1

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

3

4 u'''Class L{Fsum} for precision floating point summation and I{running}

5 summation based on, respectively similar to Python's C{math.fsum}.

6

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

8 C{residual} value, see property L{Fsum.residual}. However, there are several

9 C{integer} L{Fsum} cases, for example the result of C{ceil}, C{floor},

10 C{Fsum.__floordiv__} and methods L{Fsum.fint} and L{Fsum.fint2}.

11

12 Also, L{Fsum} methods L{Fsum.pow}, L{Fsum.__ipow__}, L{Fsum.__pow__} and

13 L{Fsum.__rpow__} return a (very long) C{int} if invoked with optional argument

14 C{mod} set to C{None}. The C{residual} of an C{integer} L{Fsum} may be between

15 C{-1.0} and C{+1.0}, including C{INT0} if considered to be I{exact}.

16

17 Set env variable C{PYGEODESY_FSUM_PARTIALS} to an empty string (or anything

18 other than C{"fsum"}) for backward compatible summation of L{Fsum} partials.

19

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

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

22 exponention of an L{Fsum} instance with a I{relative} C{residual} exceeding

23 the threshold, see methods L{Fsum.RESIDUAL}, L{Fsum.pow}, L{Fsum.__ipow__}

24 and L{Fsum.__itruediv__}.

25'''

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

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

28

29 from pygeodesy.basics import iscomplex, isint, isscalar, map1, signOf, _signOf

30 from pygeodesy.constants import INT0, _isfinite, isinf, isnan, \

31 _0_0, _1_0, _N_1_0

32 from pygeodesy.errors import itemsorted, _OverflowError, _TypeError, \

33 _ValueError, _xError2, _xkwds_get, _xkwds_get_, \

34 _ZeroDivisionError

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

36 _exceeds_, _from_, _iadd_, _LANGLE_, _negative_, \

37 _NOTEQUAL_, _not_finite_, _not_scalar_, \

38 _PERCENT_, _PLUS_, _R_, _RANGLE_, _SLASH_, \

39 _SPACE_, _STAR_, _UNDER_

40 from pygeodesy.lazily import _ALL_LAZY, _getenv, _sys, _sys_version_info2

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

42 from pygeodesy.props import _allPropertiesOf_n, deprecated_property_RO, \

43 Property_RO, property_RO

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

45 from pygeodesy.units import Float, Int

46

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

48

49__all__ = _ALL_LAZY.fsums

50__version__ = '23.05.18'

51

52_add_op_ = _PLUS_

53_eq_op_ = _EQUAL_ * 2 # _DEQUAL_

54_COMMASPACE_R_ = _COMMASPACE_ + _R_

55_exceeds_R_ = _SPACE_ + _exceeds_(_R_)

56_floordiv_op_ = _SLASH_ * 2 # _DSLASH_

57_fset_op_ = _EQUAL_

58_ge_op_ = _RANGLE_ + _EQUAL_

59_gt_op_ = _RANGLE_

60_integer_ = 'integer'

61_le_op_ = _LANGLE_ + _EQUAL_

62_lt_op_ = _LANGLE_

63_mod_op_ = _PERCENT_

64_mul_op_ = _STAR_

65_ne_op_ = _NOTEQUAL_

66_non_zero_ = 'non-zero'

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

68_sub_op_ = _DASH_

69_truediv_op_ = _SLASH_

70_divmod_op_ = _floordiv_op_ + _mod_op_

71

72_pos_self = _1_0.__pos__() is _1_0 # in .vector3dBase

73

74

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

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

77 '''

78 n, v = name_value.popitem() # _xkwds_popitem(name_value)

79 try:

80 v = float(v)

81 if _isfinite(v):

82 return v

83 E, t = _ValueError, _not_finite_

84 except Exception as e:

85 E, t = _xError2(e)

86 if index is not None:

87 n = Fmt.SQUARE(n, index)

88 raise E(n, v, txt=t)

89

90

91 def _2floats(xs, origin=0, sub=False):

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

93 '''

94 try:

95 i, x = origin, None

96 _fin = _isfinite

97 _Fsum = Fsum

98 for x in xs:

99 if isinstance(x, _Fsum):

100 for p in x._ps:

101 yield (-p) if sub else p

102 else:

103 f = float(x)

104 if not _fin(f):

105 raise ValueError(_not_finite_)

106 if f:

107 yield (-f) if sub else f

108 i += 1

109 except Exception as e:

110 E, t = _xError2(e)

111 n = Fmt.SQUARE(xs=i)

112 raise E(n, x, txt=t)

113

114

115 def _Powers(power, xs, origin=1): # in .fmath

116 '''(INTERNAL) Yield each C{xs} as C{float(x**power)}.

117 '''

118 if not isscalar(power):

119 raise _TypeError(power=power)

120 try:

121 i, x = origin, None

122 _fin = _isfinite

123 _Fsum = Fsum

124 _pow = pow # XXX math.pow

125 for x in xs:

126 if isinstance(x, _Fsum):

127 P = x.pow(power)

128 for p in P._ps:

129 yield p

130 else:

131 p = _pow(float(x), power)

132 if not _fin(p):

133 raise ValueError(_not_finite_)

134 yield p

135 i += 1

136 except Exception as e:

137 E, t = _xError2(e)

138 n = Fmt.SQUARE(xs=i)

139 raise E(n, x, txt=t)

140

141

142 def _1primed(xs):

143 '''(INTERNAL) 1-Prime the summation of C{xs}

144 arguments I{known} to be C{finite float}.

145 '''

146 yield _1_0

147 for x in xs:

148 if x:

149 yield x

150 yield _N_1_0

151

152

153 def _psum(ps): # PYCHOK used!

154 '''(INTERNAL) Partials summation updating C{ps}, I{overridden below}.

155 '''

156 i = len(ps) - 1 # len(ps) > 2

157 s = ps[i]

158 _2s = _2sum

159 while i > 0:

160 i -= 1

161 s, r = _2s(s, ps[i])

162 if r: # sum(ps) became inexact

163 ps[i:] = [s, r] if s else [r]

164 if i > 0:

165 p = ps[i-1] # round half-even

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

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

168 r *= 2

169 t = s + r

170 if r == (t - s):

171 s = t

172 break

173 ps[i:] = [s]

174 return s

175

176

177 def _2scalar(other, _raiser=None):

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

179 '''

180 if isinstance(other, Fsum):

181 s, r = other._fint2

182 if r:

183 s, r = other._fprs2

184 if r: # PYCHOK no cover

185 if _raiser and _raiser(r, s):

186 raise ValueError(_stresidual(_non_zero_, r))

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

188 else:

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

190 if isint(s, both=True):

191 s = int(s)

192 return s

193

194

195 def _strcomplex(s, *args):

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

197 '''

198 c = iscomplex.__name__[2:]

199 n = _DASH_(len(args), _arg_)

200 t = _SPACE_(c, s, _from_, n, pow.__name__)

201 return unstr(t, *args)

202

203

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

205 '''(INTERNAL) Residual error C{str}.

206 '''

207 p = _SPACE_(prefix, Fsum.residual.name)

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

209 for n, v in itemsorted(name_values):

210 n = n.replace(_UNDER_, _SPACE_)

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

212 t = _COMMASPACE_(t, p)

213 return t

214

215

216 def _2sum(a, b): # by .testFmath

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

218 '''

219 s = a + b

220 if not _isfinite(s):

221 u = unstr(_2sum.__name__, a, b)

222 t = Fmt.PARENSPACED(_not_finite_, s)

223 raise _OverflowError(u, txt=t)

224 r = (a - (s - b)) if fabs(a) < fabs(b) else \

225 (b - (s - a)) # fabs(a) >= fabs(b)

226 return s, r

227

228

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

230 '''Precision floating point I{running} summation.

