1# -*- coding: utf-8 -*-

2

3 u'''Single-instance C{float} and C{int} constants across C{pygeodesy}

4 modules and related functions L{pygeodesy.float_}, L{pygeodesy.isclose},

5 L{pygeodesy.isfinite}, L{pygeodesy.isinf}, L{pygeodesy.isint0},

6 L{pygeodesy.isnan}, L{pygeodesy.isnear0}, L{pygeodesy.isnear1},

7 L{pygeodesy.isnear90}, L{pygeodesy.isneg0}, L{pygeodesy.isninf},

8 L{pygeodesy.isnon0} and L{pygeodesy.remainder}.

9'''

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

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

12

13 from pygeodesy.basics import _0_0, _copysign, isbool, iscomplex, \

14 isint, _ALL_LAZY

15 from pygeodesy.errors import _xError, _xError2, _xkwds_get

16 from pygeodesy.interns import _INF_, _NAN_, _UNDER_

17# from pygeodesy.lazily import _ALL_LAZY # from .basics

18# from pygeodesy.streprs import Fmt # from .unitsBase

19 from pygeodesy.unitsBase import Float, Int, Radius, Fmt

20

21 from math import fabs, isinf, isnan, pi as _PI, sqrt

22 try:

23 from math import inf as _inf, nan as _nan # PYCHOK Python 3+

24 except ImportError: # Python 2-

25 _inf, _nan = float(_INF_), float(_NAN_)

26

27__all__ = _ALL_LAZY.constants

28__version__ = '23.10.19'

29

30

31 def _copysign_0_0(y):

32 '''(INTERNAL) copysign(0.0, y), only C{float}.

33 '''

34 return _N_0_0 if y < 0 else _0_0

35

36

37 def _copysign_1_0(y):

38 '''(INTERNAL) copysign(1.0, y), only C{float}.

39 '''

40 return _N_1_0 if y < 0 else _1_0

41

42

43 def _copysignINF(y):

44 '''(INTERNAL) copysign(INF, y), only C{float}.

45 '''

46 return NINF if y < 0 else INF

47

48

49 def _Float(**name_arg):

50 '''(INTERNAL) New named, cached C{Float}.

51 '''

52 n, arg = name_arg.popitem()

53 return Float(_float(arg), name=n)

54

55

56 def _Radius(**name_arg):

57 '''(INTERNAL) New named, cached C{Radius}.

58 '''

59 n, arg = name_arg.popitem()

60 return Radius(_float(arg), name=n)

61

62

63 def float_(*fs, **sets): # sets=False

64 '''Get scalars as C{float} or I{intern}'ed C{float}.

65

66 @arg fs: One more values (C{scalar}), all positional.

67 @kwarg sets: Use C{B{sets}=True} to C{intern} each

68 B{C{fs}}, otherwise don't C{intern}.

69

70 @return: A single C{float} if only one B{C{fs}} is

71 given, otherwise a tuple of C{float}s.

72

73 @raise TypeError: Some B{C{fs}} is not C{scalar}.

74 '''

75 fl = []

76 _a = fl.append

77 _f = _floats.setdefault if _xkwds_get(sets, sets=False) else \

78 _floats.get

79 try:

80 for i, f in enumerate(fs):

81 f = float(f)

82 _a(_f(f, f))

83 except Exception as x:

84 E, t = _xError2(x)

85 fs_i = Fmt.SQUARE(fs=i)

86 raise E(fs_i, f, txt=t)

87 return fl[0] if len(fl) == 1 else tuple(fl)

88

89

90 def _float(f): # in .datums, .ellipsoids, ...

91 '''(INTERNAL) Cache initial C{float}s.

92 '''

93 f = float(f)

94 return _floats.setdefault(f, f) # PYCHOK del _floats

95

96

97 def float0_(*xs):

98 '''Yield C{B{x}s} as a non-NEG0 C{float}.

99 '''

100 for x in xs:

101 yield float(x) if x else _0_0

102

103

104 def _float0(f): # in .resections, .vector3dBase, ...

105 '''(INTERNAL) Return C{float(B{f})} or C{INT0}.

