Average Error: 34.6 → 9.9
Time: 15.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\ \;\;\;\;\frac{\frac{\frac{1.5 \cdot a}{\frac{b}{c}} + -2 \cdot b}{3}}{a}\\ \mathbf{elif}\;b \le 8.085265835057349842233247168077451568119 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]
\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 -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\
\;\;\;\;\frac{\frac{\frac{1.5 \cdot a}{\frac{b}{c}} + -2 \cdot b}{3}}{a}\\

\mathbf{elif}\;b \le 8.085265835057349842233247168077451568119 \cdot 10^{-63}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r93041 = b;
        double r93042 = -r93041;
        double r93043 = r93041 * r93041;
        double r93044 = 3.0;
        double r93045 = a;
        double r93046 = r93044 * r93045;
        double r93047 = c;
        double r93048 = r93046 * r93047;
        double r93049 = r93043 - r93048;
        double r93050 = sqrt(r93049);
        double r93051 = r93042 + r93050;
        double r93052 = r93051 / r93046;
        return r93052;
}
↓
double f(double a, double b, double c) {
        double r93053 = b;
        double r93054 = -7.943482039519134e+75;
        bool r93055 = r93053 <= r93054;
        double r93056 = 1.5;
        double r93057 = a;
        double r93058 = r93056 * r93057;
        double r93059 = c;
        double r93060 = r93053 / r93059;
        double r93061 = r93058 / r93060;
        double r93062 = -2.0;
        double r93063 = r93062 * r93053;
        double r93064 = r93061 + r93063;
        double r93065 = 3.0;
        double r93066 = r93064 / r93065;
        double r93067 = r93066 / r93057;
        double r93068 = 8.08526583505735e-63;
        bool r93069 = r93053 <= r93068;
        double r93070 = r93053 * r93053;
        double r93071 = r93065 * r93057;
        double r93072 = r93071 * r93059;
        double r93073 = r93070 - r93072;
        double r93074 = sqrt(r93073);
        double r93075 = r93074 - r93053;
        double r93076 = r93075 / r93065;
        double r93077 = r93076 / r93057;
        double r93078 = r93059 / r93053;
        double r93079 = -0.5;
        double r93080 = r93078 * r93079;
        double r93081 = r93069 ? r93077 : r93080;
        double r93082 = r93055 ? r93067 : r93081;
        return r93082;
}

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 < -7.943482039519134e+75

    1. Initial program 42.8

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

    3. Applied associate-/r*42.8

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

    4. Simplified42.8

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

    5. Taylor expanded around -inf 9.7

      \[\leadsto \frac{\frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3}}{a}\]

    6. Simplified4.5

      \[\leadsto \frac{\frac{\color{blue}{b \cdot -2 + \frac{1.5 \cdot a}{\frac{b}{c}}}}{3}}{a}\]

    if -7.943482039519134e+75 < b < 8.08526583505735e-63

    1. Initial program 13.7

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

    3. Applied associate-/r*13.7

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

    4. Simplified13.7

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

    if 8.08526583505735e-63 < b

    1. Initial program 53.6

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

    2. Taylor expanded around inf 8.3

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

    3. Simplified8.3

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\ \;\;\;\;\frac{\frac{\frac{1.5 \cdot a}{\frac{b}{c}} + -2 \cdot b}{3}}{a}\\ \mathbf{elif}\;b \le 8.085265835057349842233247168077451568119 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))