*gawk:* Try To Round

15.4.3 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