Average Error: 28.1 → 0.5
Time: 6.3s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{4}}{b \cdot b + \left(4 \cdot a\right) \cdot c} - \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{4}}{b \cdot b + \left(4 \cdot a\right) \cdot c} - \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}\right)}
double code(double a, double b, double c) {
	return ((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a));
}
↓
double code(double a, double b, double c) {
	return ((1.0 / 2.0) * ((4.0 * (a * c)) / (a * (-b - sqrt(((pow(b, 4.0) / ((b * b) + ((4.0 * a) * c))) - (((4.0 * (a * c)) * (4.0 * (a * c))) / ((b * b) + ((4.0 * 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 28.1

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
  2. Using strategy rm

  3. Applied flip-+28.1

    \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

  4. Simplified0.5

    \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
  5. Using strategy rm

  6. Applied flip--0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}\]

  7. Simplified0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{\color{blue}{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]
  8. Using strategy rm

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

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}\right)}}}{2 \cdot a}\]

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

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}\right)}}{2 \cdot a}\]

  11. Applied times-frac0.5

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}\]

  12. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{2} \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}{a}}\]

  13. Simplified0.5

    \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}{a}\]

  14. Simplified0.5

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{4 \cdot \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}\right)}}\]
  15. Using strategy rm

  16. Applied div-sub0.5

    \[\leadsto \frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{\color{blue}{\frac{{b}^{4}}{b \cdot b + \left(4 \cdot a\right) \cdot c} - \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}}\right)}\]

  17. Final simplification0.5

    \[\leadsto \frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{4}}{b \cdot b + \left(4 \cdot a\right) \cdot c} - \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}\right)}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))