Average Error: 44.0 → 11.3
Time: 7.1s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.114113106997123639:\\ \;\;\;\;\frac{\frac{{b}^{2} - \left({b}^{2} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 0.114113106997123639:\\
\;\;\;\;\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 r76670 = b;
        double r76671 = -r76670;
        double r76672 = r76670 * r76670;
        double r76673 = 3.0;
        double r76674 = a;
        double r76675 = r76673 * r76674;
        double r76676 = c;
        double r76677 = r76675 * r76676;
        double r76678 = r76672 - r76677;
        double r76679 = sqrt(r76678);
        double r76680 = r76671 + r76679;
        double r76681 = r76680 / r76675;
        return r76681;
}
↓
double f(double a, double b, double c) {
        double r76682 = b;
        double r76683 = 0.11411310699712364;
        bool r76684 = r76682 <= r76683;
        double r76685 = 2.0;
        double r76686 = pow(r76682, r76685);
        double r76687 = 3.0;
        double r76688 = a;
        double r76689 = r76687 * r76688;
        double r76690 = c;
        double r76691 = r76689 * r76690;
        double r76692 = r76686 - r76691;
        double r76693 = r76686 - r76692;
        double r76694 = -r76682;
        double r76695 = r76682 * r76682;
        double r76696 = r76695 - r76691;
        double r76697 = sqrt(r76696);
        double r76698 = r76694 - r76697;
        double r76699 = r76693 / r76698;
        double r76700 = r76699 / r76689;
        double r76701 = -0.5;
        double r76702 = r76690 / r76682;
        double r76703 = r76701 * r76702;
        double r76704 = r76684 ? r76700 : r76703;
        return r76704;
}

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 < 0.11411310699712364

    1. Initial program 23.5

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

    3. Applied flip-+23.5

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

    4. Simplified22.5

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

    if 0.11411310699712364 < b

    1. Initial program 46.9

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

    2. Taylor expanded around inf 9.7

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.114113106997123639:\\ \;\;\;\;\frac{\frac{{b}^{2} - \left({b}^{2} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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)))