TITLE: Representing IF ... THEN ... ELSE ... in math notation
QUESTION [35 upvotes]: How do I correctly represent the following pseudocode in math notation?
EDIT1: Formula expanded.
EDIT2: Clarification.
(a,b) represents a line segment on a 1D line. a <= b for each segment. The division show below is done as per the following T-SQL code (which I suppose could be represented as a function in the formula?):
Input:  @a1 real, @b1 real, @an real, @bn real
DECLARE @Result real

if @a1 <= @an begin
    SET @Result = @an - @b1

    if @Result <= 0 RETURN 0

    RETURN @Result / @an
end

SET @Result = @a1 - @bn

if @Result <= 0 RETURN 0

RETURN @Result / @a1

Formula:
if m = 1 then
  if (a,b)_1 intersects (a,b)_n then
    r = 1
  else if (a,b)_1 < (a,b)_n then
    r = (a,b)_1 / (a,b)_n
  else
    r = (a,b)_n / (a,b)_1
else if m = 2 then
  if (a,b)_1 intersects (a,b)_n then
    r = 1
  else if (a,b)_1 < (a,b)_n then
    r = (a,b)_1 / (a,b)_n
  else
    r = (a,b)_n / (a,b)_1

The m = 2 block is shown as being the same as the m = 1 one for simplicity's sake.
The divisions are against the two points that are closets to each other, unless the segments intersect, at which point r = 1.

REPLY [7 votes]: This was a rejected edit on the accepted answer so I'm posting it as a new answer instead. I just wanted to point out that "If $\varphi$ then $\psi$, else $\tau$" is equivalent to $(\varphi\wedge\psi)\vee(\neg\varphi\wedge\tau)$. Since $P \to Q$ is equivalent to $\neg P \vee Q$, we can expand $(\varphi\rightarrow\psi)\wedge(\neg\varphi\rightarrow\tau)$ as follows:
$\begin{align*}
 (\varphi\rightarrow\psi)\wedge(\neg\varphi\rightarrow\tau) &\iff (\neg\varphi\vee\psi)\wedge(\varphi\vee\tau) \\ 
 &\iff \left((\neg\varphi\vee\psi)\wedge\varphi\right)\vee\left((\neg\varphi\vee\psi)\wedge\tau\right) \\ 
 &\iff (\varphi\wedge\psi)\vee(\neg\varphi\wedge\tau)\vee(\psi\wedge\tau)
\end{align*}$
The last term, $(\psi\wedge\tau)$, is redundant. This can be corroborated with a truth table but it should be intuitive as the first two terms cover all cases due to the presence of $\varphi$ and $\neg\varphi$. Thus, the concept of "If $\varphi$ then $\psi$, else $\tau$" is mathematically equivalent to the sentential logic formula $(\varphi\wedge\psi)\vee(\neg\varphi\wedge\tau)$.