231

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

233 I{running} precision floating point summation. Accumulation may continue after

234 intermediate, I{running} summuation.

235

236 @note: Accumulated values may be L{Fsum} or C{scalar} instances with C{scalar} meaning

237 type C{float}, C{int} or any C{type} convertible to a single C{float}, having

238 method C{__float__}.

239

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

241 Python's C{math.fsum}.

242

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

244 393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py>}, U{Kahan

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

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

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

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

249 '''

250 _math_fsum = None

251 _n = 0

252# _ps = [] # partial sums

253# _px = 0

254 _ratio = None

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

256

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

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

259

260 @arg xs: No, one or more initial values (each C{scalar} or an

261 L{Fsum} instance).

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

263 (C{str}) and C{B{RESIDUAL}=None} for the

264 L{ResidualError} threshold.

265

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

267 '''

268 if name_RESIDUAL:

269 n, r = _xkwds_get_(name_RESIDUAL, name=NN, RESIDUAL=None)

270 if n: # set name ...

271 self.name = n

272 if r is not None:

273 self.RESIDUAL(r) # ... for ResidualError

274# self._n = 0

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

276 if len(xs) > 1:

277 self._facc(_2floats(xs, origin=1), up=False) # PYCHOK yield

278 elif xs: # len(xs) == 1

279 self._ps = [_2float(x=xs[0])]

280 self._n = 1

281

282 def __abs__(self):

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

284 '''

285 s = _fsum(self._ps_1()) # == self._cmp_0(0, ...)

286 return self._copy_n(self.__abs__) if s < 0 else \

287 self._copy_2(self.__abs__)

288

289 def __add__(self, other):

290 '''Return the C{Fsum(B{self}, B{other})}.

291

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

293

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

295

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

297 '''

298 f = self._copy_2(self.__add__)

299 return f._fadd(other, _add_op_)

300

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

302 '''Return C{True} if this instance is non-zero.

303 '''

304 return self != 0

305

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

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

308

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

310

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

312 '''

313 return self.ceil

314

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

316 '''Compare this with an other instance or scalar.

317

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

319

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

321 '''

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

323 return -1 if s < 0 else (+1 if s > 0 else 0)

324

325 cmp = __cmp__

326

327 def __divmod__(self, other):

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

329 remainder)}, an C{int} in Python 3+ or C{float} in Python 2-

330 and an L{Fsum}.

331

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

333

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

335 '''

336 f = self._copy_2(self.__divmod__)

337 return f._fdivmod2(other, _divmod_op_)

338

339 def __eq__(self, other):

340 '''Compare this with an other instance or scalar.

341 '''

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

343

344 def __float__(self):

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

346

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

348 '''

349 return float(self._fprs)

350

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

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

353

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

355

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

357 '''

358 return self.floor

359

360 def __floordiv__(self, other):

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

362

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

364

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

366

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

368 '''

369 f = self._copy_2(self.__floordiv__)

370 return f._floordiv(other, _floordiv_op_)

371

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

373 '''Not implemented.'''

374 return _NotImplemented(self, *other)

375

376 def __ge__(self, other):

377 '''Compare this with an other instance or scalar.

378 '''

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

380

381 def __gt__(self, other):

382 '''Compare this with an other instance or scalar.

383 '''

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

385

386 def __hash__(self): # PYCHOK no cover

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

388 '''

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

390

391 def __iadd__(self, other):

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

393

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

395

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

397

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

399 C{scalar} nor L{Fsum}.

400

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

402 '''

403 return self._fadd(other, _iadd_)

404

405 def __ifloordiv__(self, other):

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

407

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

409

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

411

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

413

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

415

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

417

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

419

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

421 '''

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

423

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

425 '''Not implemented.'''

426 return _NotImplemented(self, other)

427

428 def __imod__(self, other):

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

430

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

432

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

434

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

436 '''

437 self._fdivmod2(other, _mod_op_ + _fset_op_)

438 return self

439

440 def __imul__(self, other):

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

442

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

444

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

446

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

448

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

450

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

452 '''

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

454

455 def __int__(self):

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

457

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

459 and L{Fsum.__floor__} and properties

460 L{Fsum.ceil} and L{Fsum.floor}.

461 '''

462 i, _ = self._fint2

463 return i

464

465 def __invert__(self): # PYCHOK no cover

466 '''Not implemented.'''

467 # Luciano Ramalho, "Fluent Python", 2nd Ed, page 567, O'Reilly, 2022

468 return _NotImplemented(self)

469

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

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

472

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

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

475 3-argument C{pow(B{self}, B{other}, B{mod})}

476 version.

477

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

479

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

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

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

483

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

485

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

487 env var C{PYGEODESY_FSUM_RESIDUAL}

488 set or this instance has a non-zero

489 residual and either B{C{mod}} is

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

491 is a negative or fractional C{scalar}.

492

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

494 C{pow} invocation failed.

495

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

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

498 is a fractional C{scalar} and this

499 instance is negative or has a non-zero

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

501

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

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

504 '''

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

506

507 def __isub__(self, other):

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

509

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

511

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

513

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

515

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

517 '''

518 return self._fsub(other, _sub_op_ + _fset_op_)

519

520 def __iter__(self):

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

522 '''

523 return iter(self.partials)

524

525 def __itruediv__(self, other):

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

527

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

529

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

531

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

533

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

535 env var C{PYGEODESY_FSUM_RESIDUAL} set.

536

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

538

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

540

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

542

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

544 '''

545 return self._ftruediv(other, _truediv_op_ + _fset_op_)

546

547 def __le__(self, other):

548 '''Compare this with an other instance or scalar.

549 '''

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

551

552 def __len__(self):

553 '''Return the I{total} number accumulations (C{int}).

554 '''

555 return self._n

556

557 def __lt__(self, other):

558 '''Compare this with an other instance or scalar.

559 '''

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

561

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

563 '''Not implemented.'''

564 return _NotImplemented(self, other)

565

566 def __mod__(self, other):

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

568

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

570 '''

571 f = self._copy_2(self.__mod__)

572 return f._fdivmod2(other, _mod_op_)[1]

573

574 def __mul__(self, other):

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

576

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

578 '''

579 f = self._copy_2(self.__mul__)

580 return f._fmul(other, _mul_op_)

581

582 def __ne__(self, other):

583 '''Compare this with an other instance or scalar.

584 '''

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

586

587 def __neg__(self):

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

589 '''

590 return self._copy_n(self.__neg__)

591

592 def __pos__(self):

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

594 '''

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

596

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

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

599

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

601 '''

602 f = self._copy_2(self.__pow__)

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

604

605 def __radd__(self, other):

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

607

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

609 '''

610 f = self._copy_r2(other, self.__radd__)

611 return f._fadd(self, _add_op_)

612

613 def __rdivmod__(self, other):

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

615 remainder)}.

616

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

618 '''

619 f = self._copy_r2(other, self.__rdivmod__)

620 return f._fdivmod2(self, _divmod_op_)

621

622# def __repr__(self):

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

624# '''

625# return self.toRepr(lenc=True)

626

627 def __rfloordiv__(self, other):

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

629

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

631 '''

632 f = self._copy_r2(other, self.__rfloordiv__)

633 return f._floordiv(self, _floordiv_op_)

634

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

636 '''Not implemented.'''

