Average Error: 33.8 → 9.1
Time: 18.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.970010565552108757188050455448622102575 \cdot 10^{58}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \mathbf{elif}\;b_2 \le -4.767857970173808386258495003523252842217 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 3.628799960716311990444092539387346352569 \cdot 10^{50}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.970010565552108757188050455448622102575 \cdot 10^{58}:\\
\;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\

\mathbf{elif}\;b_2 \le -4.767857970173808386258495003523252842217 \cdot 10^{-309}:\\
\;\;\;\;\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 3.628799960716311990444092539387346352569 \cdot 10^{50}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r54887 = b_2;
        double r54888 = -r54887;
        double r54889 = r54887 * r54887;
        double r54890 = a;
        double r54891 = c;
        double r54892 = r54890 * r54891;
        double r54893 = r54889 - r54892;
        double r54894 = sqrt(r54893);
        double r54895 = r54888 - r54894;
        double r54896 = r54895 / r54890;
        return r54896;
}
↓
double f(double a, double b_2, double c) {
        double r54897 = b_2;
        double r54898 = -1.9700105655521088e+58;
        bool r54899 = r54897 <= r54898;
        double r54900 = c;
        double r54901 = r54900 / r54897;
        double r54902 = -0.5;
        double r54903 = r54901 * r54902;
        double r54904 = -4.76785797017381e-309;
        bool r54905 = r54897 <= r54904;
        double r54906 = a;
        double r54907 = r54906 * r54900;
        double r54908 = r54897 * r54897;
        double r54909 = r54908 - r54907;
        double r54910 = sqrt(r54909);
        double r54911 = r54910 - r54897;
        double r54912 = r54907 / r54911;
        double r54913 = r54912 / r54906;
        double r54914 = 3.628799960716312e+50;
        bool r54915 = r54897 <= r54914;
        double r54916 = 1.0;
        double r54917 = -r54897;
        double r54918 = r54917 - r54910;
        double r54919 = r54906 / r54918;
        double r54920 = r54916 / r54919;
        double r54921 = -2.0;
        double r54922 = r54897 / r54906;
        double r54923 = r54921 * r54922;
        double r54924 = r54915 ? r54920 : r54923;
        double r54925 = r54905 ? r54913 : r54924;
        double r54926 = r54899 ? r54903 : r54925;
        return r54926;
}

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 < -1.9700105655521088e+58

    1. Initial program 57.5

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

    2. Taylor expanded around -inf 3.4

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

    3. Simplified3.4

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

    if -1.9700105655521088e+58 < b_2 < -4.76785797017381e-309

    1. Initial program 29.3

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

    3. Applied flip--29.4

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

    4. Simplified16.6

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

    5. Simplified16.6

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

    if -4.76785797017381e-309 < b_2 < 3.628799960716312e+50

    1. Initial program 9.7

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

    3. Applied clear-num9.8

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

    if 3.628799960716312e+50 < b_2

    1. Initial program 38.1

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

    3. Applied clear-num38.2

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

    4. Taylor expanded around 0 6.3

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

    5. Simplified6.3

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.970010565552108757188050455448622102575 \cdot 10^{58}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \mathbf{elif}\;b_2 \le -4.767857970173808386258495003523252842217 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 3.628799960716311990444092539387346352569 \cdot 10^{50}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))