Cubic critical, medium range

Average Error: 43.9 → 0.2

Time: 4.8s

Precision: binary64

\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]

Math

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

↓

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

C

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 (c / ((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (3.0 * ((double) (c * a)))))))))));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

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. 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. Simplified 0.5

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

7. Applied *-un-lft-identity 0.5

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

8. Applied times-frac 0.5

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

9. Applied associate-/l* 0.5

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

10. Simplified 0.4

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

12. Applied associate-*r/ 0.4

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

13. Applied associate-*r/ 0.4

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

14. Applied associate-/r/ 0.4

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

15. Simplified 0.3

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

17. Applied associate-*l/ 0.2

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

18. Simplified 0.2

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

19. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020196 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))