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The result is between 0 and pi.acosh($module, x, /) -- Return the inverse hyperbolic cosine of x.asin($module, x, /) -- Return the arc sine (measured in radians) of x. The result is between -pi/2 and pi/2.asinh($module, x, /) -- Return the inverse hyperbolic sine of x.atan($module, x, /) -- Return the arc tangent (measured in radians) of x. The result is between -pi/2 and pi/2.atan2($module, y, x, /) -- Return the arc tangent (measured in radians) of y/x. Unlike atan(y/x), the signs of both x and y are considered.atanh($module, x, /) -- Return the inverse hyperbolic tangent of x.ceil($module, x, /) -- Return the ceiling of x as an Integral. This is the smallest integer >= x.copysign($module, x, y, /) -- Return a float with the magnitude (absolute value) of x but the sign of y. On platforms that support signed zeros, copysign(1.0, -0.0) returns -1.0. cos($module, x, /) -- Return the cosine of x (measured in radians).cosh($module, x, /) -- Return the hyperbolic cosine of x.degrees($module, x, /) -- Convert angle x from radians to degrees.dist($module, p, q, /) -- Return the Euclidean distance between two points p and q. The points should be specified as sequences (or iterables) of coordinates. Both inputs must have the same dimension. Roughly equivalent to: sqrt(sum((px - qx) ** 2.0 for px, qx in zip(p, q)))erf($module, x, /) -- Error function at x.erfc($module, x, /) -- Complementary error function at x.exp($module, x, /) -- Return e raised to the power of x.expm1($module, x, /) -- Return exp(x)-1. This function avoids the loss of precision involved in the direct evaluation of exp(x)-1 for small x.fabs($module, x, /) -- Return the absolute value of the float x.factorial($module, x, /) -- Find x!. Raise a ValueError if x is negative or non-integral.floor($module, x, /) -- Return the floor of x as an Integral. This is the largest integer <= x.fmod($module, x, y, /) -- Return fmod(x, y), according to platform C. x % y may differ.frexp($module, x, /) -- Return the mantissa and exponent of x, as pair (m, e). m is a float and e is an int, such that x = m * 2.**e. If x is 0, m and e are both 0. Else 0.5 <= abs(m) < 1.0.fsum($module, seq, /) -- Return an accurate floating point sum of values in the iterable seq. Assumes IEEE-754 floating point arithmetic.gamma($module, x, /) -- Gamma function at x.gcd($module, *integers) -- Greatest Common Divisor.hypot(*coordinates) -> value Multidimensional Euclidean distance from the origin to a point. Roughly equivalent to: sqrt(sum(x**2 for x in coordinates)) For a two dimensional point (x, y), gives the hypotenuse using the Pythagorean theorem: sqrt(x*x + y*y). For example, the hypotenuse of a 3/4/5 right triangle is: >>> hypot(3.0, 4.0) 5.0 isclose($module, /, a, b, *, rel_tol=1e-09, abs_tol=0.0) -- Determine whether two floating point numbers are close in value. rel_tol maximum difference for being considered "close", relative to the magnitude of the input values abs_tol maximum difference for being considered "close", regardless of the magnitude of the input values Return True if a is close in value to b, and False otherwise. For the values to be considered close, the difference between them must be smaller than at least one of the tolerances. -inf, inf and NaN behave similarly to the IEEE 754 Standard. That is, NaN is not close to anything, even itself. inf and -inf are only close to themselves.isfinite($module, x, /) -- Return True if x is neither an infinity nor a NaN, and False otherwise.isinf($module, x, /) -- Return True if x is a positive or negative infinity, and False otherwise.isnan($module, x, /) -- Return True if x is a NaN (not a number), and False otherwise.isqrt($module, n, /) -- Return the integer part of the square root of the input.lcm($module, *integers) -- Least Common Multiple.ldexp($module, x, i, /) -- Return x * (2**i). This is essentially the inverse