Average Error: 34.4 → 7.1
Time: 21.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.30101840923646093 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -9.97798043992307827 \cdot 10^{-261}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.94192058439483012 \cdot 10^{74}:\\ \;\;\;\;\frac{1}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.30101840923646093 \cdot 10^{98}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -9.97798043992307827 \cdot 10^{-261}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 5.94192058439483012 \cdot 10^{74}:\\
\;\;\;\;\frac{1}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}{\sqrt[3]{a}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r88960 = b_2;
        double r88961 = -r88960;
        double r88962 = r88960 * r88960;
        double r88963 = a;
        double r88964 = c;
        double r88965 = r88963 * r88964;
        double r88966 = r88962 - r88965;
        double r88967 = sqrt(r88966);
        double r88968 = r88961 + r88967;
        double r88969 = r88968 / r88963;
        return r88969;
}
↓
double f(double a, double b_2, double c) {
        double r88970 = b_2;
        double r88971 = -4.301018409236461e+98;
        bool r88972 = r88970 <= r88971;
        double r88973 = 0.5;
        double r88974 = c;
        double r88975 = r88974 / r88970;
        double r88976 = r88973 * r88975;
        double r88977 = 2.0;
        double r88978 = a;
        double r88979 = r88970 / r88978;
        double r88980 = r88977 * r88979;
        double r88981 = r88976 - r88980;
        double r88982 = -9.977980439923078e-261;
        bool r88983 = r88970 <= r88982;
        double r88984 = -r88970;
        double r88985 = r88970 * r88970;
        double r88986 = r88978 * r88974;
        double r88987 = r88985 - r88986;
        double r88988 = sqrt(r88987);
        double r88989 = r88984 + r88988;
        double r88990 = 1.0;
        double r88991 = r88990 / r88978;
        double r88992 = r88989 * r88991;
        double r88993 = 5.94192058439483e+74;
        bool r88994 = r88970 <= r88993;
        double r88995 = cbrt(r88978);
        double r88996 = r88995 * r88995;
        double r88997 = r88990 / r88996;
        double r88998 = r88996 * r88997;
        double r88999 = r88990 / r88998;
        double r89000 = r88984 - r88988;
        double r89001 = r89000 / r88995;
        double r89002 = r89001 / r88974;
        double r89003 = r88990 / r89002;
        double r89004 = r89003 / r88995;
        double r89005 = r88999 * r89004;
        double r89006 = -0.5;
        double r89007 = r89006 * r88975;
        double r89008 = r88994 ? r89005 : r89007;
        double r89009 = r88983 ? r88992 : r89008;
        double r89010 = r88972 ? r88981 : r89009;
        return r89010;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -4.301018409236461e+98

    1. Initial program 47.2

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 3.9

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    if -4.301018409236461e+98 < b_2 < -9.977980439923078e-261

    1. Initial program 8.4

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv8.6

      \[\leadsto \color{blue}{\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]

    if -9.977980439923078e-261 < b_2 < 5.94192058439483e+74

    1. Initial program 29.3

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip-+29.4

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Simplified16.1

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
    5. Using strategy rm

    6. Applied clear-num16.3

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{0 + a \cdot c}}}}{a}\]

    7. Simplified14.8

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}}{a}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt15.6

      \[\leadsto \frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

    10. Applied *-un-lft-identity15.6

      \[\leadsto \frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{\color{blue}{1 \cdot c}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    11. Applied add-cube-cbrt14.9

      \[\leadsto \frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}}{1 \cdot c}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    12. Applied *-un-lft-identity14.9

      \[\leadsto \frac{\frac{1}{\frac{\frac{\color{blue}{1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{1 \cdot c}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    13. Applied times-frac14.9

      \[\leadsto \frac{\frac{1}{\frac{\color{blue}{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}}{1 \cdot c}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    14. Applied times-frac14.4

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{1} \cdot \frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    15. Applied add-sqr-sqrt14.4

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{1} \cdot \frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    16. Applied times-frac14.0

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{1}}{\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{1}} \cdot \frac{\sqrt{1}}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    17. Applied times-frac10.7

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{1}}{\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{1}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\sqrt{1}}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}{\sqrt[3]{a}}}\]

    18. Simplified10.7

      \[\leadsto \color{blue}{\frac{1}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}} \cdot \frac{\frac{\sqrt{1}}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}{\sqrt[3]{a}}\]

    19. Simplified10.7

      \[\leadsto \frac{1}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \color{blue}{\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}{\sqrt[3]{a}}}\]

    if 5.94192058439483e+74 < b_2

    1. Initial program 59.0

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 3.3

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.30101840923646093 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -9.97798043992307827 \cdot 10^{-261}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.94192058439483012 \cdot 10^{74}:\\ \;\;\;\;\frac{1}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\sqrt[3]{a}}}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))