Next: Characters, Previous: Packages, Up: Top [Contents][Index]
| • Numeric Operations | ||
| • Implementation-Dependent Numeric Constants | ||
| • Rational Computations | ||
| • Floating-point Computations | ||
| • Complex Computations | ||
| • Interval Designators | ||
| • Random-State Operations |
Common Lisp provides a large variety of operations related to numbers. This section provides an overview of those operations by grouping them into categories that emphasize some of the relationships among them.
The next figure shows operators relating to arithmetic operations.
|
Figure 12.1: Operators relating to Arithmetic.
The next figure shows defined names relating to exponential, logarithmic, and trigonometric operations.
|
Figure 12.2: Defined names relating to Exponentials, Logarithms, and Trigonometry.
The next figure shows operators relating to numeric comparison and predication.
|
Figure 12.3: Operators for numeric comparison and predication.
The next figure shows defined names relating to numeric type manipulation and coercion.
|
Figure 12.4: Defined names relating to numeric type manipulation and coercion.
For functions that are mathematically associative (and possibly commutative), a conforming implementation may process the arguments in any manner consistent with associative (and possibly commutative) rearrangement. This does not affect the order in which the argument forms are evaluated; for a discussion of evaluation order, see Section 3.1.2.1.2.3 (Function Forms). What is unspecified is only the order in which the parameter values are processed. This implies that implementations may differ in which automatic coercions are applied; see Section 12.1.1.2 (Contagion in Numeric Operations).
A conforming program can control the order of processing explicitly by separating the operations into separate (possibly nested) function forms, or by writing explicit calls to functions that perform coercions.
Consider the following expression, in which we assume that 1.0 and
1.0e-15 both denote single floats:
(+ 1/3 2/3 1.0d0 1.0 1.0e-15)
One conforming implementation might
process the arguments from left to right,
first adding 1/3 and 2/3 to get 1,
then converting that to a double float
for combination with 1.0d0,
then successively converting and adding 1.0 and 1.0e-15.
Another conforming implementation might process the arguments from
right to left, first performing a single float addition of 1.0 and
1.0e-15 (perhaps losing accuracy in the process), then converting the sum to
a double float and adding 1.0d0, then converting 2/3 to a
double float and adding it, and then converting 1/3 and adding that.
A third conforming implementation might first scan all the arguments, process all the rationals first to keep that part of the computation exact, then find an argument of the largest floating-point format among all the arguments and add that, and then add in all other arguments, converting each in turn (all in a perhaps misguided attempt to make the computation as accurate as possible).
In any case, all three strategies are legitimate.
A conforming program could control the order by writing, for example,
(+ (+ 1/3 2/3) (+ 1.0d0 1.0e-15) 1.0)
For information about the contagion rules for implicit coercions of arguments in numeric operations, see Section 12.1.4.4 (Rule of Float Precision Contagion), Section 12.1.4.1 (Rule of Float and Rational Contagion), and Section 12.1.5.2 (Rule of Complex Contagion).
Logical operations require integers as arguments;
an error of type type-error should be signaled
if an argument is supplied that is not an integer.
Integer arguments to logical operations are treated as if
they were represented in two’s-complement notation.
The next figure shows defined names relating to logical operations on numbers.
|
Figure 12.5: Defined names relating to logical operations on numbers.
The byte-manipulation functions use objects
called byte specifiers to designate the size and position
of a specific byte within an integer.
The representation of a byte specifier is implementation-dependent;
it might or might not be a number.
The function byte will construct a byte specifier,
which various other byte-manipulation functions will accept.
The next figure shows defined names relating to manipulating bytes of numbers.
|
Figure 12.6: Defined names relating to byte manipulation.
Next: Rational Computations, Previous: Numeric Operations, Up: Number Concepts [Contents][Index]
The next figure shows defined names relating to implementation-dependent details about numbers.
|
Figure 12.7: Defined names relating to implementation-dependent details about numbers.
Next: Floating-point Computations, Previous: Implementation-Dependent Numeric Constants, Up: Number Concepts [Contents][Index]
The rules in this section apply to rational computations.
Rational computations cannot overflow in the usual sense (though there may not be enough storage to represent a result), since integers and ratios may in principle be of any magnitude.
If any computation produces a result that is a mathematical ratio of two integers such that the denominator evenly divides the numerator, then the result is converted to the equivalent integer.
If the denominator does not evenly divide the numerator, the canonical representation of a rational number is as the ratio that numerator and that denominator, where the greatest common divisor of the numerator and denominator is one, and where the denominator is positive and greater than one.
When used as input (in the default syntax),
the notation -0 always denotes the integer 0.
A conforming implementation must not have a
representation of “minus zero” for integers
that is distinct from its representation of zero for integers.
However, such a distinction is possible for floats;
see the type float.
When the arguments to an irrational mathematical function
are all rational and the true mathematical result is also (mathematically) rational, then unless otherwise noted an implementation is free to return either an accurate rational result or a single float approximation. If the arguments are all rational but the result cannot be expressed as a rational number, then a single float approximation is always returned.
If the arguments to
an irrational
mathematical function are all of type
(or rational (complex rational))
and the true mathematical result is
(mathematically) a complex number with rational real and imaginary
parts, then unless otherwise noted an implementation is free to return
either an accurate result of type (or rational (complex rational))
or
a single float
(permissible only if the imaginary part of the true mathematical
result is zero) or (complex single-float). If the arguments are
all of type (or rational (complex rational))
but the result cannot be
expressed as a rational or complex rational,
then the returned
value will be of type single-float
(permissible only if the imaginary
part of the true mathematical result is zero) or (complex single-float).
Float substitutability applies neither to the rational functions
+,
-,
*,
and /
nor to the related operators
1+,
1-,
incf,
decf,
and conjugate.
For rational functions,
if all arguments are rational,
then the result is rational;
if all arguments are of type (or rational (complex rational)),
then the result is of type (or rational (complex rational)).
|
Figure 12.8: Functions Affected by Rule of Float Substitutability
Next: Complex Computations, Previous: Rational Computations, Up: Number Concepts [Contents][Index]
The following rules apply to floating point computations.
When rationals and floats are combined by a numerical function,
the rational is first converted to a float of the same format.
For functions such as + that take more than two arguments,
it is permitted that part of the operation be carried out exactly using
rationals and the rest be done using floating-point arithmetic.
When rationals and floats are compared by a numerical function,
the function rational is effectively called to convert the float
to a rational and then an exact
comparison is performed. In the case of complex numbers,
the real and imaginary parts are effectively handled individually.
;;;; Combining rationals with floats. ;;; This example assumes an implementation in which ;;; (float-radix 0.5) is 2 (as in IEEE) or 16 (as in IBM/360), ;;; or else some other implementation in which 1/2 has an exact ;;; representation in floating point. (+ 1/2 0.5) → 1.0 (- 1/2 0.5d0) → 0.0d0 (+ 0.5 -0.5 1/2) → 0.5 ;;;; Comparing rationals with floats. ;;; This example assumes an implementation in which the default float ;;; format is IEEE single-float, IEEE double-float, or some other format ;;; in which 5/7 is rounded upwards by FLOAT. (< 5/7 (float 5/7)) → true (< 5/7 (rational (float 5/7))) → true (< (float 5/7) (float 5/7)) → false
Computations with floats are only approximate, although they are described as if the results were mathematically accurate. Two mathematically identical expressions may be computationally different because of errors inherent in the floating-point approximation process. The precision of a float is not necessarily correlated with the accuracy of that number. For instance, 3.142857142857142857 is a more precise approximation to π than 3.14159, but the latter is more accurate. The precision refers to the number of bits retained in the representation. When an operation combines a short float with a long float, the result will be a long float. Common Lisp functions assume that the accuracy of arguments to them does not exceed their precision. Therefore when two small floats are combined, the result is a small float. Common Lisp functions never convert automatically from a larger size to a smaller one.
An error of type floating-point-overflow
or floating-point-underflow should be signaled if a
floating-point computation causes exponent overflow or underflow, respectively.
The result of a numerical function is a float of the largest format among all the floating-point arguments to the function.
Next: Interval Designators, Previous: Floating-point Computations, Up: Number Concepts [Contents][Index]
The following rules apply to complex computations:
Except during the execution of irrational and transcendental functions, no numerical function ever yields a complex unless one or more of its arguments is a complex.
When a
real
and
a complex are both part of a computation,
the
real
is first converted to a complex by providing an imaginary part of 0.
If the result of any computation would be a complex
number whose real part is of type rational and whose imaginary
part is zero, the result is converted to the rational
which is the real part.
This rule does not apply to complex numbers whose parts
are floats.
For example, #C(5 0) and 5 are not different objects in Common Lisp
(they are always the same under eql);
#C(5.0 0.0) and 5.0 are always different objects in Common Lisp
(they are never the same under eql,
although they are the same under equalp and =).
#c(1.0 1.0) → #C(1.0 1.0) #c(0.0 0.0) → #C(0.0 0.0) #c(1.0 1) → #C(1.0 1.0) #c(0.0 0) → #C(0.0 0.0) #c(1 1) → #C(1 1) #c(0 0) → 0 (typep #c(1 1) '(complex (eql 1))) → true (typep #c(0 0) '(complex (eql 0))) → false
Many of the irrational and transcendental functions are multiply defined in the complex domain; for example, there are in general an infinite number of complex values for the logarithm function. In each such case, a principal value must be chosen for the function to return. In general, such values cannot be chosen so as to make the range continuous; lines in the domain called branch cuts must be defined, which in turn define the discontinuities in the range. Common Lisp defines the branch cuts, principal values, and boundary conditions for the complex functions following “Principal Values and Branch Cuts in Complex APL.” The branch cut rules that apply to each function are located with the description of that function.
The next figure lists the identities that are obeyed throughout the applicable portion of the complex domain, even on the branch cuts:
|
Figure 12.9: Trigonometric Identities for Complex Domain
The quadrant numbers referred to in the discussions of branch cuts are as illustrated in the next figure.
Positive Imaginary Axis : II : I : Negative Real Axis ....................... Positive Real Axis : III : IV : Negative Imaginary Axis
Figure 12.10: Quadrant Numbering for Branch Cuts
Next: Random-State Operations, Previous: Complex Computations, Up: Number Concepts [Contents][Index]
The compound type specifier form of the numeric type specifiers permit the user to specify an interval on the real number line which describe a subtype of the type which would be described by the corresponding atomic type specifier. A subtype of some type T is specified using an ordered pair of objects called interval designators for type T.
The first of the two interval designators for type T can be any of the following:
This denotes a lower inclusive bound of N. That is, elements of the subtype of T will be greater than or equal to N.
This denotes a lower exclusive bound of M. That is, elements of the subtype of T will be greater than M.
This denotes the absence of a lower bound on the interval.
The second of the two interval designators for type T can be any of the following:
This denotes an upper inclusive bound of N. That is, elements of the subtype of T will be less than or equal to N.
This denotes an upper exclusive bound of M. That is, elements of the subtype of T will be less than M.
This denotes the absence of an upper bound on the interval.
Previous: Interval Designators, Up: Number Concepts [Contents][Index]
The next figure lists some defined names that are applicable to random states.
|
Figure 12.11: Random-state defined names
Next: complex (System Class), Previous: Number Concepts, Up: Numbers [Contents][Index]
number,
t
The type number contains objects which represent
mathematical numbers.
The types real and complex are disjoint
subtypes of number.
The function = tests for numerical equality.
The function eql, when its arguments are both numbers,
tests that they have both the same type and numerical value.
Two numbers that are the same under eql or =
are not necessarily the same under eq.
