If   f,   a,   and   b   are values with uncertainties   σf,   σa,   and   σb,   and   c   is a constant;

then if   f   is derived from   a,   b,   and   c   in the following ways,

then   σf   can be calculated as follows:

Addition/Subtraction
 If   f = a ± c,   or   f = c ± a   then   σf = σa
 If   f = a ± b   then   σf2 = σa2 + σb2
Multiplication/Division
 If   f = ca   or   f = ac       then   σf = |cσa|
 If   f = ab   or   f = a / b   then   σf2 = f2( (σa / a)2 + (σb / b)2)
Exponentiation
 If   f = ac   then   σf = |fc(σa / a)|


Caution:

This implementation of error propagation does not address issues of dependent and independent values.   It is assumed that   a   and   b   are independent and so the formula for multiplication should not be applied to   a*a   for example.   See   the talk page   for some of the implications of this issue.


Task details

 Add an uncertain number type to your language that can support addition, subtraction, multiplication, division, and exponentiation between numbers with an associated error term together with 'normal' floating point numbers without an associated error term. 
Implement enough functionality to perform the following calculations.
 Given coordinates and their errors:
x1 = 100 ± 1.1
y1 = 50 ± 1.2
x2 = 200 ± 2.2
y2 = 100 ± 2.3
 if point p1 is located at (x1, y1) and p2 is at (x2, y2); calculate the distance between the two points using the classic Pythagorean formula: 
  d = √   (x1 - x2)²   +   (y1 - y2)²    
 Print and display both    d    and its error.


References

 A Guide to Error Propagation B. Keeney, 2005.
 Propagation of uncertainty Wikipedia.


Related task

   Quaternion type


generic
   type Real is digits <>;
   with function Sqrt(X: Real) return Real;
   with function "**"(X: Real; Y: Real) return Real;
package Approximation is
 
   type Number is private;
 
   -- create an approximation
   function Approx(Value: Real; Sigma: Real) return Number;
 
   -- unary operations and conversion Real to Number
   function "+"(X: Real) return Number;
   function "-"(X: Real) return Number;
   function "+"(X: Number) return Number;
   function "-"(X: Number) return Number;
 
   -- addition / subtraction
   function "+"(X: Number; Y: Number) return Number;
   function "-"(X: Number; Y: Number) return Number;
 
   -- multiplication / division
   function "*"(X: Number; Y: Number) return Number;
   function "/"(X: Number; Y: Number) return Number;
 
   -- exponentiation
   function "**"(X: Number; Y: Positive) return Number;
   function "**"(X: Number; Y: Real) return Number;
 
   -- Output to Standard IO (wrapper for Ada.Text_IO and Ada.Text_IO.Float_IO)
   procedure Put_Line(Message: String;
                      Item: Number;
                      Value_Fore: Natural := 7;
                      Sigma_Fore: Natural := 4;
                      Aft:  Natural := 2;
                      Exp:  Natural := 0);
   procedure Put(Item: Number;
                 Value_Fore: Natural := 7;
                 Sigma_Fore: Natural := 3;
                 Aft:  Natural := 2;
                 Exp:  Natural := 0);
 
private
   type Number is record
      Value: Real;
      Sigma: Real;
   end record;
end Approximation;