Average Error: 33.1 → 9.3
Time: 1.0m
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.8719024022082672 \cdot 10^{+146}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.524249207244028 \cdot 10^{-127}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.890069572265506 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{a \cdot c}{-a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.8719024022082672 \cdot 10^{+146}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 5.524249207244028 \cdot 10^{-127}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 5.890069572265506 \cdot 10^{-16}:\\
\;\;\;\;\frac{\frac{a \cdot c}{-a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2\right)}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r3041813 = b_2;
        double r3041814 = -r3041813;
        double r3041815 = r3041813 * r3041813;
        double r3041816 = a;
        double r3041817 = c;
        double r3041818 = r3041816 * r3041817;
        double r3041819 = r3041815 - r3041818;
        double r3041820 = sqrt(r3041819);
        double r3041821 = r3041814 - r3041820;
        double r3041822 = r3041821 / r3041816;
        return r3041822;
}
↓
double f(double a, double b_2, double c) {
        double r3041823 = b_2;
        double r3041824 = -1.8719024022082672e+146;
        bool r3041825 = r3041823 <= r3041824;
        double r3041826 = -0.5;
        double r3041827 = c;
        double r3041828 = r3041827 / r3041823;
        double r3041829 = r3041826 * r3041828;
        double r3041830 = 5.524249207244028e-127;
        bool r3041831 = r3041823 <= r3041830;
        double r3041832 = r3041823 * r3041823;
        double r3041833 = a;
        double r3041834 = r3041833 * r3041827;
        double r3041835 = r3041832 - r3041834;
        double r3041836 = sqrt(r3041835);
        double r3041837 = r3041836 - r3041823;
        double r3041838 = r3041827 / r3041837;
        double r3041839 = 5.890069572265506e-16;
        bool r3041840 = r3041823 <= r3041839;
        double r3041841 = -r3041834;
        double r3041842 = r3041834 / r3041841;
        double r3041843 = r3041836 + r3041823;
        double r3041844 = r3041842 * r3041843;
        double r3041845 = r3041844 / r3041833;
        double r3041846 = 0.5;
        double r3041847 = r3041828 * r3041846;
        double r3041848 = 2.0;
        double r3041849 = r3041823 / r3041833;
        double r3041850 = r3041848 * r3041849;
        double r3041851 = r3041847 - r3041850;
        double r3041852 = r3041840 ? r3041845 : r3041851;
        double r3041853 = r3041831 ? r3041838 : r3041852;
        double r3041854 = r3041825 ? r3041829 : r3041853;
        return r3041854;
}

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 < -1.8719024022082672e+146

    1. Initial program 62.0

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

    2. Taylor expanded around -inf 1.7

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

    if -1.8719024022082672e+146 < b_2 < 5.524249207244028e-127

    1. Initial program 29.1

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

    3. Applied flip--30.1

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

    4. Simplified16.2

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

    5. Simplified16.2

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

    7. Applied *-un-lft-identity16.2

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

    8. Applied *-un-lft-identity16.2

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

    9. Applied times-frac16.2

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

    10. Simplified16.2

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

    11. Simplified10.4

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

    if 5.524249207244028e-127 < b_2 < 5.890069572265506e-16

    1. Initial program 6.4

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

    3. Applied flip--36.2

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

    4. Simplified36.2

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

    5. Simplified36.2

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

    7. Applied flip--36.2

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

    8. Applied associate-/r/36.2

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

    9. Simplified19.3

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

    if 5.890069572265506e-16 < b_2

    1. Initial program 28.8

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

    2. Taylor expanded around inf 9.4

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.8719024022082672 \cdot 10^{+146}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.524249207244028 \cdot 10^{-127}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.890069572265506 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{a \cdot c}{-a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))