Average Error: 33.6 → 9.1
Time: 20.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.2887136042886476 \cdot 10^{+71}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.407079315314288 \cdot 10^{-176}:\\ \;\;\;\;\frac{a \cdot \frac{c}{{\left(\sqrt{e}\right)}^{\left(\log \left(b_2 \cdot b_2 - c \cdot a\right)\right)} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 8.016779424032652 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.2887136042886476 \cdot 10^{+71}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.407079315314288 \cdot 10^{-176}:\\
\;\;\;\;\frac{a \cdot \frac{c}{{\left(\sqrt{e}\right)}^{\left(\log \left(b_2 \cdot b_2 - c \cdot a\right)\right)} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 8.016779424032652 \cdot 10^{+82}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1183852 = b_2;
        double r1183853 = -r1183852;
        double r1183854 = r1183852 * r1183852;
        double r1183855 = a;
        double r1183856 = c;
        double r1183857 = r1183855 * r1183856;
        double r1183858 = r1183854 - r1183857;
        double r1183859 = sqrt(r1183858);
        double r1183860 = r1183853 - r1183859;
        double r1183861 = r1183860 / r1183855;
        return r1183861;
}
↓
double f(double a, double b_2, double c) {
        double r1183862 = b_2;
        double r1183863 = -4.2887136042886476e+71;
        bool r1183864 = r1183862 <= r1183863;
        double r1183865 = -0.5;
        double r1183866 = c;
        double r1183867 = r1183866 / r1183862;
        double r1183868 = r1183865 * r1183867;
        double r1183869 = -3.407079315314288e-176;
        bool r1183870 = r1183862 <= r1183869;
        double r1183871 = a;
        double r1183872 = exp(1.0);
        double r1183873 = sqrt(r1183872);
        double r1183874 = r1183862 * r1183862;
        double r1183875 = r1183866 * r1183871;
        double r1183876 = r1183874 - r1183875;
        double r1183877 = log(r1183876);
        double r1183878 = pow(r1183873, r1183877);
        double r1183879 = r1183878 - r1183862;
        double r1183880 = r1183866 / r1183879;
        double r1183881 = r1183871 * r1183880;
        double r1183882 = r1183881 / r1183871;
        double r1183883 = 8.016779424032652e+82;
        bool r1183884 = r1183862 <= r1183883;
        double r1183885 = 1.0;
        double r1183886 = -r1183862;
        double r1183887 = sqrt(r1183876);
        double r1183888 = r1183886 - r1183887;
        double r1183889 = r1183871 / r1183888;
        double r1183890 = r1183885 / r1183889;
        double r1183891 = 0.5;
        double r1183892 = r1183867 * r1183891;
        double r1183893 = 2.0;
        double r1183894 = r1183862 / r1183871;
        double r1183895 = r1183893 * r1183894;
        double r1183896 = r1183892 - r1183895;
        double r1183897 = r1183884 ? r1183890 : r1183896;
        double r1183898 = r1183870 ? r1183882 : r1183897;
        double r1183899 = r1183864 ? r1183868 : r1183898;
        return r1183899;
}

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 < -4.2887136042886476e+71

    1. Initial program 57.3

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

    2. Taylor expanded around -inf 3.3

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

    if -4.2887136042886476e+71 < b_2 < -3.407079315314288e-176

    1. Initial program 36.0

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

    3. Applied flip--36.1

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

    4. Simplified15.5

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

    5. Simplified15.5

      \[\leadsto \frac{\frac{a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity15.5

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

    8. Applied times-frac12.6

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

    9. Simplified12.6

      \[\leadsto \frac{\color{blue}{a} \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\]
    10. Using strategy rm

    11. Applied add-exp-log15.3

      \[\leadsto \frac{a \cdot \frac{c}{\color{blue}{e^{\log \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}} - b_2}}{a}\]
    12. Using strategy rm

    13. Applied pow1/215.3

      \[\leadsto \frac{a \cdot \frac{c}{e^{\log \color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\frac{1}{2}}\right)}} - b_2}}{a}\]

    14. Applied log-pow15.3

      \[\leadsto \frac{a \cdot \frac{c}{e^{\color{blue}{\frac{1}{2} \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}} - b_2}}{a}\]

    15. Applied exp-prod15.7

      \[\leadsto \frac{a \cdot \frac{c}{\color{blue}{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(b_2 \cdot b_2 - a \cdot c\right)\right)}} - b_2}}{a}\]

    16. Simplified15.7

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

    if -3.407079315314288e-176 < b_2 < 8.016779424032652e+82

    1. Initial program 12.0

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

    3. Applied clear-num12.1

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

    if 8.016779424032652e+82 < b_2

    1. Initial program 42.4

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

    2. Taylor expanded around inf 3.6

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.2887136042886476 \cdot 10^{+71}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.407079315314288 \cdot 10^{-176}:\\ \;\;\;\;\frac{a \cdot \frac{c}{{\left(\sqrt{e}\right)}^{\left(\log \left(b_2 \cdot b_2 - c \cdot a\right)\right)} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 8.016779424032652 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))