File: gawk.info,  Node: Errors accumulate,  Next: Strange values,  Prev: Comparing FP Values,  Up: Inexactness of computations

16.4.1.3 Errors Accumulate
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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