Average Error: 19.6 → 6.5
Time: 20.8s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.8096996975249965 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 5.909172052458495 \cdot 10^{+80}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\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 -2.8096996975249965 \cdot 10^{+152}:\\
\;\;\;\;\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{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\

\end{array}\\

\mathbf{elif}\;b \le 5.909172052458495 \cdot 10^{+80}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(-b\right) - \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}}}{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:\\
\;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b} - 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 r737611 = b;
        double r737612 = 0.0;
        bool r737613 = r737611 >= r737612;
        double r737614 = -r737611;
        double r737615 = r737611 * r737611;
        double r737616 = 4.0;
        double r737617 = a;
        double r737618 = r737616 * r737617;
        double r737619 = c;
        double r737620 = r737618 * r737619;
        double r737621 = r737615 - r737620;
        double r737622 = sqrt(r737621);
        double r737623 = r737614 - r737622;
        double r737624 = 2.0;
        double r737625 = r737624 * r737617;
        double r737626 = r737623 / r737625;
        double r737627 = r737624 * r737619;
        double r737628 = r737614 + r737622;
        double r737629 = r737627 / r737628;
        double r737630 = r737613 ? r737626 : r737629;
        return r737630;
}
↓
double f(double a, double b, double c) {
        double r737631 = b;
        double r737632 = -2.8096996975249965e+152;
        bool r737633 = r737631 <= r737632;
        double r737634 = 0.0;
        bool r737635 = r737631 >= r737634;
        double r737636 = -r737631;
        double r737637 = r737631 * r737631;
        double r737638 = 4.0;
        double r737639 = a;
        double r737640 = r737638 * r737639;
        double r737641 = c;
        double r737642 = r737640 * r737641;
        double r737643 = r737637 - r737642;
        double r737644 = sqrt(r737643);
        double r737645 = r737636 - r737644;
        double r737646 = 2.0;
        double r737647 = r737646 * r737639;
        double r737648 = r737645 / r737647;
        double r737649 = r737641 * r737646;
        double r737650 = r737639 / r737631;
        double r737651 = r737641 * r737650;
        double r737652 = r737651 - r737631;
        double r737653 = r737646 * r737652;
        double r737654 = r737649 / r737653;
        double r737655 = r737635 ? r737648 : r737654;
        double r737656 = 5.909172052458495e+80;
        bool r737657 = r737631 <= r737656;
        double r737658 = cbrt(r737643);
        double r737659 = sqrt(r737658);
        double r737660 = r737658 * r737658;
        double r737661 = sqrt(r737660);
        double r737662 = r737659 * r737661;
        double r737663 = r737636 - r737662;
        double r737664 = r737663 / r737647;
        double r737665 = r737644 + r737636;
        double r737666 = r737649 / r737665;
        double r737667 = r737635 ? r737664 : r737666;
        double r737668 = r737653 / r737647;
        double r737669 = r737635 ? r737668 : r737666;
        double r737670 = r737657 ? r737667 : r737669;
        double r737671 = r737633 ? r737655 : r737670;
        return r737671;
}

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 < -2.8096996975249965e+152

    1. Initial program 38.8

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

    2. Taylor expanded around -inf 5.5

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

    3. Simplified1.1

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

    if -2.8096996975249965e+152 < b < 5.909172052458495e+80

    1. Initial program 8.4

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

    3. Applied add-cube-cbrt8.6

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

    4. Applied sqrt-prod8.6

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

    if 5.909172052458495e+80 < b

    1. Initial program 40.2

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

    2. Taylor expanded around inf 9.9

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

    3. Simplified4.3

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.8096996975249965 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 5.909172052458495 \cdot 10^{+80}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(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)))))))