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`gawk`

When given the `-M` option,
`gawk`

performs all integer arithmetic using GMP arbitrary-precision
integers. Any number that looks like an integer in a source
or data file is stored as an arbitrary-precision integer. The size
of the integer is limited only by the available memory. For example,
the following computes
5^4^3^2,
the result of which is beyond the
limits of ordinary hardware double-precision floating-point values:

$gawk -M 'BEGIN {>x = 5^4^3^2>print "number of digits =", length(x)>print substr(x, 1, 20), "...", substr(x, length(x) - 19, 20)>}'-| number of digits = 183231 -| 62060698786608744707 ... 92256259918212890625

If instead you were to compute the same value using arbitrary-precision
floating-point values, the precision needed for correct output (using
the formula
‘`prec = 3.322 * dps`’)
would be 3.322 x 183231,
or 608693.

The result from an arithmetic operation with an integer and a floating-point value
is a floating-point value with a precision equal to the working precision.
The following program calculates the eighth term in
Sylvester’s sequence^{96}
using a recurrence:

$gawk -M 'BEGIN {>s = 2.0>for (i = 1; i <= 7; i++)>s = s * (s - 1) + 1>print s>}'-| 113423713055421845118910464

The output differs from the actual number, 113,423,713,055,421,844,361,000,443,
because the default precision of 53 bits is not enough to represent the
floating-point results exactly. You can either increase the precision
(100 bits is enough in this case), or replace the floating-point constant
‘`2.0`’ with an integer, to perform all computations using integer
arithmetic to get the correct output.

Sometimes `gawk`

must implicitly convert an arbitrary-precision
integer into an arbitrary-precision floating-point value. This is
primarily because the MPFR library does not always provide the relevant
interface to process arbitrary-precision integers or mixed-mode numbers
as needed by an operation or function. In such a case, the precision is
set to the minimum value necessary for exact conversion, and the working
precision is not used for this purpose. If this is not what you need or
want, you can employ a subterfuge and convert the integer to floating
point first, like this:

gawk -M 'BEGIN { n = 13; print (n + 0.0) % 2.0 }'

You can avoid this issue altogether by specifying the number as a floating-point value to begin with:

gawk -M 'BEGIN { n = 13.0; print n % 2.0 }'

Note that for this particular example, it is likely best to just use the following:

gawk -M 'BEGIN { n = 13; print n % 2 }'

Weisstein, Eric W. Sylvester’s Sequence. From MathWorld—A Wolfram Web Resource (http://mathworld.wolfram.com/SylvestersSequence.html).