Average Error: 21.5 → 0.3
Time: 30.5s
Precision: 64
Internal Precision: 320
\[{\left(\left|x\right|\right)}^2 \cdot {\left(\left|y\right|\right)}^2\]
\[{\left(\left(\left|x\right| \cdot \left|y\right|\right) \cdot \left(\left|x\right| \cdot \left|y\right|\right)\right)}^{1}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 21.5

    \[{\left(\left|x\right|\right)}^2 \cdot {\left(\left|y\right|\right)}^2\]
  2. Using strategy rm
  3. Applied add-cube-cbrt21.9

    \[\leadsto {\left(\left|x\right|\right)}^2 \cdot {\color{blue}{\left(\left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right) \cdot \sqrt[3]{\left|y\right|}\right)}}^2\]
  4. Applied square-prod21.9

    \[\leadsto {\left(\left|x\right|\right)}^2 \cdot \color{blue}{\left(\left(\left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right) \cdot \left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)\right) \cdot \left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)\right)}\]
  5. Applied associate-*r*17.9

    \[\leadsto \color{blue}{\left({\left(\left|x\right|\right)}^2 \cdot \left(\left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right) \cdot \left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)\right)\right) \cdot \left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)}\]
  6. Applied simplify3.3

    \[\leadsto \color{blue}{\left(\left(\left|x\right| \cdot \left|y\right|\right) \cdot \left(\sqrt[3]{\left|y\right|} \cdot \left|x\right|\right)\right)} \cdot \left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)\]
  7. Using strategy rm
  8. Applied pow13.3

    \[\leadsto \left(\left(\left|x\right| \cdot \left|y\right|\right) \cdot \left(\sqrt[3]{\left|y\right|} \cdot \left|x\right|\right)\right) \cdot \left(\sqrt[3]{\left|y\right|} \cdot \color{blue}{{\left(\sqrt[3]{\left|y\right|}\right)}^{1}}\right)\]
  9. Applied pow13.3

    \[\leadsto \left(\left(\left|x\right| \cdot \left|y\right|\right) \cdot \left(\sqrt[3]{\left|y\right|} \cdot \left|x\right|\right)\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{\left|y\right|}\right)}^{1}} \cdot {\left(\sqrt[3]{\left|y\right|}\right)}^{1}\right)\]
  10. Applied pow-prod-down3.3

    \[\leadsto \left(\left(\left|x\right| \cdot \left|y\right|\right) \cdot \left(\sqrt[3]{\left|y\right|} \cdot \left|x\right|\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)}^{1}}\]
  11. Applied pow13.3

    \[\leadsto \left(\left(\left|x\right| \cdot \left|y\right|\right) \cdot \left(\sqrt[3]{\left|y\right|} \cdot \color{blue}{{\left(\left|x\right|\right)}^{1}}\right)\right) \cdot {\left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)}^{1}\]
  12. Applied pow13.3

    \[\leadsto \left(\left(\left|x\right| \cdot \left|y\right|\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{\left|y\right|}\right)}^{1}} \cdot {\left(\left|x\right|\right)}^{1}\right)\right) \cdot {\left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)}^{1}\]
  13. Applied pow-prod-down3.3

    \[\leadsto \left(\left(\left|x\right| \cdot \left|y\right|\right) \cdot \color{blue}{{\left(\sqrt[3]{\left|y\right|} \cdot \left|x\right|\right)}^{1}}\right) \cdot {\left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)}^{1}\]
  14. Applied pow13.3

    \[\leadsto \left(\left(\left|x\right| \cdot \color{blue}{{\left(\left|y\right|\right)}^{1}}\right) \cdot {\left(\sqrt[3]{\left|y\right|} \cdot \left|x\right|\right)}^{1}\right) \cdot {\left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)}^{1}\]
  15. Applied pow13.3

    \[\leadsto \left(\left(\color{blue}{{\left(\left|x\right|\right)}^{1}} \cdot {\left(\left|y\right|\right)}^{1}\right) \cdot {\left(\sqrt[3]{\left|y\right|} \cdot \left|x\right|\right)}^{1}\right) \cdot {\left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)}^{1}\]
  16. Applied pow-prod-down3.3

    \[\leadsto \left(\color{blue}{{\left(\left|x\right| \cdot \left|y\right|\right)}^{1}} \cdot {\left(\sqrt[3]{\left|y\right|} \cdot \left|x\right|\right)}^{1}\right) \cdot {\left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)}^{1}\]
  17. Applied pow-prod-down3.3

    \[\leadsto \color{blue}{{\left(\left(\left|x\right| \cdot \left|y\right|\right) \cdot \left(\sqrt[3]{\left|y\right|} \cdot \left|x\right|\right)\right)}^{1}} \cdot {\left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)}^{1}\]
  18. Applied pow-prod-down3.3

    \[\leadsto \color{blue}{{\left(\left(\left(\left|x\right| \cdot \left|y\right|\right) \cdot \left(\sqrt[3]{\left|y\right|} \cdot \left|x\right|\right)\right) \cdot \left(\sqrt[3]{\left|y\right|} \cdot \sqrt[3]{\left|y\right|}\right)\right)}^{1}}\]
  19. Applied simplify0.3

    \[\leadsto {\color{blue}{\left(\left(\left|x\right| \cdot \left|y\right|\right) \cdot \left(\left|x\right| \cdot \left|y\right|\right)\right)}}^{1}\]

Runtime

Time bar (total: 30.5s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x y)
  :name "sqr(abs(x)) * sqr(abs(y))"
  (* (sqr (fabs x)) (sqr (fabs y))))