Common Lisp differs from mathematics on some naming issues. In mathematics,
the set of real numbers is traditionally described as a subset of the
complex numbers, but in Common Lisp, the type real and the type complex are
disjoint. The Common Lisp type which includes all mathematical complex
numbers is called number. The reasons for these differences
include historical precedent, compatibility with most other popular
computer languages, and various issues of time and space efficiency.
complex,
number,
t
The type complex includes all mathematical complex numbers
other than those included in the type rational.
Complexes are
expressed
in Cartesian form with a
real part and an imaginary part, each of which is a real.
The real part and imaginary part are either both
rational or both of the same float type.
The imaginary part can be a float zero, but can never
be a rational zero, for such a number is always represented
by Common Lisp as a rational rather than a complex.
Specializing.
(complex [typespec | *])
typespec—a type specifier that denotes a subtype of type real.
Every element of this type is a complex whose
real part and imaginary part are each of type
(upgraded-complex-part-type typespec).
This type encompasses those complexes
that can result by giving numbers of type typespec
to complex.
(complex type-specifier)
refers to all complexes that can result from giving
numbers of type type-specifier to the function complex,
plus all other complexes of the same specialized representation.
Section 12.1.5.3 (Rule of Canonical Representation for Complex Rationals), Section 2.3.2 (Constructing Numbers from Tokens), Section 22.1.3.1.4 (Printing Complexes)
The input syntax for a complex with real part r and
imaginary part i is #C(r i).
For further details, see Section 2.4 (Standard Macro Characters).
For every float, n, there is a complex
which represents the same mathematical number
and which can be obtained by (COERCE n 'COMPLEX).
Next: float (System Class), Previous: complex (System Class), Up: Numbers [Contents][Index]
real,
number,
t
The type real includes all numbers that
represent mathematical real numbers, though there are mathematical real
numbers (e.g., irrational numbers) that do not have an exact representation
in Common Lisp. Only reals can be ordered using the
<, >, <=, and >= functions.
The types rational and float are disjoint
subtypes of type real.
Abbreviating.
(real [lower-limit [upper-limit]])
lower-limit, upper-limit—interval designators
for type real.
The defaults for each of lower-limit and upper-limit is the symbol *.
This denotes the reals on the interval described by lower-limit and upper-limit.
Next: short-float; single-float; double-float; long-float, Previous: real, Up: Numbers [Contents][Index]
float,
real,
number,
t
A float
is a mathematical rational (but not a Common Lisp rational)
of the form
s· f· b^e-p,
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 float;
f is a positive integer
between b^p-1 and
b^p-1 (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 float
within that implementation. In addition, there is a floating-point zero;
depending on the implementation, there can 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.
(= 0.0 -0.0) is always true.
If there is a minus zero, (eql -0.0 0.0) is false,
otherwise it is true.
The types short-float, single-float, double-float,
and long-float are subtypes of type float. Any two of them must be
either disjoint types or the same type;
if the same type, then any other types between them in the
above ordering must also be the same type. For example,
if the type single-float and the type long-float are the same type,
then the type double-float must be the same type also.
Abbreviating.
(float [lower-limit [upper-limit]])
lower-limit, upper-limit—interval designators
for type float.
The defaults for each of lower-limit and upper-limit is the symbol *.
This denotes the floats on the interval described by lower-limit and upper-limit.
Figure 2.9, Section 2.3.2 (Constructing Numbers from Tokens), Section 22.1.3.1.3 (Printing Floats)
Note that all mathematical integers are representable not only as
Common Lisp reals, but also as complex floats. For example,
possible representations of the mathematical number 1
include the integer 1,
the float 1.0,
or the complex #C(1.0 0.0).
Next: rational (System Class), Previous: float (System Class), Up: Numbers [Contents][Index]
short-float:
short-float,
float,
real,
number,
t
single-float:
single-float,
float,
real,
number,
t
double-float:
double-float,
float,
real,
number,
t
long-float:
long-float,
float,
real,
number,
t
For the four defined subtypes of type float, it is true that
intermediate between the type short-float and the type long-float are
the type single-float and the type double-float.
The precise definition of these categories is
implementation-defined.
The precision (measured in “bits”, computed as plog2
and the exponent size (also measured in ``bits,'' computed as
log2
to be at least as great
as the values in the next figure.
Each of the defined subtypes of type float might or might not have a minus zero.
|
Figure 12.12: Recommended Minimum Floating-Point Precision and Exponent Size
single-float.
In this representation, an object is simultaneously of types
single-float, double-float, short-float,
and long-float.
single-float and
short-float. An object is simultaneously
of types single-float, double-float, and long-float.
single-float and
double-float. An object is simultaneously of types
single-float and short-float, or
double-float and long-float.
short-float,
single-float, and double-float.
An object can simultaneously be of type double-float
and long-float.
single-float, double-float,
and long-float. An object can simultaneously
be of types single-float and short-float.
short-float.
The defaults for each of lower-limit and upper-limit is the symbol *.
single-lower-limit, single-upper-limit---interval designators
for type single-float.
The defaults for each of lower-limit and upper-limit is the symbol *.
double-lower-limit, double-upper-limit---interval designators
for type double-float.
The defaults for each of lower-limit and upper-limit is the symbol *.
long-lower-limit, long-upper-limit---interval designators
for type long-float.
The defaults for each of lower-limit and upper-limit is the symbol *.
Next: ratio, Previous: short-float; single-float; double-float; long-float, Up: Numbers [Contents][Index]
rational,
real,
number,
t
The canonical representation of a rational is as an integer if its value is integral, and otherwise as a ratio.
The types integer and ratio
are disjoint subtypes of type rational.
Abbreviating.
(rational [lower-limit [upper-limit]])
lower-limit, upper-limit—interval designators
for type rational.
The defaults for each of lower-limit and upper-limit is the symbol *.
This denotes the rationals on the interval described by lower-limit and upper-limit.
Next: integer, Previous: rational (System Class), Up: Numbers [Contents][Index]
ratio,
rational,
real,
number,
t
A ratio is a number representing the mathematical ratio of two non-zero integers, the numerator and denominator, whose greatest common divisor is one, and of which the denominator is positive and greater than one.
Figure 2.9, Section 2.3.2 (Constructing Numbers from Tokens), Section 22.1.3.1.2 (Printing Ratios)
Next: signed-byte, Previous: ratio, Up: Numbers [Contents][Index]
integer,
rational,
real,
number,
t
An integer is a mathematical integer. There is no limit on the magnitude of an integer.
The types fixnum and bignum
form an exhaustive partition of type integer.
Abbreviating.
(integer [lower-limit [upper-limit]])
lower-limit, upper-limit—interval designators
for type integer.
The defaults for each of lower-limit and upper-limit is the symbol *.
This denotes the integers on the interval described by lower-limit and upper-limit.
Figure 2.9, Section 2.3.2 (Constructing Numbers from Tokens), Section 22.1.3.1.1 (Printing Integers)
The type (integer lower upper),
where lower and upper
are most-negative-fixnum and most-positive-fixnum, respectively,
is also called fixnum.
The type (integer 0 1) is also called bit.
The type (integer 0 *) is also called unsigned-byte.
Next: unsigned-byte, Previous: integer, Up: Numbers [Contents][Index]
signed-byte,
integer,
rational,
real,
number,
t
The atomic type specifier signed-byte denotes the same
type as is denoted by the type specifier integer;
however, the list forms of these two type specifiers have different semantics.
Abbreviating.
(signed-byte [s | *])
s—a positive integer.
This denotes the set of integers that can be represented
in two’s-complement form in a byte of s bits. This is
equivalent to (integer -2^s-1 2^s-1-1). The type
signed-byte or the type (signed-byte *) is the same
as the type integer.
Next: mod (Type Specifier), Previous: signed-byte, Up: Numbers [Contents][Index]
unsigned-byte,
signed-byte,
integer,
rational,
real,
number,
t
The atomic type specifier unsigned-byte denotes the same
type as is denoted by the type specifier (integer 0 *).
Abbreviating.
(unsigned-byte [s | *])
s—a positive integer.
This denotes the set of non-negative integers that can be
represented in a byte of size s (bits).
This is equivalent
to (mod m) for m=2^s, or
to (integer 0 n) for n=2^s-1.
The type unsigned-byte or
the type (unsigned-byte *) is the same as
the type (integer 0 *), the set of non-negative integers.
The type (unsigned-byte 1) is also called bit.
Next: bit (Type), Previous: unsigned-byte, Up: Numbers [Contents][Index]
Abbreviating.
(mod n)
n—a positive integer.
This denotes the set of non-negative integers less than n.
This is equivalent to
(integer 0 (n))
or to
(integer 0 m),
where m=n-1.
The argument is required, and cannot be *.
The symbol mod is not valid as a type specifier.
Next: fixnum, Previous: mod (Type Specifier), Up: Numbers [Contents][Index]
bit,
unsigned-byte,
signed-byte,
integer,
rational,
real,
number,
t
The type bit is equivalent to the type (integer 0 1)
and (unsigned-byte 1).
Next: bignum, Previous: bit (Type), Up: Numbers [Contents][Index]
fixnum,
integer,
rational,
real,
number,
t
A fixnum is an integer whose value is between
most-negative-fixnum and most-positive-fixnum inclusive.
Exactly which integers are fixnums is
implementation-defined.
The type fixnum is required to be a supertype of
(signed-byte 16).
Next: =; /=; <; >; <=; >=, Previous: fixnum, Up: Numbers [Contents][Index]
bignum,
integer,
rational,
real,
number,
t
The type bignum is defined to be exactly (and integer (not fixnum)).
number—for <, >, <=, >=: a real;
for =, /=: a number.
generalized-boolean—a generalized boolean.
=, /=, <, >, <=, and >=
perform arithmetic comparisons on their arguments as follows:
=The value of = is true if all numbers are the same in value;
otherwise it is false.
Two complexes are considered equal by =
if their real and imaginary parts are equal according to =.
/=The value of /= is true if no two numbers are the same in value;
otherwise it is false.
<The value of < is true if the numbers are in monotonically increasing order;
otherwise it is false.
>The value of > is true if the numbers are in monotonically decreasing order;
otherwise it is false.
<=The value of <= is true if the numbers are in monotonically
nondecreasing order;
otherwise it is false.
>=The value of >= is true if the numbers are in monotonically
nonincreasing order;
otherwise it is false.
=, /=, <, >, <=, and >=
perform necessary type conversions.
The uses of these functions are illustrated in the next figure.
|
Figure 12.13: Uses of /=, =, <, >, <=, and >=
Might signal type-error if some argument is not a real.
Might signal arithmetic-error if otherwise unable to fulfill its contract.
= differs from eql in that
(= 0.0 -0.0) is always true,
because = compares the mathematical values of its operands,
whereas eql compares the representational values, so to speak.
Next: minusp; plusp, Previous: =; /=; <; >; <=; >=, Up: Numbers [Contents][Index]
real—a real.
max-real, min-real—a real.
max returns the real that is greatest (closest to positive infinity).
min returns the real that is least (closest to negative infinity).
For max,
the implementation has the choice of returning the largest
argument as is or applying the rules of floating-point contagion,
taking all the arguments into consideration for contagion purposes.
Also, if one or more of the arguments are =, then any one
of them may be chosen as the value to return.
For example, if the reals are a mixture of rationals and floats,
and the largest argument is a rational,
then the implementation is free to
produce either that rational
or its float approximation;
if the largest argument is a
float of a smaller format
than the largest format of any float argument,
then the implementation is free to
return the argument in its given format or expanded to the larger format.
Similar remarks apply to min
(replacing “largest argument” by “smallest argument”).
