Average Error: 28.7 → 16.2
Time: 22.5s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2134.20029068235863:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5}{3} \cdot \frac{c}{b}\\ \end{array}\]
\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 2134.20029068235863:\\
\;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1.5}{3} \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r62336 = b;
        double r62337 = -r62336;
        double r62338 = r62336 * r62336;
        double r62339 = 3.0;
        double r62340 = a;
        double r62341 = r62339 * r62340;
        double r62342 = c;
        double r62343 = r62341 * r62342;
        double r62344 = r62338 - r62343;
        double r62345 = sqrt(r62344);
        double r62346 = r62337 + r62345;
        double r62347 = r62346 / r62341;
        return r62347;
}
↓
double f(double a, double b, double c) {
        double r62348 = b;
        double r62349 = 2134.2002906823586;
        bool r62350 = r62348 <= r62349;
        double r62351 = r62348 * r62348;
        double r62352 = 3.0;
        double r62353 = a;
        double r62354 = c;
        double r62355 = r62353 * r62354;
        double r62356 = r62352 * r62355;
        double r62357 = r62351 - r62356;
        double r62358 = r62351 - r62357;
        double r62359 = -r62348;
        double r62360 = r62352 * r62353;
        double r62361 = r62360 * r62354;
        double r62362 = r62351 - r62361;
        double r62363 = sqrt(r62362);
        double r62364 = r62359 - r62363;
        double r62365 = r62358 / r62364;
        double r62366 = r62365 / r62360;
        double r62367 = -1.5;
        double r62368 = r62367 / r62352;
        double r62369 = r62354 / r62348;
        double r62370 = r62368 * r62369;
        double r62371 = r62350 ? r62366 : r62370;
        return r62371;
}

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 < 2134.2002906823586

    1. Initial program 18.0

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

    3. Applied flip-+18.0

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

    4. Simplified16.9

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

    if 2134.2002906823586 < b

    1. Initial program 37.0

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

    2. Taylor expanded around inf 15.8

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    3. Using strategy rm

    4. Applied times-frac15.7

      \[\leadsto \color{blue}{\frac{-1.5}{3} \cdot \frac{\frac{a \cdot c}{b}}{a}}\]

    5. Taylor expanded around 0 15.6

      \[\leadsto \frac{-1.5}{3} \cdot \color{blue}{\frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2134.20029068235863:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5}{3} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))