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gawk: Arbitrary Precision Integers

 15.5 Arbitrary-Precision Integer Arithmetic with '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(1) 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 }'
    When dividing two arbitrary precision integers with either '/' or
 '%', the result is typically an arbitrary precision floating point value
 (unless the denominator evenly divides into the numerator).
    ---------- Footnotes ----------
    (1) Weisstein, Eric W. 'Sylvester's Sequence'.  From MathWorld--A
 Wolfram Web Resource
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