Average Error: 34.1 → 7.0
Time: 7.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.32792716898209886 \cdot 10^{29}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -5.498188913150987 \cdot 10^{-229}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 1.67192499022346934 \cdot 10^{111}:\\ \;\;\;\;{\left(\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.32792716898209886 \cdot 10^{29}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -5.498188913150987 \cdot 10^{-229}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{elif}\;b_2 \le 1.67192499022346934 \cdot 10^{111}:\\
\;\;\;\;{\left(\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{1}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r22909 = b_2;
        double r22910 = -r22909;
        double r22911 = r22909 * r22909;
        double r22912 = a;
        double r22913 = c;
        double r22914 = r22912 * r22913;
        double r22915 = r22911 - r22914;
        double r22916 = sqrt(r22915);
        double r22917 = r22910 + r22916;
        double r22918 = r22917 / r22912;
        return r22918;
}
↓
double f(double a, double b_2, double c) {
        double r22919 = b_2;
        double r22920 = -1.3279271689820989e+29;
        bool r22921 = r22919 <= r22920;
        double r22922 = 0.5;
        double r22923 = c;
        double r22924 = r22923 / r22919;
        double r22925 = r22922 * r22924;
        double r22926 = 2.0;
        double r22927 = a;
        double r22928 = r22919 / r22927;
        double r22929 = r22926 * r22928;
        double r22930 = r22925 - r22929;
        double r22931 = -5.498188913150987e-229;
        bool r22932 = r22919 <= r22931;
        double r22933 = -r22919;
        double r22934 = r22919 * r22919;
        double r22935 = r22927 * r22923;
        double r22936 = r22934 - r22935;
        double r22937 = sqrt(r22936);
        double r22938 = r22933 + r22937;
        double r22939 = r22938 / r22927;
        double r22940 = 1.6719249902234693e+111;
        bool r22941 = r22919 <= r22940;
        double r22942 = r22933 - r22937;
        double r22943 = r22923 / r22942;
        double r22944 = 1.0;
        double r22945 = pow(r22943, r22944);
        double r22946 = -0.5;
        double r22947 = r22946 * r22924;
        double r22948 = r22941 ? r22945 : r22947;
        double r22949 = r22932 ? r22939 : r22948;
        double r22950 = r22921 ? r22930 : r22949;
        return r22950;
}

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 < -1.3279271689820989e+29

    1. Initial program 35.8

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

    2. Taylor expanded around -inf 6.7

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

    if -1.3279271689820989e+29 < b_2 < -5.498188913150987e-229

    1. Initial program 8.3

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

    if -5.498188913150987e-229 < b_2 < 1.6719249902234693e+111

    1. Initial program 30.8

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

    3. Applied flip-+30.8

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

    4. Simplified16.4

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

    6. Applied *-un-lft-identity16.4

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

    7. Applied associate-/r*16.4

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

    8. Simplified14.7

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

    10. Applied add-sqr-sqrt39.1

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

    11. Applied div-inv39.1

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

    12. Applied add-sqr-sqrt39.0

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

    13. Applied times-frac39.3

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

    14. Applied times-frac37.8

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

    15. Simplified37.8

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

    16. Simplified9.8

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

    18. Applied pow19.8

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

    19. Applied pow19.8

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

    20. Applied pow-prod-down9.8

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

    21. Simplified9.6

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

    if 1.6719249902234693e+111 < b_2

    1. Initial program 60.0

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

    2. Taylor expanded around inf 2.1

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.32792716898209886 \cdot 10^{29}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -5.498188913150987 \cdot 10^{-229}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 1.67192499022346934 \cdot 10^{111}:\\ \;\;\;\;{\left(\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))