637 return _NotImplemented(self, other)

638

639 def __rmod__(self, other):

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

641

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

643 '''

644 f = self._copy_r2(other, self.__rmod__)

645 return f._fdivmod2(self, _mod_op_)[1]

646

647 def __rmul__(self, other):

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

649

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

651 '''

652 f = self._copy_r2(other, self.__rmul__)

653 return f._fmul(self, _mul_op_)

654

655 def __round__(self, ndigits=None): # PYCHOK no cover

656 '''Not implemented.'''

657 return _NotImplemented(self, ndigits=ndigits)

658

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

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

661

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

663 '''

664 f = self._copy_r2(other, self.__rpow__)

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

666

667 def __rsub__(self, other):

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

669

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

671 '''

672 f = self._copy_r2(other, self.__rsub__)

673 return f._fsub(self, _sub_op_)

674

675 def __rtruediv__(self, other):

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

677

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

679 '''

680 f = self._copy_r2(other, self.__rtruediv__)

681 return f._ftruediv(self, _truediv_op_)

682

683 def __sizeof__(self): # PYCHOK not special in Python 2-

684 '''Return the current size of this instance in C{bytes}.

685 '''

686 return sum(map1(_sys.getsizeof, self._fint2,

687 self._fint2[0],

688 self._fint2[1],

689 self._fprs,

690 self._fprs2,

691 self._fprs2.fsum,

692 self._fprs2.residual,

693 self._n,

694 self._ps, *self._ps))

695

696 def __str__(self):

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

698 '''

699 return self.toStr(lenc=True)

700

701 def __sub__(self, other):

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

703

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

705

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

707

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

709 '''

710 f = self._copy_2(self.__sub__)

711 return f._fsub(other, _sub_op_)

712

713 def __truediv__(self, other):

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

715

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

717

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

719

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

721 '''

722 f = self._copy_2(self.__truediv__)

723 return f._ftruediv(other, _truediv_op_)

724

725 __trunc__ = __int__

726

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

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

729 __div__ = __truediv__

730 __idiv__ = __itruediv__

731 __long__ = __int__

732 __nonzero__ = __bool__

733 __rdiv__ = __rtruediv__

734

735 def as_integer_ratio(self):

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

737

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

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

740

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

742 '''

743 n, r = self._fint2

744 if r:

745 i, d = r.as_integer_ratio()

746 n *= d

747 n += i

748 else: # PYCHOK no cover

749 d = 1

750 return n, d

751

752 @property_RO

753 def ceil(self):

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

755 but C{float} in Python 2-).

756

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

758

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

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

761 '''

762 s, r = self._fprs2

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

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

765 c += 1

766 return c

767

768 def _cmp_0(self, other, op):

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

770 '''

771 if isscalar(other):

772 s = _fsum(self._ps_1(other))

773 elif isinstance(other, Fsum):

774 s = _fsum(self._ps_1(*other._ps))

775 else:

776 raise self._TypeError(op, other) # txt=_invalid_

777 return s

778

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

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

781

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

783 '''

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

785 f._n = self._n if deep else 1

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

787 return f

788

789 def _copy_0(self, *xs):

790 '''(INTERNAL) Copy with/-out overriding C{partials}.

791 '''

792 # for x in xs:

793 # assert isscalar(x)

794 f = self._Fsum(self._n + len(xs), *xs)

795 if self.name:

796 f._name = self.name # .rename calls _update_attrs

797 return f

798

799 def _copy_2(self, which):

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

801 '''

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

803 f = _Named.copy(self, deep=False, name=which.__name__)

804 # assert f._n == self._n

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

806 return f

807

808 def _copy_n(self, which):

809 '''(INTERNAL) Negated copy for I{monadic} C{__abs__} and C{__neg__}.

810 '''

811 if self._ps:

812 f = self._Fsum(self._n)

813 f._ps[:] = self._ps_n()

814# f._facc_up(up=False)

815 else:

816 f = self._Fsum(self._n, _0_0)

817 f._name = which.__name__ # .rename calls _update_attrs

818 return f

819

820 def _copy_r2(self, other, which):

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

822 '''

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

824 Fsum(other, name=which.__name__) # see ._copy_2

825

826 def _copy_RESIDUAL(self, other):

827 '''(INTERNAL) Copy C{other._RESIDUAL}.

828 '''

829 R = other._RESIDUAL

830 if R is not Fsum._RESIDUAL:

831 self._RESIDUAL = R

832

833 def _copy_up(self, _fprs2=False):

834 '''(INTERNAL) Minimal, anonymous copy.

835 '''

836 f = self._Fsum(self._n, *self._ps)

837 if _fprs2: # only the ._fprs2 2-tuple

838 Fsum._fprs2._update_from(f, self)

839 return f

840

841 def divmod(self, other):

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

843 remainder)}.

844

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

846

847 @return: 2-Tuple C{(quotient, remainder)}, with the C{quotient}

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

849 the C{remainder} an L{Fsum} instance.

850

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

852 '''

853 f = self._copy_2(self.divmod)

854 return f._fdivmod2(other, _divmod_op_)

855

856 def _Error(self, op, other, Error, **txt):

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

858 '''

859 return Error(_SPACE_(self.toRepr(), op, repr(other)), **txt)

860

861 def _ErrorX(self, X, xs, **kwds): # in .fmath

862 '''(INTERNAL) Format a caught exception.

863 '''

864 E, t = _xError2(X)

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

866 return E(n, txt=t, cause=X)

867

868 def _facc(self, xs, up=True): # from .elliptic._Defer.Fsum

869 '''(INTERNAL) Accumulate more known C{scalar}s.

870 '''

871 n, ps, _2s = 0, self._ps, _2sum

872 for x in xs: # _iter()

873 # assert isscalar(x) and isfinite(x)

874 i = 0

875 for p in ps:

876 x, p = _2s(x, p)

877 if p:

878 ps[i] = p

879 i += 1

880 ps[i:] = [x]

881 n += 1

882 # assert self._ps is ps

883 if n:

884 self._n += n

885 # Fsum._px = max(Fsum._px, len(ps))

886 if up:

887 self._update()

888 return self

889

890 def _facc_(self, *xs, **up):

891 '''(INTERNAL) Accumulate all positional C{scalar}s.

892 '''

893 return self._facc(xs, **up) if xs else self

894

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

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

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

898# '''

899# while len(self._ps) > 1:

900# p = self._ps.pop()

901# if p:

902# n = self._n

903# self._facc_(p, up=False)

904# self._n = n

905# break

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

907

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

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

910 to this instance.

911

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

913 L{Fsum} instances).

914

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

916

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

918

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

920 nor L{Fsum}.

921

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

923 '''

924 if isinstance(xs, Fsum):

925 self._facc(xs._ps)

926 elif isscalar(xs): # for backward compatibility

927 self._facc_(_2float(x=xs)) # PYCHOK no cover

928 elif xs:

929 self._facc(_2floats(xs)) # PYCHOK yield

930 return self

931

932 def fadd_(self, *xs):

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

934 to this instance.

935

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

937 all positional.

938

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

940

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

942

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

944 nor L{Fsum}.

945

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

947 '''

948 return self._facc(_2floats(xs, origin=1)) # PYCHOK yield

949

950 def _fadd(self, other, op): # in .fmath.Fhorner

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

952 '''

953 if isinstance(other, Fsum):

954 if other is self:

955 self._facc_(*other._ps) # == ._facc(tuple(other._ps))

956 elif other._ps:

957 self._facc(other._ps)

958 elif not isscalar(other):

959 raise self._TypeError(op, other) # txt=_invalid_

960 elif other:

961 self._facc_(other)

962 return self

963

964 fcopy = copy # for backward compatibility

965 fdiv = __itruediv__ # for backward compatibility

966 fdivmod = __divmod__ # for backward compatibility

967

968 def _fdivmod2(self, other, op):

969 '''(INTERNAL) C{divmod(B{self}, B{other})} as 2-tuple

970 (C{int} or C{float}, remainder C{self}).

