Average Error: 33.4 → 15.4
Time: 22.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.277637730923319 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \le 3.32629031803127 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{a \cdot 3}\\ \end{array}\]
\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 -1.277637730923319 \cdot 10^{+112}:\\
\;\;\;\;\frac{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}{a \cdot 3}\\

\mathbf{elif}\;b \le 3.32629031803127 \cdot 10^{-71}:\\
\;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b}{a \cdot 3}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4107742 = b;
        double r4107743 = -r4107742;
        double r4107744 = r4107742 * r4107742;
        double r4107745 = 3.0;
        double r4107746 = a;
        double r4107747 = r4107745 * r4107746;
        double r4107748 = c;
        double r4107749 = r4107747 * r4107748;
        double r4107750 = r4107744 - r4107749;
        double r4107751 = sqrt(r4107750);
        double r4107752 = r4107743 + r4107751;
        double r4107753 = r4107752 / r4107747;
        return r4107753;
}
↓
double f(double a, double b, double c) {
        double r4107754 = b;
        double r4107755 = -1.277637730923319e+112;
        bool r4107756 = r4107754 <= r4107755;
        double r4107757 = 1.5;
        double r4107758 = a;
        double r4107759 = c;
        double r4107760 = r4107758 * r4107759;
        double r4107761 = r4107760 / r4107754;
        double r4107762 = r4107757 * r4107761;
        double r4107763 = 2.0;
        double r4107764 = r4107763 * r4107754;
        double r4107765 = r4107762 - r4107764;
        double r4107766 = 3.0;
        double r4107767 = r4107758 * r4107766;
        double r4107768 = r4107765 / r4107767;
        double r4107769 = 3.32629031803127e-71;
        bool r4107770 = r4107754 <= r4107769;
        double r4107771 = r4107754 * r4107754;
        double r4107772 = r4107766 * r4107760;
        double r4107773 = r4107771 - r4107772;
        double r4107774 = sqrt(r4107773);
        double r4107775 = sqrt(r4107774);
        double r4107776 = r4107775 * r4107775;
        double r4107777 = r4107776 - r4107754;
        double r4107778 = r4107777 / r4107767;
        double r4107779 = -1.5;
        double r4107780 = r4107779 * r4107761;
        double r4107781 = r4107780 / r4107767;
        double r4107782 = r4107770 ? r4107778 : r4107781;
        double r4107783 = r4107756 ? r4107768 : r4107782;
        return r4107783;
}

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 < -1.277637730923319e+112

    1. Initial program 47.4

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

    2. Simplified47.4

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

    3. Taylor expanded around -inf 47.4

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

    4. Taylor expanded around -inf 9.9

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

    if -1.277637730923319e+112 < b < 3.32629031803127e-71

    1. Initial program 13.0

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

    2. Simplified13.0

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

    3. Taylor expanded around -inf 13.0

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

    5. Applied add-sqr-sqrt13.2

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

    if 3.32629031803127e-71 < b

    1. Initial program 52.6

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

    2. Simplified52.6

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

    3. Taylor expanded around -inf 52.6

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

    4. Taylor expanded around inf 19.9

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

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.277637730923319 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \le 3.32629031803127 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{a \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))