Average Error: 28.5 → 16.6
Time: 16.4s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 781.9086092205042:\\ \;\;\;\;\frac{\frac{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right) \cdot \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 781.9086092205042:\\
\;\;\;\;\frac{\frac{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right) \cdot \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b}\right)}}{a \cdot 3}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4392391 = b;
        double r4392392 = -r4392391;
        double r4392393 = r4392391 * r4392391;
        double r4392394 = 3.0;
        double r4392395 = a;
        double r4392396 = r4392394 * r4392395;
        double r4392397 = c;
        double r4392398 = r4392396 * r4392397;
        double r4392399 = r4392393 - r4392398;
        double r4392400 = sqrt(r4392399);
        double r4392401 = r4392392 + r4392400;
        double r4392402 = r4392401 / r4392396;
        return r4392402;
}
↓
double f(double a, double b, double c) {
        double r4392403 = b;
        double r4392404 = 781.9086092205042;
        bool r4392405 = r4392403 <= r4392404;
        double r4392406 = c;
        double r4392407 = a;
        double r4392408 = -3.0;
        double r4392409 = r4392407 * r4392408;
        double r4392410 = r4392406 * r4392409;
        double r4392411 = r4392403 * r4392403;
        double r4392412 = r4392410 + r4392411;
        double r4392413 = sqrt(r4392412);
        double r4392414 = r4392412 * r4392413;
        double r4392415 = r4392403 * r4392411;
        double r4392416 = r4392414 - r4392415;
        double r4392417 = r4392403 * r4392413;
        double r4392418 = r4392411 + r4392417;
        double r4392419 = r4392412 + r4392418;
        double r4392420 = r4392416 / r4392419;
        double r4392421 = 3.0;
        double r4392422 = r4392407 * r4392421;
        double r4392423 = r4392420 / r4392422;
        double r4392424 = -0.5;
        double r4392425 = r4392406 / r4392403;
        double r4392426 = r4392424 * r4392425;
        double r4392427 = r4392405 ? r4392423 : r4392426;
        return r4392427;
}

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 < 781.9086092205042

    1. Initial program 17.1

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

    2. Simplified17.1

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

    4. Applied flip3--17.1

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

    5. Simplified16.4

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

    6. Simplified16.4

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

    if 781.9086092205042 < b

    1. Initial program 35.8

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

    2. Simplified35.8

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

    3. Taylor expanded around inf 16.7

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 781.9086092205042:\\ \;\;\;\;\frac{\frac{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right) \cdot \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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 a) c)))) (* 3 a)))