4.5 Numerical operations

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

procedure: number? object

procedure: complex? object

procedure: real? object

procedure: rational? object

procedure: integer? object

These numerical type predicates can be applied to any kind of argument, including non-numbers. They return #t if the object is of the named type, and otherwise they return #f. In general, if a type predicate is true of a number then all higher type predicates are also true of that number. Consequently, if a type predicate is false of a number, then all lower type predicates are also false of that number.³

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

(complex? 3+4i)         ⇒  #t
(complex? 3)            ⇒  #t
(real? 3)               ⇒  #t
(real? -2.5+0.0i)       ⇒  #t
(real? #e1e10)          ⇒  #t
(rational? 6/10)        ⇒  #t
(rational? 6/3)         ⇒  #t
(integer? 3+0i)         ⇒  #t
(integer? 3.0)          ⇒  #t
(integer? 8/4)          ⇒  #t

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

procedure: exact? z

procedure: inexact? z

These numerical predicates provide tests for the exactness of a quantity. For any Scheme number, precisely one of these predicates is true.

procedure: exact-integer? object

procedure: exact-nonnegative-integer? object

procedure: exact-rational? object

These procedures test for some very common types of numbers. These tests could be written in terms of simpler predicates, but are more efficient.

procedure: = z1 z2 z3 …

procedure: < x1 x2 x3 …

procedure: > x1 x2 x3 …

procedure: <= x1 x2 x3 …

procedure: >= x1 x2 x3 …

These procedures return #t if their arguments are (respectively): equal, monotonically increasing, monotonically decreasing, monotonically nondecreasing, or monotonically nonincreasing.

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

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

procedure: zero? z

procedure: positive? x

procedure: negative? x

procedure: odd? x

procedure: even? x

These numerical predicates test a number for a particular property, returning #t or #f. See note above regarding inexact numbers.

procedure: max x1 x2 …

procedure: min x1 x2 …

These procedures return the maximum or minimum of their arguments.

(max 3 4)              ⇒  4    ; exact
(max 3.9 4)            ⇒  4.0  ; inexact

Note: If any argument is inexact, then the result will also be inexact (unless the procedure can prove that the inaccuracy is not large enough to affect the result, which is possible only in unusual implementations). If min or max is used to compare numbers of mixed exactness, and the numerical value of the result cannot be represented as an inexact number without loss of accuracy, then the procedure may report a violation of an implementation restriction.⁴

procedure: + z1 …

procedure: * z1 …

These procedures return the sum or product of their arguments.

(+ 3 4)                 ⇒  7
(+ 3)                   ⇒  3
(+)                     ⇒  0
(* 4)                   ⇒  4
(*)                     ⇒  1

procedure: - z1 z2 …

procedure: / z1 z2 …

With two or more arguments, these procedures return the difference or quotient of their arguments, associating to the left. With one argument, however, they return the additive or multiplicative inverse of their argument.

(- 3 4)                 ⇒  -1
(- 3 4 5)               ⇒  -6
(- 3)                   ⇒  -3
(/ 3 4 5)               ⇒  3/20
(/ 3)                   ⇒  1/3

procedure: 1+ z

procedure: -1+ z

(1+ z) is equivalent to (+ z 1); (-1+ z) is equivalent to (- z 1).

procedure: abs x

abs returns the magnitude of its argument.

(abs -7)                ⇒  7

procedure: quotient n1 n2

procedure: remainder n1 n2

procedure: modulo n1 n2

These procedures implement number-theoretic (integer) division: for positive integers n1 and n2, if n3 and n4 are integers such that then

(quotient n1 n2)        ⇒  n3
(remainder n1 n2)       ⇒  n4
(modulo n1 n2)          ⇒  n4

For integers n1 and n2 with n2 not equal to 0,

(= n1
   (+ (* n2 (quotient n1 n2))
      (remainder n1 n2)))
                                    ⇒  #t

provided all numbers involved in that computation are exact.

