Average Error: 33.8 → 15.6
Time: 17.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.77269298205175246694556857744985395776 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 4.106481463860356990792798090588830555114 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \sqrt[3]{c} \cdot \left(\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \left(3 \cdot a\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\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 -9.77269298205175246694556857744985395776 \cdot 10^{154}:\\
\;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\

\mathbf{elif}\;b \le 4.106481463860356990792798090588830555114 \cdot 10^{-26}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \sqrt[3]{c} \cdot \left(\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \left(3 \cdot a\right)\right)} - 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 r4941726 = b;
        double r4941727 = -r4941726;
        double r4941728 = r4941726 * r4941726;
        double r4941729 = 3.0;
        double r4941730 = a;
        double r4941731 = r4941729 * r4941730;
        double r4941732 = c;
        double r4941733 = r4941731 * r4941732;
        double r4941734 = r4941728 - r4941733;
        double r4941735 = sqrt(r4941734);
        double r4941736 = r4941727 + r4941735;
        double r4941737 = r4941736 / r4941731;
        return r4941737;
}
↓
double f(double a, double b, double c) {
        double r4941738 = b;
        double r4941739 = -9.772692982051752e+154;
        bool r4941740 = r4941738 <= r4941739;
        double r4941741 = 1.5;
        double r4941742 = a;
        double r4941743 = c;
        double r4941744 = r4941742 * r4941743;
        double r4941745 = r4941744 / r4941738;
        double r4941746 = r4941741 * r4941745;
        double r4941747 = r4941746 - r4941738;
        double r4941748 = r4941747 - r4941738;
        double r4941749 = 3.0;
        double r4941750 = r4941749 * r4941742;
        double r4941751 = r4941748 / r4941750;
        double r4941752 = 4.106481463860357e-26;
        bool r4941753 = r4941738 <= r4941752;
        double r4941754 = r4941738 * r4941738;
        double r4941755 = cbrt(r4941743);
        double r4941756 = r4941755 * r4941755;
        double r4941757 = r4941756 * r4941750;
        double r4941758 = r4941755 * r4941757;
        double r4941759 = r4941754 - r4941758;
        double r4941760 = sqrt(r4941759);
        double r4941761 = r4941760 - r4941738;
        double r4941762 = r4941761 / r4941750;
        double r4941763 = -1.5;
        double r4941764 = r4941763 * r4941745;
        double r4941765 = r4941764 / r4941750;
        double r4941766 = r4941753 ? r4941762 : r4941765;
        double r4941767 = r4941740 ? r4941751 : r4941766;
        return r4941767;
}

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 < -9.772692982051752e+154

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around -inf 10.7

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

    if -9.772692982051752e+154 < b < 4.106481463860357e-26

    1. Initial program 14.2

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

    2. Simplified14.2

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

    4. Applied add-cube-cbrt14.4

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

    5. Applied associate-*r*14.4

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

    if 4.106481463860357e-26 < b

    1. Initial program 54.3

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

    2. Simplified54.3

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

    3. Taylor expanded around inf 18.8

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

  4. Final simplification15.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.77269298205175246694556857744985395776 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 4.106481463860356990792798090588830555114 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \sqrt[3]{c} \cdot \left(\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \left(3 \cdot a\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))