Average Error: 20.4 → 7.7
Time: 6.2s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3459740091915103 \cdot 10^{154} \lor \neg \left(b \le -1.078731019298903 \cdot 10^{-310} \lor \neg \left(b \le 3.6883396060119998 \cdot 10^{69}\right)\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\left(\frac{a}{\sqrt{b}} \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}}\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.3459740091915103 \cdot 10^{154} \lor \neg \left(b \le -1.078731019298903 \cdot 10^{-310} \lor \neg \left(b \le 3.6883396060119998 \cdot 10^{69}\right)\right):\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\left(\frac{a}{\sqrt{b}} \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}}\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right) - 2 \cdot b}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r35753 = b;
        double r35754 = 0.0;
        bool r35755 = r35753 >= r35754;
        double r35756 = 2.0;
        double r35757 = c;
        double r35758 = r35756 * r35757;
        double r35759 = -r35753;
        double r35760 = r35753 * r35753;
        double r35761 = 4.0;
        double r35762 = a;
        double r35763 = r35761 * r35762;
        double r35764 = r35763 * r35757;
        double r35765 = r35760 - r35764;
        double r35766 = sqrt(r35765);
        double r35767 = r35759 - r35766;
        double r35768 = r35758 / r35767;
        double r35769 = r35759 + r35766;
        double r35770 = r35756 * r35762;
        double r35771 = r35769 / r35770;
        double r35772 = r35755 ? r35768 : r35771;
        return r35772;
}
↓
double f(double a, double b, double c) {
        double r35773 = b;
        double r35774 = -1.3459740091915103e+154;
        bool r35775 = r35773 <= r35774;
        double r35776 = -1.0787310192989e-310;
        bool r35777 = r35773 <= r35776;
        double r35778 = 3.688339606012e+69;
        bool r35779 = r35773 <= r35778;
        double r35780 = !r35779;
        bool r35781 = r35777 || r35780;
        double r35782 = !r35781;
        bool r35783 = r35775 || r35782;
        double r35784 = 0.0;
        bool r35785 = r35773 >= r35784;
        double r35786 = 2.0;
        double r35787 = c;
        double r35788 = r35786 * r35787;
        double r35789 = -r35773;
        double r35790 = r35773 * r35773;
        double r35791 = 4.0;
        double r35792 = a;
        double r35793 = r35791 * r35792;
        double r35794 = r35793 * r35787;
        double r35795 = r35790 - r35794;
        double r35796 = sqrt(r35795);
        double r35797 = r35789 - r35796;
        double r35798 = r35788 / r35797;
        double r35799 = r35792 * r35787;
        double r35800 = r35799 / r35773;
        double r35801 = r35786 * r35800;
        double r35802 = 2.0;
        double r35803 = r35802 * r35773;
        double r35804 = r35801 - r35803;
        double r35805 = r35786 * r35792;
        double r35806 = r35804 / r35805;
        double r35807 = r35785 ? r35798 : r35806;
        double r35808 = sqrt(r35773);
        double r35809 = r35792 / r35808;
        double r35810 = 0.5;
        double r35811 = pow(r35773, r35810);
        double r35812 = r35787 / r35811;
        double r35813 = cbrt(r35812);
        double r35814 = cbrt(r35813);
        double r35815 = r35814 * r35814;
        double r35816 = r35815 * r35814;
        double r35817 = r35816 * r35813;
        double r35818 = r35809 * r35817;
        double r35819 = r35818 * r35813;
        double r35820 = r35786 * r35819;
        double r35821 = r35820 - r35803;
        double r35822 = r35788 / r35821;
        double r35823 = r35789 + r35796;
        double r35824 = r35823 / r35805;
        double r35825 = r35785 ? r35822 : r35824;
        double r35826 = r35783 ? r35807 : r35825;
        return r35826;
}

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 2 regimes

  2. if b < -1.3459740091915103e+154 or -1.0787310192989e-310 < b < 3.688339606012e+69

    1. Initial program 24.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Taylor expanded around -inf 9.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\]

    if -1.3459740091915103e+154 < b < -1.0787310192989e-310 or 3.688339606012e+69 < b

    1. Initial program 17.5

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Taylor expanded around inf 8.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt8.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    5. Applied times-frac6.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \color{blue}{\left(\frac{a}{\sqrt{b}} \cdot \frac{c}{\sqrt{b}}\right)} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    6. Simplified6.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \color{blue}{\frac{c}{{b}^{\frac{1}{2}}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt6.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}} \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    9. Applied associate-*r*6.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \color{blue}{\left(\left(\frac{a}{\sqrt{b}} \cdot \left(\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}} \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt6.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\left(\frac{a}{\sqrt{b}} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}}\right)} \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3459740091915103 \cdot 10^{154} \lor \neg \left(b \le -1.078731019298903 \cdot 10^{-310} \lor \neg \left(b \le 3.6883396060119998 \cdot 10^{69}\right)\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\left(\frac{a}{\sqrt{b}} \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}}\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right)\right) \cdot \sqrt[3]{\frac{c}{{b}^{\frac{1}{2}}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))