Average Error: 34.5 → 6.6
Time: 18.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.602441184943772642330945248646923860899 \cdot 10^{118}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.897979690537010916247637791104885449418 \cdot 10^{-281}:\\ \;\;\;\;\frac{c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.602441184943772642330945248646923860899 \cdot 10^{118}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.897979690537010916247637791104885449418 \cdot 10^{-281}:\\
\;\;\;\;\frac{c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\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 r45853 = b_2;
        double r45854 = -r45853;
        double r45855 = r45853 * r45853;
        double r45856 = a;
        double r45857 = c;
        double r45858 = r45856 * r45857;
        double r45859 = r45855 - r45858;
        double r45860 = sqrt(r45859);
        double r45861 = r45854 - r45860;
        double r45862 = r45861 / r45856;
        return r45862;
}
↓
double f(double a, double b_2, double c) {
        double r45863 = b_2;
        double r45864 = -5.602441184943773e+118;
        bool r45865 = r45863 <= r45864;
        double r45866 = -0.5;
        double r45867 = c;
        double r45868 = r45867 / r45863;
        double r45869 = r45866 * r45868;
        double r45870 = -3.897979690537011e-281;
        bool r45871 = r45863 <= r45870;
        double r45872 = 2.0;
        double r45873 = pow(r45863, r45872);
        double r45874 = a;
        double r45875 = r45874 * r45867;
        double r45876 = r45873 - r45875;
        double r45877 = sqrt(r45876);
        double r45878 = r45877 - r45863;
        double r45879 = r45867 / r45878;
        double r45880 = 2.1255630798514387e+135;
        bool r45881 = r45863 <= r45880;
        double r45882 = -r45863;
        double r45883 = r45863 * r45863;
        double r45884 = r45883 - r45875;
        double r45885 = sqrt(r45884);
        double r45886 = r45882 - r45885;
        double r45887 = r45886 / r45874;
        double r45888 = 0.5;
        double r45889 = r45888 * r45868;
        double r45890 = r45863 / r45874;
        double r45891 = r45872 * r45890;
        double r45892 = r45889 - r45891;
        double r45893 = r45881 ? r45887 : r45892;
        double r45894 = r45871 ? r45879 : r45893;
        double r45895 = r45865 ? r45869 : r45894;
        return r45895;
}

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 < -5.602441184943773e+118

    1. Initial program 61.1

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

    2. Taylor expanded around -inf 1.9

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

    if -5.602441184943773e+118 < b_2 < -3.897979690537011e-281

    1. Initial program 34.7

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

    3. Applied flip--34.7

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

    4. Simplified16.1

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

    5. Simplified16.1

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

    7. Applied *-un-lft-identity16.1

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

    8. Applied times-frac15.0

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

    9. Applied associate-/l*10.5

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

    10. Simplified8.4

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

    11. Taylor expanded around 0 8.4

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

    if -3.897979690537011e-281 < b_2 < 2.1255630798514387e+135

    1. Initial program 9.3

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

    if 2.1255630798514387e+135 < b_2

    1. Initial program 58.2

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

    2. Taylor expanded around inf 3.0

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.602441184943772642330945248646923860899 \cdot 10^{118}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.897979690537010916247637791104885449418 \cdot 10^{-281}:\\ \;\;\;\;\frac{c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))