971 '''

972 # result mostly follows CPython function U{float_divmod

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

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

975 q = self._copy_up(_fprs2=True)._ftruediv(other, op).floor

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

977 self -= other * q

978

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

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

981 self += other

982 q -= 1

983

984# t = self.signOf()

985# if t and t != s:

986# from pygeodesy.errors import _AssertionError

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

988 return q, self # q is C{int} in Python 3+, but C{float} in Python 2-

989

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

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

992 '''

993 if _isfinite(other):

994 return other

995 raise ValueError(_not_finite_) if not op else \

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

997

998 def fint(self, raiser=True, name=NN):

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

1000

1001 @kwarg raiser: If C{True} throw a L{ResidualError} if the

1002 I{integer} residual is non-zero.

1003 @kwarg name: Optional name (C{str}), overriding C{"fint"}.

1004

1005 @return: The C{integer} (L{Fsum}).

1006

1007 @raise ResidualError: Non-zero I{integer} residual.

1008

1009 @see: Methods L{Fsum.int_float} and L{Fsum.is_integer}.

1010 '''

1011 i, r = self._fint2

1012 if r and raiser:

1013 t = _stresidual(_integer_, r)

1014 raise ResidualError(_integer_, i, txt=t)

1015 n = name or self.fint.__name__

1016 return Fsum(name=n)._fset(i, asis=True)

1017

1018 def fint2(self, **name):

1019 '''Return this instance' current running sum as C{int} and

1020 the I{integer} residual.

1021

1022 @kwarg name: Optional name (C{str}).

1023

1024 @return: An L{Fsum2Tuple}C{(fsum, residual)} with C{fsum}

1025 an C{int} and I{integer} C{residual} a C{float} or

1026 C{INT0} if the C{fsum} is considered to be I{exact}.

1027 '''

1028 return Fsum2Tuple(*self._fint2, **name)

1029

1030 @Property_RO

1031 def _fint2(self): # see ._fset

1032 '''(INTERNAL) Get 2-tuple (C{int}, I{integer} residual).

1033 '''

1034 i = int(self._fprs) # int(self)

1035 r = _fsum(self._ps_1(i)) if len(self._ps) > 1 else (

1036 (self._ps[0] - i) if self._ps else -i)

1037 return i, (r or INT0)

1038

1039 @deprecated_property_RO

1040 def float_int(self): # PYCHOK no cover

1041 '''DEPRECATED, use method C{Fsum.int_float}.'''

1042 return self.int_float() # raiser=False

1043

1044 @property_RO

1045 def floor(self):

1046 '''Get this instance' C{floor} (C{int} in Python 3+, but

1047 C{float} in Python 2-).

1048

1049 @note: The C{floor} takes the C{residual} into account.

1050

1051 @see: Method L{Fsum.int_float} and properties L{Fsum.ceil},

1052 L{Fsum.imag} and L{Fsum.real}.

1053 '''

1054 s, r = self._fprs2

1055 f = _floor(s) + _floor(r) + 1

1056 while r < (f - s): # (s + r) < f

1057 f -= 1

1058 return f

1059

1060# floordiv = __floordiv__ # for naming consistency

1061

1062 def _floordiv(self, other, op): # rather _ffloordiv?

1063 '''Apply C{B{self} //= B{other}}.

1064 '''

1065 q = self._ftruediv(other, op) # == self

1066 return self._fset(q.floor, asis=True) # floor(q)

1067

1068 fmul = __imul__ # for backward compatibility

1069

1070 def _fmul(self, other, op):

1071 '''(INTERNAL) Apply C{B{self} *= B{other}}.

1072 '''

1073 if isscalar(other):

1074 f = self._mul_scalar(other, op)

1075 elif not isinstance(other, Fsum):

1076 raise self._TypeError(op, other) # txt=_invalid_

1077 elif len(self._ps) != 1:

1078 f = self._mul_Fsum(other, op)

1079 elif len(other._ps) != 1: # len(self._ps) == 1

1080 f = other._copy_up()._mul_scalar(self._ps[0], op)

1081 else: # len(other._ps) == len(self._ps) == 1

1082 s = self._finite(self._ps[0] * other._ps[0])

1083 return self._fset(s, asis=True, n=len(self) + 1)

1084 return self._fset(f)

1085

1086 def fover(self, over):

1087 '''Apply C{B{self} /= B{over}} and summate.

1088

1089 @arg over: An L{Fsum} or C{scalar} denominator.

1090

1091 @return: Precision running sum (C{float}).

1092

1093 @see: Methods L{Fsum.fsum} and L{Fsum.__itruediv__}.

1094 '''

1095 return float(self.fdiv(over)._fprs)

1096

1097 fpow = __ipow__ # for backward compatibility

1098

1099 def _fpow(self, other, op, *mod):

1100 '''Apply C{B{self} **= B{other}}, optional B{C{mod}} or C{None}.

1101 '''

1102 if mod and mod[0] is not None: # == 3-arg C{pow}

1103 s = self._pow_3(other, mod[0], op)

1104 elif mod and mod[0] is None and self.is_integer():

1105 # return an exact C{int} for C{int}**C{int}

1106 i = self._copy_0(self._fint2[0]) # assert _fint2[1] == 0

1107 x = _2scalar(other) # C{int}, C{float} or other

1108 s = i._pow_2(x, other, op) if isscalar(x) else i._fpow(x, op)

1109 else: # pow(self, other) == pow(self, other, None)

1110 p = None

1111 if isinstance(other, Fsum):

1112 x, r = other._fprs2

1113 if r:

1114 if self._raiser(r, x):

1115 raise self._ResidualError(op, other, r)

1116 p = self._pow_scalar(r, other, op)

1117# p = _2scalar(p) # _raiser = None

1118 elif not isscalar(other):

1119 raise self._TypeError(op, other) # txt=_invalid_

1120 else:

1121 x = self._finite(other, op)

1122 s = self._pow_scalar(x, other, op)

1123 if p is not None:

1124 s *= p

1125 return self._fset(s, asis=isint(s), n=max(len(self), 1))

1126

1127 @Property_RO

1128 def _fprs(self):

1129 '''(INTERNAL) Get and cache this instance' precision

1130 running sum (C{float} or C{int}), ignoring C{residual}.

1131

1132 @note: The precision running C{fsum} after a C{//=} or

1133 C{//} C{floor} division is C{int} in Python 3+.

1134 '''

1135 ps = self._ps

1136 n = len(ps) - 1

1137 if n > 1:

1138 s = _psum(ps)

1139 elif n > 0: # len(ps) == 2

1140 s, p = _2sum(*ps) if ps[1] else ps

1141 ps[:] = ([p, s] if s else [p]) if p else [s]

1142 elif n < 0: # see L{Fsum.__init__}

1143 s = _0_0

1144 ps[:] = [s]

1145 else: # len(ps) == 1

1146 s = ps[0]

1147 # assert self._ps is ps

1148 # assert Fsum._fprs2.name not in self.__dict__

1149 return s

1150

1151 @Property_RO

1152 def _fprs2(self):

1153 '''(INTERNAL) Get and cache this instance' precision

1154 running sum and residual (L{Fsum2Tuple}).

1155 '''

1156 s = self._fprs

1157 r = _fsum(self._ps_1(s)) if len(self._ps) > 1 else INT0

1158 return Fsum2Tuple(s, r) # name=Fsum.fsum2.__name__

1159

1160# def _fpsqz(self):

1161# '''(INTERNAL) Compress, squeeze the C{partials}.

