Average Error: 34.2 → 10.1
Time: 5.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\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{\frac{1}{a}}{\frac{1}{\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 r88854 = b_2;
        double r88855 = -r88854;
        double r88856 = r88854 * r88854;
        double r88857 = a;
        double r88858 = c;
        double r88859 = r88857 * r88858;
        double r88860 = r88856 - r88859;
        double r88861 = sqrt(r88860);
        double r88862 = r88855 - r88861;
        double r88863 = r88862 / r88857;
        return r88863;
}
↓
double f(double a, double b_2, double c) {
        double r88864 = b_2;
        double r88865 = -4.011579732710567e-81;
        bool r88866 = r88864 <= r88865;
        double r88867 = -0.5;
        double r88868 = c;
        double r88869 = r88868 / r88864;
        double r88870 = r88867 * r88869;
        double r88871 = 1.3176462918432122e+99;
        bool r88872 = r88864 <= r88871;
        double r88873 = 1.0;
        double r88874 = a;
        double r88875 = r88873 / r88874;
        double r88876 = -r88864;
        double r88877 = r88864 * r88864;
        double r88878 = r88874 * r88868;
        double r88879 = r88877 - r88878;
        double r88880 = sqrt(r88879);
        double r88881 = r88876 - r88880;
        double r88882 = r88873 / r88881;
        double r88883 = r88875 / r88882;
        double r88884 = 0.5;
        double r88885 = r88884 * r88869;
        double r88886 = 2.0;
        double r88887 = r88864 / r88874;
        double r88888 = r88886 * r88887;
        double r88889 = r88885 - r88888;
        double r88890 = r88872 ? r88883 : r88889;
        double r88891 = r88866 ? r88870 : r88890;
        return r88891;
}

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

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

    5. Applied div-inv13.1

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

    6. Applied associate-/r*13.1

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

    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

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))