Average Error: 33.8 → 9.1
Time: 7.3s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.175532362255663 \cdot 10^{+108}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 5.378674763733228 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} - \frac{b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 8215236509114848:\\ \;\;\;\;\frac{a \cdot \left(c \cdot -3\right)}{b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.175532362255663 \cdot 10^{+108}:\\
\;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\

\mathbf{elif}\;b \leq 5.378674763733228 \cdot 10^{-110}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} - \frac{b}{3 \cdot a}\\

\mathbf{elif}\;b \leq 8215236509114848:\\
\;\;\;\;\frac{a \cdot \left(c \cdot -3\right)}{b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \frac{0.3333333333333333}{a}\\

\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 -1.175532362255663e+108)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 5.378674763733228e-110)
     (- (/ (sqrt (- (* b b) (* (* 3.0 a) c))) (* 3.0 a)) (/ b (* 3.0 a)))
     (if (<= b 8215236509114848.0)
       (*
        (/ (* a (* c -3.0)) (+ b (sqrt (- (* b b) (* 3.0 (* a c))))))
        (/ 0.3333333333333333 a))
       (* -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 <= -1.175532362255663e+108) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 5.378674763733228e-110) {
		tmp = (sqrt((b * b) - ((3.0 * a) * c)) / (3.0 * a)) - (b / (3.0 * a));
	} else if (b <= 8215236509114848.0) {
		tmp = ((a * (c * -3.0)) / (b + sqrt((b * b) - (3.0 * (a * c))))) * (0.3333333333333333 / a);
	} 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 < -1.17553236225566309e108

    1. Initial program 47.7

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

    2. Simplified47.7

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

    3. Taylor expanded around -inf 4.2

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

    if -1.17553236225566309e108 < b < 5.37867476373322791e-110

    1. Initial program 11.9

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

    2. Simplified11.9

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

    4. Applied div-sub_binary6411.9

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

    if 5.37867476373322791e-110 < b < 8215236509114848

    1. Initial program 36.3

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

    2. Simplified36.3

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

    4. Applied div-inv_binary6436.3

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

    5. Simplified36.4

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

    7. Applied flip--_binary6436.4

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

    8. Simplified15.8

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

    9. Simplified15.8

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

    if 8215236509114848 < b

    1. Initial program 56.6

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

    2. Simplified56.6

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

    3. Taylor expanded around inf 5.3

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.175532362255663 \cdot 10^{+108}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 5.378674763733228 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} - \frac{b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 8215236509114848:\\ \;\;\;\;\frac{a \cdot \left(c \cdot -3\right)}{b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))