Average Error: 34.7 → 9.3
Time: 6.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.3202538172935113 \cdot 10^{68}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.5368857650143505 \cdot 10^{-218}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.3602536904640645 \cdot 10^{97}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.3202538172935113 \cdot 10^{68}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 4.5368857650143505 \cdot 10^{-218}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 3.3602536904640645 \cdot 10^{97}:\\
\;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r49878 = b;
        double r49879 = -r49878;
        double r49880 = r49878 * r49878;
        double r49881 = 4.0;
        double r49882 = a;
        double r49883 = r49881 * r49882;
        double r49884 = c;
        double r49885 = r49883 * r49884;
        double r49886 = r49880 - r49885;
        double r49887 = sqrt(r49886);
        double r49888 = r49879 + r49887;
        double r49889 = 2.0;
        double r49890 = r49889 * r49882;
        double r49891 = r49888 / r49890;
        return r49891;
}
↓
double f(double a, double b, double c) {
        double r49892 = b;
        double r49893 = -2.3202538172935113e+68;
        bool r49894 = r49892 <= r49893;
        double r49895 = 1.0;
        double r49896 = c;
        double r49897 = r49896 / r49892;
        double r49898 = a;
        double r49899 = r49892 / r49898;
        double r49900 = r49897 - r49899;
        double r49901 = r49895 * r49900;
        double r49902 = 4.53688576501435e-218;
        bool r49903 = r49892 <= r49902;
        double r49904 = -r49892;
        double r49905 = r49892 * r49892;
        double r49906 = 4.0;
        double r49907 = r49906 * r49898;
        double r49908 = r49907 * r49896;
        double r49909 = r49905 - r49908;
        double r49910 = sqrt(r49909);
        double r49911 = r49904 + r49910;
        double r49912 = 1.0;
        double r49913 = 2.0;
        double r49914 = r49913 * r49898;
        double r49915 = r49912 / r49914;
        double r49916 = r49911 * r49915;
        double r49917 = 3.3602536904640645e+97;
        bool r49918 = r49892 <= r49917;
        double r49919 = 0.0;
        double r49920 = r49898 * r49896;
        double r49921 = r49906 * r49920;
        double r49922 = r49919 + r49921;
        double r49923 = r49904 - r49910;
        double r49924 = r49922 / r49923;
        double r49925 = r49924 / r49914;
        double r49926 = -1.0;
        double r49927 = r49926 * r49897;
        double r49928 = r49918 ? r49925 : r49927;
        double r49929 = r49903 ? r49916 : r49928;
        double r49930 = r49894 ? r49901 : r49929;
        return r49930;
}

Error

Bits error versus a

Bits error versus b

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 < -2.3202538172935113e+68

    1. Initial program 40.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around -inf 5.1

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

    3. Simplified5.1

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

    if -2.3202538172935113e+68 < b < 4.53688576501435e-218

    1. Initial program 11.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv11.4

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}}\]

    if 4.53688576501435e-218 < b < 3.3602536904640645e+97

    1. Initial program 35.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+35.8

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    4. Simplified16.2

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]

    if 3.3602536904640645e+97 < b

    1. Initial program 59.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around inf 2.5

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.3202538172935113 \cdot 10^{68}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.5368857650143505 \cdot 10^{-218}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.3602536904640645 \cdot 10^{97}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))