Average Error: 34.3 → 8.8
Time: 5.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.7863758169125638 \cdot 10^{138}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.5482564668283562 \cdot 10^{-112}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{elif}\;b \le 8.28272902225294789 \cdot 10^{102}:\\ \;\;\;\;\frac{\frac{1 \cdot \left(\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)\right)}{3 \cdot a}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.7863758169125638 \cdot 10^{138}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 2.5482564668283562 \cdot 10^{-112}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\

\mathbf{elif}\;b \le 8.28272902225294789 \cdot 10^{102}:\\
\;\;\;\;\frac{\frac{1 \cdot \left(\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)\right)}{3 \cdot a}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\

\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 <= -2.7863758169125638e+138)) {
		VAR = ((0.5 * (c / b)) - (0.6666666666666666 * (b / a)));
	} else {
		double VAR_1;
		if ((b <= 2.548256466828356e-112)) {
			VAR_1 = ((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a));
		} else {
			double VAR_2;
			if ((b <= 8.282729022252948e+102)) {
				VAR_2 = (((1.0 * ((pow(b, 2.0) - pow(b, 2.0)) + (3.0 * (a * c)))) / (3.0 * a)) / (-b - sqrt(((b * b) - ((3.0 * a) * c)))));
			} else {
				VAR_2 = (-0.5 * (c / b));
			}
			VAR_1 = VAR_2;
		}
		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 4 regimes

  2. if b < -2.7863758169125638e+138

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 3.0

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

    if -2.7863758169125638e+138 < b < 2.548256466828356e-112

    1. Initial program 11.3

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

    if 2.548256466828356e-112 < b < 8.282729022252948e+102

    1. Initial program 41.9

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

    3. Applied clear-num41.9

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

    5. Applied *-un-lft-identity41.9

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

    6. Applied times-frac41.9

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

    7. Simplified41.9

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

    9. Applied associate-/r*41.9

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

    11. Applied flip-+41.9

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

    12. Applied associate-/r/42.0

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

    13. Applied associate-/r*41.9

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

    14. Simplified14.6

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)\right)}{3 \cdot a}}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\]

    if 8.282729022252948e+102 < b

    1. Initial program 60.1

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

    2. Taylor expanded around inf 2.3

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.7863758169125638 \cdot 10^{138}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.5482564668283562 \cdot 10^{-112}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{elif}\;b \le 8.28272902225294789 \cdot 10^{102}:\\ \;\;\;\;\frac{\frac{1 \cdot \left(\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)\right)}{3 \cdot a}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))