Average Error: 34.0 → 10.5
Time: 4.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.61279552926933292 \cdot 10^{93}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.76145822117380694 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -9.61279552926933292 \cdot 10^{93}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 1.76145822117380694 \cdot 10^{-90}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a}}{3}\\

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

\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 VAR;
	if ((b <= -9.612795529269333e+93)) {
		VAR = ((0.5 * (c / b)) - (0.6666666666666666 * (b / a)));
	} else {
		double VAR_1;
		if ((b <= 1.761458221173807e-90)) {
			VAR_1 = (((sqrt(((b * b) - (3.0 * (a * c)))) - b) / a) / 3.0);
		} else {
			VAR_1 = (-0.5 * (c / b));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 < -9.612795529269333e+93

    1. Initial program 45.2

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

    2. Taylor expanded around -inf 4.5

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

    if -9.612795529269333e+93 < b < 1.761458221173807e-90

    1. Initial program 13.2

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

    3. Applied associate-/r*13.3

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

    5. Applied clear-num13.3

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

    6. Simplified13.4

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

    8. Applied associate-/r*13.3

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

    9. Simplified13.3

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

    11. Applied associate-*l*13.3

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

    if 1.761458221173807e-90 < b

    1. Initial program 52.1

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

    2. Taylor expanded around inf 9.9

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.61279552926933292 \cdot 10^{93}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.76145822117380694 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))