Average Error: 34.7 → 15.2
Time: 17.4s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -2.1711788034215313 \cdot 10^{+152}:\\ \;\;\;\;\frac{\left(-b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 12377142785381904000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array} \]
\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 -2.1711788034215313 \cdot 10^{+152}:\\
\;\;\;\;\frac{\left(-b\right) - b}{3 \cdot a}\\

\mathbf{elif}\;b \leq 12377142785381904000:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\


\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 -2.1711788034215313e+152)
   (/ (- (- b) b) (* 3.0 a))
   (if (<= b 12377142785381904000.0)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (/ (* -1.5 (/ (* a c) b)) (* 3.0 a)))))
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 <= -2.1711788034215313e+152) {
		tmp = (-b - b) / (3.0 * a);
	} else if (b <= 12377142785381904000.0) {
		tmp = (sqrt((b * b) - ((3.0 * a) * c)) - b) / (3.0 * a);
	} else {
		tmp = (-1.5 * ((a * c) / b)) / (3.0 * a);
	}
	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 3 regimes

  2. if b < -2.1711788034215313e152

    1. Initial program 63.2

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

    2. Taylor expanded in b around -inf 2.9

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

    3. Simplified2.9

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

    if -2.1711788034215313e152 < b < 12377142785381904000

    1. Initial program 17.1

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

    2. Applied neg-sub0_binary6417.1

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

    3. Applied associate-+l-_binary6417.1

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

    if 12377142785381904000 < b

    1. Initial program 56.0

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

    2. Taylor expanded in b around inf 16.1

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{c \cdot a}{b}}}{3 \cdot a} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1711788034215313 \cdot 10^{+152}:\\ \;\;\;\;\frac{\left(-b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 12377142785381904000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))