Average Error: 34.0 → 6.8
Time: 4.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.00500656176984215351659893827263540922 \cdot 10^{132}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.054528764146387149688914666009662801656 \cdot 10^{-247}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.00500656176984215351659893827263540922 \cdot 10^{132}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.054528764146387149688914666009662801656 \cdot 10^{-247}:\\
\;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{elif}\;b_2 \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\
\;\;\;\;\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 r13769 = b_2;
        double r13770 = -r13769;
        double r13771 = r13769 * r13769;
        double r13772 = a;
        double r13773 = c;
        double r13774 = r13772 * r13773;
        double r13775 = r13771 - r13774;
        double r13776 = sqrt(r13775);
        double r13777 = r13770 - r13776;
        double r13778 = r13777 / r13772;
        return r13778;
}
↓
double f(double a, double b_2, double c) {
        double r13779 = b_2;
        double r13780 = -5.005006561769842e+132;
        bool r13781 = r13779 <= r13780;
        double r13782 = -0.5;
        double r13783 = c;
        double r13784 = r13783 / r13779;
        double r13785 = r13782 * r13784;
        double r13786 = 1.0545287641463871e-247;
        bool r13787 = r13779 <= r13786;
        double r13788 = -r13779;
        double r13789 = r13779 * r13779;
        double r13790 = a;
        double r13791 = r13790 * r13783;
        double r13792 = r13789 - r13791;
        double r13793 = sqrt(r13792);
        double r13794 = r13788 + r13793;
        double r13795 = r13783 / r13794;
        double r13796 = 1.0273828621120979e+63;
        bool r13797 = r13779 <= r13796;
        double r13798 = 1.0;
        double r13799 = r13788 - r13793;
        double r13800 = r13790 / r13799;
        double r13801 = r13798 / r13800;
        double r13802 = 0.5;
        double r13803 = r13802 * r13784;
        double r13804 = 2.0;
        double r13805 = r13779 / r13790;
        double r13806 = r13804 * r13805;
        double r13807 = r13803 - r13806;
        double r13808 = r13797 ? r13801 : r13807;
        double r13809 = r13787 ? r13795 : r13808;
        double r13810 = r13781 ? r13785 : r13809;
        return r13810;
}

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 < -5.005006561769842e+132

    1. Initial program 61.7

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

    2. Taylor expanded around -inf 1.7

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

    if -5.005006561769842e+132 < b_2 < 1.0545287641463871e-247

    1. Initial program 31.9

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

    3. Applied clear-num31.9

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

    5. Applied flip--32.0

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

    6. Applied associate-/r/32.0

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

    7. Applied associate-/r*32.0

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

    8. Simplified14.6

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(\left({b_2}^{2} - {b_2}^{2}\right) + a \cdot c\right)}{a}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    9. Taylor expanded around 0 9.2

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

    if 1.0545287641463871e-247 < b_2 < 1.0273828621120979e+63

    1. Initial program 8.1

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

    3. Applied clear-num8.3

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

    if 1.0273828621120979e+63 < b_2

    1. Initial program 39.8

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

    2. Taylor expanded around inf 5.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.00500656176984215351659893827263540922 \cdot 10^{132}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.054528764146387149688914666009662801656 \cdot 10^{-247}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))