Average Error: 35.0 → 7.0
Time: 4.4s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -6.881450868076512 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq -2.6232763968795353 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.9927405752514035 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{\left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right) \cdot \frac{-1}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]
\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 -6.881450868076512 \cdot 10^{+99}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \leq -2.6232763968795353 \cdot 10^{-303}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\

\mathbf{elif}\;b \leq 2.9927405752514035 \cdot 10^{+106}:\\
\;\;\;\;\frac{1}{\left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right) \cdot \frac{-1}{c}}\\

\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 -6.881450868076512e+99)
   (- (* 0.5 (/ c b)) (* 0.6666666666666666 (/ b a)))
   (if (<= b -2.6232763968795353e-303)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (if (<= b 2.9927405752514035e+106)
       (/ 1.0 (* (+ b (sqrt (- (* b b) (* c (* a 3.0))))) (/ -1.0 c)))
       (* (/ 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 <= -6.881450868076512e+99) {
		tmp = (0.5 * (c / b)) - (0.6666666666666666 * (b / a));
	} else if (b <= -2.6232763968795353e-303) {
		tmp = (sqrt((b * b) - (c * (a * 3.0))) - b) / (a * 3.0);
	} else if (b <= 2.9927405752514035e+106) {
		tmp = 1.0 / ((b + sqrt((b * b) - (c * (a * 3.0)))) * (-1.0 / c));
	} 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 < -6.88145086807651196e99

    1. Initial program 47.6

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

    2. Simplified47.6

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

    3. Taylor expanded around -inf 4.6

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

    if -6.88145086807651196e99 < b < -2.62327639687953528e-303

    1. Initial program 9.0

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

    2. Simplified9.0

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

    if -2.62327639687953528e-303 < b < 2.9927405752514035e106

    1. Initial program 34.0

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

    2. Simplified34.0

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

    4. Applied flip--_binary6434.0

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

    5. Simplified17.2

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

    6. Simplified17.2

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

    8. Applied clear-num_binary6417.3

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

    9. Simplified9.9

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

    if 2.9927405752514035e106 < 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 simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.881450868076512 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq -2.6232763968795353 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.9927405752514035 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{\left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right) \cdot \frac{-1}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))