Average Error: 28.7 → 17.1
Time: 6.2s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 17.714642954298647:\\ \;\;\;\;\frac{\frac{{b}^{2} - \left({b}^{2} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 17.714642954298647:\\
\;\;\;\;\frac{\frac{{b}^{2} - \left({b}^{2} - \left(3 \cdot a\right) \cdot c\right)}{\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 r89325 = b;
        double r89326 = -r89325;
        double r89327 = r89325 * r89325;
        double r89328 = 3.0;
        double r89329 = a;
        double r89330 = r89328 * r89329;
        double r89331 = c;
        double r89332 = r89330 * r89331;
        double r89333 = r89327 - r89332;
        double r89334 = sqrt(r89333);
        double r89335 = r89326 + r89334;
        double r89336 = r89335 / r89330;
        return r89336;
}
↓
double f(double a, double b, double c) {
        double r89337 = b;
        double r89338 = 17.714642954298647;
        bool r89339 = r89337 <= r89338;
        double r89340 = 2.0;
        double r89341 = pow(r89337, r89340);
        double r89342 = 3.0;
        double r89343 = a;
        double r89344 = r89342 * r89343;
        double r89345 = c;
        double r89346 = r89344 * r89345;
        double r89347 = r89341 - r89346;
        double r89348 = r89341 - r89347;
        double r89349 = -r89337;
        double r89350 = r89337 * r89337;
        double r89351 = r89350 - r89346;
        double r89352 = sqrt(r89351);
        double r89353 = r89349 - r89352;
        double r89354 = r89348 / r89353;
        double r89355 = r89354 / r89344;
        double r89356 = -0.5;
        double r89357 = r89345 / r89337;
        double r89358 = r89356 * r89357;
        double r89359 = r89339 ? r89355 : r89358;
        return r89359;
}

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 2 regimes

  2. if b < 17.714642954298647

    1. Initial program 14.0

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

    3. Applied flip-+14.0

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

    4. Simplified13.0

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

    if 17.714642954298647 < b

    1. Initial program 33.6

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

    2. Taylor expanded around inf 18.4

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 17.714642954298647:\\ \;\;\;\;\frac{\frac{{b}^{2} - \left({b}^{2} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))