Gauche supports the following types of numbers
There’s no limit of the size of number except the memory of the machine.
Both denominator and numerator are represented by exact integers. There’s no limit of the size of number except the memory of the machine.
double-type of underlying C compiler, usually IEEE 64-bit
floating point number.
Real part and imaginary part are represented by inexact floating-point real numbers.
|• Number classes:|
|• Numerical predicates:|
|• Numerical comparison:|
|• Numerical conversions:|
|• Bitwise operations:|
These classes consist a class hierarchy of number objects.
Note that these classes do not exactly correspond to the
number hierarchy defined in R7RS. Especially,
only exact integers are the instances of the
class. That is,
(integer? 1) ⇒ #t (is-a? 1 <integer>) ⇒ #t (is-a? 1 <real>) ⇒ #t (integer? 1.0) ⇒ #t (is-a? 1.0 <integer>) ⇒ #f (is-a? 1.0 <real>) ⇒ #t (class-of (expt 2 100)) ⇒ #<class <integer>> (class-of (sqrt -3)) ⇒ #<class <complex>>
#t if obj is a number, a complex number, a real number,
a rational number or an integer, respectively. In Gauche, a set of
numbers is the same as a set of complex numbers.
A set of rational numbers is the same as a set of real numbers,
(since we have only limited-precision floating numbers).
(complex? 3+4i) ⇒ #t (complex? 3) ⇒ #t (real? 3) ⇒ #t (real? -2.5+0.0i) ⇒ #t (real? #e1e10) ⇒ #t (integer? 3+0i) ⇒ #t (integer? 3.0) ⇒ #t (real? +inf.0) ⇒ #t (real? +nan.0) ⇒ #t (rational? +inf.0) ⇒ #f (rational? +nan.0) ⇒ #f
Note: R6RS adopts more strict definition on exactness,
and notably, it defines a complex number with non-exact zero imaginary
part is not a real number. Currently Gauche doesn’t have
exact complex numbers, and automatically coerces complex
numbers with zero imaginary part to a real number.
Thus R6RS code that relies on the fact that
(real? 1+0.0i) is
#f won’t work with Gauche.
In Gauche these are just an alias of
integer?. They are provided for R6RS compatibility.
The difference of those and non
-valued versions in R6RS is
that these returns
#t if obj is a complex number
with nonexact zero imaginary part. Since Gauche doesn’t distinguish
complex numbers with zero imaginary part and real numbers, we don’t
have the difference.
#t if obj is an exact number and an inexact number,
(exact? 1) ⇒ #t (exact? 1.0) ⇒ #f (inexact? 1) ⇒ #f (inexact? 1.0) ⇒ #t (exact? (modulo 5 3)) ⇒ #t (inexact? (modulo 5 3.0)) ⇒ #f
(and (exact? obj) (integer? obj)), but more efficient.
#t if a number z equals to zero.
(zero? 1) ⇒ #f (zero? 0) ⇒ #t (zero? 0.0) ⇒ #t (zero? 0.0+0.0i) ⇒ #t
#t if a real number x is positive and negative,
respectively. It is an error to pass a non-real number.
For real numbers, returns
#f iff the given number
is finite, infinite, or NaN, respectively.
For non-real complex numbers,
both real and imaginary components are finite,
#t if at least either real or imaginary component
is infinite, and
#t if at least either
real or imaginary component is NaN. (Note: It is incompatible
to R6RS, in which these procedures must raise an error if
the given argument is non-real number.)
In R7RS, these procedures are in
(scheme inexact) library.
#t if an integer n is odd and even,
respectively. It is an error to pass a non-integral number.
(odd? 3) ⇒ #t (even? 3) ⇒ #f (odd? 3.0) ⇒ #t
#t iff n is an exact integer whose internal
representation is fixnum and bignum, respectively.
Portable Scheme programs don’t need to care about the internal
representation of integer. These are for certain low-level
routines that does particular optimization.
