL) Add each C{xs} as C{float(x**power)}.

1187 '''

1188 def _Pow4(p):

1189 r = 0

1190 if _isFsum_2Tuple(p):

1191 s, r = p._fprs2

1192 if r:

1193 m = Fsum._pow

1194 else: # scalar

1195 return _Pow4(s)

1196 elif isint(p, both=True) and int(p) >= 0:

1197 p = s = int(p)

1198 m = Fsum._pow_int

1199 else:

1200 p = s = _2float(power=p, **self._isfine)

1201 m = Fsum._pow_scalar

1202 return m, p, s, r

1203

1204 _Pow, p, s, r = _Pow4(power)

1205 if p: # and xs:

1206 op = which.__name__

1207 _FsT = _Fsum_2Tuple_types

1208 _pow = self._pow_2_3

1209

1210 def _P(X):

1211 f = _Pow(X, p, power, op, **raiser_RESIDUAL)

1212 return f._ps if isinstance(f, _FsT) else (f,)

1213

1214 def _p(x):

1215 x = float(x)

1216 f = _pow(x, s, power, op, **raiser_RESIDUAL)

1217 if f and r:

1218 f *= _pow(x, r, power, op, **raiser_RESIDUAL)

1219 return f

1220

1221 f = self._facc(xs, _X=_P, _x=_p) # origin=1?

1222 else:

1223 f = self._facc_scalar_(float(len(xs))) # x**0 == 1

1224 return f

1225

1226 def _facc_scalar(self, xs, **up):

1227 '''(INTERNAL) Accumulate all C{xs}, each C{scalar}.

1228 '''

1229 if xs:

1230 ps = self._ps

1231 ps[:] = self._ps_acc(list(ps), xs, **up)

1232 return self

1233

1234 def _facc_scalar_(self, *xs, **up):

1235 '''(INTERNAL) Accumulate all positional C{xs}, each C{scalar}.

1236 '''

1237 return self._facc_scalar(xs, **up)

1238

1239 def _facc_scalarf(self, xs, up=True, **origin_which):

1240 '''(INTERNAL) Accumulate all C{xs}, each C{scalar}, an L{Fsum} or

1241 L{Fsum2Tuple}, like function C{_xsum}.

1242 '''

1243 _C = self.__class__

1244 fs = _xs(xs, **_x_isfine(self.nonfinitesOK, _Cdot=_C,

1245 **origin_which)) # PYCHOK yield

1246 return self._facc_scalar(fs, up=up)

1247

1248# def _facc_up(self, up=True):

1249# '''(INTERNAL) Update the C{partials}, by removing

1250# and re-accumulating the final C{partial}.

1251# '''

1252# ps = self._ps

1253# while len(ps) > 1:

1254# p = ps.pop()

1255# if p:

1256# n = self._n

1257# _ = self._ps_acc(ps, (p,), up=False)

1258# self._n = n

1259# break

1260# return self._update() if up else self

1261

1262 def fadd(self, xs=()):

1263 '''Add an iterable's items to this instance.

1264

1265 @arg xs: Iterable of items to add (each C{scalar},

1266 an L{Fsum} or L{Fsum2Tuple}).

1267

1268 @return: This instance (L{Fsum}).

1269

1270 @raise OverflowError: Partial C{2sum} overflow.

1271

1272 @raise TypeError: An invalid B{C{xs}} item.

1273

1274 @raise ValueError: Invalid or I{non-finite} B{C{xs}} value.

1275 '''

1276 if _isFsum_2Tuple(xs):

1277 self._facc_scalar(xs._ps)

1278 elif isscalar(xs): # for backward compatibility # PYCHOK no cover

1279 x = _2float(x=xs, **self._isfine)

1280 self._facc_scalar_(x)

1281 elif xs: # _xiterable(xs)

1282 self._facc(xs)

1283 return self

1284

1285 def fadd_(self, *xs):

1286 '''Add all positional items to this instance.

1287

1288 @arg xs: Values to add (each C{scalar}, an L{Fsum}

1289 or L{Fsum2Tuple}), all positional.

1290

1291 @see: Method L{Fsum.fadd} for further details.

1292 '''

1293 return self._facc_args(xs)

1294

1295 def _fadd(self, other, op=_add_op_, **up):

1296 '''(INTERNAL) Apply C{B{self} += B{other}}.

1297 '''

1298 if _isFsum_2Tuple(other):

1299 self._facc_scalar(other._ps, **up)

1300 elif self._scalar(other, op):

1301 self._facc_scalar_(other, **up)

1302 return self

1303

1304 fcopy = copy # for backward compatibility

1305 fdiv = __itruediv__

1306 fdivmod = __divmod__

1307

1308 def _fdivmod2(self, other, op, **raiser_RESIDUAL):

1309 '''(INTERNAL) Apply C{B{self} %= B{other}} and return a L{DivMod2Tuple}.

1310 '''

1311 # result mostly follows CPython function U{float_divmod

1312 # <https://GitHub.com/python/cpython/blob/main/Objects/floatobject.c>},

1313 # but at least divmod(-3, 2) equals Cpython's result (-2, 1).

1314 q = self._truediv(other, op, **raiser_RESIDUAL).floor

1315 if q: # == float // other == floor(float / other)

1316 self -= self._Fsum_as(q) * other # NOT other * q!

1317

1318 s = signOf(other) # make signOf(self) == signOf(other)

1319 if s and self.signOf() == -s: # PYCHOK no cover

1320 self += other

1321 q -= 1

1322# t = self.signOf()

1323# if t and t != s:

1324# raise self._Error(op, other, _AssertionError, txt__=signOf)

1325 return DivMod2Tuple(q, self) # q is C{int} in Python 3+, but C{float} in Python 2-

1326

1327 def _fhorner(self, x, cs, where, incx=True): # in .fmath

1328 '''(INTERNAL) Add an L{Fhorner} evaluation of polynomial

1329 C{sum(cs[i] * B{x}**i for i=0..len(cs)-1) if B{incx}

1330 else sum(... i=len(cs)-1..0)}.

1331 '''

1332 # assert _xiterablen(cs)

1333 try:

1334 n = len(cs)

1335 H = self._Fsum_as(name__=self._fhorner)

1336 _m = H._mul_Fsum if _isFsum_2Tuple(x) else \

1337 H._mul_scalar

1338 if _2finite(x, **self._isfine) and n > 1:

1339 for c in (reversed(cs) if incx else cs):

1340 H._fset(_m(x, _mul_op_), up=False)

1341 H._fadd(c, up=False)

1342 else: # x == 0

1343 H = cs[0] if n else 0

1344 self._fadd(H)

1345 except Exception as X:

1346 t = unstr(where, x, *cs, _ELLIPSIS=4, incx=incx)

1347 raise self._ErrorX(X, _add_op_, t)

1348 return self

1349

1350 def _finite(self, other, op=None):

1351 '''(INTERNAL) Return B{C{other}} if C{finite}.

1352 '''

1353 if _isOK_or_finite(other, **self._isfine):

1354 return other

1355 E = _NonfiniteError(other)

1356 raise self._Error(op, other, E, txt=_not_finite_)

1357

1358 def fint(self, name=NN, **raiser_RESIDUAL):

1359 '''Return this instance' current running sum as C{integer}.

1360

1361 @kwarg name: Optional, overriding C{B{name}="fint"} (C{str}).

1362 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore

1363 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}

1364 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

1365

1366 @return: The C{integer} sum (L{Fsum}) if this instance C{is_integer}

1367 with a zero or insignificant I{integer} residual.

1368

1369 @raise ResidualError: Non-zero, significant residual or invalid

1370 B{C{RESIDUAL}}.

1371

1372 @see: Methods L{Fsum.fint2}, L{Fsum.int_float} and L{Fsum.is_integer}.

1373 '''

1374 i, r = self._fint2

1375 if r:

1376 R = self._raiser(r, i, **raiser_RESIDUAL)

1377 if R:

1378 t = _stresidual(_integer_, r, **R)

1379 raise ResidualError(_integer_, i, txt=t)

1380 return self._Fsum_as(i, name=_name__(name, name__=self.fint))

1381

1382 def fint2(self, **name):

1383 '''Return this instance' current running sum as C{int} and the

1384 I{integer} residual.

1385

1386 @kwarg name: Optional name (C{str}).

1387

1388 @return: An L{Fsum2Tuple}C{(fsum, residual)} with C{fsum}

1389 an C{int} and I{integer} C{residual} a C{float} or

1390 C{INT0} if the C{fsum} is considered to be I{exact}.

1391 The C{fsum} is I{non-finite} if this instance is.

1392 '''

1393 return Fsum2Tuple(*self._fint2, **name)

1394

1395 @Property

1396 def _fint2(self): # see ._fset

1397 '''(INTERNAL) Get 2-tuple (C{int}, I{integer} residual).

