Average Error: 34.2 → 13.8
Time: 24.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\frac{\frac{b}{2} \cdot -2}{a}\\ \mathbf{elif}\;b \le 1.627160326743933989296751920246101845745 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{2}}} \cdot \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\sqrt{\sqrt{2}}}}{\sqrt{2}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \end{array}\]
\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 -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\
\;\;\;\;\frac{\frac{b}{2} \cdot -2}{a}\\

\mathbf{elif}\;b \le 1.627160326743933989296751920246101845745 \cdot 10^{-46}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{2}}} \cdot \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\sqrt{\sqrt{2}}}}{\sqrt{2}}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2434774 = b;
        double r2434775 = -r2434774;
        double r2434776 = r2434774 * r2434774;
        double r2434777 = 4.0;
        double r2434778 = a;
        double r2434779 = r2434777 * r2434778;
        double r2434780 = c;
        double r2434781 = r2434779 * r2434780;
        double r2434782 = r2434776 - r2434781;
        double r2434783 = sqrt(r2434782);
        double r2434784 = r2434775 + r2434783;
        double r2434785 = 2.0;
        double r2434786 = r2434785 * r2434778;
        double r2434787 = r2434784 / r2434786;
        return r2434787;
}
↓
double f(double a, double b, double c) {
        double r2434788 = b;
        double r2434789 = -3.7108875578650606e+138;
        bool r2434790 = r2434788 <= r2434789;
        double r2434791 = 2.0;
        double r2434792 = r2434788 / r2434791;
        double r2434793 = -2.0;
        double r2434794 = r2434792 * r2434793;
        double r2434795 = a;
        double r2434796 = r2434794 / r2434795;
        double r2434797 = 1.627160326743934e-46;
        bool r2434798 = r2434788 <= r2434797;
        double r2434799 = 1.0;
        double r2434800 = sqrt(r2434791);
        double r2434801 = sqrt(r2434800);
        double r2434802 = r2434799 / r2434801;
        double r2434803 = r2434788 * r2434788;
        double r2434804 = 4.0;
        double r2434805 = r2434804 * r2434795;
        double r2434806 = c;
        double r2434807 = r2434805 * r2434806;
        double r2434808 = r2434803 - r2434807;
        double r2434809 = sqrt(r2434808);
        double r2434810 = r2434809 - r2434788;
        double r2434811 = r2434810 / r2434801;
        double r2434812 = r2434811 / r2434800;
        double r2434813 = r2434802 * r2434812;
        double r2434814 = r2434813 / r2434795;
        double r2434815 = -1.0;
        double r2434816 = r2434795 * r2434806;
        double r2434817 = r2434816 / r2434788;
        double r2434818 = r2434815 * r2434817;
        double r2434819 = r2434818 / r2434795;
        double r2434820 = r2434798 ? r2434814 : r2434819;
        double r2434821 = r2434790 ? r2434796 : r2434820;
        return r2434821;
}

Error

Bits error versus a

Bits error versus b

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 < -3.7108875578650606e+138

    1. Initial program 58.5

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

    2. Simplified58.5

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

    4. Applied add-sqr-sqrt58.6

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

    5. Applied associate-/r*58.6

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

    6. Taylor expanded around -inf 3.3

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

    7. Simplified2.1

      \[\leadsto \frac{\color{blue}{\frac{b}{2} \cdot -2}}{a}\]

    if -3.7108875578650606e+138 < b < 1.627160326743934e-46

    1. Initial program 13.0

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

    2. Simplified13.0

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

    4. Applied add-sqr-sqrt13.8

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

    5. Applied associate-/r*13.5

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

    7. Applied *-un-lft-identity13.5

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

    8. Applied sqrt-prod13.5

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

    9. Applied add-sqr-sqrt13.5

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

    10. Applied sqrt-prod13.1

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

    11. Applied *-un-lft-identity13.1

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

    12. Applied times-frac13.2

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

    13. Applied times-frac13.2

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

    14. Simplified13.2

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

    if 1.627160326743934e-46 < b

    1. Initial program 54.2

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

    2. Simplified54.2

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

    3. Taylor expanded around inf 18.5

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

  4. Final simplification13.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\frac{\frac{b}{2} \cdot -2}{a}\\ \mathbf{elif}\;b \le 1.627160326743933989296751920246101845745 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{2}}} \cdot \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\sqrt{\sqrt{2}}}}{\sqrt{2}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))