Average Error: 33.3 → 10.3
Time: 30.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2922524578506893 \cdot 10^{+141}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 4.35366028427847 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.2922524578506893 \cdot 10^{+141}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\

\mathbf{elif}\;b \le 4.35366028427847 \cdot 10^{-60}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r21709977 = b;
        double r21709978 = -r21709977;
        double r21709979 = r21709977 * r21709977;
        double r21709980 = 3.0;
        double r21709981 = a;
        double r21709982 = r21709980 * r21709981;
        double r21709983 = c;
        double r21709984 = r21709982 * r21709983;
        double r21709985 = r21709979 - r21709984;
        double r21709986 = sqrt(r21709985);
        double r21709987 = r21709978 + r21709986;
        double r21709988 = r21709987 / r21709982;
        return r21709988;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r21709989 = b;
        double r21709990 = -2.2922524578506893e+141;
        bool r21709991 = r21709989 <= r21709990;
        double r21709992 = 0.5;
        double r21709993 = c;
        double r21709994 = r21709993 / r21709989;
        double r21709995 = r21709992 * r21709994;
        double r21709996 = a;
        double r21709997 = r21709989 / r21709996;
        double r21709998 = 0.6666666666666666;
        double r21709999 = r21709997 * r21709998;
        double r21710000 = r21709995 - r21709999;
        double r21710001 = 4.35366028427847e-60;
        bool r21710002 = r21709989 <= r21710001;
        double r21710003 = r21709989 * r21709989;
        double r21710004 = 3.0;
        double r21710005 = r21709993 * r21709996;
        double r21710006 = r21710004 * r21710005;
        double r21710007 = r21710003 - r21710006;
        double r21710008 = sqrt(r21710007);
        double r21710009 = r21710008 - r21709989;
        double r21710010 = r21710009 / r21710004;
        double r21710011 = r21710010 / r21709996;
        double r21710012 = -0.5;
        double r21710013 = r21710012 * r21709994;
        double r21710014 = r21710002 ? r21710011 : r21710013;
        double r21710015 = r21709991 ? r21710000 : r21710014;
        return r21710015;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -2.2922524578506893e+141

    1. Initial program 55.9

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

    2. Simplified55.9

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

    3. Taylor expanded around -inf 2.5

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b} - \frac{2}{3} \cdot \frac{b}{a}}\]

    if -2.2922524578506893e+141 < b < 4.35366028427847e-60

    1. Initial program 13.2

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

    2. Simplified13.2

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

    4. Applied associate-/r*13.2

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

    6. Applied associate-*l*13.1

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

    if 4.35366028427847e-60 < b

    1. Initial program 52.8

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

    2. Simplified52.8

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

    3. Taylor expanded around inf 8.8

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2922524578506893 \cdot 10^{+141}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 4.35366028427847 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))