Average Error: 34.5 → 15.5
Time: 20.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.700064339395893085146742845772084716487 \cdot 10^{61}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 3.564941316298825958159967506173783765672 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\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 -4.700064339395893085146742845772084716487 \cdot 10^{61}:\\
\;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\

\mathbf{elif}\;b \le 3.564941316298825958159967506173783765672 \cdot 10^{-72}:\\
\;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} - b}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r4181774 = b;
        double r4181775 = -r4181774;
        double r4181776 = r4181774 * r4181774;
        double r4181777 = 3.0;
        double r4181778 = a;
        double r4181779 = r4181777 * r4181778;
        double r4181780 = c;
        double r4181781 = r4181779 * r4181780;
        double r4181782 = r4181776 - r4181781;
        double r4181783 = sqrt(r4181782);
        double r4181784 = r4181775 + r4181783;
        double r4181785 = r4181784 / r4181779;
        return r4181785;
}
↓
double f(double a, double b, double c) {
        double r4181786 = b;
        double r4181787 = -4.700064339395893e+61;
        bool r4181788 = r4181786 <= r4181787;
        double r4181789 = 1.5;
        double r4181790 = a;
        double r4181791 = c;
        double r4181792 = r4181790 * r4181791;
        double r4181793 = r4181792 / r4181786;
        double r4181794 = r4181789 * r4181793;
        double r4181795 = r4181794 - r4181786;
        double r4181796 = r4181795 - r4181786;
        double r4181797 = 3.0;
        double r4181798 = r4181797 * r4181790;
        double r4181799 = r4181796 / r4181798;
        double r4181800 = 3.564941316298826e-72;
        bool r4181801 = r4181786 <= r4181800;
        double r4181802 = r4181786 * r4181786;
        double r4181803 = r4181798 * r4181791;
        double r4181804 = r4181802 - r4181803;
        double r4181805 = sqrt(r4181804);
        double r4181806 = sqrt(r4181805);
        double r4181807 = r4181806 * r4181806;
        double r4181808 = r4181807 - r4181786;
        double r4181809 = r4181808 / r4181798;
        double r4181810 = -1.5;
        double r4181811 = r4181810 * r4181793;
        double r4181812 = r4181811 / r4181798;
        double r4181813 = r4181801 ? r4181809 : r4181812;
        double r4181814 = r4181788 ? r4181799 : r4181813;
        return r4181814;
}

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 < -4.700064339395893e+61

    1. Initial program 39.2

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

    2. Simplified39.2

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

    3. Taylor expanded around -inf 11.1

      \[\leadsto \frac{\color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)} - b}{3 \cdot a}\]

    if -4.700064339395893e+61 < b < 3.564941316298826e-72

    1. Initial program 13.5

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

    2. Simplified13.5

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

    4. Applied add-sqr-sqrt13.5

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

    5. Applied sqrt-prod13.8

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

    if 3.564941316298826e-72 < b

    1. Initial program 53.9

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

    2. Simplified53.9

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

    3. Taylor expanded around inf 19.3

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.700064339395893085146742845772084716487 \cdot 10^{61}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 3.564941316298825958159967506173783765672 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))