QPI v4.2			Mika Heiskanen
			--------			mheiskan@delta.hut.fi

	Based on QPirac by A.Coulom and MuPAD rationalize command.

QPI approximates any floating point numbers by a rational number, square
root, multiple of PI, exponential or a logarithm depending on which
approximation seems best.

QPI should work in all ports of any HP48 SX/GX model.

Allowed argument types are

	real number		--> symbolic / real number
	complex number		--> symbolic / real number
	identifier		--> identifier
	local name		--> local name
	symbolic		--> symbolic
	array of real/complex	--> list of symbolic/real
	constants (PI,e)	--> constant
	list of above		--> list of above

The rational approximation algorithm is taken from MuPAD, and works quite
well for numbers of any scale. The decision procedure for the minimal
approximation of real numbers is as follows

	If x=zero then return (x)
	nom,den=approximate(x)
	If den < 100 then return (nom/den)		% Early abort
	nom2,den2=approximate(x*x)
	If (x*x<5E5) & (den2<1000) & (den2<den) then	% Choose 'smaller'
		nom,den=sign(x)*nom2,den2
		If nom<10 then return (sqrt(nom,den))	% Early abort
	nom2,den2=approximate(x/pi)
	If (|x/pi<100) & (den2<1000) & (den2<den) then	% Choose 'smaller'
		nom,den=nom2,den2
		If nom<10 then return (nom/den*pi)	% Early abort
	nom2,den2=approximate(exp(x))
	If (den2<50) & (den2<den) then			% Choose 'smaller'
		nom,den=nom2,den2
		If nom<10 then return (ln(nom/den))	% Early abort
	nom2,den2=approximate(ln(x))
	If (x>0) & (den2<50) & (den2<den) then		% Choose 'smaller'
		nom,den=nom2,den2
		If nom<10 then return (exp(nom/den))	% Early abort
	Return (nom/den) in the found minimal form

The rational approximation algorithm itself uses 9-digit accuracy in
the approximation by default. The accuracy can be overridden by using
the FIX/SCI/ENG modes where the accuracy is taken to be the accuracy
of the display (1-11).

Example:

	[ -.714285714286  -1.41421356237  ]
	[ -.405465108108   1.11751906874  ]
	[ -2.44346095279    .657047293577 ]

	==>

	{ {   '-5/7'      '-SQRT(2)'    }
	  { 'LN(2/3)'     'EXP(1/9)'    }
	  { '-7/9*PI'  '5/7*SQRT(11/13' } }