Average Error: 34.4 → 8.2
Time: 1.7m
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.617481894797657186488541194791400766101 \cdot 10^{121}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.486902549052064353297992509183366635369 \cdot 10^{-284}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.086842933652666425044939416105268609611 \cdot 10^{97}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{\frac{c}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}}}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -7.617481894797657186488541194791400766101 \cdot 10^{121}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -1.486902549052064353297992509183366635369 \cdot 10^{-284}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 1.086842933652666425044939416105268609611 \cdot 10^{97}:\\
\;\;\;\;\frac{\frac{1}{\frac{1}{\frac{c}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}}}}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r227868 = b_2;
        double r227869 = -r227868;
        double r227870 = r227868 * r227868;
        double r227871 = a;
        double r227872 = c;
        double r227873 = r227871 * r227872;
        double r227874 = r227870 - r227873;
        double r227875 = sqrt(r227874);
        double r227876 = r227869 + r227875;
        double r227877 = r227876 / r227871;
        return r227877;
}
↓
double f(double a, double b_2, double c) {
        double r227878 = b_2;
        double r227879 = -7.617481894797657e+121;
        bool r227880 = r227878 <= r227879;
        double r227881 = 0.5;
        double r227882 = c;
        double r227883 = r227882 / r227878;
        double r227884 = r227881 * r227883;
        double r227885 = 2.0;
        double r227886 = a;
        double r227887 = r227878 / r227886;
        double r227888 = r227885 * r227887;
        double r227889 = r227884 - r227888;
        double r227890 = -1.4869025490520644e-284;
        bool r227891 = r227878 <= r227890;
        double r227892 = -r227878;
        double r227893 = r227878 * r227878;
        double r227894 = r227886 * r227882;
        double r227895 = r227893 - r227894;
        double r227896 = sqrt(r227895);
        double r227897 = r227892 + r227896;
        double r227898 = 1.0;
        double r227899 = r227898 / r227886;
        double r227900 = r227897 * r227899;
        double r227901 = 1.0868429336526664e+97;
        bool r227902 = r227878 <= r227901;
        double r227903 = r227882 * r227886;
        double r227904 = r227893 - r227903;
        double r227905 = sqrt(r227904);
        double r227906 = r227892 - r227905;
        double r227907 = r227906 / r227886;
        double r227908 = r227882 / r227907;
        double r227909 = r227898 / r227908;
        double r227910 = r227898 / r227909;
        double r227911 = r227910 / r227886;
        double r227912 = -0.5;
        double r227913 = r227912 * r227883;
        double r227914 = r227902 ? r227911 : r227913;
        double r227915 = r227891 ? r227900 : r227914;
        double r227916 = r227880 ? r227889 : r227915;
        return r227916;
}

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 < -7.617481894797657e+121

    1. Initial program 52.4

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

    2. Taylor expanded around -inf 2.8

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

    if -7.617481894797657e+121 < b_2 < -1.4869025490520644e-284

    1. Initial program 8.8

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

    3. Applied div-inv8.9

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

    if -1.4869025490520644e-284 < b_2 < 1.0868429336526664e+97

    1. Initial program 31.1

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

    3. Applied flip-+31.2

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

    4. Simplified15.4

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

    6. Applied clear-num15.6

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

    7. Simplified14.3

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

    9. Applied *-un-lft-identity14.3

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

    10. Applied *-un-lft-identity14.3

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

    11. Applied times-frac14.3

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

    12. Applied associate-/l*14.3

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

    if 1.0868429336526664e+97 < b_2

    1. Initial program 59.6

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

    2. Taylor expanded around inf 2.6

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.617481894797657186488541194791400766101 \cdot 10^{121}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.486902549052064353297992509183366635369 \cdot 10^{-284}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.086842933652666425044939416105268609611 \cdot 10^{97}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{\frac{c}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}}}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))