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

(modulo 13 4)           ⇒  1
(remainder 13 4)        ⇒  1

(modulo -13 4)          ⇒  3
(remainder -13 4)       ⇒  -1

(modulo 13 -4)          ⇒  -3
(remainder 13 -4)       ⇒  1

(modulo -13 -4)         ⇒  -1
(remainder -13 -4)      ⇒  -1

(remainder -13 -4.0)    ⇒  -1.0  ; inexact

Note that quotient is the same as integer-truncate.

procedure: integer-floor n1 n2

procedure: integer-ceiling n1 n2

procedure: integer-truncate n1 n2

procedure: integer-round n1 n2

These procedures combine integer division with rounding. For example, the following are equivalent:

(integer-floor n1 n2)
(floor (/ n1 n2))

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

Note that integer-truncate is the same as quotient.

procedure: integer-divide n1 n2

procedure: integer-divide-quotient qr

procedure: integer-divide-remainder qr

integer-divide is equivalent to performing both quotient and remainder at once. The result of integer-divide is an object with two components; the procedures integer-divide-quotient and integer-divide-remainder select those components. These procedures are useful when both the quotient and remainder are needed; often computing both of these numbers simultaneously is much faster than computing them separately.

For example, the following are equivalent:

(lambda (n d)
  (cons (quotient n d)
        (remainder n d)))

(lambda (n d)
  (let ((qr (integer-divide n d)))
    (cons (integer-divide-quotient qr)
          (integer-divide-remainder qr))))

procedure: gcd n1 …

procedure: lcm n1 …

These procedures return the greatest common divisor or least common multiple of their arguments. The result is always non-negative.

(gcd 32 -36)            ⇒  4
(gcd)                   ⇒  0

(lcm 32 -36)            ⇒  288
(lcm 32.0 -36)          ⇒  288.0  ; inexact
(lcm)                   ⇒  1

procedure: numerator q

procedure: denominator q

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

(numerator (/ 6 4))  ⇒  3
(denominator (/ 6 4))  ⇒  2
(denominator (inexact (/ 6 4))) ⇒ 2.0

procedure: floor x

procedure: ceiling x

procedure: truncate x

procedure: round x

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

Rationale: round rounds to even for consistency with the rounding modes required by the IEEE floating point standard.

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

(floor -4.3)          ⇒  -5.0
(ceiling -4.3)        ⇒  -4.0
(truncate -4.3)       ⇒  -4.0
(round -4.3)          ⇒  -4.0

(floor 3.5)           ⇒  3.0
(ceiling 3.5)         ⇒  4.0
(truncate 3.5)        ⇒  3.0
(round 3.5)           ⇒  4.0  ; inexact

(round 7/2)           ⇒  4    ; exact
(round 7)             ⇒  7

procedure: floor->exact x

procedure: ceiling->exact x

procedure: truncate->exact x

procedure: round->exact x

These procedures are similar to the preceding procedures except that they always return an exact result. For example, the following are equivalent

(floor->exact x)
(exact (floor x))

except that the former is faster and has fewer range restrictions.

procedure: rationalize x y

procedure: rationalize->exact x y

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

(rationalize (exact .3) 1/10)  ⇒ 1/3    ; exact
(rationalize .3 1/10)          ⇒ #i1/3  ; inexact

rationalize->exact is similar to rationalize except that it always returns an exact result.

procedure: simplest-rational x y

procedure: simplest-exact-rational x y

simplest-rational returns the simplest rational number between x and y inclusive; simplest-exact-rational is similar except that it always returns an exact result.

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

procedure: exp z

procedure: log z

procedure: sin z

procedure: cos z

procedure: tan z

procedure: asin z

procedure: acos z

procedure: atan z

procedure: atan y x

These procedures compute the usual transcendental functions. log computes the natural logarithm of z (not the base ten logarithm). asin, acos, and atan compute arcsine, arccosine, and arctangent, respectively. The two-argument variant of atan computes (angle (make-rectangular x y)) (see below).

In general, the mathematical functions log, arcsine, arccosine, and arctangent are multiply defined. For nonzero real x, the value of log x is defined to be the one whose imaginary part lies in the range minus pi (exclusive) to pi (inclusive). log 0 is undefined. The value of log z when z is complex is defined according to the formula With log defined this way, the values of arcsine, arccosine, and arctangent are according to the following formulae: The above specification follows Common Lisp: the Language, which in turn cites Principal Values and Branch Cuts in Complex APL; refer to these sources for more detailed discussion of branch cuts, boundary conditions, and implementation of these functions. When it is possible these procedures produce a real result from a real argument.

procedure: log1p z

procedure: expm1 z

Equivalent to:

log1p z = log(1 + z).
expm1 z = exp(z) - 1,

However, for real numbers close to zero, these provide better approximations than (log (+ 1 z)) or (- (exp z) 1):

• Floating-point numbers have much higher density around 0 than around 1, so naive translation from near 0 to near 1 loses precision, and naive computation of a number near 1 loses precision even if it is followed by translation to near 0.
• The condition number of log near 1 is unbounded, while the condition number of log1p near 0 is near 1:

  x f'(x)/f(x) = [x/(1 + x)]/log(1 + x).
  

