Floating point summary (The GNU Awk User’s Guide)

16.8 Summary

Most computer arithmetic is done using either integers or floating-point values. Standard awk uses double-precision floating-point values.
In the early 1990s Barbie mistakenly said, “Math class is tough!” Although math isn’t tough, floating-point arithmetic isn’t the same as pencil-and-paper math, and care must be taken: - Not all numbers can be represented exactly. - Comparing values should use a delta, instead of being done directly with ‘==’ and ‘!=’. - Errors accumulate. - Operations are not always truly associative or distributive.
Increasing the accuracy can help, but it is not a panacea.
Often, increasing the accuracy and then rounding to the desired number of digits produces reasonable results.
Use -M (or --bignum) to enable MPFR arithmetic. Use PREC to set the precision in bits, and ROUNDMODE to set the IEEE 754 rounding mode.
With -M, gawk performs arbitrary-precision integer arithmetic using the GMP library. This is faster and more space-efficient than using MPFR for the same calculations.
There are several areas with respect to floating-point numbers where gawk disagrees with the POSIX standard. It pays to be aware of them.
Overall, there is no need to be unduly suspicious about the results from floating-point arithmetic. The lesson to remember is that floating-point arithmetic is always more complex than arithmetic using pencil and paper. In order to take advantage of the power of floating-point arithmetic, you need to know its limitations and work within them. For most casual use of floating-point arithmetic, you will often get the expected result if you simply round the display of your final results to the correct number of significant decimal digits.
As general advice, avoid presenting numerical data in a manner that implies better precision than is actually the case.