Cubic critical, wide range

Average Error: 52.6 → 0.1

Time: 4.1s

Precision: 64

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

Math

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

↓

\[\frac{c}{1 \cdot \left(\left(-b\right) - \sqrt{\frac{\left(-\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) + {b}^{4}}{b \cdot b + \left(3 \cdot a\right) \cdot c}}\right)}\]

C

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 (c / (1.0 * (-b - sqrt(((-(((3.0 * a) * c) * ((3.0 * a) * c)) + pow(b, 4.0)) / ((b * b) + ((3.0 * a) * c)))))));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.6

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

3. Applied flip-+ 52.6

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

4. Simplified 0.5

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

6. Applied clear-num 0.6

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

7. Simplified 0.3

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

9. Applied clear-num 0.2

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

10. Applied associate-/r/ 0.1

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

11. Simplified 0.1

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

13. Applied flip-- 0.1

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

14. Simplified 0.1

    \[\leadsto 1 \cdot \frac{c}{1 \cdot \left(\left(-b\right) - \sqrt{\frac{\color{blue}{\left(-\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) + {b}^{4}}}{b \cdot b + \left(3 \cdot a\right) \cdot c}}\right)}\]

15. Final simplification 0.1

    \[\leadsto \frac{c}{1 \cdot \left(\left(-b\right) - \sqrt{\frac{\left(-\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) + {b}^{4}}{b \cdot b + \left(3 \cdot a\right) \cdot c}}\right)}\]

Reproduce

herbie shell --seed 2020071 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))