Average Error: 34.4 → 10.4
Time: 4.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.0674124610604968 \cdot 10^{-82}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 5.96876625840091586 \cdot 10^{107}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{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 r73844 = b_2;
        double r73845 = -r73844;
        double r73846 = r73844 * r73844;
        double r73847 = a;
        double r73848 = c;
        double r73849 = r73847 * r73848;
        double r73850 = r73846 - r73849;
        double r73851 = sqrt(r73850);
        double r73852 = r73845 - r73851;
        double r73853 = r73852 / r73847;
        return r73853;
}
↓
double f(double a, double b_2, double c) {
        double r73854 = b_2;
        double r73855 = -1.0674124610604968e-82;
        bool r73856 = r73854 <= r73855;
        double r73857 = -0.5;
        double r73858 = c;
        double r73859 = r73858 / r73854;
        double r73860 = r73857 * r73859;
        double r73861 = 5.968766258400916e+107;
        bool r73862 = r73854 <= r73861;
        double r73863 = -r73854;
        double r73864 = cbrt(r73863);
        double r73865 = r73864 * r73864;
        double r73866 = r73854 * r73854;
        double r73867 = a;
        double r73868 = r73867 * r73858;
        double r73869 = r73866 - r73868;
        double r73870 = sqrt(r73869);
        double r73871 = -r73870;
        double r73872 = fma(r73865, r73864, r73871);
        double r73873 = r73872 / r73867;
        double r73874 = 0.5;
        double r73875 = r73874 * r73859;
        double r73876 = 2.0;
        double r73877 = r73854 / r73867;
        double r73878 = r73876 * r73877;
        double r73879 = r73875 - r73878;
        double r73880 = r73862 ? r73873 : r73879;
        double r73881 = r73856 ? r73860 : r73880;
        return r73881;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -1.0674124610604968e-82

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 8.8

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

    if -1.0674124610604968e-82 < b_2 < 5.968766258400916e+107

    1. Initial program 13.7

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

    3. Applied add-cube-cbrt13.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}\right) \cdot \sqrt[3]{-b_2}} - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    4. Applied fma-neg13.9

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a}\]

    if 5.968766258400916e+107 < b_2

    1. Initial program 50.0

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

    2. Taylor expanded around inf 3.8

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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))