If all the numbers z are equal, returns
(= 2 2) ⇒ #t (= 2 3) ⇒ #f (= 2 2.0) ⇒ #t (= 2 2.0 2.0+0i) ⇒ #t (= 2/4 1/2) ⇒ #t
#t If all the real numbers x are
monotonically nondecreasing, monotonically decreasing, or monotonically
[R7RS] Returns a maximum or minimum number in the given real numbers, respectively. If any of the arguments are NaN, NaN is returned.
Selection and searching in collection.
Returns a maximum and minimum number in the given real numbers.
Selection and searching in collection.
Returns the sum or the product of given numbers, respectively.
If no argument is given,
(+) yields 0 and
(*) yields 1.
[R7RS] If only one number z1 is given, returns its negation and reciprocal, respectively.
If more than one number are given, returns:
z1 - z2 - z3 … z1 / z2 / z3 …
(- 3) ⇒ -3 (- -3.0) ⇒ 3.0 (- 5+2i) ⇒ -5.0-2.0i (/ 3) ⇒ 1/3 (/ 5+2i) ⇒ 0.172413793103448-0.0689655172413793i (- 5 2 1) ⇒ 2 (- 5 2.0 1) ⇒ 2.0 (- 5+3i -i) ⇒ 5.0+2.0i (/ 14 6) ⇒ 7/3 (/ 6+2i 2) ⇒ 3.0+1.0i
Note: Gauche didn’t have exact rational number support until 0.8.8;
/ coerced the result to inexact even if both
divisor and dividend were exact numbers, when the result wasn’t
a whole number. It is not the case anymore.
If the existing code relies on the old behavior, it runs
very slowly on the newer versions of Gauche, since the calculation
proceeds with exact rational arithmetics that is much slower than
floating point arithmetics. You want to use
to use fast inexact arithmetics (unless you
need exact results).
/, but the arguments
are coerced to inexact number. So they always return inexact number.
These are useful when you know you don’t need exact calculation
and want to avoid accidental overhead of bignums and/or exact
[R7RS+] For real number z, returns an absolute value of it. For complex number z, returns the magnitude of the number. The complex part is Gauche extension.
(abs -1) ⇒ 1 (abs -1.0) ⇒ 1.0 (abs 1+i) ⇒ 1.4142135623731
[R7RS] Returns the quotient, remainder and modulo of dividing an integer n1 by an integer n2. The result is an exact number only if both n1 and n2 are exact numbers.
Remainder and modulo differ when either one of the arguments is negative. Remainder R and quotient Q have the following relationship.
n1 = Q * n2 + R
abs(Q) = floor(abs(n1)/abs(n2)).
Consequently, R’s sign is always the same as n1’s.
On the other hand, modulo works as expected for positive n2,
regardless of the sign of n1
(modulo -1 n2) == n2 - 1).
If n2 is negative, it is mapped to the positive case by
the following relationship.
modulo(n1, n2) = -modulo(-n1, -n2)
Consequently, modulo’s sign is always the same as n2’s.
(remainder 10 3) ⇒ 1 (modulo 10 3) ⇒ 1 (remainder -10 3) ⇒ -1 (modulo -10 3) ⇒ 2 (remainder 10 -3) ⇒ 1 (modulo 10 -3) ⇒ -2 (remainder -10 -3) ⇒ -1 (modulo -10 -3) ⇒ -1
Calculates the quotient and the remainder of dividing integer n1 by integer n2 simultaneously, and returns them as two values.
These are integer division procedures introduced in R6RS.
these procedures can take non-integral values.
The dividend x can be an arbitrary real number,
and the divisor y can be non-zero real number.
div returns an integer n, and
a real number m, such that:
(div 123 10) ⇒ 12 (mod 123 10) ⇒ 3 (div 123 -10) ⇒ -12 (mod 123 -10) ⇒ 3 (div -123 10) ⇒ -13 (mod -123 10) ⇒ 7 (div -123 -10) ⇒ 13 (mod -123 -10) ⇒ 7 (div 123/7 10/9) ⇒ 15 (mod 123/7 10/9) ⇒ 19/21 ;; 123/7 = 10/9 * 15 + 19/21 (div 14.625 3.75) ⇒ 3.0 (mod 14.625 3.75) ⇒ 3.375 ;; 14.625 = 3.75 * 3.0 + 3.375
For a nonnegative integer x and an integer y,
The results of
remainder. If x
is negative, they differ, though.