(max 3) → 3 (min 3) → 3 (max 6 12) → 12 (min 6 12) → 6 (max -6 -12) → -6 (min -6 -12) → -12 (max 1 3 2 -7) → 3 (min 1 3 2 -7) → -7 (max -2 3 0 7) → 7 (min -2 3 0 7) → -2 (max 5.0 2) → 5.0 (min 5.0 2) → 2 or→ 2.0 (max 3.0 7 1) → 7 or→ 7.0 (min 3.0 7 1) → 1 or→ 1.0 (max 1.0s0 7.0d0) → 7.0d0 (min 1.0s0 7.0d0) → 1.0s0 or→ 1.0d0 (max 3 1 1.0s0 1.0d0) → 3 or→ 3.0d0 (min 3 1 1.0s0 1.0d0) → 1 or→ 1.0s0 or→ 1.0d0
Should signal an error of type type-error if number is not
a real.
real—a real.
generalized-boolean—a generalized boolean.
minusp returns true if real is less than zero;
otherwise, returns false.
plusp returns true if real is greater than zero;
otherwise, returns false.
Regardless of whether an implementation provides distinct
representations for positive and negative float zeros,
(minusp -0.0) always returns false.
(minusp -1) → true (plusp 0) → false (plusp least-positive-single-float) → true
Should signal an error of type type-error if real is not a real.
Next: floor; ffloor; ceiling; fceiling; truncate; ftruncate; round; fround, Previous: minusp; plusp, Up: Numbers [Contents][Index]
[ˈzē(ˌ)rō(ˌ)pē]
number—a number.
generalized-boolean—a generalized boolean.
Returns true if number is zero (integer, float, or complex); otherwise, returns false.
Regardless of whether an implementation provides distinct representations
for positive and negative floating-point zeros, (zerop -0.0)
always returns true.
(zerop 0) → true (zerop 1) → false (zerop -0.0) → true (zerop 0/100) → true (zerop #c(0 0.0)) → true
Should signal an error of type type-error if number is not a number.
(zerop number) ≡ (= number 0)
Next: sin; cos; tan, Previous: zerop, Up: Numbers [Contents][Index]
number—a real.
divisor—a non-zero real.
The default is the integer 1.
quotient—for floor, ceiling,
truncate, and round: an integer;
for ffloor, fceiling,
ftruncate, and fround: a float.
remainder—a real.
These functions divide number by divisor, returning a quotient and remainder, such that
quotient· divisor+remainder=number
The quotient always represents a mathematical integer. When more than one mathematical integer might be possible (i.e., when the remainder is not zero), the kind of rounding or truncation depends on the operator:
floor, ffloorfloor and ffloor produce a quotient
that has been truncated toward negative infinity;
that is, the quotient represents the largest mathematical integer
that is not larger than the mathematical quotient.
ceiling, fceilingceiling and fceiling produce a quotient
that has been truncated toward positive infinity;
that is, the quotient represents the smallest mathematical integer
that is not smaller than the mathematical result.
truncate, ftruncatetruncate and ftruncate produce a quotient
that has been truncated towards zero;
that is, the quotient represents the mathematical integer
of the same sign as the mathematical quotient, and
that has the greatest integral magnitude not greater than that of the mathematical quotient.
round, froundround and fround produce a quotient
that has been rounded to the nearest mathematical integer;
if the mathematical quotient is exactly halfway between two integers,
(that is, it has the form integer+1/2),
then the quotient has been rounded to the even (divisible by two) integer.
All of these functions perform type conversion operations on numbers.
The remainder
is an integer if both x and y are integers,
is a rational if both x and y are rationals, and
is a float if either x or y is a float.
ffloor, fceiling, ftruncate, and fround
handle arguments of different types in the following way:
If number is a float,
and divisor is not a float of longer format,
then the first result is a float of the same type as number.
Otherwise, the first result is of the type determined by contagion rules;
see Section 12.1.1.2 (Contagion in Numeric Operations).
(floor 3/2) → 1, 1/2 (ceiling 3 2) → 2, -1 (ffloor 3 2) → 1.0, 1 (ffloor -4.7) → -5.0, 0.3 (ffloor 3.5d0) → 3.0d0, 0.5d0 (fceiling 3/2) → 2.0, -1/2 (truncate 1) → 1, 0 (truncate .5) → 0, 0.5 (round .5) → 0, 0.5 (ftruncate -7 2) → -3.0, -1 (fround -7 2) → -4.0, 1 (dolist (n '(2.6 2.5 2.4 0.7 0.3 -0.3 -0.7 -2.4 -2.5 -2.6)) (format t "~&~4,1@F ~2,' D ~2,' D ~2,' D ~2,' D" n (floor n) (ceiling n) (truncate n) (round n))) ▷ +2.6 2 3 2 3 ▷ +2.5 2 3 2 2 ▷ +2.4 2 3 2 2 ▷ +0.7 0 1 0 1 ▷ +0.3 0 1 0 0 ▷ -0.3 -1 0 0 0 ▷ -0.7 -1 0 0 -1 ▷ -2.4 -3 -2 -2 -2 ▷ -2.5 -3 -2 -2 -2 ▷ -2.6 -3 -2 -2 -3 → NIL
When only number is given, the two results are exact; the mathematical sum of the two results is always equal to the mathematical value of number.
(function number divisor)
and (function (/ number divisor))
(where function is any of one
of floor, ceiling, ffloor,
fceiling, truncate,
round, ftruncate, and fround)
return the same first value, but
they return different remainders as the second value. For example:
(floor 5 2) → 2, 1 (floor (/ 5 2)) → 2, 1/2
If an effect is desired that is similar to round,
but that always rounds up or down (rather than toward the nearest even integer)
if the mathematical quotient is exactly halfway between two integers,
the programmer should consider a construction such as
(floor (+ x 1/2))
or (ceiling (- x 1/2)).
Next: asin; acos; atan, Previous: floor; ffloor; ceiling; fceiling; truncate; ftruncate; round; fround, Up: Numbers [Contents][Index]
radians—a number given in radians.
number—a number.
sin, cos, and tan
return the sine, cosine, and tangent, respectively, of radians.
(sin 0) → 0.0 (cos 0.7853982) → 0.707107 (tan #c(0 1)) → #C(0.0 0.761594)
Should signal an error of type type-error if radians is not a number.
Might signal arithmetic-error.
asin, acos, atan, Section 12.1.3.3 (Rule of Float Substitutability)
Next: pi, Previous: sin; cos; tan, Up: Numbers [Contents][Index]
number—a number.
number1—a number if number2 is not supplied, or a real if number2 is supplied.
number2—a real.
radians—a number (of radians).
asin, acos, and atan
compute the arc sine, arc cosine, and arc tangent respectively.
The arc sine, arc cosine, and arc tangent (with only number1 supplied) functions can be defined mathematically for number or number1 specified as x as in the next figure.
|
Figure 12.14: Mathematical definition of arc sine, arc cosine, and arc tangent
These formulae are mathematically correct, assuming completely accurate computation. They are not necessarily the simplest ones for real-valued computations.
If both number1 and number2 are supplied
for atan, the result is the arc tangent
of number1/number2.
The value of atan is always between
-π (exclusive) and π (inclusive)
when minus zero is not supported.
The range of the two-argument arc tangent when minus zero is supported
includes -π.
For a real number1, the result is a real and lies between -π/2 and π/2 (both exclusive). number1 can be a complex if number2 is not supplied. If both are supplied, number2 can be zero provided number1 is not zero.
The following definition for arc sine determines the range and branch cuts:
arcsin z = -i log (iz+sqrt(1-z^2))
The branch cut for the arc sine function is in two pieces: one along the negative real axis to the left of -1 (inclusive), continuous with quadrant II, and one along the positive real axis to the right of 1 (inclusive), continuous with quadrant IV. The range is that strip of the complex plane containing numbers whose real part is between -π/2 and π/2. A number with real part equal to -π/2 is in the range if and only if its imaginary part is non-negative; a number with real part equal to π/2 is in the range if and only if its imaginary part is non-positive.
The following definition for arc cosine determines the range and branch cuts:
arccos z = π/2- arcsin z
or, which are equivalent,
arccos z = -i log (z+i sqrt(1-z^2))
arccos z = 2 log (sqrt((1+z)/2) + i sqrt((1-z)/2))/(i)
The branch cut for the arc cosine function is in two pieces: one along the negative real axis to the left of -1 (inclusive), continuous with quadrant II, and one along the positive real axis to the right of 1 (inclusive), continuous with quadrant IV. This is the same branch cut as for arc sine. The range is that strip of the complex plane containing numbers whose real part is between 0 and π. A number with real part equal to 0 is in the range if and only if its imaginary part is non-negative; a number with real part equal to π is in the range if and only if its imaginary part is non-positive.
The following definition for (one-argument) arc tangent determines the range and branch cuts:
arctan z = (log (1+iz) - log (1-iz))/2i
Beware of simplifying this formula; “obvious” simplifications are likely to alter the branch cuts or the values on the branch cuts incorrectly. The branch cut for the arc tangent function is in two pieces: one along the positive imaginary axis above i (exclusive), continuous with quadrant II, and one along the negative imaginary axis below -i (exclusive), continuous with quadrant IV. The points i and -i are excluded from the domain. The range is that strip of the complex plane containing numbers whose real part is between -π/2 and π/2. A number with real part equal to -π/2 is in the range if and only if its imaginary part is strictly positive; a number with real part equal to π/2 is in the range if and only if its imaginary part is strictly negative. Thus the range of arc tangent is identical to that of arc sine with the points -π/2 and π/2 excluded.
For atan,
the signs of number1 (indicated as x)
and number2 (indicated as y) are used to derive quadrant
information. The next figure details various special cases.
The asterisk (*) indicates that the entry in the figure applies to
implementations that support minus zero.
|
Figure 12.15: Quadrant information for arc tangent
(asin 0) → 0.0 (acos #c(0 1)) → #C(1.5707963267948966 -0.8813735870195432) (/ (atan 1 (sqrt 3)) 6) → 0.087266 (atan #c(0 2)) → #C(-1.5707964 0.54930615)
acos and asin should signal an error of type type-error if number is not a number.
atan should signal type-error if
one argument is supplied and that argument is not a number,
or if two arguments are supplied and both of those arguments are not reals.
acos, asin, and atan might signal arithmetic-error.
log, sqrt, Section 12.1.3.3 (Rule of Float Substitutability)
The result of either asin or acos can be a complex
even if number is not a complex; this occurs when the
absolute value of number is greater than one.
Next: sinh; cosh; tanh; asinh; acosh; atanh, Previous: asin; acos; atan, Up: Numbers [Contents][Index]
an implementation-dependent long float.
The best long float approximation to the mathematical constant π.
;; In each of the following computations, the precision depends ;; on the implementation. Also, if `long float' is treated by ;; the implementation as equivalent to some other float format ;; (e.g., `double float') the exponent marker might be the marker ;; for that equivalent (e.g., `D' instead of `L'). pi → 3.141592653589793L0 (cos pi) → -1.0L0 (defun sin-of-degrees (degrees) (let ((x (if (floatp degrees) degrees (float degrees pi)))) (sin (* x (/ (float pi x) 180)))))
An approximation to π in some other precision can be obtained
by writing (float pi x), where x is a float of the
desired precision, or by writing (coerce pi type),
where type is the desired type, such as short-float.
Next: * (Function), Previous: pi, Up: Numbers [Contents][Index]
number—a number.
result—a number.
These functions compute the hyperbolic sine, cosine, tangent, arc sine, arc cosine, and arc tangent functions, which are mathematically defined for an argument x as given in the next figure.
|
Figure 12.16: Mathematical definitions for hyperbolic functions
The following definition for the inverse hyperbolic cosine determines the range and branch cuts:
arccosh z = 2 log (sqrt((z+1)/2) + sqrt((z-1)/2)).
The branch cut for the inverse hyperbolic cosine function lies along the real axis to the left of 1 (inclusive), extending indefinitely along the negative real axis, continuous with quadrant II and (between 0 and 1) with quadrant I. The range is that half-strip of the complex plane containing numbers whose real part is non-negative and whose imaginary part is between -π (exclusive) and π (inclusive). A number with real part zero is in the range if its imaginary part is between zero (inclusive) and π (inclusive).
