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gawk: Math Definitions

 15.2 Other Stuff to Know
 The rest of this major node uses a number of terms.  Here are some
 informal definitions that should help you work your way through the
 material here:
      A floating-point calculation's accuracy is how close it comes to
      the real (paper and pencil) value.
      The difference between what the result of a computation "should be"
      and what it actually is.  It is best to minimize error as much as
      The order of magnitude of a value; some number of bits in a
      floating-point value store the exponent.
      A special value representing infinity.  Operations involving
      another number and infinity produce infinity.
      "Not a number."(1)  A special value that results from attempting a
      calculation that has no answer as a real number.  In such a case,
      programs can either receive a floating-point exception, or get
      'NaN' back as the result.  The IEEE 754 standard recommends that
      systems return 'NaN'.  Some examples:
           This makes sense in the range of complex numbers, but not in
           the range of real numbers, so the result is 'NaN'.
           -8 is out of the domain of 'log()', so the result is 'NaN'.
      How the significand (see later in this list) is usually stored.
      The value is adjusted so that the first bit is one, and then that
      leading one is assumed instead of physically stored.  This provides
      one extra bit of precision.
      The number of bits used to represent a floating-point number.  The
      more bits, the more digits you can represent.  Binary and decimal
      precisions are related approximately, according to the formula:
           PREC = 3.322 * DPS
      Here, _prec_ denotes the binary precision (measured in bits) and
      _dps_ (short for decimal places) is the decimal digits.
 "Rounding mode"
      How numbers are rounded up or down when necessary.  More details
      are provided later.
      A floating-point value consists of the significand multiplied by 10
      to the power of the exponent.  For example, in '1.2345e67', the
      significand is '1.2345'.
      From the Wikipedia article on numerical stability
      ( "Calculations
      that can be proven not to magnify approximation errors are called
      "numerically stable"."
    See the Wikipedia article on accuracy and precision
 ( for more
 information on some of those terms.
    On modern systems, floating-point hardware uses the representation
 and operations defined by the IEEE 754 standard.  Three of the standard
 IEEE 754 types are 32-bit single precision, 64-bit double precision, and
 128-bit quadruple precision.  The standard also specifies extended
 precision formats to allow greater precisions and larger exponent
 ranges.  ('awk' uses only the 64-bit double-precision format.)
    ⇒Table 15.2 table-ieee-formats. lists the precision and
 exponent field values for the basic IEEE 754 binary formats.
 Name           Total bits     Precision      Minimum        Maximum
                                              exponent       exponent
 Single         32             24             -126           +127
 Double         64             53             -1022          +1023
 Quadruple      128            113            -16382         +16383
 Table 15.2: Basic IEEE format values
      NOTE: The precision numbers include the implied leading one that
      gives them one extra bit of significand.
    ---------- Footnotes ----------
    (1) Thanks to Michael Brennan for this description, which we have
 paraphrased, and for the examples.
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