106 '''

107 if f:

108 f = float(f)

109 f = _floats.get(f, f)

110 elif f is not INT0:

111 f = _0_0

112 return f

113

114

115 def _floatuple(*fs):

116 '''(INTERNAL) Cache a tuple of C{float}s.

117 '''

118 return tuple(map(_float, fs))

119

120

121 try:

122 from math import log2 as _log2

123 except ImportError: # Python 3.3-

124 from math import log as _log

125

126 def _log2(x): # in .rhumbaux, .auxilats.auxLat

127 return _log(x, 2)

128

129

130 def _naninf(p, *xs):

131 '''(INTERNAL) Return C{NAN}, C{NINF} or C{INF}.

132 '''

133 for x in xs:

134 p *= x # fmath.fprod(xs)

135 return _copysignINF(p) if isfinite(p) else p

136

137

138 def _over(p, q):

139 '''(INTERNAL) Return C{B{p} / B{q}} avoiding C{ZeroDivisionError} exceptions.

140 '''

141 try:

142 return (p / q)

143 except ZeroDivisionError:

144 return (_copysignINF(p) if isfinite(p) else NAN) if p else p

145

146

147 def _1_over(x):

148 '''(INTERNAL) Return reciprocal C{1 / B{x}} avoiding C{ZeroDivisionError} exceptions.

149 '''

150 try:

151 return _1_0 / float(x)

152 except ZeroDivisionError:

153 return NINF if isneg0(x) else INF

154

155

156_floats = {} # PYCHOK floats cache, in .__main__

157# _float = float # PYCHOK expected

158# del _floats # XXX zap floats cache never

159

160_0_0 = _float( _0_0) # PYCHOK expected

161_0_0_1T = _0_0, # PYCHOK 1-tuple

162_0_0_9T = _0_0_1T * 9 # PYCHOK 9-tuple

163_0_0001 = _float( 0.0001) # PYCHOK expected

164_0_001 = _float( 0.001) # PYCHOK expected

165_0_01 = _float( 0.01) # PYCHOK expected

166_0_1 = _float( 0.1) # PYCHOK expected

167_0_125 = _float( 0.125) # PYCHOK expected

168_0_25 = _float( 0.25) # PYCHOK expected

169_0_26 = _float( 0.26) # PYCHOK expected

170_0_5 = _float( 0.5) # PYCHOK expected

171_1_0 = _float( 1) # PYCHOK expected

172_1_0_1T = _1_0, # PYCHOK 1-tuple

173_1_5 = _float( 1.5) # PYCHOK expected

174_1_75 = _float( 1.75) # PYCHOK expected

175_2_0 = _float( 2) # PYCHOK expected

176_3_0 = _float( 3) # PYCHOK expected

177_4_0 = _float( 4) # PYCHOK expected

178_5_0 = _float( 5) # PYCHOK expected

179_6_0 = _float( 6) # PYCHOK expected

180_8_0 = _float( 8) # PYCHOK expected

181_9_0 = _float( 9) # PYCHOK expected

182_10_0 = _float( 10) # PYCHOK expected

183_16_0 = _float( 16) # PYCHOK expected

184_32_0 = _float( 32) # PYCHOK expected

185_60_0 = _float( 60) # PYCHOK expected

186_90_0 = _float( 90) # PYCHOK expected

187_100_0 = _float( 100) # PYCHOK expected

188_180_0 = _float( 180) # PYCHOK expected

189_270_0 = _float( 270) # PYCHOK expected

190_360_0 = _float( 360) # PYCHOK expected

191_400_0 = _float( 400) # PYCHOK expected

192_720_0 = _float( 720) # PYCHOK expected

193_1000_0 = _float(1000) # PYCHOK expected

194_3600_0 = _float(3600) # PYCHOK expected

195

196_N_0_0 = float( '-0.0') # PYCHOK NOT _float!