The following definition for the inverse hyperbolic sine determines the range and branch cuts:
arcsinh z = log (z+sqrt(1+z^2)).
The branch cut for the inverse hyperbolic sine function is in two pieces: one along the positive imaginary axis above i (inclusive), continuous with quadrant I, and one along the negative imaginary axis below -i (inclusive), continuous with quadrant III. The range is that strip of the complex plane containing numbers whose imaginary part is between -π/2 and π/2. A number with imaginary part equal to -π/2 is in the range if and only if its real part is non-positive; a number with imaginary part equal to π/2 is in the range if and only if its imaginary part is non-negative.
The following definition for the inverse hyperbolic tangent determines the range and branch cuts:
arctanh z = (log (1+z) - log (1-z))/2.
Note that:
i arctan z = arctanh iz.
The branch cut for the inverse hyperbolic tangent function is in two pieces: one along the negative real axis to the left of -1 (inclusive), continuous with quadrant III, and one along the positive real axis to the right of 1 (inclusive), continuous with quadrant I. The points -1 and 1 are excluded from the domain. The range is that strip of the complex plane containing numbers whose imaginary part is between -π/2 and π/2. A number with imaginary part equal to -π/2 is in the range if and only if its real part is strictly negative; a number with imaginary part equal to π/2 is in the range if and only if its imaginary part is strictly positive. Thus the range of the inverse hyperbolic tangent function is identical to that of the inverse hyperbolic sine function with the points -π i/2 and π i/2 excluded.
(sinh 0) → 0.0 (cosh (complex 0 -1)) → #C(0.540302 -0.0)
Should signal an error of type type-error if number is not a number.
Might signal arithmetic-error.
log, sqrt, Section 12.1.3.3 (Rule of Float Substitutability)
The result of acosh may be a complex even if number
is not a complex; this occurs when number is less than one.
Also, the result of atanh may be a complex even if number
is not a complex; this occurs when the absolute value of number
is greater than one.
The branch cut formulae are mathematically correct, assuming completely accurate computation. Implementors should consult a good text on numerical analysis. The formulae given above are not necessarily the simplest ones for real-valued computations; they are chosen to define the branch cuts in desirable ways for the complex case.
Next: + (Function), Previous: sinh; cosh; tanh; asinh; acosh; atanh, Up: Numbers [Contents][Index]
number—a number.
product—a number.
Returns the product of numbers,
performing any necessary type conversions in the process.
If no numbers are supplied, 1 is returned.
(*) → 1 (* 3 5) → 15 (* 1.0 #c(22 33) 55/98) → #C(12.346938775510203 18.520408163265305)
Might signal type-error if some argument is not a number.
Might signal arithmetic-error.
Section 12.1.1 (Numeric Operations), Section 12.1.3 (Rational Computations), Section 12.1.4 (Floating-point Computations), Section 12.1.5 (Complex Computations)
Next: - (Function), Previous: * (Function), Up: Numbers [Contents][Index]
number—a number.
sum—a number.
Returns the sum of numbers,
performing any necessary type conversions in the process.
If no numbers are supplied, 0 is returned.
(+) → 0 (+ 1) → 1 (+ 31/100 69/100) → 1 (+ 1/5 0.8) → 1.0
Might signal type-error if some argument is not a number.
Might signal arithmetic-error.
Section 12.1.1 (Numeric Operations), Section 12.1.3 (Rational Computations), Section 12.1.4 (Floating-point Computations), Section 12.1.5 (Complex Computations)
Next: / (Function), Previous: + (Function), Up: Numbers [Contents][Index]
number, minuend, subtrahend—a number.
negation, difference—a number.
The function - performs arithmetic subtraction and negation.
If only one number is supplied, the negation of that number is returned.
If more than one argument is given, it subtracts all of the subtrahends from the minuend and returns the result.
The function - performs necessary type conversions.
(- 55.55) → -55.55 (- #c(3 -5)) → #C(-3 5) (- 0) → 0 (eql (- 0.0) -0.0) → true (- #c(100 45) #c(0 45)) → 100 (- 10 1 2 3 4) → 0
Might signal type-error if some argument is not a number.
Might signal arithmetic-error.
Section 12.1.1 (Numeric Operations), Section 12.1.3 (Rational Computations), Section 12.1.4 (Floating-point Computations), Section 12.1.5 (Complex Computations)
Next: 1+; 1-, Previous: - (Function), Up: Numbers [Contents][Index]
number, denominator—a non-zero number.
numerator, quotient, reciprocal—a number.
The function / performs division or reciprocation.
If no denominators are supplied,
the function / returns the reciprocal of number.
If at least one denominator is supplied,
the function / divides the numerator by all of the denominators
and returns the resulting quotient.
If each argument is either an integer or a ratio, and the result is not an integer, then it is a ratio.
The function / performs necessary type conversions.
If any argument is a float then the rules of floating-point contagion apply; see Section 12.1.4 (Floating-point Computations).
(/ 12 4) → 3 (/ 13 4) → 13/4 (/ -8) → -1/8 (/ 3 4 5) → 3/20 (/ 0.5) → 2.0 (/ 20 5) → 4 (/ 5 20) → 1/4 (/ 60 -2 3 5.0) → -2.0 (/ 2 #c(2 2)) → #C(1/2 -1/2)
The consequences are unspecified if any argument other than the first is zero. If there is only one argument, the consequences are unspecified if it is zero.
Might signal type-error if some argument is not a number.
Might signal division-by-zero if division by zero is attempted.
Might signal arithmetic-error.
floor, ceiling, truncate, round
Next: abs, Previous: / (Function), Up: Numbers [Contents][Index]
number—a number.
successor, predecessor—a number.
1+ returns a number that is one more than its argument number.
1- returns a number that is one less than its argument number.
(1+ 99) → 100 (1- 100) → 99 (1+ (complex 0.0)) → #C(1.0 0.0) (1- 5/3) → 2/3
Might signal type-error if its argument is not a number.
Might signal arithmetic-error.
(1+ number) ≡ (+ number 1) (1- number) ≡ (- number 1)
Implementors are encouraged to make the performance of both the previous expressions be the same.
Next: evenp; oddp, Previous: 1+; 1-, Up: Numbers [Contents][Index]
number—a number.
absolute-value—a non-negative real.
abs returns the absolute value of number.
If number is a real, the result is of the same type as number.
If number is a complex, the result is a positive real with the same magnitude as number. The result can be a float
even if number’s components are rationals
and an exact rational result
would have been possible.
Thus the result of (abs #c(3 4)) can be either 5 or 5.0,
depending on the implementation.
(abs 0) → 0 (abs 12/13) → 12/13 (abs -1.09) → 1.09 (abs #c(5.0 -5.0)) → 7.071068 (abs #c(5 5)) → 7.071068 (abs #c(3/5 4/5)) → 1 or approximately 1.0 (eql (abs -0.0) -0.0) → true
Section 12.1.3.3 (Rule of Float Substitutability)
If number is a complex, the result is equivalent to the following:
(sqrt (+ (expt (realpart number) 2) (expt (imagpart number) 2)))
An implementation should not use this formula directly for all complexes but should handle very large or very small components specially to avoid intermediate overflow or underflow.
integer—an integer.
generalized-boolean—a generalized boolean.
evenp returns true if integer is even (divisible by two);
otherwise, returns false.
oddp returns true if integer is odd (not divisible by two);
otherwise, returns false.
(evenp 0) → true (oddp 10000000000000000000000) → false (oddp -1) → true
Should signal an error of type type-error if integer is not an integer.
(evenp integer) ≡ (not (oddp integer)) (oddp integer) ≡ (not (evenp integer))
Next: gcd, Previous: evenp; oddp, Up: Numbers [Contents][Index]
number—a number.
base-number—a number.
power-number—a number.
result—a number.
exp and expt perform exponentiation.
exp returns e raised to the power number,
where e is the base of the natural logarithms.
exp has no branch cut.
expt returns base-number
raised to the power power-number.
If the base-number is a rational
and power-number is
an integer,
the calculation is exact and the result will be of type rational;
otherwise a floating-point approximation might result.
For expt of a complex rational to an integer power,
the calculation must be exact and the result is
of type (or rational (complex rational)).
The result of expt can be a complex,
even when neither argument is a complex,
if base-number is negative and power-number
is not an integer.
The result is always the principal complex value.
For example, (expt -8 1/3) is not permitted to return -2,
even though -2 is one of the cube roots of -8.
The principal cube root is a complex
approximately equal to #C(1.0 1.73205), not -2.
expt is defined
as b^x = e^x log b.
This defines the principal values precisely. The range of
expt is the entire complex plane. Regarded
as a function of x, with b fixed, there is no branch cut.
Regarded as a function of b, with x fixed, there is in general
a branch cut along the negative real axis, continuous with quadrant II.
The domain excludes the origin.
By definition, 0^0=1. If b=0 and the real part of x is strictly
positive, then
b^x=0. For all other values of x, 0^x
is an error.
When power-number is an integer 0,
then the result is always the value one in the type
of base-number,
even if the base-number is zero (of any type). That is:
(expt x 0) ≡ (coerce 1 (type-of x))
If power-number is a zero of any other type,
then the result is also the value one, in the type of the arguments
after the application of the contagion rules in Section 12.1.1.2 (Contagion in Numeric Operations),
with one exception:
the consequences are undefined if base-number is zero when power-number
is zero and not of type integer.
(exp 0) → 1.0 (exp 1) → 2.718282 (exp (log 5)) → 5.0 (expt 2 8) → 256 (expt 4 .5) → 2.0 (expt #c(0 1) 2) → -1 (expt #c(2 2) 3) → #C(-16 16) (expt #c(2 2) 4) → -64
log, Section 12.1.3.3 (Rule of Float Substitutability)
Implementations of expt are permitted to use different algorithms
for the cases of a power-number of type rational
and a power-number of type float.
Note that by the following logic,
(sqrt (expt x 3)) is not equivalent to
(expt x 3/2).
(setq x (exp (/ (* 2 pi #c(0 1)) 3))) ;exp(2.pi.i/3) (expt x 3) → 1 ;except for round-off error (sqrt (expt x 3)) → 1 ;except for round-off error (expt x 3/2) → -1 ;except for round-off error
Next: incf; decf, Previous: exp; expt, Up: Numbers [Contents][Index]
integer—an integer.
greatest-common-denominator—a non-negative integer.
Returns the greatest common divisor of integers.
If only one integer is supplied, its absolute value is returned.
If no integers are given, gcd returns 0,
which is an identity for this operation.
(gcd) → 0 (gcd 60 42) → 6 (gcd 3333 -33 101) → 1 (gcd 3333 -33 1002001) → 11 (gcd 91 -49) → 7 (gcd 63 -42 35) → 7 (gcd 5) → 5 (gcd -4) → 4
Should signal an error of type type-error if integer is not
an integer.
For three or more arguments,
(gcd b c ... z) ≡ (gcd (gcd a b) c ... z)
place—a place.
delta-form—a form; evaluated to produce a delta.
The default is 1.
delta—a number.
new-value—a number.
incf and decf are used for incrementing and
decrementing the value of place, respectively.
The delta is
added to (in the case of incf)
or subtracted from (in the case of decf)
the number in place and the result is stored in place.
Any necessary type conversions are performed automatically.
For information about the evaluation of subforms of places, see Section 5.1.1.1 (Evaluation of Subforms to Places).
(setq n 0) (incf n) → 1 n → 1 (decf n 3) → -2 n → -2 (decf n -5) → 3 (decf n) → 2 (incf n 0.5) → 2.5 (decf n) → 1.5 n → 1.5
Place is modified.