div-and-mod calculates both
and returns their results in two values.
div0 and mod0 are similar, except the range of m:
(div0 123 10) ⇒ 12 (mod0 123 10) ⇒ 3 (div0 127 10) ⇒ 13 (mod0 127 10) ⇒ -3 (div0 127 -10) ⇒ -13 (mod0 127 -10) ⇒ -3 (div0 -127 10) ⇒ -13 (mod0 -127 10) ⇒ 3 (div0 -127 -10) ⇒ 13 (mod0 -127 -10) ⇒ 3
div0-and-mod0 calculates both
and returns their results in two values.
Here’s a visualization of R6RS and R7RS division and modulo operations: http://blog.practical-scheme.net/gauche/20100618-integer-divisions It might help to grasp how they works.
[R7RS] Returns the greatest common divisor or the least common multiplier of the given integers, respectively
Arguments must be integers, but doesn’t need to be exact. If any of arguments is inexact, the result is inexact.
Returns a lazy sequence of regular continued fraction expansion of finite real number x. An error is raised if x is infinite or NaN, or not a real number. The returned sequence is lazy, so the terms are calculated as needed.
(continued-fraction 13579/2468) ⇒ (5 1 1 122 1 9) (+ 5 (/ (+ 1 (/ (+ 1 (/ (+ 122 (/ (+ 1 (/ 9)))))))))) ⇒ 13579/2468 (continued-fraction (exact 3.141592653589793)) ⇒ (3 7 15 1 292 1 1 1 2 1 3 1 14 3 3 2 1 3 3 7 2 1 1 3 2 42 2) (continued-fraction 1.5625) ⇒ (1.0 1.0 1.0 3.0 2.0)
[R7RS] Returns the numerator and denominator of a rational number q.
[R7RS] Returns the simplest rational approximation q of a real number x, such that the difference between x and q is no more than the error bound ebound.
Note that Gauche doesn’t have inexact rational number, so if x and/or ebound is inexact, the result is coerced to floating point representation.
(rationalize 1234/5678 1/1000) ⇒ 5/23 (rationalize 3.141592653589793 1/10000) ⇒ 3.141509433962264 (rationalize (exact 3.141592653589793) 1/10000) ⇒ 333/106 (rationalize (exact 3.141592653589793) 1/10000000) ⇒ 75948/24175 ;; Some edge cases (rationalize 2 +inf.0) ⇒ 0 (rationalize +inf.0 0) ⇒ +inf.0 (rationalize +inf.0 +inf.0) ⇒ +nan.0
The argument x must be a real number.
ceiling return a maximum integer that
isn’t greater than x and a minimum integer that isn’t less
than x, respectively.
Truncate returns an integer that truncates
x towards zero. Round returns an integer that is closest
to x. If fractional part of x is exactly 0.5, round
returns the closest even integer.
Following Scheme’s general rule, the result is inexact if x is an
inexact number; e.g.
(round 2.3) is
2.0. If you need
an exact integer by rounding an inexact number, you have to use
on the result, or use one of the following procedure (
These are convenience procedures of the popular
(exact (floor x)) etc.
min if x
<min x if min
<=max max if max
If min or max is omitted or
#f, it is regarded
Returns an exact integer only if all the given numbers are exact integers.
(clamp 3.1 0.0 1.0) ⇒ 1.0 (clamp 0.5 0.0 1.0) ⇒ 0.5 (clamp -0.3 0.0 1.0) ⇒ 0.0 (clamp -5 0) ⇒ 0 (clamp 3724 #f 256) ⇒ 256
Transcendental functions. Work for complex numbers as well.
In R7RS, these procedures are in the
(scheme inexact) module.
The two-argument version of
log is added in R6RS, and returns
base-z2 logarithm of z1.
