Average Error: 33.9 → 11.3
Time: 13.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.869662346631121401645595393947635525169 \cdot 10^{101}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.869662346631121401645595393947635525169 \cdot 10^{101}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\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 r52845 = b_2;
        double r52846 = -r52845;
        double r52847 = r52845 * r52845;
        double r52848 = a;
        double r52849 = c;
        double r52850 = r52848 * r52849;
        double r52851 = r52847 - r52850;
        double r52852 = sqrt(r52851);
        double r52853 = r52846 - r52852;
        double r52854 = r52853 / r52848;
        return r52854;
}
↓
double f(double a, double b_2, double c) {
        double r52855 = b_2;
        double r52856 = -1.8696623466311214e+101;
        bool r52857 = r52855 <= r52856;
        double r52858 = -0.5;
        double r52859 = c;
        double r52860 = r52859 / r52855;
        double r52861 = r52858 * r52860;
        double r52862 = 7.455592343308264e-170;
        bool r52863 = r52855 <= r52862;
        double r52864 = r52855 * r52855;
        double r52865 = a;
        double r52866 = r52865 * r52859;
        double r52867 = r52864 - r52866;
        double r52868 = sqrt(r52867);
        double r52869 = r52868 - r52855;
        double r52870 = r52859 / r52869;
        double r52871 = 0.5;
        double r52872 = r52871 * r52860;
        double r52873 = 2.0;
        double r52874 = r52855 / r52865;
        double r52875 = r52873 * r52874;
        double r52876 = r52872 - r52875;
        double r52877 = r52863 ? r52870 : r52876;
        double r52878 = r52857 ? r52861 : r52877;
        return r52878;
}

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 < -1.8696623466311214e+101

    1. Initial program 59.8

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

    2. Taylor expanded around -inf 2.5

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

    if -1.8696623466311214e+101 < b_2 < 7.455592343308264e-170

    1. Initial program 28.9

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

    3. Applied flip--29.1

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

    4. Simplified16.6

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

    5. Simplified16.6

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

    7. Applied *-un-lft-identity16.6

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

    8. Applied times-frac15.4

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

    9. Applied associate-/l*12.2

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

    10. Simplified11.1

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

    if 7.455592343308264e-170 < b_2

    1. Initial program 23.0

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

    2. Taylor expanded around inf 17.1

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.869662346631121401645595393947635525169 \cdot 10^{101}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))