197_N_0_5 = _float( -_0_5) # PYCHOK expected

198_N_1_0 = _float( -_1_0) # PYCHOK expected

199_N_2_0 = _float( -_2_0) # PYCHOK expected

200_N_90_0 = _float( -_90_0) # PYCHOK expected

201_N_180_0 = _float(-_180_0) # PYCHOK expected

202

203_M_KM = _1000_0 # meter per Kilo meter, see .utily

204_M_NM = _float(1852.0) # meter per Nautical Mile

205_M_SM = _float(1609.344) # meter per Statute Mile

206

207 try:

208 from sys import float_info as _f_i

209 # @see: <https://NumPy.org/doc/stable/reference/generated/numpy.finfo.html>

210 DIG = Int( DIG =_f_i.dig) # PYCHOK system's float decimal digits

211 EPS = _Float(EPS =_f_i.epsilon) # PYCHOK system's EPSilon

212 MANT_DIG = Int( MANT_DIG=_f_i.mant_dig) # PYCHOK system's float mantissa bits

213 MAX = _Float(MAX =_f_i.max) # PYCHOK system's MAX float 1.7976931348623157e+308

214 MAX_EXP = Int( MAX_EXP =_f_i.max_exp) # PYTHON system's max base 2 exponent

215 MIN = _Float(MIN =_f_i.min) # PYCHOK system's MIN float 2.2250738585072014e-308

216 MIN_EXP = Int( MIN_EXP =_f_i.min_exp) # PYTHON system's min base 2 exponent

217# RADIX = Int( RADIX =_f_i.radix) # PYTHON system's float base

218 del _f_i

219 except ImportError: # PYCHOK no cover

220 DIG = Int( DIG =15) # PYCHOK system's 64-bit float decimal digits

221 EPS = _Float(EPS =2.220446049250313e-16) # PYCHOK EPSilon 2**-52, M{EPS +/- 1 != 1}

222 MANT_DIG = Int( MANT_DIG=53) # PYCHOK float mantissa bits ≈ 53 (C{int})

223 MAX = _Float(MAX =pow(_2_0, 1023) * (_2_0 - EPS)) # PYCHOK ≈ 10**308

224 MAX_EXP = Int( MAX_ESP =_log2(MAX)) # 308 base 10

225 MIN = _Float(MIN =pow(_2_0, -1021)) # PYCHOK ≈ 10**-308

226 MIN_EXP = Int(MIN_EXP =_log2(MIN)) # -307 base 10

227# RADIX = Int(Radix =2) # base

228

229 EPS0 = _Float( EPS0 = EPS**2) # PYCHOK near-/non-zero comparison 4.930381e-32, or EPS or EPS_2

230 EPS02 = _Float( EPS02 = EPS**4) # PYCHOK near-zero-squared comparison 2.430865e-63

231 EPS_2 = _Float( EPS_2 = EPS / _2_0) # PYCHOK ≈ 1.110223024625e-16

232 EPS1 = _Float( EPS1 =_1_0 - EPS) # PYCHOK ≈ 0.9999999999999998

233 EPS2 = _Float( EPS2 = EPS * _2_0) # PYCHOK ≈ 4.440892098501e-16

234 EPS4 = _Float( EPS4 = EPS * _4_0) # PYCHOK ≈ 8.881784197001e-16

235# _1EPS = _Float(_1EPS =_1_0 + EPS) # PYCHOK ≈ 1.0000000000000002

236_1_EPS = _Float(_1_EPS =_1_0 / EPS) # PYCHOK = 4503599627370496.0

237# _2_EPS = _Float(_2_EPS =_2_0 / EPS) # PYCHOK = 9007199254740992.0

238_EPS2e4 = _Float(_EPS2e4 = EPS2 * 1.e4) # PYCHOK ≈ 4.440892098501e-12

239_EPS4e8 = _Float(_EPS4e8 = EPS4 * 1.e8) # PYCHOK ≈ 8.881784197001e-08

240_EPSmin = _Float(_EPSmin = sqrt(MIN)) # PYCHOK = 1.49166814624e-154

241_EPSqrt = _Float(_EPSqrt = sqrt(EPS)) # PYCHOK = 1.49011611938e5-08

242_EPStol = _Float(_EPStol =_EPSqrt * _0_1) # PYCHOK = 1.49011611938e5-09 == sqrt(EPS * _0_01)

243

244_89_999_ = _Float(_89_999_= EPS1 * _90_0) # just below 90.0

245# <https://Numbers.Computation.Free.FR/Constants/Miscellaneous/digits.html>

246_1__90 = _Float(_1__90 =_1_0 / _90_0) # PYCHOK = 0.011_111_111_111_111_111_111_111_111_111_111_111_111_111_111_11111

