Average Error: 34.1 → 9.9
Time: 6.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.260729798395838 \cdot 10^{-43}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.67643144401154069 \cdot 10^{104}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.260729798395838 \cdot 10^{-43}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.67643144401154069 \cdot 10^{104}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r22832 = b_2;
        double r22833 = -r22832;
        double r22834 = r22832 * r22832;
        double r22835 = a;
        double r22836 = c;
        double r22837 = r22835 * r22836;
        double r22838 = r22834 - r22837;
        double r22839 = sqrt(r22838);
        double r22840 = r22833 - r22839;
        double r22841 = r22840 / r22835;
        return r22841;
}
↓
double f(double a, double b_2, double c) {
        double r22842 = b_2;
        double r22843 = -8.260729798395838e-43;
        bool r22844 = r22842 <= r22843;
        double r22845 = -0.5;
        double r22846 = c;
        double r22847 = r22846 / r22842;
        double r22848 = r22845 * r22847;
        double r22849 = 1.6764314440115407e+104;
        bool r22850 = r22842 <= r22849;
        double r22851 = -r22842;
        double r22852 = a;
        double r22853 = r22851 / r22852;
        double r22854 = r22842 * r22842;
        double r22855 = r22852 * r22846;
        double r22856 = r22854 - r22855;
        double r22857 = sqrt(r22856);
        double r22858 = r22857 / r22852;
        double r22859 = r22853 - r22858;
        double r22860 = 0.5;
        double r22861 = r22860 * r22847;
        double r22862 = 2.0;
        double r22863 = r22842 / r22852;
        double r22864 = r22862 * r22863;
        double r22865 = r22861 - r22864;
        double r22866 = r22850 ? r22859 : r22865;
        double r22867 = r22844 ? r22848 : r22866;
        return r22867;
}

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 3 regimes

  2. if b_2 < -8.260729798395838e-43

    1. Initial program 54.7

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

    2. Taylor expanded around -inf 7.3

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

    if -8.260729798395838e-43 < b_2 < 1.6764314440115407e+104

    1. Initial program 14.0

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

    3. Applied div-sub14.0

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

    if 1.6764314440115407e+104 < b_2

    1. Initial program 47.9

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

    3. Applied div-sub47.9

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

    5. Applied *-un-lft-identity47.9

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

    6. Applied add-sqr-sqrt47.9

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

    7. Applied sqrt-prod48.0

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

    8. Applied times-frac48.0

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

    9. Simplified48.0

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

    10. Taylor expanded around inf 3.3

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.260729798395838 \cdot 10^{-43}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.67643144401154069 \cdot 10^{104}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))