The Standard ML Basis Library

The `MATH` signature

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.

Synopsis

signature MATH structure Math : MATH

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

type real

pi

denotes the constant pi (3.141592653...).

e

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

sqrt x

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

sin x

cos x

tan x

return the sine, cosine and tangent, respectively, of x, measured in radians. If x is an infinity, 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

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 of +-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 (y,x). When x = 0, this corresponds to an angle of 90 degrees, and the result is sign(y * pi/2). 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	`x`, 0 < x	+-0
+-0	+0	+-0
+-0	`x`, 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	`x`, finite x	+-pi/2
+-infinity	+infinity	+-pi/4
+-infinity	-infinity	+-3pi/4

exp x

returns e^(x), i.e., e raised to the xth 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 yth 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

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

sinh x

cosh x

tanh x

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