1398 '''

1399 s, r = self._nfprs2

1400 if _isfinite(s):

1401 i = int(s)

1402 r = (self._ps_1sum(i) if len(self._ps) > 1 else

1403 float(s - i)) or INT0

1404 else: # INF, NAN, NINF

1405 i = float(s)

1406# r = _NONFINITEr

1407 return i, r # Fsum2Tuple?

1408

1409 @_fint2.setter_ # PYCHOK setter_UNDERscore!

1410 def _fint2(self, s): # in _fset

1411 '''(INTERNAL) Replace the C{_fint2} value.

1412 '''

1413 if _isfinite(s):

1414 i = int(s)

1415 r = (s - i) or INT0

1416 else: # INF, NAN, NINF

1417 i = float(s)

1418 r = _NONFINITEr

1419 return i, r # like _fint2.getter

1420

1421 @deprecated_property_RO

1422 def float_int(self): # PYCHOK no cover

1423 '''DEPRECATED, use method C{Fsum.int_float}.'''

1424 return self.int_float() # raiser=False

1425

1426 @property_RO

1427 def floor(self):

1428 '''Get this instance' C{floor} (C{int} in Python 3+, but

1429 C{float} in Python 2-).

1430

1431 @note: This C{floor} takes the C{residual} into account.

1432

1433 @see: Method L{Fsum.int_float} and properties L{Fsum.ceil},

1434 L{Fsum.imag} and L{Fsum.real}.

1435 '''

1436 s, r = self._fprs2

1437 f = _floor(s) + _floor(r) + 1

1438 while (f - s) > r: # f > (s + r)

1439 f -= 1

1440 return f # _floor(self._n_d)

1441

1442# ffloordiv = __ifloordiv__ # for naming consistency?

1443# floordiv = __floordiv__ # for naming consistency?

1444

1445 def _floordiv(self, other, op, **raiser_RESIDUAL): # rather _ffloordiv?

1446 '''Apply C{B{self} //= B{other}}.

1447 '''

1448 q = self._ftruediv(other, op, **raiser_RESIDUAL) # == self

1449 return self._fset(q.floor) # floor(q)

1450

1451 def fma(self, other1, other2, **nonfinites): # in .fmath.fma

1452 '''Fused-multiply-add C{self *= B{other1}; self += B{other2}}.

1453

1454 @arg other1: Multiplier (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1455 @arg other2: Addend (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

1456 @kwarg nonfinites: Use C{B{nonfinites}=True} or C{False}, to

1457 override L{nonfinites<Fsum.nonfinites>} and

1458 L{nonfiniterrors} default (C{bool}).

1459 '''

1460 op = self.fma.__name__

1461 _fs = self._ps_other

1462 try:

1463 s, r = self._fprs2

1464 if r:

1465 f = self._f2mul(self.fma, other1, **nonfinites)

1466 f += other2

1467 elif _residue(other1) or _residue(other2):

1468 fs = _2split3s(_fs(op, other1))

1469 fs = _2products(s, fs, *_fs(op, other2))

1470 f = _Psum(self._ps_acc([], fs, up=False), name=op)

1471 else:

1472 f = _fma(s, other1, other2)

1473 f = _2finite(f, **self._isfine)

1474 except TypeError as X:

1475 raise self._ErrorX(X, op, (other1, other2))

1476 except (OverflowError, ValueError) as X: # from math.fma

1477 f = self._mul_reduce(s, other1) # INF, NAN, NINF

1478 f += sum(_fs(op, other2))

1479 f = self._nonfiniteX(X, op, f, **nonfinites)

1480 return self._fset(f)

1481

1482 fmul = __imul__

1483

1484 def _fmul(self, other, op):

1485 '''(INTERNAL) Apply C{B{self} *= B{other}}.

1486 '''

1487 if _isFsum_2Tuple(other):

1488 if len(self._ps) != 1:

1489 f = self._mul_Fsum(other, op)

1490 elif len(other._ps) != 1: # and len(self._ps) == 1

1491 f = self._ps_mul(op, *other._ps) if other._ps else _0_0

1492 elif self._f2product: # len(other._ps) == 1

1493 f = self._mul_scalar(other._ps[0], op)

1494 else: # len(other._ps) == len(self._ps) == 1

1495 f = self._finite(self._ps[0] * other._ps[0], op=op)

1496 else:

1497 s = self._scalar(other, op)

1498 f = self._mul_scalar(s, op)

1499 return self._fset(f) # n=len(self) + 1

1500

1501 @deprecated_method

1502 def f2mul(self, *others, **raiser):

1503 '''DEPRECATED on 2024.09.13, use method L{f2mul_<Fsum.f2mul_>}.'''

1504 return self._fset(self.f2mul_(*others, **raiser))

1505

1506 def f2mul_(self, *others, **nonfinites): # in .fmath.f2mul

1507 '''Return C{B{self} * B{other} * B{other} ...} for all B{C{others}} using cascaded,

1508 accurate multiplication like with L{f2product<Fsum.f2product>} set to C{True}.

1509

1510 @arg others: Multipliers (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all

1511 positional.

1512 @kwarg nonfinites: Use C{B{nonfinites}=True} or C{False}, to override both

1513 L{nonfinites<Fsum.nonfinites>} and the L{nonfiniterrors}

1514 default (C{bool}).

1515

1516 @return: The cascaded I{TwoProduct} (L{Fsum} or C{float}).

1517

1518 @see: U{Equations 2.3<https://www.TUHH.De/ti3/paper/rump/OzOgRuOi06.pdf>}

1519 '''

1520 return self._f2mul(self.f2mul_, *others, **nonfinites)

1521

1522 def _f2mul(self, where, *others, **nonfinites_raiser):

1523 '''(INTERNAL) See methods C{fma} and C{f2mul_}.

1524 '''

1525 f = self._copy_2(where)

1526 ps = f._ps

1527 if ps and others:

1528 op = where.__name__

1529 try:

1530 for other in others: # to pinpoint errors

1531 for p in self._ps_other(op, other):

1532 pfs = _2products(p, _2split3s(ps))

1533 ps[:] = f._ps_acc([], pfs, up=False)

1534 f._update()

1535 except TypeError as X:

1536 raise self._ErrorX(X, op, other)

1537 except (OverflowError, ValueError) as X:

1538 r = self._mul_reduce(sum(ps), other) # INF, NAN, NINF

1539 r = self._nonfiniteX(X, op, r, **nonfinites_raiser)

1540 f._fset(r)

1541 return f

1542

1543 def fover(self, over, **raiser_RESIDUAL):

1544 '''Apply C{B{self} /= B{over}} and summate.

1545

1546 @arg over: An L{Fsum} or C{scalar} denominator.

1547 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore

1548 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}

1549 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

1550

1551 @return: Precision running sum (C{float}).

1552

1553 @raise ResidualError: Non-zero, significant residual or invalid

1554 B{C{RESIDUAL}}.

1555

1556 @see: Methods L{Fsum.fsum} and L{Fsum.__itruediv__}.

1557 '''

1558 return float(self.fdiv(over, **raiser_RESIDUAL)._fprs)

1559

1560 fpow = __ipow__

1561

1562 def _fpow(self, other, op, *mod, **raiser_RESIDUAL):

1563 '''Apply C{B{self} **= B{other}}, optional B{C{mod}} or C{None}.

1564 '''

1565 if mod:

1566 if mod[0] is not None: # == 3-arg C{pow}

1567 f = self._pow_2_3(self, other, other, op, *mod, **raiser_RESIDUAL)

1568 elif self.is_integer():

1569 # return an exact C{int} for C{int}**C{int}

1570 i, _ = self._fint2 # assert _ == 0

1571 x, r = _2s_r(other) # C{int}, C{float} or other

1572 f = self._Fsum_as(i)._pow_Fsum(other, op, **raiser_RESIDUAL) if r else \

1573 self._pow_2_3(i, x, other, op, **raiser_RESIDUAL)

1574 else: # mod[0] is None, power(self, other)

1575 f = self._pow(other, other, op, **raiser_RESIDUAL)

1576 else: # pow(self, other)

1577 f = self._pow(other, other, op, **raiser_RESIDUAL)

1578 return self._fset(f) # n=max(len(self), 1)

1579

1580 def f2product(self, *two):

1581 '''Get and set accurate I{TwoProduct} multiplication for this

1582 L{Fsum}, overriding the L{f2product} default.

1583

1584 @arg two: If omitted, leave the override unchanged, if C{True},

1585 turn I{TwoProduct} on, if C{False} off, if C{None}e

1586 remove th override (C{bool} or C{None}).

1587

1588 @return: The previous setting (C{bool} or C{None} if not set).

1589

1590 @see: Function L{f2product<fsums.f2product>}.

1591

1592 @note: Use C{f.f2product() or f2product()} to determine whether

1593 multiplication is accurate for L{Fsum} C{f}.

