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

Login
FlightAware bounty program for improvements to Tcl and certain Tcl packages.
Author:         Arjen Markus <arjen.markus@wldelft.nl>
Author:         Donal K. Fellows <donal.k.fellows@man.ac.uk>
State:          Final
Type:           Project
Vote:           Done
Created:        16-Dec-2002
Post-History:   
Keywords:       mathematics,evaluation
Tcl-Version:    8.5
Tcl-Ticket:     655176

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.

Sample Implementation

http://sf.net/tracker/?func=detail&aid=655176&group\_id=10894&atid=310894

Copyright

This document is placed in the public domain.