Average Error: 34.8 → 7.3
Time: 6.6s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -6.738042698067906 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq -1.3703763978612727 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.677485094739758 \cdot 10^{+106}:\\ \;\;\;\;\frac{-c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]
\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 -6.738042698067906 \cdot 10^{+47}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \leq -1.3703763978612727 \cdot 10^{-221}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\

\mathbf{elif}\;b \leq 4.677485094739758 \cdot 10^{+106}:\\
\;\;\;\;\frac{-c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}\\

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

\end{array}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b -6.738042698067906e+47)
   (- (* 0.5 (/ c b)) (* 0.6666666666666666 (/ b a)))
   (if (<= b -1.3703763978612727e-221)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (if (<= b 4.677485094739758e+106)
       (/ (- c) (+ b (sqrt (- (* b b) (* c (* a 3.0))))))
       (* (/ c b) -0.5)))))
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 tmp;
	if (b <= -6.738042698067906e+47) {
		tmp = (0.5 * (c / b)) - (0.6666666666666666 * (b / a));
	} else if (b <= -1.3703763978612727e-221) {
		tmp = (sqrt((b * b) - (c * (a * 3.0))) - b) / (a * 3.0);
	} else if (b <= 4.677485094739758e+106) {
		tmp = -c / (b + sqrt((b * b) - (c * (a * 3.0))));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

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 < -6.73804269806790567e47

    1. Initial program 39.2

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

    2. Simplified39.2

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

    3. Taylor expanded around -inf 6.3

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

    if -6.73804269806790567e47 < b < -1.37037639786127275e-221

    1. Initial program 8.1

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

    if -1.37037639786127275e-221 < b < 4.6774850947397581e106

    1. Initial program 30.7

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

    2. Simplified30.7

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

    4. Applied flip--_binary64_141730.8

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

    5. Simplified17.3

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

    6. Simplified17.3

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

    8. Applied *-un-lft-identity_binary64_144217.3

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

    9. Applied times-frac_binary64_144817.3

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

    10. Simplified17.3

      \[\leadsto \color{blue}{0.3333333333333333} \cdot \frac{\frac{a \cdot \left(c \cdot -3\right)}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}\]

    11. Simplified10.6

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(1 \cdot \frac{c \cdot -3}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right)}\]
    12. Using strategy rm

    13. Applied pow1_binary64_150310.6

      \[\leadsto 0.3333333333333333 \cdot \left(1 \cdot \color{blue}{{\left(\frac{c \cdot -3}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right)}^{1}}\right)\]

    14. Applied pow1_binary64_150310.6

      \[\leadsto 0.3333333333333333 \cdot \left(\color{blue}{{1}^{1}} \cdot {\left(\frac{c \cdot -3}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right)}^{1}\right)\]

    15. Applied pow-prod-down_binary64_151310.6

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left(1 \cdot \frac{c \cdot -3}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right)}^{1}}\]

    16. Applied pow1_binary64_150310.6

      \[\leadsto \color{blue}{{0.3333333333333333}^{1}} \cdot {\left(1 \cdot \frac{c \cdot -3}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right)}^{1}\]

    17. Applied pow-prod-down_binary64_151310.6

      \[\leadsto \color{blue}{{\left(0.3333333333333333 \cdot \left(1 \cdot \frac{c \cdot -3}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right)\right)}^{1}}\]

    18. Simplified10.3

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

    if 4.6774850947397581e106 < 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. Simplified60.1

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

    3. Taylor expanded around inf 2.8

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.738042698067906 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq -1.3703763978612727 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.677485094739758 \cdot 10^{+106}:\\ \;\;\;\;\frac{-c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2020352 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))