+ (Function), - (Function), 1+, 1- (Function), setf
Next: log, Previous: incf; decf, Up: Numbers [Contents][Index]
integer—an integer.
least-common-multiple—a non-negative integer.
lcm returns the least common multiple of the integers.
If no integer is supplied, the integer 1 is returned.
If only one integer is supplied, the absolute value of that integer is returned.
For two arguments that are not both zero,
(lcm a b) ≡ (/ (abs (* a b)) (gcd a b))
If one or both arguments are zero,
(lcm a 0) ≡ (lcm 0 a) ≡ 0
For three or more arguments,
(lcm a b c ... z) ≡ (lcm (lcm a b) c ... z)
(lcm 10) → 10 (lcm 25 30) → 150 (lcm -24 18 10) → 360 (lcm 14 35) → 70 (lcm 0 5) → 0 (lcm 1 2 3 4 5 6) → 60
Should signal type-error if any argument is not an integer.
number—a non-zero number.
base—a number.
logarithm—a number.
log returns the logarithm of number in base base.
If base is not supplied its value is e,
the base of the natural logarithms.
log may return a complex when given a
real
negative number.
(log -1.0) ≡ (complex 0.0 (float pi 0.0))
If base is zero,
log returns zero.
The result of (log 8 2) may be either 3 or 3.0, depending on the
implementation. An implementation can use floating-point calculations
even if an exact integer result is possible.
The branch cut for the logarithm function of one argument (natural logarithm) lies along the negative real axis, continuous with quadrant II. The domain excludes the origin.
The mathematical definition of a complex logarithm is as follows, whether or not minus zero is supported by the implementation:
(log x) ≡ (complex (log (abs x)) (phase x))
Therefore the range of the one-argument logarithm function is that strip of the complex plane containing numbers with imaginary parts between -π (exclusive) and π (inclusive) if minus zero is not supported, or -π (inclusive) and π (inclusive) if minus zero is supported.
The two-argument logarithm function is defined as
(log base number) ≡ (/ (log number) (log base))
This defines the principal values precisely. The range of the two-argument logarithm function is the entire complex plane.
(log 100 10) → 2.0 → 2 (log 100.0 10) → 2.0 (log #c(0 1) #c(0 -1)) → #C(-1.0 0.0) or→ #C(-1 0) (log 8.0 2) → 3.0
(log #c(-16 16) #c(2 2)) → 3 or approximately #c(3.0 0.0)
or approximately 3.0 (unlikely)
The implementation.
exp, expt, Section 12.1.3.3 (Rule of Float Substitutability)
number—a real.
divisor—a real.
modulus, remainder—a real.
mod and rem are generalizations of the modulus
and remainder functions respectively.
mod performs the operation floor
on number and divisor
and returns the remainder of the floor operation.
rem performs the operation truncate
on number and divisor
and returns the remainder of the truncate operation.
mod and rem are
the modulus and remainder functions
when number and divisor are integers.
(rem -1 5) → -1 (mod -1 5) → 4 (mod 13 4) → 1 (rem 13 4) → 1 (mod -13 4) → 3 (rem -13 4) → -1 (mod 13 -4) → -3 (rem 13 -4) → 1 (mod -13 -4) → -1 (rem -13 -4) → -1 (mod 13.4 1) → 0.4 (rem 13.4 1) → 0.4 (mod -13.4 1) → 0.6 (rem -13.4 1) → -0.4
The result of mod is either zero or a
real
with the same sign as divisor.
Next: sqrt; isqrt, Previous: mod; rem, Up: Numbers [Contents][Index]
number—a number.
signed-prototype—a number.
signum determines a numerical value that indicates whether
number is negative, zero, or positive.
For a rational,
signum returns one of -1, 0, or 1
according to whether number is negative, zero, or positive.
For a float,
the result is a float of the same format
whose value is minus one, zero, or one.
For a complex number z,
(signum z) is a complex number of the same phase but with unit magnitude,
unless z is a complex zero, in which case the result is z.
For rational arguments, signum is a rational function,
but it may be irrational for complex arguments.
If number is a float, the result is a float. If number is a rational, the result is a rational. If number is a complex float, the result is a complex float. If number is a complex rational, the result is a complex, but it is implementation-dependent whether that result is a complex rational or a complex float.
(signum 0) → 0 (signum 99) → 1 (signum 4/5) → 1 (signum -99/100) → -1 (signum 0.0) → 0.0 (signum #c(0 33)) → #C(0.0 1.0) (signum #c(7.5 10.0)) → #C(0.6 0.8) (signum #c(0.0 -14.7)) → #C(0.0 -1.0) (eql (signum -0.0) -0.0) → true
Section 12.1.3.3 (Rule of Float Substitutability)
(signum x) ≡ (if (zerop x) x (/ x (abs x)))
Next: random-state, Previous: signum, Up: Numbers [Contents][Index]
number, root—a number.
natural, natural-root—a non-negative integer.
sqrt and isqrt compute square roots.
sqrt returns the principal square root of number.
If the number is not a complex but is negative,
then the result is a complex.
isqrt returns the greatest integer
less than or equal to the exact positive square root of natural.
If number is a positive rational, it is implementation-dependent whether root is a rational or a float. If number is a negative rational, it is implementation-dependent whether root is a complex rational or a complex float.
The mathematical definition of complex square root (whether or not minus zero is supported) follows:
(sqrt x) = (exp (/ (log x) 2))
The branch cut for square root lies along the negative real axis, continuous with quadrant II. The range consists of the right half-plane, including the non-negative imaginary axis and excluding the negative imaginary axis.
(sqrt 9.0) → 3.0 (sqrt -9.0) → #C(0.0 3.0) (isqrt 9) → 3 (sqrt 12) → 3.4641016 (isqrt 12) → 3 (isqrt 300) → 17 (isqrt 325) → 18 (sqrt 25) → 5 or→ 5.0 (isqrt 25) → 5 (sqrt -1) → #C(0.0 1.0) (sqrt #c(0 2)) → #C(1.0 1.0)
The function sqrt should signal type-error if its argument
is not a number.
The function isqrt should signal type-error if its argument
is not a non-negative integer.
The functions sqrt and isqrt might signal arithmetic-error.
exp, log, Section 12.1.3.3 (Rule of Float Substitutability)
(isqrt x) ≡ (values (floor (sqrt x)))
but it is potentially more efficient.
Next: make-random-state, Previous: sqrt; isqrt, Up: Numbers [Contents][Index]
random-state,
t
A random state object contains state information used by the pseudo-random number generator. The nature of a random state object is implementation-dependent. It can be printed out and successfully read back in by the same implementation, but might not function correctly as a random state in another implementation.
Implementations are required to provide a read syntax for
objects of type random-state, but the specific nature of that syntax
is implementation-dependent.
*random-state*, random, Section 22.1.3.10 (Printing Random States)
Next: random, Previous: random-state, Up: Numbers [Contents][Index]
state—a random state, or nil, or t.
The default is nil.
new-state—a random state object.
Creates a fresh object of type random-state suitable
for use as the value of *random-state*.
If state is a random state object,
the new-state is a copy5
If state is nil,
the new-state is a copy5
If state is t,
the new-state is a fresh random state object
that has been randomly initialized by some means.
(let* ((rs1 (make-random-state nil))
(rs2 (make-random-state t))
(rs3 (make-random-state rs2))
(rs4 nil))
(list (loop for i from 1 to 10
collect (random 100)
when (= i 5)
do (setq rs4 (make-random-state)))
(loop for i from 1 to 10 collect (random 100 rs1))
(loop for i from 1 to 10 collect (random 100 rs2))
(loop for i from 1 to 10 collect (random 100 rs3))
(loop for i from 1 to 10 collect (random 100 rs4))))
→ ((29 25 72 57 55 68 24 35 54 65)
(29 25 72 57 55 68 24 35 54 65)
(93 85 53 99 58 62 2 23 23 59)
(93 85 53 99 58 62 2 23 23 59)
(68 24 35 54 65 54 55 50 59 49))
Should signal an error of type type-error if state is not a random state, or nil, or t.
One important use of make-random-state is to allow the same
series of pseudo-random numbers to be generated many times within a
single program.
Next: random-state-p, Previous: make-random-state, Up: Numbers [Contents][Index]
limit—a positive integer, or a positive float.
random-state—a random state. The default is the current random state.
random-number—a non-negative number less than limit and of the same type as limit.
Returns a pseudo-random number that is a non-negative number less than limit and of the same type as limit.
The random-state, which is modified by this function, encodes the internal state maintained by the random number generator.
An approximately uniform choice distribution is used. If limit is an integer, each of the possible results occurs with (approximate) probability 1/limit.
(<= 0 (random 1000) 1000) → true (let ((state1 (make-random-state)) (state2 (make-random-state))) (= (random 1000 state1) (random 1000 state2))) → true
The random-state is modified.
Should signal an error of type type-error if limit is not a positive integer or a positive real.
make-random-state, *random-state*
See Common Lisp: The Language for information about generating random numbers.
Next: *random-state*, Previous: random, Up: Numbers [Contents][Index]
object—an object.
generalized-boolean—a generalized boolean.
Returns true if object is of type random-state;
otherwise, returns false.
(random-state-p *random-state*) → true (random-state-p (make-random-state)) → true (random-state-p 'test-function) → false
make-random-state, *random-state*
(random-state-p object) ≡ (typep object 'random-state)
Next: numberp, Previous: random-state-p, Up: Numbers [Contents][Index]
a random state.
implementation-dependent.
The current random state, which is used, for example,
by the function random when a random state is not explicitly supplied.
(random-state-p *random-state*) → true (setq snap-shot (make-random-state)) ;; The series from any given point is random, ;; but if you backtrack to that point, you get the same series. (list (loop for i from 1 to 10 collect (random)) (let ((*random-state* snap-shot)) (loop for i from 1 to 10 collect (random))) (loop for i from 1 to 10 collect (random)) (let ((*random-state* snap-shot)) (loop for i from 1 to 10 collect (random)))) → ((19 16 44 19 96 15 76 96 13 61) (19 16 44 19 96 15 76 96 13 61) (16 67 0 43 70 79 58 5 63 50) (16 67 0 43 70 79 58 5 63 50))
The implementation.
random.
make-random-state, random, random-state
Binding *random-state* to a different
random state object correctly saves and
restores the old random state object.
Next: cis, Previous: *random-state*, Up: Numbers [Contents][Index]
object—an object.
generalized-boolean—a generalized boolean.
Returns true if object is of type number;
otherwise, returns false.
(numberp 12) → true (numberp (expt 2 130)) → true (numberp #c(5/3 7.2)) → true (numberp nil) → false (numberp (cons 1 2)) → false
(numberp object) ≡ (typep object 'number)
Next: complex (Function), Previous: numberp, Up: Numbers [Contents][Index]
radians—a real.
number—a complex.
cis returns the value of e^i· radians,
which is a complex in which the
real part is equal to the cosine of radians, and the
imaginary part is equal to the sine of radians.
(cis 0) → #C(1.0 0.0)
Section 12.1.3.3 (Rule of Float Substitutability)
realpart—a real.
imagpart—a real.
complex—a rational or a complex.
complex returns a number
whose real part is realpart
and whose imaginary part is imagpart.
If realpart is a rational
and imagpart is the rational number zero,
the result of complex is realpart, a rational.
Otherwise, the result is a complex.
If either realpart or imagpart is a float,
the non-float is converted to a float
before the complex is created.
If imagpart is not supplied, the imaginary part is a
zero of the same type as realpart; i.e.,
(coerce 0 (type-of realpart)) is
effectively used.
Type upgrading implies a movement upwards in the type hierarchy lattice. In the case of complexes, the type-specifier
must be a subtype of
(upgraded-complex-part-type type-specifier).
