Average Error: 34.2 → 10.4
Time: 5.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.4270058556435274 \cdot 10^{-117}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.49922826628406174 \cdot 10^{84}:\\ \;\;\;\;1 \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.4270058556435274 \cdot 10^{-117}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.49922826628406174 \cdot 10^{84}:\\
\;\;\;\;1 \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r67869 = b_2;
        double r67870 = -r67869;
        double r67871 = r67869 * r67869;
        double r67872 = a;
        double r67873 = c;
        double r67874 = r67872 * r67873;
        double r67875 = r67871 - r67874;
        double r67876 = sqrt(r67875);
        double r67877 = r67870 - r67876;
        double r67878 = r67877 / r67872;
        return r67878;
}
↓
double f(double a, double b_2, double c) {
        double r67879 = b_2;
        double r67880 = -4.4270058556435274e-117;
        bool r67881 = r67879 <= r67880;
        double r67882 = -0.5;
        double r67883 = c;
        double r67884 = r67883 / r67879;
        double r67885 = r67882 * r67884;
        double r67886 = 2.4992282662840617e+84;
        bool r67887 = r67879 <= r67886;
        double r67888 = 1.0;
        double r67889 = -r67879;
        double r67890 = r67879 * r67879;
        double r67891 = a;
        double r67892 = r67891 * r67883;
        double r67893 = r67890 - r67892;
        double r67894 = sqrt(r67893);
        double r67895 = r67889 - r67894;
        double r67896 = r67895 / r67891;
        double r67897 = r67888 * r67896;
        double r67898 = 0.5;
        double r67899 = r67898 * r67884;
        double r67900 = 2.0;
        double r67901 = r67879 / r67891;
        double r67902 = r67900 * r67901;
        double r67903 = r67899 - r67902;
        double r67904 = r67887 ? r67897 : r67903;
        double r67905 = r67881 ? r67885 : r67904;
        return r67905;
}

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 < -4.4270058556435274e-117

    1. Initial program 51.5

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

    2. Taylor expanded around -inf 11.0

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

    if -4.4270058556435274e-117 < b_2 < 2.4992282662840617e+84

    1. Initial program 12.4

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

    3. Applied div-inv12.5

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

    5. Applied *-un-lft-identity12.5

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

    6. Applied associate-*l*12.5

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

    7. Simplified12.4

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

    if 2.4992282662840617e+84 < b_2

    1. Initial program 43.1

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

    2. Taylor expanded around inf 4.1

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.4270058556435274 \cdot 10^{-117}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.49922826628406174 \cdot 10^{84}:\\ \;\;\;\;1 \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))