úÎT^Q‹&      !"#$%portable experimentalbos@serpentine.com Safe-Inferred A very large number.  The largest & x such that 2**(x-1) is approximately  representable as a '. Positive infinity. Negative infinity. Not a number.  sqrt 2  sqrt (2 * pi) 2 / sqrt pi 1 / sqrt 2  The smallest ' µ such that 1 + µ "` 1. log(sqrt((2*pi)) +Euler Mascheroni constant (³ = 0.57721...)       portable experimentalbos@serpentine.com Safe-Inferred Evaluate the deviance term x log(x/ np) + np - x.   x np   portable experimentalbos@serpentine.comNone8Evaluate a Chebyshev polynomial of the first kind. Uses  Clenshaw' s algorithm. ?Evaluate a Chebyshev polynomial of the first kind. Uses Broucke's B ECHEB algorithm, and his convention for coefficient handling. It $ treat 0th coefficient different so ? chebyshev x [a0,a1,a2...] == chebyshevBroucke [2*a0,a1,a2...] ()*+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.comNone Evaluate polynomial using Horner's method. Coefficients starts  from lowest. In pseudocode: 3 evaluateOddPolynomial x [1,2,3] = 1 + 2*x + 3*x^2 6Evaluate polynomial with only even powers using Horner' s method. 1 Coefficients starts from lowest. In pseudocode: 5 evaluateOddPolynomial x [1,2,3] = 1 + 2*x^2 + 3*x^4 5Evaluate polynomial with only odd powers using Horner' s method. 1 Coefficients starts from lowest. In pseudocode: 7 evaluateOddPolynomial x [1,2,3] = 1*x + 2*x^3 + 3*x^5 x  Coefficients x  Coefficients x  Coefficients portable experimentalbos@serpentine.comNoneError function.  erf -" = -1  erf 0 = 0  erf +" = 1 Complementary error function.  erfc -" = 2  erfc 0 = 1  errc +" = 0  Inverse of .  Inverse of . .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 . 2Returns " 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). ,,Compute the log gamma correction factor for x "e 10. This : correction factor is suitable for an alternate (but less % numerically accurate) definition of : Elgg x = 0.5 * log(2*pi) + (x-0.5) * log x - x + logGammaCorrection x 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 4Compute the natural logarithm of the beta function. =Regularized incomplete beta function. Uses algorithm AS63 by B Majumder and Bhattachrjee and quadrature approximation for large  p and q. .Regularized incomplete beta function. Same as  9 but also takes logarithm of beta function as parameter. >Compute inverse of regularized incomplete beta function. Uses C initial approximation from AS109, AS64 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) -)Quickly compute the natural logarithm of n $ k, with  no checking. $!Compute the binomial coefficient n `$` 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 % Compute È0(x1), the first logarithmic derivative of the gamma < function. Uses Algorithm AS 103 by Bernardo, based on Minka's C  implementation. ./p " [-1,1] p " [0,2] ,s " (0,") x " (0,") s " (0,") p " [0,1] p > 0 q > 0 x, must lie in [0,1] range %logarithm of beta function for given p and q p > 0 q > 0 x, must lie in [0,1] range 01p > 0 q > 0 a " [0,1] 2 !"#-$%3456 !"#$%% !"#$./,012 !"#-$%34567      !"#$%&'()*+,-.,-/0011234456789:;<math-functions-0.1.3.0Numeric.MathFunctions.ConstantsNumeric.SpecFunctions.ExtraNumeric.Polynomial.ChebyshevNumeric.PolynomialNumeric.SpecFunctionsm_hugem_tiny m_max_exp m_pos_inf m_neg_infm_NaNm_sqrt_2 m_sqrt_2_pi m_2_sqrt_pi m_1_sqrt_2 m_epsilonm_ln_sqrt_2_pim_eulerMascheronibd0 chebyshevchebyshevBrouckeevaluatePolynomialevaluateEvenPolynomialevaluateOddPolynomialerferfcinvErfinvErfclogGamma logGammaLincompleteGammainvIncompleteGammalogBetaincompleteBetaincompleteBeta_invIncompleteBetalog1plog2 factorial logFactorial stirlingErrorchoosedigammaghc-prim GHC.TypesIntDoubleBClogGammaCorrection logChooseFastLincompleteBetaApproxincompleteBetaWorkerinvIncompleteBetaWorkercoefWcoefY trigamma1modErr