Common Lisp the Language 2nd Edition

2.1.3. Floating-Point Numbers

Common Lisp allows an implementation to provide one or more kinds of floating-point number which collectively make up the type float. Now a floating-point number is a (mathematical) rational number of the form where s is +1 or -1 the sign; b is an integer greater than 1 the base or radix of the representation; p is a positive integer the precision (in base-b digits) of the floating-point number; f is a positive integer between and (inclusive) the significand; and e is an integer the exponent. The value of p and the range of e depends on the implementation and on the type of floating-point number within that implementation. In addition there is a floating-point zero; depending on the implementation there may also be a ``minus zero.'' If there is no minus zero then 0.0 and -0.0 are both interpreted as simply a floating-point zero.

Floating-point numbers may be provided in a variety of precisions and sizes depending on the implementation. High-quality floating-point software tends to depend critically on the precise nature of the floating-point arithmetic and so may not always be completely portable. As an aid in writing programs that are moderately portable however certain definitions are made here:

• A short floating-point number (type short-float) is of the representation of smallest fixed precision provided by an implementation.

• A long floating-point number (type long-float) is of the representation of the largest fixed precision provided by an implementation.

• Intermediate between short and long formats are two others arbitrarily called single and double (types single-float and double-float).

Floating-point numbers are written in either decimal fraction or computerized scientific notation: an optional sign then a non-empty sequence of digits with an embedded decimal point then an optional decimal exponent specification. If there is no exponent specifier then the decimal point is required and there must be digits after it. The exponent specifier consists of an exponent marker an optional sign and a non-empty sequence of digits. For preciseness here is a modified-BNF description of floating-point notation.

floating-point-number ::= [sign] {digit}* decimal-point {digit}* [exponent]
| [sign] {digit}+ [decimal-point {digit}*] exponent
sign ::= + | -
decimal-point ::= .
digit ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
exponent ::= exponent-marker [sign] {digit}+
exponent-marker ::= e | s | f | d | l | E | S | F | D | L

The letters s f d and l (or their respective uppercase equivalents) explicitly specify the use of short single double and long format respectively.

Examples of floating-point numbers:

0.0                         ;Floating-point zero in default format
0E0                         ;Also floating-point zero in default format
-.0                         ;This may be a zero or a minus zero

; depending on the implementation
0.                          ;The integer zero
not a floating-point zero!
0.0s0                       ;A floating-point zero in short format
0s0                         ;Also a floating-point zero in short format
3.1415926535897932384d0     ;A double-format approximation to 
6.02E+23                    ;Avogadro's number
in default format
602E+21                     ;Also Avogadro's number
in default format
3.010299957f-1              ;
in single format
-0.000000001s9              ; in short format
the hard way

Notice of correction. The first edition unfortunately listed an incorrect value (3.1010299957f-1) for the base-10 logarithm of 2.

The internal format used for an external representation depends only on the exponent marker and not on the number of decimal digits in the external representation.

While Common Lisp provides terminology and notation sufficient to accommodate four distinct floating-point formats not all implementations will have the means to support that many distinct formats. An implementation is therefore permitted to provide fewer than four distinct internal floating-point formats in which case at least one of them will be ``shared'' by more than one of the external format names short single double and long according to the following rules:

• If one internal format is provided then it is considered to be single but serves also as short double and long. The data types short-float single-float double-float and long-float are considered to be identical. An expression such as (eql 1.0s0 1.0d0) will be true in such an implementation because the two numbers 1.0s0 and 1.0d0 will be converted into the same internal format and therefore be considered to have the same data type despite the differing external syntax. Similarly (typep 1.0L0 'short-float) will be true in such an implementation. For output purposes all floating-point numbers are assumed to be of single format and thus will print using the exponent letter E or F.

• If two internal formats are provided then either of two correspondences may be used depending on which is the more appropriate:
  • One format is short; the other is single and serves also as double and long. The data types single-float double-float and long-float are considered to be identical but short-float is distinct. An expression such as (eql 1.0s0 1.0d0) will be false but (eql 1.0f0 1.0d0) will be true. Similarly (typep 1.0L0 'short-float) will be false but (typep 1.0L0 'single-float) will be true. For output purposes all floating-point numbers are assumed to be of short or single format.

  • One format is single and serves also as short; the other is double and serves also as long. The data types short-float and single-float are considered to be identical and the data types double-float and long-float are considered to be identical. An expression such as (eql 1.0s0 1.0d0) will be false as will (eql 1.0f0 1.0d0); but (eql 1.0d0 1.0L0) will be true. Similarly (typep 1.0L0 'short-float) will be false but (typep 1.0L0 'double-float) will be true. For output purposes all floating-point numbers are assumed to be of single or double format.

• If three internal formats are provided then either of two correspondences may be used depending on which is the more appropriate:
  • One format is short; another format is single; and the third format is double and serves also as long. Similar constraints apply.

  • One format is single and serves also as short; another is double; and the third format is long.