Average Error: 33.2 → 9.9
Time: 22.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.8774910265390396 \cdot 10^{-73}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.5703497435733685 \cdot 10^{+102}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1302822 = b_2;
        double r1302823 = -r1302822;
        double r1302824 = r1302822 * r1302822;
        double r1302825 = a;
        double r1302826 = c;
        double r1302827 = r1302825 * r1302826;
        double r1302828 = r1302824 - r1302827;
        double r1302829 = sqrt(r1302828);
        double r1302830 = r1302823 - r1302829;
        double r1302831 = r1302830 / r1302825;
        return r1302831;
}
↓
double f(double a, double b_2, double c) {
        double r1302832 = b_2;
        double r1302833 = -1.8774910265390396e-73;
        bool r1302834 = r1302832 <= r1302833;
        double r1302835 = -0.5;
        double r1302836 = c;
        double r1302837 = r1302836 / r1302832;
        double r1302838 = r1302835 * r1302837;
        double r1302839 = 2.5703497435733685e+102;
        bool r1302840 = r1302832 <= r1302839;
        double r1302841 = -r1302832;
        double r1302842 = r1302832 * r1302832;
        double r1302843 = a;
        double r1302844 = r1302843 * r1302836;
        double r1302845 = r1302842 - r1302844;
        double r1302846 = sqrt(r1302845);
        double r1302847 = r1302841 - r1302846;
        double r1302848 = 1.0;
        double r1302849 = r1302848 / r1302843;
        double r1302850 = r1302847 * r1302849;
        double r1302851 = 0.5;
        double r1302852 = r1302837 * r1302851;
        double r1302853 = 2.0;
        double r1302854 = r1302832 / r1302843;
        double r1302855 = r1302853 * r1302854;
        double r1302856 = r1302852 - r1302855;
        double r1302857 = r1302840 ? r1302850 : r1302856;
        double r1302858 = r1302834 ? r1302838 : r1302857;
        return r1302858;
}

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 < -1.8774910265390396e-73

    1. Initial program 52.5

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

    2. Taylor expanded around -inf 8.6

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

    if -1.8774910265390396e-73 < b_2 < 2.5703497435733685e+102

    1. Initial program 13.1

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

    3. Applied div-inv13.2

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

    if 2.5703497435733685e+102 < b_2

    1. Initial program 43.9

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

    2. Taylor expanded around inf 2.9

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))