### 4Mathematical Functions

This chapter describes the basic mathematical constants and functions provided by the Science Collection.

The constants and functions described in this chapter are defined in the "math.ss" file in the Science Collection and are made available using the form:

 (require (planet williams/science/math))

#### 4.1Mathematical Constants

The following are the mathematical constants defined by the Science Collection:

 syntax
The base of the exponentials, e
 syntax
The base two logarithm of e, log2 e
 syntax
The base ten logarithm of e, log10 e
 syntax
The square root of two, 2
 syntax
The square root of one half, √½
 syntax
The square root of three, 3

The constant pi was added to racket/math, which is included in the racket language. Therefore, it has been removed from the Science Collection.

 syntax
Pi time two, 2π
 syntax
Pi time four, 4π
 syntax
Pi divided by two, π/2
 syntax
Pi divided by four, π/4
 syntax
The square root of pi, π)
 syntax
Two divided by the square root of pi, 2/√π
 syntax
The reciprocal of pi, 1/π
 syntax
Twice the reciprocal of pi, 2/π
 syntax
The natural log of ten, ln 10 or loge 10
 syntax
The inverse of the natural log of ten, 1/ln 10
 syntax
The natural log of two, ln 2 or loge 2
 syntax
The inverse of the natural log of two, 1/ln 2
 syntax
The natural log of pi, ln π or loge π
 syntax
Euler’s constant, γ

#### 4.2Testing for Infinities and Not-a-Number

Racket provides +inf.0 (positive infinity), -inf.0 (negative infinity), +nan.0 (not a number), and +nan.0 (same as +nan.0) as inexact numerical constants. The following functions are provided as a convenience for checking for infinities and not-a-number.

Racket now provides the predicate functions nan? and infinite?. These have been removed from the Science Collection.

 procedure(infinite x) → (or/c -1 #f 1) x : any/c
Returns 1 if x is positive infinity (i.e., equivalent to +inf.0), -1 if x is negative infinity (i.e., equivalent to -inf.0), and false, #f, otherwise. Note that (finite? x) is not equivalent to (not (infinite? x)), since both finite? and infinite? return false, #f for anything that is not a real number.

 procedure(finite? x) → boolean? x : any/c
Returns true, #t, if x is a finite real number and false, #f, otherwise. Note that (finite? x) is not equivalent to (not (infinite? x)), since both finite? and infinite? return false, #f for anything that is not a real number.

#### 4.3Elementary Functions

The following functions provide some elementary mathematical functions that are not provide by Racket.

 procedure(log1p x) → number? x : real?
Computes the value of log(1 + x) in a way that is accurate for small x.

 procedure(expm1 x) → real? x : real?
Computes the value of exp(x - 1) in a way that is accurate for small x.

 procedure(hypot x y) → real? x : real? y : real?
Computes the value of (x2 + y2)½ in a way that avoids overflow.

 procedure(acosh x) → real? x : real?
Computes the value of the hyperbolic arccosine, arccosh, of x.

 procedure(asinh x) → real? x : real?
Computes the value of the hyperbolic arcsine, arcsinh, of x.

 procedure(atahh x) → real? x : real?
Computes the value of the hyperbolic arctangent, arctanh, of x.

 procedure(ldexp x e) → real? x : real? e : integer?
Computes the value of x × 2e.

procedure

(frexp x)
 real? integer?
x : real?
Splits the real number x into a normalized fraction f and exponent e such that x = f × 2e and 0.5 ≤ f < 1. The function returns f and e as multiple values. If x is zero, both f and e are returned as zero.

#### 4.4Testing the Sign of Numbers

 procedure(sign x) → (integer-in -1 1) x : real?
Returns the sign of x as 1 if x ≥ 0 and -1 if x < 0. Note that the sign of zero is positive, regardless of its floating-point sign bit.

#### 4.5Approximate Comparisons of Real Numbers

It is sometimes useful to be able to compare two real (in particular, floating-point) numbers approximately to allow for rounding and truncation errors. The following functions implements the approximate floating-point comparison algorithm proposed by D.E. Knuth in Section 4.2.2 of Seminumerical Algorithms (3rd edition) [Knuth].

 procedure(fcmp x y epsilon) → (integer-in -1 1) x : real? y : real? epsilon : real?
Determines whether x and y are approximately equal to within a relative accuracy, epsilon. The relative accuracy is measured using an interval of 2 × delta, where delta = 2k × epsilon and k is the maximum base 2 exponent of x and y as computed by the function frexp. If x and y lie within this interval, they are considered equal and the function returns 0. Otherwise, if x < y, the function returns -1, or if x > y>, the function returns 1.

The implementation of this function is based on the packege fcmp by T.C. Belding.

#### 4.6Log10 and Decibels (dB)

 procedure(log10 x) → real? x : real?
Returns the log base 10 of x, log10(x).

 procedure(dB x) → real? x : real?
Returns the value of x in decibels, 10*log10(x).