#### 16.4.3 Try a Few Extra Bits of Precision and Rounding ¶

Instead of arbitrary-precision floating-point arithmetic,
often all you need is an adjustment of your logic
or a different order for the operations in your calculation.
The stability and the accuracy of the computation of *pi*
in the earlier example can be enhanced by using the following
simple algebraic transformation:

(sqrt(x * x + 1) - 1) / x ≡ x / (sqrt(x * x + 1) + 1)

After making this change, the program converges to
*pi* in under 30 iterations:

$ `gawk -f pi2.awk`
-| 3.215390309173473
-| 3.159659942097501
-| 3.146086215131436
-| 3.142714599645370
-| 3.141873049979825
...
-| 3.141592653589797
-| 3.141592653589797