Average Error: 19.9 → 6.6
Time: 16.8s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.098467977454582102316048322967968795279 \cdot 10^{76}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2 + \frac{2 \cdot a}{\frac{b}{c}}}\\ \end{array}\\ \mathbf{elif}\;b \le 2.477477027570354726331144001887489320672 \cdot 10^{83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{b}{c}}\\ \end{array}\]
\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 -3.098467977454582102316048322967968795279 \cdot 10^{76}:\\
\;\;\;\;\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}{b \cdot -2 + \frac{2 \cdot a}{\frac{b}{c}}}\\

\end{array}\\

\mathbf{elif}\;b \le 2.477477027570354726331144001887489320672 \cdot 10^{83}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{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:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

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

\end{array}
double f(double a, double b, double c) {
        double r32631 = b;
        double r32632 = 0.0;
        bool r32633 = r32631 >= r32632;
        double r32634 = -r32631;
        double r32635 = r32631 * r32631;
        double r32636 = 4.0;
        double r32637 = a;
        double r32638 = r32636 * r32637;
        double r32639 = c;
        double r32640 = r32638 * r32639;
        double r32641 = r32635 - r32640;
        double r32642 = sqrt(r32641);
        double r32643 = r32634 - r32642;
        double r32644 = 2.0;
        double r32645 = r32644 * r32637;
        double r32646 = r32643 / r32645;
        double r32647 = r32644 * r32639;
        double r32648 = r32634 + r32642;
        double r32649 = r32647 / r32648;
        double r32650 = r32633 ? r32646 : r32649;
        return r32650;
}
↓
double f(double a, double b, double c) {
        double r32651 = b;
        double r32652 = -3.098467977454582e+76;
        bool r32653 = r32651 <= r32652;
        double r32654 = 0.0;
        bool r32655 = r32651 >= r32654;
        double r32656 = -r32651;
        double r32657 = r32651 * r32651;
        double r32658 = 4.0;
        double r32659 = a;
        double r32660 = r32658 * r32659;
        double r32661 = c;
        double r32662 = r32660 * r32661;
        double r32663 = r32657 - r32662;
        double r32664 = sqrt(r32663);
        double r32665 = r32656 - r32664;
        double r32666 = 2.0;
        double r32667 = r32666 * r32659;
        double r32668 = r32665 / r32667;
        double r32669 = r32661 * r32666;
        double r32670 = -2.0;
        double r32671 = r32651 * r32670;
        double r32672 = r32651 / r32661;
        double r32673 = r32667 / r32672;
        double r32674 = r32671 + r32673;
        double r32675 = r32669 / r32674;
        double r32676 = r32655 ? r32668 : r32675;
        double r32677 = 2.4774770275703547e+83;
        bool r32678 = r32651 <= r32677;
        double r32679 = cbrt(r32663);
        double r32680 = fabs(r32679);
        double r32681 = sqrt(r32679);
        double r32682 = r32680 * r32681;
        double r32683 = r32656 - r32682;
        double r32684 = r32683 / r32667;
        double r32685 = r32664 + r32656;
        double r32686 = r32669 / r32685;
        double r32687 = r32655 ? r32684 : r32686;
        double r32688 = r32661 / r32651;
        double r32689 = r32651 / r32659;
        double r32690 = r32688 - r32689;
        double r32691 = 1.0;
        double r32692 = r32690 * r32691;
        double r32693 = -2.0;
        double r32694 = r32693 / r32672;
        double r32695 = r32655 ? r32692 : r32694;
        double r32696 = r32678 ? r32687 : r32695;
        double r32697 = r32653 ? r32676 : r32696;
        return r32697;
}

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 < -3.098467977454582e+76

    1. Initial program 27.3

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

    2. Taylor expanded around -inf 6.3

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

    3. Simplified2.9

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

    if -3.098467977454582e+76 < b < 2.4774770275703547e+83

    1. Initial program 9.1

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

    3. Applied add-cube-cbrt9.3

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

    4. Applied sqrt-prod9.3

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

    5. Simplified9.3

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

    if 2.4774770275703547e+83 < b

    1. Initial program 43.8

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

    2. Taylor expanded around inf 8.8

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

    3. Simplified3.7

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

    4. Taylor expanded around 0 3.5

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

    5. Simplified3.5

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

    6. Taylor expanded around -inf 3.5

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

    7. Simplified3.5

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.098467977454582102316048322967968795279 \cdot 10^{76}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2 + \frac{2 \cdot a}{\frac{b}{c}}}\\ \end{array}\\ \mathbf{elif}\;b \le 2.477477027570354726331144001887489320672 \cdot 10^{83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{b}{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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)))))))