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

↓

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

double f(double a, double b, double c) {
        double r18932860 = b;
        double r18932861 = -r18932860;
        double r18932862 = r18932860 * r18932860;
        double r18932863 = 4.0;
        double r18932864 = a;
        double r18932865 = r18932863 * r18932864;
        double r18932866 = c;
        double r18932867 = r18932865 * r18932866;
        double r18932868 = r18932862 - r18932867;
        double r18932869 = sqrt(r18932868);
        double r18932870 = r18932861 + r18932869;
        double r18932871 = 2.0;
        double r18932872 = r18932871 * r18932864;
        double r18932873 = r18932870 / r18932872;
        return r18932873;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r18932874 = c;
        double r18932875 = b;
        double r18932876 = r18932874 / r18932875;
        double r18932877 = -r18932876;
        return r18932877;
}

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

↓

-\frac{c}{b}

Derivation

Initial program 52.4
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
Simplified52.4
\[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}}\]
Taylor expanded around inf 6.3
\[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
Simplified6.3
\[\leadsto \color{blue}{-\frac{c}{b}}\]
Final simplification6.3
\[\leadsto -\frac{c}{b}\]

Reproduce

herbie shell --seed 2019102 
(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