@node Numbers, Characters, Equivalence Predicates, Top
@chapter Numbers
@cindex number

(This section is largely taken from the @cite{Revised^4 Report on the
Algorithmic Language Scheme}.)

Numerical computation has traditionally been neglected by the Lisp
community.  Until Common Lisp there was no carefully thought out
strategy for organizing numerical computation, and with the exception of
the MacLisp system little effort was made to execute numerical code
efficiently.  This report recognizes the excellent work of the Common
Lisp committee and accepts many of their recommendations.  In some ways
this report simplifies and generalizes their proposals in a manner
consistent with the purposes of Scheme.

It is important to distinguish between the mathematical numbers, the
Scheme numbers that attempt to model them, the machine representations
used to implement the Scheme numbers, and notations used to write
numbers.  This report uses the types @emph{number}, @emph{complex},
@emph{real}, @emph{rational}, and @emph{integer} to refer to both
mathematical numbers and Scheme numbers.  Machine representations such
as fixed point and floating point are referred to by names such as
@emph{fixnum} and @emph{flonum}.

@menu
* Numerical types::
* Exactness::
* Implementation restrictions::
* Syntax of numerical constants::
* Numerical operations::
* Numerical input and output::
* Fixnum and Flonum Operations::
* Random Numbers::
@end menu

@node Numerical types, Exactness, Numbers, Numbers
@section Numerical types
@cindex numerical types

Mathematically, numbers may be arranged into a tower of subtypes in
which each level is a subset of the level above it:

@display
@group
number
complex
real
rational
integer
@end group
@end display

For example, 3 is an integer.  Therefore 3 is also a rational, a real,
and a complex.  The same is true of the Scheme numbers that model 3.
For Scheme numbers, these types are defined by the predicates
@code{number?}, @code{complex?}, @code{real?}, @code{rational?}, and
@code{integer?}.

There is no simple relationship between a number's type and its
representation inside a computer.  Although most implementations of
Scheme will offer at least two different representations of 3, these
different representations denote the same integer.

Scheme's numerical operations treat numbers as abstract data, as
independent of their representation as possible.  Although an
implementation of Scheme may use fixnum, flonum, and perhaps other
representations for numbers, this should not be apparent to a casual
programmer writing simple programs.

It is necessary, however, to distinguish between numbers that are
represented exactly and those that may not be.  For example, indexes
into data structures must be known exactly, as must some polynomial
coefficients in a symbolic algebra system.  On the other hand, the
results of measurements are inherently inexact, and irrational numbers
may be approximated by rational and therefore inexact approximations.
In order to catch uses of inexact numbers where exact numbers are
required, Scheme explicitly distinguishes exact from inexact numbers.
This distinction is orthogonal to the dimension of type.

@node Exactness, Implementation restrictions, Numerical types, Numbers
@section Exactness
@cindex exactness

Scheme numbers are either @emph{exact} or @emph{inexact}.  A number is
exact if it was written as an exact constant or was derived from exact
numbers using only exact operations.  A number is inexact if it was
written as an inexact constant, if it was derived using inexact
ingredients, or if it was derived using inexact operations. Thus
inexactness is a contagious property of a number.

If two implementations produce exact results for a computation that did
not involve inexact intermediate results, the two ultimate results will
be mathematically equivalent.  This is generally not true of
computations involving inexact numbers since approximate methods such as
floating point arithmetic may be used, but it is the duty of each
implementation to make the result as close as practical to the
mathematically ideal result.

@findex +
Rational operations such as @code{+} should always produce exact results
when given exact arguments.  If the operation is unable to produce an
exact result, then it may either report the violation of an
implementation restriction or it may silently coerce its result to an
inexact value.  @xref{Implementation restrictions}.

@findex exact
With the exception of @code{exact}, the operations described in
this section must generally return inexact results when given any
inexact arguments.  An operation may, however, return an exact result if
it can prove that the value of the result is unaffected by the
inexactness of its arguments.  For example, multiplication of any number
by an exact zero may produce an exact zero result, even if the other
argument is inexact.

@node Implementation restrictions, Syntax of numerical constants, Exactness, Numbers
@section Implementation restrictions
@cindex implementation restriction

Implementations of Scheme are not required to implement the whole tower
of subtypes (@pxref{Numerical types}), but they must implement a
coherent subset consistent with both the purposes of the implementation
and the spirit of the Scheme language.  For example, an implementation
in which all numbers are real may still be quite useful.@footnote{MIT/GNU
Scheme implements the whole tower of numerical types.  It has
unlimited-precision exact integers and exact rationals.  Flonums are
used to implement all inexact reals; on machines that support @sc{ieee}
floating-point arithmetic these are double-precision floating-point
numbers.}

Implementations may also support only a limited range of numbers of any
type, subject to the requirements of this section.  The supported range
for exact numbers of any type may be different from the supported range
for inexact numbers of that type.  For example, an implementation that
uses flonums to represent all its inexact real numbers may support a
practically unbounded range of exact integers and rationals while
limiting the range of inexact reals (and therefore the range of inexact
integers and rationals) to the dynamic range of the flonum format.
Furthermore the gaps between the representable inexact integers and
rationals are likely to be very large in such an implementation as the
limits of this range are approached.

@findex length
@findex vector-length
@findex string-length
An implementation of Scheme must support exact integers throughout the
range of numbers that may be used for indexes of lists, vectors, and
strings or that may result from computing the length of a list, vector,
or string.  The @code{length}, @code{vector-length}, and
@code{string-length} procedures must return an exact integer, and it is
an error to use anything but an exact integer as an index.  Furthermore
any integer constant within the index range, if expressed by an exact
integer syntax, will indeed be read as an exact integer, regardless of
any implementation restrictions that may apply outside this range.
Finally, the procedures listed below will always return an exact integer
result provided all their arguments are exact integers and the
mathematically expected result is representable as an exact integer
within the implementation:

@example
@group
*                gcd                modulo
+                imag-part          numerator
-                exact              quotient
abs              lcm                rationalize
angle            magnitude          real-part
ceiling          make-polar         remainder
denominator      make-rectangular   round
expt             max                truncate
floor            min
@end group
@end example

@findex /
Implementations are encouraged, but not required, to support exact
integers and exact rationals of practically unlimited size and
precision, and to implement the above procedures and the @code{/}
procedure in such a way that they always return exact results when given
exact arguments.  If one of these procedures is unable to deliver an
exact result when given exact arguments, then it may either report a
violation of an implementation restriction or it may silently coerce its
result to an inexact number.  Such a coercion may cause an error
later.

