User Manual

148 Appendix: Accuracy of Numerical Calculations
148
0 < x < 1 above. When R(x) is squared 50 times to produce F(x) = S(R(x)), the result is clearly
1 for x ≥ 1, but why is F(x) = 0 for 0 ≤ x < 1? When x <1,
    
.
48898
1014.6
50
2
10
1019999999999.0









 sxRs
This value is so small that the calculated value F(x) = S(R(x)) underflows to 0. So the
HP-15C isn't broken; it is doing the best that can be done with 10 significant digits of
precision and 2 exponent digits.
We have explained example 1 using no more information about the HP-15C than that it
performs each arithmetic operation ¤ and x and fully as accurately as is possible within
the limitations of 10 significant digits and 2 exponent digits. The rest of the information we
needed was mathematical knowledge about the functions f, r, and s. For instance, the value
r(10
100
) above was evaluated as
 
 













21313
13
50
50100
2
1
100100
)10045.2(
2
1
)10045.2(1
)10045.2exp(
2)10ln(100exp
2)10ln(exp
)10()10(
50
r
by using the series
.
3
6
1
2
2
1
1)exp(  zzzz
Similarly, the binomial theorem was used for
 
   
.
2
1010
2
1
10
10
8
1
10
2
1
1
1019999999999.0




These mathematical facts lie well beyond the kind of knowledge that might have been
considered adequate to cope with a calculation containing only a handful of multiplications
and square roots. In this respect, example 1 illustrates an unhappy truism: Errors make
computation very much harder to analyze. That is why a well-designed calculator, like the
HP-15C, will introduce errors of its own as sparingly as is possible at a tolerable cost. Much
more error than that would turn an already difficult task into something hopeless.
Example 1 should lay two common misconceptions to rest:
 Rounding errors can overwhelm a computation only if vast numbers of them
accumulate.
 A few rounding errors can overwhelm a computation only if accompanied by massive
cancellation.