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

2

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

4 related functions L{pygeodesy.float_}, L{pygeodesy.isclose}, L{pygeodesy.isfinite},

5 L{pygeodesy.isinf}, L{pygeodesy.isint0}, L{pygeodesy.isnan}, L{pygeodesy.isnear0},

6 L{pygeodesy.isnear1}, L{pygeodesy.isnear90}, L{pygeodesy.isneg0}, L{pygeodesy.isninf},

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

8'''

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

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

11

12 from pygeodesy.basics import _0_0, _copysign, isbool, iscomplex, isint, _ALL_LAZY

13 from pygeodesy.errors import _xError, _xError2, _xkwds_get, _xkwds_item2

14 from pygeodesy.interns import _INF_, _NAN_, _UNDER_

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

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

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

18

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

20 try:

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

22 except ImportError: # Python 2-

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

24

25__all__ = _ALL_LAZY.constants

26__version__ = '24.04.23'

27

28

29 def _copysign_0_0(y):

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

31 '''

32 return _N_0_0 if y < 0 else _0_0

33

34

35 def _copysign_1_0(y):

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

37 '''

38 return _N_1_0 if y < 0 else _1_0

39

40

41 def _copysignINF(y):

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

43 '''

44 return NINF if y < 0 else INF

45

46

47 def _Float(**name_arg):

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

49 '''

50 n, arg = _xkwds_item2(name_arg)

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

52

53

54 def _Radius(**name_arg):

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

56 '''

57 n, arg = _xkwds_item2(name_arg)

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

59

60

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

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

63

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

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

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

67

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

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

70

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

72 '''

73 fl = []

74 _a = fl.append

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

76 _floats.get

77 try:

78 for i, f in enumerate(fs):

79 f = float(f)

80 _a(_f(f, f))

81 except Exception as x:

82 E, t = _xError2(x)

83 fs_i = Fmt.SQUARE(fs=i)

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

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

86

87

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

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

90 '''

91 f = float(f)

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

93

94

95 def float0_(*xs):

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

97 '''

98 for x in xs:

99 yield float(x) if x else _0_0

100

101

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

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

104 '''

105 if f:

106 f = float(f)

107 f = _floats.get(f, f)

108 elif f is not INT0:

109 f = float(f) or _0_0 # force None, NN error

110 return f

111

112

113 def _floatuple(*fs):

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

115 '''

116 return tuple(map(_float, fs))

117

118

119 try:

120 from math import log2 as _log2

121 except ImportError: # Python 3.3-

122 from math import log as _log

123

124 def _log2(x): # in .rhumb.aux_, .auxilats.auxLat

125 return _log(x, 2)

126

127

128 def _naninf(p, *xs):

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

130 '''

131 for x in xs:

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

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

134

135

136 def _over(p, q):

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

138 '''

139 try:

140 return (p / q) # if p else _copysign_0_0(q)

141 except ZeroDivisionError:

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

143

144

145 def _1_over(x):

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

147 '''

148 try:

149 return _1_0 / float(x)

150 except ZeroDivisionError:

151 return NINF if isneg0(x) else INF

152

153

154_floats = {} # PYCHOK floats cache, in .__main__

155# _float = float # PYCHOK expected

156# del _floats # XXX zap floats cache never

157

158_0_0 = _float( _0_0) # PYCHOK expected

159_0_0_1T = _0_0, # PYCHOK 1-tuple

160_0_0_9T = _0_0_1T * 9 # PYCHOK 9-tuple

161_0_0001 = _float( 0.0001) # PYCHOK expected

162_0_001 = _float( 0.001) # PYCHOK expected

163_0_01 = _float( 0.01) # PYCHOK expected

164_0_1 = _float( 0.1) # PYCHOK expected

165_0_125 = _float( 0.125) # PYCHOK expected

166_0_25 = _float( 0.25) # PYCHOK expected

167_0_5 = _float( 0.5) # PYCHOK expected

168_1_0 = _float( 1) # PYCHOK expected

169_1_0_1T = _1_0, # PYCHOK 1-tuple

170_1_5 = _float( 1.5) # PYCHOK expected

171_1_75 = _float( 1.75) # PYCHOK expected

172_2_0 = _float( 2) # PYCHOK expected

173_3_0 = _float( 3) # PYCHOK expected

174_4_0 = _float( 4) # PYCHOK expected

175_5_0 = _float( 5) # PYCHOK expected

176_6_0 = _float( 6) # PYCHOK expected

177_8_0 = _float( 8) # PYCHOK expected

178_9_0 = _float( 9) # PYCHOK expected

179_10_0 = _float( 10) # PYCHOK expected

180_16_0 = _float( 16) # PYCHOK expected

181_32_0 = _float( 32) # PYCHOK expected

182_60_0 = _float( 60) # PYCHOK expected

183_90_0 = _float( 90) # PYCHOK expected

184_100_0 = _float( 100) # PYCHOK expected

185_180_0 = _float( 180) # PYCHOK expected

186_270_0 = _float( 270) # PYCHOK expected

187_360_0 = _float( 360) # PYCHOK expected

188_400_0 = _float( 400) # PYCHOK expected

189_720_0 = _float( 720) # PYCHOK expected

190_1000_0 = _float(1000) # PYCHOK expected

191_3600_0 = _float(3600) # PYCHOK expected

192

193_N_0_0 = float( '-0.0') # PYCHOK NOT _float!

