User Formulas

XaoS supports entering your own custom formulas:

1. Select User formula from the Fractal menu to define a custom fractal formula.
2. (Optional) Select User initialization to calculate a different starting point z(0) for each pixel. By default z(0) is set to 0.

User formulas should be interpreted as functions in the form z(n+1) = formula (z(n), z(n-1), c).

Variables

In user formulas, as shown above, three variables are available:

variable	meaning
z	current sequence point z(n)
c	current plane 'point'
p	previous sequence point z(n-1)
n	current iteration number n (beginning with 1)

Constants

Format for complex numbers is {re,im}, e.g. {3,2} is complex number 3+2i. You must write constants for the coordinates. You can also use a semicolon instead of a comma. Also, parameters in functions must be separated by commas or semicolons (see below).

The number {0,1} can also be written as i. If you need to express a complex number with a non-constant, split it to real and imaginary parts like this: (re(z)+im(z)*i)^2+c.

Real numbers can be used as usual.

Functions

Available functions are listed below:

Basic operations

+, -, *, /, ^

The multiplication sign may be omitted in most cases.

Real and imaginary part

• re(x) - real part of x
• im(x) - imaginary part of x

Power functions

• powi(x,y) - power of x with integer exponent y (fastest)
• powd(x,y) - power of x with real exponent y
• powdc(x,y) - power of x with complex exponent y
• pow(x,y) - same as powd(x,y)

The symbol ^ can also be used (see above) without explicitly defining the number set for the exponent.

Basic trigonometry

• sin(x), cos(x), tan(x), cot(x), asin(x), acos(x), atan(x), acot(x), sinh(x), cosh(x), tanh(x), coth(x) can be used for computing trigonometrical functions

Exponential and logarithmic functions

• exp(x), log(x) - exponential and natural logarithm of x
• log2(x), log10(x) - logarithm of x in base 2 and 10
• logn(x,y) - logarithm of y in base x (where x is integer)
• logcn(x,y) - logarithm of y in base x (where x is complex)
• sqrt(x) - square root of x

Other functions

• rand(x) - random real number in range [0, x)
• abs(x) - absolute value of x

Examples

• z^2+c and z^3+c produce the quadratic and cubic Mandelbrot sets. z^2.5+c produces another fractal "between them".
• (re(z)+im(z)i)^2+c also produces the quadratic Mandelbrot set.
• powd(z,2)+c creates the quadratic Mandelbrot set. A minor change in the formula, by using powdc(z,{2,0.1})+c is equivalent to z^{2,0.1}+c.
• (abs(re(z))+i*abs(im(z)))^2+c produces the Burning Ship fractal.
• z^2+c*(-1)^n expresses the iteration (((z(0)^2-c)^2+c)^2-c)^2+....

Notes

• The entered formula is case insensitive. But it will always be rewritten in lowercased letters if the parsing was successful.
• Handling user formulas is still in experimental phase. Please expect random crashes or malfunctioning in some cases.
• Fast computation of user formulas is based on Mateusz Malczak's SFFE library.