Simplify

A complicated mathematical expression can often be transformed into a form that is easier to understand.

The expr.simplify() function tries expanding, factoring and applying many other transformations to find a simple a simpler form of a symbolic expression.

Before the transformation rules are applied, the expression is put into a canonical form.

When a function is simplified, its arguments are simplified as well, unless the argument is “held”. Which arguments are held is specified by the hold property of the function definition. In addition, any argument wrapped with a Hold function will be held, that is, not simplified. Conversely, a held argument wrapped with a ReleaseHold function will not be held, and it will be simplified.

Defining “Simpler”

An expression may be represented by several equivalent forms.

For example \( (x + 4)(x-5) \) and \(x^2 -x -20\) represent the same expression.

Determining which is “the simplest” depends on how the complexity is measured.

By default, the complexity of an expression is measured by counting the number of operations in the expression, and giving an increasing cost to:

  • integers with fewer digits
  • integers with more digits
  • other numeric values
  • add, multiply, divide
  • subtract and negate
  • square root and root
  • exp
  • power and log
  • trigonometric function
  • inverse trigonometric function
  • hyperbolic functions
  • inverse hyperbolic functions
  • other functions

To influence how the complexity of an expression is measured, set the costFunction property of the compute engine to a function assigning a cost to an expression.

Numeric Simplifications

The expr.simplify() function will apply some numeric simplifications, such as combining small integer and rational values, simplifying division by 1, addition or subtraction of 0, etc…

It avoids making any simplification that could result in a loss of precision.

For example, \( 10^{300} + 1\) cannot be simplified without losing the least significant digit, so expr.simplify() will return the expression unmodified.

Using Assumptions

Assumptions are additional information available about some symbols, for example \( x > 0 \) or \(n \in \N\).

Some transformations are only applicable if some assumptions can be verified.

For example, if no assumptions about \(x \) is available the expression \( \sqrt{x^2} \) cannot be simplified. However, if an assumption that \( x \geq 0 \) is available, then the expression can be simplified to \( x \).

Read more about Assumptions