1594 '''

1595 if two: # delattrof(self, _f2product=None)

1596 t = _xkwds_pop(self.__dict__, _f2product=None)

1597 if two[0] is not None:

1598 self._f2product = bool(two[0])

1599 else: # getattrof(self, _f2product=None)

1600 t = _xkwds_get(self.__dict__, _f2product=None)

1601 return t

1602

1603 @Property

1604 def _fprs(self):

1605 '''(INTERNAL) Get and cache this instance' precision

1606 running sum (C{float} or C{int}), ignoring C{residual}.

1607

1608 @note: The precision running C{fsum} after a C{//=} or

1609 C{//} C{floor} division is C{int} in Python 3+.

1610 '''

1611 s, _ = self._fprs2

1612 return s # ._fprs2.fsum

1613

1614 @_fprs.setter_ # PYCHOK setter_UNDERscore!

1615 def _fprs(self, s):

1616 '''(INTERNAL) Replace the C{_fprs} value.

1617 '''

1618 return s

1619

1620 @Property

1621 def _fprs2(self):

1622 '''(INTERNAL) Get and cache this instance' precision

1623 running sum and residual (L{Fsum2Tuple}).

1624 '''

1625 ps = self._ps

1626 n = len(ps)

1627 try:

1628 if n > 2:

1629 s = _psum(ps, **self._isfine)

1630 if not _isfinite(s):

1631 ps[:] = s, # collapse ps

1632 return Fsum2Tuple(s, _NONFINITEr)

1633 n = len(ps)

1634# Fsum._ps_max = max(Fsum._ps_max, n)

1635 if n > 2:

1636 r = self._ps_1sum(s)

1637 return Fsum2Tuple(*_s_r(s, r))

1638 if n > 1: # len(ps) == 2

1639 s, r = _s_r(*_2sum(*ps, **self._isfine))

1640 ps[:] = (r, s) if r else (s,)

1641 elif ps: # len(ps) == 1

1642 s = ps[0]

1643 r = INT0 if _isfinite(s) else _NONFINITEr

1644 else: # len(ps) == 0

1645 s = _0_0

1646 r = INT0 if _isfinite(s) else _NONFINITEr

1647 ps[:] = s,

1648 except (OverflowError, ValueError) as X:

1649 op = _fset_op_ # INF, NAN, NINF

1650 ps[:] = sum(ps), # collapse ps

1651 s = self._nonfiniteX(X, op, ps[0])

1652 r = _NONFINITEr

1653 # assert self._ps is ps

1654 return Fsum2Tuple(s, r)

1655

1656 @_fprs2.setter_ # PYCHOK setter_UNDERscore!

1657 def _fprs2(self, s_r):

1658 '''(INTERNAL) Replace the C{_fprs2} value.

1659 '''

1660 return Fsum2Tuple(s_r)

1661

1662 def fset_(self, *xs):

1663 '''Apply C{B{self}.partials = Fsum(*B{xs}).partials}.

1664

1665 @arg xs: Optional, new values (each C{scalar} or

1666 an L{Fsum} or L{Fsum2Tuple} instance), all

1667 positional.

1668

1669 @return: This instance, replaced (C{Fsum}).

1670

1671 @see: Method L{Fsum.fadd} for further details.

1672 '''

1673 f = (xs[0] if xs else _0_0) if len(xs) < 2 else \

1674 Fsum(*xs, nonfinites=self.nonfinites()) # self._Fsum_as(*xs)

1675 return self._fset(f, op=_fset_op_)

1676

1677 def _fset(self, other, n=0, up=True, **op):

1678 '''(INTERNAL) Overwrite this instance with an other or a C{scalar}.

1679 '''

1680 if other is self:

1681 pass # from ._fmul, ._ftruediv and ._pow_0_1

1682 elif _isFsum_2Tuple(other):

1683 if op: # and not self.nonfinitesOK:

1684 self._finite(other._fprs, **op)

1685 self._ps[:] = other._ps

1686 self._n = n or other._n

1687 if up: # use or zap the C{Property_RO} values

1688 Fsum._fint2._update_from(self, other)

1689 Fsum._fprs ._update_from(self, other)

1690 Fsum._fprs2._update_from(self, other)

1691 elif isscalar(other):

1692 s = float(self._finite(other, **op)) if op else other

1693 self._ps[:] = s,

1694 self._n = n or 1

1695 if up: # Property _fint2, _fprs and _fprs2 all have

1696 # @.setter_underscore and NOT @.setter because the

1697 # latter's _fset zaps the value set by @.setter

1698 self._fint2 = s

1699 self._fprs = s

1700 self._fprs2 = s, INT0

1701 # assert self._fprs is s

1702 else:

1703 op = _xkwds_get1(op, op=_fset_op_)

1704 raise self._Error(op, other, _TypeError)

1705 return self

1706

1707 def fsub(self, xs=()):

1708 '''Subtract an iterable's items from this instance.

1709

1710 @see: Method L{Fsum.fadd} for further details.

1711 '''

1712 return self._facc_neg(xs)

1713

1714 def fsub_(self, *xs):

1715 '''Subtract all positional items from this instance.

1716

1717 @see: Method L{Fsum.fadd_} for further details.

1718 '''

1719 return self._fsub(xs[0], _sub_op_) if len(xs) == 1 else \

1720 self._facc_neg(xs) # origin=1?

1721

1722 def _fsub(self, other, op):

1723 '''(INTERNAL) Apply C{B{self} -= B{other}}.

1724 '''

1725 if _isFsum_2Tuple(other):

1726 if other is self: # or other._fprs2 == self._fprs2:

1727 self._fset(_0_0, n=len(self) * 2)

1728 elif other._ps:

1729 self._facc_scalar(other._ps_neg)

1730 elif self._scalar(other, op):

1731 self._facc_scalar_(-other)

1732 return self

1733

1734 def fsum(self, xs=()):

1735 '''Add an iterable's items, summate and return the current

1736 precision running sum.

1737

1738 @arg xs: Iterable of items to add (each item C{scalar},

1739 an L{Fsum} or L{Fsum2Tuple}).

1740

1741 @return: Precision running sum (C{float} or C{int}).

1742

1743 @see: Method L{Fsum.fadd}.

1744

1745 @note: Accumulation can continue after summation.

1746 '''

1747 return self._facc(xs)._fprs

1748

1749 def fsum_(self, *xs):

1750 '''Add any positional items, summate and return the current

1751 precision running sum.

1752

1753 @arg xs: Items to add (each C{scalar}, an L{Fsum} or

1754 L{Fsum2Tuple}), all positional.

1755

1756 @return: Precision running sum (C{float} or C{int}).

1757

1758 @see: Methods L{Fsum.fsum}, L{Fsum.Fsum_} and L{Fsum.fsumf_}.

1759 '''

1760 return self._facc_args(xs)._fprs

1761

1762 def Fsum_(self, *xs, **name):

1763 '''Like method L{Fsum.fsum_} but returning a named L{Fsum}.

1764

1765 @kwarg name: Optional name (C{str}).

1766

1767 @return: Copy of this updated instance (L{Fsum}).

1768 '''

1769 return self._facc_args(xs)._copy_2(self.Fsum_, **name)

1770

1771 def Fsum2Tuple_(self, *xs, **name):

1772 '''Like method L{Fsum.fsum_} but returning a named L{Fsum2Tuple}.

1773

1774 @kwarg name: Optional name (C{str}).

1775

1776 @return: Precision running sum (L{Fsum2Tuple}).

1777 '''

1778 return Fsum2Tuple(self._facc_args(xs)._nfprs2, **name)

1779

1780 @property_RO

1781 def _Fsum(self): # like L{Fsum2Tuple._Fsum}, in .fstats

1782 return self # NOT @Property_RO, see .copy and ._copy_2

1783

1784 def _Fsum_as(self, *xs, **name_f2product_nonfinites_RESIDUAL):

1785 '''(INTERNAL) Return an C{Fsum} with this C{Fsum}'s C{.f2product},

1786 C{.nonfinites} and C{.RESIDUAL} setting, optionally

1787 overridden with C{name_f2product_nonfinites_RESIDUAL} and

1788 with any C{xs} accumulated.

1789 '''

1790 kwds = _xkwds_not(None, Fsum._RESIDUAL, f2product =self.f2product(),

1791 nonfinites=self.nonfinites(),

1792 RESIDUAL =self.RESIDUAL())

1793 if name_f2product_nonfinites_RESIDUAL: # overwrites

1794 kwds.update(name_f2product_nonfinites_RESIDUAL)

1795 f = Fsum(**kwds)

1796 # assert all(v == self.__dict__[n] for n, v in f.__dict__.items())

1797 return f._fset(xs[0], op=_fset_op_) if len(xs) == 1 else (

1798 f._facc(xs, up=False) if xs else f)

1799

1800 def fsum2(self, xs=(), **name):

1801 '''Add an iterable's items, summate and return the

1802 current precision running sum I{and} the C{residual}.

1803

1804 @arg xs: Iterable of items to add (each item C{scalar},

1805 an L{Fsum} or L{Fsum2Tuple}).

1806 @kwarg name: Optional C{B{name}=NN} (C{str}).