An implementation may use floating point and other approximate
representation strategies for inexact numbers.  This report recommends,
but does not require, that the @sc{ieee} 32-bit and 64-bit floating
point standards be followed by implementations that use flonum
representations, and that implementations using other representations
should match or exceed the precision achievable using these floating
point standards.

@findex sqrt
In particular, implementations that use flonum representations must
follow these rules: A flonum result must be represented with at least as
much precision as is used to express any of the inexact arguments to
that operation.  It is desirable (but not required) for potentially
inexact operations such as @code{sqrt}, when applied to exact arguments,
to produce exact answers whenever possible (for example the square root
of an exact 4 ought to be an exact 2).  If, however, an exact number is
operated upon so as to produce an inexact result (as by @code{sqrt}),
and if the result is represented as a flonum, then the most precise
flonum format available must be used; but if the result is represented
in some other way then the representation must have at least as much
precision as the most precise flonum format available.

Although Scheme allows a variety of written notations for numbers, any
particular implementation may support only some of them.@footnote{MIT/GNU
Scheme implements all of the written notations for numbers.} For
example, an implementation in which all numbers are real need not
support the rectangular and polar notations for complex numbers.  If an
implementation encounters an exact numerical constant that it cannot
represent as an exact number, then it may either report a violation of
an implementation restriction or it may silently represent the constant
by an inexact number.

@node Syntax of numerical constants, Numerical operations, Implementation restrictions, Numbers
@section Syntax of numerical constants
@cindex number, external representation
@cindex external representation, for number

@findex #b
@findex #o
@findex #d
@findex #x
@cindex #b as external representation
@cindex #o as external representation
@cindex #d as external representation
@cindex #x as external representation
A number may be written in binary, octal, decimal, or hexadecimal by the
use of a radix prefix.  The radix prefixes are @code{#b} (binary),
@code{#o} (octal), @code{#d} (decimal), and @code{#x} (hexadecimal).
With no radix prefix, a number is assumed to be expressed in
decimal.

@findex #e
@findex #i
@findex #
@cindex #e as external representation
@cindex #i as external representation
@cindex # in external representation of number
A numerical constant may be specified to be either exact or inexact by a
prefix.  The prefixes are @code{#e} for exact, and @code{#i} for
inexact.  An exactness prefix may appear before or after any radix
prefix that is used.  If the written representation of a number has no
exactness prefix, the constant may be either inexact or exact.  It is
inexact if it contains a decimal point, an exponent, or a @code{#}
character in the place of a digit, otherwise it is exact.

@cindex s, as exponent marker in number
@cindex f, as exponent marker in number
@cindex d, as exponent marker in number
@cindex l, as exponent marker in number
@cindex e, as exponent marker in number
@cindex exponent marker (defn)
@cindex precision, of inexact number
@cindex numeric precision, inexact
@cindex internal representation, for inexact number
@cindex short precision, of inexact number
@cindex single precision, of inexact number
@cindex double precision, of inexact number
@cindex long precision, of inexact number
In systems with inexact numbers of varying precisions it may be useful
to specify the precision of a constant.  For this purpose, numerical
constants may be written with an @dfn{exponent marker} that indicates
the desired precision of the inexact representation.  The letters
@code{s}, @code{f}, @code{d}, and @code{l} specify the use of
@emph{short}, @emph{single}, @emph{double}, and @emph{long} precision,
respectively.  (When fewer than four internal inexact representations
exist, the four size specifications are mapped onto those available.
For example, an implementation with two internal representations may map
short and single together and long and double together.)  In addition,
the exponent marker @code{e} specifies the default precision for the
implementation.  The default precision has at least as much precision as
@emph{double}, but implementations may wish to allow this default to be
set by the user.

@example
@group
3.14159265358979F0
       @r{Round to single ---} 3.141593
0.6L0
       @r{Extend to long ---} .600000000000000
@end group
@end example

@node Numerical operations, Numerical input and output, Syntax of numerical constants, Numbers
@section Numerical operations
@cindex numerical operations

@xref{Entry Format}, for a summary of the naming conventions used to
specify restrictions on the types of arguments to numerical routines.
The examples used in this section assume that any numerical constant
written using an exact notation is indeed represented as an exact
number.  Some examples also assume that certain numerical constants
written using an inexact notation can be represented without loss of
accuracy; the inexact constants were chosen so that this is likely to be
true in implementations that use flonums to represent inexact
numbers.

@deffn procedure number? object
@deffnx procedure complex? object
@deffnx procedure real? object
@deffnx procedure rational? object
@deffnx procedure integer? object
@cindex type predicate, for number
These numerical type predicates can be applied to any kind of argument,
including non-numbers.  They return @code{#t} if the object is of the
named type, and otherwise they return @code{#f}.  In general, if a type
predicate is true of a number then all higher type predicates are also
true of that number.  Consequently, if a type predicate is false of a
number, then all lower type predicates are also false of that
number.@footnote{In MIT/GNU Scheme the @code{rational?} procedure is the
same as @code{real?}, and the @code{complex?} procedure is the same as
@code{number?}.}

@findex zero?
@findex =
If @var{z} is an inexact complex number, then @code{(real? @var{z})} is
true if and only if @code{(zero? (imag-part @var{z}))} is true.  If
@var{x} is an inexact real number, then @code{(integer? @var{x})} is
true if and only if @code{(= @var{x} (round @var{x}))}.

@example
@group
(complex? 3+4i)         @result{}  #t
(complex? 3)            @result{}  #t
(real? 3)               @result{}  #t
(real? -2.5+0.0i)       @result{}  #t
(real? #e1e10)          @result{}  #t
(rational? 6/10)        @result{}  #t
(rational? 6/3)         @result{}  #t
(integer? 3+0i)         @result{}  #t
(integer? 3.0)          @result{}  #t
(integer? 8/4)          @result{}  #t
@end group
@end example

Note: The behavior of these type predicates on inexact numbers is
unreliable, since any inaccuracy may affect the result.

