Average Error: 43.9 → 11.5
Time: 17.8s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.2283921088226662:\\ \;\;\;\;\frac{\frac{\left(-3 \cdot \left(a \cdot c\right) + b \cdot b\right) \cdot \sqrt{-3 \cdot \left(a \cdot c\right) + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(b \cdot b + \left(-3 \cdot \left(a \cdot c\right) + b \cdot b\right)\right) + b \cdot \sqrt{-3 \cdot \left(a \cdot c\right) + b \cdot b}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 0.2283921088226662:\\
\;\;\;\;\frac{\frac{\left(-3 \cdot \left(a \cdot c\right) + b \cdot b\right) \cdot \sqrt{-3 \cdot \left(a \cdot c\right) + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(b \cdot b + \left(-3 \cdot \left(a \cdot c\right) + b \cdot b\right)\right) + b \cdot \sqrt{-3 \cdot \left(a \cdot c\right) + b \cdot b}}}{a \cdot 3}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4505809 = b;
        double r4505810 = -r4505809;
        double r4505811 = r4505809 * r4505809;
        double r4505812 = 3.0;
        double r4505813 = a;
        double r4505814 = r4505812 * r4505813;
        double r4505815 = c;
        double r4505816 = r4505814 * r4505815;
        double r4505817 = r4505811 - r4505816;
        double r4505818 = sqrt(r4505817);
        double r4505819 = r4505810 + r4505818;
        double r4505820 = r4505819 / r4505814;
        return r4505820;
}
↓
double f(double a, double b, double c) {
        double r4505821 = b;
        double r4505822 = 0.2283921088226662;
        bool r4505823 = r4505821 <= r4505822;
        double r4505824 = -3.0;
        double r4505825 = a;
        double r4505826 = c;
        double r4505827 = r4505825 * r4505826;
        double r4505828 = r4505824 * r4505827;
        double r4505829 = r4505821 * r4505821;
        double r4505830 = r4505828 + r4505829;
        double r4505831 = sqrt(r4505830);
        double r4505832 = r4505830 * r4505831;
        double r4505833 = r4505821 * r4505829;
        double r4505834 = r4505832 - r4505833;
        double r4505835 = r4505829 + r4505830;
        double r4505836 = r4505821 * r4505831;
        double r4505837 = r4505835 + r4505836;
        double r4505838 = r4505834 / r4505837;
        double r4505839 = 3.0;
        double r4505840 = r4505825 * r4505839;
        double r4505841 = r4505838 / r4505840;
        double r4505842 = -0.5;
        double r4505843 = r4505826 / r4505821;
        double r4505844 = r4505842 * r4505843;
        double r4505845 = r4505823 ? r4505841 : r4505844;
        return r4505845;
}

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 < 0.2283921088226662

    1. Initial program 23.9

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

    2. Simplified23.9

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

    4. Applied flip3--24.0

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

    5. Simplified23.3

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

    6. Simplified23.3

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

    if 0.2283921088226662 < b

    1. Initial program 47.3

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

    2. Simplified47.3

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

    3. Taylor expanded around inf 9.5

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.2283921088226662:\\ \;\;\;\;\frac{\frac{\left(-3 \cdot \left(a \cdot c\right) + b \cdot b\right) \cdot \sqrt{-3 \cdot \left(a \cdot c\right) + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(b \cdot b + \left(-3 \cdot \left(a \cdot c\right) + b \cdot b\right)\right) + b \cdot \sqrt{-3 \cdot \left(a \cdot c\right) + b \cdot b}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b c)
  :name "Cubic critical, 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)))