Average Error: 33.6 → 10.3
Time: 31.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.6044431639032268 \cdot 10^{+147}:\\ \;\;\;\;(\left(\frac{c}{b}\right) \cdot \frac{1}{2} + \left(b \cdot \frac{\frac{-2}{3}}{a}\right))_*\\ \mathbf{elif}\;b \le 4.999603533426357 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{elif}\;b \le 8.668665792614052 \cdot 10^{-80}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \mathbf{elif}\;b \le 1.0203561418891653 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r15160733 = b;
        double r15160734 = -r15160733;
        double r15160735 = r15160733 * r15160733;
        double r15160736 = 3.0;
        double r15160737 = a;
        double r15160738 = r15160736 * r15160737;
        double r15160739 = c;
        double r15160740 = r15160738 * r15160739;
        double r15160741 = r15160735 - r15160740;
        double r15160742 = sqrt(r15160741);
        double r15160743 = r15160734 + r15160742;
        double r15160744 = r15160743 / r15160738;
        return r15160744;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r15160745 = b;
        double r15160746 = -2.6044431639032268e+147;
        bool r15160747 = r15160745 <= r15160746;
        double r15160748 = c;
        double r15160749 = r15160748 / r15160745;
        double r15160750 = 0.5;
        double r15160751 = -0.6666666666666666;
        double r15160752 = a;
        double r15160753 = r15160751 / r15160752;
        double r15160754 = r15160745 * r15160753;
        double r15160755 = fma(r15160749, r15160750, r15160754);
        double r15160756 = 4.999603533426357e-105;
        bool r15160757 = r15160745 <= r15160756;
        double r15160758 = r15160745 * r15160745;
        double r15160759 = 3.0;
        double r15160760 = r15160759 * r15160752;
        double r15160761 = r15160748 * r15160760;
        double r15160762 = r15160758 - r15160761;
        double r15160763 = sqrt(r15160762);
        double r15160764 = -r15160745;
        double r15160765 = r15160763 + r15160764;
        double r15160766 = r15160765 / r15160759;
        double r15160767 = r15160766 / r15160752;
        double r15160768 = 8.668665792614052e-80;
        bool r15160769 = r15160745 <= r15160768;
        double r15160770 = -0.5;
        double r15160771 = r15160749 * r15160770;
        double r15160772 = 1.0203561418891653e-53;
        bool r15160773 = r15160745 <= r15160772;
        double r15160774 = r15160773 ? r15160767 : r15160771;
        double r15160775 = r15160769 ? r15160771 : r15160774;
        double r15160776 = r15160757 ? r15160767 : r15160775;
        double r15160777 = r15160747 ? r15160755 : r15160776;
        return r15160777;
}


\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}:\\
\;\;\;\;(\left(\frac{c}{b}\right) \cdot \frac{1}{2} + \left(b \cdot \frac{\frac{-2}{3}}{a}\right))_*\\

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

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

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

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

\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. Taylor expanded around -inf 3.1

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

    3. Simplified3.0

      \[\leadsto \color{blue}{(\left(\frac{c}{b}\right) \cdot \frac{1}{2} + \left(\frac{\frac{-2}{3}}{a} \cdot b\right))_*}\]

    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. Using strategy rm

    3. Applied associate-/r*12.6

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

    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. 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

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

Reproduce

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