Average Error: 33.9 → 8.9
Time: 24.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.594947000714855189120603839967237527365 \cdot 10^{98}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.605240993580696637928945849484483239083 \cdot 10^{-181}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{elif}\;b \le 1.407972248542575850769292731823305040926 \cdot 10^{61}:\\ \;\;\;\;\frac{\frac{\left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.594947000714855189120603839967237527365 \cdot 10^{98}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 2.605240993580696637928945849484483239083 \cdot 10^{-181}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\

\mathbf{elif}\;b \le 1.407972248542575850769292731823305040926 \cdot 10^{61}:\\
\;\;\;\;\frac{\frac{\left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r131524 = b;
        double r131525 = -r131524;
        double r131526 = r131524 * r131524;
        double r131527 = 3.0;
        double r131528 = a;
        double r131529 = r131527 * r131528;
        double r131530 = c;
        double r131531 = r131529 * r131530;
        double r131532 = r131526 - r131531;
        double r131533 = sqrt(r131532);
        double r131534 = r131525 + r131533;
        double r131535 = r131534 / r131529;
        return r131535;
}
↓
double f(double a, double b, double c) {
        double r131536 = b;
        double r131537 = -8.594947000714855e+98;
        bool r131538 = r131536 <= r131537;
        double r131539 = 0.5;
        double r131540 = c;
        double r131541 = r131540 / r131536;
        double r131542 = r131539 * r131541;
        double r131543 = 0.6666666666666666;
        double r131544 = a;
        double r131545 = r131536 / r131544;
        double r131546 = r131543 * r131545;
        double r131547 = r131542 - r131546;
        double r131548 = 2.6052409935806966e-181;
        bool r131549 = r131536 <= r131548;
        double r131550 = r131536 * r131536;
        double r131551 = 3.0;
        double r131552 = r131551 * r131544;
        double r131553 = r131552 * r131540;
        double r131554 = r131550 - r131553;
        double r131555 = sqrt(r131554);
        double r131556 = r131555 - r131536;
        double r131557 = r131556 / r131551;
        double r131558 = r131557 / r131544;
        double r131559 = 1.4079722485425759e+61;
        bool r131560 = r131536 <= r131559;
        double r131561 = -r131536;
        double r131562 = r131561 - r131555;
        double r131563 = r131553 / r131562;
        double r131564 = r131563 / r131552;
        double r131565 = -0.5;
        double r131566 = r131565 * r131541;
        double r131567 = r131560 ? r131564 : r131566;
        double r131568 = r131549 ? r131558 : r131567;
        double r131569 = r131538 ? r131547 : r131568;
        return r131569;
}

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 < -8.594947000714855e+98

    1. Initial program 46.3

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

    2. Taylor expanded around -inf 4.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}}\]

    if -8.594947000714855e+98 < b < 2.6052409935806966e-181

    1. Initial program 11.0

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

    2. Simplified11.0

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

    if 2.6052409935806966e-181 < b < 1.4079722485425759e+61

    1. Initial program 34.8

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

    3. Applied flip-+34.9

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

    4. Simplified15.6

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

    if 1.4079722485425759e+61 < b

    1. Initial program 57.4

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

    2. Taylor expanded around inf 4.0

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.594947000714855189120603839967237527365 \cdot 10^{98}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.605240993580696637928945849484483239083 \cdot 10^{-181}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{elif}\;b \le 1.407972248542575850769292731823305040926 \cdot 10^{61}:\\ \;\;\;\;\frac{\frac{\left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))