Average Error: 28.8 → 0.4
Time: 4.0s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{\left(-b\right) - \sqrt{\frac{\left(-\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) + {b}^{4}}{b \cdot b + \left(3 \cdot a\right) \cdot c}}} \cdot c\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{1}{\left(-b\right) - \sqrt{\frac{\left(-\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) + {b}^{4}}{b \cdot b + \left(3 \cdot a\right) \cdot c}}} \cdot c
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 ((1.0 / (-b - sqrt(((-(((3.0 * a) * c) * ((3.0 * a) * c)) + pow(b, 4.0)) / ((b * b) + ((3.0 * a) * c)))))) * c);
}

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 28.8

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

  3. Applied flip-+28.9

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

  4. Simplified0.6

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

  6. Applied associate-*r*0.4

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

  7. Applied associate-/l*0.5

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

  8. Applied associate-/l/0.5

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

  10. Applied associate-*r/0.5

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

  11. Applied associate-/r/0.5

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

  12. Simplified0.4

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

  14. Applied flip--0.4

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

  15. Simplified0.4

    \[\leadsto \frac{1}{\left(-b\right) - \sqrt{\frac{\color{blue}{\left(-\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) + {b}^{4}}}{b \cdot b + \left(3 \cdot a\right) \cdot c}}} \cdot c\]

  16. Final simplification0.4

    \[\leadsto \frac{1}{\left(-b\right) - \sqrt{\frac{\left(-\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) + {b}^{4}}{b \cdot b + \left(3 \cdot a\right) \cdot c}}} \cdot c\]

Reproduce

herbie shell --seed 2020071 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))