Average Error: 0.1 → 0.2
Time: 57.5s
Precision: 64
\[x - \frac{3}{8} \cdot y\]
↓
\[\left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \mathsf{fma}\left(-\frac{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{1}}, \left(\frac{\sqrt[3]{\sqrt[3]{3}}}{\sqrt[3]{8}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}, {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)\]
x - \frac{3}{8} \cdot y
↓
\left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \mathsf{fma}\left(-\frac{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{1}}, \left(\frac{\sqrt[3]{\sqrt[3]{3}}}{\sqrt[3]{8}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}, {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)
double code(double x, double y) {
	return ((double) (x - ((double) (((double) (3.0 / 8.0)) * y))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (x - ((double) (((double) (((double) (((double) cbrt(3.0)) / ((double) cbrt(8.0)))) * y)) * ((double) (((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))) / ((double) (((double) cbrt(8.0)) * ((double) cbrt(8.0)))))))))) + ((double) (y * ((double) fma(((double) -(((double) (((double) cbrt(((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))) / ((double) cbrt(1.0)))))), ((double) (((double) (((double) (((double) cbrt(((double) cbrt(3.0)))) / ((double) cbrt(8.0)))) * ((double) (((double) cbrt(3.0)) / ((double) cbrt(8.0)))))) * ((double) (((double) cbrt(3.0)) / ((double) cbrt(8.0)))))), ((double) pow(((double) (((double) cbrt(3.0)) / ((double) cbrt(8.0)))), 3.0))))))));
}

Error

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Results

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Derivation

  1. Initial program 0.1

    \[x - \frac{3}{8} \cdot y\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.1

    \[\leadsto x - \frac{3}{\color{blue}{\left(\sqrt[3]{8} \cdot \sqrt[3]{8}\right) \cdot \sqrt[3]{8}}} \cdot y\]

  4. Applied add-cube-cbrt0.1

    \[\leadsto x - \frac{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}{\left(\sqrt[3]{8} \cdot \sqrt[3]{8}\right) \cdot \sqrt[3]{8}} \cdot y\]

  5. Applied times-frac0.1

    \[\leadsto x - \color{blue}{\left(\frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)} \cdot y\]

  6. Applied associate-*l*0.2

    \[\leadsto x - \color{blue}{\frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}} \cdot \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right)}\]

  7. Applied add-sqr-sqrt31.6

    \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x}} - \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}} \cdot \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right)\]

  8. Applied prod-diff31.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, -\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + \mathsf{fma}\left(-\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y, \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}, \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right)}\]

  9. Simplified0.2

    \[\leadsto \color{blue}{\left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right)} + \mathsf{fma}\left(-\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y, \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}, \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right)\]

  10. Simplified0.2

    \[\leadsto \left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + \color{blue}{y \cdot \left(\left(-{\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right) + {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)}\]
  11. Using strategy rm

  12. Applied add-cbrt-cube1.0

    \[\leadsto \left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \left(\left(-{\color{blue}{\left(\sqrt[3]{\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}}\right)}}^{3}\right) + {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)\]

  13. Applied rem-cube-cbrt0.3

    \[\leadsto \left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \left(\left(-\color{blue}{\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}}\right) + {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)\]
  14. Using strategy rm

  15. Applied *-un-lft-identity0.3

    \[\leadsto \left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \left(\left(-\left(\frac{\sqrt[3]{3}}{\sqrt[3]{\color{blue}{1 \cdot 8}}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) + {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)\]

  16. Applied cbrt-prod0.3

    \[\leadsto \left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \left(\left(-\left(\frac{\sqrt[3]{3}}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{8}}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) + {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)\]

  17. Applied add-cube-cbrt0.3

    \[\leadsto \left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \left(\left(-\left(\frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}}{\sqrt[3]{1} \cdot \sqrt[3]{8}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) + {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)\]

  18. Applied cbrt-prod0.3

    \[\leadsto \left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \left(\left(-\left(\frac{\color{blue}{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}}{\sqrt[3]{1} \cdot \sqrt[3]{8}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) + {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)\]

  19. Applied times-frac0.3

    \[\leadsto \left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \left(\left(-\left(\color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{1}} \cdot \frac{\sqrt[3]{\sqrt[3]{3}}}{\sqrt[3]{8}}\right)} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) + {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)\]

  20. Applied associate-*l*0.3

    \[\leadsto \left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \left(\left(-\color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{1}} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{3}}}{\sqrt[3]{8}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)\right)} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) + {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)\]

  21. Applied associate-*l*0.2

    \[\leadsto \left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \left(\left(-\color{blue}{\frac{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{1}} \cdot \left(\left(\frac{\sqrt[3]{\sqrt[3]{3}}}{\sqrt[3]{8}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}\right) + {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)\]

  22. Applied distribute-lft-neg-in0.2

    \[\leadsto \left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \left(\color{blue}{\left(-\frac{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{1}}\right) \cdot \left(\left(\frac{\sqrt[3]{\sqrt[3]{3}}}{\sqrt[3]{8}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)} + {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)\]

  23. Applied fma-def0.2

    \[\leadsto \left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \color{blue}{\mathsf{fma}\left(-\frac{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{1}}, \left(\frac{\sqrt[3]{\sqrt[3]{3}}}{\sqrt[3]{8}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}, {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)}\]

  24. Final simplification0.2

    \[\leadsto \left(x - \left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}} \cdot y\right) \cdot \frac{\sqrt[3]{3} \cdot \sqrt[3]{3}}{\sqrt[3]{8} \cdot \sqrt[3]{8}}\right) + y \cdot \mathsf{fma}\left(-\frac{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{1}}, \left(\frac{\sqrt[3]{\sqrt[3]{3}}}{\sqrt[3]{8}} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right) \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{8}}, {\left(\frac{\sqrt[3]{3}}{\sqrt[3]{8}}\right)}^{3}\right)\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3 8) y)))