1807

1808 @return: L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} the

1809 current precision running sum and C{residual}, the

1810 (precision) sum of the remaining C{partials}. The

1811 C{residual is INT0} if the C{fsum} is considered

1812 to be I{exact}.

1813

1814 @see: Methods L{Fsum.fint2}, L{Fsum.fsum} and L{Fsum.fsum2_}

1815 '''

1816 t = self._facc(xs)._fprs2

1817 return t.dup(name=name) if name else t

1818

1819 def fsum2_(self, *xs):

1820 '''Add any positional items, summate and return the current

1821 precision running sum and the I{differential}.

1822

1823 @arg xs: Values to add (each C{scalar}, an L{Fsum} or

1824 L{Fsum2Tuple}), all positional.

1825

1826 @return: 2Tuple C{(fsum, delta)} with the current, precision

1827 running C{fsum} like method L{Fsum.fsum} and C{delta},

1828 the difference with previous running C{fsum}, C{float}.

1829

1830 @see: Methods L{Fsum.fsum_} and L{Fsum.fsum}.

1831 '''

1832 return self._fsum2(xs, self._facc_args)

1833

1834 def _fsum2(self, xs, _facc, **facc_kwds):

1835 '''(INTERNAL) Helper for L{Fsum.fsum2_} and L{Fsum.fsum2f_}.

1836 '''

1837 p, q = self._fprs2

1838 if xs:

1839 s, r = _facc(xs, **facc_kwds)._fprs2

1840 if _isfinite(s): # _fsum(_1primed((s, -p, r, -q))

1841 d, r = _2sum(s - p, r - q, _isfine=_isOK)

1842 r, _ = _s_r(d, r)

1843 return s, (r if _isfinite(r) else _NONFINITEr)

1844 else:

1845 return p, _0_0

1846

1847 def fsumf_(self, *xs):

1848 '''Like method L{Fsum.fsum_} iff I{all} C{B{xs}}, each I{known to be}

1849 C{scalar}, an L{Fsum} or L{Fsum2Tuple}.

1850 '''

1851 return self._facc_scalarf(xs, which=self.fsumf_)._fprs # origin=1?

1852

1853 def Fsumf_(self, *xs):

1854 '''Like method L{Fsum.Fsum_} iff I{all} C{B{xs}}, each I{known to be}

1855 C{scalar}, an L{Fsum} or L{Fsum2Tuple}.

1856 '''

1857 return self._facc_scalarf(xs, which=self.Fsumf_)._copy_2(self.Fsumf_) # origin=1?

1858

1859 def fsum2f_(self, *xs):

1860 '''Like method L{Fsum.fsum2_} iff I{all} C{B{xs}}, each I{known to be}

1861 C{scalar}, an L{Fsum} or L{Fsum2Tuple}.

1862 '''

1863 return self._fsum2(xs, self._facc_scalarf, which=self.fsum2f_) # origin=1?

1864

1865# ftruediv = __itruediv__ # for naming consistency?

1866

1867 def _ftruediv(self, other, op, **raiser_RESIDUAL):

1868 '''(INTERNAL) Apply C{B{self} /= B{other}}.

1869 '''

1870 n = _1_0

1871 if _isFsum_2Tuple(other):

1872 if other is self or self == other:

1873 return self._fset(n, n=len(self))

1874 d, r = other._fprs2

1875 if r:

1876 R = self._raiser(r, d, **raiser_RESIDUAL)

1877 if R:

1878 raise self._ResidualError(op, other, r, **R)

1879 d, n = other.as_integer_ratio()

1880 else:

1881 d = self._scalar(other, op)

1882 try:

1883 s = n / d

1884 except Exception as X:

1885 raise self._ErrorX(X, op, other)

1886 f = self._mul_scalar(s, _mul_op_) # handles 0, INF, NAN

1887 return self._fset(f)

1888

1889 @property_RO

1890 def imag(self):

1891 '''Get the C{imaginary} part of this instance (C{0.0}, always).

1892

1893 @see: Property L{Fsum.real}.

1894 '''

1895 return _0_0

1896

1897 def int_float(self, **raiser_RESIDUAL):

1898 '''Return this instance' current running sum as C{int} or C{float}.

1899

1900 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore

1901 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}

1902 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

1903

1904 @return: This C{int} sum if this instance C{is_integer} and

1905 I{finite}, otherwise the C{float} sum if the residual

1906 is zero or not significant.

1907

1908 @raise ResidualError: Non-zero, significant residual or invalid

1909 B{C{RESIDUAL}}.

1910

1911 @see: Methods L{Fsum.fint}, L{Fsum.fint2}, L{Fsum.is_integer},

1912 L{Fsum.RESIDUAL} and property L{Fsum.as_iscalar}.

1913 '''

1914 s, r = self._fint2

1915 if r:

1916 s, r = self._fprs2

1917 if r: # PYCHOK no cover

1918 R = self._raiser(r, s, **raiser_RESIDUAL)

1919 if R:

1920 t = _stresidual(_non_zero_, r, **R)

1921 raise ResidualError(int_float=s, txt=t)

1922 s = float(s)

1923 return s

1924

1925 def is_exact(self):

1926 '''Is this instance' running C{fsum} considered to be exact?

1927 (C{bool}), C{True} only if the C{residual is }L{INT0}.

1928 '''

1929 return self.residual is INT0

1930

1931 def is_finite(self): # in .constants

1932 '''Is this instance C{finite}? (C{bool}).

1933

1934 @see: Function L{isfinite<pygeodesy.isfinite>}.

1935 '''

1936 return _isfinite(sum(self._ps)) # == sum(self)

1937

1938 def is_integer(self):

1939 '''Is this instance' running sum C{integer}? (C{bool}).

1940

1941 @see: Methods L{Fsum.fint}, L{Fsum.fint2} and L{Fsum.is_scalar}.

1942 '''

1943 s, r = self._fint2

1944 return False if r else (_isfinite(s) and isint(s))

1945

1946 def is_math_fma(self):

1947 '''Is accurate L{f2product} multiplication based on Python's C{math.fma}?

1948

1949 @return: C{True} if accurate multiplication uses C{math.fma}, C{False}

1950 an C{fma} implementation as C{math.fma} or C{None}, a previous

1951 C{PyGeodesy} implementation.

1952 '''

1953 return (_2split3s is _passarg) or (False if _2n_d is None else None)

1954

1955 def is_math_fsum(self):

1956 '''Are the summation functions L{fsum}, L{fsum_}, L{fsumf_}, L{fsum1},

1957 L{fsum1_} and L{fsum1f_} based on Python's C{math.fsum}?

1958

1959 @return: C{True} if summation functions use C{math.fsum}, C{False}

1960 otherwise.

1961 '''

1962 return _sum is _fsum # _fsum.__module__ is fabs.__module__

1963

1964 def is_scalar(self, **raiser_RESIDUAL):

1965 '''Is this instance' running sum C{scalar} with C{0} residual or with

1966 a residual I{ratio} not exceeding the RESIDUAL threshold?

1967

1968 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore

1969 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}

1970 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

1971

1972 @return: C{True} if this instance' residual is C{0} or C{insignificant},

1973 i.e. its residual C{ratio} doesn't exceed the L{RESIDUAL

1974 <Fsum.RESIDUAL>} threshold (C{bool}).

1975

1976 @raise ResidualError: Non-zero, significant residual or invalid

1977 B{C{RESIDUAL}}.

1978

1979 @see: Methods L{Fsum.RESIDUAL} and L{Fsum.is_integer} and property

1980 L{Fsum.as_iscalar}.

1981 '''

1982 s, r = self._fprs2

1983 return False if r and self._raiser(r, s, **raiser_RESIDUAL) else True

1984

1985 def _mul_Fsum(self, other, op):

1986 '''(INTERNAL) Return C{B{self} * B{other}} as L{Fsum} or C{0}.

1987 '''

1988 # assert _isFsum_2Tuple(other)

1989 if self._ps and other._ps:

1990 try:

1991 f = self._ps_mul(op, *other._ps) # NO .as_iscalar!

1992 except Exception as X:

1993 raise self._ErrorX(X, op, other)

1994 else:

1995 f = _0_0

1996 return f

1997

1998 def _mul_reduce(self, *others):

1999 '''(INTERNAL) Like fmath.fprod for I{non-finite} C{other}s.

2000 '''

2001 r = _1_0

2002 for f in others:

2003 r *= sum(f._ps) if _isFsum_2Tuple(f) else float(f)

2004 return r

2005

2006 def _mul_scalar(self, factor, op):

2007 '''(INTERNAL) Return C{B{self} * scalar B{factor}} as L{Fsum}, C{0.0} or C{self}.

