Average Error: 28.7 → 0.3
Time: 5.8s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\left(3 \cdot a\right) \cdot \frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\left(3 \cdot a\right) \cdot \frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot 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) {
	return ((3.0 * a) * ((c / (-b - sqrt(((b * b) - ((3.0 * a) * c))))) / (3.0 * a)));
}

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 28.7

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

  3. Applied flip-+28.7

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

  4. Simplified0.6

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

  6. Applied frac-2neg0.6

    \[\leadsto \color{blue}{\frac{-\frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{-3 \cdot a}}\]

  7. Simplified0.5

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

  9. Applied neg-mul-10.5

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

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

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

  11. Applied distribute-lft-neg-in0.5

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

  12. Applied times-frac0.3

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

  13. Applied times-frac0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2020091 
(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)))