2-ancestry mixing, zero discriminant

Average Error: 16.0 → 0.8

Time: 3.9s

Precision: binary64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus g

Bits error versus a

Derivation

1. Initial program 16.0

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

3. Applied div-inv 16.0

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

4. Applied cbrt-prod 0.9

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

6. Applied *-un-lft-identity 0.9

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

7. Applied times-frac 0.8

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

8. Applied cbrt-prod 0.8

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

9. Applied associate-*r* 0.8

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

10. Simplified 0.8

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

12. Applied cbrt-unprod 0.8

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

13. Simplified 0.8

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

15. Applied cbrt-div 0.8

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

16. Simplified 0.8

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

17. Final simplification 0.8

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

Reproduce

herbie shell --seed 2020182 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))