Average Error: 28.6 → 0.3
Time: 5.4s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{a}{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \frac{\frac{\frac{c}{\sqrt[3]{1}}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{a}{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \frac{\frac{\frac{c}{\sqrt[3]{1}}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}
double code(double a, double b, double c) {
	return ((double) (((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) (((double) (a / ((double) (((double) cbrt(1.0)) * ((double) cbrt(1.0)))))) * ((double) (((double) (((double) (c / ((double) cbrt(1.0)))) / ((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (3.0 * a)) * c)))))))))) / 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 28.6

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

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

  6. Applied associate-/r*0.6

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

  7. Simplified0.5

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

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

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

  10. Applied associate-/r*0.5

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

  11. Simplified0.4

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

  13. Applied *-un-lft-identity0.4

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

  14. Applied *-un-lft-identity0.4

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

  15. Applied add-cube-cbrt0.4

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

  16. Applied times-frac0.4

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

  17. Applied times-frac0.3

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

  18. Applied times-frac0.3

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

  19. Simplified0.3

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

  20. Final simplification0.3

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

Reproduce

herbie shell --seed 2020121 
(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)))