1162# '''

1163# if len(self._ps) > 2:

1164# _ = self._fprs

1165# return self

1166

1167 def _fset(self, other, asis=False, n=1):

1168 '''(INTERNAL) Overwrite this instance with an other or a C{scalar}.

1169 '''

1170 if other is self:

1171 pass # from ._fmul, ._ftruediv and ._pow_scalar

1172 elif isinstance(other, Fsum):

1173 self._n = other._n

1174 self._ps[:] = other._ps

1175 self._copy_RESIDUAL(other)

1176 # use or zap the C{Property_RO} values

1177 Fsum._fint2._update_from(self, other)

1178 Fsum._fprs ._update_from(self, other)

1179 Fsum._fprs2._update_from(self, other)

1180 elif isscalar(other):

1181 s = other if asis else float(other)

1182 i = int(s) # see ._fint2

1183 t = i, ((s - i) or INT0)

1184 self._n = n

1185 self._ps[:] = [s]

1186 # Property_RO _fint2, _fprs and _fprs2 can't be a Property:

1187 # Property's _fset zaps the value just set by the @setter

1188 self.__dict__.update(_fint2=t, _fprs=s, _fprs2=Fsum2Tuple(s, INT0))

1189 else: # PYCHOK no cover

1190 raise self._TypeError(_fset_op_, other) # txt=_invalid_

1191 return self

1192

1193 def fsub(self, xs=()):

1194 '''Subtract an iterable of C{scalar} or L{Fsum} instances

1195 from this instance.

1196

1197 @arg xs: Iterable, list, tuple. etc. (C{scalar}

1198 or L{Fsum} instances).

1199

1200 @return: This instance, updated (L{Fsum}).

1201

1202 @see: Method L{Fsum.fadd}.

1203 '''

1204 return self._facc(_2floats(xs, sub=True)) if xs else self # PYCHOK yield

1205

1206 def fsub_(self, *xs):

1207 '''Subtract all positional C{scalar} or L{Fsum} instances

1208 from this instance.

1209

1210 @arg xs: Values to subtract (C{scalar} or

1211 L{Fsum} instances), all positional.

1212

1213 @return: This instance, updated (L{Fsum}).

1214

1215 @see: Method L{Fsum.fadd}.

1216 '''

1217 return self._facc(_2floats(xs, origin=1, sub=True)) if xs else self # PYCHOK yield

1218

1219 def _fsub(self, other, op):

1220 '''(INTERNAL) Apply C{B{self} -= B{other}}.

1221 '''

1222 if isinstance(other, Fsum):

1223 if other is self: # or other._fprs2 == self._fprs2:

1224 self._fset(_0_0, asis=True, n=len(self) * 2) # self -= self

1225 elif other._ps:

1226 self._facc(other._ps_n())

1227 elif not isscalar(other):

1228 raise self._TypeError(op, other) # txt=_invalid_

1229 elif self._finite(other, op):

1230 self._facc_(-other)

1231 return self

1232

1233 def _Fsum(self, n, *ps):

1234 '''(INTERNAL) New L{Fsum} instance.

1235 '''

1236 f = Fsum()

1237 f._n = n

1238 if ps:

1239 f._ps[:] = ps

1240 f._copy_RESIDUAL(self)

1241 return f

1242

1243 def fsum(self, xs=()):

1244 '''Add more C{scalar} or L{Fsum} instances and summate.

1245

1246 @kwarg xs: Iterable, list, tuple, etc. (C{scalar} or

1247 L{Fsum} instances).

1248

1249 @return: Precision running sum (C{float} or C{int}).

1250

1251 @see: Method L{Fsum.fadd}.

1252

1253 @note: Accumulation can continue after summation.

1254 '''

1255 f = self._facc(_2floats(xs)) if xs else self # PYCHOK yield

1256 return f._fprs

1257

1258 def fsum_(self, *xs):

1259 '''Add all positional C{scalar} or L{Fsum} instances and summate.

1260

1261 @arg xs: Values to add (C{scalar} or L{Fsum} instances),

1262 all positional.

1263

1264 @return: Precision running sum (C{float} or C{int}).

1265

1266 @see: Method L{Fsum.fsum}.

1267 '''

1268 f = self._facc(_2floats(xs, origin=1)) if xs else self # PYCHOK yield

1269 return f._fprs

1270

1271 def fsum2(self, xs=(), **name):

1272 '''Add more C{scalar} or L{Fsum} instances and return the

1273 current precision running sum and the C{residual}.

1274

1275 @kwarg xs: Iterable, list, tuple, etc. (C{scalar} or

1276 L{Fsum} instances).

1277 @kwarg name: Optional name (C{str}).

1278

1279 @return: L{Fsum2Tuple}C{(fsum, residual)} with C{fsum} the

1280 current precision running sum and C{residual}, the

1281 (precision) sum of the remaining C{partials}. The

1282 C{residual is INT0} if the C{fsum} is considered

1283 to be I{exact}.

1284

1285 @see: Methods L{Fsum.fint2}, L{Fsum.fsum} and L{Fsum.fsum2_}

1286 '''

1287 f = self._facc(_2floats(xs)) if xs else self # PYCHOK yield

1288 t = f._fprs2

1289 if name:

1290 t = t.dup(name=_xkwds_get(name, name=NN))

1291 return t

1292

1293 def fsum2_(self, *xs):

1294 '''Add any positional C{scalar} or L{Fsum} instances and return

1295 the precision running sum and the C{differential}.

1296

1297 @arg xs: Values to add (C{scalar} or L{Fsum} instances),

1298 all positional.

1299

1300 @return: 2-Tuple C{(fsum, delta)} with the current precision

1301 running C{fsum} and C{delta}, the difference with

1302 the previous running C{fsum} (C{float}s).

1303

1304 @see: Methods L{Fsum.fsum_} and L{Fsum.fsum}.

1305 '''

1306 p, r = self._fprs2

1307 if xs:

1308 s, t = self._facc(_2floats(xs, origin=1))._fprs2 # PYCHOK yield

1309 return s, _fsum((s, -p, r, -t)) # ((s - p) + (r - t))

1310 else: # PYCHOK no cover

1311 return p, _0_0

1312

1313# ftruediv = __itruediv__ # for naming consistency

1314

1315 def _ftruediv(self, other, op):

1316 '''(INTERNAL) Apply C{B{self} /= B{other}}.

1317 '''

1318 n = _1_0

1319 if isinstance(other, Fsum):

1320 if other is self or other._fprs2 == self._fprs2:

1321 return self._fset(_1_0, asis=True, n=len(self))

1322 d, r = other._fprs2

1323 if r:

1324 if not d: # PYCHOK no cover

1325 d = r

1326 elif self._raiser(r, d):

1327 raise self._ResidualError(op, other, r)

1328 else:

1329 d, n = other.as_integer_ratio()

1330 elif isscalar(other):

1331 d = other

1332 else: # PYCHOK no cover

1333 raise self._TypeError(op, other) # txt=_invalid_

1334 try:

1335 s = 0 if isinf(d) else (

1336 d if isnan(d) else self._finite(n / d))

1337 except Exception as x:

1338 E, t = _xError2(x)

1339 raise self._Error(op, other, E, txt=t)

1340 f = self._mul_scalar(s, _mul_op_) # handles 0, NAN, etc.

1341 return self._fset(f)

1342

1343 @property_RO

1344 def imag(self):

1345 '''Get the C{imaginary} part of this instance (C{0.0}, always).

1346

1347 @see: Properties L{Fsum.ceil}, L{Fsum.floor} and L{Fsum.real}.

1348 '''

1349 return _0_0

1350

1351 def int_float(self, raiser=False):

1352 '''Return this instance' current running sum as C{int} or C{float}.

