Average Error: 33.9 → 9.7
Time: 12.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.028459812939689 \cdot 10^{+151}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 5.850891614847679 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{elif}\;b \le 3.3885604098642834 \cdot 10^{-26}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \mathbf{elif}\;b \le 2.518503630513393 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)}} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]
\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 -7.028459812939689 \cdot 10^{+151}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\

\mathbf{elif}\;b \le 5.850891614847679 \cdot 10^{-83}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b}{a \cdot 3}\\

\mathbf{elif}\;b \le 3.3885604098642834 \cdot 10^{-26}:\\
\;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\

\mathbf{elif}\;b \le 2.518503630513393 \cdot 10^{-16}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)}} - b}{3}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r2446427 = b;
        double r2446428 = -r2446427;
        double r2446429 = r2446427 * r2446427;
        double r2446430 = 3.0;
        double r2446431 = a;
        double r2446432 = r2446430 * r2446431;
        double r2446433 = c;
        double r2446434 = r2446432 * r2446433;
        double r2446435 = r2446429 - r2446434;
        double r2446436 = sqrt(r2446435);
        double r2446437 = r2446428 + r2446436;
        double r2446438 = r2446437 / r2446432;
        return r2446438;
}
↓
double f(double a, double b, double c) {
        double r2446439 = b;
        double r2446440 = -7.028459812939689e+151;
        bool r2446441 = r2446439 <= r2446440;
        double r2446442 = 0.5;
        double r2446443 = c;
        double r2446444 = r2446443 / r2446439;
        double r2446445 = r2446442 * r2446444;
        double r2446446 = a;
        double r2446447 = r2446439 / r2446446;
        double r2446448 = 0.6666666666666666;
        double r2446449 = r2446447 * r2446448;
        double r2446450 = r2446445 - r2446449;
        double r2446451 = 5.850891614847679e-83;
        bool r2446452 = r2446439 <= r2446451;
        double r2446453 = r2446439 * r2446439;
        double r2446454 = r2446443 * r2446446;
        double r2446455 = 3.0;
        double r2446456 = r2446454 * r2446455;
        double r2446457 = r2446453 - r2446456;
        double r2446458 = sqrt(r2446457);
        double r2446459 = r2446458 - r2446439;
        double r2446460 = r2446446 * r2446455;
        double r2446461 = r2446459 / r2446460;
        double r2446462 = 3.3885604098642834e-26;
        bool r2446463 = r2446439 <= r2446462;
        double r2446464 = -0.5;
        double r2446465 = r2446444 * r2446464;
        double r2446466 = 2.518503630513393e-16;
        bool r2446467 = r2446439 <= r2446466;
        double r2446468 = r2446455 * r2446443;
        double r2446469 = r2446446 * r2446468;
        double r2446470 = r2446453 - r2446469;
        double r2446471 = sqrt(r2446470);
        double r2446472 = r2446470 * r2446471;
        double r2446473 = cbrt(r2446472);
        double r2446474 = r2446473 - r2446439;
        double r2446475 = r2446474 / r2446455;
        double r2446476 = r2446475 / r2446446;
        double r2446477 = r2446467 ? r2446476 : r2446465;
        double r2446478 = r2446463 ? r2446465 : r2446477;
        double r2446479 = r2446452 ? r2446461 : r2446478;
        double r2446480 = r2446441 ? r2446450 : r2446479;
        return r2446480;
}

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 < -7.028459812939689e+151

    1. Initial program 60.4

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

    2. Simplified60.4

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

    3. Taylor expanded around -inf 2.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b} - \frac{2}{3} \cdot \frac{b}{a}}\]

    if -7.028459812939689e+151 < b < 5.850891614847679e-83

    1. Initial program 11.6

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

    2. Simplified11.6

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

    3. Taylor expanded around 0 11.7

      \[\leadsto \frac{\sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(a \cdot c\right)}} - b}{3 \cdot a}\]

    4. Simplified11.7

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

    if 5.850891614847679e-83 < b < 3.3885604098642834e-26 or 2.518503630513393e-16 < b

    1. Initial program 53.0

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

    2. Simplified53.0

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

    3. Taylor expanded around inf 8.5

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]

    if 3.3885604098642834e-26 < b < 2.518503630513393e-16

    1. Initial program 42.2

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

    2. Simplified42.2

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

    4. Applied associate-/r*42.2

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

    6. Applied add-cbrt-cube45.4

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

    7. Simplified45.4

      \[\leadsto \frac{\frac{\sqrt[3]{\color{blue}{\left(b \cdot b - \left(3 \cdot c\right) \cdot a\right) \cdot \sqrt{b \cdot b - \left(3 \cdot c\right) \cdot a}}} - b}{3}}{a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.028459812939689 \cdot 10^{+151}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 5.850891614847679 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{elif}\;b \le 3.3885604098642834 \cdot 10^{-26}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \mathbf{elif}\;b \le 2.518503630513393 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)}} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))