The Common Lisp Cookbook – Numbers

Introduction

Common Lisp has a rich set of numerical types, including integer, rational, floating point, and complex.

Some sources:

Integer types

Common Lisp provides a true integer type, called bignum, limited only by the total memory available (not the machine word size). For example this would overflow a 64 bit integer by some way:

* (expt 2 200)
1606938044258990275541962092341162602522202993782792835301376

For efficiency, integers can be limited to a fixed number of bits, called a fixnum type. The range of integers which can be represented is given by:

* most-positive-fixnum
4611686018427387903
* most-negative-fixnum
-4611686018427387904

Functions which operate on or evaluate to integers include:

  • isqrt, which returns the greatest integer less than or equal to the exact positive square root of natural.
* (isqrt 10)
3
* (isqrt 4)
2
  • gcd to find the Greatest Common Denominator
  • lcm for the Least Common Multiple.

Rational types

Rational numbers of type RATIO consist of two bignums, the numerator and denominator. Both can therefore be arbitrarily large:

* (/ (1+ (expt 2 100)) (expt 2 100))
1267650600228229401496703205377/1267650600228229401496703205376

It is a subtype of the rational class, along with integer.

Floating point types

See Common Lisp the Language, 2nd Edition, section 2.1.3.

Floating point types attempt to represent the continuous real numbers using a finite number of bits. This means that many real numbers cannot be represented, but are approximated. This can lead to some nasty surprises, particularly when converting between base-10 and the base-2 internal representation. If you are working with floating point numbers then reading What Every Computer Scientist Should Know About Floating-Point Arithmetic is highly recommended.

The Common Lisp standard allows for several floating point types. In order of increasing precision these are: short-float, single-float, double-float, and long-float. Their precisions are implementation dependent, and it is possible for an implementation to have only one floating point precision for all types.

The constants short-float-epsilon, single-float-epsilon, double-float-epsilon and long-float-epsilon give a measure of the precision of the floating point types, and are implementation dependent.

Floating point literals

When reading floating point numbers, the default type is set by the special variable read-default-float-format. By default this is SINGLE-FLOAT, so if you want to ensure that a number is read as double precision then put a d0 suffix at the end

* (type-of 1.24)
SINGLE-FLOAT

* (type-of 1.24d0)
DOUBLE-FLOAT

Other suffixes are s (short), f (single float), d (double float), l (long float) and e (default; usually single float).

The default type can be changed, but note that this may break packages which assume single-float type.

* (setq *read-default-float-format* 'double-float)
* (type-of 1.24)
DOUBLE-FLOAT

Note that unlike in some languages, appending a single decimal point to the end of a number does not make it a float:

* (type-of 10.)
(INTEGER 0 4611686018427387903)

* (type-of 10.0)
SINGLE-FLOAT

Floating point errors

If the result of a floating point calculation is too large then a floating point overflow occurs. By default in SBCL (and other implementations) this results in an error condition:

* (exp 1000)
; Evaluation aborted on #<FLOATING-POINT-OVERFLOW {10041720B3}>.

The error can be caught and handled, or this behaviour can be changed, to return +infinity. In SBCL this is:

* (sb-int:set-floating-point-modes :traps '(:INVALID :DIVIDE-BY-ZERO))

* (exp 1000)
#.SB-EXT:SINGLE-FLOAT-POSITIVE-INFINITY

* (/ 1 (exp 1000))
0.0

The calculation now silently continues, without an error condition.

A similar functionality to disable floating overflow errors exists in CCL:

* (set-fpu-mode :overflow nil)

In SBCL the floating point modes can be inspected:

* (sb-int:get-floating-point-modes)
(:TRAPS (:OVERFLOW :INVALID :DIVIDE-BY-ZERO) :ROUNDING-MODE :NEAREST
 :CURRENT-EXCEPTIONS NIL :ACCRUED-EXCEPTIONS NIL :FAST-MODE NIL)

Arbitrary precision

For arbitrary high precision calculations there is the computable-reals library on QuickLisp:

* (ql:quickload :computable-reals)
* (use-package :computable-reals)

* (sqrt-r 2)
+1.41421356237309504880...

* (sin-r (/r +pi-r+ 2))
+1.00000000000000000000...

The precision to print is set by *PRINT-PREC*, by default 20

* (setq *PRINT-PREC* 50)
* (sqrt-r 2)
+1.41421356237309504880168872420969807856967187537695...

Complex types

There are 5 types of complex number: The real and imaginary parts must be of the same type, and can be rational, or one of the floating point types (short, single, double or long).

Complex values can be created using the #C reader macro or the complex. The reader macro does not allow the use of expressions as real and imaginary parts:

* #C(1 1)
#C(1 1)

* #C((+ 1 2) 5)
; Evaluation aborted on #<TYPE-ERROR expected-type: REAL datum: (+ 1 2)>.

* (complex (+ 1 2) 5)
#C(3 5)

If constructed with mixed types then the higher precision type will be used for both parts.

* (type-of #C(1 1))
(COMPLEX (INTEGER 1 1))

* (type-of #C(1.0 1))
(COMPLEX (SINGLE-FLOAT 1.0 1.0))

* (type-of #C(1.0 1d0))
(COMPLEX (DOUBLE-FLOAT 1.0d0 1.0d0))

The real and imaginary parts of a complex number can be extracted using realpart and imagpart:

* (realpart #C(7 9))
7
* (imagpart #C(4.2 9.5))
9.5

Reading numbers from strings

The parse-integer function reads an integer from a string.

