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16.2 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:

"Accuracy"
     A floating-point calculation's accuracy is how close it comes to
     the real (paper and pencil) value.

"Error"
     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
     possible.

"Exponent"
     The order of magnitude of a value; some number of bits in a
     floating-point value store the exponent.

"Inf"
     A special value representing infinity.  Operations involving
     another number and infinity produce infinity.

"NaN"
     "Not a number."  A special value that results from attempting a
     calculation that has no answer as a real number.  *Note Strange
     values::, for more information about infinity and not-a-number
     values.

"Normalized"
     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.

"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.

"Significand"
     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'.

"Stability"
     From the Wikipedia article on numerical stability
     (https://en.wikipedia.org/wiki/Numerical_stability): "Calculations
     that can be proven not to magnify approximation errors are called
     "numerically stable"."

   See the Wikipedia article on accuracy and precision
(https://en.wikipedia.org/wiki/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.)

   *note Table 16.3: 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 16.3: Basic IEEE format values

     NOTE: The precision numbers include the implied leading one that
     gives them one extra bit of significand.