Average Error: 34.1 → 9.9
Time: 6.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.260729798395838 \cdot 10^{-43}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.67643144401154069 \cdot 10^{104}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a} - \left(\frac{b_2}{a} - \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.260729798395838 \cdot 10^{-43}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.67643144401154069 \cdot 10^{104}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-b_2}{a} - \left(\frac{b_2}{a} - \frac{1}{2} \cdot \frac{c}{b_2}\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r88819 = b_2;
        double r88820 = -r88819;
        double r88821 = r88819 * r88819;
        double r88822 = a;
        double r88823 = c;
        double r88824 = r88822 * r88823;
        double r88825 = r88821 - r88824;
        double r88826 = sqrt(r88825);
        double r88827 = r88820 - r88826;
        double r88828 = r88827 / r88822;
        return r88828;
}
↓
double f(double a, double b_2, double c) {
        double r88829 = b_2;
        double r88830 = -8.260729798395838e-43;
        bool r88831 = r88829 <= r88830;
        double r88832 = -0.5;
        double r88833 = c;
        double r88834 = r88833 / r88829;
        double r88835 = r88832 * r88834;
        double r88836 = 1.6764314440115407e+104;
        bool r88837 = r88829 <= r88836;
        double r88838 = -r88829;
        double r88839 = a;
        double r88840 = r88838 / r88839;
        double r88841 = r88829 * r88829;
        double r88842 = r88839 * r88833;
        double r88843 = r88841 - r88842;
        double r88844 = sqrt(r88843);
        double r88845 = r88844 / r88839;
        double r88846 = r88840 - r88845;
        double r88847 = r88829 / r88839;
        double r88848 = 0.5;
        double r88849 = r88848 * r88834;
        double r88850 = r88847 - r88849;
        double r88851 = r88840 - r88850;
        double r88852 = r88837 ? r88846 : r88851;
        double r88853 = r88831 ? r88835 : r88852;
        return r88853;
}

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 < -8.260729798395838e-43

    1. Initial program 54.7

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

    2. Taylor expanded around -inf 7.3

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

    if -8.260729798395838e-43 < b_2 < 1.6764314440115407e+104

    1. Initial program 14.0

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

    3. Applied div-sub14.0

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

    if 1.6764314440115407e+104 < b_2

    1. Initial program 47.9

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

    3. Applied div-sub47.9

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

    4. Taylor expanded around inf 3.3

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.260729798395838 \cdot 10^{-43}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.67643144401154069 \cdot 10^{104}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a} - \left(\frac{b_2}{a} - \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))