If type-specifier1 is a subtype of type-specifier2, then
(upgraded-complex-element-type 'type-specifier1)
must also be a subtype of
(upgraded-complex-element-type 'type-specifier2).
Two disjoint types can be upgraded into
the same thing.
(complex 0) → 0 (complex 0.0) → #C(0.0 0.0) (complex 1 1/2) → #C(1 1/2) (complex 1 .99) → #C(1.0 0.99) (complex 3/2 0.0) → #C(1.5 0.0)
realpart, imagpart, Section 2.4.8.11 (Sharpsign C)
Next: conjugate, Previous: complex (Function), Up: Numbers [Contents][Index]
object—an object.
generalized-boolean—a generalized boolean.
Returns true if object is of type complex;
otherwise, returns false.
(complexp 1.2d2) → false (complexp #c(5/3 7.2)) → true
complex (Function) (function and type), typep
(complexp object) ≡ (typep object 'complex)
number—a number.
conjugate—a number.
Returns the complex conjugate of number. The conjugate of a real number is itself.
(conjugate #c(0 -1)) → #C(0 1) (conjugate #c(1 1)) → #C(1 -1) (conjugate 1.5) → 1.5 (conjugate #C(3/5 4/5)) → #C(3/5 -4/5) (conjugate #C(0.0D0 -1.0D0)) → #C(0.0D0 1.0D0) (conjugate 3.7) → 3.7
For a complex number z,
(conjugate z) ≡ (complex (realpart z) (- (imagpart z)))
Next: realpart; imagpart, Previous: conjugate, Up: Numbers [Contents][Index]
number—a number.
phase—a number.
phase
returns the phase
of number (the angle part of its polar representation)
in radians, in the range
-π (exclusive) if minus zero is not supported, or
-π (inclusive) if minus zero is supported,
to π (inclusive). The phase of a positive
real
number
is zero; that of a negative
real
number is π.
The phase of zero is defined to be zero.
If number is a complex float, the result is a float of the same type as the components of number. If number is a float, the result is a float of the same type. If number is a rational or a complex rational, the result is a single float.
The branch cut for phase lies along the negative real
axis, continuous with quadrant II. The range consists of that portion of
the real axis between -π (exclusive) and π (inclusive).
The mathematical definition of phase is as follows:
(phase x) = (atan (imagpart x) (realpart x))
(phase 1) → 0.0s0 (phase 0) → 0.0s0 (phase (cis 30)) → -1.4159266 (phase #c(0 1)) → 1.5707964
Should signal type-error if its argument is not a number.
Might signal arithmetic-error.
Section 12.1.3.3 (Rule of Float Substitutability)
Next: upgraded-complex-part-type, Previous: phase, Up: Numbers [Contents][Index]
number—a number.
real—a real.
realpart and imagpart return the real and
imaginary parts of number respectively.
If number is
real,
then realpart returns number and imagpart
returns (* 0 number), which has the effect that the
imaginary part of a rational is 0 and that of
a float is a floating-point zero of the same format.
(realpart #c(23 41)) → 23 (imagpart #c(23 41.0)) → 41.0 (realpart #c(23 41.0)) → 23.0 (imagpart 23.0) → 0.0
Should signal an error of type type-error if number is not a number.
Next: realp, Previous: realpart; imagpart, Up: Numbers [Contents][Index]
typespec—a type specifier.
environment—an environment object.
The default is nil, denoting the null lexical environment
and the and current global environment.
upgraded-typespec—a type specifier.
upgraded-complex-part-type returns the part type of the
most specialized complex number representation that can
hold parts of type typespec.
The typespec is a subtype of (and possibly type equivalent to) the upgraded-typespec.
The purpose of upgraded-complex-part-type
is to reveal how an implementation does its upgrading.
complex (Function) (function and type)
Next: numerator; denominator, Previous: upgraded-complex-part-type, Up: Numbers [Contents][Index]
object—an object.
generalized-boolean—a generalized boolean.
Returns true if object is of type real;
otherwise, returns false.
(realp 12) → true (realp #c(5/3 7.2)) → false (realp nil) → false (realp (cons 1 2)) → false
(realp object) ≡ (typep object 'real)
Next: rational; rationalize, Previous: realp, Up: Numbers [Contents][Index]
rational—a rational.
numerator—an integer.
denominator—a positive integer.
numerator and denominator reduce rational
to canonical form and compute the numerator or denominator of that number.
numerator and denominator return the numerator
or denominator of the canonical form of rational.
If rational is an integer,
numerator returns rational
and denominator returns 1.
(numerator 1/2) → 1 (denominator 12/36) → 3 (numerator -1) → -1 (denominator (/ -33)) → 33 (numerator (/ 8 -6)) → -4 (denominator (/ 8 -6)) → 3
(gcd (numerator x) (denominator x)) → 1
Next: rationalp, Previous: numerator; denominator, Up: Numbers [Contents][Index]
number—a real.
rational—a rational.
rational and rationalize convert
reals
to rationals.
If number is already rational, it is returned.
If number is a float,
rational returns a rational
that is mathematically equal in value to the float.
rationalize returns a rational that
approximates the float to the accuracy of
the underlying floating-point representation.
rational assumes that the float is completely accurate.
rationalize assumes that the
float is accurate only to the precision of the
floating-point representation.
(rational 0) → 0 (rationalize -11/100) → -11/100 (rational .1) → 13421773/134217728 ;implementation-dependent (rationalize .1) → 1/10
The implementation.
Should signal an error of type type-error if number is not a real.
Might signal arithmetic-error.
It is always the case that
(float (rational x) x) ≡ x
and
(float (rationalize x) x) ≡ x
That is, rationalizing a float by either method and then converting it back to a float of the same format produces the original number.
Next: ash, Previous: rational; rationalize, Up: Numbers [Contents][Index]
object—an object.
generalized-boolean—a generalized boolean.
Returns true if object is of type rational;
otherwise, returns false.
(rationalp 12) → true (rationalp 6/5) → true (rationalp 1.212) → false
(rationalp object) ≡ (typep object 'rational)
Next: integer-length, Previous: rationalp, Up: Numbers [Contents][Index]
integer—an integer.
count—an integer.
shifted-integer—an integer.
ash performs the arithmetic shift operation on the binary
representation of integer, which is treated as if it were binary.
ash shifts integer arithmetically left by count bit
positions if count is positive,
or right count bit positions if count is negative.
The shifted value of the same sign
as integer is returned.
Mathematically speaking, ash performs the computation
floor(integer· 2^count).
Logically, ash
moves all of the bits in integer to the left,
adding zero-bits at the right, or moves them to the right,
discarding bits.
ash is defined to behave as if integer were
represented in two’s complement form, regardless of
how integers are represented internally.
(ash 16 1) → 32 (ash 16 0) → 16 (ash 16 -1) → 8 (ash -100000000000000000000000000000000 -100) → -79
Should signal an error of type type-error if integer is not an integer.
Should signal an error of type type-error if count is not an integer.
Might signal arithmetic-error.
(logbitp j (ash n k)) ≡ (and (>= j k) (logbitp (- j k) n))
integer—an integer.
number-of-bits—a non-negative integer.
Returns the number of bits needed to represent integer in binary two’s-complement format.
(integer-length 0) → 0 (integer-length 1) → 1 (integer-length 3) → 2 (integer-length 4) → 3 (integer-length 7) → 3 (integer-length -1) → 0 (integer-length -4) → 2 (integer-length -7) → 3 (integer-length -8) → 3 (integer-length (expt 2 9)) → 10 (integer-length (1- (expt 2 9))) → 9 (integer-length (- (expt 2 9))) → 9 (integer-length (- (1+ (expt 2 9)))) → 10
Should signal an error of type type-error if integer is not an integer.
This function could have been defined by:
(defun integer-length (integer)
(ceiling (log (if (minusp integer)
(- integer)
(1+ integer))
2)))
If integer is non-negative, then its value can be represented
in unsigned binary form in a field whose width in bits is
no smaller than (integer-length integer).
Regardless of the sign of integer, its value can be
represented in signed binary two’s-complement form in a field
whose width in bits is no smaller than (+ (integer-length integer) 1).
Next: parse-integer, Previous: integer-length, Up: Numbers [Contents][Index]
object—an object.
generalized-boolean—a generalized boolean.
Returns true if object is of type integer;
otherwise, returns false.
(integerp 1) → true (integerp (expt 2 130)) → true (integerp 6/5) → false (integerp nil) → false
(integerp object) ≡ (typep object 'integer)
string—a string.
start, end—bounding index designators of string.
The defaults for start and end are 0 and nil, respectively.
radix—a radix.
The default is 10.
junk-allowed—a generalized boolean. The default is false.
integer—an integer or false.
pos—a bounding index of string.
parse-integer parses an integer in the specified radix
from the substring of string delimited by start and end.
parse-integer expects an optional sign (+ or -) followed by
a a non-empty sequence of digits to be interpreted in the specified radix.
Optional leading and trailing whitespace1
parse-integer does not recognize the syntactic radix-specifier
prefixes #O, #B, #X, and #nR,
nor does it recognize a trailing decimal point.
If junk-allowed is false, an error of type parse-error is
signaled if substring does not consist entirely of the representation of a
signed integer, possibly surrounded on either side by whitespace1
characters.
The first value returned is either
the integer that was parsed,
or else nil if no syntactically correct integer
was seen but junk-allowed was true.
The second value is either the index into the string of the delimiter that terminated the parse, or the upper bounding index of the substring if the parse terminated at the end of the substring (as is always the case if junk-allowed is false).
(parse-integer "123") → 123, 3 (parse-integer "123" :start 1 :radix 5) → 13, 3 (parse-integer "no-integer" :junk-allowed t) → NIL, 0
If junk-allowed is false, an error is signaled if substring does not consist entirely of the representation of an integer, possibly surrounded on either side by whitespace1
Next: boole-1; boole-2; boole-and; boole-andc1; boole-andc2; boole-c1; boole-+, Previous: parse-integer, Up: Numbers [Contents][Index]
Op—a bit-wise logical operation specifier.
integer-1—an integer.
integer-2—an integer.
result-integer—an integer.
boole performs bit-wise logical operations on
integer-1 and integer-2, which are treated as if
they were binary and in two’s complement representation.
The operation to be performed and the return value are determined by op.
boole returns the values
specified for any op in the next figure.
|
Figure 12.17: Bit-Wise Logical Operations
(boole boole-ior 1 16) → 17 (boole boole-and -2 5) → 4 (boole boole-eqv 17 15) → -31 ;;; These examples illustrate the result of applying BOOLE and each ;;; of the possible values of OP to each possible combination of bits. (progn (format t "~&Results of (BOOLE <op> #b0011 #b0101) ...~ ~%---Op-------Decimal-----Binary----Bits---~%") (dolist (symbol '(boole-1 boole-2 boole-and boole-andc1 boole-andc2 boole-c1 boole-c2 boole-clr boole-eqv boole-ior boole-nand boole-nor boole-orc1 boole-orc2 boole-set boole-xor)) (let ((result (boole (symbol-value symbol) #b0011 #b0101))) (format t "~& ~A~13T~3,' D~23T~:*~5,' B~31T ...~4,'0B~%" symbol result (logand result #b1111))))) ▷ Results of (BOOLE <op> #b0011 #b0101) ... ▷ ---Op-------Decimal-----Binary----Bits--- ▷ BOOLE-1 3 11 ...0011 ▷ BOOLE-2 5 101 ...0101 ▷ BOOLE-AND 1 1 ...0001 ▷ BOOLE-ANDC1 4 100 ...0100 ▷ BOOLE-ANDC2 2 10 ...0010 ▷ BOOLE-C1 -4 -100 ...1100 ▷ BOOLE-C2 -6 -110 ...1010 ▷ BOOLE-CLR 0 0 ...0000 ▷ BOOLE-EQV -7 -111 ...1001 ▷ BOOLE-IOR 7 111 ...0111 ▷ BOOLE-NAND -2 -10 ...1110 ▷ BOOLE-NOR -8 -1000 ...1000 ▷ BOOLE-ORC1 -3 -11 ...1101 ▷ BOOLE-ORC2 -5 -101 ...1011 ▷ BOOLE-SET -1 -1 ...1111 ▷ BOOLE-XOR 6 110 ...0110 → NIL
Should signal type-error if its first argument is not a
bit-wise logical operation specifier or if any subsequent argument is not
an integer.
