Average Error: 34.5 → 6.8
Time: 4.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.96852488717399238 \cdot 10^{152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -7.5932715112131794 \cdot 10^{-252}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 9.9656763960867421 \cdot 10^{45}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.96852488717399238 \cdot 10^{152}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -7.5932715112131794 \cdot 10^{-252}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 9.9656763960867421 \cdot 10^{45}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\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 r71825 = b_2;
        double r71826 = -r71825;
        double r71827 = r71825 * r71825;
        double r71828 = a;
        double r71829 = c;
        double r71830 = r71828 * r71829;
        double r71831 = r71827 - r71830;
        double r71832 = sqrt(r71831);
        double r71833 = r71826 - r71832;
        double r71834 = r71833 / r71828;
        return r71834;
}
↓
double f(double a, double b_2, double c) {
        double r71835 = b_2;
        double r71836 = -6.968524887173992e+152;
        bool r71837 = r71835 <= r71836;
        double r71838 = -0.5;
        double r71839 = c;
        double r71840 = r71839 / r71835;
        double r71841 = r71838 * r71840;
        double r71842 = -7.593271511213179e-252;
        bool r71843 = r71835 <= r71842;
        double r71844 = r71835 * r71835;
        double r71845 = a;
        double r71846 = r71845 * r71839;
        double r71847 = r71844 - r71846;
        double r71848 = sqrt(r71847);
        double r71849 = r71848 - r71835;
        double r71850 = r71839 / r71849;
        double r71851 = 9.965676396086742e+45;
        bool r71852 = r71835 <= r71851;
        double r71853 = -r71835;
        double r71854 = r71853 / r71845;
        double r71855 = r71848 / r71845;
        double r71856 = r71854 - r71855;
        double r71857 = 0.5;
        double r71858 = r71857 * r71840;
        double r71859 = 2.0;
        double r71860 = r71835 / r71845;
        double r71861 = r71859 * r71860;
        double r71862 = r71858 - r71861;
        double r71863 = r71852 ? r71856 : r71862;
        double r71864 = r71843 ? r71850 : r71863;
        double r71865 = r71837 ? r71841 : r71864;
        return r71865;
}

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 < -6.968524887173992e+152

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 1.5

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

    if -6.968524887173992e+152 < b_2 < -7.593271511213179e-252

    1. Initial program 36.7

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

    3. Applied flip--36.7

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

    4. Simplified16.0

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

    5. Simplified16.0

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity16.0

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{1 \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}}}{a}\]

    8. Applied *-un-lft-identity16.0

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + a \cdot c\right)}}{1 \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}}{a}\]

    9. Applied times-frac16.0

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

    10. Applied associate-/l*16.2

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{a}{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}}\]

    11. Simplified14.5

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

    13. Applied associate-/r*14.2

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

    14. Simplified7.6

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

    if -7.593271511213179e-252 < b_2 < 9.965676396086742e+45

    1. Initial program 10.6

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

    3. Applied div-sub10.6

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

    if 9.965676396086742e+45 < b_2

    1. Initial program 36.8

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

    2. Taylor expanded around inf 5.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.96852488717399238 \cdot 10^{152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -7.5932715112131794 \cdot 10^{-252}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 9.9656763960867421 \cdot 10^{45}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))