Average Error: 33.0 → 10.8
Time: 20.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{1}{a} \cdot \sqrt{b_2 \cdot b_2 - c \cdot a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.348931433494438 \cdot 10^{+39}:\\
\;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 1.3353078790738604 \cdot 10^{-121}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.6168702840263923 \cdot 10^{-79}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.546013236023957 \cdot 10^{-67}:\\
\;\;\;\;\frac{1}{a} \cdot \sqrt{b_2 \cdot b_2 - c \cdot a} - \frac{b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r961878 = b_2;
        double r961879 = -r961878;
        double r961880 = r961878 * r961878;
        double r961881 = a;
        double r961882 = c;
        double r961883 = r961881 * r961882;
        double r961884 = r961880 - r961883;
        double r961885 = sqrt(r961884);
        double r961886 = r961879 + r961885;
        double r961887 = r961886 / r961881;
        return r961887;
}
↓
double f(double a, double b_2, double c) {
        double r961888 = b_2;
        double r961889 = -9.348931433494438e+39;
        bool r961890 = r961888 <= r961889;
        double r961891 = c;
        double r961892 = r961891 / r961888;
        double r961893 = 2.0;
        double r961894 = r961892 / r961893;
        double r961895 = a;
        double r961896 = r961888 / r961895;
        double r961897 = r961896 * r961893;
        double r961898 = r961894 - r961897;
        double r961899 = 1.3353078790738604e-121;
        bool r961900 = r961888 <= r961899;
        double r961901 = r961888 * r961888;
        double r961902 = r961891 * r961895;
        double r961903 = r961901 - r961902;
        double r961904 = sqrt(r961903);
        double r961905 = r961904 / r961895;
        double r961906 = r961905 - r961896;
        double r961907 = 1.6168702840263923e-79;
        bool r961908 = r961888 <= r961907;
        double r961909 = -0.5;
        double r961910 = r961909 * r961892;
        double r961911 = 1.546013236023957e-67;
        bool r961912 = r961888 <= r961911;
        double r961913 = 1.0;
        double r961914 = r961913 / r961895;
        double r961915 = r961914 * r961904;
        double r961916 = r961915 - r961896;
        double r961917 = r961912 ? r961916 : r961910;
        double r961918 = r961908 ? r961910 : r961917;
        double r961919 = r961900 ? r961906 : r961918;
        double r961920 = r961890 ? r961898 : r961919;
        return r961920;
}

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 < -9.348931433494438e+39

    1. Initial program 34.0

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

    2. Simplified34.0

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

    4. Applied div-sub34.0

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

    5. Taylor expanded around -inf 6.2

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

    6. Simplified6.2

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

    if -9.348931433494438e+39 < b_2 < 1.3353078790738604e-121

    1. Initial program 12.2

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

    2. Simplified12.2

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

    4. Applied div-sub12.2

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

    6. Applied *-un-lft-identity12.2

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

    7. Applied associate-/r*12.2

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

    8. Simplified12.2

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

    if 1.3353078790738604e-121 < b_2 < 1.6168702840263923e-79 or 1.546013236023957e-67 < 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 11.2

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

    if 1.6168702840263923e-79 < b_2 < 1.546013236023957e-67

    1. Initial program 35.9

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

    2. Simplified35.9

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

    4. Applied div-sub35.9

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

    6. Applied div-inv36.0

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{1}{a} \cdot \sqrt{b_2 \cdot b_2 - c \cdot a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))