Average Error: 33.2 → 10.3
Time: 26.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2183894153494036 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 0.031080860458343948:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b + -3 \cdot \left(c \cdot a\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.2183894153494036 \cdot 10^{+146}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\

\mathbf{elif}\;b \le 0.031080860458343948:\\
\;\;\;\;\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 r17791978 = b;
        double r17791979 = -r17791978;
        double r17791980 = r17791978 * r17791978;
        double r17791981 = 3.0;
        double r17791982 = a;
        double r17791983 = r17791981 * r17791982;
        double r17791984 = c;
        double r17791985 = r17791983 * r17791984;
        double r17791986 = r17791980 - r17791985;
        double r17791987 = sqrt(r17791986);
        double r17791988 = r17791979 + r17791987;
        double r17791989 = r17791988 / r17791983;
        return r17791989;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r17791990 = b;
        double r17791991 = -2.2183894153494036e+146;
        bool r17791992 = r17791990 <= r17791991;
        double r17791993 = 0.5;
        double r17791994 = c;
        double r17791995 = r17791994 / r17791990;
        double r17791996 = r17791993 * r17791995;
        double r17791997 = a;
        double r17791998 = r17791990 / r17791997;
        double r17791999 = 0.6666666666666666;
        double r17792000 = r17791998 * r17791999;
        double r17792001 = r17791996 - r17792000;
        double r17792002 = 0.031080860458343948;
        bool r17792003 = r17791990 <= r17792002;
        double r17792004 = r17791990 * r17791990;
        double r17792005 = -3.0;
        double r17792006 = r17791994 * r17791997;
        double r17792007 = r17792005 * r17792006;
        double r17792008 = r17792004 + r17792007;
        double r17792009 = sqrt(r17792008);
        double r17792010 = r17792009 - r17791990;
        double r17792011 = 3.0;
        double r17792012 = r17792010 / r17792011;
        double r17792013 = r17792012 / r17791997;
        double r17792014 = -0.5;
        double r17792015 = r17792014 * r17791995;
        double r17792016 = r17792003 ? r17792013 : r17792015;
        double r17792017 = r17791992 ? r17792001 : r17792016;
        return r17792017;
}

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 < -2.2183894153494036e+146

    1. Initial program 57.6

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

    2. Simplified57.6

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

    3. Taylor expanded around 0 57.6

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

    4. Simplified57.6

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

    5. Taylor expanded around -inf 1.7

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

    if -2.2183894153494036e+146 < b < 0.031080860458343948

    1. Initial program 14.9

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

    2. Simplified14.9

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

    3. Taylor expanded around 0 14.9

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

    4. Simplified14.9

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

    6. Applied associate-/r*14.8

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

    7. Taylor expanded around -inf 14.8

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

    if 0.031080860458343948 < b

    1. Initial program 55.2

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

    2. Simplified55.2

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

    3. Taylor expanded around 0 55.2

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

    4. Simplified55.2

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

    5. Taylor expanded around inf 5.7

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2183894153494036 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 0.031080860458343948:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b + -3 \cdot \left(c \cdot a\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))