The Standard ML Basis Library

The `MATH` signature

Synopsis

signature MATH structure Math :> MATH where type real = Real.real

The signature MATH specifies basic mathematical constants, the square root function, and trigonometric, hyperbolic, exponential, and logarithmic functions based on a real type. The functions defined here have roughly the same semantics as their counterparts in ISO C's math.h.

The top-level structure Math provides these functions for the default real type Real.real.

In the functions below, unless specified otherwise, if any argument is a NaN, the return value is a NaN. In a list of rules specifying the behavior of a function in special cases, the first matching rule defines the semantics.

Interface

type real val pi : real val e : real val sqrt : real -> real val sin : real -> real val cos : real -> real val tan : real -> real val asin : real -> real val acos : real -> real val atan : real -> real val atan2 : real * real -> real val exp : real -> real val pow : real * real -> real val ln : real -> real val log10 : real -> real val sinh : real -> real val cosh : real -> real val tanh : real -> real

Description

val pi : real

The constant pi (3.141592653...).

val e : real

The base e (2.718281828...) of the natural logarithm.

sqrt x

returns the square root of x. sqrt (~0.0) = ~0.0. If x < 0, it returns NaN.

sin x

cos x

tan x

These return the sine, cosine, and tangent, respectively, of x, measured in radians. If x is an infinity, these functions return NaN. Note that tan will produce infinities at various finite values, roughly corresponding to the singularities of the tangent function.

asin x

acos x

These return the arc sine and arc cosine, respectively, of x. asin is the inverse of sin. Its result is guaranteed to be in the closed interval [-pi/2,pi/2]. acos is the inverse of cos. Its result is guaranteed to be in the closed interval [0,pi]. If the magnitude of x exceeds 1.0, they return NaN.

atan x

returns the arc tangent of x. atan is the inverse of tan. For finite arguments, the result is guaranteed to be in the open interval (-pi/2,pi/2). If x is +infinity, it returns pi/2; if x is -infinity, it returns -pi/2.

atan2 (y, x)

returns the arc tangent of (y/x) in the closed interval [-pi,pi], corresponding to angles within +-180 degrees. The quadrant of the resulting angle is determined using the signs of both x and y, and is the same as the quadrant of the point (x,y). When x = 0, this corresponds to an angle of 90 degrees, and the result is (real (sign y)) * pi/2.0. It holds that

sign ( cos ( atan2 (y,x))) = sign(x)

and

sign ( sin ( atan2 (y,x))) = sign(y)

except for inaccuracies incurred by the finite precision of real and the approximation algorithms used to compute the mathematical functions.

Rules for exceptional cases are specified in the following table.

`y`	`x`	`atan2(y,x)`
+-0	0 < `x`	+-0
+-0	+0	+-0
+-0	`x` < 0	+-pi
+-0	-0	+-pi
`y`, 0 < `y`	+-0	pi/2
`y`, `y` < 0	+-0	-pi/2
+-`y`, finite `y` > 0	+infinity	+-0
+-`y`, finite `y` > 0	-infinity	+-pi
+-infinity	finite `x`	+-pi/2
+-infinity	+infinity	+-pi/4
+-infinity	-infinity	+-3pi/4

exp x

returns e^(x), i.e., e raised to the x^(th) power. If x is +infinity, it returns +infinity; if x is -infinity, it returns 0.

pow (x, y)

returns x^(y), i.e., x raised to the y^(th) power. For finite x and y, this is well-defined when x > 0, or when x < 0 and y is integral. Rules for exceptional cases are specified below.

`x`	`y`	`pow(x,y)`
`x`, including NaN	0	1
\|`x`\| > 1	+infinity	+infinity
\|`x`\| < 1	+infinity	+0
\|`x`\| > 1	-infinity	+0
\|`x`\| < 1	-infinity	+infinity
+infinity	`y` > 0	+infinity
+infinity	`y` < 0	+0
-infinity	`y` > 0, odd integer	-infinity
-infinity	`y` > 0, not odd integer	+infinity
-infinity	`y` < 0, odd integer	-0
-infinity	`y` < 0, not odd integer	+0
`x`	NaN	NaN
NaN	`y` <> 0	NaN
+-1	+-infinity	NaN
finite `x` < 0	finite non-integer `y`	NaN
+-0	`y` < 0, odd integer	+-infinity
+-0	finite `y` < 0, not odd integer	+infinity
+-0	`y` > 0, odd integer	+-0
+-0	`y` > 0, not odd integer	+0

ln x

log10 r

These return the natural logarithm (base e) and decimal logarithm (base 10), respectively, of x. If x < 0, they return NaN; if x = 0, they return -infinity; if x is infinity, they return infinity.

sinh x

cosh x

tanh x

These return the hyperbolic sine, hyperbolic cosine, and hyperbolic tangent, respectively, of x, that is, the values (e^(x) - e^(-x)) / 2, (e^(x) + e^(-x)) / 2, and (sinh x)/(cosh x).

For infinities, we have sinh +-infinity = +-infinity, cosh +-infinity = +infinity, and tanh +-infinity = +-1.

The Standard ML Basis Library

The MATH signature

Synopsis

Interface

Description

See Also

The `MATH` signature