Average Error: 34.5 → 8.9
Time: 5.1s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.72893889301538444 \cdot 10^{27}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -9.19851418750357702 \cdot 10^{-275}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.174676155214135 \cdot 10^{112}:\\ \;\;\;\;\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}} \cdot \frac{\sqrt[3]{1}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.72893889301538444 \cdot 10^{27}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -9.19851418750357702 \cdot 10^{-275}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 5.174676155214135 \cdot 10^{112}:\\
\;\;\;\;\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}} \cdot \frac{\sqrt[3]{1}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r17894 = b_2;
        double r17895 = -r17894;
        double r17896 = r17894 * r17894;
        double r17897 = a;
        double r17898 = c;
        double r17899 = r17897 * r17898;
        double r17900 = r17896 - r17899;
        double r17901 = sqrt(r17900);
        double r17902 = r17895 + r17901;
        double r17903 = r17902 / r17897;
        return r17903;
}
↓
double f(double a, double b_2, double c) {
        double r17904 = b_2;
        double r17905 = -1.7289388930153844e+27;
        bool r17906 = r17904 <= r17905;
        double r17907 = 0.5;
        double r17908 = c;
        double r17909 = r17908 / r17904;
        double r17910 = r17907 * r17909;
        double r17911 = 2.0;
        double r17912 = a;
        double r17913 = r17904 / r17912;
        double r17914 = r17911 * r17913;
        double r17915 = r17910 - r17914;
        double r17916 = -9.198514187503577e-275;
        bool r17917 = r17904 <= r17916;
        double r17918 = -r17904;
        double r17919 = r17904 * r17904;
        double r17920 = r17912 * r17908;
        double r17921 = r17919 - r17920;
        double r17922 = sqrt(r17921);
        double r17923 = r17918 + r17922;
        double r17924 = 1.0;
        double r17925 = r17924 / r17912;
        double r17926 = r17923 * r17925;
        double r17927 = 5.174676155214135e+112;
        bool r17928 = r17904 <= r17927;
        double r17929 = r17918 - r17922;
        double r17930 = r17929 / r17908;
        double r17931 = r17912 / r17930;
        double r17932 = cbrt(r17924);
        double r17933 = r17932 / r17912;
        double r17934 = r17931 * r17933;
        double r17935 = -0.5;
        double r17936 = r17935 * r17909;
        double r17937 = r17928 ? r17934 : r17936;
        double r17938 = r17917 ? r17926 : r17937;
        double r17939 = r17906 ? r17915 : r17938;
        return r17939;
}

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 < -1.7289388930153844e+27

    1. Initial program 35.3

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

    2. Taylor expanded around -inf 6.6

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

    if -1.7289388930153844e+27 < b_2 < -9.198514187503577e-275

    1. Initial program 9.8

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

    3. Applied div-inv9.9

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

    if -9.198514187503577e-275 < b_2 < 5.174676155214135e+112

    1. Initial program 32.0

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

    3. Applied flip-+32.0

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

    4. Simplified16.7

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
    5. Using strategy rm

    6. Applied div-inv16.8

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

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

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

    9. Applied add-cube-cbrt16.8

      \[\leadsto \frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot a}\]

    10. Applied times-frac16.8

      \[\leadsto \frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{a}\right)}\]

    11. Applied associate-*r*16.8

      \[\leadsto \color{blue}{\left(\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right) \cdot \frac{\sqrt[3]{1}}{a}}\]

    12. Simplified14.6

      \[\leadsto \color{blue}{\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}} \cdot \frac{\sqrt[3]{1}}{a}\]

    if 5.174676155214135e+112 < b_2

    1. Initial program 60.3

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

    2. Taylor expanded around inf 1.8

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.72893889301538444 \cdot 10^{27}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -9.19851418750357702 \cdot 10^{-275}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.174676155214135 \cdot 10^{112}:\\ \;\;\;\;\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}} \cdot \frac{\sqrt[3]{1}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))