NumericFuns

Numerical functions and functors.

Note: This package was originally part of the NumericExtensions package. I realized later that the functors and the type inference machinery can be useful in other packages. Hence, I separate this part to construct a standalone package.

This package provides:

• Additional numerical functions, such as sqr, rsqrt, xlogx, sigmoid, logit, etc.
• Vectorized methods of the additional numerical functions.
• Typed functors.

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New Numeric Functions

This package provides several commonly used numerical functions that are not in the Julia Base.

function	equivalent expression
sqr(x)	x * x
rcp(x)	1 / x
rsqrt(x)	1 / sqrt(x)
rcbrt(x)	1 / cbrt(x)
xlogx(x)	ifelse(x > 0, x * log(x), 0)
xlogy(x, y)	ifelse(x > 0, x * log(y), 0)
sigmoid(x)	1 / (1 + exp(-x))
logit(x)	log(x / (1 - x))
softplus(x)	log(1 + exp(x))
invsoftplus(x)	log(exp(x) - 1)
logsumexp(x, y)	log(exp(x) + exp(y))

Note that the equivalent expressions above are just for the purpose to conveying the semantics. The actual implementation might be different, which would takes a more optimal route that takes care of risk of overflow, type stability, and computational efficiency.

Functors

Functors are typed instances used in indicate a particular function. Since Julia is not able to specialize on functions (yet), functors provide an effective way that allow mutliple dispatch and functional programming to work together.

The package defines an abstract type Functor as

abstract Functor{N}

where, N is an integer indicating the number of arguments. All functor types are subtypes of Functor.

Each functor type comes with an evaluate method, which evaluates the corresponding function given arguments.

Define a functor

Here is an example that illustrates how one can define a functor

type Add <: Functor{2} end
evaluate{T1<:Number,T2<:Number}(::Add, x::Number, y::Number) = x + y

Two macros @functor1 and @functor2 are provided for simplifying the definition of unary and binary functors:

@functor1(Cbrt, cbrt, Real)
@functor2(Add, +, Number)

These macros accept three arguments: the functor type name, the corresponding function, and the super type of all acceptable argument types.

Note: The packages also defines a large collection of functors for various mathematical operations (so you don't have to define them yourself).

Functors for operators

Here is a table of functor types for operators:

functor type	operator	domain
Negate	-	Number
Add	+	Number
Subtract	-	Number
Multiply	*	Number
Divide	/	Number
RDivide	\	Number
Pow	^	Number
And	&	Bool
Or	|	Bool
Not	!	Bool
BitwiseAnd	&	Integer
BitwiseOr	|	Integer
BitwiseNot	~	Integer
BitwiseXor	$	Integer
LT	<	Real
GT	>	Real
LE	<=	Real
GE	>=	Real
EQ	==	Number
NE	!=	Number

Functors for math functions

The package also defined functors for named functions. The naming of functor types follows the $(capitalize(funname))Fun rule. For example, the functor type for sqrt is SqrtFun, and that for lgamma is LgammaFun, etc.

In particular, the package defines functors for the following functions:

• arithmetic functions

  abs, abs2, real, imag, sqr, rcp,
  sign, signbit, div, fld, rem, mod
  

• rounding functions

  floor, ceil, trunc, round, 
  ifloor, iceil, itrunc, iround
  

• number classification functions

  isnan, isinf, isfinite

• algebraic functions

  sqrt, rsqrt, cbrt, rcbrt, hypot

• exponential & logarithm

  exp, exp2, exp10, expm1, 
  log, log2, log10, log1p,
  
  sigmoid, logit, xlogx, xlogy, 
  softplus, invsoftplus, logsumexp
  

• trigonometric functions

  sin, cos, tan, cot, sec, csc, 
  asin, acos, atan, acot, asec, acsc, atan2, 
  sinc, cosc, sinpi, cospi, 
  sind, cosd, tand, cotd, secd, cscd, 
  asind, acosd, atand, acotd, asecd, acscd
  

• hyperbolic functions

  sinh, cosh, tanh, coth, sech, csch,
  asinh, acosh, atanh, acoth, asech, acsch
  

• special functions

  erf, erfc, erfinv, erfcinv, erfi, erfcx,
  gamma, lgamma, digamma,
  eta, zeta, beta, lbeta,
  airy, airyprime, airyai, airyaiprime, airybi, airybiprime,
  besselj0, besselj1, bessely0, bessely1
  besseli, besselj, besselk, bessely,
  hankelh1, hankelh2
  

Result Type Inference

Each functor defined in this package comes with result_type methods that return the type of the result, given the argument types. These methods are thoroughly tested to ensure correctness. For example,

result_type(Add(), Int, Float64)  # --> returns Float64
result_type(SqrtFun(), Int)   # --> returns Float64

The package also provides other convenient methods for type inference, which include fptype and arithtype. Particularly, we have

fptype{T<:Real}(::Type{T}) == typeof(Convert(FloatingPoint, one(T)))
fptype{T<:Real}(::Type{Complex{T}}) == Complex{fptype(T)}

arithtype{T1<:Number, T2<:Number} == typeof(one(T1) + one(T2))

The internal implementation of these functions are very efficient, usually without actually evaluating the expressions.