Average Error: 33.8 → 10.3
Time: 19.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.7185065337941975 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 7.9784983646909015 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{1}{3} \cdot \left(\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.7185065337941975 \cdot 10^{+66}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\

\mathbf{elif}\;b \le 7.9784983646909015 \cdot 10^{-53}:\\
\;\;\;\;\frac{\frac{1}{3} \cdot \left(\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b\right)}{a}\\

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

\end{array}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r3953334 = b;
        double r3953335 = -r3953334;
        double r3953336 = r3953334 * r3953334;
        double r3953337 = 3.0;
        double r3953338 = a;
        double r3953339 = r3953337 * r3953338;
        double r3953340 = c;
        double r3953341 = r3953339 * r3953340;
        double r3953342 = r3953336 - r3953341;
        double r3953343 = sqrt(r3953342);
        double r3953344 = r3953335 + r3953343;
        double r3953345 = r3953344 / r3953339;
        return r3953345;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r3953346 = b;
        double r3953347 = -2.7185065337941975e+66;
        bool r3953348 = r3953346 <= r3953347;
        double r3953349 = 0.5;
        double r3953350 = c;
        double r3953351 = r3953350 / r3953346;
        double r3953352 = r3953349 * r3953351;
        double r3953353 = a;
        double r3953354 = r3953346 / r3953353;
        double r3953355 = 0.6666666666666666;
        double r3953356 = r3953354 * r3953355;
        double r3953357 = r3953352 - r3953356;
        double r3953358 = 7.9784983646909015e-53;
        bool r3953359 = r3953346 <= r3953358;
        double r3953360 = 0.3333333333333333;
        double r3953361 = r3953346 * r3953346;
        double r3953362 = 3.0;
        double r3953363 = r3953353 * r3953362;
        double r3953364 = r3953350 * r3953363;
        double r3953365 = r3953361 - r3953364;
        double r3953366 = sqrt(r3953365);
        double r3953367 = r3953366 - r3953346;
        double r3953368 = r3953360 * r3953367;
        double r3953369 = r3953368 / r3953353;
        double r3953370 = -0.5;
        double r3953371 = r3953370 * r3953351;
        double r3953372 = r3953359 ? r3953369 : r3953371;
        double r3953373 = r3953348 ? r3953357 : r3953372;
        return r3953373;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -2.7185065337941975e+66

    1. Initial program 39.5

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

    2. Simplified39.5

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

    3. Taylor expanded around -inf 5.8

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

    if -2.7185065337941975e+66 < b < 7.9784983646909015e-53

    1. Initial program 14.3

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

    2. Simplified14.3

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

    4. Applied *-un-lft-identity14.3

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

    5. Applied times-frac14.5

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

    6. Simplified14.5

      \[\leadsto \color{blue}{\frac{1}{3}} \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{a}\]
    7. Using strategy rm

    8. Applied associate-*r/14.4

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

    if 7.9784983646909015e-53 < b

    1. Initial program 53.7

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

    2. Simplified53.7

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

    4. Applied *-un-lft-identity53.7

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

    5. Applied times-frac53.7

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

    6. Simplified53.7

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

    7. Taylor expanded around inf 7.8

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.7185065337941975 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 7.9784983646909015 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{1}{3} \cdot \left(\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))