Average Error: 43.9 → 11.1
Time: 6.3s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2.56800976414688911 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{{b}^{2} - \left({b}^{2} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 2.56800976414688911 \cdot 10^{-4}:\\
\;\;\;\;\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 r67888 = b;
        double r67889 = -r67888;
        double r67890 = r67888 * r67888;
        double r67891 = 3.0;
        double r67892 = a;
        double r67893 = r67891 * r67892;
        double r67894 = c;
        double r67895 = r67893 * r67894;
        double r67896 = r67890 - r67895;
        double r67897 = sqrt(r67896);
        double r67898 = r67889 + r67897;
        double r67899 = r67898 / r67893;
        return r67899;
}
↓
double f(double a, double b, double c) {
        double r67900 = b;
        double r67901 = 0.0002568009764146889;
        bool r67902 = r67900 <= r67901;
        double r67903 = 2.0;
        double r67904 = pow(r67900, r67903);
        double r67905 = 3.0;
        double r67906 = a;
        double r67907 = r67905 * r67906;
        double r67908 = c;
        double r67909 = r67907 * r67908;
        double r67910 = r67904 - r67909;
        double r67911 = r67904 - r67910;
        double r67912 = -r67900;
        double r67913 = r67900 * r67900;
        double r67914 = r67913 - r67909;
        double r67915 = sqrt(r67914);
        double r67916 = r67912 - r67915;
        double r67917 = r67911 / r67916;
        double r67918 = r67917 / r67907;
        double r67919 = -0.5;
        double r67920 = r67908 / r67900;
        double r67921 = r67919 * r67920;
        double r67922 = r67902 ? r67918 : r67921;
        return r67922;
}

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 < 0.0002568009764146889

    1. Initial program 19.6

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

    3. Applied flip-+19.5

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

    4. Simplified18.5

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

    if 0.0002568009764146889 < b

    1. Initial program 46.0

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

    2. Taylor expanded around inf 10.4

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

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.56800976414688911 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{{b}^{2} - \left({b}^{2} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))