Average Error: 15.8 → 0.9
Time: 3.5s
Precision: binary64
\[\sqrt[3]{\frac{g}{2 \cdot a}} \]
↓
\[\left(\sqrt[3]{g} \cdot \sqrt[3]{{0.25}^{0.3333333333333333}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{0.5}\right)}^{0.6666666666666666}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{0.5}}}{a}}\right) \]
\sqrt[3]{\frac{g}{2 \cdot a}}
↓
\left(\sqrt[3]{g} \cdot \sqrt[3]{{0.25}^{0.3333333333333333}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{0.5}\right)}^{0.6666666666666666}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{0.5}}}{a}}\right)

(FPCore (g a) :precision binary64 (cbrt (/ g (* 2.0 a))))
↓
(FPCore (g a)
 :precision binary64
 (*
  (* (cbrt g) (cbrt (pow 0.25 0.3333333333333333)))
  (*
   (cbrt (pow (cbrt 0.5) 0.6666666666666666))
   (cbrt (/ (cbrt (cbrt 0.5)) a)))))
double code(double g, double a) {
	return cbrt(g / (2.0 * a));
}
↓
double code(double g, double a) {
	return (cbrt(g) * cbrt(pow(0.25, 0.3333333333333333))) * (cbrt(pow(cbrt(0.5), 0.6666666666666666)) * cbrt(cbrt(cbrt(0.5)) / a));
}

Error

Bits error versus g

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.8

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

  2. Applied div-inv_binary6415.8

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

  3. Applied cbrt-prod_binary640.9

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

  4. Simplified0.8

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

  5. Applied *-un-lft-identity_binary640.8

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

  6. Applied add-cube-cbrt_binary640.8

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

  7. Applied times-frac_binary640.8

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

  8. Applied cbrt-prod_binary640.9

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

  9. Applied associate-*r*_binary640.9

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

  10. Simplified0.9

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

  11. Applied *-un-lft-identity_binary640.9

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

  12. Applied add-cube-cbrt_binary640.9

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

  13. Applied times-frac_binary640.9

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

  14. Applied cbrt-prod_binary640.9

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

  15. Simplified0.9

    \[\leadsto \left(\sqrt[3]{g} \cdot \sqrt[3]{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}\right) \cdot \left(\color{blue}{\sqrt[3]{{\left(\sqrt[3]{0.5}\right)}^{0.6666666666666666}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{0.5}}}{a}}\right) \]

  16. Applied pow1/3_binary640.9

    \[\leadsto \left(\sqrt[3]{g} \cdot \sqrt[3]{\sqrt[3]{0.5} \cdot \color{blue}{{0.5}^{0.3333333333333333}}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{0.5}\right)}^{0.6666666666666666}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{0.5}}}{a}}\right) \]

  17. Applied pow1/3_binary640.9

    \[\leadsto \left(\sqrt[3]{g} \cdot \sqrt[3]{\color{blue}{{0.5}^{0.3333333333333333}} \cdot {0.5}^{0.3333333333333333}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{0.5}\right)}^{0.6666666666666666}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{0.5}}}{a}}\right) \]

  18. Applied pow-prod-down_binary640.9

    \[\leadsto \left(\sqrt[3]{g} \cdot \sqrt[3]{\color{blue}{{\left(0.5 \cdot 0.5\right)}^{0.3333333333333333}}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{0.5}\right)}^{0.6666666666666666}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{0.5}}}{a}}\right) \]

  19. Simplified0.9

    \[\leadsto \left(\sqrt[3]{g} \cdot \sqrt[3]{{\color{blue}{0.25}}^{0.3333333333333333}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{0.5}\right)}^{0.6666666666666666}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{0.5}}}{a}}\right) \]

  20. Final simplification0.9

    \[\leadsto \left(\sqrt[3]{g} \cdot \sqrt[3]{{0.25}^{0.3333333333333333}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{0.5}\right)}^{0.6666666666666666}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{0.5}}}{a}}\right) \]

Reproduce

herbie shell --seed 2022039 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary64
  (cbrt (/ g (* 2.0 a))))