Mathematical Functions Reference

Every function and constant available in ToFUL expression fields is listed here. The same namespace applies to both the PMF/PDF field and — where relevant — to the range field.

Constants 

Name	Value	Notes
`pi`	3.14159265…	Mathematical constant π. Also accepted as `π` (Unicode).
`e`	2.71828182…	Euler’s number. Do not use `e` as a variable name.
`inf`	∞	Positive infinity. Use in bounds fields only.
`nan`	Not a Number	Returned by undefined operations. Avoid in PMF/PDF.

Exponential and Logarithmic 

Function	Description
`exp(x)`	Exponential function \(e^x\). Alias: `e^x`, `Exp(x)`.
`log(x)`	Natural logarithm \(\ln x\). Alias: `ln(x)`, `Ln(x)`. Undefined for \(x \leq 0\).
`log2(x)`	Base-2 logarithm \(\log_2 x\).
`log10(x)`	Base-10 logarithm \(\log_{10} x\). Alias: `log10(x)`.
`sqrt(x)`	Square root \(\sqrt{x}\). Alias: `√(x)`, `√x`. Undefined for \(x < 0\).

Trigonometric 

Function	Description
`sin(x)`	Sine (argument in radians).
`cos(x)`	Cosine (argument in radians).
`tan(x)`	Tangent (argument in radians). Undefined at \(x = \pi/2 + k\pi\).
`asin(x)`	Arcsine. Range \([-\pi/2, \pi/2]\). Alias: `arcsin(x)`.
`acos(x)`	Arccosine. Range \([0, \pi]\). Alias: `arccos(x)`.
`atan(x)`	Arctangent. Range \((-\pi/2, \pi/2)\). Alias: `arctan(x)`.
`atan2(y, x)`	Four-quadrant arctangent. Returns angle in \((-\pi, \pi]\).
`sinh(x)`	Hyperbolic sine.
`cosh(x)`	Hyperbolic cosine.
`tanh(x)`	Hyperbolic tangent.

Rounding and Sign 

Function	Description
`abs(x)`	Absolute value \(\|x\|\). Alias: `\|x\|`.
`ceil(x)`	Ceiling — smallest integer ≥ x.
`floor(x)`	Floor — largest integer ≤ x.
`sign(x)`	Sign function: −1 for negative, 0 for zero, +1 for positive.
`round(x)`	Round to nearest integer (banker’s rounding for .5).
`min(a, b)`	Minimum of two values.
`max(a, b)`	Maximum of two values.

Special Functions 

Function	Description
`factorial(n)`	Factorial \(n!\). Integer argument only. For real-valued generalisation, use `gamma(n+1)`.
`gamma(x)`	Gamma function \(\Gamma(x) = \int_0^\infty t^{x-1}e^{-t}\,dt\). Satisfies \(\Gamma(n) = (n-1)!\) for positive integers.
`erf(x)`	Error function \(\text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt\).
`erfc(x)`	Complementary error function \(\text{erfc}(x) = 1 - \text{erf}(x)\).

Operator Reference 

Operator	Unicode alias	Meaning
`+`	—	Addition
`-`	`−` (U+2212)	Subtraction / negation
`*`	`×`, `·`	Multiplication
`/`	`÷`	Division (true division, always returns float)
`**`	`^`, superscripts	Exponentiation
`<=`	`≤`	Less than or equal
`>=`	`≥`	Greater than or equal
`==`	—	Equal (use in conditionals: `if x == 0`)
`!=`	`≠`	Not equal
`and`	`∧`	Logical and (for compound conditions)
`or`	`∨`	Logical or
`not`	`¬`	Logical not

Examples 

# Gamma(alpha=3, beta=2) PDF
(x**2 * exp(-x/2)) / (2**3 * gamma(3))  if x >= 0  else 0

# Standard Normal PDF
exp(-(x**2)/2) / sqrt(2*pi)

# Poisson(lambda=4) PMF
(exp(-4) * 4**x) / factorial(x)  if x >= 0  else 0

# Beta(2,3) PDF — check: normalising constant = B(2,3)^-1 = 12
12 * x * (1-x)**2  if  0 <= x <= 1  else  0

# Laplace(mu=0, b=1) PDF
exp(-abs(x)) / 2