math - Quadratic equation in Ada

Question

I just came around and decided to try some Ada. The downside is that the syntax and function strays quite away from C++. So I had to like cram various stuff to get this thing to work.

My question is if there are some better way to do this calculation that what I have done here

   IF(B < 0.0) THEN
      B := ABS(B);
      X1 := (B / 2.0) + Sqrt( (B / 2.0) ** 2.0 + ABS(C));
      X2 := (B / 2.0) - Sqrt( (B / 2.0) ** 2.0 + ABS(C));
   ELSE
      X1 := -(B / 2.0) + Sqrt( (B / 2.0) ** 2.0 - C);
      X2 := -(B / 2.0) - Sqrt( (B / 2.0) ** 2.0 - C);
   END IF;

I had some problem with negative numbers, that's why I did a IF statement and used ABS() to turn those into positive. But the weird thing is that it works perfectly for the other case, which is strange...

Answer 1

Solving quadratic equations is not as simple as most people think.

The standard formula for solving a x^2 + b x + c = 0 is

delta = b^2 - 4 a c
x1 = (-b + sqrt(delta)) / (2 a)   (*)
x2 = (-b - sqrt(delta)) / (2 a)

but when 4 a c << b^2, computing x1 involves subtracting close numbers, and makes you lose accuracy, so you use the following instead

delta as above
x1 = 2 c / (-b - sqrt(delta))     (**)
x2 = 2 c / (-b + sqrt(delta))

which yields a better x1, but whose x2 has the same problem as x1 had above.

The correct way to compute the roots is therefore

q = -0.5 (b + sign(b) sqrt(delta))

and use x1 = q / a and x2 = c / q, which I find very efficient. If you want to handle the case when delta is negative, or complex coefficients, then you must use complex arithmetic (which is quite tricky to get right too).

Edit: With Ada code:

DELTA := B * B - 4.0 * A * C;

IF(B > 0.0) THEN
    Q := -0.5 * (B + SQRT(DELTA));
ELSE
    Q := -0.5 * (B - SQRT(DELTA));
END IF;

X1 := Q / A;
X2 := C / Q;