Average Error: 34.2 → 9.5
Time: 15.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.159787972815590779244742122377140970754 \cdot 10^{74}:\\ \;\;\;\;\frac{\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \le 7.476271090407513089911775252003734724482 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)\right)\\ \mathbf{elif}\;b \le 5.494985267978682357459684619586098494544 \cdot 10^{45}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right) \cdot 3}{3}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.159787972815590779244742122377140970754 \cdot 10^{74}:\\
\;\;\;\;\frac{\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a}}{3}\\

\mathbf{elif}\;b \le 7.476271090407513089911775252003734724482 \cdot 10^{-284}:\\
\;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)\right)\\

\mathbf{elif}\;b \le 5.494985267978682357459684619586098494544 \cdot 10^{45}:\\
\;\;\;\;\frac{\frac{a \cdot c}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right) \cdot 3}{3}}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} \cdot -0.5\\

\end{array}
double f(double a, double b, double c) {
        double r88555 = b;
        double r88556 = -r88555;
        double r88557 = r88555 * r88555;
        double r88558 = 3.0;
        double r88559 = a;
        double r88560 = r88558 * r88559;
        double r88561 = c;
        double r88562 = r88560 * r88561;
        double r88563 = r88557 - r88562;
        double r88564 = sqrt(r88563);
        double r88565 = r88556 + r88564;
        double r88566 = r88565 / r88560;
        return r88566;
}
↓
double f(double a, double b, double c) {
        double r88567 = b;
        double r88568 = -2.1597879728155908e+74;
        bool r88569 = r88567 <= r88568;
        double r88570 = a;
        double r88571 = 1.5;
        double r88572 = r88570 * r88571;
        double r88573 = c;
        double r88574 = r88567 / r88573;
        double r88575 = r88572 / r88574;
        double r88576 = r88575 - r88567;
        double r88577 = r88576 - r88567;
        double r88578 = r88577 / r88570;
        double r88579 = 3.0;
        double r88580 = r88578 / r88579;
        double r88581 = 7.476271090407513e-284;
        bool r88582 = r88567 <= r88581;
        double r88583 = 1.0;
        double r88584 = r88579 * r88570;
        double r88585 = r88583 / r88584;
        double r88586 = r88567 * r88567;
        double r88587 = r88584 * r88573;
        double r88588 = r88586 - r88587;
        double r88589 = sqrt(r88588);
        double r88590 = -r88567;
        double r88591 = r88589 + r88590;
        double r88592 = r88585 * r88591;
        double r88593 = 5.4949852679786824e+45;
        bool r88594 = r88567 <= r88593;
        double r88595 = r88570 * r88573;
        double r88596 = r88579 * r88595;
        double r88597 = r88586 - r88596;
        double r88598 = sqrt(r88597);
        double r88599 = r88590 - r88598;
        double r88600 = r88599 * r88579;
        double r88601 = r88600 / r88579;
        double r88602 = r88595 / r88601;
        double r88603 = r88602 / r88570;
        double r88604 = r88573 / r88567;
        double r88605 = -0.5;
        double r88606 = r88604 * r88605;
        double r88607 = r88594 ? r88603 : r88606;
        double r88608 = r88582 ? r88592 : r88607;
        double r88609 = r88569 ? r88580 : r88608;
        return r88609;
}

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 4 regimes

  2. if b < -2.1597879728155908e+74

    1. Initial program 41.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Simplified41.5

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{a}}{3}}\]

    3. Taylor expanded around -inf 11.4

      \[\leadsto \frac{\frac{\color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)} - b}{a}}{3}\]

    4. Simplified5.9

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right)} - b}{a}}{3}\]

    if -2.1597879728155908e+74 < b < 7.476271090407513e-284

    1. Initial program 10.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv10.3

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}}\]

    if 7.476271090407513e-284 < b < 5.4949852679786824e+45

    1. Initial program 30.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+30.7

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

    4. Simplified17.8

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

    5. Simplified17.7

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 3}{\color{blue}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}}{3 \cdot a}\]
    6. Using strategy rm

    7. Applied associate-/r*17.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + \left(a \cdot c\right) \cdot 3}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{3}}{a}}\]

    8. Simplified17.7

      \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right) \cdot 3}{3}}}}{a}\]

    if 5.4949852679786824e+45 < b

    1. Initial program 57.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around inf 4.3

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.159787972815590779244742122377140970754 \cdot 10^{74}:\\ \;\;\;\;\frac{\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \le 7.476271090407513089911775252003734724482 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)\right)\\ \mathbf{elif}\;b \le 5.494985267978682357459684619586098494544 \cdot 10^{45}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right) \cdot 3}{3}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))