Average Error: 33.9 → 15.0
Time: 17.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.0519886074223697 \cdot 10^{+155}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.3635892865650846 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.0519886074223697 \cdot 10^{+155}:\\
\;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\

\mathbf{elif}\;b \le 1.3635892865650846 \cdot 10^{-93}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r4124700 = b;
        double r4124701 = -r4124700;
        double r4124702 = r4124700 * r4124700;
        double r4124703 = 3.0;
        double r4124704 = a;
        double r4124705 = r4124703 * r4124704;
        double r4124706 = c;
        double r4124707 = r4124705 * r4124706;
        double r4124708 = r4124702 - r4124707;
        double r4124709 = sqrt(r4124708);
        double r4124710 = r4124701 + r4124709;
        double r4124711 = r4124710 / r4124705;
        return r4124711;
}
↓
double f(double a, double b, double c) {
        double r4124712 = b;
        double r4124713 = -2.0519886074223697e+155;
        bool r4124714 = r4124712 <= r4124713;
        double r4124715 = 1.5;
        double r4124716 = a;
        double r4124717 = c;
        double r4124718 = r4124716 * r4124717;
        double r4124719 = r4124718 / r4124712;
        double r4124720 = r4124715 * r4124719;
        double r4124721 = r4124720 - r4124712;
        double r4124722 = r4124721 - r4124712;
        double r4124723 = 3.0;
        double r4124724 = r4124723 * r4124716;
        double r4124725 = r4124722 / r4124724;
        double r4124726 = 1.3635892865650846e-93;
        bool r4124727 = r4124712 <= r4124726;
        double r4124728 = r4124712 * r4124712;
        double r4124729 = r4124724 * r4124717;
        double r4124730 = r4124728 - r4124729;
        double r4124731 = sqrt(r4124730);
        double r4124732 = r4124731 - r4124712;
        double r4124733 = r4124732 / r4124724;
        double r4124734 = -1.5;
        double r4124735 = r4124734 * r4124719;
        double r4124736 = r4124735 / r4124724;
        double r4124737 = r4124727 ? r4124733 : r4124736;
        double r4124738 = r4124714 ? r4124725 : r4124737;
        return r4124738;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -2.0519886074223697e+155

    1. Initial program 60.9

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

    2. Simplified60.9

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

    3. Taylor expanded around -inf 11.7

      \[\leadsto \frac{\color{blue}{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right)} - b}{3 \cdot a}\]

    if -2.0519886074223697e+155 < b < 1.3635892865650846e-93

    1. Initial program 11.6

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

    2. Simplified11.6

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

    if 1.3635892865650846e-93 < b

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 19.6

      \[\leadsto \frac{\color{blue}{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.0519886074223697 \cdot 10^{+155}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.3635892865650846 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))