Average Error: 43.6 → 0.4
Time: 4.2s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{3}{-\left(\left(-b\right) - \sqrt{\frac{\left(-\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) + {b}^{4}}{b \cdot b + \left(3 \cdot a\right) \cdot c}}\right)} \cdot \frac{-c}{3}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{3}{-\left(\left(-b\right) - \sqrt{\frac{\left(-\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) + {b}^{4}}{b \cdot b + \left(3 \cdot a\right) \cdot c}}\right)} \cdot \frac{-c}{3}
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 / -(-b - sqrt(((-(((3.0 * a) * c) * ((3.0 * a) * c)) + pow(b, 4.0)) / ((b * b) + ((3.0 * a) * c)))))) * (-c / 3.0));
}

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 43.6

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

  3. Applied flip-+43.6

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

  4. Simplified0.5

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

  6. Applied associate-*r*0.4

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

  7. Applied associate-/l*0.4

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

  9. Applied *-commutative0.4

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

  10. Applied frac-2neg0.4

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

  11. Applied associate-/r/0.4

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

  12. Applied times-frac0.5

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

  13. Simplified0.4

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

  15. Applied flip--0.4

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

  16. Simplified0.4

    \[\leadsto \frac{3}{-\left(\left(-b\right) - \sqrt{\frac{\color{blue}{\left(-\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) + {b}^{4}}}{b \cdot b + \left(3 \cdot a\right) \cdot c}}\right)} \cdot \frac{-c}{3}\]

  17. Final simplification0.4

    \[\leadsto \frac{3}{-\left(\left(-b\right) - \sqrt{\frac{\left(-\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) + {b}^{4}}{b \cdot b + \left(3 \cdot a\right) \cdot c}}\right)} \cdot \frac{-c}{3}\]

Reproduce

herbie shell --seed 2020071 
(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)))