of frexp().lgamma($module, x, /) -- Natural logarithm of absolute value of Gamma function at x.log(x, [base=math.e]) Return the logarithm of x to the given base. If the base not specified, returns the natural logarithm (base e) of x.log1p($module, x, /) -- Return the natural logarithm of 1+x (base e). The result is computed in a way which is accurate for x near zero.log10($module, x, /) -- Return the base 10 logarithm of x.log2($module, x, /) -- Return the base 2 logarithm of x.modf($module, x, /) -- Return the fractional and integer parts of x. Both results carry the sign of x and are floats.pow($module, x, y, /) -- Return x**y (x to the power of y).radians($module, x, /) -- Convert angle x from degrees to radians.remainder($module, x, y, /) -- Difference between x and the closest integer multiple of y. Return x - n*y where n*y is the closest integer multiple of y. In the case where x is exactly halfway between two multiples of y, the nearest even value of n is used. The result is always exact.sin($module, x, /) -- Return the sine of x (measured in radians).sinh($module, x, /) -- Return the hyperbolic sine of x.sqrt($module, x, /) -- Return the square root of x.tan($module, x, /) -- Return the tangent of x (measured in radians).tanh($module, x, /) -- Return the hyperbolic tangent of x.trunc($module, x, /) -- Truncates the Real x to the nearest Integral toward 0. Uses the __trunc__ magic method.prod($module, iterable, /, *, start=1) -- Calculate the product of all the elements in the input iterable. The default start value for the product is 1. When the iterable is empty, return the start value. This function is intended specifically for use with numeric values and may reject non-numeric types.perm($module, n, k=None, /) -- Number of ways to choose k items from n items without repetition and with order. Evaluates to n! / (n - k)! when k <= n and evaluates to zero when k > n. 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@bL720210628153654Z0 S{G83 00tb M0  *H  0|100.U 'Apple Timestamp Certification Authority1&0$U Apple Certification Authority10U Apple Inc.1 0 UUS0 210623184502Z 210804184501Z0A10U Timestamp Signer MA110U Apple Inc.1 0 UUS0"0  *H 0 OɪYΜ?ݻFS}[)pPgH.IiQ+₲(p A226Ei6@>\7CtY [ 䗋j|eM0%~Պ9onviIeU;NztDTh(#hK̤ CB$4LyJ\gBzi@ŝsymmY9} )ז`Oc,ZpQ 00 U00U#04%N78X&)0U 00 *Hcd00(+http://www.apple.com/appleca0+0 Reliance on this certificate by any party assumes acceptance of the then applicable standard terms and conditions of use, certificate policy and certification practice statements.0U% 0 +03U,0*0(&$"http://crl.apple.com/timestamp.crl0U,|=ɕDXq'$0U0  *H  /U/ݱ,~bj$wڂt%L>C.u /v&Ț[51FnWy9ԡhZqTt^-[['.0)7xǘ31FH[ -O3ХD~ 'rEB } s@/P'4p <No97ڞecIm] 9 :JidF5ea. =o%L x"!{[ѣ~I; r3,JZr 9Pt(ny_~VVlĮ (tM&By^`EXQI^JMs{Rˣ]klt-D>a>E~4u&@u0Kw00U4%N78X&)0U00U#0+iGv k.@GM^0.U'0%0#!http://crl.apple.com/root.crl0U0 *Hcd 0  *H  6qS#xe[#Ǣϴ(p!93rC=BQu]ˌjj:T}k>mzT_Cv\ N<>r4\g 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NAN`?N ` ` Tg`!`T@(5* ` !igta@`AjgtaT@5HR@h4@qT hX@wP @qATn`!`%TN{COB#Am+l X@{COB#Am+l_ (X@at x !jfT(Rs@(5 a&T`"`g` d T `gDcTnoBndN$n cTon N_g@ cTnoBndN$n caT !~\n N_\ |\oBnn N_ Ձz\n N_ ա{\n N_g__WO{)(A?(T }?T T~ T{COBWA_Ĩ@ *@@@aT4@T{COBWA_Ĩ_@T#mO{@B9(7R{COB#Am @TJN ~ iaT0  ըX@"4#N iaT"N`?N``?@!a^a)`N`?# ը|X@YP {COB#Am_#m{CN`!hgtaT!`T(RgN{A#l_!`TN{A#l(R!`gg( `N{A#l_#m{CN`!hgtaT!`Tn(RgN{A#l_!`TN{A#l_(R!`gg( `N{A#l_#m{CN`!hgtaT!`TF(RgN{A#l_!`TN{A#lw7(R!`gg( `N{A#l_+m#m{` `gub*`!agDub!T `T aTg!aga N{C#Bm+Am_( `Tg=NAN/ N@9` !`dTmT @a9ilaAN#(`)9`no!n=@niNO{C `O\WAW0 |7 ՀP\NAV s7 ՠO\E!UP j7R<!TP a7R3!SP X|{AO¨_{AO¨_ `AT_ հ[X Ր[X p[X P[X 0[X [X ZX ZX հZX ՐZX pZX PZX 0ZX ZX YX YX հYX ՐYX pYX PYX 0YX YX XX XX հXX ՐXX pXX PXX 0XX XX WX WX հWX ՐWX pWX PWX 0WX WX VX VX հVX ՐVX pVX PVX 0VX VX UX UX հUX ՐUX pUX PUX 0UX UX TX TX հTX ՐTX pTX PTX 0TX TX SX SX հSX ՐSX pSX PSX 0SX SX RX RX հRX ՐRX pRX PRX 0RX RX QX QX հQX ՐQX pQX PQX 0QX QX PX PXPPP/PBP`PtPPPPPPPP*PAPaP}PzPwPtPqPnPk0PhEPe[PbtP_P\PYPVPSPP PM&PJBPGXPDqPAP>P;P8P5P2 P/!P,@P)^P&sP#P PPPPPP5P UPiPP; G Ր(XPPPPPPP PP(P6PBPOP[PhPtPPPPPPPPPPPPP&P3P?cܥL@#B ;E@EE@CQBWLup#B2 BiA&"BA补ApqA?tAA*_{ AqqiA]v}ALPEA뇇BAX@R;{`Zj@' @-DT! @-DT! & .