Average Error: 19.6 → 6.7
Time: 17.4s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.456332528378644 \cdot 10^{+162}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 1.0084026083208836 \cdot 10^{+97}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot 2\right) \cdot \frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 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 -2.456332528378644 \cdot 10^{+162}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{2 \cdot a}\\

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

\end{array}\\

\mathbf{elif}\;b \le 1.0084026083208836 \cdot 10^{+97}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}}}{2 \cdot a}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{else}:\\
\;\;\;\;\left(c \cdot 2\right) \cdot \frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\

\end{array}
double f(double a, double b, double c) {
        double r1331631 = b;
        double r1331632 = 0.0;
        bool r1331633 = r1331631 >= r1331632;
        double r1331634 = -r1331631;
        double r1331635 = r1331631 * r1331631;
        double r1331636 = 4.0;
        double r1331637 = a;
        double r1331638 = r1331636 * r1331637;
        double r1331639 = c;
        double r1331640 = r1331638 * r1331639;
        double r1331641 = r1331635 - r1331640;
        double r1331642 = sqrt(r1331641);
        double r1331643 = r1331634 - r1331642;
        double r1331644 = 2.0;
        double r1331645 = r1331644 * r1331637;
        double r1331646 = r1331643 / r1331645;
        double r1331647 = r1331644 * r1331639;
        double r1331648 = r1331634 + r1331642;
        double r1331649 = r1331647 / r1331648;
        double r1331650 = r1331633 ? r1331646 : r1331649;
        return r1331650;
}
↓
double f(double a, double b, double c) {
        double r1331651 = b;
        double r1331652 = -2.456332528378644e+162;
        bool r1331653 = r1331651 <= r1331652;
        double r1331654 = 0.0;
        bool r1331655 = r1331651 >= r1331654;
        double r1331656 = 2.0;
        double r1331657 = a;
        double r1331658 = c;
        double r1331659 = r1331651 / r1331658;
        double r1331660 = r1331657 / r1331659;
        double r1331661 = r1331660 - r1331651;
        double r1331662 = r1331656 * r1331661;
        double r1331663 = r1331656 * r1331657;
        double r1331664 = r1331662 / r1331663;
        double r1331665 = r1331658 * r1331656;
        double r1331666 = r1331657 / r1331651;
        double r1331667 = r1331666 * r1331658;
        double r1331668 = r1331667 - r1331651;
        double r1331669 = r1331656 * r1331668;
        double r1331670 = r1331665 / r1331669;
        double r1331671 = r1331655 ? r1331664 : r1331670;
        double r1331672 = 1.0084026083208836e+97;
        bool r1331673 = r1331651 <= r1331672;
        double r1331674 = -r1331651;
        double r1331675 = r1331651 * r1331651;
        double r1331676 = 4.0;
        double r1331677 = r1331657 * r1331676;
        double r1331678 = r1331677 * r1331658;
        double r1331679 = r1331675 - r1331678;
        double r1331680 = cbrt(r1331679);
        double r1331681 = r1331680 * r1331680;
        double r1331682 = sqrt(r1331681);
        double r1331683 = sqrt(r1331680);
        double r1331684 = r1331682 * r1331683;
        double r1331685 = r1331674 - r1331684;
        double r1331686 = r1331685 / r1331663;
        double r1331687 = sqrt(r1331679);
        double r1331688 = r1331687 + r1331674;
        double r1331689 = r1331665 / r1331688;
        double r1331690 = r1331655 ? r1331686 : r1331689;
        double r1331691 = r1331658 / r1331651;
        double r1331692 = r1331651 / r1331657;
        double r1331693 = r1331691 - r1331692;
        double r1331694 = 1.0;
        double r1331695 = -4.0;
        double r1331696 = r1331657 * r1331695;
        double r1331697 = r1331658 * r1331696;
        double r1331698 = r1331675 + r1331697;
        double r1331699 = sqrt(r1331698);
        double r1331700 = r1331699 - r1331651;
        double r1331701 = r1331694 / r1331700;
        double r1331702 = r1331665 * r1331701;
        double r1331703 = r1331655 ? r1331693 : r1331702;
        double r1331704 = r1331673 ? r1331690 : r1331703;
        double r1331705 = r1331653 ? r1331671 : r1331704;
        return r1331705;
}

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 < -2.456332528378644e+162

    1. Initial program 38.6

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Taylor expanded around inf 38.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    3. Simplified38.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    4. Taylor expanded around -inf 7.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \end{array}\]

    5. Simplified1.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}}\\ \end{array}\]

    if -2.456332528378644e+162 < b < 1.0084026083208836e+97

    1. Initial program 8.6

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt8.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    4. Applied sqrt-prod8.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    if 1.0084026083208836e+97 < b

    1. Initial program 43.7

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Taylor expanded around inf 10.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    3. Simplified3.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    4. Taylor expanded around 0 3.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} - \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    5. Using strategy rm

    6. Applied div-inv3.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot c\right) \cdot \frac{1}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    7. Simplified3.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\left(2 \cdot c\right) \cdot \frac{1}{\sqrt{b \cdot b + \left(a \cdot -4\right) \cdot c} - b}}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.456332528378644 \cdot 10^{+162}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 1.0084026083208836 \cdot 10^{+97}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot 2\right) \cdot \frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))