In general,
(boole boole-and x y) ≡ (logand x y)
Programmers who would prefer to use numeric indices rather than bit-wise logical operation specifiers can get an equivalent effect by a technique such as the following:
;; The order of the values in this `table' are such that
;; (logand (boole (elt boole-n-vector n) #b0101 #b0011) #b1111) => n
(defconstant boole-n-vector
(vector boole-clr boole-and boole-andc1 boole-2
boole-andc2 boole-1 boole-xor boole-ior
boole-nor boole-eqv boole-c1 boole-orc1
boole-c2 boole-orc2 boole-nand boole-set))
→ BOOLE-N-VECTOR
(proclaim '(inline boole-n))
→ implementation-dependent
(defun boole-n (n integer &rest more-integers)
(apply #'boole (elt boole-n-vector n) integer more-integers))
→ BOOLE-N
(boole-n #b0111 5 3) → 7
(boole-n #b0001 5 3) → 1
(boole-n #b1101 5 3) → -3
(loop for n from #b0000 to #b1111 collect (boole-n n 5 3))
→ (0 1 2 3 4 5 6 7 -8 -7 -6 -5 -4 -3 -2 -1)
Next: logand; logandc1; logandc2; logeqv; logior; lognand; lognor; lognot; lo+, Previous: boole, Up: Numbers [Contents][Index]
The identity and nature of the values of each of these variables
is implementation-dependent,
except that it must be distinct from each of the values of the others,
and it must be a valid first argument to the function boole.
Each of these constants has a value which is one of the sixteen possible bit-wise logical operation specifiers.
(boole boole-ior 1 16) → 17 (boole boole-and -2 5) → 4 (boole boole-eqv 17 15) → -31
Next: logbitp, Previous: boole-1; boole-2; boole-and; boole-andc1; boole-andc2; boole-c1; boole-+, Up: Numbers [Contents][Index]
integers—integers.
integer—an integer.
integer-1—an integer.
integer-2—an integer.
result-integer—an integer.
The functions
logandc1,
logandc2,
logand,
logeqv,
logior,
lognand,
lognor,
lognot,
logorc1,
logorc2,
and logxor
perform bit-wise logical operations on their arguments,
that are treated as if they were binary.
The next figure lists the meaning of each of the functions. Where an ‘identity’ is shown, it indicates the value yielded by the function when no arguments are supplied.
|
Figure 12.18: Bit-wise Logical Operations on Integers
Negative integers are treated as if they were in two’s-complement notation.
(logior 1 2 4 8) → 15 (logxor 1 3 7 15) → 10 (logeqv) → -1 (logand 16 31) → 16 (lognot 0) → -1 (lognot 1) → -2 (lognot -1) → 0 (lognot (1+ (lognot 1000))) → 999 ;;; In the following example, m is a mask. For each bit in ;;; the mask that is a 1, the corresponding bits in x and y are ;;; exchanged. For each bit in the mask that is a 0, the ;;; corresponding bits of x and y are left unchanged. (flet ((show (m x y) (format t "~%m = #o~6,'0O~%x = #o~6,'0O~%y = #o~6,'0O~%" m x y))) (let ((m #o007750) (x #o452576) (y #o317407)) (show m x y) (let ((z (logand (logxor x y) m))) (setq x (logxor z x)) (setq y (logxor z y)) (show m x y)))) ▷ m = #o007750 ▷ x = #o452576 ▷ y = #o317407 ▷ ▷ m = #o007750 ▷ x = #o457426 ▷ y = #o312557 → NIL
Should signal type-error if any argument is not an integer.
(logbitp k -1) returns true for all values of k.
Because the following functions are not associative, they take exactly two arguments rather than any number of arguments.
(lognand n1 n2) ≡ (lognot (logand n1 n2)) (lognor n1 n2) ≡ (lognot (logior n1 n2)) (logandc1 n1 n2) ≡ (logand (lognot n1) n2) (logandc2 n1 n2) ≡ (logand n1 (lognot n2)) (logiorc1 n1 n2) ≡ (logior (lognot n1) n2) (logiorc2 n1 n2) ≡ (logior n1 (lognot n2)) (logbitp j (lognot x)) ≡ (not (logbitp j x))
Next: logcount, Previous: logand; logandc1; logandc2; logeqv; logior; lognand; lognor; lognot; lo+, Up: Numbers [Contents][Index]
index—a non-negative integer.
integer—an integer.
generalized-boolean—a generalized boolean.
logbitp is used to test the value of a particular bit
in integer, that is treated as if it were binary.
The value of logbitp is true if the bit in integer
whose index is index (that is, its weight is 2^index)
is a one-bit; otherwise it is false.
Negative integers are treated as if they were in two’s-complement notation.
(logbitp 1 1) → false (logbitp 0 1) → true (logbitp 3 10) → true (logbitp 1000000 -1) → true (logbitp 2 6) → true (logbitp 0 6) → false
Should signal an error of type type-error if index is not a non-negative integer.
Should signal an error of type type-error if integer is not an integer.
(logbitp k n) ≡ (ldb-test (byte 1 k) n)
integer—an integer.
number-of-on-bits—a non-negative integer.
Computes and returns the number of bits
in the two’s-complement binary representation of integer
that are ‘on’ or ‘set’.
If integer is negative, the 0 bits are counted;
otherwise, the 1 bits are counted.
(logcount 0) → 0 (logcount -1) → 0 (logcount 7) → 3 (logcount 13) → 3 ;Two's-complement binary: ...0001101 (logcount -13) → 2 ;Two's-complement binary: ...1110011 (logcount 30) → 4 ;Two's-complement binary: ...0011110 (logcount -30) → 4 ;Two's-complement binary: ...1100010 (logcount (expt 2 100)) → 1 (logcount (- (expt 2 100))) → 100 (logcount (- (1+ (expt 2 100)))) → 1
Should signal type-error if its argument is not an integer.
Even if the implementation does not represent integers internally
in two’s complement binary, logcount behaves as if it did.
The following identity always holds:
(logcount x) ≡ (logcount (- (+ x 1))) ≡ (logcount (lognot x))
Next: byte; byte-size; byte-position, Previous: logcount, Up: Numbers [Contents][Index]
integer-1—an integer.
integer-2—an integer.
generalized-boolean—a generalized boolean.
Returns true if any of the bits designated by the 1’s in integer-1 is 1 in integer-2; otherwise it is false. integer-1 and integer-2 are treated as if they were binary.
Negative integer-1 and integer-2 are treated as if they were represented in two’s-complement binary.
(logtest 1 7) → true (logtest 1 2) → false (logtest -2 -1) → true (logtest 0 -1) → false
Should signal an error of type type-error if integer-1 is not an integer.
Should signal an error of type type-error if integer-2 is not an integer.
(logtest x y) ≡ (not (zerop (logand x y)))
Next: deposit-field, Previous: logtest, Up: Numbers [Contents][Index]
size, position—a non-negative integer.
bytespec—a byte specifier.
byte returns a byte specifier that indicates
a byte of width size and whose bits have weights
2^position + size - 1 through 2^position,
and whose representation is
implementation-dependent.
byte-size returns the number of bits specified by bytespec.
byte-position returns the position specified by bytespec.
(setq b (byte 100 200)) → #<BYTE-SPECIFIER size 100 position 200> (byte-size b) → 100 (byte-position b) → 200
(byte-size (byte j k)) ≡ j (byte-position (byte j k)) ≡ k
A byte of size of 0 is permissible;
it refers to a byte of width zero. For example,
(ldb (byte 0 3) #o7777) → 0 (dpb #o7777 (byte 0 3) 0) → 0
Next: dpb, Previous: byte; byte-size; byte-position, Up: Numbers [Contents][Index]
newbyte—an integer.
bytespec—a byte specifier.
integer—an integer.
result-integer—an integer.
Replaces a field of bits within integer; specifically, returns an integer that contains the bits of newbyte within the byte specified by bytespec, and elsewhere contains the bits of integer.
(deposit-field 7 (byte 2 1) 0) → 6 (deposit-field -1 (byte 4 0) 0) → 15 (deposit-field 0 (byte 2 1) -3) → -7
(logbitp j (deposit-field m (byte s p) n))
≡ (if (and (>= j p) (< j (+ p s)))
(logbitp j m)
(logbitp j n))
deposit-field is to mask-field
as dpb is to ldb.
Next: ldb, Previous: deposit-field, Up: Numbers [Contents][Index]
[ˌdə ˈpib] or [ˌdə ˈpə b] or [ˈdēˈpēˈbē]
newbyte—an integer.
bytespec—a byte specifier.
integer—an integer.
result-integer—an integer.
dpb (deposit byte) is used to
replace a field of bits within integer.
dpb returns an integer that is
the same as integer except in the bits specified by bytespec.
Let s be the size specified
by bytespec; then the low s bits of newbyte appear in
the result in the byte specified by bytespec.
Newbyte is interpreted as
being right-justified, as if it were the result of ldb.
(dpb 1 (byte 1 10) 0) → 1024 (dpb -2 (byte 2 10) 0) → 2048 (dpb 1 (byte 2 10) 2048) → 1024
(logbitp j (dpb m (byte s p) n))
≡ (if (and (>= j p) (< j (+ p s)))
(logbitp (- j p) m)
(logbitp j n))
In general,
(dpb x (byte 0 y) z) → z
for all valid values of x, y, and z.
Historically, the name “dpb” comes from a DEC PDP-10 assembly language instruction meaning “deposit byte.”
(setf (ldb bytespec place) new-byte)
[ˈlidib] or [ˈlidə b] or [ˈelˈdēˈbē]
bytespec—a byte specifier.
integer—an integer.
byte, new-byte—a non-negative integer.
ldb extracts and returns the byte of integer
specified by bytespec.
ldb returns an integer in which the bits with weights
2^(s-1) through 2^0 are the same as those in
integer with weights 2^(p+s-1)
through 2^p, and all other bits zero; s is
(byte-size bytespec)
and p is (byte-position bytespec).
setf may be used with ldb to modify
a byte within the integer that is stored
in a given place.
The order of evaluation, when an ldb form is supplied
to setf, is exactly left-to-right.
The effect is to perform a dpb operation
and then store the result back into the place.
(ldb (byte 2 1) 10) → 1 (setq a (list 8)) → (8) (setf (ldb (byte 2 1) (car a)) 1) → 1 a → (10)
byte, byte-position, byte-size, dpb
(logbitp j (ldb (byte s p) n))
≡ (and (< j s) (logbitp (+ j p) n))
In general,
(ldb (byte 0 x) y) → 0
for all valid values of x and y.
Historically, the name “ldb” comes from a DEC PDP-10 assembly language instruction meaning “load byte.”
Next: mask-field, Previous: ldb, Up: Numbers [Contents][Index]
bytespec—a byte specifier.
integer—an integer.
generalized-boolean—a generalized boolean.
Returns true if any of the bits of the byte in integer specified by bytespec is non-zero; otherwise returns false.
(ldb-test (byte 4 1) 16) → true (ldb-test (byte 3 1) 16) → false (ldb-test (byte 3 2) 16) → true
(ldb-test bytespec n) ≡ (not (zerop (ldb bytespec n))) ≡ (logtest (ldb bytespec -1) n)
Next: most-positive-fixnum; most-negative-fixnum, Previous: ldb-test, Up: Numbers [Contents][Index]
(setf (mask-field bytespec place) new-masked-integer)
bytespec—a byte specifier.
integer—an integer.
masked-integer, new-masked-integer—a non-negative integer.
mask-field performs a “mask” operation on integer.
