Input Syntax Guide

ToFUL accepts natural mathematical notation and automatically corrects common variants before passing expressions to the computation engine. This page documents everything the parser understands.

Basic Syntax Rules 

Expressions are written as single-line Python-style conditionals:

<value_when_true>  if  <condition>  else  <value_when_false>

The variable is always x. The condition defines the support of the distribution.

# Correct
2 * x  if  0 <= x <= 1  else  0

# Also correct — no guard needed if the formula is already zero outside
exp(-x)

Auto-Corrections Applied by the Parser 

The parser normalises your input before evaluation. Corrections are shown in a collapsible notice above the results.

Exponentiation 

You type	Interpreted as	Notes
`x^2`	`x**2`	Caret → Python power
`x²`	`x**2`	Unicode superscript
`2^n`	`2**n`	Any base/exponent

Implicit Multiplication 

You type	Interpreted as	Notes
`2x`	`2*x`	Digit × letter
`(x+1)(x-1)`	`(x+1)*(x-1)`	Adjacent brackets
`2(x+1)`	`2*(x+1)`	Digit × bracket

Exponential Notation 

You type	Interpreted as	Notes
`e^x`	`exp(x)`	Euler’s number raised
`e^(-2*x)`	`exp(-2*x)`	Negative exponent
`Exp(x)`	`exp(x)`	Capitalised alias

Function Aliases 

You type	Interpreted as	Notes
`ln(x)`	`log(x)`	Natural logarithm
`arcsin(x)`	`asin(x)`	Inverse trig
`√(x)`	`sqrt(x)`	Unicode root sign
`√x`	`sqrt(x)`	Without brackets

Unicode Operators 

You type	Interpreted as	Notes
`≤`	`<=`	Less-than-or-equal
`≥`	`>=`	Greater-than-or-equal
`≠`	`!=`	Not equal
`×`	`*`	Multiplication sign
`÷`	`/`	Division sign
`−`	`-`	Unicode minus (U+2212)

Absolute Value 

|x - 1|           ->   abs(x - 1)
|x|               ->   abs(x)

Subscript Variables 

You type	Interpreted as	Notes
`x₁`	`x1`	Unicode subscript
`x_1`	`x1`	Underscore notation
`y₂`	`y2`	Any letter + digit

Greek Letters 

Unicode Greek letters are mapped to their conventional parameter names:

Symbol	Name in eval context	Typical use
`π`	`pi`	Mathematical constant
`λ`	`lam`	Rate parameter (Poisson, Exponential)
`μ`	`mu`	Mean parameter
`σ`	`sigma`	Standard deviation
`α`	`alpha`	Shape parameter
`β`	`beta`	Scale parameter
`θ`	`theta`	General parameter

Available Functions 

All standard mathematical functions are available by name:

Exponential & Logarithmic

exp(x)      natural exponential e^x
log(x)      natural logarithm ln(x)
log2(x)     base-2 logarithm
log10(x)    base-10 logarithm
sqrt(x)     square root

Trigonometric

sin(x)   cos(x)   tan(x)
asin(x)  acos(x)  atan(x)   atan2(y, x)
sinh(x)  cosh(x)  tanh(x)

Rounding & Sign

abs(x)     absolute value
ceil(x)    ceiling
floor(x)   floor
sign(x)    sign function (+1, 0, -1)
round(x)   round to nearest integer

Special Functions

factorial(n)   n!  (integer n only, DRV expressions)
gamma(x)       Gamma function Γ(x)
erf(x)         error function
erfc(x)        complementary error function

Constants

pi    π ≈ 3.14159265…
e     Euler's number ≈ 2.71828182…
inf   positive infinity (use in bounds only)

Piecewise Expressions 

All piecewise functions use Python’s ternary conditional:

<result_if_true>  if  <condition>  else  <result_if_false>

Multiple pieces can be chained:

# Triangular distribution on [0, 2] with peak at 1
x  if  0 <= x <= 1  else  (2 - x)  if  1 < x <= 2  else  0

Chained comparisons work natively:

0 <= x <= 1        # equivalent to  0 <= x and x <= 1

Range Input Syntax 

Discrete ranges:

1,2,3,4,5          Finite — exactly these values
0,1,2,3,...        Infinite arithmetic sequence (step detected automatically)
1,2,4,8,...        Infinite geometric sequence (ratio detected automatically)

Continuous bounds:

0         numeric lower/upper bound
inf       positive infinity
-inf      negative infinity
∞         Unicode infinity (auto-converted)
-∞        Unicode negative infinity

Common Mistakes 

`factorial` with float arguments 

factorial only accepts non-negative integers. For continuous distributions, use gamma(x+1) (which equals factorial(x) for integers but is defined on the reals).

# Wrong — raises TypeError for non-integer x
factorial(x)

# Correct for continuous distributions
gamma(x + 1)

Missing `else 0`

If the function is undefined outside the stated range, always include a guard:

# Risky — may evaluate to a non-zero value outside the intended support
exp(-x)

# Correct
exp(-x) if x >= 0 else 0

Syntax check before computing 

After entering a function, the interpreted form is shown (in teal monospace) below the function field before you click Compute. Check this to verify the parser understood your intent correctly.