The two-argument version of
(angle (make-rectangular x y)) for the real numbers
Hyperbolic trigonometric functions. Work for complex numbers as well.
[R7RS] Returns a square root of a complex number z. The branch cut scheme is the same as Common Lisp. For real numbers, it returns a positive root.
If z is the square of an exact real number, the return value is also an exact number.
(sqrt 2) ⇒ 1.4142135623730951 (sqrt -2) ⇒ 0.0+1.4142135623730951i (sqrt 256) ⇒ 16 (sqrt 256.0) ⇒ 16.0 (sqrt 81/169) ⇒ 9/13
[R7RS] Given an exact nonnegative integer k, returns two exact nonnegative integer s and r that satisfy the following equations:
k = (+ (* s s) r) k < (* (+ s 1) (+ s 1))
(exact-integer-sqrt 782763574) ⇒ 27977 and 51045
[R7RS] Returns z1^z2 (z1 powered by z2), where z1 and z2 are complex numbers.
(modulo (expt base exponent) mod) efficiently.
The next example shows the last 10 digits of a mersenne prime M_74207281 (2^74207281 - 1)
(- (expt-mod 2 74207281 #e1e10) 1) ⇒ 1086436351
Gamma function and natural logarithmic of absolute value of Gamma function.
NB: Mathematically these functions are defined in complex domain, but currently we only supports real number argument.
These procedures return the width of fixnum (w),
the greatest integer representable by fixnum (
2^(w-1) - 1),
and the least integer representable by fixnum (
respectively. You might want to care the fixnum range when
you are writing a performance-critical section.
These names are defined in R6RS. Common Lisp and ChezScheme have
NB: Before 0.9.5,
fixnum-width had a bug
to return one smaller than the supposed value.
Creates a complex number from two real numbers, x1 and x2.
make-rectangular returns x1 + ix2.
make-polar returns x1e^(ix2).
In R7RS, these procedures are in the
(scheme complex) library.
Decompose a complex number z and returns a real number.
imag-part return z’s real and imaginary
z’s magnitude and angle, respectively.
In R7RS, these procedures are in the
(scheme complex) library.
For a given finite floating-point number, returns
a vector of three exact integers,
#(m, e, sign),
x = (* sign m (expt 2.0 e)) sign is either 1, 0 or -1.
If x is
-inf.0, m is
If x is
+nan.0, m is
The API is taken from ChezScheme.
(decode-float 3.1415926) ⇒ #(7074237631354954 -51 1) (* 7074237631354954 (expt 2.0 -51)) ⇒ 3.1415926 (decode-float +nan.0) ⇒ #(#f 0 -1)
This is an inverse of
decode-float. Vector must be
a three-element vector as returned from
(encode-float '#(7074237631354954 -51 1)) ⇒ 3.1415926 (encode-float '#(#t 0 1)) ⇒ +inf.0
These procedures can be used to compose and decompose floating
Fmod computes the remainder of dividing x
by y, that is, it returns x-n*y where
n is the quotient of x/y rounded towards zero
to an integer.
Modf returns two values; a fractional
part of x and an integral part of x.
returns two values, fraction and exponent of x,
where x = fraction * 2^exponent, and
0.5 <= |fraction| < 1.0, unless x is zero.
(When x is zero, both fraction and exponent are zero).
Ldexp is a reverse operation of
frexp; it returns a real number x * 2^n.
(fmod 32.1 10.0) ⇒ 2.1 (fmod 1.5 1.4) ⇒ 0.1 (modf 12.5) ⇒ 0.5 and 12.0 (frexp 3.14) ⇒ 0.785 and 2 (ldexp 0.785 2) ⇒ 3.14
Returns an exact or an inexact representation of the
given number z, respectively. Passing an exact number
exact, and an inexact number to
inexact, are no-op.
Gauche doesn’t have exact complex number with non-zero imaginary part,
nor exact infinites and NaNs, so passing those to
(inexact 1) ⇒ 1.0 (inexact 1/10) ⇒ 0.1
If an inexact finite real number is passed to
the simplest exact rational number within the precision of the
floating point representation is returned.
