Average Error: 34.8 → 10.3
Time: 5.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.19823863200283504 \cdot 10^{153}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.2392895115955295 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.19823863200283504 \cdot 10^{153}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 1.2392895115955295 \cdot 10^{-66}:\\
\;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r119031 = b;
        double r119032 = -r119031;
        double r119033 = r119031 * r119031;
        double r119034 = 3.0;
        double r119035 = a;
        double r119036 = r119034 * r119035;
        double r119037 = c;
        double r119038 = r119036 * r119037;
        double r119039 = r119033 - r119038;
        double r119040 = sqrt(r119039);
        double r119041 = r119032 + r119040;
        double r119042 = r119041 / r119036;
        return r119042;
}
↓
double f(double a, double b, double c) {
        double r119043 = b;
        double r119044 = -4.198238632002835e+153;
        bool r119045 = r119043 <= r119044;
        double r119046 = 0.5;
        double r119047 = c;
        double r119048 = r119047 / r119043;
        double r119049 = r119046 * r119048;
        double r119050 = 0.6666666666666666;
        double r119051 = a;
        double r119052 = r119043 / r119051;
        double r119053 = r119050 * r119052;
        double r119054 = r119049 - r119053;
        double r119055 = 1.2392895115955295e-66;
        bool r119056 = r119043 <= r119055;
        double r119057 = -r119043;
        double r119058 = r119043 * r119043;
        double r119059 = 3.0;
        double r119060 = r119059 * r119051;
        double r119061 = r119060 * r119047;
        double r119062 = r119058 - r119061;
        double r119063 = sqrt(r119062);
        double r119064 = r119057 + r119063;
        double r119065 = r119064 / r119059;
        double r119066 = r119065 / r119051;
        double r119067 = -0.5;
        double r119068 = r119067 * r119048;
        double r119069 = r119056 ? r119066 : r119068;
        double r119070 = r119045 ? r119054 : r119069;
        return r119070;
}

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 < -4.198238632002835e+153

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 3.3

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}}\]

    if -4.198238632002835e+153 < b < 1.2392895115955295e-66

    1. Initial program 13.5

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

    3. Applied associate-/r*13.5

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

    if 1.2392895115955295e-66 < b

    1. Initial program 54.0

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

    2. Taylor expanded around inf 8.1

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.19823863200283504 \cdot 10^{153}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.2392895115955295 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))