2008 '''

2009 # assert isscalar(factor)

2010 if self._ps and self._finite(factor, op=op):

2011 f = self if factor == _1_0 else (

2012 self._neg if factor == _N_1_0 else

2013 self._ps_mul(op, factor).as_iscalar)

2014 else:

2015 f = _0_0

2016 return f

2017

2018# @property_RO

2019# def _n_d(self):

2020# n, d = self.as_integer_ratio()

2021# return n / d

2022

2023 @property_RO

2024 def _neg(self):

2025 '''(INTERNAL) Return C{Fsum(-self)} or scalar C{NEG0}.

2026 '''

2027 return _Psum(self._ps_neg) if self._ps else NEG0

2028

2029 @property_RO

2030 def _nfprs2(self):

2031 '''(INTERNAL) Handle I{non-finite} C{_fprs2}.

2032 '''

2033 try: # to handle nonfiniterrors, etc.

2034 t = self._fprs2

2035 except (OverflowError, ValueError):

2036 t = Fsum2Tuple(sum(self._ps), _NONFINITEr)

2037 return t

2038

2039 def nonfinites(self, *OK):

2040 '''Handle I{non-finite} C{float}s as C{inf}, C{INF}, C{NINF}, C{nan}

2041 and C{NAN} for this L{Fsum} or throw C{OverflowError} respectively

2042 C{ValueError} exceptions, overriding the L{nonfiniterrors} default.

2043

2044 @arg OK: If omitted, leave the override unchanged, if C{True},

2045 I{non-finites} are C{OK}, if C{False} throw exceptions

2046 or if C{None} remove the override (C{bool} or C{None}).

2047

2048 @return: The previous setting (C{bool} or C{None} if not set).

2049

2050 @see: Function L{nonfiniterrors<fsums.nonfiniterrors>}.

2051

2052 @note: Use property L{nonfinitesOK<Fsum.nonfinitesOK>} to determine

2053 whether I{non-finites} are C{OK} for this L{Fsum} and by the

2054 L{nonfiniterrors} default.

2055 '''

2056 _ks = Fsum._nonfinites_isfine_kwds

2057 if OK: # delattrof(self, _isfine=None)

2058 k = _xkwds_pop(self.__dict__, _isfine=None)

2059 if OK[0] is not None:

2060 self._isfine = _ks[bool(OK[0])]

2061 self._update()

2062 else: # getattrof(self, _isfine=None)

2063 k = _xkwds_get(self.__dict__, _isfine=None)

2064 # dict(map(reversed, _ks.items())).get(k, None)

2065 # raises a TypeError: unhashable type: 'dict'

2066 return True if k is _ks[True] else (

2067 False if k is _ks[False] else None)

2068

2069 _nonfinites_isfine_kwds = {True: dict(_isfine=_isOK),

2070 False: dict(_isfine=_isfinite)}

2071

2072 @property_RO

2073 def nonfinitesOK(self):

2074 '''Are I{non-finites} C{OK} for this L{Fsum} or by default? (C{bool}).

2075 '''

2076# nf = self.nonfinites()

2077# if nf is None:

2078# nf = not nonfiniterrors()

2079 return _isOK_or_finite(INF, **self._isfine)

2080

2081 def _nonfiniteX(self, X, op, f, nonfinites=None, raiser=None):

2082 '''(INTERNAL) Handle a I{non-finite} exception.

2083 '''

2084 if nonfinites is None:

2085 nonfinites = _isOK_or_finite(f, **self._isfine) if raiser is None else (not raiser)

2086 if not nonfinites:

2087 raise self._ErrorX(X, op, f)

2088 return f

2089

2090 def _optionals(self, f2product=None, nonfinites=None, **name_RESIDUAL):

2091 '''(INTERNAL) Re/set options from keyword arguments.

2092 '''

2093 if f2product is not None:

2094 self.f2product(f2product)

2095 if nonfinites is not None:

2096 self.nonfinites(nonfinites)

2097 if name_RESIDUAL: # MUST be last

2098 n, kwds = _name2__(**name_RESIDUAL)

2099 if kwds:

2100 R = Fsum._RESIDUAL

2101 t = _threshold(R, **kwds)

2102 if t != R:

2103 self._RESIDUAL = t

2104 if n:

2105 self.name = n # self.rename(n)

2106

2107 def _1_Over(self, x, op, **raiser_RESIDUAL): # vs _1_over

2108 '''(INTERNAL) Return C{Fsum(1) / B{x}}.

2109 '''

2110 return self._Fsum_as(_1_0)._ftruediv(x, op, **raiser_RESIDUAL)

2111

2112 @property_RO

2113 def partials(self):

2114 '''Get this instance' current, partial sums (C{tuple} of C{float}s).

2115 '''

2116 return tuple(self._ps)

2117

2118 def pow(self, x, *mod, **raiser_RESIDUAL):

2119 '''Return C{B{self}**B{x}} as L{Fsum}.

2120

2121 @arg x: The exponent (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

2122 @arg mod: Optional modulus (C{int} or C{None}) for the 3-argument

2123 C{pow(B{self}, B{other}, B{mod})} version.

2124 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore

2125 L{ResidualError}s (C{bool}) and C{B{RESIDUAL}=scalar}

2126 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

2127

2128 @return: The C{pow(self, B{x})} or C{pow(self, B{x}, *B{mod})}

2129 result (L{Fsum}).

2130

2131 @raise ResidualError: Non-zero, significant residual or invalid

2132 B{C{RESIDUAL}}.

2133

2134 @note: If B{C{mod}} is given and C{None}, the result will be an

2135 C{integer} L{Fsum} provided this instance C{is_integer}

2136 or set to C{integer} by an L{Fsum.fint} call.

2137

2138 @see: Methods L{Fsum.__ipow__}, L{Fsum.fint}, L{Fsum.is_integer}

2139 and L{Fsum.root}.

2140 '''

2141 f = self._copy_2(self.pow)

2142 return f._fpow(x, _pow_op_, *mod, **raiser_RESIDUAL) # f = pow(f, x, *mod)

2143

2144 def _pow(self, other, unused, op, **raiser_RESIDUAL):

2145 '''Return C{B{self} ** B{other}}.

2146 '''

2147 if _isFsum_2Tuple(other):

2148 f = self._pow_Fsum(other, op, **raiser_RESIDUAL)

2149 elif self._scalar(other, op):

2150 x = self._finite(other, op=op)

2151 f = self._pow_scalar(x, other, op, **raiser_RESIDUAL)

2152 else:

2153 f = self._pow_0_1(0, other)

2154 return f

2155

2156 def _pow_0_1(self, x, other):

2157 '''(INTERNAL) Return B{C{self}**1} or C{B{self}**0 == 1.0}.

2158 '''

2159 return self if x else (1 if isint(other) and self.is_integer() else _1_0)

2160

2161 def _pow_2_3(self, b, x, other, op, *mod, **raiser_RESIDUAL):

2162 '''(INTERNAL) 2-arg C{pow(B{b}, scalar B{x})} and 3-arg C{pow(B{b},

2163 B{x}, int B{mod} or C{None})}, embellishing errors.

2164 '''

2165

2166 if mod: # b, x, mod all C{int}, unless C{mod} is C{None}

2167 m = mod[0]

2168 # assert _isFsum_2Tuple(b)

2169

2170 def _s(s, r):

2171 R = self._raiser(r, s, **raiser_RESIDUAL)

2172 if R:

2173 raise self._ResidualError(op, other, r, mod=m, **R)

2174 return s

2175

2176 b = _s(*(b._fprs2 if m is None else b._fint2))

2177 x = _s(*_2s_r(x))

2178

2179 try:

2180 # 0**INF == 0.0, 1**INF == 1.0, -1**2.3 == -(1**2.3)

2181 s = pow(b, x, *mod)

2182 if iscomplex(s):

2183 # neg**frac == complex in Python 3+, but ValueError in 2-

2184 raise ValueError(_strcomplex(s, b, x, *mod))

2185 _ = _2finite(s, **self._isfine) # ignore float

2186 return s

2187 except Exception as X:

2188 raise self._ErrorX(X, op, other, *mod)

2189

2190 def _pow_Fsum(self, other, op, **raiser_RESIDUAL):

2191 '''(INTERNAL) Return C{B{self} **= B{other}} for C{_isFsum_2Tuple(other)}.

2192 '''

2193 # assert _isFsum_2Tuple(other)

2194 x, r = other._fprs2

2195 f = self._pow_scalar(x, other, op, **raiser_RESIDUAL)

2196 if f and r:

2197 f *= self._pow_scalar(r, other, op, **raiser_RESIDUAL)

2198 return f

2199

2200 def _pow_int(self, x, other, op, **raiser_RESIDUAL):

2201 '''(INTERNAL) Return C{B{self} **= B{x}} for C{int B{x} >= 0}.

