2-ancestry mixing, zero discriminant

Average Error: 15.3 → 1.1

Time: 3.9s

Precision: 64

Math

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

↓

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

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(-1.0)) * (cbrt(-2.0) / cbrt(g))));
}

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 frac-2neg 15.3

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

4. Applied cbrt-div 0.8

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

6. Applied clear-num 0.8

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

8. Applied neg-mul-1 0.8

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

9. Applied cbrt-prod 0.8

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

10. Applied *-commutative 0.8

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

11. Applied distribute-rgt-neg-in 0.8

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

12. Applied cbrt-prod 1.1

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

13. Applied times-frac 1.1

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

14. Final simplification 1.1

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

Reproduce

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