Average Error: 33.6 → 10.5
Time: 20.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.874603183983119 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 1.9030999523937384 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.874603183983119 \cdot 10^{+152}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\

\mathbf{elif}\;b \le 1.9030999523937384 \cdot 10^{-67}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r5956117 = b;
        double r5956118 = -r5956117;
        double r5956119 = r5956117 * r5956117;
        double r5956120 = 3.0;
        double r5956121 = a;
        double r5956122 = r5956120 * r5956121;
        double r5956123 = c;
        double r5956124 = r5956122 * r5956123;
        double r5956125 = r5956119 - r5956124;
        double r5956126 = sqrt(r5956125);
        double r5956127 = r5956118 + r5956126;
        double r5956128 = r5956127 / r5956122;
        return r5956128;
}
↓
double f(double a, double b, double c) {
        double r5956129 = b;
        double r5956130 = -2.874603183983119e+152;
        bool r5956131 = r5956129 <= r5956130;
        double r5956132 = 0.5;
        double r5956133 = c;
        double r5956134 = r5956133 / r5956129;
        double r5956135 = r5956132 * r5956134;
        double r5956136 = a;
        double r5956137 = r5956129 / r5956136;
        double r5956138 = 0.6666666666666666;
        double r5956139 = r5956137 * r5956138;
        double r5956140 = r5956135 - r5956139;
        double r5956141 = 1.9030999523937384e-67;
        bool r5956142 = r5956129 <= r5956141;
        double r5956143 = r5956129 * r5956129;
        double r5956144 = 3.0;
        double r5956145 = r5956144 * r5956136;
        double r5956146 = r5956145 * r5956133;
        double r5956147 = r5956143 - r5956146;
        double r5956148 = sqrt(r5956147);
        double r5956149 = r5956148 - r5956129;
        double r5956150 = r5956149 / r5956144;
        double r5956151 = r5956150 / r5956136;
        double r5956152 = -0.5;
        double r5956153 = r5956152 * r5956134;
        double r5956154 = r5956142 ? r5956151 : r5956153;
        double r5956155 = r5956131 ? r5956140 : r5956154;
        return r5956155;
}

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 < -2.874603183983119e+152

    1. Initial program 60.4

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

    2. Simplified60.4

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

    3. Taylor expanded around -inf 2.2

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

    if -2.874603183983119e+152 < b < 1.9030999523937384e-67

    1. Initial program 13.3

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

    2. Simplified13.3

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

    4. Applied associate-/r*13.3

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

    if 1.9030999523937384e-67 < b

    1. Initial program 52.0

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

    2. Simplified52.0

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

    3. Taylor expanded around inf 9.3

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.874603183983119 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 1.9030999523937384 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))