  

  (Conversely, the condition number of log near 0 approaches 0, while the condition number of log1p near -1 is unbounded, so for inputs near 0 it is better to compute them via log rather than via log1p.)

• Similarly, although the condition number of exp near 0 is near 0, its value near 0 is near 1, and the condition number of y - 1 is unbounded for y near 1, so the intermediate error introduced by (exp z) may be amplified arbitrarily by then computing (- (exp z) 1). In contrast, the condition number of expm1 itself near 0 is near 1, so it does not inherently amplify errors:

  x f'(x)/f(x) = x e^x/(e^x - 1).
  

  

The forward relative error of this implementation is determined by the system’s math library, usually below 1ulp.

procedure: log1mexp x

procedure: log1pexp x

Equivalent to:

log1mexp x = log (1 - e^x),
log1pexp x = log (1 + e^x).

Like log1p and expm1, these avoid numerical pathologies with the intermediate quantities 1 - e^x and 1 + e^x and inputs to log near 1.

• log1mexp computes the complement of a probability p in log-space \log p, and as such is a self-inverse. It is finite when x < 0; negative infinity when x = 0; and invalid otherwise.
• log1pexp is related to the logistic function 1/(1 + e^{-x}) — specifically, it differs from the logarithm of the logistic function only by the sign of the input and the output.

This implementation gives forward relative error bounded by ten times the forward relative error bound of the system math library’s log and exp, which is usually below 1ulp.

Beware that although the forward relative error of the MIT/GNU Scheme implementations of these functions is bounded, these functions are ill-conditioned for large negative inputs:

x f'(x)/f(x) = (+/- x exp(x))/((1 +/- e^x) log(1 +/- e^x)),
  --> x,  for x << 0.

procedure: logistic x

procedure: logit x

Logistic and logit functions. Equivalent to:

logistic x = exp(x)/[1 + exp(x)] = 1/[1 + exp(-x)],
logit p = log p/(1 - p).

These functions are inverses of one another. The logit function maps a probablity p in [0, 1] into log-odds x in the extended real line, and the logistic function maps back from log-odds to probabilities.

• The logistic function is defined on the entire real line, but is ill-conditioned for large negative x, with condition number

  x f'(x)/f(x) = x exp(-x)/[1 + exp(-x)].
  

  

  The identity

  logistic(-x) = 1 - logistic(x)
  

  may help to rearrange a computation, along with the logistic-1/2 function which ranges from -1/2 to +1/2 and centered at zero rather than from 0 to 1 and centered at 1/2.

  This implementation gives forward relative error bounded by at most seven times the forward relative error bound of the system math library’s exp, which is usually below 1ulp.

• The logit function is defined on the closed unit interval [0, 1], but is ill-conditioned near 1/2 and 1, with condition number

  x f'(x)/f(x) = 1/[(1 - p) log(p/(1 - p))].
  

  

  The identity

  logit(1 - p) = -logit(p)
  

  may help to rearrange a computation, along with the logit1/2+ function which is defined on -1/2 to +1/2 and centered at zero rather than on 0 to 1 and centered at 1/2.

  This implementation gives forward relative error bounded by at most ten times the forward relative error bound of the system math library’s log, which is usually below 1ulp.

procedure: logistic-1/2 x

procedure: logit1/2+ x

Equivalent to:

logistic-1/2 x = logistic(x) - 1/2,
logit1/2+ p = logit(1/2 + p).

Like logistic and logit, these functions are inverses of one another; unlike logistic and logit, their domains and codomains are both centered at zero.

• The logistic-1/2 function is well-conditioned on the entire real line, with maximum condition number 1 at 0:

  x f'(x)/f(x) = 2 x e^-x / (1 - (e^-x)^2).
  

  

  This implementation gives forward relative error bounded by 5 times the forward relative error bound of the system math library’s exp, which is usually below 1ulp.

