Average Error: 34.8 → 8.9
Time: 5.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.7201296316436681 \cdot 10^{98}:\\ \;\;\;\;\frac{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \le -5.5917199964036573 \cdot 10^{-179}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.1612125867005037 \cdot 10^{127}:\\ \;\;\;\;\frac{c}{1} \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}\\ \end{array}\]
\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 -2.7201296316436681 \cdot 10^{98}:\\
\;\;\;\;\frac{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{3 \cdot a}\\

\mathbf{elif}\;b \le -5.5917199964036573 \cdot 10^{-179}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

\mathbf{elif}\;b \le 1.1612125867005037 \cdot 10^{127}:\\
\;\;\;\;\frac{c}{1} \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}\\

\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 <= -2.720129631643668e+98)) {
		VAR = (((1.5 * ((a * c) / b)) - (2.0 * b)) / (3.0 * a));
	} else {
		double VAR_1;
		if ((b <= -5.591719996403657e-179)) {
			VAR_1 = ((-b + (sqrt(sqrt(((b * b) - ((3.0 * a) * c)))) * sqrt(sqrt(((b * b) - ((3.0 * a) * c)))))) / (3.0 * a));
		} else {
			double VAR_2;
			if ((b <= 1.1612125867005037e+127)) {
				VAR_2 = ((c / 1.0) * (1.0 / (-b - sqrt(((b * b) - ((3.0 * a) * c))))));
			} else {
				VAR_2 = (1.0 / ((1.0 / c) * ((1.5 * ((a * c) / b)) - (2.0 * 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 < -2.720129631643668e+98

    1. Initial program 46.3

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

    2. Taylor expanded around -inf 10.0

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

    if -2.720129631643668e+98 < b < -5.591719996403657e-179

    1. Initial program 7.4

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

    3. Applied add-sqr-sqrt7.4

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

    4. Applied sqrt-prod7.6

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

    if -5.591719996403657e-179 < b < 1.1612125867005037e+127

    1. Initial program 30.5

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

    3. Applied flip-+30.7

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

    4. Simplified16.4

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

    6. Applied clear-num16.6

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}\]

    7. Simplified15.6

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

    8. Taylor expanded around 0 10.2

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

    10. Applied add-cube-cbrt10.2

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

    11. Applied times-frac10.0

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

    12. Simplified9.9

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

    13. Simplified9.9

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

    if 1.1612125867005037e+127 < b

    1. Initial program 61.6

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

    3. Applied flip-+61.6

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

    4. Simplified35.4

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

    6. Applied clear-num35.4

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}\]

    7. Simplified34.5

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

    8. Taylor expanded around 0 34.1

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

    9. Taylor expanded around inf 7.4

      \[\leadsto \frac{1}{\frac{1}{c} \cdot \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.7201296316436681 \cdot 10^{98}:\\ \;\;\;\;\frac{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \le -5.5917199964036573 \cdot 10^{-179}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.1612125867005037 \cdot 10^{127}:\\ \;\;\;\;\frac{c}{1} \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))