Average Error: 34.2 → 10.1
Time: 5.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.01157973271056712 \cdot 10^{-81}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.3176462918432122 \cdot 10^{99}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

\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 r17884 = b_2;
        double r17885 = -r17884;
        double r17886 = r17884 * r17884;
        double r17887 = a;
        double r17888 = c;
        double r17889 = r17887 * r17888;
        double r17890 = r17886 - r17889;
        double r17891 = sqrt(r17890);
        double r17892 = r17885 - r17891;
        double r17893 = r17892 / r17887;
        return r17893;
}
↓
double f(double a, double b_2, double c) {
        double r17894 = b_2;
        double r17895 = -4.011579732710567e-81;
        bool r17896 = r17894 <= r17895;
        double r17897 = -0.5;
        double r17898 = c;
        double r17899 = r17898 / r17894;
        double r17900 = r17897 * r17899;
        double r17901 = 1.3176462918432122e+99;
        bool r17902 = r17894 <= r17901;
        double r17903 = 1.0;
        double r17904 = a;
        double r17905 = -r17894;
        double r17906 = r17894 * r17894;
        double r17907 = r17904 * r17898;
        double r17908 = r17906 - r17907;
        double r17909 = sqrt(r17908);
        double r17910 = r17905 - r17909;
        double r17911 = r17904 / r17910;
        double r17912 = r17903 / r17911;
        double r17913 = 0.5;
        double r17914 = r17913 * r17899;
        double r17915 = 2.0;
        double r17916 = r17894 / r17904;
        double r17917 = r17915 * r17916;
        double r17918 = r17914 - r17917;
        double r17919 = r17902 ? r17912 : r17918;
        double r17920 = r17896 ? r17900 : r17919;
        return r17920;
}

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 < -4.011579732710567e-81

    1. Initial program 52.8

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

    2. Taylor expanded around -inf 9.4

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

    if -4.011579732710567e-81 < b_2 < 1.3176462918432122e+99

    1. Initial program 12.9

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

    3. Applied clear-num13.0

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

    if 1.3176462918432122e+99 < b_2

    1. Initial program 46.8

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))