247_2__PI = _Float(_2__PI =_2_0 / _PI) # PYCHOK = 0.636_619_772_367_581_343_075_535_053_490_057_448_137_838_582_96182

248

249_1_16th = _Float(_1_16th =_1_0 / _16_0) # PYCHOK in .ellipsoids, .karney

250_1_64th = _Float(_1_64th =_1_0 / 64) # PYCHOK in .elliptic, pow(2.0, -6)

251_1_3rd = _Float(_1_3rd =_1_0 / _3_0) # PYCHOK in .fmath

252_1_6th = _Float(_1_6th =_1_0 / _6_0) # PYCHOK in .fmath

253_2_3rd = _Float(_2_3rd =_2_0 / _3_0) # PYCHOK in .fmath

254

255_K0_UTM = _Float(_K0_UTM = 0.9996) # PYCHOK in .etm, .ktm, .utm, UTM scale at central meridian

256# sqrt(2) <https://WikiPedia.org/wiki/Square_root_of_2>

257# 1.414213562373095_048_801_688_724_209_698_078_569_671_875_376_948_073_176_679_737_99

258# _1SQRT2= _Float(_1SQRT2 =sqrt(_2_0) + 1)

259_SQRT2_2 = _Float(_SQRT2_2=sqrt(_0_5)) # PYCHOK = 0.707106781186547_6 == sqrt(2) / 2

260

261 INF = Float(INF =_inf) # PYCHOK INFinity, see function L{isinf}, L{isfinite}, NOT _Float!

262 INT0 = Int( INT0= 0) # PYCHOK unique int(0) instance, see .fsums, useZ=False

263 NAN = Float(NAN =_nan) # PYCHOK Not-A-Number, see function L{isnan}, NOT _Float!

264 NEG0 = Float(NEG0=_N_0_0) # PYCHOK NEGative 0.0, see function L{isneg0}, NOT _Float!

265 NINF = Float(NINF=-INF) # PYCHOK Negative INFinity, NOT _Float!

266

267 PI = _Float(PI =_PI)

268 PI2 = _Float(PI2 =_PI * _2_0) # PYCHOK Two PI, M{PI * 2} aka I{Tau}

269 PI_2 = _Float(PI_2 =_PI / _2_0) # PYCHOK Half PI, M{PI / 2}

270 PI3 = _Float(PI3 =_PI * _3_0) # PYCHOK Three PI, M{PI * 3}

271 PI3_2 = _Float(PI3_2=_PI * _1_5) # PYCHOK PI and a half, M{PI * 3 / 2}

272 PI_3 = _Float(PI_3 =_PI / _3_0) # PYCHOK One third PI, M{PI / 3}

273 PI4 = _Float(PI4 =_PI * _4_0) # PYCHOK Four PI, M{PI * 4}

274 PI_4 = _Float(PI_4 =_PI / _4_0) # PYCHOK Quarter PI, M{PI / 4}

275

276 R_MA = _Radius(R_MA=6378137.0) # PYCHOK equatorial earth radius (C{meter}), WGS84, EPSG:3785

277 R_MB = _Radius(R_MB=6356752.3) # PYCHOK polar earth radius (C{meter}), WGS84, EPSG:3785

278 R_M = _Radius(R_M =6371008.771415) # PYCHOK mean, spherical earth radius (C{meter})

279 R_KM = _Radius(R_KM=R_M / _M_KM) # PYCHOK mean, spherical earth radius (C{KM}, Kilo meter)

280 R_NM = _Radius(R_NM=R_M / _M_NM) # PYCHOK mean, spherical earth radius (C{NM}, nautical miles)

281 R_SM = _Radius(R_SM=R_M / _M_SM) # PYCHOK mean, spherical earth radius (C{SM}, statute miles)

282# See <https://www.EdWilliams.org/avform.htm>, <https://www.DTIC.mil/dtic/tr/fulltext/u2/a216843.pdf>

283# and <https://GitHub.com/NASA/MultiDop/blob/master/src/share/man/man3/geog_lib.3> based on the

284# International Standard Nautical Mile of 1,852 meter (1' latitude)

