Average Error: 32.8 → 9.8
Time: 17.9s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.3599228730895225 \cdot 10^{+90}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.3599228730895225 \cdot 10^{+90}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 3.1295384133612364 \cdot 10^{-73}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r351895 = b_2;
        double r351896 = -r351895;
        double r351897 = r351895 * r351895;
        double r351898 = a;
        double r351899 = c;
        double r351900 = r351898 * r351899;
        double r351901 = r351897 - r351900;
        double r351902 = sqrt(r351901);
        double r351903 = r351896 + r351902;
        double r351904 = r351903 / r351898;
        return r351904;
}
↓
double f(double a, double b_2, double c) {
        double r351905 = b_2;
        double r351906 = -1.3599228730895225e+90;
        bool r351907 = r351905 <= r351906;
        double r351908 = 0.5;
        double r351909 = c;
        double r351910 = r351909 / r351905;
        double r351911 = r351908 * r351910;
        double r351912 = a;
        double r351913 = r351905 / r351912;
        double r351914 = 2.0;
        double r351915 = r351913 * r351914;
        double r351916 = r351911 - r351915;
        double r351917 = 3.1295384133612364e-73;
        bool r351918 = r351905 <= r351917;
        double r351919 = r351905 * r351905;
        double r351920 = r351909 * r351912;
        double r351921 = r351919 - r351920;
        double r351922 = sqrt(r351921);
        double r351923 = r351922 - r351905;
        double r351924 = r351923 / r351912;
        double r351925 = -0.5;
        double r351926 = r351910 * r351925;
        double r351927 = r351918 ? r351924 : r351926;
        double r351928 = r351907 ? r351916 : r351927;
        return r351928;
}

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 < -1.3599228730895225e+90

    1. Initial program 41.6

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

    2. Simplified41.6

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

    3. Taylor expanded around inf 41.6

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

    4. Simplified41.6

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

    5. Taylor expanded around -inf 3.8

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

    if -1.3599228730895225e+90 < b_2 < 3.1295384133612364e-73

    1. Initial program 12.8

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

    2. Simplified12.8

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

    3. Taylor expanded around inf 12.8

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

    4. Simplified12.8

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

    if 3.1295384133612364e-73 < b_2

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 52.3

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

    4. Simplified52.3

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

    5. Taylor expanded around inf 9.0

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.3599228730895225 \cdot 10^{+90}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))