EPS [(x)]


Math Package

Since the precision of the QL is limited, a number may not change if a very small value is added. The function EPS(x) returns the smallest value which can be added to x so that the sum of x and EPS(x) will be different from x. This only makes sense for floating point numbers. The default parameter is 0. EPS(x) attains its smallest value at x=0, so EPS(0) returns the smallest absolute number which can be handled by SuperBASIC. EPS(x) is always greater than zero and EPS(x)=EPS(-x).


An approximation of PI/4 as proposed by Leibniz:

100 x = 0: d = 1
110 t0 = DATE
120 FOR i=1 TO 1E100
130   IF ABS(1/d) < EPS(x) THEN EXIT i
140   x = x + 1/d
150   d = - SGN(d) \* (ABS(d)+2)
160 END FOR i
170 t = DATE - t0
180 PRINT "Iterations ="!i!" Runtime ="!t;"s"
190 PRINT "Iterations per Second ="!i/t
200 PRINT "PI ="!4\*x!"(";PI;")"

Unfortunately, the algorithm is not efficient enough to compete with the QL’s precision, so that about 2E9 iterations are necessary to get a suitable result. Since this will take a while (ages!), you can reduce precision by a factor of one million, by modifying line 130:

130 IF ABS(1/d) < 1E6 * EPS(x) THEN EXIT i

The program will then finish after 1075 iterations with 4*x = 3.140662, not bad compared to 3.141593 when taking the drastic reduction of precision into account.


EPS does not recognise the higher precision used by Minerva. Minerva’s higher precision may have an effect on fractals and similar esoteric calculations.