Average Error: 33.3 → 9.1
Time: 17.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.2810366229709043 \cdot 10^{+70}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.2889142225980239 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 1.4483715500512764 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.2810366229709043 \cdot 10^{+70}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.2889142225980239 \cdot 10^{-280}:\\
\;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 1.4483715500512764 \cdot 10^{+101}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r413773 = b_2;
        double r413774 = -r413773;
        double r413775 = r413773 * r413773;
        double r413776 = a;
        double r413777 = c;
        double r413778 = r413776 * r413777;
        double r413779 = r413775 - r413778;
        double r413780 = sqrt(r413779);
        double r413781 = r413774 - r413780;
        double r413782 = r413781 / r413776;
        return r413782;
}
↓
double f(double a, double b_2, double c) {
        double r413783 = b_2;
        double r413784 = -1.2810366229709043e+70;
        bool r413785 = r413783 <= r413784;
        double r413786 = -0.5;
        double r413787 = c;
        double r413788 = r413787 / r413783;
        double r413789 = r413786 * r413788;
        double r413790 = 1.2889142225980239e-280;
        bool r413791 = r413783 <= r413790;
        double r413792 = a;
        double r413793 = r413787 * r413792;
        double r413794 = r413783 * r413783;
        double r413795 = r413794 - r413793;
        double r413796 = sqrt(r413795);
        double r413797 = r413796 - r413783;
        double r413798 = r413793 / r413797;
        double r413799 = r413798 / r413792;
        double r413800 = 1.4483715500512764e+101;
        bool r413801 = r413783 <= r413800;
        double r413802 = 1.0;
        double r413803 = r413802 / r413792;
        double r413804 = -r413783;
        double r413805 = r413804 - r413796;
        double r413806 = r413803 * r413805;
        double r413807 = 0.5;
        double r413808 = r413807 * r413788;
        double r413809 = r413783 / r413792;
        double r413810 = 2.0;
        double r413811 = r413809 * r413810;
        double r413812 = r413808 - r413811;
        double r413813 = r413801 ? r413806 : r413812;
        double r413814 = r413791 ? r413799 : r413813;
        double r413815 = r413785 ? r413789 : r413814;
        return r413815;
}

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 4 regimes

  2. if b_2 < -1.2810366229709043e+70

    1. Initial program 56.8

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

    2. Taylor expanded around -inf 3.0

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

    if -1.2810366229709043e+70 < b_2 < 1.2889142225980239e-280

    1. Initial program 30.3

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

    3. Applied flip--30.4

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

    4. Simplified17.2

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified17.2

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

    if 1.2889142225980239e-280 < b_2 < 1.4483715500512764e+101

    1. Initial program 8.9

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

    3. Applied div-inv9.1

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

    if 1.4483715500512764e+101 < b_2

    1. Initial program 43.7

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

    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 4 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.2810366229709043 \cdot 10^{+70}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.2889142225980239 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 1.4483715500512764 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))