2-ancestry mixing, zero discriminant

Average Error: 15.3 → 0.8

Time: 4.1s

Precision: 64

Math

\[\sqrt[3]{\frac{g}{2 \cdot a}}\]

↓

\[\frac{1}{\sqrt[3]{a}} \cdot \sqrt[3]{\frac{g}{2}}\]

C

double code(double g, double a) {
	return cbrt((g / (2.0 * a)));
}

↓

double code(double g, double a) {
	return ((1.0 / cbrt(a)) * cbrt((g / 2.0)));
}

Error

Bits error versus g

Bits error versus a

Derivation

1. Initial program 15.3

   \[\sqrt[3]{\frac{g}{2 \cdot a}}\]
2. Using strategy rm

3. Applied associate-/r* 15.3

   \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{2}}{a}}}\]

4. Applied cbrt-div 0.8

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}}\]
5. Using strategy rm

6. Applied clear-num 0.8

   \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{1}{\frac{2}{g}}}}}{\sqrt[3]{a}}\]

7. Applied cbrt-div 0.9

   \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{2}{g}}}}}{\sqrt[3]{a}}\]

8. Simplified 0.9

   \[\leadsto \frac{\frac{\color{blue}{1}}{\sqrt[3]{\frac{2}{g}}}}{\sqrt[3]{a}}\]
9. Using strategy rm

10. Applied div-inv 0.9

    \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{2}{g}}} \cdot \frac{1}{\sqrt[3]{a}}}\]
11. Using strategy rm

12. Applied clear-num 0.9

    \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\frac{1}{\frac{g}{2}}}}} \cdot \frac{1}{\sqrt[3]{a}}\]

13. Applied cbrt-div 0.8

    \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{g}{2}}}}} \cdot \frac{1}{\sqrt[3]{a}}\]

14. Applied associate-/r/ 0.8

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{1}} \cdot \sqrt[3]{\frac{g}{2}}\right)} \cdot \frac{1}{\sqrt[3]{a}}\]

15. Simplified 0.8

    \[\leadsto \left(\color{blue}{1} \cdot \sqrt[3]{\frac{g}{2}}\right) \cdot \frac{1}{\sqrt[3]{a}}\]

16. Final simplification 0.8

    \[\leadsto \frac{1}{\sqrt[3]{a}} \cdot \sqrt[3]{\frac{g}{2}}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2 a))))