Average Error: 28.6 → 0.4
Time: 3.9s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{-\frac{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{1}{-\frac{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{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(((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 28.6

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

  3. Applied flip-+28.6

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

  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. Using strategy rm

  8. Applied *-commutative0.4

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

  9. Applied associate-/l*0.5

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

  10. Applied associate-/l/0.3

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

  11. Simplified0.3

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

  13. Applied clear-num0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020078 
(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)))