Average Error: 34.2 → 10.1
Time: 4.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.2389466313579672 \cdot 10^{127}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.6670468245058271 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -5.2389466313579672 \cdot 10^{127}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 1.6670468245058271 \cdot 10^{-85}:\\
\;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r139772 = b;
        double r139773 = -r139772;
        double r139774 = r139772 * r139772;
        double r139775 = 3.0;
        double r139776 = a;
        double r139777 = r139775 * r139776;
        double r139778 = c;
        double r139779 = r139777 * r139778;
        double r139780 = r139774 - r139779;
        double r139781 = sqrt(r139780);
        double r139782 = r139773 + r139781;
        double r139783 = r139782 / r139777;
        return r139783;
}
↓
double f(double a, double b, double c) {
        double r139784 = b;
        double r139785 = -5.238946631357967e+127;
        bool r139786 = r139784 <= r139785;
        double r139787 = 0.5;
        double r139788 = c;
        double r139789 = r139788 / r139784;
        double r139790 = r139787 * r139789;
        double r139791 = 0.6666666666666666;
        double r139792 = a;
        double r139793 = r139784 / r139792;
        double r139794 = r139791 * r139793;
        double r139795 = r139790 - r139794;
        double r139796 = 1.667046824505827e-85;
        bool r139797 = r139784 <= r139796;
        double r139798 = -r139784;
        double r139799 = r139784 * r139784;
        double r139800 = 3.0;
        double r139801 = r139800 * r139792;
        double r139802 = r139801 * r139788;
        double r139803 = r139799 - r139802;
        double r139804 = sqrt(r139803);
        double r139805 = r139798 + r139804;
        double r139806 = r139805 / r139800;
        double r139807 = r139806 / r139792;
        double r139808 = -0.5;
        double r139809 = r139808 * r139789;
        double r139810 = r139797 ? r139807 : r139809;
        double r139811 = r139786 ? r139795 : r139810;
        return r139811;
}

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 < -5.238946631357967e+127

    1. Initial program 54.3

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

    2. Taylor expanded around -inf 3.6

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}}\]

    if -5.238946631357967e+127 < b < 1.667046824505827e-85

    1. Initial program 12.3

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

    3. Applied associate-/r*12.3

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

    if 1.667046824505827e-85 < b

    1. Initial program 52.8

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

    2. Taylor expanded around inf 9.7

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.2389466313579672 \cdot 10^{127}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.6670468245058271 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))