• The logit1/2+ function is defined on [-1/2, +1/2], and is ill-conditioned near -1/2 and +1/2:

  x f'(x)/f(x) = x/[(1 - 4 x^2) logit(1/2 + x)].
  

  

  For points near -1/2 or +1/2, it may be better to compute logit of a point near 0 instead. This implementation gives forward relative error bounded by 34 times the forward relative error bound of the system math library’s log, which is usually below 1ulp.

procedure: log-logistic x

procedure: logit-exp x

Equivalent to:

log-logistic x = log(logistic(x)) = log [1/(1 + exp(-x))]
logit-exp x = logit(exp(x)) = log [exp(x)/(1 - exp(x))]

Like logistic and logit, these functions are inverses of one another.

• The loglogistic function maps log-odds on the extended real line to logprobability on the nonpositive half of the extended real line, and is illconditioned for large positive x:

  x f'(x)/f(x) = (-x exp(-x))/[(1 + exp(-x)) log(1 + exp(-x))]
  

  
• The logitexp function maps log-probability on the nonpositive half of the extended real line to log-odds on the extended real line, and is illconditioned near \log(1/2):

  x f'(x)/f(x) = x/[(1 - exp(x)) log(exp(x)/(1 - exp(x)))]
  

  

This implementation gives forward relative error bounded by ten times the forward relative error bound of the system math library’s log and exp, which is usually below 1ulp.

procedure: logsumexp list

List must be a list of real numbers x1, x2, …, xn. Returns an approximation to:

log(exp(x1) + exp(x2) + … + exp(xn)).

The approximation avoids intermediate overflow and underflow. To minimize error, the caller should arrange for the numbers to be sorted from least to greatest.

Edge cases:

• If list is empty, the result is -inf, as if the intermediate sum were zero.
• If list contains only finite numbers and -inf, the -inf elements are ignored, since the exponential of -inf is zero.
• If list contains only finite numbers and +inf, the result is +inf as if the sum had overflowed. (Otherwise, overflow is not possible.)
• If list contains both -inf and +inf, or if list contains any NaNs, the result is a NaN.

Logsumexp never raises any of the standard IEEE 754-2008 floating-point exceptions other than invalid-operation.

procedure: sqrt z

Returns the principal square root of z. The result will have either positive real part, or zero real part and non-negative imaginary part.

procedure: expt z1 z2

Returns z1 raised to the power z2:

procedure: make-rectangular x1 x2

procedure: make-polar x3 x4

procedure: real-part z

procedure: imag-part z

procedure: magnitude z

procedure: angle z

procedure: conjugate z

Suppose x1, x2, x3, and x4 are real numbers and z is a complex number such that Then make-rectangular and make-polar return z, real-part returns x1, imag-part returns x2, magnitude returns x3, and angle returns x4. In the case of angle, whose value is not uniquely determined by the preceding rule, the value returned will be the one in the range minus pi (exclusive) to pi (inclusive).

conjugate returns the complex conjugate of z.

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

procedure: inexact z

procedure: exact->inexact z

inexact returns an inexact representation of z. The value returned is the inexact number that is numerically closest to the argument. For inexact arguments, the result is the same as the argument. For exact complex numbers, the result is a complex number whose real and imaginary parts are the result of applying inexact to the real and imaginary parts of the argument, respectively. If an exact argument has no reasonably close inexact equivalent (in the sense of =), then a violation of an implementation restriction may be reported.

The procedure exact->inexact has been deprecated by R7RS.

procedure: exact z

procedure: inexact->exact z

exact returns an exact representation of z. The value returned is the exact number that is numerically closest to the argument. For exact arguments, the result is the same as the argument. For inexact non-integral real arguments, the implementation may return a rational approximation, or may report an implementation violation. For inexact complex arguments, the result is a complex number whose real and imaginary parts are the result of applying exact to the real and imaginary parts of the argument, respectively. If an inexact argument has no reasonably close exact equivalent (in the sense of =), then a violation of an implementation restriction may be reported.

The procedure inexact->exact has been deprecated by R7RS.

procedure: copysign x1 x2

Returns a real number with the magnitude of x1 and the sign of x2.

(copysign 123 -1)              ⇒ -123
(copysign 0. -1)               ⇒ -0.
(copysign -0. 0.)              ⇒ 0.
(copysign -nan.123 0.)         ⇒ +nan.123