Average Error: 33.9 → 6.9
Time: 2.6m
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.157094219357017 \cdot 10^{+135}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 5.369711498123029 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.743659918667874 \cdot 10^{+75}:\\ \;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.157094219357017 \cdot 10^{+135}:\\
\;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 5.369711498123029 \cdot 10^{-186}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.743659918667874 \cdot 10^{+75}:\\
\;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r4368875 = b_2;
        double r4368876 = -r4368875;
        double r4368877 = r4368875 * r4368875;
        double r4368878 = a;
        double r4368879 = c;
        double r4368880 = r4368878 * r4368879;
        double r4368881 = r4368877 - r4368880;
        double r4368882 = sqrt(r4368881);
        double r4368883 = r4368876 + r4368882;
        double r4368884 = r4368883 / r4368878;
        return r4368884;
}
↓
double f(double a, double b_2, double c) {
        double r4368885 = b_2;
        double r4368886 = -3.157094219357017e+135;
        bool r4368887 = r4368885 <= r4368886;
        double r4368888 = 0.5;
        double r4368889 = c;
        double r4368890 = r4368889 / r4368885;
        double r4368891 = r4368888 * r4368890;
        double r4368892 = a;
        double r4368893 = r4368885 / r4368892;
        double r4368894 = r4368891 - r4368893;
        double r4368895 = r4368894 - r4368893;
        double r4368896 = 5.369711498123029e-186;
        bool r4368897 = r4368885 <= r4368896;
        double r4368898 = r4368885 * r4368885;
        double r4368899 = r4368892 * r4368889;
        double r4368900 = r4368898 - r4368899;
        double r4368901 = sqrt(r4368900);
        double r4368902 = r4368901 / r4368892;
        double r4368903 = r4368902 - r4368893;
        double r4368904 = 1.743659918667874e+75;
        bool r4368905 = r4368885 <= r4368904;
        double r4368906 = -r4368889;
        double r4368907 = r4368901 + r4368885;
        double r4368908 = r4368906 / r4368907;
        double r4368909 = -0.5;
        double r4368910 = r4368909 * r4368890;
        double r4368911 = r4368905 ? r4368908 : r4368910;
        double r4368912 = r4368897 ? r4368903 : r4368911;
        double r4368913 = r4368887 ? r4368895 : r4368912;
        return r4368913;
}

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 < -3.157094219357017e+135

    1. Initial program 54.4

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

    2. Simplified54.4

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

    4. Applied div-sub54.4

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

    5. Taylor expanded around -inf 2.8

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

    if -3.157094219357017e+135 < b_2 < 5.369711498123029e-186

    1. Initial program 10.8

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

    2. Simplified10.8

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

    4. Applied div-sub10.8

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

    if 5.369711498123029e-186 < b_2 < 1.743659918667874e+75

    1. Initial program 36.2

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

    2. Simplified36.2

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

    4. Applied flip--36.4

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

    5. Applied associate-/l/40.1

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

    6. Simplified20.9

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

    8. Applied distribute-frac-neg20.9

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

    9. Simplified6.8

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

    if 1.743659918667874e+75 < b_2

    1. Initial program 57.3

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

    2. Simplified57.3

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

    3. Taylor expanded around inf 3.2

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.157094219357017 \cdot 10^{+135}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 5.369711498123029 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.743659918667874 \cdot 10^{+75}:\\ \;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))