@end deffn

@deffn procedure exact? z
@deffnx procedure inexact? z
These numerical predicates provide tests for the exactness of a
quantity.  For any Scheme number, precisely one of these predicates is
true.
@end deffn

@deffn procedure exact-integer? object
@deffnx procedure exact-nonnegative-integer? object
@deffnx procedure exact-rational? object
These procedures test for some very common types of numbers.  These
tests could be written in terms of simpler predicates, but are more
efficient.
@end deffn

@deffn procedure = z1 z2 z3 @dots{}
@deffnx procedure < x1 x2 x3 @dots{}
@deffnx procedure > x1 x2 x3 @dots{}
@deffnx procedure <= x1 x2 x3 @dots{}
@deffnx procedure >= x1 x2 x3 @dots{}
@cindex ordering, of numbers
@cindex comparison, of numbers
@cindex equivalence predicate, for numbers
These procedures return @code{#t} if their arguments are (respectively):
equal, monotonically increasing, monotonically decreasing, monotonically
nondecreasing, or monotonically nonincreasing.

These predicates are transitive.  Note that the traditional
implementations of these predicates in Lisp-like languages are not
transitive.

Note: While it is not an error to compare inexact numbers using these
predicates, the results may be unreliable because a small inaccuracy may
affect the result; this is especially true of @code{=} and @code{zero?}.
When in doubt, consult a numerical analyst.
@end deffn

@deffn procedure zero? z
@deffnx procedure positive? x
@deffnx procedure negative? x
@deffnx procedure odd? x
@deffnx procedure even? x
@cindex zero
@cindex positive number
@cindex negative number
@cindex odd number
@cindex even number
These numerical predicates test a number for a particular property,
returning @code{#t} or @code{#f}.  See note above regarding inexact
numbers.
@end deffn

@deffn procedure max x1 x2 @dots{}
@deffnx procedure min x1 x2 @dots{}
@cindex minimum, of numbers
@cindex maximum, of numbers
These procedures return the maximum or minimum of their
arguments.

@example
@group
(max 3 4)              @result{}  4    @r{; exact}
(max 3.9 4)            @result{}  4.0  @r{; inexact}
@end group
@end example

Note: If any argument is inexact, then the result will also be inexact
(unless the procedure can prove that the inaccuracy is not large enough
to affect the result, which is possible only in unusual
implementations).  If @code{min} or @code{max} is used to compare
numbers of mixed exactness, and the numerical value of the result cannot
be represented as an inexact number without loss of accuracy, then the
procedure may report a violation of an implementation
restriction.@footnote{MIT/GNU Scheme signals an error of type
@code{condition-type:bad-range-argument} in this case.}
@findex condition-type:bad-range-argument
@end deffn

@deffn procedure + z1 @dots{}
@deffnx procedure * z1 @dots{}
@cindex addition, of numbers
@cindex sum, of numbers
@cindex identity, additive
@cindex multiplication, of numbers
@cindex product, of numbers
@cindex identity, multiplicative
These procedures return the sum or product of their arguments.

@example
@group
(+ 3 4)                 @result{}  7
(+ 3)                   @result{}  3
(+)                     @result{}  0
(* 4)                   @result{}  4
(*)                     @result{}  1
@end group
@end example
@end deffn

@deffn procedure - z1 z2 @dots{}
@deffnx procedure / z1 z2 @dots{}
@cindex subtraction, of numbers
@cindex difference, of numbers
@cindex inverse, additive, of number
@cindex division, of numbers
@cindex quotient, of numbers
@cindex inverse, multiplicative, of number
With two or more arguments, these procedures return the difference or
quotient of their arguments, associating to the left.  With one
argument, however, they return the additive or multiplicative inverse of
their argument.

@example
@group
(- 3 4)                 @result{}  -1
(- 3 4 5)               @result{}  -6
(- 3)                   @result{}  -3
(/ 3 4 5)               @result{}  3/20
(/ 3)                   @result{}  1/3
@end group
@end example
@end deffn

@deffn procedure 1+ z
@deffnx procedure -1+ z
@code{(1+ z)} is equivalent to @code{(+ z 1)}; @code{(-1+ z)} is
equivalent to @code{(- z 1)}.
@end deffn

@deffn procedure abs x
@cindex absolute value, of number
@cindex magnitude, of real number
@code{abs} returns the magnitude of its argument.

@example
(abs -7)                @result{}  7
@end example
@end deffn

@deffn procedure quotient n1 n2
@deffnx procedure remainder n1 n2
@deffnx procedure modulo n1 n2
@cindex division, of integers
@cindex quotient, of integers
@cindex remainder, of integers
@cindex modulus, of integers
@cindex integer division
These procedures implement number-theoretic (integer) division: for
positive integers @var{n1} and @var{n2}, if @var{n3} and @var{n4} are
integers such that
@tex
$$n_1=n_2n_3+n_4$$
$$0\leq n_4<n_2$$
@end tex
@ifinfo

@example
@group
@var{n1} = (@var{n2} * @var{n3}) + @var{n4}

0 <= @var{n4} < @var{n2}
@end group
@end example

@end ifinfo
@noindent
then

@example
@group
(quotient @var{n1} @var{n2})        @result{}  @var{n3}
(remainder @var{n1} @var{n2})       @result{}  @var{n4}
(modulo @var{n1} @var{n2})          @result{}  @var{n4}
@end group
@end example

@noindent
For integers @var{n1} and @var{n2} with @var{n2} not equal to 0,

@example
@group
(= @var{n1}
   (+ (* @var{n2} (quotient @var{n1} @var{n2}))
      (remainder @var{n1} @var{n2})))
                                    @result{}  #t
@end group
@end example

@noindent
provided all numbers involved in that computation are exact.

The value returned by @code{quotient} always has the sign of the product
of its arguments.  @code{remainder} and @code{modulo} differ on negative
arguments --- the @code{remainder} always has the sign of the dividend,
the @code{modulo} always has the sign of the divisor:

@example
@group
(modulo 13 4)           @result{}  1
(remainder 13 4)        @result{}  1

(modulo -13 4)          @result{}  3
(remainder -13 4)       @result{}  -1

(modulo 13 -4)          @result{}  -3
(remainder 13 -4)       @result{}  1

(modulo -13 -4)         @result{}  -1
(remainder -13 -4)      @result{}  -1

(remainder -13 -4.0)    @result{}  -1.0  @r{; inexact}
@end group
@end example

@findex integer-truncate
Note that @code{quotient} is the same as @code{integer-truncate}.
@end deffn

@deffn procedure integer-floor n1 n2
@deffnx procedure integer-ceiling n1 n2
@deffnx procedure integer-truncate n1 n2
@deffnx procedure integer-round n1 n2
These procedures combine integer division with rounding.  For example,
the following are equivalent:

@example
@group
(integer-floor @var{n1} @var{n2})
(floor (/ @var{n1} @var{n2}))
@end group
@end example

@noindent
However, the former is faster and does not produce an intermediate
result.