(exact 1.0) ⇒ 1 (exact 0.1) ⇒ 1/10 (exact (/ 3.0)) ⇒ 1/3
For all finite inexact real number x,
(inexact (exact x)) is always
the original number x.
(Note that the inverse doesn’t hold, that is, an exact number n
(exact (inexact n)) aren’t necessarily the same.
It’s because many (actually, infinite number of) exact numbers
can be mapped to one inexact number.)
To specify the error tolerance when converting inexact real numbers
to exact rational numbers, use
[R5RS] Converts exact number to inexact one, and vice versa.
exact->inexact returns the argument as is
if an inexact number is passed, and
returns the argument if an exact number is passed, so
in Gauche they are equivalent to
respectively. Note that other R5RS implementation may raise
an error if passing an inexact number to
inexact are preferred,
for they are more concise, and you don’t need to care
whether the argument is exact or inexact numbers.
These procedures are for compatibility with R5RS programs.
Find the simplest rational representation of a finite real
number x within the specified error bounds. This is the low-level
routine called by
The result rational value r satisfies the following condition:
(<= (- x lo) r (+ x hi)) ; when open? is #f (< (- x lo) r (+ x hi)) ; otherwise
Note that both hi and lo must be nonnegative.
If hi and/or lo is omitted, it is determined by x:
if x is exact, hi and lo are defaulted to zero; if
x is inexact, hi and lo depend on
the precision of the floating point representation of x.
In the latter case, the open? also depends on x—it is true
if the mantissa of x is odd, and false otherwise, reflecting
the round-to-even rule. So, if you call
one finite number, you’ll get the same result as
(real->rational 0.1) ⇒ 1/10
Passing zeros to the error bounds makes it return the exact
conversion of the floating number itself (that is, the exact
(* sign mantissa (expt 2 exponent))).
(real->rational 0.1 0 0) ⇒ 3602879701896397/36028797018963968
(If you give both hi and lo, but omit open?, we assume closed range.)
[R7RS+] These procedures convert a number and its string representation in radix radix system. radix must be between 2 and 36 inclusive. If radix is omitted, 10 is assumed.
Number->string takes a number z and returns a string.
If z is not an exact integer, radix must be 10.
For the numbers with radix more than 10, lower case alphabet
character is used for digits, unless the optional argument
use-upper? is true, in that case upper case characters are used.
The argument use-upper? is Gauche’s extension.
String->number takes a string string and parses it
as a number in radix radix system. If the number looks like
non-exact number, only radix 10 is allowed. If the given string
can’t be a number,
#f is returned.
Generic coercion functions. Returns ‘natural’ interpretation of obj
as a number or an exact integer, respectively.
The default methods are defined for numbers and strings; a string is
interpreted by string->number, and if the string can’t be
interpreted as a number, 0 is returned.
Other obj is simply converted to 0.
If obj is naturally interpreted
as a number that is not an exact integer,
inexact->exact to obtain an integer.
Other class may provide a method to customize the behavior.
These procedures treat integers as half-open bit vectors. If an integer is positive, it is regarded as if infinite number of zeros are padded to the left. If an integer is negative, it is regarded in 2’s complement form, and infinite number of 1’s are padded to the left.
In regard to the names of those operations, there are two groups
in the Scheme world; Gauche follows the names of the
original SLIB’s “logical” module, which was rooted in CL.
Another group uses a bit long but descriptive name such as
SRFI-60 (see Integers as bits) defines both names, and also some additional procedures. If you’re porting libraries written for other Scheme, you might want to check it.
Shifts integer n left with count bits.
If count is negative,
ash shifts n right with
; Note: 6 ≡ [...00110], and ; -6 ≡ [...11010] (ash 6 2) ⇒ 24 ;[...0011000] (ash 6 -2) ⇒ 1 ;[...0000001] (ash -6 2) ⇒ -24 ;[...1101000] (ash -6 -2) ⇒ -2 ;[...1111110]
Returns bitwise and, bitwise inclusive or and bitwise exclusive or
of integers n1 …. If no arguments are given,
[SRFI-60] Returns bitwise not of an integer n.