1353

1354 @kwarg raiser: If C{True} throw a L{ResidualError} if the

1355 residual is non-zero.

1356

1357 @return: This C{integer} sum if this instance C{is_integer},

1358 otherwise return the C{float} sum if the residual

1359 is zero or if C{B{raiser}=False}.

1360

1361 @raise ResidualError: Non-zero residual and C{B{raiser}=True}.

1362

1363 @see: Methods L{Fsum.fint} and L{Fsum.fint2}.

1364 '''

1365 s, r = self._fint2

1366 if r:

1367 s, r = self._fprs2

1368 if r and raiser: # PYCHOK no cover

1369 t = _stresidual(_non_zero_, r)

1370 raise ResidualError(int_float=s, txt=t)

1371 s = float(s) # redundant

1372 return s

1373

1374 def is_exact(self):

1375 '''Is this instance' current running C{fsum} considered to

1376 be exact? (C{bool}).

1377 '''

1378 return self.residual is INT0

1379

1380 def is_integer(self):

1381 '''Is this instance' current running sum C{integer}? (C{bool}).

1382

1383 @see: Methods L{Fsum.fint} and L{Fsum.fint2}.

1384 '''

1385 _, r = self._fint2

1386 return False if r else True

1387

1388 def is_math_fsum(self):

1389 '''Return whether functions L{fsum}, L{fsum_}, L{fsum1}

1390 and L{fsum1_} plus partials summation are based on

1391 Python's C{math.fsum} or not.

1392

1393 @return: C{2} if all functions and partials summation

1394 are based on C{math.fsum}, C{True} if only

1395 the functions are based on C{math.fsum} (and

1396 partials summation is not) or C{False} if

1397 none are.

1398 '''

1399 f = Fsum._math_fsum

1400 return 2 if _psum is f else bool(f)

1401

1402 def _mul_Fsum(self, other, op=_mul_op_):

1403 '''(INTERNAL) Return C{B{self} * Fsum B{other}} as L{Fsum}.

1404 '''

1405 # assert isinstance(other, Fsum)

1406 return self._copy_0()._facc(self._ps_x(op, *other._ps), up=False)

1407

1408 def _mul_scalar(self, factor, op):

1409 '''(INTERNAL) Return C{B{self} * scalar B{factor}} as L{Fsum} or C{0}.

1410 '''

1411 # assert isscalar(factor)

1412 if self._finite(factor, op) and self._ps:

1413 if factor == _1_0:

1414 return self

1415 f = self._copy_0()._facc(self._ps_x(op, factor), up=False)

1416 else:

1417 f = self._copy_0(_0_0)

1418 return f

1419

1420 @property_RO

1421 def partials(self):

1422 '''Get this instance' current partial sums (C{tuple} of C{float}s and/or C{int}s).

1423 '''

1424 return tuple(self._ps)

1425

1426 def pow(self, x, *mod):

1427 '''Return C{B{self}**B{x}} as L{Fsum}.

1428

1429 @arg x: The exponent (L{Fsum} or C{scalar}).

1430 @arg mod: Optional modulus (C{int} or C{None}) for the 3-argument

1431 C{pow(B{self}, B{other}, B{mod})} version.

1432

1433 @return: The C{pow(self, B{x})} or C{pow(self, B{x}, *B{mod})}

1434 result (L{Fsum}).

1435

1436 @note: If B{C{mod}} is given as C{None}, the result will be an

1437 C{integer} L{Fsum} provided this instance C{is_integer}

1438 or set C{integer} with L{Fsum.fint}.

1439

1440 @see: Methods L{Fsum.__ipow__}, L{Fsum.fint} and L{Fsum.is_integer}.

1441 '''

1442 f = self._copy_2(self.pow)

1443 if f and isint(x) and x >= 0 and not mod:

1444 f._pow_int(x, x, _pow_op_) # f **= x

1445 else:

1446 f._fpow(x, _pow_op_, *mod) # f = pow(f, x, *mod)

1447 return f

1448

1449 def _pow_0_1(self, x, other):

1450 '''(INTERNAL) Return B{C{self}**1} or C{B{self}**0 == 1.0}.

1451 '''

1452 return self if x else (1 if self.is_integer() and isint(other) else _1_0)

1453

1454 def _pow_2(self, x, other, op):

1455 '''(INTERNAL) 2-arg C{pow(B{self}, scalar B{x})} embellishing errors.

1456 '''

1457 # assert len(self._ps) == 1 and isscalar(x)

1458 b = self._ps[0] # assert isscalar(b)

1459 try: # type(s) == type(x) if x in (_1_0, 1)

1460 s = pow(b, x) # -1**2.3 == -(1**2.3)

1461 if not iscomplex(s):

1462 return self._finite(s) # 0**INF == 0.0, 1**INF==1.0

1463 # neg**frac == complex in Python 3+, but ValueError in 2-

1464 E, t = _ValueError, _strcomplex(s, b, x) # PYCHOK no cover

1465 except Exception as x:

1466 E, t = _xError2(x)

1467 raise self._Error(op, other, E, txt=t)

1468

1469 def _pow_3(self, other, mod, op):

1470 '''(INTERNAL) 3-arg C{pow(B{self}, B{other}, int B{mod} or C{None})}.

1471 '''

1472 b, r = self._fprs2 if mod is None else self._fint2

1473 if r and self._raiser(r, b):

1474 t = _non_zero_ if mod is None else _integer_

1475 E, t = ResidualError, _stresidual(t, r, mod=mod)

1476 else:

1477 try: # b, other, mod all C{int}, unless C{mod} is C{None}

1478 x = _2scalar(other, _raiser=self._raiser)

1479 s = pow(b, x, mod)

1480 if not iscomplex(s):

1481 return self._finite(s)

1482 # neg**frac == complex in Python 3+, but ValueError in 2-

1483 E, t = _ValueError, _strcomplex(s, b, x, mod) # PYCHOK no cover

1484 except Exception as x:

1485 E, t = _xError2(x)

1486 t = _COMMASPACE_(Fmt.PARENSPACED(mod=mod), t)

1487 raise self._Error(op, other, E, txt=t)

1488

1489 def _pow_int(self, x, other, op):

1490 '''(INTERNAL) Return C{B{self} **= B{x}} for C{int B{x} >= 0}.

1491 '''

1492 # assert isint(x) and x >= 0

1493 if len(self._ps) > 1:

1494 if x > 2:

1495 p = self._copy_up()

1496 m = 1 # single-bit mask

1497 if x & m:

1498 x -= m # x ^= m

1499 f = p._copy_up()

1500 else:

1501 f = self._copy_0(_1_0)

1502 while x:

1503 p = p._mul_Fsum(p, op) # p **= 2

1504 m += m # m <<= 1

1505 if x & m:

1506 x -= m # x ^= m

1507 f = f._mul_Fsum(p, op) # f *= p

1508 elif x > 1: # self**2

1509 f = self._mul_Fsum(self, op)

1510 else: # self**1 or self**0

1511 f = self._pow_0_1(x, other)

1512 elif self._ps: # self._ps[0]**x

1513 f = self._pow_2(x, other, op)

1514 else: # PYCHOK no cover

1515 # 0**pos_int == 0, but 0**0 == 1

1516 f = 0 if x else 1 # like ._fprs

1517 return self._fset(f, asis=isint(f), n=len(self))

1518

1519 def _pow_scalar(self, x, other, op):

1520 '''(INTERNAL) Return C{self**B{x}} for C{scalar B{x}}.

