Average Error: 34.2 → 10.8
Time: 17.5s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.828737593568666707297780122417133955299 \cdot 10^{109}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{\frac{a}{2}}\\ \mathbf{elif}\;b_2 \le 1.423163788076292919361412483479690249604 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} - b_2}}{\frac{a}{\sqrt{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.828737593568666707297780122417133955299 \cdot 10^{109}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{\frac{a}{2}}\\

\mathbf{elif}\;b_2 \le 1.423163788076292919361412483479690249604 \cdot 10^{-136}:\\
\;\;\;\;\frac{\sqrt{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} - b_2}}{\frac{a}{\sqrt{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r24875 = b_2;
        double r24876 = -r24875;
        double r24877 = r24875 * r24875;
        double r24878 = a;
        double r24879 = c;
        double r24880 = r24878 * r24879;
        double r24881 = r24877 - r24880;
        double r24882 = sqrt(r24881);
        double r24883 = r24876 + r24882;
        double r24884 = r24883 / r24878;
        return r24884;
}
↓
double f(double a, double b_2, double c) {
        double r24885 = b_2;
        double r24886 = -3.828737593568667e+109;
        bool r24887 = r24885 <= r24886;
        double r24888 = 0.5;
        double r24889 = c;
        double r24890 = r24889 / r24885;
        double r24891 = r24888 * r24890;
        double r24892 = a;
        double r24893 = 2.0;
        double r24894 = r24892 / r24893;
        double r24895 = r24885 / r24894;
        double r24896 = r24891 - r24895;
        double r24897 = 1.423163788076293e-136;
        bool r24898 = r24885 <= r24897;
        double r24899 = r24885 * r24885;
        double r24900 = r24892 * r24889;
        double r24901 = r24899 - r24900;
        double r24902 = sqrt(r24901);
        double r24903 = sqrt(r24902);
        double r24904 = r24903 * r24903;
        double r24905 = r24904 - r24885;
        double r24906 = sqrt(r24905);
        double r24907 = pow(r24885, r24893);
        double r24908 = r24907 - r24900;
        double r24909 = sqrt(r24908);
        double r24910 = r24909 - r24885;
        double r24911 = sqrt(r24910);
        double r24912 = r24892 / r24911;
        double r24913 = r24906 / r24912;
        double r24914 = -0.5;
        double r24915 = r24890 * r24914;
        double r24916 = r24898 ? r24913 : r24915;
        double r24917 = r24887 ? r24896 : r24916;
        return r24917;
}

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 3 regimes

  2. if b_2 < -3.828737593568667e+109

    1. Initial program 49.5

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

    2. Simplified49.5

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

    3. Taylor expanded around -inf 3.7

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

    4. Simplified3.7

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

    if -3.828737593568667e+109 < b_2 < 1.423163788076293e-136

    1. Initial program 11.4

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

    2. Simplified11.4

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

    4. Applied clear-num11.5

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

    5. Simplified11.5

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt11.8

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

    8. Applied *-un-lft-identity11.8

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

    9. Applied times-frac11.8

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

    10. Applied associate-/r*11.8

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

    11. Simplified11.8

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}}{\frac{a}{\sqrt{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}\]
    12. Using strategy rm

    13. Applied add-sqr-sqrt11.8

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

    14. Applied sqrt-prod11.8

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

    15. Simplified11.8

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

    16. Simplified11.8

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

    if 1.423163788076293e-136 < b_2

    1. Initial program 50.8

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

    2. Simplified50.8

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

    3. Taylor expanded around inf 12.2

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

    4. Simplified12.2

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.828737593568666707297780122417133955299 \cdot 10^{109}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{\frac{a}{2}}\\ \mathbf{elif}\;b_2 \le 1.423163788076292919361412483479690249604 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} - b_2}}{\frac{a}{\sqrt{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))