Average Error: 34.2 → 8.6
Time: 35.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.99946224548089213456388959139204668765 \cdot 10^{73}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.296272708131498829504916428849430668856 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{a \cdot c}{a}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.748502676649782580252214156933339561376 \cdot 10^{143}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.99946224548089213456388959139204668765 \cdot 10^{73}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -4.296272708131498829504916428849430668856 \cdot 10^{-127}:\\
\;\;\;\;\frac{\frac{a \cdot c}{a}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 5.748502676649782580252214156933339561376 \cdot 10^{143}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r3579783 = b_2;
        double r3579784 = -r3579783;
        double r3579785 = r3579783 * r3579783;
        double r3579786 = a;
        double r3579787 = c;
        double r3579788 = r3579786 * r3579787;
        double r3579789 = r3579785 - r3579788;
        double r3579790 = sqrt(r3579789);
        double r3579791 = r3579784 - r3579790;
        double r3579792 = r3579791 / r3579786;
        return r3579792;
}
↓
double f(double a, double b_2, double c) {
        double r3579793 = b_2;
        double r3579794 = -5.999462245480892e+73;
        bool r3579795 = r3579793 <= r3579794;
        double r3579796 = -0.5;
        double r3579797 = c;
        double r3579798 = r3579797 / r3579793;
        double r3579799 = r3579796 * r3579798;
        double r3579800 = -4.296272708131499e-127;
        bool r3579801 = r3579793 <= r3579800;
        double r3579802 = a;
        double r3579803 = r3579802 * r3579797;
        double r3579804 = r3579803 / r3579802;
        double r3579805 = r3579793 * r3579793;
        double r3579806 = r3579805 - r3579803;
        double r3579807 = sqrt(r3579806);
        double r3579808 = r3579807 - r3579793;
        double r3579809 = r3579804 / r3579808;
        double r3579810 = 5.748502676649783e+143;
        bool r3579811 = r3579793 <= r3579810;
        double r3579812 = -r3579793;
        double r3579813 = r3579812 / r3579802;
        double r3579814 = r3579807 / r3579802;
        double r3579815 = r3579813 - r3579814;
        double r3579816 = r3579793 / r3579802;
        double r3579817 = -2.0;
        double r3579818 = r3579816 * r3579817;
        double r3579819 = r3579811 ? r3579815 : r3579818;
        double r3579820 = r3579801 ? r3579809 : r3579819;
        double r3579821 = r3579795 ? r3579799 : r3579820;
        return r3579821;
}

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 < -5.999462245480892e+73

    1. Initial program 58.3

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

    2. Taylor expanded around -inf 3.4

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

    if -5.999462245480892e+73 < b_2 < -4.296272708131499e-127

    1. Initial program 40.0

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

    3. Applied flip--40.0

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

    4. Simplified15.6

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

    5. Simplified15.6

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

    7. Applied *-un-lft-identity15.6

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

    8. Applied times-frac13.7

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

    9. Simplified13.7

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

    11. Applied div-inv13.8

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

    13. Applied associate-*r/15.7

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

    14. Applied associate-*l/14.8

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

    15. Simplified14.7

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

    if -4.296272708131499e-127 < b_2 < 5.748502676649783e+143

    1. Initial program 11.3

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

    3. Applied div-sub11.2

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

    if 5.748502676649783e+143 < b_2

    1. Initial program 59.9

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

    3. Applied flip--63.9

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

    4. Simplified62.7

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

    5. Simplified62.7

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

    7. Applied *-un-lft-identity62.7

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

    8. Applied times-frac62.6

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

    9. Simplified62.6

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

    11. Applied div-inv62.6

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

    12. Taylor expanded around 0 2.2

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.99946224548089213456388959139204668765 \cdot 10^{73}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.296272708131498829504916428849430668856 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{a \cdot c}{a}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.748502676649782580252214156933339561376 \cdot 10^{143}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))