>EHP?9RFߑ?-DT!?!3|@-DT!?iW @-DT!@(&7*-5 This module provides access to the mathematical functions defined by the C standard.acos($module, x, /) -- Return the arc cosine (measured in radians) of x. The result is between 0 and pi.acosh($module, x, /) -- Return the inverse hyperbolic cosine of x.asin($module, x, /) -- Return the arc sine (measured in radians) of x. The result is between -pi/2 and pi/2.asinh($module, x, /) -- Return the inverse hyperbolic sine of x.atan($module, x, /) -- Return the arc tangent (measured in radians) of x. The result is between -pi/2 and pi/2.atan2($module, y, x, /) -- Return the arc tangent (measured in radians) of y/x. Unlike atan(y/x), the signs of both x and y are considered.atanh($module, x, /) -- Return the inverse hyperbolic tangent of x.ceil($module, x, /) -- Return the ceiling of x as an Integral. This is the smallest integer >= x.copysign($module, x, y, /) -- Return a float with the magnitude (absolute value) of x but the sign of y. On platforms that support signed zeros, copysign(1.0, -0.0) returns -1.0. cos($module, x, /) -- Return the cosine of x (measured in radians).cosh($module, x, /) -- Return the hyperbolic cosine of x.degrees($module, x, /) -- Convert angle x from radians to degrees.dist($module, p, q, /) -- Return the Euclidean distance between two points p and q. The points should be specified as sequences (or iterables) of coordinates. Both inputs must have the same dimension. Roughly equivalent to: sqrt(sum((px - qx) ** 2.0 for px, qx in zip(p, q)))erf($module, x, /) -- Error function at x.erfc($module, x, /) -- Complementary error function at x.exp($module, x, /) -- Return e raised to the power of x.expm1($module, x, /) -- Return exp(x)-1. This function avoids the loss of precision involved in the direct evaluation of exp(x)-1 for small x.fabs($module, x, /) -- Return the absolute value of the float x.factorial($module, x, /) -- Find x!. Raise a ValueError if x is negative or non-integral.floor($module, x, /) -- Return the floor of x as an Integral. This is the largest integer <= x.fmod($module, x, y, /) -- Return fmod(x, y), according to platform C. x % y may differ.frexp($module, x, /) -- Return the mantissa and exponent of x, as pair (m, e). m is a float and e is an int, such that x = m * 2.**e. If x is 0, m and e are both 0. Else 0.5 <= abs(m) < 1.0.fsum($module, seq, /) -- Return an accurate floating point sum of values in the iterable seq. Assumes IEEE-754 floating point arithmetic.gamma($module, x, /) -- Gamma function at x.gcd($module, *integers) -- Greatest Common Divisor.hypot(*coordinates) -> value Multidimensional Euclidean distance from the origin to a point. Roughly equivalent to: sqrt(sum(x**2 for x in coordinates)) For a two dimensional point (x, y), gives the hypotenuse using the Pythagorean theorem: sqrt(x*x + y*y). For example, the hypotenuse of a 3/4/5 right triangle is: >>> hypot(3.0, 4.0) 5.0 isclose($module, /, a, b, *, rel_tol=1e-09, abs_tol=0.0) -- Determine whether two floating point numbers are close in value. rel_tol maximum difference for being considered "close", relative to the magnitude of the input values abs_tol maximum difference for being considered "close", regardless of the magnitude of the input values Return True if a is close in value to b, and False otherwise. For the values to be considered close, the difference between them must be smaller than at least one of the tolerances. -inf, inf and NaN behave similarly to the IEEE 754 Standard. That is, NaN is not close to anything, even itself. inf and -inf are only close to themselves.isfinite($module, x, /) -- Return True if x is neither an infinity nor a NaN, and False otherwise.isinf($module, x, /) -- Return