Average Error: 34.9 → 8.8
Time: 20.7s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.806935659273273367110965907543014627108 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.882280638219398649947750208343762878178 \cdot 10^{-149}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 811154.20607897103764116764068603515625:\\ \;\;\;\;\frac{\frac{a \cdot c}{a}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.806935659273273367110965907543014627108 \cdot 10^{98}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.882280638219398649947750208343762878178 \cdot 10^{-149}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{elif}\;b_2 \le 811154.20607897103764116764068603515625:\\
\;\;\;\;\frac{\frac{a \cdot c}{a}}{\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 r27920 = b_2;
        double r27921 = -r27920;
        double r27922 = r27920 * r27920;
        double r27923 = a;
        double r27924 = c;
        double r27925 = r27923 * r27924;
        double r27926 = r27922 - r27925;
        double r27927 = sqrt(r27926);
        double r27928 = r27921 + r27927;
        double r27929 = r27928 / r27923;
        return r27929;
}
↓
double f(double a, double b_2, double c) {
        double r27930 = b_2;
        double r27931 = -1.8069356592732734e+98;
        bool r27932 = r27930 <= r27931;
        double r27933 = 0.5;
        double r27934 = c;
        double r27935 = r27934 / r27930;
        double r27936 = r27933 * r27935;
        double r27937 = 2.0;
        double r27938 = a;
        double r27939 = r27930 / r27938;
        double r27940 = r27937 * r27939;
        double r27941 = r27936 - r27940;
        double r27942 = 1.8822806382193986e-149;
        bool r27943 = r27930 <= r27942;
        double r27944 = 1.0;
        double r27945 = r27930 * r27930;
        double r27946 = r27938 * r27934;
        double r27947 = r27945 - r27946;
        double r27948 = sqrt(r27947);
        double r27949 = r27948 - r27930;
        double r27950 = r27938 / r27949;
        double r27951 = r27944 / r27950;
        double r27952 = 811154.206078971;
        bool r27953 = r27930 <= r27952;
        double r27954 = r27946 / r27938;
        double r27955 = -r27930;
        double r27956 = r27955 - r27948;
        double r27957 = r27954 / r27956;
        double r27958 = -0.5;
        double r27959 = r27958 * r27935;
        double r27960 = r27953 ? r27957 : r27959;
        double r27961 = r27943 ? r27951 : r27960;
        double r27962 = r27932 ? r27941 : r27961;
        return r27962;
}

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 < -1.8069356592732734e+98

    1. Initial program 46.8

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

    2. Taylor expanded around -inf 4.1

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

    if -1.8069356592732734e+98 < b_2 < 1.8822806382193986e-149

    1. Initial program 11.5

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

    3. Applied clear-num11.6

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

    4. Simplified11.6

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

    if 1.8822806382193986e-149 < b_2 < 811154.206078971

    1. Initial program 34.2

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

    3. Applied flip-+34.2

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

    4. Simplified16.7

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

    6. Applied div-inv16.8

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

    8. Applied *-un-lft-identity16.8

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

    9. Applied associate-*l*16.8

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

    10. Simplified16.6

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

    if 811154.206078971 < b_2

    1. Initial program 56.8

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

    2. Taylor expanded around inf 4.9

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.806935659273273367110965907543014627108 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.882280638219398649947750208343762878178 \cdot 10^{-149}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 811154.20607897103764116764068603515625:\\ \;\;\;\;\frac{\frac{a \cdot c}{a}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))