Average Error: 52.5 → 6.2
Time: 51.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{b} \cdot \frac{-1}{2}\]
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r14149074 = b;
        double r14149075 = -r14149074;
        double r14149076 = r14149074 * r14149074;
        double r14149077 = 3.0;
        double r14149078 = a;
        double r14149079 = r14149077 * r14149078;
        double r14149080 = c;
        double r14149081 = r14149079 * r14149080;
        double r14149082 = r14149076 - r14149081;
        double r14149083 = sqrt(r14149082);
        double r14149084 = r14149075 + r14149083;
        double r14149085 = r14149084 / r14149079;
        return r14149085;
}
↓
double f(double __attribute__((unused)) a, double b, double c, double __attribute__((unused)) d) {
        double r14149086 = c;
        double r14149087 = b;
        double r14149088 = r14149086 / r14149087;
        double r14149089 = -0.5;
        double r14149090 = r14149088 * r14149089;
        return r14149090;
}


\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{b} \cdot \frac{-1}{2}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 52.5

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

  2. Simplified52.5

    \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}}\]

  3. Taylor expanded around inf 6.2

    \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]

  4. Final simplification6.2

    \[\leadsto \frac{c}{b} \cdot \frac{-1}{2}\]

Reproduce

herbie shell --seed 2019102 
(FPCore (a b c d)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))