Average Error: 34.0 → 8.0
Time: 5.3s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.7431685240570133 \cdot 10^{102}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.0417939395900796 \cdot 10^{-259}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 4263616953.59318209:\\ \;\;\;\;\frac{1 \cdot \frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.7431685240570133 \cdot 10^{102}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.0417939395900796 \cdot 10^{-259}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{elif}\;b_2 \le 4263616953.59318209:\\
\;\;\;\;\frac{1 \cdot \frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r17846 = b_2;
        double r17847 = -r17846;
        double r17848 = r17846 * r17846;
        double r17849 = a;
        double r17850 = c;
        double r17851 = r17849 * r17850;
        double r17852 = r17848 - r17851;
        double r17853 = sqrt(r17852);
        double r17854 = r17847 + r17853;
        double r17855 = r17854 / r17849;
        return r17855;
}
↓
double f(double a, double b_2, double c) {
        double r17856 = b_2;
        double r17857 = -1.7431685240570133e+102;
        bool r17858 = r17856 <= r17857;
        double r17859 = 0.5;
        double r17860 = c;
        double r17861 = r17860 / r17856;
        double r17862 = r17859 * r17861;
        double r17863 = 2.0;
        double r17864 = a;
        double r17865 = r17856 / r17864;
        double r17866 = r17863 * r17865;
        double r17867 = r17862 - r17866;
        double r17868 = 1.0417939395900796e-259;
        bool r17869 = r17856 <= r17868;
        double r17870 = -r17856;
        double r17871 = r17856 * r17856;
        double r17872 = r17864 * r17860;
        double r17873 = r17871 - r17872;
        double r17874 = sqrt(r17873);
        double r17875 = r17870 + r17874;
        double r17876 = r17875 / r17864;
        double r17877 = 4263616953.593182;
        bool r17878 = r17856 <= r17877;
        double r17879 = 1.0;
        double r17880 = r17870 - r17874;
        double r17881 = r17880 / r17860;
        double r17882 = r17864 / r17881;
        double r17883 = r17879 * r17882;
        double r17884 = r17883 / r17864;
        double r17885 = -0.5;
        double r17886 = r17885 * r17861;
        double r17887 = r17878 ? r17884 : r17886;
        double r17888 = r17869 ? r17876 : r17887;
        double r17889 = r17858 ? r17867 : r17888;
        return r17889;
}

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 < -1.7431685240570133e+102

    1. Initial program 47.5

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

    2. Taylor expanded around -inf 3.0

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

    if -1.7431685240570133e+102 < b_2 < 1.0417939395900796e-259

    1. Initial program 9.6

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

    if 1.0417939395900796e-259 < b_2 < 4263616953.593182

    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-+29.3

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

    4. Simplified18.2

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

    6. Applied *-un-lft-identity18.2

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

    7. Applied *-un-lft-identity18.2

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

    8. Applied times-frac18.2

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

    9. Simplified18.2

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

    10. Simplified14.5

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

    if 4263616953.593182 < b_2

    1. Initial program 56.2

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

    2. Taylor expanded around inf 4.9

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

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.7431685240570133 \cdot 10^{102}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.0417939395900796 \cdot 10^{-259}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 4263616953.59318209:\\ \;\;\;\;\frac{1 \cdot \frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))