Average Error: 33.2 → 10.5
Time: 18.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.525652579795996 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 3.821014310434392 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{3}}{a} \cdot \left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.525652579795996 \cdot 10^{+106}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\

\mathbf{elif}\;b \le 3.821014310434392 \cdot 10^{-21}:\\
\;\;\;\;\frac{\frac{1}{3}}{a} \cdot \left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r3846093 = b;
        double r3846094 = -r3846093;
        double r3846095 = r3846093 * r3846093;
        double r3846096 = 3.0;
        double r3846097 = a;
        double r3846098 = r3846096 * r3846097;
        double r3846099 = c;
        double r3846100 = r3846098 * r3846099;
        double r3846101 = r3846095 - r3846100;
        double r3846102 = sqrt(r3846101);
        double r3846103 = r3846094 + r3846102;
        double r3846104 = r3846103 / r3846098;
        return r3846104;
}
↓
double f(double a, double b, double c) {
        double r3846105 = b;
        double r3846106 = -4.525652579795996e+106;
        bool r3846107 = r3846105 <= r3846106;
        double r3846108 = 0.5;
        double r3846109 = c;
        double r3846110 = r3846109 / r3846105;
        double r3846111 = r3846108 * r3846110;
        double r3846112 = a;
        double r3846113 = r3846105 / r3846112;
        double r3846114 = 0.6666666666666666;
        double r3846115 = r3846113 * r3846114;
        double r3846116 = r3846111 - r3846115;
        double r3846117 = 3.821014310434392e-21;
        bool r3846118 = r3846105 <= r3846117;
        double r3846119 = 0.3333333333333333;
        double r3846120 = r3846119 / r3846112;
        double r3846121 = r3846105 * r3846105;
        double r3846122 = 3.0;
        double r3846123 = r3846122 * r3846112;
        double r3846124 = r3846123 * r3846109;
        double r3846125 = r3846121 - r3846124;
        double r3846126 = sqrt(r3846125);
        double r3846127 = r3846126 - r3846105;
        double r3846128 = r3846120 * r3846127;
        double r3846129 = -0.5;
        double r3846130 = r3846129 * r3846110;
        double r3846131 = r3846118 ? r3846128 : r3846130;
        double r3846132 = r3846107 ? r3846116 : r3846131;
        return r3846132;
}

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 < -4.525652579795996e+106

    1. Initial program 46.5

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

    2. Simplified46.5

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

    3. Taylor expanded around -inf 4.0

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

    if -4.525652579795996e+106 < b < 3.821014310434392e-21

    1. Initial program 14.8

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

    2. Simplified14.8

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

    4. Applied div-inv14.9

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

    5. Taylor expanded around 0 14.9

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

    if 3.821014310434392e-21 < b

    1. Initial program 54.7

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

    2. Simplified54.7

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

    3. Taylor expanded around inf 6.8

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.525652579795996 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 3.821014310434392 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{3}}{a} \cdot \left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))