It returns an integer that has the same bits as integer in
the byte specified by bytespec, but that has zero-bits everywhere else.
setf may be used with mask-field
to modify a byte within the integer that is stored
in a given place.
The effect is to perform a deposit-field operation
and then store the result back into the place.
(mask-field (byte 1 5) -1) → 32 (setq a 15) → 15 (mask-field (byte 2 0) a) → 3 a → 15 (setf (mask-field (byte 2 0) a) 1) → 1 a → 13
(ldb bs (mask-field bs n)) ≡ (ldb bs n) (logbitp j (mask-field (byte s p) n)) ≡ (and (>= j p) (< j s) (logbitp j n)) (mask-field bs n) ≡ (logand n (dpb -1 bs 0))
Next: decode-float; scale-float; float-radix; float-sign; float-digits; float+, Previous: mask-field, Up: Numbers [Contents][Index]
implementation-dependent.
most-positive-fixnum is that fixnum closest in value
to positive infinity provided by the implementation,
and greater than or equal to both 2^15 - 1 and
array-dimension-limit.
most-negative-fixnum is that fixnum closest in value
to negative infinity provided by the implementation,
and less than or equal to -2^15.
Next: float (Function), Previous: most-positive-fixnum; most-negative-fixnum, Up: Numbers [Contents][Index]
digits1—a non-negative integer.
digits2—a non-negative integer.
exponent—an integer.
float—a float.
float-1—a float.
float-2—a float.
float-radix—an integer.
integer—a non-negative integer.
integer-sign—the integer -1,
or the integer 1.
scaled-float—a float.
sign—A float of the same type as float
but numerically equal to 1.0 or -1.0.
signed-float—a float.
significand—a float.
decode-float computes three values that characterize
float.
The first value is of the same type
as float and
represents the significand.
The second value represents the exponent
to which the radix (notated in this description by b) must
be raised to obtain the value that, when multiplied with the first
result, produces the absolute value of float.
If float is zero, any integer value may be returned,
provided that the identity shown for scale-float holds.
The third value
is of the same type as float
and is 1.0 if float is greater
than or equal to zero or -1.0 otherwise.
decode-float
divides float by an integral power of b
so as to bring its value between 1/b (inclusive) and 1 (exclusive),
and returns the quotient as the first value.
If float is zero, however, the result
equals the absolute value of float (that is, if there is a negative
zero, its significand is considered to be a positive zero).
scale-float returns
(* float (expt (float b float)
integer)), where b is the radix of the floating-point
representation. float is not necessarily between 1/b and 1.
float-radix returns
the radix of float.
float-sign returns a number z such
that z and float-1 have the same sign and also such that
z and float-2 have the same absolute value.
If float-2 is not supplied, its value is (float 1 float-1).
If an implementation
has distinct representations for negative zero and positive zero,
then (float-sign -0.0) → -1.0.
float-digits returns
the number of radix b digits
used in the representation of float (including any implicit
digits, such as a “hidden bit”).
float-precision
returns
the number of significant radix b digits present in float;
if float is a float
zero, then the result is an integer zero.
For normalized floats,
the results of float-digits and float-precision are the same,
but the precision is less than the number of representation digits
for a denormalized or zero number.
integer-decode-float computes three values that characterize
float -
the significand scaled so as to be an integer,
and the same last two
values that are returned by decode-float.
If float is zero, integer-decode-float returns
zero as the first value.
The second value bears the same relationship to the first value
as for decode-float:
(multiple-value-bind (signif expon sign)
(integer-decode-float f)
(scale-float (float signif f) expon)) ≡ (abs f)
;; Note that since the purpose of this functionality is to expose ;; details of the implementation, all of these examples are necessarily ;; very implementation-dependent. Results may vary widely. ;; Values shown here are chosen consistently from one particular implementation. (decode-float .5) → 0.5, 0, 1.0 (decode-float 1.0) → 0.5, 1, 1.0 (scale-float 1.0 1) → 2.0 (scale-float 10.01 -2) → 2.5025 (scale-float 23.0 0) → 23.0 (float-radix 1.0) → 2 (float-sign 5.0) → 1.0 (float-sign -5.0) → -1.0 (float-sign 0.0) → 1.0 (float-sign 1.0 0.0) → 0.0 (float-sign 1.0 -10.0) → 10.0 (float-sign -1.0 10.0) → -10.0 (float-digits 1.0) → 24 (float-precision 1.0) → 24 (float-precision least-positive-single-float) → 1 (integer-decode-float 1.0) → 8388608, -23, 1
The implementation’s representation for floats.
The functions decode-float, float-radix, float-digits,
float-precision, and integer-decode-float should signal an error
if their only argument is not a float.
The function scale-float should signal an error if its first argument
is not a float or if its second argument is not an integer.
The function float-sign should signal an error if its first argument
is not a float or if its second argument is supplied but is
not a float.
The product of the first result of decode-float or integer-decode-float,
of the radix raised to the power of the second result, and of the third result
is exactly equal to the value of float.
(multiple-value-bind (signif expon sign)
(decode-float f)
(scale-float signif expon))
≡ (abs f)
and
(multiple-value-bind (signif expon sign)
(decode-float f)
(* (scale-float signif expon) sign))
≡ f
Next: floatp, Previous: decode-float; scale-float; float-radix; float-sign; float-digits; float+, Up: Numbers [Contents][Index]
number—a real.
prototype—a float.
float—a float.
float converts a
real
number to a float.
If a prototype is supplied, a float is returned that is mathematically equal to number but has the same format as prototype.
If prototype is not supplied, then if the number is already a float, it is returned; otherwise, a float is returned that is mathematically equal to number but is a single float.
(float 0) → 0.0 (float 1 .5) → 1.0 (float 1.0) → 1.0 (float 1/2) → 0.5 → 1.0d0 or→ 1.0 (eql (float 1.0 1.0d0) 1.0d0) → true
Next: most-positive-short-float; least-positive-short-float; least-positive-n+, Previous: float (Function), Up: Numbers [Contents][Index]
object—an object.
generalized-boolean—a generalized boolean.
Returns true if object is of type float;
otherwise, returns false.
(floatp 1.2d2) → true (floatp 1.212) → true (floatp 1.2s2) → true (floatp (expt 2 130)) → false
(floatp object) ≡ (typep object 'float)
Next: short-float-epsilon; short-float-negative-epsilon; single-float-epsilon+, Previous: floatp, Up: Numbers [Contents][Index]
implementation-dependent.
These constant variables provide a way for programs to examine the implementation-defined limits for the various float formats.
Of these variables,
each which has “-normalized” in its name
must have a value which is a normalized float, and
each which does not have “-normalized” in its name
may have a value which is either a normalized float
or a denormalized float, as appropriate.
Of these variables,
each which has “short-float” in its name
must have a value which is a short float,
each which has “single-float” in its name
must have a value which is a single float,
each which has “double-float” in its name
must have a value which is a double float, and
each which has “long-float” in its name
must have a value which is a long float.
most-positive-short-float, most-positive-single-float,
most-positive-double-float, most-positive-long-float
Each of these constant variables has as its value the positive float of the largest magnitude (closest in value to, but not equal to, positive infinity) for the float format implied by its name.
least-positive-short-float, least-positive-normalized-short-float,
least-positive-single-float, least-positive-normalized-single-float,
least-positive-double-float, least-positive-normalized-double-float,
least-positive-long-float, least-positive-normalized-long-float
Each of these constant variables has as its value the smallest positive (nonzero) float for the float format implied by its name.
least-negative-short-float, least-negative-normalized-short-float,
least-negative-single-float, least-negative-normalized-single-float,
least-negative-double-float, least-negative-normalized-double-float,
least-negative-long-float, least-negative-normalized-long-float
Each of these constant variables has as its value the negative (nonzero) float of the smallest magnitude for the float format implied by its name. (If an implementation supports minus zero as a different object from positive zero, this value must not be minus zero.)
most-negative-short-float, most-negative-single-float,
most-negative-double-float, most-negative-long-float
Each of these constant variables has as its value the negative float of the largest magnitude (closest in value to, but not equal to, negative infinity) for the float format implied by its name.
Next: arithmetic-error, Previous: most-positive-short-float; least-positive-short-float; least-positive-n+, Up: Numbers [Contents][Index]
implementation-dependent.
The value of each of the constants short-float-epsilon,
single-float-epsilon,
double-float-epsilon, and long-float-epsilon is
the smallest positive float ε of the given format,
such that the following expression is true when evaluated:
(not (= (float 1 ε) (+ (float 1 ε) ε)))
The value of each of the constants short-float-negative-epsilon,
single-float-negative-epsilon,
double-float-negative-epsilon, and
long-float-negative-epsilon is the smallest positive
float ε of the given format, such that the following
expression is true when evaluated:
(not (= (float 1 ε) (- (float 1 ε) ε)))
Next: arithmetic-error-operands; arithmetic-error-operation, Previous: short-float-epsilon; short-float-negative-epsilon; single-float-epsilon+, Up: Numbers [Contents][Index]
arithmetic-error,
error,
serious-condition,
condition,
t
The type arithmetic-error consists of error conditions
that occur during arithmetic operations.
The operation and operands are initialized with
the initialization arguments named :operation and :operands to make-condition,
and are accessed by
the functions arithmetic-error-operation and
arithmetic-error-operands.
arithmetic-error-operation, arithmetic-error-operands
Next: division-by-zero, Previous: arithmetic-error, Up: Numbers [Contents][Index]
condition—a condition of type arithmetic-error.
operands—a list.
operation—a function designator.
arithmetic-error-operands returns a list of the operands
which were used in the offending call to the operation that signaled
the condition.
arithmetic-error-operation returns a list of
the offending operation in the offending call that signaled the condition.
arithmetic-error, Chapter 9 (Conditions)
Next: floating-point-invalid-operation, Previous: arithmetic-error-operands; arithmetic-error-operation, Up: Numbers [Contents][Index]
division-by-zero,
arithmetic-error,
error,
serious-condition,
condition,
t
The type division-by-zero consists of error conditions that
occur because of division by zero.
Next: floating-point-inexact, Previous: division-by-zero, Up: Numbers [Contents][Index]
floating-point-invalid-operation,
arithmetic-error,
error,
serious-condition,
condition,
t
The type floating-point-invalid-operation consists of
error conditions that occur because of certain
floating point traps.
It is implementation-dependent whether floating point traps occur, and whether or how they may be enabled or disabled. Therefore, conforming code may establish handlers for this condition, but must not depend on its being signaled.
Next: floating-point-overflow, Previous: floating-point-invalid-operation, Up: Numbers [Contents][Index]
floating-point-inexact,
arithmetic-error,
error,
serious-condition,
condition,
t
The type floating-point-inexact consists of
error conditions that occur because of certain
floating point traps.
It is implementation-dependent whether floating point traps occur, and whether or how they may be enabled or disabled. Therefore, conforming code may establish handlers for this condition, but must not depend on its being signaled.
Next: floating-point-underflow, Previous: floating-point-inexact, Up: Numbers [Contents][Index]
floating-point-overflow,
arithmetic-error,
error,
serious-condition,
condition,
t
The type floating-point-overflow consists of error
conditions that occur because of floating-point overflow.
Previous: floating-point-overflow, Up: Numbers [Contents][Index]
floating-point-underflow,
arithmetic-error,
error,
serious-condition,
condition,
t
The type floating-point-underflow consists of
error conditions that occur because of floating-point underflow.
Previous: floating-point-overflow, Up: Numbers [Contents][Index]