True if x is a positive or negative infinity, and False otherwise.isnan($module, x, /) -- Return True if x is a NaN (not a number), and False otherwise.isqrt($module, n, /) -- Return the integer part of the square root of the input.lcm($module, *integers) -- Least Common Multiple.ldexp($module, x, i, /) -- Return x * (2**i). This is essentially the inverse of frexp().lgamma($module, x, /) -- Natural logarithm of absolute value of Gamma function at x.log(x, [base=math.e]) Return the logarithm of x to the given base. If the base not specified, returns the natural logarithm (base e) of x.log1p($module, x, /) -- Return the natural logarithm of 1+x (base e). The result is computed in a way which is accurate for x near zero.log10($module, x, /) -- Return the base 10 logarithm of x.log2($module, x, /) -- Return the base 2 logarithm of x.modf($module, x, /) -- Return the fractional and integer parts of x. Both results carry the sign of x and are floats.pow($module, x, y, /) -- Return x**y (x to the power of y).radians($module, x, /) -- Convert angle x from degrees to radians.remainder($module, x, y, /) -- Difference between x and the closest integer multiple of y. Return x - n*y where n*y is the closest integer multiple of y. In the case where x is exactly halfway between two multiples of y, the nearest even value of n is used. The result is always exact.sin($module, x, /) -- Return the sine of x (measured in radians).sinh($module, x, /) -- Return the hyperbolic sine of x.sqrt($module, x, /) -- Return the square root of x.tan($module, x, /) -- Return the tangent of x (measured in radians).tanh($module, x, /) -- Return the hyperbolic tangent of x.trunc($module, x, /) -- Truncates the Real x to the nearest Integral toward 0. Uses the __trunc__ magic method.prod($module, iterable, /, *, start=1) -- Calculate the product of all the elements in the input iterable. The default start value for the product is 1. When the iterable is empty, return the start value. This function is intended specifically for use with numeric values and may reject non-numeric types.perm($module, n, k=None, /) -- Number of ways to choose k items from n items without repetition and with order. Evaluates to n! / (n - k)! when k <= n and evaluates to zero when k > n. If k is not specified or is None, then k defaults to n and the function returns n!. Raises TypeError if either of the arguments are not integers. Raises ValueError if either of the arguments are negative.comb($module, n, k, /) -- Number of ways to choose k items from n items without repetition and without order. Evaluates to n! / (k! * (n - k)!) when k <= n and evaluates to zero when k > n. Also called the binomial coefficient because it is equivalent to the coefficient of k-th term in polynomial expansion of the expression (1 + x)**n. Raises TypeError if either of the arguments are not integers. Raises ValueError if either of the arguments are negative.nextafter($module, x, y, /) -- Return the next floating-point value after x towards y.ulp($module, x, /) -- Return the value of the least significant bit of the float x.x_7a(s(;LXww0uw~Cs+|g!??@@8@^@@@@&AKAAA2A(;L4BuwsBuwB7Bs6Ch0{CZAC Ƶ;(DlYaRwNDmathacosacoshasinasinhatanatan2atanhceilcopysigncoscoshdegreesdisterferfcexpexpm1fabsfactorialfloorfmodfrexpfsumgammagcdhypotiscloseisfiniteisinfisnanisqrtlcmldexplgammaloglog1plog10log2modfpowradiansremaindersinsinhsqrttantanhtruncprodpermcombnextafterulpmath domain errormath range error__ceil__both points must have the same number of dimensionsUsing factorial() with floats is deprecatedfactorial() only accepts integral valuesfactorial() argument should not exceed %ldfactorial() not defined for negative values__floor__(di)intermediate overflow in fsum-inf + inf in fsummath.fsum partialsabrel_tolabs_toltolerances must be non-negativeisqrt() argument must be nonnegativeExpected an int as 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