Average Error: 43.9 → 0.5
Time: 5.3s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left({b}^{2} - {b}^{2}\right) \cdot \left({b}^{2} - {b}^{2}\right) - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)}{\left(3 \cdot a\right) \cdot \left(\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(-\left(3 \cdot a\right) \cdot c\right)\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\left({b}^{2} - {b}^{2}\right) \cdot \left({b}^{2} - {b}^{2}\right) - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)}{\left(3 \cdot a\right) \cdot \left(\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(-\left(3 \cdot a\right) \cdot c\right)\right)}
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 ((((pow(b, 2.0) - pow(b, 2.0)) * (pow(b, 2.0) - pow(b, 2.0))) - (((3.0 * a) * c) * ((3.0 * a) * c))) / ((3.0 * a) * ((-b - sqrt(((b * b) - ((3.0 * a) * c)))) * -((3.0 * 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 43.9

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

  3. Applied flip-+43.9

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

  4. Simplified0.5

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

  6. Applied associate-*r*0.4

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

  8. Applied flip-+0.4

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

  9. Applied associate-/l/0.5

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

  10. Simplified0.5

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

  12. Applied div-inv0.6

    \[\leadsto \frac{\color{blue}{\left(\left({b}^{2} - {b}^{2}\right) \cdot \left({b}^{2} - {b}^{2}\right) - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) \cdot \frac{1}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(-\left(3 \cdot a\right) \cdot c\right)}}}{3 \cdot a}\]

  13. Applied associate-/l*0.6

    \[\leadsto \color{blue}{\frac{\left({b}^{2} - {b}^{2}\right) \cdot \left({b}^{2} - {b}^{2}\right) - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)}{\frac{3 \cdot a}{\frac{1}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(-\left(3 \cdot a\right) \cdot c\right)}}}}\]

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2020075 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))