Average Error: 34.2 → 9.9
Time: 18.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.008402374157600307221015587992064225208 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.008402374157600307221015587992064225208 \cdot 10^{154}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r80439 = b;
        double r80440 = -r80439;
        double r80441 = r80439 * r80439;
        double r80442 = 3.0;
        double r80443 = a;
        double r80444 = r80442 * r80443;
        double r80445 = c;
        double r80446 = r80444 * r80445;
        double r80447 = r80441 - r80446;
        double r80448 = sqrt(r80447);
        double r80449 = r80440 + r80448;
        double r80450 = r80449 / r80444;
        return r80450;
}
↓
double f(double a, double b, double c) {
        double r80451 = b;
        double r80452 = -1.0084023741576003e+154;
        bool r80453 = r80451 <= r80452;
        double r80454 = 0.5;
        double r80455 = c;
        double r80456 = r80455 / r80451;
        double r80457 = r80454 * r80456;
        double r80458 = 0.6666666666666666;
        double r80459 = a;
        double r80460 = r80451 / r80459;
        double r80461 = r80458 * r80460;
        double r80462 = r80457 - r80461;
        double r80463 = 1.611450844781215e-34;
        bool r80464 = r80451 <= r80463;
        double r80465 = r80451 * r80451;
        double r80466 = 3.0;
        double r80467 = r80466 * r80459;
        double r80468 = r80467 * r80455;
        double r80469 = r80465 - r80468;
        double r80470 = sqrt(r80469);
        double r80471 = r80470 - r80451;
        double r80472 = r80471 / r80466;
        double r80473 = r80472 / r80459;
        double r80474 = -0.5;
        double r80475 = r80474 * r80456;
        double r80476 = r80464 ? r80473 : r80475;
        double r80477 = r80453 ? r80462 : r80476;
        return r80477;
}

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 < -1.0084023741576003e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 2.1

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

    if -1.0084023741576003e+154 < b < 1.611450844781215e-34

    1. Initial program 13.6

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

    3. Applied associate-/r*13.7

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

    4. Simplified13.7

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

    if 1.611450844781215e-34 < b

    1. Initial program 55.0

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

    2. Taylor expanded around inf 7.0

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.008402374157600307221015587992064225208 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))