Average Error: 28.6 → 0.4
Time: 4.7s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[c \cdot \frac{-1}{b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
c \cdot \frac{-1}{b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}
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) (c * (-1.0 / ((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (3.0 * ((double) (a * 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. Simplified28.6

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

  4. Applied flip--28.6

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

  5. Simplified0.6

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

  6. Simplified0.6

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

  8. Applied *-un-lft-identity0.6

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

  9. Applied times-frac0.6

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

  10. Applied associate-/l*0.6

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

  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 add-sqr-sqrt0.4

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

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

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

  15. Applied times-frac0.4

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

  16. Applied times-frac0.4

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

  17. Simplified0.4

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

  18. Simplified0.4

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

  19. Final simplification0.4

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

Reproduce

herbie shell --seed 2020196 
(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) (* (* 3.0 a) c)))) (* 3.0 a)))