úÎ<:m     portable experimentalbos@serpentine.com Safe-Infered A very large number.  The largest  x such that 2**(x-1) is approximately  representable as a .  sqrt 2  sqrt (2 * pi) 2 / sqrt pi 1 / sqrt 2 The smallest  µ such that 1 + µ "` 1.  log(sqrt((2*pi)) Positive infinity. Negative infinity. Not a number.     portable experimentalbos@serpentine.com Safe-Infered Evaluate the deviance term x log(x/ np) + np - x.   x np   portable experimentalbos@serpentine.com Safe-Infered 8Evaluate a Chebyshev polynomial of the first kind. Uses  Clenshaw' s algorithm. ?Evaluate a Chebyshev polynomial of the first kind. Uses Broucke's F ECHEB algorithm, and his convention for coefficient handling, and so  gives different results than   for the same inputs.  Parameter of each function. ;Coefficients of each polynomial term, in increasing order. Parameter of each function. ;Coefficients of each polynomial term, in increasing order.    portable experimentalbos@serpentine.com Safe-Infered,Compute the logarithm of the gamma function “(x ). Uses  Algorithm AS 245 by Macleod. Gives an accuracy of 10 &12 significant decimal digits, except  for small regions around x = 1 and x = 2, where the function * goes to zero. For greater accuracy, use . Returns ") if the input is outside of the range (0 < x  "d 1e305). -Compute the logarithm of the gamma function, “(x ). Uses a  Lanczos approximation. This function is slower than , but gives 14 or more 7 significant decimal digits of accuracy, except around x = 1 and  x' = 2, where the function goes to zero. Returns ") if the input is outside of the range (0 < x  "d 1e305). 7Compute the normalized lower incomplete gamma function  ³(s,x). Normalization means that  ³(s,"$)=1. Uses Algorithm AS 239 by Shea. %Inverse incomplete gamma function. It's approximately inverse of   for the same s. So following equality  approximately holds:  / invIncompleteGamma s . incompleteGamma s = id For invIncompleteGamma s p s must be positive and p must be  in [0,1] range. 4Compute the natural logarithm of the beta function. =Regularized incomplete beta function. Uses algorithm AS63 by  Majumder abd Bhattachrjee. .Regularized incomplete beta function. Same as  9 but also takes logarithm of beta function as parameter. >Compute inverse of regularized incomplete beta function. Uses G initial approximation from AS109 and Halley method to solve equation. %Compute the natural logarithm of 1 + x. This is accurate even  for values of x near zero, where use of log(1+x) would lose  precision. O(log n)5 Compute the logarithm in base 2 of the given value. Compute the factorial function n !. Returns " if the E input is above 170 (above which the result cannot be represented by  a 64-bit ). @Compute the natural logarithm of the factorial function. Gives ! 16 decimal digits of precision. ACalculate the error term of the Stirling approximation. This is ' only defined for non-negative values. : stirlingError @n@ = @log(n!) - log(sqrt(2*pi*n)*(n/e)^n) !Compute the binomial coefficient n `` k. For  values of k2 > 30, this uses an approximation for performance E reasons. The approximation is accurate to 12 decimal places in the  worst case  Example:  7 `choose` 3 == 35 s x p > 0 q > 0 x, must lie in [0,1] range logarithm of beta function p > 0 q > 0 x, must lie in [0,1] range p q a       !"#$"#%&math-functions-0.1.1.1Numeric.MathFunctions.ConstantsNumeric.SpecFunctions.ExtraNumeric.Polynomial.ChebyshevNumeric.SpecFunctionsm_hugem_tiny m_max_expm_sqrt_2 m_sqrt_2_pi m_2_sqrt_pi m_1_sqrt_2 m_epsilonm_ln_sqrt_2_pi m_pos_inf m_neg_infm_NaNbd0 chebyshevchebyshevBrouckelogGamma logGammaLincompleteGammainvIncompleteGammalogBetaincompleteBetaincompleteBeta_invIncompleteBetalog1plog2 factorial logFactorial stirlingErrorchooseghc-prim GHC.TypesIntDouble