Average Error: 19.3 → 7.7
Time: 10.7s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;-2 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \le -1.2159818839962 \cdot 10^{-310} \lor \neg \left(b \le 2.8436139815832302 \cdot 10^{57}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -3.12428337420519208 \cdot 10^{57}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;-2 \cdot \frac{c}{b}\\

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

\end{array}\\

\mathbf{elif}\;b \le -1.2159818839962 \cdot 10^{-310} \lor \neg \left(b \le 2.8436139815832302 \cdot 10^{57}\right):\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r53555 = b;
        double r53556 = 0.0;
        bool r53557 = r53555 >= r53556;
        double r53558 = 2.0;
        double r53559 = c;
        double r53560 = r53558 * r53559;
        double r53561 = -r53555;
        double r53562 = r53555 * r53555;
        double r53563 = 4.0;
        double r53564 = a;
        double r53565 = r53563 * r53564;
        double r53566 = r53565 * r53559;
        double r53567 = r53562 - r53566;
        double r53568 = sqrt(r53567);
        double r53569 = r53561 - r53568;
        double r53570 = r53560 / r53569;
        double r53571 = r53561 + r53568;
        double r53572 = r53558 * r53564;
        double r53573 = r53571 / r53572;
        double r53574 = r53557 ? r53570 : r53573;
        return r53574;
}
↓
double f(double a, double b, double c) {
        double r53575 = b;
        double r53576 = -3.124283374205192e+57;
        bool r53577 = r53575 <= r53576;
        double r53578 = 0.0;
        bool r53579 = r53575 >= r53578;
        double r53580 = -2.0;
        double r53581 = c;
        double r53582 = r53581 / r53575;
        double r53583 = r53580 * r53582;
        double r53584 = 1.0;
        double r53585 = a;
        double r53586 = r53575 / r53585;
        double r53587 = r53582 - r53586;
        double r53588 = r53584 * r53587;
        double r53589 = r53579 ? r53583 : r53588;
        double r53590 = -1.2159818839962e-310;
        bool r53591 = r53575 <= r53590;
        double r53592 = 2.8436139815832302e+57;
        bool r53593 = r53575 <= r53592;
        double r53594 = !r53593;
        bool r53595 = r53591 || r53594;
        double r53596 = 2.0;
        double r53597 = r53596 * r53581;
        double r53598 = r53585 * r53581;
        double r53599 = r53598 / r53575;
        double r53600 = r53596 * r53599;
        double r53601 = 2.0;
        double r53602 = r53601 * r53575;
        double r53603 = r53600 - r53602;
        double r53604 = r53597 / r53603;
        double r53605 = r53575 * r53575;
        double r53606 = 4.0;
        double r53607 = r53606 * r53585;
        double r53608 = r53607 * r53581;
        double r53609 = r53605 - r53608;
        double r53610 = sqrt(r53609);
        double r53611 = r53610 - r53575;
        double r53612 = r53611 / r53596;
        double r53613 = r53612 / r53585;
        double r53614 = r53579 ? r53604 : r53613;
        double r53615 = 1.0;
        double r53616 = -r53575;
        double r53617 = r53606 * r53598;
        double r53618 = r53605 - r53617;
        double r53619 = sqrt(r53618);
        double r53620 = r53616 - r53619;
        double r53621 = r53620 / r53597;
        double r53622 = r53615 / r53621;
        double r53623 = r53579 ? r53622 : r53588;
        double r53624 = r53595 ? r53614 : r53623;
        double r53625 = r53577 ? r53589 : r53624;
        return r53625;
}

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 < -3.124283374205192e+57

    1. Initial program 39.5

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

    2. Simplified39.5

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

    3. Taylor expanded around -inf 11.5

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

    4. Taylor expanded around 0 5.4

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

    5. Simplified5.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
    6. Using strategy rm

    7. Applied associate-*l*5.4

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

    8. Taylor expanded around inf 5.4

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

    if -3.124283374205192e+57 < b < -1.2159818839962e-310 or 2.8436139815832302e+57 < b

    1. Initial program 17.4

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

    2. Simplified17.4

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

    3. Taylor expanded around inf 7.5

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

    if -1.2159818839962e-310 < b < 2.8436139815832302e+57

    1. Initial program 9.4

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

    2. Simplified9.4

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

    3. Taylor expanded around -inf 9.4

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

    4. Taylor expanded around 0 9.4

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

    5. Simplified9.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
    6. Using strategy rm

    7. Applied associate-*l*9.4

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

    9. Applied clear-num9.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;-2 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \le -1.2159818839962 \cdot 10^{-310} \lor \neg \left(b \le 2.8436139815832302 \cdot 10^{57}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))