285 R_FM = _Radius(R_FM=6371000.0) # PYCHOK former FAI Sphere earth radius (C{meter})

286 R_GM = _Radius(R_GM=6371230.0) # PYCHOK avg. radius, distance to geoid surface (C{meter})

287# <http://Wiki.GIS.com/wiki/index.php/Ellipsoidal_quadratic_mean_radius>

288 R_QM = _Radius(R_QM=6372797.560856) # PYCHOK earth' quadratic mean radius (C{meter})

289# Rtri= _Radius(Rtri=6372797.5559594) # PYCHOK Rtriaxial quadratic mean radius (C{meter}), WGS84

290# Rbi = _Radius(Rbi =6367453.6345163) # PYCHOK Rbiaxial quadratic mean radius (C{meter}), WGS84

291 R_VM = _Radius(R_VM=6366707.0194937) # PYCHOK aViation/naVigation earth radius (C{meter})

292# R_AU= Meter( R_AU=149597870700.0) # PYCHOK <https://WikiPedia.org/wiki/Astronomical_unit>

293

294_INF_NAN_NINF = INF, NAN, NINF

295_pos_self = _1_0.__pos__() is _1_0 # PYCHOK in .fsums, .vector3dBase

296

297

298 def _0_0s(n):

299 '''(INTERNAL) Return an C{B{n}-tuple} of C{_0_0} zeros.

300 '''

301 return _0_0_9T[:n] if 0 <= n <= len(_0_0_9T) else (_0_0_1T * n)

302

303

304 try:

305 from math import isclose as _isclose

306 except ImportError: # Python 3.4-

307

308 def _isclose(a, b, rel_tol=1e-9, abs_tol=0):

309 '''Mimick Python 3.5+ C{math.isclose}.

310 '''

311 t, d = abs_tol, fabs(a - b)

312 if d > t:

313 r = max(fabs(a), fabs(b)) * rel_tol

314 t = max(r, t)

315 return d <= t

316

317

318 def isclose(a, b, rel_tol=1e-12, abs_tol=EPS0):

319 '''Like C{math.isclose}, but with defaults such

320 that C{isclose(0, EPS0)} is C{True} by default.

321 '''

322 return _isclose(a, b, rel_tol=rel_tol, abs_tol=abs_tol)

323

324

325 try:

326 from math import isfinite as _isfinite # in .ellipsoids, ...

327 except ImportError: # Python 3.1-

328

329 def _isfinite(x):

330 '''Mimick Python 3.2+ C{math.isfinite}.

331 '''

332 return not (isinf(x) or isnan(x))

333

334

335 def isfinite(obj):

336 '''Check a finite C{scalar} or C{complex} value.

337

338 @arg obj: Value (C{scalar} or C{complex}).

339

340 @return: C{False} if B{C{obj}} is C{INF}, C{NINF}

341 or C{NAN}, C{True} otherwise.

342

343 @raise TypeError: Non-scalar and non-complex B{C{obj}}.

344 '''

345 try:

346 return (obj not in _INF_NAN_NINF) and _isfinite(obj)

347 except Exception as x:

348 if iscomplex(obj): # _isfinite(complex) thows TypeError

349 return isfinite(obj.real) and isfinite(obj.imag)

350 raise _xError(x, Fmt.PAREN(isfinite.__name__, obj))

351

352

353 def isint0(obj, both=False):

354 '''Check for L{INT0} or C{int(0)} value.

355

356 @arg obj: The object (any C{type}).

357 @kwarg both: If C{true}, also check C{float(0)} (C{bool}).

358

359 @return: C{True} if B{C{obj}} is L{INT0}, C{int(0)} or

360 C{float(0)}, C{False} otherwise.

361 '''

362 return (obj is INT0 or obj is int(0) or bool(both and

363 (not obj) and isint(obj, both=True))) and not isbool(obj)

364

365

366 def isnear0(x, eps0=EPS0):

367 '''Is B{C{x}} near I{zero} within a tolerance?

368

369 @arg x: Value (C{scalar}).

370 @kwarg eps0: Near-I{zero} tolerance (C{EPS0}).

