TIP #123: Adding an Exponentiation Operator to the [expr] Command
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TIP:	123
Title:	Adding an Exponentiation Operator to the [expr] Command
Version:	$Revision: 1.6 $
Authors:	Arjen Markus <arjen.markus@wldelft.nl>
	Donal K. Fellows <donal.k.fellows@man.ac.uk>
State:	Final
Type:	Project
Tcl-Version:	8.5
Vote:	Done
Created:	Mon Dec 16 00:00:00 UTC 2002
Keywords:	mathematics, evaluation
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.
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