Average Error: 33.9 → 6.9
Time: 8.9s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.442369070306521925925945300368519903805 \cdot 10^{91}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le \frac{8176673959656255}{5.8652475225036720546649669496166069229 \cdot 10^{253}}:\\ \;\;\;\;\left(\frac{1}{a} \cdot 0 + c\right) \cdot \frac{1}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 1.530904404386409423412434180578840981346 \cdot 10^{64}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.442369070306521925925945300368519903805 \cdot 10^{91}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le \frac{8176673959656255}{5.8652475225036720546649669496166069229 \cdot 10^{253}}:\\
\;\;\;\;\left(\frac{1}{a} \cdot 0 + c\right) \cdot \frac{1}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{elif}\;b_2 \le 1.530904404386409423412434180578840981346 \cdot 10^{64}:\\
\;\;\;\;\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 r22771 = b_2;
        double r22772 = -r22771;
        double r22773 = r22771 * r22771;
        double r22774 = a;
        double r22775 = c;
        double r22776 = r22774 * r22775;
        double r22777 = r22773 - r22776;
        double r22778 = sqrt(r22777);
        double r22779 = r22772 - r22778;
        double r22780 = r22779 / r22774;
        return r22780;
}
↓
double f(double a, double b_2, double c) {
        double r22781 = b_2;
        double r22782 = -2.442369070306522e+91;
        bool r22783 = r22781 <= r22782;
        double r22784 = -0.5;
        double r22785 = c;
        double r22786 = r22785 / r22781;
        double r22787 = r22784 * r22786;
        double r22788 = 8176673959656255.0;
        double r22789 = 5.865247522503672e+253;
        double r22790 = r22788 / r22789;
        bool r22791 = r22781 <= r22790;
        double r22792 = 1.0;
        double r22793 = a;
        double r22794 = r22792 / r22793;
        double r22795 = 0.0;
        double r22796 = r22794 * r22795;
        double r22797 = r22796 + r22785;
        double r22798 = -r22781;
        double r22799 = r22781 * r22781;
        double r22800 = r22793 * r22785;
        double r22801 = r22799 - r22800;
        double r22802 = sqrt(r22801);
        double r22803 = r22798 + r22802;
        double r22804 = r22792 / r22803;
        double r22805 = r22797 * r22804;
        double r22806 = 1.5309044043864094e+64;
        bool r22807 = r22781 <= r22806;
        double r22808 = r22798 - r22802;
        double r22809 = r22808 / r22793;
        double r22810 = 0.5;
        double r22811 = r22810 * r22786;
        double r22812 = 2.0;
        double r22813 = r22781 / r22793;
        double r22814 = r22812 * r22813;
        double r22815 = r22811 - r22814;
        double r22816 = r22807 ? r22809 : r22815;
        double r22817 = r22791 ? r22805 : r22816;
        double r22818 = r22783 ? r22787 : r22817;
        return r22818;
}

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 < -2.442369070306522e+91

    1. Initial program 59.0

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

    2. Taylor expanded around -inf 2.7

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

    if -2.442369070306522e+91 < b_2 < 1.3940884725297859e-238

    1. Initial program 30.0

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

    3. Applied clear-num30.0

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

    5. Applied flip--30.1

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

    6. Applied associate-/r/30.2

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

    7. Applied *-un-lft-identity30.2

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

    8. Applied times-frac30.2

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

    9. Simplified15.8

      \[\leadsto \color{blue}{\left(\frac{1}{a} \cdot 0 + \frac{1}{a} \cdot \left(a \cdot c\right)\right)} \cdot \frac{1}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    10. Taylor expanded around 0 9.9

      \[\leadsto \left(\frac{1}{a} \cdot 0 + \color{blue}{c}\right) \cdot \frac{1}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    if 1.3940884725297859e-238 < b_2 < 1.5309044043864094e+64

    1. Initial program 8.1

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

    if 1.5309044043864094e+64 < 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 5.5

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.442369070306521925925945300368519903805 \cdot 10^{91}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le \frac{8176673959656255}{5.8652475225036720546649669496166069229 \cdot 10^{253}}:\\ \;\;\;\;\left(\frac{1}{a} \cdot 0 + c\right) \cdot \frac{1}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 1.530904404386409423412434180578840981346 \cdot 10^{64}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))