371

372 @return: C{True} if C{abs(B{x}) < B{eps0}},

373 C{False} otherwise.

374

375 @see: Function L{isnon0}.

376 '''

377 return bool(eps0 > x > -eps0)

378

379

380 def isnear1(x, eps1=EPS0):

381 '''Is B{C{x}} near I{one} within a tolerance?

382

383 @arg x: Value (C{scalar}).

384 @kwarg eps1: Near-I{one} tolerance (C{EPS0}).

385

386 @return: C{isnear0(B{x} - 1, eps0=B{eps1})}.

387

388 @see: Function L{isnear0}.

389 '''

390 return bool(eps1 > (x - _1_0) > -eps1)

391

392

393 def isnear90(x, eps90=EPS0):

394 '''Is B{C{x}} near I{90} within a tolerance?

395

396 @arg x: Value (C{scalar}).

397 @kwarg eps90: Near-I{90} tolerance (C{EPS0}).

398

399 @return: C{isnear0(B{x} - 90)}.

400

401 @see: Function L{isnear0}.

402 '''

403 return bool(eps90 > (x - _90_0) > -eps90)

404

405

406 def isneg0(x):

407 '''Check for L{NEG0}, negative C{0.0}.

408

409 @arg x: Value (C{scalar}).

410

411 @return: C{True} if B{C{x}} is C{NEG0} or C{-0.0},

412 C{False} otherwise.

413 '''

414 return (not x) and _copysign(1, x) < 0

415# and str(x).startswith(_MINUS_)

416

417

418 def isninf(x):

419 '''Check for L{NINF}, negative C{INF}.

420

421 @arg x: Value (C{scalar}).

422

423 @return: C{True} if B{C{x}} is C{NINF} or C{-inf},

424 C{False} otherwise.

425 '''

426 return x is NINF or (x < 0 and not isfinite(x))

427

428

429 def isnon0(x, eps0=EPS0):

430 '''Is B{C{x}} non-zero with a tolerance?

431

432 @arg x: Value (C{scalar}).

433 @kwarg eps0: Non-zero tolerance (C{EPS0}).

434

435 @return: C{True} if C{abs(B{x}) > B{eps0}},

436 C{False} otherwise.

437

438 @see: Function L{isnear0}.

439 '''

440 return not bool(eps0 > x > -eps0) # not isnear0

441

442

443 def _off90(lat):

444 '''(INTERNAL) Off 90.0 for .gars and .wgrs.

445 '''

446 return max(min(lat, _89_999_), -_89_999_)

447

448

449 try:

450 from math import remainder

451 except ImportError: # Python 3.6-

452 from math import fmod as _fmod

453

454 def remainder(x, y):

455 '''Mimick Python 3.7+ C{math.remainder}.

456 '''

457 if isnan(y):

458 x = NAN

459 elif x and not isnan(x):

460 y = fabs(y)

461 x = _fmod(x, y)

462 h = _0_5 * y

463 if x >= h:

464 x -= y

465 elif x < -h:

466 x += y

467 return x # keep signed 0.0

468

469

470 def _umod_360(deg):

471 '''(INTERNAL) Non-negative C{deg} modulo 360, basic C{.utily.wrap360}.

472 '''

473 return (deg % _360_0) or _0_0

474

475

476 def _umod_PI2(rad):

477 '''(INTERNAL) Non-negative C{rad} modulo PI2, basic C{.utily.wrapPI2}.

478 '''

479 return (rad % PI2) or _0_0

480

481

482 if __name__ == '__main__':

483

484 from pygeodesy.errors import itemsorted

485 from pygeodesy.lazily import printf

486

487 t = n = v = []

488 for n, v in itemsorted(locals()):

489 if isinstance(v, (Float, Int, Radius)):

490 printf('%9s: %r', n, v.toRepr(std=False))

491 if v.name != n:

492 raise AssertionError('%r != %r' % (n, v))

493 if v.name is not n:

494 raise AssertionError('%r is not %r' % (n, v))

495 if not n.startswith(_UNDER_):

496 t.append(n)

497 t.append(float_.__name__)

498 printf('__all__ = %r', tuple(t))

499

500# **) MIT License

501#

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

503#

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

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

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

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

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

510#

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

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

513#

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

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

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