Average Error: 43.8 → 11.1
Time: 6.3s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1.7936624356974993 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 1.7936624356974993 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r74984 = b;
        double r74985 = -r74984;
        double r74986 = r74984 * r74984;
        double r74987 = 3.0;
        double r74988 = a;
        double r74989 = r74987 * r74988;
        double r74990 = c;
        double r74991 = r74989 * r74990;
        double r74992 = r74986 - r74991;
        double r74993 = sqrt(r74992);
        double r74994 = r74985 + r74993;
        double r74995 = r74994 / r74989;
        return r74995;
}
↓
double f(double a, double b, double c) {
        double r74996 = b;
        double r74997 = 0.00017936624356974993;
        bool r74998 = r74996 <= r74997;
        double r74999 = 2.0;
        double r75000 = pow(r74996, r74999);
        double r75001 = 3.0;
        double r75002 = a;
        double r75003 = c;
        double r75004 = r75002 * r75003;
        double r75005 = r75001 * r75004;
        double r75006 = r75000 - r75005;
        double r75007 = r75006 - r75000;
        double r75008 = r74996 * r74996;
        double r75009 = r75001 * r75002;
        double r75010 = r75009 * r75003;
        double r75011 = r75008 - r75010;
        double r75012 = sqrt(r75011);
        double r75013 = r75012 + r74996;
        double r75014 = r75007 / r75013;
        double r75015 = r75014 / r75009;
        double r75016 = -0.5;
        double r75017 = r75003 / r74996;
        double r75018 = r75016 * r75017;
        double r75019 = r74998 ? r75015 : r75018;
        return r75019;
}

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 < 0.00017936624356974993

    1. Initial program 18.4

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

    2. Simplified18.4

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

    4. Applied flip--18.4

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

    5. Simplified17.4

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

    if 0.00017936624356974993 < b

    1. Initial program 45.7

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

    2. Simplified45.7

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

    3. Taylor expanded around inf 10.6

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

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.7936624356974993 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))