HP-17BII Precision: Epsilon



#2

Calculating Epsilon on the HP-17BII

I wrote a program to compute 'Epsilon' which---in computational context---means the
smallest number (in the computer) that may be added to one and yield a result
that is greater than one. It is a measure of the precision that the
computer uses to represent numbers (really the number of bits used for the
mantissa).

I first wrote a traditional "Epsilon" equation for the HP-17BII, but I
realized that the number storage was not the standard binary floating point,
but rather some form of binary-coded-decimal. As a result, the original
"Epsilon" program yielded an incorrect result. The general algorithm
may be used on any machine that stores floating point numbers in an IEEE-like
format. So, I re-wrote the equation for a binary-coded-decimal machine.
The new equation yields the correct result for the HP-17BII; the Epsilon
is 5.00000000001e-12 (twelve digits in the mantissa).

Binary Floating-Point Epsilon algorithm---INCORRECT for HP-17BII:

Epsilon = 1;
while ((1 + Epsilon) > 1) {
Epsilon /= 2;
}
Epsilon *= 2;

Below is the HP-17BII equation. Please note that in the listing below
"SIGMA" means the sigma symbol:

In the ALPHA menu:

[WXYZ][OTHER][MORE],

then the second button

Binary-Coded-Decimal equation for the HP-17BII:

EPSILON: 0*L(E:1)*
SIGMA(N:1:25:1:(
SIGMA(I:1:9:1:
IF(1+L(X:(G(E)-I*10^(-N)))>1:
L(NEWE:G(X))
:
0
)
)
+L(OLDE:G(E))
+L(E:G(NEWE))
)
)
+
IF(G(OLDE)>G(E):
0
:
G(E)
)
=EPSILON

The extra code concerning the "OLDE" variable is just to make sure
that we have looped far enough. If Epsilon comes back as zero, then we
need to search more powers of ten in the "N" loop (i.e. increasing
the value of 25 here).


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