Quadratic equation in Ada

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;