Average Error: 34.1 → 13.5
Time: 4.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.56941706508999029 \cdot 10^{163}:\\ \;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le -8.7337006906970859 \cdot 10^{-271}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.8041467096464137 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{3 \cdot \left(a \cdot c\right) + 0}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\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 -1.56941706508999029 \cdot 10^{163}:\\
\;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\

\mathbf{elif}\;b \le -8.7337006906970859 \cdot 10^{-271}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\

\mathbf{elif}\;b \le 1.8041467096464137 \cdot 10^{153}:\\
\;\;\;\;\frac{\frac{3 \cdot \left(a \cdot c\right) + 0}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\

\end{array}
double code(double a, double b, double c) {
	return ((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a));
}
↓
double code(double a, double b, double c) {
	double temp;
	if ((b <= -1.5694170650899903e+163)) {
		temp = ((-b + ((1.5 * ((a * c) / b)) - b)) / (3.0 * a));
	} else {
		double temp_1;
		if ((b <= -8.733700690697086e-271)) {
			temp_1 = ((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a));
		} else {
			double temp_2;
			if ((b <= 1.8041467096464137e+153)) {
				temp_2 = ((((3.0 * (a * c)) + 0.0) / (-b - sqrt(((b * b) - (3.0 * (a * c)))))) / (3.0 * a));
			} else {
				temp_2 = ((-1.5 * ((a * c) / b)) / (3.0 * a));
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

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 4 regimes

  2. if b < -1.5694170650899903e+163

    1. Initial program 64.0

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

    3. Applied associate-*l*64.0

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

    4. Taylor expanded around -inf 13.1

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}}{3 \cdot a}\]

    if -1.5694170650899903e+163 < b < -8.733700690697086e-271

    1. Initial program 9.8

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

    if -8.733700690697086e-271 < b < 1.8041467096464137e+153

    1. Initial program 33.4

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

    3. Applied associate-*l*33.4

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

    5. Applied flip-+33.4

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

    6. Simplified16.3

      \[\leadsto \frac{\frac{\color{blue}{3 \cdot \left(a \cdot c\right) + 0}}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}\]

    if 1.8041467096464137e+153 < b

    1. Initial program 63.8

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

    2. Taylor expanded around inf 14.3

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.56941706508999029 \cdot 10^{163}:\\ \;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le -8.7337006906970859 \cdot 10^{-271}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.8041467096464137 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{3 \cdot \left(a \cdot c\right) + 0}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))