TIP #123 Version 1.1: Adding an Exponentiation Operator to the [expr] Command

This is not necessarily the current version of this TIP.

Abstract

This TIP proposes to add a new operator to the operators recognised by the [expr] command: the exponentiation operator. This operator will enhance the functionality of the current pow() function by returning a result that depends on the type of its operands. It will also make complicated formulae more readable.

Introduction

Currently Tcl's [expr] command uses the exponentiation function pow() to calculate such expressions as "2 to the power 10". The drawback of this is twofold:

• Expressions using several exponentiations become difficult to read. For instance, a third-degree polynomial looks like:

   2.0*pow($x,3) - 1.2*pow($x,2) + 3.0*$x + 4.0
  

  or:

   2.0*$x*$x*$x - 1.2*$x*$x + 3.0*$x + 4.0
  

* The result of raising an integer to an integer power is a double

2 to the power 10 is 1024.0, not 1024.

Other languages, like for instance FORTRAN, use an operator instead of a function. Two operators are commonly found: ** and ^. As the latter already has a meaning within the [expr] command, we propose to add the "**" operator instead. The above example would become:

 2.0*$x**3 - 1.2*$x**2 + 3.0*$x + 4.0

Mathematical Details

The implementation of the exponentiation operator will have the following properties (below we refer to the expression $x**$y):

If x and y are both integers (ordinary or wide):

• The result is of the same type as the widest operand

• An error is raised if the operation makes no mathematical sense, 0**(-1) for instance.

• If x has the value 0, then:

  • if y > 0, the result is 0

  • if y < 0, the result is an error

  • if y == 0, the result is 1

• If x has the value 1, then the result is always 1

• If y has the value 0, the result is always 1

• If x has a negative value lower than -1 and y < 0, the result is 0

• If x has the value -1, then depending on whether y is even or odd, the result is 1 or -1 (respectively.)

• For all other combinations, the value is "x raised to the power y"

• When evaluating this, no attention is paid to overflow, even though the result might fit into a wide integer (though of course the result will be a wide integer if either operand was wide.) This is in accordance with the type model used in other [expr] operators.

If either x or y is a double, the C function pow() is used to compute the result.

The following expressions are parsed and evaluated in accordance with all other operators:

 $x ** $y ** $z ==> ($x ** $y ) ** $z
 $x ** -1       ==> ($x ** (-1))

The precedence of the exponentiation operator is thus higher than the multiplication, division and remainder operations and lower than the unary operations, in accordance with common definitions.

Copyright

This document is placed in the public domain.

This is not necessarily the current version of this TIP.