1521 '''

1522 s, r = self._fprs2

1523 if isint(x, both=True):

1524 x = int(x) # Fsum**int

1525 y = abs(x)

1526 if y > 1:

1527 if r:

1528 f = self._copy_up()._pow_int(y, other, op)

1529 if x > 0: # > 1

1530 return f

1531 # assert x < 0 # < -1

1532 s, r = f._fprs2

1533 if r:

1534 return self._copy_0(_1_0)._ftruediv(f, op)

1535 # use **= -1 for the CPython float_pow

1536 # error if s is zero, and not s = 1 / s

1537 x = -1

1538# elif y > 1: # self**2 or self**-2

1539# f = self._mul_Fsum(self, op)

1540# if x < 0:

1541# f = f._copy_0(_1_0)._ftruediv(f, op)

1542# return f

1543 elif x < 0: # self**-1 == 1 / self

1544 if r:

1545 return self._copy_0(_1_0)._ftruediv(self, op)

1546 else: # self**1 or self**0

1547 return self._pow_0_1(x, other) # self or 0.0

1548 elif not isscalar(x): # assert ...

1549 raise self._TypeError(op, other, txt=_not_scalar_)

1550 elif r and self._raiser(r, s): # non-zero residual**fractional

1551 # raise self._ResidualError(op, other, r, fractional_power=x)

1552 t = _stresidual(_non_zero_, r, fractional_power=x)

1553 raise self._Error(op, other, ResidualError, txt=t)

1554 # assert isscalar(s) and isscalar(x)

1555 return self._copy_0(s)._pow_2(x, other, op)

1556

1557 def _ps_1(self, *less):

1558 '''(INTERNAL) Yield partials, 1-primed and subtract any C{less}.

1559 '''

1560 yield _1_0

1561 for p in self._ps:

1562 if p:

1563 yield p

1564 for p in less:

1565 if p:

1566 yield -p

1567 yield _N_1_0

1568

1569 def _ps_n(self):

1570 '''(INTERNAL) Yield partials, negated.

1571 '''

1572 for p in self._ps:

1573 if p:

1574 yield -p

1575

1576 def _ps_x(self, op, *factors): # see .fmath.Fhorner

1577 '''(INTERNAL) Yield all C{partials} times each B{C{factor}},

1578 in total, up to C{len(partials) * len(factors)} items.

1579 '''

1580 ps = self._ps

1581 if len(ps) < len(factors):

1582 ps, factors = factors, ps

1583 _f = _isfinite

1584 for f in factors:

1585 for p in ps:

1586 p *= f

1587 if _f(p):

1588 yield p

1589 else: # PYCHOK no cover

1590 self._finite(p, op) # throw ValueError

1591

1592 @property_RO

1593 def real(self):

1594 '''Get the C{real} part of this instance (C{float}).

1595

1596 @see: Methods L{Fsum.__float__} and L{Fsum.fsum}

1597 and properties L{Fsum.ceil}, L{Fsum.floor},

1598 L{Fsum.imag} and L{Fsum.residual}.

1599 '''

1600 return float(self._fprs)

1601

1602 @property_RO

1603 def residual(self):

1604 '''Get this instance' residual (C{float} or C{int}), the

1605 C{sum(partials)} less the precision running sum C{fsum}.

1606

1607 @note: If the C{residual is INT0}, the precision running

1608 C{fsum} is considered to be I{exact}.

1609

1610 @see: Methods L{Fsum.fsum}, L{Fsum.fsum2} and L{Fsum.is_exact}.

1611 '''

1612 return self._fprs2.residual

1613

1614 def _raiser(self, r, s):

1615 '''(INTERNAL) Does the ratio C{r / s} exceed threshold?

1616 '''

1617 self._ratio = t = fabs((r / s) if s else r)

1618 return t > self._RESIDUAL

1619

1620 def RESIDUAL(self, *threshold):

1621 '''Get and set this instance' I{ratio} for raising L{ResidualError}s,

1622 overriding the default from env variable C{PYGEODESY_FSUM_RESIDUAL}.

1623

1624 @arg threshold: If C{scalar}, the I{ratio} to exceed for raising

1625 L{ResidualError}s in division and exponention, if

1626 C{None} restore the default set with env variable

1627 C{PYGEODESY_FSUM_RESIDUAL} or if omitted, keep the

1628 current setting.

1629

1630 @return: The previous C{RESIDUAL} setting (C{float}).

1631

1632 @raise ValueError: Negative B{C{threshold}}.

1633

1634 @note: A L{ResidualError} is thrown if the non-zero I{ratio}

1635 C{residual} / C{fsum} exceeds the B{C{threshold}}.

1636 '''

1637 r = self._RESIDUAL

1638 if threshold:

1639 t = threshold[0]

1640 t = Fsum._RESIDUAL if t is None else (

1641 float(t) if isscalar(t) else ( # for backward ...

1642 _0_0 if bool(t) else _1_0)) # ... compatibility

1643 if t < 0:

1644 u = self._unstr(self.RESIDUAL, *threshold)

1645 raise _ValueError(u, RESIDUAL=t, txt=_negative_)

1646 self._RESIDUAL = t

1647 return r

1648

1649 def _ResidualError(self, op, other, residual):

1650 '''(INTERNAL) Non-zero B{C{residual}} etc.

1651 '''

1652 t = _stresidual(_non_zero_, residual, ratio=self._ratio,

1653 RESIDUAL=self._RESIDUAL)

1654 t = t.replace(_COMMASPACE_R_, _exceeds_R_)

1655 return self._Error(op, other, ResidualError, txt=t)

1656

1657 def signOf(self, res=True):

1658 '''Determine the sign of this instance.

1659

1660 @kwarg res: If C{True} consider, otherwise

1661 ignore the residual (C{bool}).

1662

1663 @return: The sign (C{int}, -1, 0 or +1).

1664 '''

1665 s, r = self._fprs2 if res else (self._fprs, 0)

1666 return _signOf(r, -s)

1667

1668 def toRepr(self, **prec_sep_fmt_lenc): # PYCHOK signature

1669 '''Return this C{Fsum} instance as representation.

1670

1671 @kwarg prec_sep_fmt_lenc: Optional keyword arguments for

1672 method L{Fsum2Tuple.toRepr} plus C{B{lenc}=True}

1673 (C{bool}) to in-/exclude the current C{[len]}

1674 of this L{Fsum} enclosed in I{[brackets]}.

1675

1676 @return: This instance (C{repr}).

1677 '''

1678 return self._toT(self._fprs2.toRepr, **prec_sep_fmt_lenc)

1679

1680 def toStr(self, **prec_sep_fmt_lenc): # PYCHOK signature

1681 '''Return this C{Fsum} instance as string.

1682

1683 @kwarg prec_sep_fmt_lenc: Optional keyword arguments for

1684 method L{Fsum2Tuple.toStr} plus C{B{lenc}=True}

1685 (C{bool}) to in-/exclude the current C{[len]}

1686 of this L{Fsum} enclosed in I{[brackets]}.

1687

1688 @return: This instance (C{str}).

1689 '''

1690 return self._toT(self._fprs2.toStr, **prec_sep_fmt_lenc)

1691

1692 def _toT(self, toT, fmt=Fmt.g, lenc=True, **kwds):

1693 '''(INTERNAL) Helper for C{toRepr} and C{toStr}.

1694 '''

1695 n = self.named3

1696 if lenc:

1697 n = Fmt.SQUARE(n, len(self))

1698 return _SPACE_(n, toT(fmt=fmt, **kwds))

1699

1700 def _TypeError(self, op, other, **txt): # PYCHOK no cover

1701 '''(INTERNAL) Return a C{TypeError}.

1702 '''

1703 return self._Error(op, other, _TypeError, **txt)

1704

1705 def _update(self): # see ._fset

1706 '''(INTERNAL) Zap all cached C{Property_RO} values.

