Average Error: 34.1 → 10.4
Time: 21.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.5144039826007546 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 2.223474256889498 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} - \frac{b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.5144039826007546 \cdot 10^{+152}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\

\mathbf{elif}\;b \le 2.223474256889498 \cdot 10^{-114}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} - \frac{b}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4624108 = b;
        double r4624109 = -r4624108;
        double r4624110 = r4624108 * r4624108;
        double r4624111 = 3.0;
        double r4624112 = a;
        double r4624113 = r4624111 * r4624112;
        double r4624114 = c;
        double r4624115 = r4624113 * r4624114;
        double r4624116 = r4624110 - r4624115;
        double r4624117 = sqrt(r4624116);
        double r4624118 = r4624109 + r4624117;
        double r4624119 = r4624118 / r4624113;
        return r4624119;
}
↓
double f(double a, double b, double c) {
        double r4624120 = b;
        double r4624121 = -1.5144039826007546e+152;
        bool r4624122 = r4624120 <= r4624121;
        double r4624123 = 0.5;
        double r4624124 = c;
        double r4624125 = r4624124 / r4624120;
        double r4624126 = r4624123 * r4624125;
        double r4624127 = a;
        double r4624128 = r4624120 / r4624127;
        double r4624129 = 0.6666666666666666;
        double r4624130 = r4624128 * r4624129;
        double r4624131 = r4624126 - r4624130;
        double r4624132 = 2.223474256889498e-114;
        bool r4624133 = r4624120 <= r4624132;
        double r4624134 = r4624120 * r4624120;
        double r4624135 = 3.0;
        double r4624136 = r4624135 * r4624127;
        double r4624137 = r4624136 * r4624124;
        double r4624138 = r4624134 - r4624137;
        double r4624139 = sqrt(r4624138);
        double r4624140 = r4624139 / r4624136;
        double r4624141 = r4624120 / r4624136;
        double r4624142 = r4624140 - r4624141;
        double r4624143 = -0.5;
        double r4624144 = r4624143 * r4624125;
        double r4624145 = r4624133 ? r4624142 : r4624144;
        double r4624146 = r4624122 ? r4624131 : r4624145;
        return r4624146;
}

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 < -1.5144039826007546e+152

    1. Initial program 59.8

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

    2. Simplified59.8

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

    3. Taylor expanded around -inf 2.4

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

    if -1.5144039826007546e+152 < b < 2.223474256889498e-114

    1. Initial program 11.9

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

    2. Simplified11.9

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

    4. Applied div-sub11.9

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

    if 2.223474256889498e-114 < b

    1. Initial program 51.6

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

    2. Simplified51.6

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

    3. Taylor expanded around inf 10.8

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.5144039826007546 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 2.223474256889498 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} - \frac{b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))