Average Error: 34.3 → 8.9
Time: 39.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -212906428822562352817263083520:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.85046734525134507507504026935831729408 \cdot 10^{-161}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 0.173897874048477174557802982235443778336:\\ \;\;\;\;\frac{\left(-b_2\right) - e^{\log \left(\sqrt{b_2 \cdot b_2 - c \cdot a}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -212906428822562352817263083520:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.85046734525134507507504026935831729408 \cdot 10^{-161}:\\
\;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 0.173897874048477174557802982235443778336:\\
\;\;\;\;\frac{\left(-b_2\right) - e^{\log \left(\sqrt{b_2 \cdot b_2 - c \cdot a}\right)}}{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 r993867 = b_2;
        double r993868 = -r993867;
        double r993869 = r993867 * r993867;
        double r993870 = a;
        double r993871 = c;
        double r993872 = r993870 * r993871;
        double r993873 = r993869 - r993872;
        double r993874 = sqrt(r993873);
        double r993875 = r993868 - r993874;
        double r993876 = r993875 / r993870;
        return r993876;
}
↓
double f(double a, double b_2, double c) {
        double r993877 = b_2;
        double r993878 = -2.1290642882256235e+29;
        bool r993879 = r993877 <= r993878;
        double r993880 = -0.5;
        double r993881 = c;
        double r993882 = r993881 / r993877;
        double r993883 = r993880 * r993882;
        double r993884 = 2.850467345251345e-161;
        bool r993885 = r993877 <= r993884;
        double r993886 = a;
        double r993887 = r993877 * r993877;
        double r993888 = r993881 * r993886;
        double r993889 = r993887 - r993888;
        double r993890 = sqrt(r993889);
        double r993891 = r993890 - r993877;
        double r993892 = r993881 / r993891;
        double r993893 = r993886 * r993892;
        double r993894 = r993893 / r993886;
        double r993895 = 0.17389787404847717;
        bool r993896 = r993877 <= r993895;
        double r993897 = -r993877;
        double r993898 = log(r993890);
        double r993899 = exp(r993898);
        double r993900 = r993897 - r993899;
        double r993901 = r993900 / r993886;
        double r993902 = 0.5;
        double r993903 = r993882 * r993902;
        double r993904 = 2.0;
        double r993905 = r993877 / r993886;
        double r993906 = r993904 * r993905;
        double r993907 = r993903 - r993906;
        double r993908 = r993896 ? r993901 : r993907;
        double r993909 = r993885 ? r993894 : r993908;
        double r993910 = r993879 ? r993883 : r993909;
        return r993910;
}

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 < -2.1290642882256235e+29

    1. Initial program 56.4

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

    2. Taylor expanded around -inf 4.3

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

    if -2.1290642882256235e+29 < b_2 < 2.850467345251345e-161

    1. Initial program 24.8

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

    3. Applied flip--25.1

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

    4. Simplified16.5

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

    5. Simplified16.5

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

    7. Applied *-un-lft-identity16.5

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

    8. Applied times-frac14.0

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

    9. Simplified14.0

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

    if 2.850467345251345e-161 < b_2 < 0.17389787404847717

    1. Initial program 7.7

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

    3. Applied add-exp-log10.9

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

    if 0.17389787404847717 < b_2

    1. Initial program 31.2

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

    2. Taylor expanded around inf 7.3

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -212906428822562352817263083520:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.85046734525134507507504026935831729408 \cdot 10^{-161}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 0.173897874048477174557802982235443778336:\\ \;\;\;\;\frac{\left(-b_2\right) - e^{\log \left(\sqrt{b_2 \cdot b_2 - c \cdot a}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))