Average Error: 34.5 → 10.9
Time: 9.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.615257373542238721197930661559276546696 \cdot 10^{153}:\\ \;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(2, \frac{c}{b}, -2 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.388070047225937856958905133202240499626 \cdot 10^{-143}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.615257373542238721197930661559276546696 \cdot 10^{153}:\\
\;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(2, \frac{c}{b}, -2 \cdot \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.388070047225937856958905133202240499626 \cdot 10^{-143}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r43836 = b;
        double r43837 = -r43836;
        double r43838 = r43836 * r43836;
        double r43839 = 4.0;
        double r43840 = a;
        double r43841 = r43839 * r43840;
        double r43842 = c;
        double r43843 = r43841 * r43842;
        double r43844 = r43838 - r43843;
        double r43845 = sqrt(r43844);
        double r43846 = r43837 + r43845;
        double r43847 = 2.0;
        double r43848 = r43847 * r43840;
        double r43849 = r43846 / r43848;
        return r43849;
}
↓
double f(double a, double b, double c) {
        double r43850 = b;
        double r43851 = -2.6152573735422387e+153;
        bool r43852 = r43850 <= r43851;
        double r43853 = 1.0;
        double r43854 = 2.0;
        double r43855 = r43853 / r43854;
        double r43856 = c;
        double r43857 = r43856 / r43850;
        double r43858 = -2.0;
        double r43859 = a;
        double r43860 = r43850 / r43859;
        double r43861 = r43858 * r43860;
        double r43862 = fma(r43854, r43857, r43861);
        double r43863 = r43855 * r43862;
        double r43864 = 1.3880700472259379e-143;
        bool r43865 = r43850 <= r43864;
        double r43866 = 2.0;
        double r43867 = pow(r43850, r43866);
        double r43868 = 4.0;
        double r43869 = r43859 * r43856;
        double r43870 = r43868 * r43869;
        double r43871 = r43867 - r43870;
        double r43872 = sqrt(r43871);
        double r43873 = r43872 - r43850;
        double r43874 = r43873 / r43859;
        double r43875 = r43855 * r43874;
        double r43876 = -1.0;
        double r43877 = r43876 * r43857;
        double r43878 = r43865 ? r43875 : r43877;
        double r43879 = r43852 ? r43863 : r43878;
        return r43879;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -2.6152573735422387e+153

    1. Initial program 63.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied clear-num63.8

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\]

    4. Simplified63.8

      \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity63.8

      \[\leadsto \frac{1}{\frac{2 \cdot a}{\color{blue}{1 \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)}}}\]

    7. Applied times-frac63.8

      \[\leadsto \frac{1}{\color{blue}{\frac{2}{1} \cdot \frac{a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}\]

    8. Applied add-cube-cbrt63.8

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{2}{1} \cdot \frac{a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\]

    9. Applied times-frac63.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{2}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}\]

    10. Simplified63.8

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\]

    11. Simplified63.8

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{a}}\]

    12. Taylor expanded around -inf 2.2

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right)}\]

    13. Simplified2.2

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(2, \frac{c}{b}, -2 \cdot \frac{b}{a}\right)}\]

    if -2.6152573735422387e+153 < b < 1.3880700472259379e-143

    1. Initial program 11.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied clear-num11.6

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\]

    4. Simplified11.6

      \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity11.6

      \[\leadsto \frac{1}{\frac{2 \cdot a}{\color{blue}{1 \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)}}}\]

    7. Applied times-frac11.6

      \[\leadsto \frac{1}{\color{blue}{\frac{2}{1} \cdot \frac{a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}\]

    8. Applied add-cube-cbrt11.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{2}{1} \cdot \frac{a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\]

    9. Applied times-frac11.6

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{2}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}\]

    10. Simplified11.6

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\]

    11. Simplified11.5

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{a}}\]

    if 1.3880700472259379e-143 < b

    1. Initial program 50.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around inf 12.6

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.615257373542238721197930661559276546696 \cdot 10^{153}:\\ \;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(2, \frac{c}{b}, -2 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.388070047225937856958905133202240499626 \cdot 10^{-143}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))