File: gawk.info,  Node: Getting Accuracy,  Next: Try To Round,  Prev: Inexactness of computations,  Up: FP Math Caution

16.4.2 Getting the Accuracy You Need
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Can arbitrary-precision arithmetic give exact results?  There are no
easy answers.  The standard rules of algebra often do not apply when
using floating-point arithmetic.  Among other things, the distributive
and associative laws do not hold completely, and order of operation may
be important for your computation.  Rounding error, cumulative precision
loss, and underflow are often troublesome.

   When 'gawk' tests the expressions '0.1 + 12.2' and '12.3' for
equality using the machine double-precision arithmetic, it decides that
they are not equal!  (*Note Comparing FP Values::.)  You can get the
result you want by increasing the precision; 56 bits in this case does
the job:

     $ gawk -M -v PREC=56 'BEGIN { print (0.1 + 12.2 == 12.3) }'
     -| 1

   If adding more bits is good, perhaps adding even more bits of
precision is better?  Here is what happens if we use an even larger
value of 'PREC':

     $ gawk -M -v PREC=201 'BEGIN { print (0.1 + 12.2 == 12.3) }'
     -| 0

   This is not a bug in 'gawk' or in the MPFR library.  It is easy to
forget that the finite number of bits used to store the value is often
just an approximation after proper rounding.  The test for equality
succeeds if and only if _all_ bits in the two operands are exactly the
same.  Because this is not necessarily true after floating-point
computations with a particular precision and effective rounding mode, a
straight test for equality may not work.  Instead, compare the two
numbers to see if they are within the desirable delta of each other.

   In applications where 15 or fewer decimal places suffice, hardware
double-precision arithmetic can be adequate, and is usually much faster.
But you need to keep in mind that every floating-point operation can
suffer a new rounding error with catastrophic consequences, as
illustrated by our earlier attempt to compute the value of pi.  Extra
precision can greatly enhance the stability and the accuracy of your
computation in such cases.

   Additionally, you should understand that repeated addition is not
necessarily equivalent to multiplication in floating-point arithmetic.
In the example in *note Errors accumulate:::

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

you may or may not succeed in getting the correct result by choosing an
arbitrarily large value for 'PREC'.  Reformulation of the problem at
hand is often the correct approach in such situations.