2202 '''

2203 # assert isint(x) and x >= 0

2204 ps = self._ps

2205 if len(ps) > 1:

2206 _mul_Fsum = Fsum._mul_Fsum

2207 if x > 4:

2208 p = self

2209 f = self if (x & 1) else self._Fsum_as(_1_0)

2210 m = x >> 1 # // 2

2211 while m:

2212 p = _mul_Fsum(p, p, op) # p **= 2

2213 if (m & 1):

2214 f = _mul_Fsum(f, p, op) # f *= p

2215 m >>= 1 # //= 2

2216 elif x > 1: # self**2, 3, or 4

2217 f = _mul_Fsum(self, self, op)

2218 if x > 2: # self**3 or 4

2219 p = self if x < 4 else f

2220 f = _mul_Fsum(f, p, op)

2221 else: # self**1 or self**0 == 1 or _1_0

2222 f = self._pow_0_1(x, other)

2223 elif ps: # self._ps[0]**x

2224 f = self._pow_2_3(ps[0], x, other, op, **raiser_RESIDUAL)

2225 else: # PYCHOK no cover

2226 # 0**pos_int == 0, but 0**0 == 1

2227 f = 0 if x else 1

2228 return f

2229

2230 def _pow_scalar(self, x, other, op, **raiser_RESIDUAL):

2231 '''(INTERNAL) Return C{self**B{x}} for C{scalar B{x}}.

2232 '''

2233 s, r = self._fprs2

2234 if r:

2235 # assert s != 0

2236 if isint(x, both=True): # self**int

2237 x = int(x)

2238 y = abs(x)

2239 if y > 1:

2240 f = self._pow_int(y, other, op, **raiser_RESIDUAL)

2241 if x > 0: # i.e. > 1

2242 return f # Fsum or scalar

2243 # assert x < 0 # i.e. < -1

2244 if _isFsum(f):

2245 s, r = f._fprs2

2246 if r:

2247 return self._1_Over(f, op, **raiser_RESIDUAL)

2248 else: # scalar

2249 s = f

2250 # use s**(-1) to get the CPython

2251 # float_pow error iff s is zero

2252 x = -1

2253 elif x < 0: # self**(-1)

2254 return self._1_Over(self, op, **raiser_RESIDUAL) # 1 / self

2255 else: # self**1 or self**0

2256 return self._pow_0_1(x, other) # self, 1 or 1.0

2257 else: # self**fractional

2258 R = self._raiser(r, s, **raiser_RESIDUAL)

2259 if R:

2260 raise self._ResidualError(op, other, r, **R)

2261 n, d = self.as_integer_ratio()

2262 if abs(n) > abs(d):

2263 n, d, x = d, n, (-x)

2264 s = n / d

2265 # assert isscalar(s) and isscalar(x)

2266 return self._pow_2_3(s, x, other, op, **raiser_RESIDUAL)

2267

2268 def _ps_acc(self, ps, xs, up=True, **unused):

2269 '''(INTERNAL) Accumulate C{xs} known scalars into list C{ps}.

2270 '''

2271 n = 0

2272 _2s = _2sum

2273 _fi = self._isfine

2274 for x in (tuple(xs) if xs is ps else xs):

2275 # assert isscalar(x) and _isOK_or_finite(x, **self._isfine)

2276 if x:

2277 i = 0

2278 for p in ps:

2279 x, p = _2s(x, p, **_fi)

2280 if p:

2281 ps[i] = p

2282 i += 1

2283 ps[i:] = (x,) if x else ()

2284 n += 1

2285 if n:

2286 self._n += n

2287 # Fsum._ps_max = max(Fsum._ps_max, len(ps))

2288 if up:

2289 self._update()

2290# x = sum(ps)

2291# if not _isOK_or_finite(x, **fi):

2292# ps[:] = x, # collapse ps

2293 return ps

2294

2295 def _ps_mul(self, op, *factors):

2296 '''(INTERNAL) Multiply this instance' C{partials} with

2297 each scalar C{factor} and accumulate into an C{Fsum}.

2298 '''

2299 def _psfs(ps, fs, _isfine=_isfinite):

2300 if len(ps) < len(fs):

2301 ps, fs = fs, ps

2302 if self._f2product:

2303 fs, p = _2split3s(fs), fs

2304 if len(ps) > 1 and fs is not p:

2305 fs = tuple(fs) # several ps

2306 _pfs = _2products

2307 else:

2308 def _pfs(p, fs):

2309 return (p * f for f in fs)

2310

2311 for p in ps:

2312 for f in _pfs(p, fs):

2313 yield f if _isfine(f) else _nfError(f)

2314

2315 fs = _psfs(self._ps, factors, **self._isfine)

2316 f = _Psum(self._ps_acc([], fs, up=False), name=op)

2317 return f

2318

2319 @property_RO

2320 def _ps_neg(self):

2321 '''(INTERNAL) Yield the partials, I{negated}.

2322 '''

2323 for p in self._ps:

2324 yield -p

2325

2326 def _ps_other(self, op, other):

2327 '''(INTERNAL) Yield C{other} as C{scalar}s.

2328 '''

2329 if _isFsum_2Tuple(other):

2330 for p in other._ps:

2331 yield p

2332 else:

2333 yield self._scalar(other, op)

2334

2335 def _ps_1sum(self, *less):

2336 '''(INTERNAL) Return the partials sum, 1-primed C{less} some scalars.

2337 '''

2338 def _1psls(ps, ls):

2339 yield _1_0

2340 for p in ps:

2341 yield p

2342 for p in ls:

2343 yield -p

2344 yield _N_1_0

2345

2346 return _fsum(_1psls(self._ps, less))

2347

2348 def _raiser(self, r, s, raiser=True, **RESIDUAL):

2349 '''(INTERNAL) Does ratio C{r / s} exceed the RESIDUAL threshold

2350 I{and} is residual C{r} I{non-zero} or I{significant} (for a

2351 negative respectively positive C{RESIDUAL} threshold)?

2352 '''

2353 if r and raiser:

2354 t = self._RESIDUAL

2355 if RESIDUAL:

2356 t = _threshold(t, **RESIDUAL)

2357 if t < 0 or (s + r) != s:

2358 q = (r / s) if s else s # == 0.

2359 if fabs(q) > fabs(t):

2360 return dict(ratio=q, R=t)

2361 return {}

2362

2363 rdiv = __rtruediv__

2364

2365 @property_RO

2366 def real(self):

2367 '''Get the C{real} part of this instance (C{float}).

2368

2369 @see: Methods L{Fsum.__float__} and L{Fsum.fsum}

2370 and properties L{Fsum.ceil}, L{Fsum.floor},

2371 L{Fsum.imag} and L{Fsum.residual}.

2372 '''

2373 return float(self)

2374

2375 @property_RO

2376 def residual(self):

2377 '''Get this instance' residual or residue (C{float} or C{int}):

2378 the C{sum(partials)} less the precision running sum C{fsum}.

2379

2380 @note: The C{residual is INT0} iff the precision running

2381 C{fsum} is considered to be I{exact}.

2382

2383 @see: Methods L{Fsum.fsum}, L{Fsum.fsum2} and L{Fsum.is_exact}.

2384 '''

2385 return self._fprs2.residual

2386

2387 def RESIDUAL(self, *threshold):

2388 '''Get and set this instance' I{ratio} for raising L{ResidualError}s,

2389 overriding the default from env variable C{PYGEODESY_FSUM_RESIDUAL}.

2390

2391 @arg threshold: If C{scalar}, the I{ratio} to exceed for raising

2392 L{ResidualError}s in division and exponention, if

2393 C{None}, restore the default set with env variable

2394 C{PYGEODESY_FSUM_RESIDUAL} or if omitted, keep the

2395 current setting.

2396

2397 @return: The previous C{RESIDUAL} setting (C{float}), default C{0.0}.

2398

2399 @raise ResidualError: Invalid B{C{threshold}}.

2400

2401 @note: L{ResidualError}s may be thrown if (1) the non-zero I{ratio}

2402 C{residual / fsum} exceeds the given B{C{threshold}} and (2)

2403 the C{residual} is non-zero and (3) is I{significant} vs the

2404 C{fsum}, i.e. C{(fsum + residual) != fsum} and (4) optional

2405 keyword argument C{raiser=False} is missing. Specify a

2406 negative B{C{threshold}} for only non-zero C{residual}

2407 testing without the I{significant} case.

2408 '''

2409 r = self._RESIDUAL

2410 if threshold:

2411 t = threshold[0]

2412 self._RESIDUAL = Fsum._RESIDUAL if t is None else ( # for ...

2413 (_0_0 if t else _1_0) if isbool(t) else

2414 _threshold(t)) # ... backward compatibility

2415 return r

2416

2417 def _ResidualError(self, op, other, residual, **mod_R):

2418 '''(INTERNAL) Non-zero B{C{residual}} etc.

2419 '''

2420 def _p(mod=None, R=0, **unused): # ratio=0

2421 return (_non_zero_ if R < 0 else _significant_) \

2422 if mod is None else _integer_

2423

2424 t = _stresidual(_p(**mod_R), residual, **mod_R)

2425 return self._Error(op, other, ResidualError, txt=t)

2426

2427 def root(self, root, **raiser_RESIDUAL):

2428 '''Return C{B{self}**(1 / B{root})} as L{Fsum}.

