Average Error: 19.7 → 6.8
Time: 25.2s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.073691083157782267875644621647570574227 \cdot 10^{153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, c \cdot \frac{a}{b}, -2 \cdot b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 2.473233968279812603451846937244594428558 \cdot 10^{117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}, \sqrt[3]{-b}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(-\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right) + \mathsf{fma}\left(-\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\left(\sqrt[3]{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - \left(c \cdot 2\right) \cdot \frac{a}{b}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -4.073691083157782267875644621647570574227 \cdot 10^{153}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, c \cdot \frac{a}{b}, -2 \cdot b\right)}\\

\end{array}\\

\mathbf{elif}\;b \le 2.473233968279812603451846937244594428558 \cdot 10^{117}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}, \sqrt[3]{-b}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(-\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right) + \mathsf{fma}\left(-\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\left(\sqrt[3]{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}{2 \cdot a}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \left(b - \left(c \cdot 2\right) \cdot \frac{a}{b}\right)}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1387753 = b;
        double r1387754 = 0.0;
        bool r1387755 = r1387753 >= r1387754;
        double r1387756 = -r1387753;
        double r1387757 = r1387753 * r1387753;
        double r1387758 = 4.0;
        double r1387759 = a;
        double r1387760 = r1387758 * r1387759;
        double r1387761 = c;
        double r1387762 = r1387760 * r1387761;
        double r1387763 = r1387757 - r1387762;
        double r1387764 = sqrt(r1387763);
        double r1387765 = r1387756 - r1387764;
        double r1387766 = 2.0;
        double r1387767 = r1387766 * r1387759;
        double r1387768 = r1387765 / r1387767;
        double r1387769 = r1387766 * r1387761;
        double r1387770 = r1387756 + r1387764;
        double r1387771 = r1387769 / r1387770;
        double r1387772 = r1387755 ? r1387768 : r1387771;
        return r1387772;
}
↓
double f(double a, double b, double c) {
        double r1387773 = b;
        double r1387774 = -4.073691083157782e+153;
        bool r1387775 = r1387773 <= r1387774;
        double r1387776 = 0.0;
        bool r1387777 = r1387773 >= r1387776;
        double r1387778 = -r1387773;
        double r1387779 = r1387773 * r1387773;
        double r1387780 = 4.0;
        double r1387781 = a;
        double r1387782 = r1387780 * r1387781;
        double r1387783 = c;
        double r1387784 = r1387782 * r1387783;
        double r1387785 = r1387779 - r1387784;
        double r1387786 = sqrt(r1387785);
        double r1387787 = r1387778 - r1387786;
        double r1387788 = 2.0;
        double r1387789 = r1387788 * r1387781;
        double r1387790 = r1387787 / r1387789;
        double r1387791 = r1387783 * r1387788;
        double r1387792 = r1387781 / r1387773;
        double r1387793 = r1387783 * r1387792;
        double r1387794 = -2.0;
        double r1387795 = r1387794 * r1387773;
        double r1387796 = fma(r1387788, r1387793, r1387795);
        double r1387797 = r1387791 / r1387796;
        double r1387798 = r1387777 ? r1387790 : r1387797;
        double r1387799 = 2.4732339682798126e+117;
        bool r1387800 = r1387773 <= r1387799;
        double r1387801 = cbrt(r1387778);
        double r1387802 = r1387801 * r1387801;
        double r1387803 = cbrt(r1387785);
        double r1387804 = r1387803 * r1387803;
        double r1387805 = sqrt(r1387804);
        double r1387806 = sqrt(r1387803);
        double r1387807 = -r1387806;
        double r1387808 = r1387805 * r1387807;
        double r1387809 = fma(r1387802, r1387801, r1387808);
        double r1387810 = cbrt(r1387803);
        double r1387811 = cbrt(r1387804);
        double r1387812 = r1387810 * r1387811;
        double r1387813 = r1387812 * r1387803;
        double r1387814 = sqrt(r1387813);
        double r1387815 = r1387814 * r1387806;
        double r1387816 = fma(r1387807, r1387805, r1387815);
        double r1387817 = r1387809 + r1387816;
        double r1387818 = r1387817 / r1387789;
        double r1387819 = r1387786 + r1387778;
        double r1387820 = r1387791 / r1387819;
        double r1387821 = r1387777 ? r1387818 : r1387820;
        double r1387822 = r1387791 * r1387792;
        double r1387823 = r1387773 - r1387822;
        double r1387824 = r1387778 - r1387823;
        double r1387825 = r1387824 / r1387789;
        double r1387826 = r1387777 ? r1387825 : r1387820;
        double r1387827 = r1387800 ? r1387821 : r1387826;
        double r1387828 = r1387775 ? r1387798 : r1387827;
        return r1387828;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -4.073691083157782e+153

    1. Initial program 38.3

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

    2. Taylor expanded around -inf 7.4

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

    3. Simplified1.4

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

    if -4.073691083157782e+153 < b < 2.4732339682798126e+117

    1. Initial program 8.6

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

    3. Applied add-cube-cbrt8.8

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

    4. Applied sqrt-prod8.8

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

    5. Applied add-cube-cbrt8.9

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

    6. Applied prod-diff8.9

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

    8. Applied add-cube-cbrt8.9

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

    9. Applied cbrt-prod8.9

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

    if 2.4732339682798126e+117 < b

    1. Initial program 52.1

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

    2. Taylor expanded around inf 10.1

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

    3. Simplified3.1

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.073691083157782267875644621647570574227 \cdot 10^{153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, c \cdot \frac{a}{b}, -2 \cdot b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 2.473233968279812603451846937244594428558 \cdot 10^{117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}, \sqrt[3]{-b}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(-\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right) + \mathsf{fma}\left(-\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\left(\sqrt[3]{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - \left(c \cdot 2\right) \cdot \frac{a}{b}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))