@findex quotient
Note that @code{integer-truncate} is the same as @code{quotient}.
@end deffn

@deffn procedure integer-divide n1 n2
@deffnx procedure integer-divide-quotient qr
@deffnx procedure integer-divide-remainder qr
@findex quotient
@findex remainder
@code{integer-divide} is equivalent to performing both @code{quotient}
and @code{remainder} at once.  The result of @code{integer-divide} is an
object with two components; the procedures
@code{integer-divide-quotient} and @code{integer-divide-remainder}
select those components.  These procedures are useful when both the
quotient and remainder are needed; often computing both of these numbers
simultaneously is much faster than computing them separately.

For example, the following are equivalent:

@example
@group
(lambda (n d)
  (cons (quotient n d)
        (remainder n d)))

(lambda (n d)
  (let ((qr (integer-divide n d)))
    (cons (integer-divide-quotient qr)
          (integer-divide-remainder qr))))
@end group
@end example
@end deffn

@deffn procedure gcd n1 @dots{}
@deffnx procedure lcm n1 @dots{}
@cindex greatest common divisor, of numbers
@cindex least common multiple, of numbers
These procedures return the greatest common divisor or least common
multiple of their arguments.  The result is always non-negative.

@example
@group
(gcd 32 -36)            @result{}  4
(gcd)                   @result{}  0

(lcm 32 -36)            @result{}  288
(lcm 32.0 -36)          @result{}  288.0  @r{; inexact}
(lcm)                   @result{}  1
@end group
@end example
@end deffn

@deffn procedure numerator q
@deffnx procedure denominator q
These procedures return the numerator or denominator of their argument;
the result is computed as if the argument was represented as a fraction
in lowest terms.  The denominator is always positive.  The denominator
of 0 is defined to be 1.

@example
@group
(numerator (/ 6 4))  @result{}  3
(denominator (/ 6 4))  @result{}  2
(denominator (inexact (/ 6 4))) @result{} 2.0
@end group
@end example
@end deffn

@deffn procedure floor x
@deffnx procedure ceiling x
@deffnx procedure truncate x
@deffnx procedure round x
These procedures return integers.  @code{floor} returns the largest
integer not larger than @var{x}.  @code{ceiling} returns the smallest
integer not smaller than @var{x}.  @code{truncate} returns the integer
closest to @var{x} whose absolute value is not larger than the absolute
value of @var{x}.  @code{round} returns the closest integer to @var{x},
rounding to even when @var{x} is halfway between two integers.

Rationale: @code{round} rounds to even for consistency with the rounding
modes required by the @sc{ieee} floating point standard.

Note: If the argument to one of these procedures is inexact, then the
result will also be inexact.  If an exact value is needed, the result
should be passed to the @code{exact} procedure (or use one of the
procedures below).

@example
@group
(floor -4.3)          @result{}  -5.0
(ceiling -4.3)        @result{}  -4.0
(truncate -4.3)       @result{}  -4.0
(round -4.3)          @result{}  -4.0

(floor 3.5)           @result{}  3.0
(ceiling 3.5)         @result{}  4.0
(truncate 3.5)        @result{}  3.0
(round 3.5)           @result{}  4.0  @r{; inexact}

(round 7/2)           @result{}  4    @r{; exact}
(round 7)             @result{}  7
@end group
@end example
@end deffn

@deffn procedure floor->exact x
@deffnx procedure ceiling->exact x
@deffnx procedure truncate->exact x
@deffnx procedure round->exact x
These procedures are similar to the preceding procedures except that
they always return an exact result.  For example, the following are
equivalent

@example
@group
(floor->exact x)
(exact (floor x))
@end group
@end example

@noindent
except that the former is faster and has fewer range restrictions.
@end deffn

@deffn procedure rationalize x y
@deffnx procedure rationalize->exact x y
@cindex simplest rational (defn)
@cindex rational, simplest (defn)
@code{rationalize} returns the @emph{simplest} rational number differing
from @var{x} by no more than @var{y}.  A rational number @var{r1} is
@emph{simpler} than another rational number @var{r2} if
@t{@var{r1}=@var{p1}/@var{q1}} and @t{@var{r2}=@var{p2}/@var{q2}} (both
in lowest terms) and @t{|@var{p1}|<=|@var{p2}|} and
@t{|@var{q1}|<=|@var{q2}|}.  Thus @t{3/5} is simpler than @t{4/7}.
Although not all rationals are comparable in this ordering (consider
@t{2/7} and @t{3/5}) any interval contains a rational number that is
simpler than every other rational number in that interval (the simpler
@t{2/5} lies between @t{2/7} and @t{3/5}).  Note that @t{0=0/1} is the
simplest rational of all.

@example
@group
(rationalize (exact .3) 1/10)  @result{} 1/3    @r{; exact}
(rationalize .3 1/10)          @result{} #i1/3  @r{; inexact}
@end group
@end example

@code{rationalize->exact} is similar to @code{rationalize} except that
it always returns an exact result.
@end deffn

@deffn procedure simplest-rational x y
@deffnx procedure simplest-exact-rational x y
@code{simplest-rational} returns the simplest rational number between
@var{x} and @var{y} inclusive; @code{simplest-exact-rational} is similar
except that it always returns an exact result.

These procedures implement the same functionality as @code{rationalize}
and @code{rationalize->exact}, except that they specify the input range
by its endpoints; @code{rationalize} specifies the range by its center
point and its (half-) width.
@end deffn

@deffn procedure exp z
@deffnx procedure log z
@deffnx procedure sin z
@deffnx procedure cos z
@deffnx procedure tan z
@deffnx procedure asin z
@deffnx procedure acos z
@deffnx procedure atan z
@deffnx procedure atan y x
@findex angle
@findex make-rectangular
These procedures compute the usual transcendental functions.  @code{log}
computes the natural logarithm of @var{z} (not the base ten logarithm).
@code{asin}, @code{acos}, and @code{atan} compute arcsine, arccosine,
and arctangent, respectively.  The two-argument variant of @code{atan}
computes @code{(angle (make-rectangular @var{x} @var{y}))} (see
below).

