Average Error: 33.6 → 10.1
Time: 6.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.0323767944871679 \cdot 10^{127}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.17528679488360856 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.0323767944871679 \cdot 10^{127}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 1.17528679488360856 \cdot 10^{-69}:\\
\;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r198511 = b;
        double r198512 = -r198511;
        double r198513 = r198511 * r198511;
        double r198514 = 3.0;
        double r198515 = a;
        double r198516 = r198514 * r198515;
        double r198517 = c;
        double r198518 = r198516 * r198517;
        double r198519 = r198513 - r198518;
        double r198520 = sqrt(r198519);
        double r198521 = r198512 + r198520;
        double r198522 = r198521 / r198516;
        return r198522;
}
↓
double f(double a, double b, double c) {
        double r198523 = b;
        double r198524 = -4.032376794487168e+127;
        bool r198525 = r198523 <= r198524;
        double r198526 = 0.5;
        double r198527 = c;
        double r198528 = r198527 / r198523;
        double r198529 = r198526 * r198528;
        double r198530 = 0.6666666666666666;
        double r198531 = a;
        double r198532 = r198523 / r198531;
        double r198533 = r198530 * r198532;
        double r198534 = r198529 - r198533;
        double r198535 = 1.1752867948836086e-69;
        bool r198536 = r198523 <= r198535;
        double r198537 = -r198523;
        double r198538 = r198523 * r198523;
        double r198539 = 3.0;
        double r198540 = r198531 * r198527;
        double r198541 = r198539 * r198540;
        double r198542 = r198538 - r198541;
        double r198543 = sqrt(r198542);
        double r198544 = r198537 + r198543;
        double r198545 = r198544 / r198539;
        double r198546 = r198545 / r198531;
        double r198547 = -0.5;
        double r198548 = r198547 * r198528;
        double r198549 = r198536 ? r198546 : r198548;
        double r198550 = r198525 ? r198534 : r198549;
        return r198550;
}

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 < -4.032376794487168e+127

    1. Initial program 53.2

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

    2. Taylor expanded around -inf 3.5

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}}\]

    if -4.032376794487168e+127 < b < 1.1752867948836086e-69

    1. Initial program 12.8

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

    3. Applied associate-/r*12.8

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

    5. Applied associate-*l*12.9

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

    if 1.1752867948836086e-69 < b

    1. Initial program 53.9

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

    2. Taylor expanded around inf 8.8

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.0323767944871679 \cdot 10^{127}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.17528679488360856 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))