2429

2430 @arg root: Non-zero order (C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

2431 @kwarg raiser_RESIDUAL: Use C{B{raiser}=False} to ignore any

2432 L{ResidualError}s (C{bool}) or C{B{RESIDUAL}=scalar}

2433 to override the current L{RESIDUAL<Fsum.RESIDUAL>}.

2434

2435 @return: The C{self ** (1 / B{root})} result (L{Fsum}).

2436

2437 @raise ResidualError: Non-zero, significant residual or invalid

2438 B{C{RESIDUAL}}.

2439

2440 @see: Method L{Fsum.pow}.

2441 '''

2442 x = self._1_Over(root, _truediv_op_, **raiser_RESIDUAL)

2443 f = self._copy_2(self.root)

2444 return f._fpow(x, f.name, **raiser_RESIDUAL) # == pow(f, x)

2445

2446 def _scalar(self, other, op, **txt):

2447 '''(INTERNAL) Return scalar C{other} or throw a C{TypeError}.

2448 '''

2449 if isscalar(other):

2450 return other

2451 raise self._Error(op, other, _TypeError, **txt) # _invalid_

2452

2453 def signOf(self, res=True):

2454 '''Determine the sign of this instance.

2455

2456 @kwarg res: If C{True}, consider the residual,

2457 otherwise ignore the latter (C{bool}).

2458

2459 @return: The sign (C{int}, -1, 0 or +1).

2460 '''

2461 s, r = self._nfprs2

2462 r = (-r) if res else 0

2463 return _signOf(s, r)

2464

2465 def toRepr(self, **lenc_prec_sep_fmt): # PYCHOK signature

2466 '''Return this C{Fsum} instance as representation.

2467

2468 @kwarg lenc_prec_sep_fmt: Optional keyword arguments

2469 for method L{Fsum.toStr}.

2470

2471 @return: This instance (C{repr}).

2472 '''

2473 return Fmt.repr_at(self, self.toStr(**lenc_prec_sep_fmt))

2474

2475 def toStr(self, lenc=True, **prec_sep_fmt): # PYCHOK signature

2476 '''Return this C{Fsum} instance as string.

2477

2478 @kwarg lenc: If C{True}, include the current C{[len]} of this

2479 L{Fsum} enclosed in I{[brackets]} (C{bool}).

2480 @kwarg prec_sep_fmt: Optional keyword arguments for method

2481 L{Fsum2Tuple.toStr}.

2482

2483 @return: This instance (C{str}).

2484 '''

2485 p = self.classname

2486 if lenc:

2487 p = Fmt.SQUARE(p, len(self))

2488 n = _enquote(self.name, white=_UNDER_)

2489 t = self._nfprs2.toStr(**prec_sep_fmt)

2490 return NN(p, _SPACE_, n, t)

2491

2492 def _truediv(self, other, op, **raiser_RESIDUAL):

2493 '''(INTERNAL) Return C{B{self} / B{other}} as an L{Fsum}.

2494 '''

2495 f = self._copy_2(self.__truediv__)

2496 return f._ftruediv(other, op, **raiser_RESIDUAL)

2497

2498 def _update(self, updated=True): # see ._fset

2499 '''(INTERNAL) Zap all cached C{Property_RO} values.

2500 '''

2501 if updated:

2502 _pop = self.__dict__.pop

2503 for p in _ROs:

2504 _ = _pop(p, None)

2505# Fsum._fint2._update(self)

2506# Fsum._fprs ._update(self)

2507# Fsum._fprs2._update(self)

2508 return self # for .fset_

2509

2510_ROs = _allPropertiesOf_n(3, Fsum, Property_RO) # PYCHOK see Fsum._update

2511

2512 if _NONFINITES == _std_: # PYCHOK no cover

2513 _ = nonfiniterrors(False)

2514

2515

2516 def _Float_Int(arg, **name_Error):

2517 '''(INTERNAL) L{DivMod2Tuple}, L{Fsum2Tuple} Unit.

2518 '''

2519 U = Int if isint(arg) else Float

2520 return U(arg, **name_Error)

2521

2522

2523 class DivMod2Tuple(_NamedTuple):

2524 '''2-Tuple C{(div, mod)} with the quotient C{div} and remainder

2525 C{mod} results of a C{divmod} operation.

2526

2527 @note: Quotient C{div} an C{int} in Python 3+ but a C{float}

2528 in Python 2-. Remainder C{mod} an L{Fsum} instance.

2529 '''

2530 _Names_ = ('div', 'mod')

2531 _Units_ = (_Float_Int, Fsum)

2532

2533

2534 class Fsum2Tuple(_NamedTuple): # in .fstats

2535 '''2-Tuple C{(fsum, residual)} with the precision running C{fsum}

2536 and the C{residual}, the sum of the remaining partials. Each

2537 item is C{float} or C{int}.

2538

2539 @note: If the C{residual is INT0}, the C{fsum} is considered

2540 to be I{exact}, see method L{Fsum2Tuple.is_exact}.

2541 '''

2542 _Names_ = ( Fsum.fsum.__name__, Fsum.residual.name)

2543 _Units_ = (_Float_Int, _Float_Int)

2544

2545 def __abs__(self): # in .fmath

2546 return self._Fsum.__abs__()

2547

2548 def __bool__(self): # PYCHOK Python 3+

2549 return bool(self._Fsum)

2550

2551 def __eq__(self, other):

2552 return self._other_op(other, self.__eq__)

2553

2554 def __float__(self):

2555 return self._Fsum.__float__()

2556

2557 def __ge__(self, other):

2558 return self._other_op(other, self.__ge__)

2559

2560 def __gt__(self, other):

2561 return self._other_op(other, self.__gt__)

2562

2563 def __le__(self, other):

2564 return self._other_op(other, self.__le__)

2565

2566 def __lt__(self, other):

2567 return self._other_op(other, self.__lt__)

2568

2569 def __int__(self):

2570 return self._Fsum.__int__()

2571

2572 def __ne__(self, other):

2573 return self._other_op(other, self.__ne__)

2574

2575 def __neg__(self):

2576 return self._Fsum.__neg__()

2577

2578 __nonzero__ = __bool__ # Python 2-

2579

2580 def __pos__(self):

2581 return self._Fsum.__pos__()

2582

2583 def as_integer_ratio(self):

2584 '''Return this instance as the ratio of 2 integers.

2585

2586 @see: Method L{Fsum.as_integer_ratio} for further details.

2587 '''

2588 return self._Fsum.as_integer_ratio()

2589

2590 @property_RO

2591 def _fint2(self):

2592 return self._Fsum._fint2

2593

2594 @property_RO

2595 def _fprs2(self):

2596 return self._Fsum._fprs2

2597

2598 @Property_RO

2599 def _Fsum(self): # this C{Fsum2Tuple} as L{Fsum}, in .fstats

2600 s, r = _s_r(*self)

2601 ps = (r, s) if r else (s,)

2602 return _Psum(ps, name=self.name)

2603

2604 def Fsum_(self, *xs, **name_f2product_nonfinites_RESIDUAL):

2605 '''Return this C{Fsum2Tuple} as an L{Fsum} plus some C{xs}.

2606 '''

2607 return Fsum(self, *xs, **name_f2product_nonfinites_RESIDUAL)

2608

2609 def is_exact(self):

2610 '''Is this L{Fsum2Tuple} considered to be exact? (C{bool}).

2611 '''

2612 return self._Fsum.is_exact()

2613

2614 def is_finite(self): # in .constants

2615 '''Is this L{Fsum2Tuple} C{finite}? (C{bool}).

2616

2617 @see: Function L{isfinite<pygeodesy.isfinite>}.

2618 '''

2619 return self._Fsum.is_finite()

2620

2621 def is_integer(self):

2622 '''Is this L{Fsum2Tuple} C{integer}? (C{bool}).

2623 '''

2624 return self._Fsum.is_integer()

2625

2626 def _mul_scalar(self, other, op): # for Fsum._fmul

2627 return self._Fsum._mul_scalar(other, op)

2628

2629 @property_RO

2630 def _n(self):

2631 return self._Fsum._n

2632

2633 def _other_op(self, other, which):

2634 C, s = (tuple, self) if isinstance(other, tuple) else (Fsum, self._Fsum)

2635 return getattr(C, which.__name__)(s, other)

2636

2637 @property_RO

2638 def _ps(self):

2639 return self._Fsum._ps

2640

2641 @property_RO

2642 def _ps_neg(self):

2643 return self._Fsum._ps_neg

2644

2645 def signOf(self, **res):

2646 '''Like method L{Fsum.signOf}.

2647 '''

2648 return self._Fsum.signOf(**res)

2649

2650 def toStr(self, fmt=Fmt.g, **prec_sep): # PYCHOK signature

2651 '''Return this L{Fsum2Tuple} as string (C{str}).

2652

2653 @kwarg fmt: Optional C{float} format (C{letter}).