194_N_0_5 = _float( -_0_5) # PYCHOK expected

195_N_1_0 = _float( -_1_0) # PYCHOK expected

196_N_2_0 = _float( -_2_0) # PYCHOK expected

197_N_90_0 = _float( -_90_0) # PYCHOK expected

198_N_180_0 = _float(-_180_0) # PYCHOK expected

199

200_M_KM = _1000_0 # meter per Kilo meter, see .utily

201_M_NM = _float(1852.0) # meter per Nautical Mile, exactly

202_M_SM = _float(1609.344) # meter per Statute Mile

203

204 try:

205 from sys import float_info as _f_i

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

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

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

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

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

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

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

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

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

215 del _f_i

216 except ImportError: # PYCHOK no cover

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

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

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

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

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

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

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

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

225

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

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

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

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

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

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

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

233_1_EPS = _Float(_1_EPS =_1_0 / EPS) # PYCHOK = 4503599627370496.0

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

235_EPS2e4 = _Float(_EPS2e4 = EPS2 * 1.e4) # PYCHOK ≈ 4.440892098501e-12

236_EPS4e8 = _Float(_EPS4e8 = EPS4 * 1.e8) # PYCHOK ≈ 8.881784197001e-08

237_EPSmin = _Float(_EPSmin = sqrt(MIN)) # PYCHOK = 1.49166814624e-154

238_EPSqrt = _Float(_EPSqrt = sqrt(EPS)) # PYCHOK = 1.49011611938e5-08

239_EPStol = _Float(_EPStol =_EPSqrt * _0_1) # PYCHOK = 1.49011611938e5-09 == sqrt(EPS * _0_01)

240

241_89_999_ = _Float(_89_999_= EPS1 * _90_0) # just below 90.0

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

243_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

244_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

245

246_1_16th = _Float(_1_16th =_1_0 / _16_0) # PYCHOK in .ellipsoids, .karney

247_1_64th = _Float(_1_64th =_1_0 / 64) # PYCHOK in .elliptic, pow(2.0, -6)

248_1_3rd = _Float(_1_3rd =_1_0 / _3_0) # PYCHOK in .fmath

249_1_6th = _Float(_1_6th =_1_0 / _6_0) # PYCHOK in .fmath

250

251_K0_UTM = _Float(_K0_UTM = 0.9996) # PYCHOK in .etm, .ktm, .utm, UTM scale at central meridian

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

253# 1.414213562373095_048_801_688_724_209_698_078_569_671_875_376_948_073_176_679_737_99

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

255_SQRT2_2 = _Float(_SQRT2_2=sqrt(_0_5)) # PYCHOK = 0.707106781186547_6 == sqrt(2) / 2

256

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

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

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

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

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

262

263 PI = _Float(PI =_PI) # 3.1415_9265_3589_7932_3846_2643_3832_795

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

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

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

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

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

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

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

271

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

289

290_INF_NAN_NINF = INF, NAN, NINF

291_pos_self = _1_0.__pos__() is _1_0 # PYCHOK in .fsums, .vector3dBase

292

293

294 def _0_0s(n):

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

296 '''

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

298

299

300 try:

301 from math import isclose as _isclose

302 except ImportError: # Python 3.4-

303

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

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

306 '''

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

308 if d > t:

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

310 t = max(r, t)

311 return d <= t

312

313

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

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

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

317 '''

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

319

320

321 try:

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

323 except ImportError: # Python 3.1-

324

325 def _isfinite(x):

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

327 '''

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

329

330

331 def isfinite(obj):

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

333

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

335

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

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

338

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

340 '''

341 try:

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

343 except Exception as x:

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

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

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

347

348

349 def isint0(obj, both=False):

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

351

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

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

354

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

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

357 '''

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

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

360

361

362 def isnear0(x, eps0=EPS0):

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

364

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

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

367

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

369 C{False} otherwise.

370

371 @see: Function L{isnon0}.

372 '''

373 return bool(eps0 > x > -eps0)

374

375

376 def isnear1(x, eps1=EPS0):

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

378

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

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

381

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

383

384 @see: Function L{isnear0}.

385 '''

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

387

388

389 def isnear90(x, eps90=EPS0):

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

391

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

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

394

395 @return: C{isnear0(B{x} - 90, eps0=eps90)}.

396

397 @see: Function L{isnear0}.

398 '''

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

400

401

402 def isneg0(x):

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

404

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

406

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

408 C{False} otherwise.

409 '''

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

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

412

413

414 def isninf(x):

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

416

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

418

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

420 C{False} otherwise.

421 '''

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

423

424

425 def isnon0(x, eps0=EPS0):

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

427

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

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

430

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

432 C{False} otherwise.

433

434 @see: Function L{isnear0}.

435 '''

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

437

438

439 def _off90(lat):

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

441 '''

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

443

444

445 try:

446 from math import remainder

447 except ImportError: # Python 3.6-

448 from math import fmod as _fmod

449

450 def remainder(x, y):

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

452 '''

453 if isnan(y):

454 x = NAN

455 elif x and not isnan(x):

456 y = fabs(y)

457 x = _fmod(x, y)

458 h = _0_5 * y

459 if x >= h:

460 x -= y

461 elif x < -h:

462 x += y

463 return x # keep signed 0.0

464

465

466 def _umod_360(deg):

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

468 '''

469 return (deg % _360_0) or _0_0

470

471

472 def _umod_PI2(rad):

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

474 '''

475 return (rad % PI2) or _0_0

476

477

478 if __name__ == '__main__':

479

480 from pygeodesy import itemsorted, printf

481

482 t = n = v = []

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

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

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

486 if v.name != n:

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

488 if v.name is not n:

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

490 if not n.startswith(_UNDER_):

491 t.append(n)

492 t.append(float_.__name__)

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

494

495# **) MIT License

496#

497# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.

498#

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

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

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

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

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

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

505#

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

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

508#

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

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

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