1707 '''

1708 Fsum._fint2._update(self)

1709 Fsum._fprs ._update(self)

1710 Fsum._fprs2._update(self)

1711 return self

1712

1713 def _ValueError(self, op, other, **txt): # PYCHOK no cover

1714 '''(INTERNAL) Return a C{ValueError}.

1715 '''

1716 return self._Error(op, other, _ValueError, **txt)

1717

1718 def _ZeroDivisionError(self, op, other, **txt): # PYCHOK no cover

1719 '''(INTERNAL) Return a C{ZeroDivisionError}.

1720 '''

1721 return self._Error(op, other, _ZeroDivisionError, **txt)

1722

1723_allPropertiesOf_n(3, Fsum, Property_RO) # PYCHOK assert, see Fsum._fset, -._update

1724

1725

1726 def _Float_Int(arg, **name_Error):

1727 '''(INTERNAL) Unit of L{Fsum2Tuple} items.

1728 '''

1729 U = Int if isint(arg) else Float

1730 return U(arg, **name_Error)

1731

1732

1733 class Fsum2Tuple(_NamedTuple):

1734 '''2-Tuple C{(fsum, residual)} with the precision running C{fsum}

1735 and the C{residual}, the sum of the remaining partials. Each

1736 item is either C{float} or C{int}.

1737

1738 @note: If the C{residual is INT0}, the C{fsum} is considered

1739 to be I{exact}, see method L{Fsum2Tuple.is_exact}.

1740 '''

1741 _Names_ = ( Fsum.fsum.__name__, Fsum.residual.name)

1742 _Units_ = (_Float_Int, _Float_Int)

1743

1744 @Property_RO

1745 def Fsum(self):

1746 '''Get this L{Fsum2Tuple} as an L{Fsum}.

1747 '''

1748 f = Fsum(name=self.name)

1749 return f._copy_0(*(s for s in reversed(self) if s))

1750

1751 def is_exact(self):

1752 '''Is this L{Fsum2Tuple} considered to be exact? (C{bool}).

1753 '''

1754 return self.Fsum.is_exact()

1755

1756 def is_integer(self):

1757 '''Is this L{Fsum2Tuple} C{integer}? (C{bool}).

1758 '''

1759 return self.Fsum.is_integer()

1760

1761

1762 class ResidualError(_ValueError):

1763 '''Error raised for an operation involving a L{pygeodesy.sums.Fsum}

1764 instance with a non-zero C{residual}, I{integer} or otherwise.

1765

1766 @see: Module L{pygeodesy.fsums} and method L{Fsum.RESIDUAL}.

1767 '''

1768 pass

1769

1770

1771 try:

1772 from math import fsum as _fsum # precision IEEE-754 sum, Python 2.6+

1773

1774 # make sure _fsum works as expected (XXX check

1775 # float.__getformat__('float')[:4] == 'IEEE'?)

1776 if _fsum((1, 1e101, 1, -1e101)) != 2: # PYCHOK no cover

1777 del _fsum # nope, remove _fsum ...

1778 raise ImportError # ... use _fsum below

1779

1780 Fsum._math_fsum = _fsum

1781

1782 if _getenv('PYGEODESY_FSUM_PARTIALS', _fsum.__name__) == _fsum.__name__:

1783 _psum = _fsum # PYCHOK redef

1784

1785 except ImportError:

1786

1787 def _fsum(xs):

1788 '''(INTERNAL) Precision summation, Python 2.5-.

1789 '''

1790 return Fsum(name=_fsum.__name__)._facc(xs, up=False)._fprs

1791

1792

1793 def fsum(xs, floats=False):

1794 '''Precision floating point summation based on or like Python's C{math.fsum}.

1795

1796 @arg xs: Iterable, list, tuple, etc. of values (C{scalar} or

1797 L{Fsum} instances).

1798 @kwarg floats: Optionally, use C{B{floats}=True} iff I{all}

1799 B{C{xs}} are known to be C{float}.

1800

1801 @return: Precision C{fsum} (C{float}).

1802

1803 @raise OverflowError: Partial C{2sum} overflow.

1804

1805 @raise TypeError: Non-scalar B{C{xs}} value.

1806

1807 @raise ValueError: Invalid or non-finite B{C{xs}} value.

1808

1809 @note: Exceptions and I{non-finite} handling may differ if not

1810 based on Python's C{math.fsum}.

1811

1812 @see: Class L{Fsum} and methods L{Fsum.fsum} and L{Fsum.fadd}.

1813 '''

1814 return _fsum(xs if floats else _2floats(xs)) if xs else _0_0 # PYCHOK yield

1815

1816

1817 def fsum_(*xs, **floats):

1818 '''Precision floating point summation of all positional arguments.

1819

1820 @arg xs: Values to be added (C{scalar} or L{Fsum} instances),

1821 all positional.

1822 @kwarg floats: Optionally, use C{B{floats}=True} iff I{all}

1823 B{C{xs}} are known to be C{float}.

1824

1825 @return: Precision C{fsum} (C{float}).

1826

1827 @see: Function C{fsum}.

1828 '''

1829 return _fsum(xs if _xkwds_get(floats, floats=False) else

1830 _2floats(xs, origin=1)) if xs else _0_0 # PYCHOK yield

1831

1832

1833 def fsumf_(*xs):

1834 '''Precision floating point summation L{fsum_}C{(*xs, floats=True)}.

1835 '''

1836 return _fsum(xs) if xs else _0_0

1837

1838

1839 def fsum1(xs, floats=False):

1840 '''Precision floating point summation of a few arguments, 1-primed.

1841

1842 @arg xs: Iterable, list, tuple, etc. of values (C{scalar} or

1843 L{Fsum} instances).

1844 @kwarg floats: Optionally, use C{B{floats}=True} iff I{all}

1845 B{C{xs}} are known to be C{float}.

1846

1847 @return: Precision C{fsum} (C{float}).

1848

1849 @see: Function C{fsum}.

1850 '''

1851 return _fsum(_1primed(xs if floats else _2floats(xs))) if xs else _0_0 # PYCHOK yield

1852

1853

1854 def fsum1_(*xs, **floats):

1855 '''Precision floating point summation of a few arguments, 1-primed.

1856

1857 @arg xs: Values to be added (C{scalar} or L{Fsum} instances),

1858 all positional.

1859 @kwarg floats: Optionally, use C{B{floats}=True} iff I{all}

1860 B{C{xs}} are known to be C{float}.

1861

1862 @return: Precision C{fsum} (C{float}).

1863

1864 @see: Function C{fsum}

1865 '''

1866 return _fsum(_1primed(xs if _xkwds_get(floats, floats=False) else

1867 _2floats(xs, origin=1))) if xs else _0_0 # PYCHOK yield

1868

1869

1870 def fsum1f_(*xs):

1871 '''Precision floating point summation L{fsum1_}C{(*xs, floats=True)}.

1872 '''

1873 return _fsum(_1primed(xs)) if xs else _0_0

1874

1875

1876# **) MIT License

1877#

1878# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.

1879#

1880# Permission is hereby granted, free of charge, to any person obtaining a

1881# copy of this software and associated documentation files (the "Software"),

1882# to deal in the Software without restriction, including without limitation

1883# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1884# and/or sell copies of the Software, and to permit persons to whom the

1885# Software is furnished to do so, subject to the following conditions:

1886#

1887# The above copyright notice and this permission notice shall be included

1888# in all copies or substantial portions of the Software.

1889#

1890# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1891# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1892# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1893# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1894# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1895# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1896# OTHER DEALINGS IN THE SOFTWARE.