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Common Lisp: Numbers

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

Some sources:

Introduction

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 bignum
s, 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 function
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

Complex arithmetic

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)

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 thestrings 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 onmapping.

Operations on multidimensional arrays are discussed inthis 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.

Bit-wise Operation

Common Lisp also provides many functions to perform bit-wise arithmetic
operations. Some commonly used ones are listed below, together with their
C/C++ equivalence.

Common Lisp C/C++ Description
(logand a b c) a & b & c Bit-wise AND of multiple operands
(logior a b c) a | b | c Bit-wise OR of multiple arguments
(lognot a) ~a Bit-wise NOT of single operand
(logxor a b c) a ^ b ^ c Bit-wise exclusive or (XOR) or multiple operands
(ash a 3) a << 3 Bit-wise left shift
(ash a -3) a >> 3 Bit-wise right shift

Negative numbers are treated as two’s-complements. If you have forgotten this,
please refer to the Wiki page
.

For example:

* (logior 1 2 4 8)
15
;; Explanation:
;;   0001
;;   0010
;;   0100
;; | 1000
;; -------
;;   1111

* (logand 2 -3 4)
0

;; Explanation:
;;   0010 (2)
;;   1101 (two's complement of -3)
;; & 0100 (4)
;; ------- 
;;   0000

* (logxor 1 3 7 15)
10

;; Explanation:
;;   0001
;;   0011
;;   0111
;; ^ 1111
;; -------
;;   1010

* (lognot -1)
0
;; Explanation:
;;   11 -> 00

* (lognot -3)
2
;;   101 -> 010

* (ash 3 2)
12
;; Explanation:
;;   11 -> 1100

* (ash -5 -2)
-2
;; Explanation
;;   11011 -> 110

Please see the CLHS page
for a more detailed explanation
or other bit-wise functions.

Page source: numbers.md

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