Average Error: 43.8 → 11.2
Time: 4.0s
Precision: binary64
\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq 9.967146805285853 \cdot 10^{-06}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot -1.5\right) \cdot \frac{c}{b}}{3 \cdot a}\\ \end{array}\]
\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 9.967146805285853 \cdot 10^{-06}:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(a \cdot -1.5\right) \cdot \frac{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 9.967146805285853e-06)
   (/
    (/
     (- (- (* b b) (* (* 3.0 a) c)) (* b b))
     (+ b (sqrt (- (* b b) (* (* 3.0 a) c)))))
    (* 3.0 a))
   (/ (* (* a -1.5) (/ 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 <= 9.967146805285853e-06) {
		tmp = ((((b * b) - ((3.0 * a) * c)) - (b * b)) / (b + sqrt((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = ((a * -1.5) * (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 2 regimes

  2. if b < 9.96714680528585339e-6

    1. Initial program 16.0

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

    2. Simplified16.0

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

    4. Applied flip--_binary64_241116.1

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

    5. Simplified15.1

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

    6. Simplified15.1

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

    if 9.96714680528585339e-6 < b

    1. Initial program 45.3

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

    2. Simplified45.3

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

    3. Taylor expanded around inf 11.2

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity_binary64_243611.2

      \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{\color{blue}{1 \cdot b}}}{3 \cdot a}\]

    6. Applied times-frac_binary64_244211.2

      \[\leadsto \frac{-1.5 \cdot \color{blue}{\left(\frac{a}{1} \cdot \frac{c}{b}\right)}}{3 \cdot a}\]

    7. Applied associate-*r*_binary64_237811.0

      \[\leadsto \frac{\color{blue}{\left(-1.5 \cdot \frac{a}{1}\right) \cdot \frac{c}{b}}}{3 \cdot a}\]

    8. Simplified11.0

      \[\leadsto \frac{\color{blue}{\left(-1.5 \cdot a\right)} \cdot \frac{c}{b}}{3 \cdot a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.967146805285853 \cdot 10^{-06}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot -1.5\right) \cdot \frac{c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020280 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))