Average Error: 19.8 → 6.3
Time: 17.1s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.177741376353205990919022742706386584896 \cdot 10^{137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2 \cdot a}{\frac{b}{c}} - b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 8.991400549328153943320663038409734113411 \cdot 10^{83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}} + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b + \frac{2 \cdot a}{\frac{b}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.177741376353205990919022742706386584896 \cdot 10^{137}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}\\

\mathbf{elif}\;b \le 8.991400549328153943320663038409734113411 \cdot 10^{83}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{-2 \cdot b + \frac{2 \cdot a}{\frac{b}{c}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r31717 = b;
        double r31718 = 0.0;
        bool r31719 = r31717 >= r31718;
        double r31720 = 2.0;
        double r31721 = c;
        double r31722 = r31720 * r31721;
        double r31723 = -r31717;
        double r31724 = r31717 * r31717;
        double r31725 = 4.0;
        double r31726 = a;
        double r31727 = r31725 * r31726;
        double r31728 = r31727 * r31721;
        double r31729 = r31724 - r31728;
        double r31730 = sqrt(r31729);
        double r31731 = r31723 - r31730;
        double r31732 = r31722 / r31731;
        double r31733 = r31723 + r31730;
        double r31734 = r31720 * r31726;
        double r31735 = r31733 / r31734;
        double r31736 = r31719 ? r31732 : r31735;
        return r31736;
}
↓
double f(double a, double b, double c) {
        double r31737 = b;
        double r31738 = -1.177741376353206e+137;
        bool r31739 = r31737 <= r31738;
        double r31740 = 0.0;
        bool r31741 = r31737 >= r31740;
        double r31742 = 2.0;
        double r31743 = c;
        double r31744 = r31742 * r31743;
        double r31745 = -r31737;
        double r31746 = r31737 * r31737;
        double r31747 = 4.0;
        double r31748 = a;
        double r31749 = r31747 * r31748;
        double r31750 = r31749 * r31743;
        double r31751 = r31746 - r31750;
        double r31752 = sqrt(r31751);
        double r31753 = r31745 - r31752;
        double r31754 = r31744 / r31753;
        double r31755 = r31742 * r31748;
        double r31756 = r31737 / r31743;
        double r31757 = r31755 / r31756;
        double r31758 = r31757 - r31737;
        double r31759 = r31758 + r31745;
        double r31760 = r31759 / r31755;
        double r31761 = r31741 ? r31754 : r31760;
        double r31762 = 8.991400549328154e+83;
        bool r31763 = r31737 <= r31762;
        double r31764 = r31747 * r31743;
        double r31765 = r31764 * r31748;
        double r31766 = r31746 - r31765;
        double r31767 = sqrt(r31766);
        double r31768 = sqrt(r31767);
        double r31769 = r31768 * r31768;
        double r31770 = r31769 + r31745;
        double r31771 = r31770 / r31755;
        double r31772 = r31741 ? r31754 : r31771;
        double r31773 = -2.0;
        double r31774 = r31773 * r31737;
        double r31775 = r31774 + r31757;
        double r31776 = r31744 / r31775;
        double r31777 = r31752 + r31745;
        double r31778 = r31777 / r31755;
        double r31779 = r31741 ? r31776 : r31778;
        double r31780 = r31763 ? r31772 : r31779;
        double r31781 = r31739 ? r31761 : r31780;
        return r31781;
}

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 < -1.177741376353206e+137

    1. Initial program 57.9

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

    2. Taylor expanded around -inf 9.3

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

    3. Simplified2.3

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

    if -1.177741376353206e+137 < b < 8.991400549328154e+83

    1. Initial program 8.4

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

    3. Applied add-sqr-sqrt8.4

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

    4. Applied sqrt-prod8.6

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

    5. Simplified8.6

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

    6. Simplified8.6

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

    if 8.991400549328154e+83 < b

    1. Initial program 28.2

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

    2. Taylor expanded around inf 6.7

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

    3. Simplified2.8

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.177741376353205990919022742706386584896 \cdot 10^{137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2 \cdot a}{\frac{b}{c}} - b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 8.991400549328153943320663038409734113411 \cdot 10^{83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}} + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b + \frac{2 \cdot a}{\frac{b}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))