Average Error: 34.5 → 6.9
Time: 5.6s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.95155870856424 \cdot 10^{+129}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5 - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq -2.8405866436241524 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}} \cdot \sqrt{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.9930311221408239 \cdot 10^{+59}:\\ \;\;\;\;\frac{c}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(\sqrt[3]{1.5} \cdot \sqrt[3]{1.5}\right) \cdot \left(c \cdot \left(\sqrt[3]{1.5} \cdot \frac{a}{b}\right)\right) + b \cdot -2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -1.95155870856424 \cdot 10^{+129}:\\
\;\;\;\;\frac{c}{b} \cdot 0.5 - 0.6666666666666666 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \leq -2.8405866436241524 \cdot 10^{-198}:\\
\;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}} \cdot \sqrt{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}} - b}{a \cdot 3}\\

\mathbf{elif}\;b \leq 1.9930311221408239 \cdot 10^{+59}:\\
\;\;\;\;\frac{c}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{\left(\sqrt[3]{1.5} \cdot \sqrt[3]{1.5}\right) \cdot \left(c \cdot \left(\sqrt[3]{1.5} \cdot \frac{a}{b}\right)\right) + b \cdot -2}\\

\end{array}
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (3.0 * a)) * c)))))))) / ((double) (3.0 * a)));
}
↓
double code(double a, double b, double c) {
	double VAR;
	if ((b <= -1.95155870856424e+129)) {
		VAR = ((double) (((double) ((c / b) * 0.5)) - ((double) (0.6666666666666666 * (b / a)))));
	} else {
		double VAR_1;
		if ((b <= -2.8405866436241524e-198)) {
			VAR_1 = (((double) (((double) (((double) sqrt(((double) sqrt(((double) (((double) (b * b)) - ((double) (3.0 * ((double) (c * a)))))))))) * ((double) sqrt(((double) sqrt(((double) (((double) (b * b)) - ((double) (3.0 * ((double) (c * a)))))))))))) - b)) / ((double) (a * 3.0)));
		} else {
			double VAR_2;
			if ((b <= 1.9930311221408239e+59)) {
				VAR_2 = (c / ((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (3.0 * ((double) (c * a)))))))))));
			} else {
				VAR_2 = (c / ((double) (((double) (((double) (((double) cbrt(1.5)) * ((double) cbrt(1.5)))) * ((double) (c * ((double) (((double) cbrt(1.5)) * (a / b))))))) + ((double) (b * -2.0)))));
			}
			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 < -1.95155870856424e129

    1. Initial program 56.2

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

    2. Taylor expanded around -inf 3.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}}\]

    3. Simplified3.0

      \[\leadsto \color{blue}{\frac{c}{b} \cdot 0.5 - 0.6666666666666666 \cdot \frac{b}{a}}\]

    if -1.95155870856424e129 < b < -2.8405866436241524e-198

    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

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

    5. Simplified7.6

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

    6. Simplified7.6

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

    if -2.8405866436241524e-198 < b < 1.99303112214082389e59

    1. Initial program 27.1

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

    3. Applied flip-+27.2

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

    4. Simplified16.9

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

    5. Simplified16.8

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

    7. Applied *-un-lft-identity16.8

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

    8. Applied times-frac16.9

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

    9. Applied times-frac16.8

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

    10. Simplified16.8

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

    11. Simplified10.4

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

    if 1.99303112214082389e59 < b

    1. Initial program 57.9

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

    3. Applied flip-+57.9

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

    4. Simplified30.5

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

    5. Simplified30.5

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

    7. Applied *-un-lft-identity30.5

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

    8. Applied times-frac30.5

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

    9. Applied times-frac30.5

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

    10. Simplified30.5

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

    11. Simplified27.1

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

    12. Taylor expanded around inf 8.0

      \[\leadsto 1 \cdot \left(1 \cdot \frac{c}{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\right)\]

    13. Simplified3.9

      \[\leadsto 1 \cdot \left(1 \cdot \frac{c}{\color{blue}{1.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}}\right)\]
    14. Using strategy rm

    15. Applied add-cube-cbrt3.9

      \[\leadsto 1 \cdot \left(1 \cdot \frac{c}{\color{blue}{\left(\left(\sqrt[3]{1.5} \cdot \sqrt[3]{1.5}\right) \cdot \sqrt[3]{1.5}\right)} \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}\right)\]

    16. Applied associate-*l*3.9

      \[\leadsto 1 \cdot \left(1 \cdot \frac{c}{\color{blue}{\left(\sqrt[3]{1.5} \cdot \sqrt[3]{1.5}\right) \cdot \left(\sqrt[3]{1.5} \cdot \left(c \cdot \frac{a}{b}\right)\right)} + b \cdot -2}\right)\]

    17. Simplified3.9

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95155870856424 \cdot 10^{+129}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5 - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq -2.8405866436241524 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}} \cdot \sqrt{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.9930311221408239 \cdot 10^{+59}:\\ \;\;\;\;\frac{c}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(\sqrt[3]{1.5} \cdot \sqrt[3]{1.5}\right) \cdot \left(c \cdot \left(\sqrt[3]{1.5} \cdot \frac{a}{b}\right)\right) + b \cdot -2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))