Average Error: 33.9 → 6.8
Time: 18.3s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -3.320360656741600748358420677927629618815 \cdot 10^{-289}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{elif}\;b_2 \le 6.326287366549382745037046972324082366467 \cdot 10^{74}:\\ \;\;\;\;c \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -3.320360656741600748358420677927629618815 \cdot 10^{-289}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

\mathbf{elif}\;b_2 \le 6.326287366549382745037046972324082366467 \cdot 10^{74}:\\
\;\;\;\;c \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r26885 = b_2;
        double r26886 = -r26885;
        double r26887 = r26885 * r26885;
        double r26888 = a;
        double r26889 = c;
        double r26890 = r26888 * r26889;
        double r26891 = r26887 - r26890;
        double r26892 = sqrt(r26891);
        double r26893 = r26886 + r26892;
        double r26894 = r26893 / r26888;
        return r26894;
}
↓
double f(double a, double b_2, double c) {
        double r26895 = b_2;
        double r26896 = -1.361733299857302e+105;
        bool r26897 = r26895 <= r26896;
        double r26898 = 0.5;
        double r26899 = c;
        double r26900 = r26899 / r26895;
        double r26901 = r26898 * r26900;
        double r26902 = 2.0;
        double r26903 = a;
        double r26904 = r26895 / r26903;
        double r26905 = r26902 * r26904;
        double r26906 = r26901 - r26905;
        double r26907 = -3.3203606567416007e-289;
        bool r26908 = r26895 <= r26907;
        double r26909 = -r26895;
        double r26910 = r26895 * r26895;
        double r26911 = r26899 * r26903;
        double r26912 = r26910 - r26911;
        double r26913 = sqrt(r26912);
        double r26914 = r26909 + r26913;
        double r26915 = r26914 / r26903;
        double r26916 = 6.326287366549383e+74;
        bool r26917 = r26895 <= r26916;
        double r26918 = 1.0;
        double r26919 = r26903 * r26899;
        double r26920 = r26910 - r26919;
        double r26921 = sqrt(r26920);
        double r26922 = r26909 - r26921;
        double r26923 = r26918 / r26922;
        double r26924 = r26899 * r26923;
        double r26925 = -0.5;
        double r26926 = r26925 * r26900;
        double r26927 = r26917 ? r26924 : r26926;
        double r26928 = r26908 ? r26915 : r26927;
        double r26929 = r26897 ? r26906 : r26928;
        return r26929;
}

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 < -1.361733299857302e+105

    1. Initial program 48.6

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

    2. Taylor expanded around -inf 3.6

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

    if -1.361733299857302e+105 < b_2 < -3.3203606567416007e-289

    1. Initial program 8.5

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

    2. Taylor expanded around 0 8.5

      \[\leadsto \frac{\left(-b_2\right) + \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified8.5

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

    if -3.3203606567416007e-289 < b_2 < 6.326287366549383e+74

    1. Initial program 29.9

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

    3. Applied flip-+29.9

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

    4. Simplified16.1

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

    6. Applied *-un-lft-identity16.1

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

    7. Applied *-un-lft-identity16.1

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

    8. Applied times-frac14.2

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

    9. Applied times-frac10.5

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

    10. Simplified10.5

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

    11. Simplified9.5

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

    if 6.326287366549383e+74 < b_2

    1. Initial program 58.0

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -3.320360656741600748358420677927629618815 \cdot 10^{-289}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{elif}\;b_2 \le 6.326287366549382745037046972324082366467 \cdot 10^{74}:\\ \;\;\;\;c \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))