\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

\[-\frac{c}{b}\]

double f(double a, double b, double c) {
        double r5103603 = b;
        double r5103604 = -r5103603;
        double r5103605 = r5103603 * r5103603;
        double r5103606 = 4.0;
        double r5103607 = a;
        double r5103608 = r5103606 * r5103607;
        double r5103609 = c;
        double r5103610 = r5103608 * r5103609;
        double r5103611 = r5103605 - r5103610;
        double r5103612 = sqrt(r5103611);
        double r5103613 = r5103604 + r5103612;
        double r5103614 = 2.0;
        double r5103615 = r5103614 * r5103607;
        double r5103616 = r5103613 / r5103615;
        return r5103616;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r5103617 = c;
        double r5103618 = b;
        double r5103619 = r5103617 / r5103618;
        double r5103620 = -r5103619;
        return r5103620;
}

\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}

↓

-\frac{c}{b}

Derivation

Initial program 52.6
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
Simplified52.6
\[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}}\]
Taylor expanded around inf 6.1
\[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
Simplified6.1
\[\leadsto \color{blue}{-\frac{c}{b}}\]
Final simplification6.1
\[\leadsto -\frac{c}{b}\]

Reproduce

herbie shell --seed 2019101 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Error

Derivation

Reproduce