Average Error: 34.6 → 9.9
Time: 4.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.7958659106438658 \cdot 10^{150}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 9.39036747108992214 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -6.7958659106438658 \cdot 10^{150}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 9.39036747108992214 \cdot 10^{-69}:\\
\;\;\;\;\frac{\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 r86960 = b;
        double r86961 = -r86960;
        double r86962 = r86960 * r86960;
        double r86963 = 3.0;
        double r86964 = a;
        double r86965 = r86963 * r86964;
        double r86966 = c;
        double r86967 = r86965 * r86966;
        double r86968 = r86962 - r86967;
        double r86969 = sqrt(r86968);
        double r86970 = r86961 + r86969;
        double r86971 = r86970 / r86965;
        return r86971;
}
↓
double f(double a, double b, double c) {
        double r86972 = b;
        double r86973 = -6.795865910643866e+150;
        bool r86974 = r86972 <= r86973;
        double r86975 = 0.5;
        double r86976 = c;
        double r86977 = r86976 / r86972;
        double r86978 = r86975 * r86977;
        double r86979 = 0.6666666666666666;
        double r86980 = a;
        double r86981 = r86972 / r86980;
        double r86982 = r86979 * r86981;
        double r86983 = r86978 - r86982;
        double r86984 = 9.390367471089922e-69;
        bool r86985 = r86972 <= r86984;
        double r86986 = -r86972;
        double r86987 = r86972 * r86972;
        double r86988 = 3.0;
        double r86989 = r86988 * r86980;
        double r86990 = r86989 * r86976;
        double r86991 = r86987 - r86990;
        double r86992 = sqrt(r86991);
        double r86993 = r86986 + r86992;
        double r86994 = r86993 / r86989;
        double r86995 = -0.5;
        double r86996 = r86995 * r86977;
        double r86997 = r86985 ? r86994 : r86996;
        double r86998 = r86974 ? r86983 : r86997;
        return r86998;
}

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 3 regimes

  2. if b < -6.795865910643866e+150

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 2.1

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}}\]

    if -6.795865910643866e+150 < b < 9.390367471089922e-69

    1. Initial program 12.6

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

    if 9.390367471089922e-69 < b

    1. Initial program 53.5

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

    2. Taylor expanded around inf 8.7

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.7958659106438658 \cdot 10^{150}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 9.39036747108992214 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))