The parse-float library provides a parser which cannot evaluate arbitrary expressions, so should be safer to use on untrusted input:

* (ql:quickload :parse-float)
* (use-package :parse-float)

* (parse-float "23.4e2" :type 'double-float)
2340.0d0
6

See the strings section on converting between strings and numbers.

Converting numbers

Most numerical functions automatically convert types as needed. The coerce function converts objects from one type to another, including numeric types.

See Common Lisp the Language, 2nd Edition, section 12.6

Convert float to rational

The rational and rationalize functions convert a real numeric argument into a rational. rational assumes that floating point arguments are exact; rationalize expoits the fact that floating point numbers are only exact to their precision, so can often find a simpler rational number.

Convert rational to integer

If the result of a calculation is a rational number where the numerator is a multiple of the denominator, then it is automatically converted to an integer:

* (type-of (* 1/2 4))
(INTEGER 0 4611686018427387903)

Rounding floating-point and rational numbers

The ceiling, floor, round and truncate functions convert floating point or rational numbers to integers. The difference between the result and the input is returned as the second value, so that the input is the sum of the two outputs.

* (ceiling 1.42)
2
-0.58000004

* (floor 1.42)
1
0.41999996

* (round 1.42)
1
0.41999996

* (truncate 1.42)
1
0.41999996

There is a difference between floor and truncate for negative numbers:

* (truncate -1.42)
-1
-0.41999996

* (floor -1.42)
-2
0.58000004

* (ceiling -1.42)
-1
-0.41999996

Similar functions fceiling, ffloor, fround and ftruncate return the result as floating point, of the same type as their argument:

* (ftruncate 1.3)
1.0
0.29999995

* (type-of (ftruncate 1.3))
SINGLE-FLOAT

* (type-of (ftruncate 1.3d0))
DOUBLE-FLOAT

Comparing numbers

See Common Lisp the Language, 2nd Edition, Section 12.3.

The = predicate returns T if all arguments are numerically equal. Note that comparison of floating point numbers includes some margin for error, due to the fact that they cannot represent all real numbers and accumulate errors.

The constant single-float-epsilon is the smallest number which will cause an = comparison to fail, if it is added to 1.0:

* (= (+ 1s0 5e-8) 1s0)
T
* (= (+ 1s0 6e-8) 1s0)
NIL

Note that this does not mean that a single-float is always precise to within 6e-8:

* (= (+ 10s0 4e-7) 10s0)
T
* (= (+ 10s0 5e-7) 10s0)
NIL

Instead this means that single-float is precise to approximately seven digits. If a sequence of calculations are performed, then error can accumulate and a larger error margin may be needed. In this case the absolute difference can be compared:

* (< (abs (- (+ 10s0 5e-7) 
             10s0))
     1s-6)
T

When comparing numbers with = mixed types are allowed. To test both numerical value and type use eql:

* (= 3 3.0)
T

* (eql 3 3.0)
NIL

Operating on a series of numbers

Many Common Lisp functions operate on sequences, which can be either lists or vectors (1D arrays). See the section on mapping.

Operations on multidimensional arrays are discussed in this section.

Libraries are available for defining and operating on lazy sequences, including “infinite” sequences of numbers. For example

  • Clazy which is on QuickLisp
  • folio2 on QuickLisp. Includes an interface to the Series package for efficient sequences.
  • lazy-seq

Working with Roman numerals

The format function can convert numbers to roman numerals with the ~@r directive:

* (format nil "~@r" 42)
"XLII"

There is a gist by tormaroe for reading roman numerals.

Generating random numbers

The random function generates either integer or floating point random numbers, depending on the type of its argument.

* (random 10)
7

* (type-of (random 10))
(INTEGER 0 4611686018427387903)
* (type-of (random 10.0))
SINGLE-FLOAT
* (type-of (random 10d0))
DOUBLE-FLOAT

In SBCL a Mersenne Twister pseudo-random number generator is used. See section 7.13 of the SBCL manual for details.

The random seed is stored in random-state whose internal representation is implementation dependent. The function make-random-state can be used to make new random states, or copy existing states.

To use the same set of random numbers multiple times, (make-random-state nil) makes a copy of the current *random-state*:

* (dotimes (i 3)
    (let ((*random-state* (make-random-state nil)))
      (format t "~a~%" 
              (loop for i from 0 below 10 collecting (random 10)))))
              
(8 3 9 2 1 8 0 0 4 1)
(8 3 9 2 1 8 0 0 4 1)
(8 3 9 2 1 8 0 0 4 1)

This generates 10 random numbers in a loop, but each time the sequence is the same because the *random-state* special variable is dynamically bound to a copy of its state before the let form.

Other resources:

  • The random-state package is available on QuickLisp, and provides a number of portable random number generators.

Using complex numbers

Common Lisp’s mathematical functions generally handle complex numbers, and return complex numbers when this is the true result. For example:

* (sqrt -1)
#C(0.0 1.0)

* (exp #C(0.0 0.5))
#C(0.87758255 0.47942555)

* (sin #C(1.0 1.0))
#C(1.2984576 0.63496387)

Page source: numbers.md