Average Error: 34.0 → 8.3
Time: 8.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -24372516198231232945584128508559360:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.421067498413492401141506084834609417442 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}} \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 3.907427008490282589205280019135947132005 \cdot 10^{93}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -24372516198231232945584128508559360:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.421067498413492401141506084834609417442 \cdot 10^{-305}:\\
\;\;\;\;\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}} \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 3.907427008490282589205280019135947132005 \cdot 10^{93}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r31838 = b_2;
        double r31839 = -r31838;
        double r31840 = r31838 * r31838;
        double r31841 = a;
        double r31842 = c;
        double r31843 = r31841 * r31842;
        double r31844 = r31840 - r31843;
        double r31845 = sqrt(r31844);
        double r31846 = r31839 - r31845;
        double r31847 = r31846 / r31841;
        return r31847;
}
↓
double f(double a, double b_2, double c) {
        double r31848 = b_2;
        double r31849 = -2.4372516198231233e+34;
        bool r31850 = r31848 <= r31849;
        double r31851 = -0.5;
        double r31852 = c;
        double r31853 = r31852 / r31848;
        double r31854 = r31851 * r31853;
        double r31855 = 2.4210674984134924e-305;
        bool r31856 = r31848 <= r31855;
        double r31857 = 1.0;
        double r31858 = r31848 * r31848;
        double r31859 = a;
        double r31860 = r31859 * r31852;
        double r31861 = r31858 - r31860;
        double r31862 = sqrt(r31861);
        double r31863 = r31862 - r31848;
        double r31864 = r31863 / r31859;
        double r31865 = r31864 / r31852;
        double r31866 = r31857 / r31865;
        double r31867 = r31857 / r31859;
        double r31868 = r31866 * r31867;
        double r31869 = 3.9074270084902826e+93;
        bool r31870 = r31848 <= r31869;
        double r31871 = -r31848;
        double r31872 = r31871 - r31862;
        double r31873 = r31859 / r31872;
        double r31874 = r31857 / r31873;
        double r31875 = -2.0;
        double r31876 = r31848 / r31859;
        double r31877 = r31875 * r31876;
        double r31878 = r31870 ? r31874 : r31877;
        double r31879 = r31856 ? r31868 : r31878;
        double r31880 = r31850 ? r31854 : r31879;
        return r31880;
}

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 < -2.4372516198231233e+34

    1. Initial program 56.6

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

    2. Taylor expanded around -inf 4.8

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

    if -2.4372516198231233e+34 < b_2 < 2.4210674984134924e-305

    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. Simplified17.4

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

    5. Simplified17.4

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied clear-num17.5

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

    8. Simplified14.6

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}}}}{a}\]
    9. Using strategy rm

    10. Applied div-inv14.7

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

    if 2.4210674984134924e-305 < b_2 < 3.9074270084902826e+93

    1. Initial program 9.0

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

    3. Applied clear-num9.1

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

    if 3.9074270084902826e+93 < b_2

    1. Initial program 45.2

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

    3. Applied flip--62.9

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

    4. Simplified62.0

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

    5. Simplified62.0

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

    6. Taylor expanded around 0 3.5

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -24372516198231232945584128508559360:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.421067498413492401141506084834609417442 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}} \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 3.907427008490282589205280019135947132005 \cdot 10^{93}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))