Average Error: 43.9 → 0.3
Time: 9.1s
Precision: binary64
\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[3 \cdot \frac{\frac{c}{b + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}}{-3}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
3 \cdot \frac{\frac{c}{b + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}}{-3}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (* 3.0 (/ (/ c (+ b (sqrt (- (* b b) (* c (* 3.0 a)))))) -3.0)))
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) {
	return 3.0 * ((c / (b + sqrt((b * b) - (c * (3.0 * a))))) / -3.0);
}

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. Initial program 43.9

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

  2. Simplified43.9

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

  4. Applied flip--_binary6443.9

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

  5. Simplified43.3

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

  6. Simplified43.3

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

  8. Applied frac-2neg_binary6443.3

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  12. Applied *-un-lft-identity_binary640.4

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

  13. Applied times-frac_binary640.2

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

  14. Applied times-frac_binary640.4

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

  15. Simplified0.3

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

  16. Simplified0.3

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2020268 
(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)))