Average Error: 34.0 → 6.4
Time: 6.1s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.7674460872703884 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq -1.2677949885504426 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.432315266375116 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{a}{a} \cdot \left(-c\right)}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]
\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 -1.7674460872703884 \cdot 10^{+149}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \leq -1.2677949885504426 \cdot 10^{-305}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\

\mathbf{elif}\;b \leq 3.432315266375116 \cdot 10^{+113}:\\
\;\;\;\;\frac{\frac{a}{a} \cdot \left(-c\right)}{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 -1.7674460872703884e+149)
   (- (* 0.5 (/ c b)) (* 0.6666666666666666 (/ b a)))
   (if (<= b -1.2677949885504426e-305)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (if (<= b 3.432315266375116e+113)
       (/ (* (/ a a) (- 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 <= -1.7674460872703884e+149) {
		tmp = (0.5 * (c / b)) - (0.6666666666666666 * (b / a));
	} else if (b <= -1.2677949885504426e-305) {
		tmp = (sqrt((b * b) - (c * (a * 3.0))) - b) / (a * 3.0);
	} else if (b <= 3.432315266375116e+113) {
		tmp = ((a / a) * -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 < -1.76744608727038835e149

    1. Initial program 62.3

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

    2. Simplified62.3

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

    3. Taylor expanded around -inf 2.2

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

    if -1.76744608727038835e149 < b < -1.2677949885504426e-305

    1. Initial program 9.4

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

    if -1.2677949885504426e-305 < b < 3.4323152663751158e113

    1. Initial program 32.1

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

    2. Simplified32.1

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

    4. Applied div-inv_binary64_212132.1

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

    5. Simplified32.2

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

    7. Applied flip--_binary64_209932.2

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

    8. Simplified15.7

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

    9. Simplified15.7

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

    11. Applied associate-*l/_binary64_206714.5

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

    12. Simplified7.7

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

    if 3.4323152663751158e113 < b

    1. Initial program 60.3

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

    2. Simplified60.3

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

    3. Taylor expanded around inf 2.1

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7674460872703884 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq -1.2677949885504426 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.432315266375116 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{a}{a} \cdot \left(-c\right)}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))