(not (zero? (logand n1 n2 …)))
#t if index-th bit of integer n is 1,
[SRFI-60] Extracts start-th bit (inclusive) to end-th bit (exclusive) from an exact integer n, where start < end.
[SRFI-60] If bit is true, sets index-th bit of an exact integer n. If bit is false, resets index-th bit of an exact integer n.
Returns an exact integer, each bit of which is the same as
n except the start-th bit (inclusive) to end-th
bit (exclusive), which is a copy of the lower
(end-start)-th bits of an exact
(number->string (copy-bit-field #b10000000 -1 1 5) 2) ⇒ "10011110" (number->string (copy-bit-field #b10000000 #b010101010 1 7) 2) ⇒ "11010100"
Note: The API of this procedure was originally taken from SLIB,
and at that time,
the argument order was
(copy-bit-field n start end from).
During the discussion of SRFI-60 the argument order was changed
for the consistency, and the new versions of SLIB followed it.
We didn’t realize the change until recently - before 0.9.4,
this procedure had the old argument order. Code that is using
this procedure needs to be fixed. If you need your code to work
with both versions of Gauche, have the following definition
in your code.
(define (copy-bit-field to from start end) (if (< start end) (let1 mask (- (ash 1 (- end start)) 1) (logior (logand to (lognot (ash mask start))) (ash (logand from mask) start))) from))
If n is positive, returns the number of
1’s in the
bits of n. If n is negative,
returns the number of
0’s in the bits of 2’s complement
representation of n.
(logcount 0) ⇒ 0 (logcount #b0010) ⇒ 1 (logcount #b0110) ⇒ 2 (logcount #b1111) ⇒ 4 (logcount #b-0001) ⇒ 0 ;; 2's complement: ....111111 (logcount #b-0010) ⇒ 1 ;; 2's complement: ....111110 (logcount #b-0011) ⇒ 1 ;; 2's complement: ....111101 (logcount #b-0100) ⇒ 2 ;; 2's complement: ....111100
[SRFI-60] Returns the minimum number of bits required to represent an exact integer n. Negative integer is assumed to be in 2’s complement form. A sign bit is not considered.
(integer-length 255) ⇒ 8 (integer-length 256) ⇒ 9 (integer-length -256) ⇒ 8 (integer-length -257) ⇒ 9
If n is a power of two, that is,
(expt 2 k) and
k >= 0,
#f if n is not a power of two.
Returns maximum k such that
(expt 2 k) is a factor of n.
In other words, returns the number of consecutive zero bits from LSB
of n. If n is zero, zero is returned.
This can be calculated by the following expression; this procedure is for speed to save creating intermediate numbers when n is bignum.
(- (integer-length (logxor n (- n 1))) 1)
In the Scheme world you rarely need to know about how the numbers are represented inside the machine. However, it matters when you have to exchange data to/from the outer world in binary representation.
Gauche’s binary I/O procedures, such as in
binary.io module (see Binary I/O) and
(see Uniform vectors), take optional endian argument
to specify the endianness.
Currently Gauche recognizes the following endiannesses.
Big endian. With this endianness, a 32-bit integer
will be written out as an octet sequence
#x12 #x34 #x56 #x78.
Little endian. With this endianness, a 32-bit integer
is written out as an octet sequence
#x78 #x56 #x34 #x12.
This is a variation of
little-endian, and used in ARM
processors in some specific modes. It works just like
except reading/writing double-precision floating point number (
which is written as two little-endian 32bit words ordered by big-endian
(e.g. If machine register’s representation is
it is written as
#x04 #x03 #x02 #x01 #x08 #x07 #x06 #x05.
When the endian argument is omitted, those procedures
use the parameter
This is a dynamic parameter (see Parameters) to specify the endianness the binary I/O routines use when its endian argument is omitted. The initial value of this parameter is the system’s native endianness.
The system’s native endianness can be queried with the following procedure:
Returns a symbol representing the system’s endianness.