In general, the mathematical functions log, arcsine, arccosine, and
arctangent are multiply defined.  For nonzero real @var{x}, the value of
log @var{x} is defined to be the one whose imaginary part lies in the
range minus @var{pi} (exclusive) to @var{pi} (inclusive).  log 0 is
undefined.  The value of log @var{z} when @var{z} is complex is defined
according to the formula
@tex
$$\log z = \log {\rm magnitude}(z) + i {\rm angle} (z)$$
@end tex
@ifinfo

@example
log @var{z} = log magnitude(@var{z}) + i angle(@var{z})
@end example

@end ifinfo
With log defined this way, the values of arcsine, arccosine, and
arctangent are according to the following formulae:
@tex
$$\sin^{-1} z = -i \log (i z + \sqrt{1 - z^2})$$
$$\cos^{-1} z = \pi / 2 - \sin^{-1} z$$
$$\tan^{-1} z = (\log (1 + i z) - \log (1 - i z)) / (2 i)$$
@end tex
@ifinfo

@example
@group
arcsin(@var{z}) = -i log(i @var{z} + sqrt(1 - @var{z}^2))
arccos(@var{z}) = pi/2 - arcsin(@var{z})
arctan(@var{z}) = (log(1 + i @var{z}) + log(1 - i @var{z})) / (2 i)
@end group
@end example

@end ifinfo
The above specification follows @cite{Common Lisp: the Language}, which
in turn cites @cite{Principal Values and Branch Cuts in Complex APL};
refer to these sources for more detailed discussion of branch cuts,
boundary conditions, and implementation of these functions.  When it is
possible these procedures produce a real result from a real
argument.
@end deffn

@deffn procedure sqrt z
Returns the principal square root of @var{z}.  The result will have
either positive real part, or zero real part and non-negative imaginary
part.
@end deffn

@deffn procedure expt z1 z2
Returns @var{z1} raised to the power @var{z2}:
@tex
$${z_1}^{z_2} = e^{z_2 \log {z_1}}$$
$0^0$ is defined to be equal to 1.
@end tex
@ifinfo

@example
@var{z1}^@var{z2} = e^(@var{z2} log @var{z1})
@end example

@noindent
0^0 is defined to be equal to 1.
@end ifinfo
@end deffn

@deffn procedure make-rectangular x1 x2
@deffnx procedure make-polar x3 x4
@deffnx procedure real-part z
@deffnx procedure imag-part z
@deffnx procedure magnitude z
@deffnx procedure angle z
@deffnx procedure conjugate z
Suppose @var{x1}, @var{x2}, @var{x3}, and @var{x4} are real numbers and
@var{z} is a complex number such that
@tex
 $$ z = x_1 + x_2\hbox{$i$}
 = x_3 \cdot e^{{\displaystyle{\hbox{$i$}} x_4}}$$
@end tex
@ifinfo

@example
@var{z} = @var{x1} + @var{x2} i = @var{x3} e^(i @var{x4})
@end example

@end ifinfo
@noindent
Then @code{make-rectangular} and @code{make-polar} return @var{z},
@code{real-part} returns @var{x1}, @code{imag-part} returns @var{x2},
@code{magnitude} returns @var{x3}, and @code{angle} returns @var{x4}.
In the case of @code{angle}, whose value is not uniquely determined by
the preceding rule, the value returned will be the one in the range
minus @var{pi} (exclusive) to @var{pi} (inclusive).

@code{conjugate} returns the complex conjugate of @var{z}.
@end deffn

The procedures @code{exact} and @code{inexact} implement the natural
one-to-one correspondence between exact and inexact integers
throughout an implementation-dependent range.

@deffn procedure inexact z
@deffnx procedure exact->inexact z
@code{inexact} returns an inexact representation of @var{z}.  The
value returned is the inexact number that is numerically closest to the
argument.  If an exact argument has no reasonably close inexact
equivalent, then a violation of an implementation restriction may be
reported; MIT/GNU Scheme signals an error of type
@code{condition-type:bad-range-argument} in this case.
@findex condition-type:bad-range-argument

The procedure @code{exact->inexact} has been deprecated by @urseven{}.
@end deffn

@deffn procedure inexact z
@deffnx procedure exact->inexact z
@code{inexact} returns an inexact representation of @var{z}.  The
value returned is the inexact number that is numerically closest to the
argument.  If an exact argument has no reasonably close inexact
equivalent, then a violation of an implementation restriction may be
reported; MIT/GNU Scheme signals an error of type
@code{condition-type:bad-range-argument} in this case.
@findex condition-type:bad-range-argument

@code{exact} returns an exact representation of @var{z}.  The value
returned is the exact number that is numerically closest to the
argument.  If an inexact argument has no reasonably close exact
equivalent, then a violation of an implementation restriction may be
reported; in MIT/GNU Scheme this case does not occur because all
inexact numbers are representable as exact numbers.

The procedure @code{inexact->exact} has been deprecated by @urseven{}.
@end deffn

@node Numerical input and output, Fixnum and Flonum Operations, Numerical operations, Numbers
@section Numerical input and output
@cindex numerical input and output

@deffn procedure number->string number [radix]
@var{Radix} must be an exact integer, either 2, 8, 10, or 16.  If
omitted, @var{radix} defaults to 10.  The procedure
@code{number->string} takes a number and a radix and returns as a string
an external representation of the given number in the given radix such
that

@example
@group
(let ((number @var{number})
      (radix @var{radix}))
  (eqv? number
        (string->number (number->string number radix)
                        radix)))
@end group
@end example

@noindent
is true.  It is an error if no possible result makes this expression
true.

If @var{number} is inexact, the radix is 10, and the above expression
can be satisfied by a result that contains a decimal point, then the
result contains a decimal point and is expressed using the minimum
number of digits (exclusive of exponent and trailing zeroes) needed to
make the above expression true; otherwise the format of the result is
unspecified.

The result returned by @code{number->string} never contains an explicit
radix prefix.

Note: The error case can occur only when @var{number} is not a complex
number or is a complex number with an non-rational real or imaginary
part.

