Average Error: 19.9 → 6.7
Time: 18.8s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.132241338163917783305627833046111956144 \cdot 10^{101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \left(\left(c \cdot 2\right) \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 1.042426094136287989665052757228371789389 \cdot 10^{152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - \left(c \cdot 2\right) \cdot \frac{a}{b}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.132241338163917783305627833046111956144 \cdot 10^{101}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \left(\left(c \cdot 2\right) \cdot \frac{a}{b} - b\right)}\\

\end{array}\\

\mathbf{elif}\;b \le 1.042426094136287989665052757228371789389 \cdot 10^{152}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \left(b - \left(c \cdot 2\right) \cdot \frac{a}{b}\right)}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\

\end{array}
double f(double a, double b, double c) {
        double r1469801 = b;
        double r1469802 = 0.0;
        bool r1469803 = r1469801 >= r1469802;
        double r1469804 = -r1469801;
        double r1469805 = r1469801 * r1469801;
        double r1469806 = 4.0;
        double r1469807 = a;
        double r1469808 = r1469806 * r1469807;
        double r1469809 = c;
        double r1469810 = r1469808 * r1469809;
        double r1469811 = r1469805 - r1469810;
        double r1469812 = sqrt(r1469811);
        double r1469813 = r1469804 - r1469812;
        double r1469814 = 2.0;
        double r1469815 = r1469814 * r1469807;
        double r1469816 = r1469813 / r1469815;
        double r1469817 = r1469814 * r1469809;
        double r1469818 = r1469804 + r1469812;
        double r1469819 = r1469817 / r1469818;
        double r1469820 = r1469803 ? r1469816 : r1469819;
        return r1469820;
}
↓
double f(double a, double b, double c) {
        double r1469821 = b;
        double r1469822 = -1.1322413381639178e+101;
        bool r1469823 = r1469821 <= r1469822;
        double r1469824 = 0.0;
        bool r1469825 = r1469821 >= r1469824;
        double r1469826 = -r1469821;
        double r1469827 = r1469821 * r1469821;
        double r1469828 = 4.0;
        double r1469829 = a;
        double r1469830 = r1469828 * r1469829;
        double r1469831 = c;
        double r1469832 = r1469830 * r1469831;
        double r1469833 = r1469827 - r1469832;
        double r1469834 = sqrt(r1469833);
        double r1469835 = r1469826 - r1469834;
        double r1469836 = 2.0;
        double r1469837 = r1469836 * r1469829;
        double r1469838 = r1469835 / r1469837;
        double r1469839 = r1469831 * r1469836;
        double r1469840 = r1469829 / r1469821;
        double r1469841 = r1469839 * r1469840;
        double r1469842 = r1469841 - r1469821;
        double r1469843 = r1469826 + r1469842;
        double r1469844 = r1469839 / r1469843;
        double r1469845 = r1469825 ? r1469838 : r1469844;
        double r1469846 = 1.042426094136288e+152;
        bool r1469847 = r1469821 <= r1469846;
        double r1469848 = sqrt(r1469834);
        double r1469849 = cbrt(r1469833);
        double r1469850 = r1469849 * r1469849;
        double r1469851 = sqrt(r1469850);
        double r1469852 = sqrt(r1469849);
        double r1469853 = r1469851 * r1469852;
        double r1469854 = sqrt(r1469853);
        double r1469855 = r1469848 * r1469854;
        double r1469856 = r1469826 + r1469855;
        double r1469857 = r1469839 / r1469856;
        double r1469858 = r1469825 ? r1469838 : r1469857;
        double r1469859 = r1469821 - r1469841;
        double r1469860 = r1469826 - r1469859;
        double r1469861 = r1469860 / r1469837;
        double r1469862 = r1469834 + r1469826;
        double r1469863 = r1469839 / r1469862;
        double r1469864 = r1469825 ? r1469861 : r1469863;
        double r1469865 = r1469847 ? r1469858 : r1469864;
        double r1469866 = r1469823 ? r1469845 : r1469865;
        return r1469866;
}

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 < -1.1322413381639178e+101

    1. Initial program 30.5

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Taylor expanded around -inf 6.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}}\\ \end{array}\]

    3. Simplified2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(\left(2 \cdot c\right) \cdot \frac{a}{b} - b\right)}}\\ \end{array}\]

    if -1.1322413381639178e+101 < b < 1.042426094136288e+152

    1. Initial program 8.8

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt8.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]

    4. Applied sqrt-prod8.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt8.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot \sqrt{\sqrt{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]

    7. Applied sqrt-prod8.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \color{blue}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot \sqrt{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]

    if 1.042426094136288e+152 < b

    1. Initial program 62.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Taylor expanded around inf 11.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    3. Simplified3.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b - \left(2 \cdot c\right) \cdot \frac{a}{b}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.132241338163917783305627833046111956144 \cdot 10^{101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \left(\left(c \cdot 2\right) \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 1.042426094136287989665052757228371789389 \cdot 10^{152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - \left(c \cdot 2\right) \cdot \frac{a}{b}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))