Average Error: 19.4 → 8.5
Time: 6.9s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.324740391027198892223528484548962382928 \cdot 10^{154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 1.344573033869019141349238012800292573281 \cdot 10^{53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{\sqrt[3]{a}}{\frac{\sqrt{b}}{\sqrt[3]{a}}} \cdot \frac{\sqrt[3]{a}}{\frac{\sqrt{b}}{c}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt[3]{{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}^{3}}}{2 \cdot a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.324740391027198892223528484548962382928 \cdot 10^{154}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}\\

\mathbf{elif}\;b \le 1.344573033869019141349238012800292573281 \cdot 10^{53}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{\sqrt[3]{a}}{\frac{\sqrt{b}}{\sqrt[3]{a}}} \cdot \frac{\sqrt[3]{a}}{\frac{\sqrt{b}}{c}}\right) - 2 \cdot b}\\

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

\end{array}
double f(double a, double b, double c) {
        double r37888 = b;
        double r37889 = 0.0;
        bool r37890 = r37888 >= r37889;
        double r37891 = 2.0;
        double r37892 = c;
        double r37893 = r37891 * r37892;
        double r37894 = -r37888;
        double r37895 = r37888 * r37888;
        double r37896 = 4.0;
        double r37897 = a;
        double r37898 = r37896 * r37897;
        double r37899 = r37898 * r37892;
        double r37900 = r37895 - r37899;
        double r37901 = sqrt(r37900);
        double r37902 = r37894 - r37901;
        double r37903 = r37893 / r37902;
        double r37904 = r37894 + r37901;
        double r37905 = r37891 * r37897;
        double r37906 = r37904 / r37905;
        double r37907 = r37890 ? r37903 : r37906;
        return r37907;
}
↓
double f(double a, double b, double c) {
        double r37908 = b;
        double r37909 = -1.3247403910271989e+154;
        bool r37910 = r37908 <= r37909;
        double r37911 = 0.0;
        bool r37912 = r37908 >= r37911;
        double r37913 = 2.0;
        double r37914 = c;
        double r37915 = r37913 * r37914;
        double r37916 = -r37908;
        double r37917 = r37908 * r37908;
        double r37918 = 4.0;
        double r37919 = a;
        double r37920 = r37918 * r37919;
        double r37921 = r37920 * r37914;
        double r37922 = r37917 - r37921;
        double r37923 = sqrt(r37922);
        double r37924 = r37916 - r37923;
        double r37925 = r37915 / r37924;
        double r37926 = r37919 * r37914;
        double r37927 = r37926 / r37908;
        double r37928 = r37913 * r37927;
        double r37929 = 2.0;
        double r37930 = r37929 * r37908;
        double r37931 = r37928 - r37930;
        double r37932 = r37913 * r37919;
        double r37933 = r37931 / r37932;
        double r37934 = r37912 ? r37925 : r37933;
        double r37935 = 1.3445730338690191e+53;
        bool r37936 = r37908 <= r37935;
        double r37937 = log(r37923);
        double r37938 = exp(r37937);
        double r37939 = r37916 - r37938;
        double r37940 = r37915 / r37939;
        double r37941 = r37916 + r37923;
        double r37942 = r37941 / r37932;
        double r37943 = r37912 ? r37940 : r37942;
        double r37944 = cbrt(r37919);
        double r37945 = sqrt(r37908);
        double r37946 = r37945 / r37944;
        double r37947 = r37944 / r37946;
        double r37948 = r37945 / r37914;
        double r37949 = r37944 / r37948;
        double r37950 = r37947 * r37949;
        double r37951 = r37913 * r37950;
        double r37952 = r37951 - r37930;
        double r37953 = r37915 / r37952;
        double r37954 = 3.0;
        double r37955 = pow(r37923, r37954);
        double r37956 = cbrt(r37955);
        double r37957 = r37916 + r37956;
        double r37958 = r37957 / r37932;
        double r37959 = r37912 ? r37953 : r37958;
        double r37960 = r37936 ? r37943 : r37959;
        double r37961 = r37910 ? r37934 : r37960;
        return r37961;
}

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 < -1.3247403910271989e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 10.7

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

    if -1.3247403910271989e+154 < b < 1.3445730338690191e+53

    1. Initial program 8.8

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

    3. Applied add-exp-log10.2

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

    if 1.3445730338690191e+53 < b

    1. Initial program 25.9

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

    2. Taylor expanded around inf 7.4

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

    4. Applied associate-/l*4.1

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

    6. Applied *-un-lft-identity4.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{\color{blue}{1 \cdot c}}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    7. Applied add-sqr-sqrt4.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}{1 \cdot c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    8. Applied times-frac4.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\color{blue}{\frac{\sqrt{b}}{1} \cdot \frac{\sqrt{b}}{c}}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    9. Applied add-cube-cbrt4.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\frac{\sqrt{b}}{1} \cdot \frac{\sqrt{b}}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    10. Applied times-frac4.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \color{blue}{\left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\frac{\sqrt{b}}{1}} \cdot \frac{\sqrt[3]{a}}{\frac{\sqrt{b}}{c}}\right)} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    11. Simplified4.1

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

    13. Applied add-cbrt-cube4.1

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

    14. Simplified4.1

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.324740391027198892223528484548962382928 \cdot 10^{154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 1.344573033869019141349238012800292573281 \cdot 10^{53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{\sqrt[3]{a}}{\frac{\sqrt{b}}{\sqrt[3]{a}}} \cdot \frac{\sqrt[3]{a}}{\frac{\sqrt{b}}{c}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt[3]{{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}^{3}}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))