Rationale: If @var{number} is an inexact number represented using
flonums, and the radix is 10, then the above expression is normally
satisfied by a result containing a decimal point.  The unspecified case
allows for infinities, NaNs, and non-flonum representations.
@end deffn

@defvr variable flonum-parser-fast?
This variable controls the behavior of @code{string->number} when
parsing inexact numbers.  Specifically, it allows the user to trade off
accuracy against speed.

When set to its default value, @code{#f}, the parser provides maximal
accuracy, as required by the Scheme standard.  If set to @code{#t}, the
parser uses faster algorithms that will sometimes introduce small errors
in the result.  The errors affect a few of the least-significant bits of
the result, and consequently can be tolerated by many applications.
@end defvr

@defvr variable flonum-unparser-cutoff
This variable is @strong{deprecated}; use
@code{param:flonum-printer-cutoff} instead.
@end defvr

@defvr parameter param:flonum-printer-cutoff
This parameter controls the action of @code{number->string} when
@var{number} is a flonum (and consequently controls all printing of
flonums).  This parameter may be called with an argument to set its
value.

The value of this parameter is normally a list of three items:

@table @var
@item rounding-type
One of the following symbols: @code{normal}, @code{relative}, or
@code{absolute}.  The symbol @code{normal} means that the number should
be printed with full precision.  The symbol @code{relative} means that
the number should be rounded to a specific number of digits.  The symbol
@code{absolute} means that the number should be rounded so that there
are a specific number of digits to the right of the decimal point.

@item precision
An exact integer.  If @var{rounding-type} is @code{normal},
@var{precision} is ignored.  If @var{rounding-type} is @code{relative},
@var{precision} must be positive, and it specifies the number of digits
to which the printed representation will be rounded.  If
@var{rounding-type} is @code{absolute}, the printed representation will
be rounded @var{precision} digits to the right of the decimal point; if
@var{precision} is negative, the representation is rounded @code{(-
@var{precision})} digits to the left of the decimal point.

@item format-type
One of the symbols: @code{normal}, @code{scientific}, or
@code{engineering}.  This specifies the format in which the number will
be printed.@*  @code{scientific} specifies that the number will be printed
using scientific notation: @code{@var{x}.@var{xxx}e@var{yyy}}.  In other
words, the number is printed as a significand between zero inclusive and
ten exclusive, and an exponent.  @code{engineering} is like
@code{scientific}, except that the exponent is always a power of three,
and the significand is constrained to be between zero inclusive and 1000
exclusive.  If @code{normal} is specified, the number will be printed in
positional notation if it is ``small enough'', otherwise it is printed
in scientific notation.  A number is ``small enough'' when the number of
digits that would be printed using positional notation does not exceed
the number of digits of precision in the underlying floating-point
number representation; @sc{ieee} double-precision floating-point numbers
have 17 digits of precision.
@end table

@noindent
This three-element list may be abbreviated in two ways.  First, the
symbol @code{normal} may be used, which is equivalent to the list
@code{(normal 0 normal)}.  Second, the third element of the list,
@var{format-type}, may be omitted, in which case it defaults to
@code{normal}.

@noindent
The default value for @code{param:flonum-printer-cutoff} is @code{normal}.
If it is bound to a value different from those described here,
@code{number->string} issues a warning and acts as though the value had
been @code{normal}.
@end defvr

@noindent
Some examples of @code{param:flonum-printer-cutoff}:

@example
(number->string (* 4 (atan 1 1)))
                                    @result{} "3.141592653589793"
(parameterize ((param:flonum-printer-cutoff '(relative 5)))
  (lambda ()
    (number->string (* 4 (atan 1 1)))))
                                    @result{} "3.1416"
(parameterize ((param:flonum-printer-cutoff '(relative 5)))
  (lambda ()
    (number->string (* 4000 (atan 1 1)))))
                                    @result{} "3141.6"
(parameterize ((param:flonum-printer-cutoff '(relative 5 scientific)))
  (lambda ()
    (number->string (* 4000 (atan 1 1)))))
                                    @result{} "3.1416e3"
(parameterize ((param:flonum-printer-cutoff '(relative 5 scientific)))
  (lambda ()
    (number->string (* 40000 (atan 1 1)))))
                                    @result{} "3.1416e4"
(parameterize ((param:flonum-printer-cutoff '(relative 5 engineering)))
  (lambda ()
    (number->string (* 40000 (atan 1 1)))))
                                    @result{} "31.416e3"
(parameterize ((param:flonum-printer-cutoff '(absolute 5)))
  (lambda ()
    (number->string (* 4 (atan 1 1)))))
                                    @result{} "3.14159"
(parameterize ((param:flonum-printer-cutoff '(absolute 5)))
  (lambda ()
    (number->string (* 4000 (atan 1 1)))))
                                    @result{} "3141.59265"
(parameterize ((param:flonum-printer-cutoff '(absolute -4)))
  (lambda ()
    (number->string (* 4e10 (atan 1 1)))))
                                    @result{} "31415930000."
(parameterize ((param:flonum-printer-cutoff '(absolute -4 scientific)))
  (lambda ()
    (number->string (* 4e10 (atan 1 1)))))
                                    @result{} "3.141593e10"
(parameterize ((param:flonum-printer-cutoff '(absolute -4 engineering)))
  (lambda ()
    (number->string (* 4e10 (atan 1 1)))))
                                    @result{} "31.41593e9"
(parameterize ((param:flonum-printer-cutoff '(absolute -5)))
  (lambda ()
    (number->string (* 4e10 (atan 1 1)))))
                                    @result{} "31415900000."
@end example

@deffn procedure string->number string [radix]
Returns a number of the maximally precise representation expressed by
the given @var{string}.  @var{Radix} must be an exact integer, either 2,
8, 10, or 16.  If supplied, @var{radix} is a default radix that may be
overridden by an explicit radix prefix in @var{string} (e.g.@:
@code{"#o177"}).  If @var{radix} is not supplied, then the default radix
is 10.  If @var{string} is not a syntactically valid notation for a
number, then @code{string->number} returns @code{#f}.

@example
@group
(string->number "100")        @result{}  100
(string->number "100" 16)     @result{}  256
(string->number "1e2")        @result{}  100.0
(string->number "15##")       @result{}  1500.0
@end group
@end example

@noindent
Note that a numeric representation using a decimal point or an exponent
marker is not recognized unless @var{radix} is @code{10}.
@end deffn

@node Fixnum and Flonum Operations, Random Numbers, Numerical input and output, Numbers
@section Fixnum and Flonum Operations

This section describes numerical operations that are restricted forms of
the operations described above.  These operations are useful because
they compile very efficiently.  However, care should be exercised: if
used improperly, these operations can return incorrect answers, or even
malformed objects that confuse the garbage collector.

