Average Error: 44.7 → 11.0
Time: 30.4s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.09946845057046652:\\ \;\;\;\;\frac{\frac{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 0.09946845057046652:\\
\;\;\;\;\frac{\frac{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\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 r3890299 = b;
        double r3890300 = -r3890299;
        double r3890301 = r3890299 * r3890299;
        double r3890302 = 3.0;
        double r3890303 = a;
        double r3890304 = r3890302 * r3890303;
        double r3890305 = c;
        double r3890306 = r3890304 * r3890305;
        double r3890307 = r3890301 - r3890306;
        double r3890308 = sqrt(r3890307);
        double r3890309 = r3890300 + r3890308;
        double r3890310 = r3890309 / r3890304;
        return r3890310;
}
↓
double f(double a, double b, double c) {
        double r3890311 = b;
        double r3890312 = 0.09946845057046652;
        bool r3890313 = r3890311 <= r3890312;
        double r3890314 = r3890311 * r3890311;
        double r3890315 = -3.0;
        double r3890316 = a;
        double r3890317 = c;
        double r3890318 = r3890316 * r3890317;
        double r3890319 = r3890315 * r3890318;
        double r3890320 = r3890314 + r3890319;
        double r3890321 = sqrt(r3890320);
        double r3890322 = r3890320 * r3890321;
        double r3890323 = r3890314 * r3890311;
        double r3890324 = r3890322 - r3890323;
        double r3890325 = r3890311 * r3890321;
        double r3890326 = r3890325 + r3890314;
        double r3890327 = r3890320 + r3890326;
        double r3890328 = r3890324 / r3890327;
        double r3890329 = 3.0;
        double r3890330 = r3890316 * r3890329;
        double r3890331 = r3890328 / r3890330;
        double r3890332 = -0.5;
        double r3890333 = r3890317 / r3890311;
        double r3890334 = r3890332 * r3890333;
        double r3890335 = r3890313 ? r3890331 : r3890334;
        return r3890335;
}

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 < 0.09946845057046652

    1. Initial program 24.0

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

    2. Simplified24.0

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

    4. Applied flip3--24.1

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

    5. Simplified23.5

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

    6. Simplified23.5

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

    if 0.09946845057046652 < b

    1. Initial program 47.8

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

    2. Simplified47.8

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

    3. Taylor expanded around inf 9.2

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.09946845057046652:\\ \;\;\;\;\frac{\frac{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))