2-ancestry mixing, zero discriminant

Average Error: 16.0 → 0.9

Time: 7.4s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\sqrt[3]{g} \cdot \left(\sqrt[3]{\sqrt{0.5}} \cdot \sqrt[3]{\frac{\sqrt{0.5}}{a}}\right)\]

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)) * ((double) (((double) cbrt(((double) sqrt(0.5)))) * ((double) cbrt((((double) sqrt(0.5)) / 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. Taylor expanded around 0 0.8

   \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\color{blue}{\frac{0.5}{a}}}\]
6. Using strategy rm

7. Applied *-un-lft-identity 0.8

   \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\frac{0.5}{\color{blue}{1 \cdot a}}}\]

8. Applied add-sqr-sqrt 0.8

   \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\frac{\color{blue}{\sqrt{0.5} \cdot \sqrt{0.5}}}{1 \cdot a}}\]

9. Applied times-frac 0.8

   \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\color{blue}{\frac{\sqrt{0.5}}{1} \cdot \frac{\sqrt{0.5}}{a}}}\]

10. Applied cbrt-prod 0.9

    \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\left(\sqrt[3]{\frac{\sqrt{0.5}}{1}} \cdot \sqrt[3]{\frac{\sqrt{0.5}}{a}}\right)}\]

11. Simplified 0.9

    \[\leadsto \sqrt[3]{g} \cdot \left(\color{blue}{\sqrt[3]{\sqrt{0.5}}} \cdot \sqrt[3]{\frac{\sqrt{0.5}}{a}}\right)\]

12. Final simplification 0.9

    \[\leadsto \sqrt[3]{g} \cdot \left(\sqrt[3]{\sqrt{0.5}} \cdot \sqrt[3]{\frac{\sqrt{0.5}}{a}}\right)\]

Reproduce

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