16.4.1.3 Errors Accumulate

The loss of accuracy during a single computation with floating-point numbers usually isn’t enough to worry about. However, if you compute a value that is the result of a sequence of floating-point operations, the error can accumulate and greatly affect the computation itself. Here is an attempt to compute the value of pi using one of its many series representations:

BEGIN {
    x = 1.0 / sqrt(3.0)
    n = 6
    for (i = 1; i < 30; i++) {
        n = n * 2.0
        x = (sqrt(x * x + 1) - 1) / x
        printf("%.15f\n", n * x)
    }
}

When run, the early errors propagate through later computations, causing the loop to terminate prematurely after attempting to divide by zero:

$ gawk -f pi.awk
-| 3.215390309173475
-| 3.159659942097510
-| 3.146086215131467
-| 3.142714599645573
…
-| 3.224515243534819
-| 2.791117213058638
-| 0.000000000000000
error→ gawk: pi.awk:6: fatal: division by zero attempted

Here is an additional example where the inaccuracies in internal representations yield an unexpected result:

$ gawk 'BEGIN {
>   for (d = 1.1; d <= 1.5; d += 0.1)    # loop five times (?)
>       i++
>   print i
> }'
-| 4