@menu
* Fixnum Operations::
* Flonum Operations::
@end menu

@node Fixnum Operations, Flonum Operations, Fixnum and Flonum Operations, Fixnum and Flonum Operations
@subsection Fixnum Operations

@cindex fixnum (defn)
A @dfn{fixnum} is an exact integer that is small enough to fit in a
machine word.  In MIT/GNU Scheme, fixnums are typically 24 or 26 bits,
depending on the machine; it is reasonable to assume that fixnums are at
least 24 bits.  Fixnums are signed; they are encoded using 2's
complement.

All exact integers that are small enough to be encoded as fixnums are
always encoded as fixnums --- in other words, any exact integer that is
not a fixnum is too big to be encoded as such.  For this reason, small
constants such as @code{0} or @code{1} are guaranteed to be fixnums.

@deffn procedure fix:fixnum? object
@cindex type predicate, for fixnum
Returns @code{#t} if @var{object} is a fixnum; otherwise returns
@code{#f}.
@end deffn

Here is an expression that determines the largest fixnum:

@example
@group
(let loop ((n 1))
  (if (fix:fixnum? n)
      (loop (* n 2))
      (- n 1)))
@end group
@end example

@noindent
A similar expression determines the smallest fixnum.

@deffn procedure fix:= fixnum fixnum
@deffnx procedure fix:< fixnum fixnum
@deffnx procedure fix:> fixnum fixnum
@deffnx procedure fix:<= fixnum fixnum
@deffnx procedure fix:>= fixnum fixnum
@cindex equivalence predicate, for fixnums
These are the standard order and equality predicates on fixnums.  When
compiled, they do not check the types of their arguments.
@end deffn

@deffn procedure fix:zero? fixnum
@deffnx procedure fix:positive? fixnum
@deffnx procedure fix:negative? fixnum
These procedures compare their argument to zero.  When compiled, they do
not check the type of their argument.  The code produced by the
following expressions is identical:

@example
@group
(fix:zero? @var{fixnum})
(fix:= @var{fixnum} 0)
@end group
@end example

@noindent
Similarly, @code{fix:positive?} and @code{fix:negative?} produce code
identical to equivalent expressions using @code{fix:>} and @code{fix:<}.
@end deffn

@deffn procedure fix:+ fixnum fixnum
@deffnx procedure fix:- fixnum fixnum
@deffnx procedure fix:* fixnum fixnum
@deffnx procedure fix:quotient fixnum fixnum
@deffnx procedure fix:remainder fixnum fixnum
@deffnx procedure fix:gcd fixnum fixnum
@deffnx procedure fix:1+ fixnum
@deffnx procedure fix:-1+ fixnum
These procedures are the standard arithmetic operations on fixnums.
When compiled, they do not check the types of their arguments.
Furthermore, they do not check to see if the result can be encoded as a
fixnum.  If the result is too large to be encoded as a fixnum, a
malformed object is returned, with potentially disastrous effect on the
garbage collector.
@end deffn

@deffn procedure fix:divide fixnum fixnum
@findex integer-divide
@findex integer-divide-quotient
@findex integer-divide-remainder
This procedure is like @code{integer-divide}, except that its arguments
and its results must be fixnums.  It should be used in conjunction with
@code{integer-divide-quotient} and @code{integer-divide-remainder}.
@end deffn

@cindex logical operations, on fixnums
@cindex bitwise-logical operations, on fixnums
The following are @dfn{bitwise-logical} operations on fixnums.

@deffn procedure fix:not fixnum
This returns the bitwise-logical inverse of its argument.  When
compiled, it does not check the type of its argument.

@example
@group
(fix:not 0)                             @result{}  -1
(fix:not -1)                            @result{}  0
(fix:not 1)                             @result{}  -2
(fix:not -34)                           @result{}  33
@end group
@end example
@end deffn

@deffn procedure fix:and fixnum fixnum
This returns the bitwise-logical ``and'' of its arguments.  When
compiled, it does not check the types of its arguments.

@example
@group
(fix:and #x43 #x0f)                     @result{}  3
(fix:and #x43 #xf0)                     @result{}  #x40
@end group
@end example
@end deffn

@deffn procedure fix:andc fixnum fixnum
Returns the bitwise-logical ``and'' of the first argument with the
bitwise-logical inverse of the second argument.  When compiled, it does
not check the types of its arguments.

@example
@group
(fix:andc #x43 #x0f)                    @result{}  #x40
(fix:andc #x43 #xf0)                    @result{}  3
@end group
@end example
@end deffn

@deffn procedure fix:or fixnum fixnum
This returns the bitwise-logical ``inclusive or'' of its arguments.
When compiled, it does not check the types of its arguments.

@example
@group
(fix:or #x40 3)                         @result{} #x43
(fix:or #x41 3)                         @result{} #x43
@end group
@end example
@end deffn

@deffn procedure fix:xor fixnum fixnum
This returns the bitwise-logical ``exclusive or'' of its arguments.
When compiled, it does not check the types of its arguments.

@example
@group
(fix:xor #x40 3)                        @result{} #x43
(fix:xor #x41 3)                        @result{} #x42
@end group
@end example
@end deffn

@deffn procedure fix:lsh fixnum1 fixnum2
This procedure returns the result of logically shifting @var{fixnum1} by
@var{fixnum2} bits.  If @var{fixnum2} is positive, @var{fixnum1} is
shifted left; if negative, it is shifted right.  When compiled, it does
not check the types of its arguments, nor the validity of its result.

@example
@group
(fix:lsh 1 10)                          @result{}  #x400
(fix:lsh #x432 -10)                     @result{}  1
(fix:lsh -1 3)                          @result{}  -8
(fix:lsh -128 -4)                       @result{}  #x3FFFF8
@end group
@end example
@end deffn

@node Flonum Operations,  , Fixnum Operations, Fixnum and Flonum Operations
@subsection Flonum Operations

@cindex flonum (defn)
A @dfn{flonum} is an inexact real number that is implemented as a
floating-point number.  In MIT/GNU Scheme, all inexact real numbers are
flonums.  For this reason, constants such as @code{0.} and @code{2.3}
are guaranteed to be flonums.

