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15.2 Other Stuff to Know
========================

The rest of this major node uses a number of terms.  Here are some
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."(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:

'sqrt(-1)'
This makes sense in the range of complex numbers, but not in
the range of real numbers, so the result is 'NaN'.

'log(-8)'
-8 is out of the domain of 'log()', so the result is 'NaN'.

"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
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.)

⇒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

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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