Average Error: 19.1 → 8.0
Time: 16.9s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.662744446675922244615861935677003632826 \cdot 10^{52}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{b}} \cdot \left(\frac{\sqrt[3]{a}}{\sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right) - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 9.009088445388659991752568414850370061349 \cdot 10^{104}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \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 -2.662744446675922244615861935677003632826 \cdot 10^{52}:\\
\;\;\;\;\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) + \left(2 \cdot \left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{b}} \cdot \left(\frac{\sqrt[3]{a}}{\sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right) - b\right)}\\

\end{array}\\

\mathbf{elif}\;b \le 9.009088445388659991752568414850370061349 \cdot 10^{104}:\\
\;\;\;\;\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{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{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}
double f(double a, double b, double c) {
        double r45681 = b;
        double r45682 = 0.0;
        bool r45683 = r45681 >= r45682;
        double r45684 = -r45681;
        double r45685 = r45681 * r45681;
        double r45686 = 4.0;
        double r45687 = a;
        double r45688 = r45686 * r45687;
        double r45689 = c;
        double r45690 = r45688 * r45689;
        double r45691 = r45685 - r45690;
        double r45692 = sqrt(r45691);
        double r45693 = r45684 - r45692;
        double r45694 = 2.0;
        double r45695 = r45694 * r45687;
        double r45696 = r45693 / r45695;
        double r45697 = r45694 * r45689;
        double r45698 = r45684 + r45692;
        double r45699 = r45697 / r45698;
        double r45700 = r45683 ? r45696 : r45699;
        return r45700;
}
↓
double f(double a, double b, double c) {
        double r45701 = b;
        double r45702 = -2.662744446675922e+52;
        bool r45703 = r45701 <= r45702;
        double r45704 = 0.0;
        bool r45705 = r45701 >= r45704;
        double r45706 = -r45701;
        double r45707 = r45701 * r45701;
        double r45708 = 4.0;
        double r45709 = a;
        double r45710 = r45708 * r45709;
        double r45711 = c;
        double r45712 = r45710 * r45711;
        double r45713 = r45707 - r45712;
        double r45714 = sqrt(r45713);
        double r45715 = r45706 - r45714;
        double r45716 = 2.0;
        double r45717 = r45716 * r45709;
        double r45718 = r45715 / r45717;
        double r45719 = r45716 * r45711;
        double r45720 = cbrt(r45709);
        double r45721 = r45720 * r45720;
        double r45722 = cbrt(r45701);
        double r45723 = r45721 / r45722;
        double r45724 = r45720 / r45722;
        double r45725 = r45711 / r45722;
        double r45726 = r45724 * r45725;
        double r45727 = r45723 * r45726;
        double r45728 = r45716 * r45727;
        double r45729 = r45728 - r45701;
        double r45730 = r45706 + r45729;
        double r45731 = r45719 / r45730;
        double r45732 = r45705 ? r45718 : r45731;
        double r45733 = 9.00908844538866e+104;
        bool r45734 = r45701 <= r45733;
        double r45735 = sqrt(r45714);
        double r45736 = r45735 * r45735;
        double r45737 = r45706 + r45736;
        double r45738 = r45719 / r45737;
        double r45739 = r45705 ? r45718 : r45738;
        double r45740 = r45709 * r45711;
        double r45741 = r45740 / r45701;
        double r45742 = r45716 * r45741;
        double r45743 = 2.0;
        double r45744 = r45743 * r45701;
        double r45745 = r45742 - r45744;
        double r45746 = r45745 / r45717;
        double r45747 = r45706 + r45714;
        double r45748 = r45719 / r45747;
        double r45749 = r45705 ? r45746 : r45748;
        double r45750 = r45734 ? r45739 : r45749;
        double r45751 = r45703 ? r45732 : r45750;
        return r45751;
}

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 < -2.662744446675922e+52

    1. Initial program 24.8

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

    2. Taylor expanded around -inf 6.9

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

    4. Applied add-cube-cbrt6.9

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

    5. Applied times-frac3.7

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

    7. Applied add-cube-cbrt3.7

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

    8. Applied times-frac3.7

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

    9. Applied associate-*l*3.7

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

    if -2.662744446675922e+52 < b < 9.00908844538866e+104

    1. Initial program 9.4

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

    3. Applied add-sqr-sqrt9.4

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

    4. Applied sqrt-prod9.5

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

    if 9.00908844538866e+104 < b

    1. Initial program 45.9

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

    2. Taylor expanded around inf 10.6

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

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.662744446675922244615861935677003632826 \cdot 10^{52}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{b}} \cdot \left(\frac{\sqrt[3]{a}}{\sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right) - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 9.009088445388659991752568414850370061349 \cdot 10^{104}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))