@deffn procedure flo:flonum? object
@cindex type predicate, for flonum
Returns @code{#t} if @var{object} is a flonum; otherwise returns @code{#f}.
@end deffn

@deffn procedure flo:= flonum1 flonum2
@deffnx procedure flo:< flonum1 flonum2
@deffnx procedure flo:> flonum1 flonum2
@cindex equivalence predicate, for flonums
These procedures are the standard order and equality predicates on
flonums.  When compiled, they do not check the types of their arguments.
@end deffn

@deffn procedure flo:zero? flonum
@deffnx procedure flo:positive? flonum
@deffnx procedure flo:negative? flonum
Each of these procedures compares its argument to zero.  When compiled,
they do not check the type of their argument.
@end deffn

@deffn procedure flo:+ flonum1 flonum2
@deffnx procedure flo:- flonum1 flonum2
@deffnx procedure flo:* flonum1 flonum2
@deffnx procedure flo:/ flonum1 flonum2
These procedures are the standard arithmetic operations on flonums.
When compiled, they do not check the types of their arguments.
@end deffn

@deffn procedure flo:finite? flonum
@vindex +inf
@vindex -inf
@vindex NaN
@cindex positive infinity (@code{+inf})
@cindex negative infinity (@code{-inf})
@cindex not a number (@code{NaN})
The @acronym{IEEE} floating-point number specification supports three
special ``numbers'': positive infinity (@code{+inf}), negative infinity
(@code{-inf}), and not-a-number (@code{NaN}).  This predicate returns
@code{#f} if @var{flonum} is one of these objects, and @code{#t} if it
is any other floating-point number.
@end deffn

@deffn procedure flo:negate flonum
This procedure returns the negation of its argument.  When compiled, it
does not check the type of its argument.  Equivalent to @code{(flo:- 0.
@var{flonum})}.
@end deffn

@deffn procedure flo:abs flonum
@deffnx procedure flo:exp flonum
@deffnx procedure flo:log flonum
@deffnx procedure flo:sin flonum
@deffnx procedure flo:cos flonum
@deffnx procedure flo:tan flonum
@deffnx procedure flo:asin flonum
@deffnx procedure flo:acos flonum
@deffnx procedure flo:atan flonum
@deffnx procedure flo:sqrt flonum
@deffnx procedure flo:expt flonum1 flonum2
@deffnx procedure flo:floor flonum
@deffnx procedure flo:ceiling flonum
@deffnx procedure flo:truncate flonum
@deffnx procedure flo:round flonum
@deffnx procedure flo:floor->exact flonum
@deffnx procedure flo:ceiling->exact flonum
@deffnx procedure flo:truncate->exact flonum
@deffnx procedure flo:round->exact flonum
These procedures are flonum versions of the corresponding procedures.
When compiled, they do not check the types of their arguments.
@end deffn

@deffn procedure flo:atan2 flonum1 flonum2
@findex atan
This is the flonum version of @code{atan} with two arguments.  When
compiled, it does not check the types of its arguments.
@end deffn

@node Random Numbers,  , Fixnum and Flonum Operations, Numbers
@section Random Numbers
@cindex random number
@cindex pseudo-random number
@cindex number, pseudo-random

MIT/GNU Scheme provides a facility for generating pseudo-random numbers.
The current implementation is a ``subtract-with-carry'' random-number
generator, based on the algorithm from @cite{A New Class of Random
Number Generators}, George Marsaglia and Arif Zaman, @cite{The Annals of
Applied Probability}, Vol.@: 1, No.@: 3, 1991.  At the time it was
implemented, this was a good algorithm for general purposes, but the
state of the art in random-number generation is constantly changing.  If
necessary, the implementation will be updated to use a new algorithm
while retaining the same interface.

The interface described here is very similar to that of Common Lisp.

@deffn procedure random modulus [state]
@var{Modulus} must be a positive real number.  @code{random} returns a
pseudo-random number between zero (inclusive) and @var{modulus}
(exclusive).  The exactness of the returned number is the same as the
exactness of @var{modulus}.  Additionally, if @var{modulus} is an exact
integer, the returned number will be also.  Usually, @var{modulus} is
either an exact integer or an inexact real; the current implementation
has been tuned to make these two cases fast.

If @var{state} is given and not @code{#f}, it must be a random-state
object; otherwise, it defaults to the @code{default-random-source}.
This object is used to maintain the state of the pseudo-random-number
generator and is altered as a side effect of the @code{random}
procedure.

@example
@group
(random 1.0)    @result{} .32744744667719056
(random 1.0)    @result{} .01668326768172354
(random 10)     @result{} 3
(random 10)     @result{} 8
(random 100)    @result{} 38
(random 100)    @result{} 63
(random 100/3)  @result{} 130501475769920525/6755399441055744
(random 100/3)  @result{} 170571694016427575/13510798882111488
@end group
@end example
@end deffn

@deffn procedure flo:random-unit state
@var{State} must be a random-state object.  @code{flo:random-unit}
returns a pseudo-random number between zero inclusive and one exclusive;
the returned number is always a flonum and therefore an inexact real
number.  @code{flo:random-unit} is equivalent to @code{random} with a
@var{modulus} of @code{1.0}, except that it is faster.
@end deffn

The next three definitions concern random-state objects.  In addition to
these definitions, it is important to know that random-state objects are
specifically designed so that they can be saved to disk using the
@code{fasdump} procedure, and later restored using the @code{fasload}
procedure.  This allows a particular random-state object to be saved in
order to replay a particular pseudo-random sequence.

@defvr variable *random-state*
This variable is @strong{deprecated}; pass an explicit state instead.
@end defvr

@deffn procedure make-random-state [state]
This procedure returns a new random-state object, suitable for use as
as the @var{state} argument to @code{random}.  If @var{state} is not
given or @code{#f}, @code{make-random-state} returns a @emph{copy} of
@code{default-random-source}.  If @var{state} is a random-state
object, a copy of that object is returned.  If @var{state} is
@code{#t}, then a new random-state object is returned that has been
``randomly'' initialized by some means (such as by a time-of-day
clock).
@end deffn

@deffn procedure random-state? object
Returns @code{#t} if @var{object} is a random-state object, otherwise
returns @code{#f}.
@end deffn
