Interval Arithmetic

TeXpr supports Interval Arithmetic, allowing you to perform calculations with ranges of values rather than single points. This is useful for error analysis, uncertainty propagation, and bounding solutions.

Syntax

Intervals are denoted using square brackets [lower, upper], where lower and upper are expressions evaluating to numbers.

[1, 2] + [3, 4]

Basic Operations

Standard arithmetic operations accept intervals and return a new interval representing all possible results.

• Addition: [a, b] + [c, d] = [a+c, b+d]
• Subtraction: [a, b] - [c, d] = [a-d, b-c]
• Multiplication: [a, b] * [c, d] = [min(ac, ad, bc, bd), max(ac, ad, bc, bd)]
• Division: [a, b] / [c, d] = [a, b] * [1/d, 1/c] (if 0 is not in [c, d])

Functions

Mathematical functions in TeXpr are extended to support intervals:

• sin([0, \pi]) → [0, 1]
• exp([0, 1]) → [1, 2.718...]
• sqrt([4, 9]) → [2, 3]

Evaluation

You can evaluate expressions containing intervals just like regular expressions:

final evaluator = Texpr();
final result = evaluator.evaluate('[1, 2] + 1');
print(result); // [2, 3]

Examples

Uncertainty Propagation

Calculate area of a square with side length $3\pm 0.1$:

let x = [2.9, 3.1]
x^2
// Result: [8.41, 9.61]