Documentation

Constant function that returns a for any value of x \[f(x) = a\]

Examples

>>> let f = constant 0.5 :: Double -> UILukasiewicz
>>> f 1
0.5

>>> f 100
0.5

Standart textbook linear function where \[f(x) = ax + b\]

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Examples

>>> let f = linear 2 1 :: Double -> UILukasiewicz
>>> f 0
1.0

>>> f 1
3.0

>>> f (-1)
-1.0

Standart logistic function Takes K which is growth value of the function and x0 a midpoint of the function. \[f(x) = \frac{1}{1 + e^{ -k(x - x_0)}} \]

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Examples

>>> let f = sigmoid 1 0 :: Double -> UILukasiewicz
>>> f 0
0.5

>>> f 1
0.7310585786300049

>>> f (-1)
0.2689414213699951

A combination of two linear functions with this specific shape. first and second arguments are interval determining where the triangle will be on the number line. \[ \operatorname{tri}(x) = \begin{cases} \frac{x - a}{b - a}, & a \leq x < b \\ \frac{c - x}{c- b}, & b \leq x \leq c \\ 0, & \text{otherwise} \end{cases} \] 0 stands for bot

         
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Examples

>>> let f = triangular 0 1 2 :: Double -> UILukasiewicz
>>> f 0
0.0

>>> f 0.5
0.5

>>> f 1
1.0

>>> f 1.5
0.5

>>> f 2
0.0

Constant function on interval [a, b], first two arguments, else returns bot of Residuated lattice - this creates a rectangle shaped function. Third argument is height of the set.

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>>> let f = rectangular 1 3 0.8 :: Double -> UILukasiewicz
>>> f 0
0.0

>>> f 2
0.8

>>> f 3
0.8

>>> f 4
0.0

Trapezoidal function is combination of triangular and rectangular functions \[ \operatorname{tra}(x) = \begin{cases} \frac{x - a}{b1 - a}, & a \leq x < b1 \\ 1, & b1 \leq x < b2 \\ \frac{c - x}{c -b2}, & b2 \leq x < c \\ 0, & \text{otherwise} \end{cases} \]

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Examples

>>> let f = trapezoidal 0 1 3 4 :: Double -> UILukasiewicz
>>> f 0
0.0

>>> f 0.5
0.5

>>> f 1
1.0

>>> f 2
1.0

>>> f 3.5
0.5

>>> f 4
0.0

Gausian function, also called Bell Curve

          
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Examples

>>> let f = gaussian 1 0 1 :: Double -> UILukasiewicz
>>> f 0
1.0

>>> f 1
0.6065306597126334

>>> f (-1)
0.6065306597126334

>>> f 2
0.1353352832366127

Exponential function eˣ

          
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Examples

>>> let f = exponential :: Double -> UILukasiewicz
>>> f 0
1.0

>>> f 1
2.718281828459045

>>> f (-1)
0.36787944117144233