Average Error: 33.6 → 6.7
Time: 8.3s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -2.991777674964526 \cdot 10^{+115}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -2.3801565570057447 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 5.197649067580201 \cdot 10^{+52}:\\ \;\;\;\;c \cdot \frac{-1}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -2.991777674964526 \cdot 10^{+115}:\\
\;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\

\mathbf{elif}\;b \leq -2.3801565570057447 \cdot 10^{-178}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\

\mathbf{elif}\;b \leq 5.197649067580201 \cdot 10^{+52}:\\
\;\;\;\;c \cdot \frac{-1}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\

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

\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 -2.991777674964526e+115)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b -2.3801565570057447e-178)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
     (if (<= b 5.197649067580201e+52)
       (* c (/ -1.0 (+ b (sqrt (- (* b b) (* (* 3.0 a) c))))))
       (* -0.5 (/ c b))))))
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 <= -2.991777674964526e+115) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -2.3801565570057447e-178) {
		tmp = (sqrt((b * b) - (3.0 * (a * c))) - b) / (3.0 * a);
	} else if (b <= 5.197649067580201e+52) {
		tmp = c * (-1.0 / (b + sqrt((b * b) - ((3.0 * a) * c))));
	} else {
		tmp = -0.5 * (c / b);
	}
	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 < -2.99177767496452593e115

    1. Initial program 50.4

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

    2. Simplified50.4

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

    3. Taylor expanded around -inf 4.1

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

    4. Simplified4.1

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

    if -2.99177767496452593e115 < b < -2.3801565570057447e-178

    1. Initial program 6.5

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

    2. Simplified6.5

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

    4. Applied associate-*l*_binary64_51346.5

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

    if -2.3801565570057447e-178 < b < 5.19764906758020092e52

    1. Initial program 25.9

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

    2. Simplified25.9

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

    4. Applied flip--_binary64_516826.2

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

    5. Simplified16.3

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

    6. Simplified16.3

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

    8. Applied clear-num_binary64_519216.4

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

    9. Simplified10.7

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

    11. Applied associate-*r/_binary64_513510.6

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

    12. Applied associate-/r/_binary64_513910.5

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

    13. Simplified10.5

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

    if 5.19764906758020092e52 < 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. Simplified57.9

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

    4. Applied flip--_binary64_516857.9

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

    5. Simplified29.0

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

    6. Simplified29.0

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

    7. Taylor expanded around 0 3.7

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.991777674964526 \cdot 10^{+115}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -2.3801565570057447 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 5.197649067580201 \cdot 10^{+52}:\\ \;\;\;\;c \cdot \frac{-1}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020357 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))