Average Error: 33.6 → 10.3
Time: 30.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.6044431639032268 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 4.999603533426357 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3}}{a}\\ \mathbf{elif}\;b \le 8.668665792614052 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.0203561418891653 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r15132817 = b;
        double r15132818 = -r15132817;
        double r15132819 = r15132817 * r15132817;
        double r15132820 = 3.0;
        double r15132821 = a;
        double r15132822 = r15132820 * r15132821;
        double r15132823 = c;
        double r15132824 = r15132822 * r15132823;
        double r15132825 = r15132819 - r15132824;
        double r15132826 = sqrt(r15132825);
        double r15132827 = r15132818 + r15132826;
        double r15132828 = r15132827 / r15132822;
        return r15132828;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r15132829 = b;
        double r15132830 = -2.6044431639032268e+147;
        bool r15132831 = r15132829 <= r15132830;
        double r15132832 = 0.5;
        double r15132833 = c;
        double r15132834 = r15132833 / r15132829;
        double r15132835 = r15132832 * r15132834;
        double r15132836 = a;
        double r15132837 = r15132829 / r15132836;
        double r15132838 = 0.6666666666666666;
        double r15132839 = r15132837 * r15132838;
        double r15132840 = r15132835 - r15132839;
        double r15132841 = 4.999603533426357e-105;
        bool r15132842 = r15132829 <= r15132841;
        double r15132843 = r15132829 * r15132829;
        double r15132844 = -3.0;
        double r15132845 = r15132833 * r15132844;
        double r15132846 = r15132836 * r15132845;
        double r15132847 = r15132843 + r15132846;
        double r15132848 = sqrt(r15132847);
        double r15132849 = r15132848 - r15132829;
        double r15132850 = 3.0;
        double r15132851 = r15132849 / r15132850;
        double r15132852 = r15132851 / r15132836;
        double r15132853 = 8.668665792614052e-80;
        bool r15132854 = r15132829 <= r15132853;
        double r15132855 = -0.5;
        double r15132856 = r15132855 * r15132834;
        double r15132857 = 1.0203561418891653e-53;
        bool r15132858 = r15132829 <= r15132857;
        double r15132859 = r15132858 ? r15132852 : r15132856;
        double r15132860 = r15132854 ? r15132856 : r15132859;
        double r15132861 = r15132842 ? r15132852 : r15132860;
        double r15132862 = r15132831 ? r15132840 : r15132861;
        return r15132862;
}


\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 -2.6044431639032268 \cdot 10^{+147}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\

\mathbf{elif}\;b \le 4.999603533426357 \cdot 10^{-105}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3}}{a}\\

\mathbf{elif}\;b \le 8.668665792614052 \cdot 10^{-80}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.0203561418891653 \cdot 10^{-53}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3}}{a}\\

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

\end{array}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 3 regimes

  2. if b < -2.6044431639032268e+147

    1. Initial program 58.9

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

    2. Simplified58.9

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

    3. Taylor expanded around -inf 3.1

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

    if -2.6044431639032268e+147 < b < 4.999603533426357e-105 or 8.668665792614052e-80 < b < 1.0203561418891653e-53

    1. Initial program 12.6

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

    2. Simplified12.6

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

    4. Applied associate-/r*12.6

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

    5. Taylor expanded around -inf 12.6

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

    6. Simplified12.6

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

    if 4.999603533426357e-105 < b < 8.668665792614052e-80 or 1.0203561418891653e-53 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 9.5

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.6044431639032268 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 4.999603533426357 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3}}{a}\\ \mathbf{elif}\;b \le 8.668665792614052 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.0203561418891653 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019102 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))