Structures | Signatures | Identifiers

Signature MATH

Mathematical floating-point values and operations.

structure Math : MATH

signature MATH =
  sig
    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
  end

[pi]

The constant pi (3.141592653...).

[e]

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

[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

[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). These functions have the following properties:

    sinh +-0            =       +-0
    sinh +-infinity     =       +-infinity
    cosh +-0            =       1
    cosh +-infinity     =       +-infinity
    tanh +-0            =       +-0