2654 @kwarg prec_sep: Optional keyword arguments for function

2655 L{fstr<streprs.fstr>}.

2656 '''

2657 return Fmt.PAREN(fstr(self, fmt=fmt, strepr=str, force=False, **prec_sep))

2658

2659_Fsum_2Tuple_types = Fsum, Fsum2Tuple # PYCHOK lines

2660

2661

2662 class ResidualError(_ValueError):

2663 '''Error raised for a division, power or root operation of

2664 an L{Fsum} instance with a C{residual} I{ratio} exceeding

2665 the L{RESIDUAL<Fsum.RESIDUAL>} threshold.

2666

2667 @see: Module L{pygeodesy.fsums} and method L{Fsum.RESIDUAL}.

2668 '''

2669 pass

2670

2671

2672 try:

2673 from math import fsum as _fsum # precision IEEE-754 sum, Python 2.6+

2674

2675 # make sure _fsum works as expected (XXX check

2676 # float.__getformat__('float')[:4] == 'IEEE'?)

2677 if _fsum((1, 1e101, 1, -1e101)) != 2: # PYCHOK no cover

2678 del _fsum # nope, remove _fsum ...

2679 raise ImportError() # ... use _fsum below

2680

2681 _sum = _fsum # in .elliptic

2682 except ImportError:

2683 _sum = sum # in .elliptic

2684

2685 def _fsum(xs):

2686 '''(INTERNAL) Precision summation, Python 2.5-.

2687 '''

2688 F = Fsum(name=_fsum.name, f2product=False, nonfinites=True)

2689 return float(F._facc(xs, up=False))

2690

2691

2692 def fsum(xs, nonfinites=None, **floats):

2693 '''Precision floating point summation from Python's C{math.fsum}.

2694

2695 @arg xs: Iterable of items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

2696 @kwarg nonfinites: Use C{B{nonfinites}=True} if I{non-finites} are C{OK}, if

2697 C{False} I{non-finites} raise an Overflow-/ValueError or if

2698 C{None}, L{nonfiniterrors} applies (C{bool} or C{None}).

2699 @kwarg floats: DEPRECATED keyword argument C{B{floats}=False} (C{bool}), use

2700 keyword argument C{B{nonfinites}=False} instead.

2701

2702 @return: Precision C{fsum} (C{float}).

2703

2704 @raise OverflowError: Infinite B{C{xs}} item or intermediate C{math.fsum} overflow.

2705

2706 @raise TypeError: Invalid B{C{xs}} item.

2707

2708 @raise ValueError: Invalid or C{NAN} B{C{xs}} item.

2709

2710 @see: Function L{nonfiniterrors}, class L{Fsum} and methods L{Fsum.nonfinites},

2711 L{Fsum.fsum}, L{Fsum.fadd} and L{Fsum.fadd_}.

2712 '''

2713 return _xsum(fsum, xs, nonfinites=nonfinites, **floats) if xs else _0_0

2714

2715

2716 def fsum_(*xs, **nonfinites):

2717 '''Precision floating point summation of all positional items.

2718

2719 @arg xs: Items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all positional.

2720 @kwarg nonfinites: Use C{B{nonfinites}=True} if I{non-finites} are C{OK} (C{bool}).

2721

2722 @see: Function L{fsum<fsums.fsum>} for further details.

2723 '''

2724 return _xsum(fsum_, xs, **nonfinites) if xs else _0_0 # origin=1?

2725

2726

2727 def fsumf_(*xs):

2728 '''Precision floating point summation of all positional items with I{non-finites} C{OK}.

2729

2730 @arg xs: Items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}),

2731 all positional.

2732

2733 @see: Function L{fsum_<fsums.fsum_>} for further details.

2734 '''

2735 return _xsum(fsumf_, xs, nonfinites=True) if xs else _0_0 # origin=1?

2736

2737

2738 def fsum1(xs, **nonfinites):

2739 '''Precision floating point summation, 1-primed.

2740

2741 @arg xs: Iterable of items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}).

2742 @kwarg nonfinites: Use C{B{nonfinites}=True} if I{non-finites} are C{OK} (C{bool}).

2743

2744 @see: Function L{fsum<fsums.fsum>} for further details.

2745 '''

2746 return _xsum(fsum1, xs, primed=1, **nonfinites) if xs else _0_0

2747

2748

2749 def fsum1_(*xs, **nonfinites):

2750 '''Precision floating point summation of all positional items, 1-primed.

2751

2752 @arg xs: Items to add (each C{scalar}, an L{Fsum} or L{Fsum2Tuple}), all positional.

2753 @kwarg nonfinites: Use C{B{nonfinites}=True} if I{non-finites} are C{OK} (C{bool}).

2754

2755 @see: Function L{fsum_<fsums.fsum_>} for further details.

2756 '''

2757 return _xsum(fsum1_, xs, primed=1, **nonfinites) if xs else _0_0 # origin=1?

2758

2759

2760 def fsum1f_(*xs):

2761 '''Precision floating point summation of all positional items, 1-primed and

2762 with I{non-finites} C{OK}.

2763

2764 @see: Function L{fsum_<fsums.fsum_>} for further details.

2765 '''

2766 return _xsum(fsum1f_, xs, nonfinites=True, primed=1) if xs else _0_0

2767

2768

2769 def _x_isfine(nfOK, **kwds): # get the C{_x} and C{_isfine} handlers.

2770 _x_kwds = dict(_x= (_passarg if nfOK else _2finite),

2771 _isfine=(_isOK if nfOK else _isfinite)) # PYCHOK kwds

2772 _x_kwds.update(kwds)

2773 return _x_kwds

2774

2775

2776 def _X_ps(X): # default C{_X} handler

2777 return X._ps # lambda X: X._ps

2778

2779

2780 def _xs(xs, _X=_X_ps, _x=float, _isfine=_isfinite, # defaults for Fsum._facc

2781 origin=0, which=None, **_Cdot):

2782 '''(INTERNAL) Yield each C{xs} item as 1 or more C{float}s.

2783 '''

2784 i, x = 0, xs

2785 try:

2786 for i, x in enumerate(_xiterable(xs)):

2787 if isinstance(x, _Fsum_2Tuple_types):

2788 for p in _X(x):

2789 yield p if _isfine(p) else _nfError(p)

2790 else:

2791 f = _x(x)

2792 yield f if _isfine(f) else _nfError(f)

2793

2794 except (OverflowError, TypeError, ValueError) as X:

2795 t = _xsError(X, xs, i + origin, x)

2796 if which: # prefix invokation

2797 w = unstr(which, *xs, _ELLIPSIS=4, **_Cdot)

2798 t = _COMMASPACE_(w, t)

2799 raise _xError(X, t, txt=None)

2800

2801

2802 def _xsum(which, xs, nonfinites=None, primed=0, **floats): # origin=0

2803 '''(INTERNAL) Precision summation of C{xs} with conditions.

2804 '''

2805 if floats: # for backward compatibility

2806 nonfinites = _xkwds_get1(floats, floats=nonfinites)

2807 elif nonfinites is None:

2808 nonfinites = not nonfiniterrors()

2809 fs = _xs(xs, **_x_isfine(nonfinites, which=which))

2810 return _fsum(_1primed(fs) if primed else fs)

2811

2812

2813# delete all decorators, etc.

2814 del _allPropertiesOf_n, deprecated_method, deprecated_property_RO, \

2815 Property, Property_RO, property_RO, _ALL_LAZY, _F2PRODUCT, \

2816 MANT_DIG, _NONFINITES, _RESIDUAL_0_0, _getPYGEODESY, _std_

2817

2818 if __name__ == '__main__':

2819

2820 # usage: python3 -m pygeodesy.fsums

2821

2822 def _test(n):

2823 # copied from Hettinger, see L{Fsum} reference

2824 from pygeodesy import frandoms, printf

2825

2826 printf(_fsum.__name__, end=_COMMASPACE_)

2827 printf(_psum.__name__, end=_COMMASPACE_)

2828

2829 F = Fsum()

2830 if F.is_math_fsum():

2831 for t in frandoms(n, seeded=True):

2832 assert float(F.fset_(*t)) == _fsum(t)

2833 printf(_DOT_, end=NN)

2834 printf(NN)

2835

2836 _test(128)

2837

2838# **) MIT License

2839#

2840# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.

2841#

2842# Permission is hereby granted, free of charge, to any person obtaining a

2843# copy of this software and associated documentation files (the "Software"),

2844# to deal in the Software without restriction, including without limitation

2845# the rights to use, copy, modify, merge, publish, distribute, sublicense,

2846# and/or sell copies of the Software, and to permit persons to whom the

2847# Software is furnished to do so, subject to the following conditions:

2848#

2849# The above copyright notice and this permission notice shall be included

2850# in all copies or substantial portions of the Software.

2851#

2852# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

2853# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

2854# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

2855# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

2856# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

2857# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

2858# OTHER DEALINGS IN THE SOFTWARE.