Average Error: 32.9 → 14.8
Time: 17.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3376190644449892 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 6.235673785124529 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\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 -1.3376190644449892 \cdot 10^{+154}:\\
\;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\

\mathbf{elif}\;b \le 6.235673785124529 \cdot 10^{-73}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r5539907 = b;
        double r5539908 = -r5539907;
        double r5539909 = r5539907 * r5539907;
        double r5539910 = 3.0;
        double r5539911 = a;
        double r5539912 = r5539910 * r5539911;
        double r5539913 = c;
        double r5539914 = r5539912 * r5539913;
        double r5539915 = r5539909 - r5539914;
        double r5539916 = sqrt(r5539915);
        double r5539917 = r5539908 + r5539916;
        double r5539918 = r5539917 / r5539912;
        return r5539918;
}
↓
double f(double a, double b, double c) {
        double r5539919 = b;
        double r5539920 = -1.3376190644449892e+154;
        bool r5539921 = r5539919 <= r5539920;
        double r5539922 = 1.5;
        double r5539923 = a;
        double r5539924 = c;
        double r5539925 = r5539923 * r5539924;
        double r5539926 = r5539925 / r5539919;
        double r5539927 = r5539922 * r5539926;
        double r5539928 = r5539927 - r5539919;
        double r5539929 = r5539928 - r5539919;
        double r5539930 = 3.0;
        double r5539931 = r5539930 * r5539923;
        double r5539932 = r5539929 / r5539931;
        double r5539933 = 6.235673785124529e-73;
        bool r5539934 = r5539919 <= r5539933;
        double r5539935 = r5539919 * r5539919;
        double r5539936 = r5539930 * r5539925;
        double r5539937 = r5539935 - r5539936;
        double r5539938 = sqrt(r5539937);
        double r5539939 = r5539938 - r5539919;
        double r5539940 = r5539939 / r5539931;
        double r5539941 = -1.5;
        double r5539942 = r5539941 * r5539926;
        double r5539943 = r5539942 / r5539931;
        double r5539944 = r5539934 ? r5539940 : r5539943;
        double r5539945 = r5539921 ? r5539932 : r5539944;
        return r5539945;
}

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 < -1.3376190644449892e+154

    1. Initial program 60.9

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

    2. Simplified60.9

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

    3. Taylor expanded around -inf 10.6

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

    if -1.3376190644449892e+154 < b < 6.235673785124529e-73

    1. Initial program 12.0

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

    2. Simplified12.0

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

    3. Taylor expanded around 0 12.1

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

    4. Simplified12.1

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

    if 6.235673785124529e-73 < b

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 19.7

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

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3376190644449892 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 6.235673785124529 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))