Average Error: 44.2 → 0.4
Time: 4.3s
Precision: binary64
\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{3}{b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \frac{-c}{3}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{3}{b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \frac{-c}{3}
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (3.0 * a)) * c)))))))) / ((double) (3.0 * a)));
}
↓
double code(double a, double b, double c) {
	return ((double) ((3.0 / ((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (3.0 * ((double) (a * c))))))))))) * (((double) -(c)) / 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 44.2

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

  2. Simplified44.2

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

  4. Applied flip--44.2

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

  5. Simplified0.5

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

  6. Simplified0.5

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

  8. Applied *-un-lft-identity0.5

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

  9. Applied times-frac0.5

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

  10. Applied associate-/l*0.5

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

  11. Simplified0.4

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

  13. Applied div-inv0.4

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

  14. Applied times-frac0.5

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2020196 
(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)))