Average Error: 0.5 → 0.4
Time: 40.4s
Precision: 64
Internal Precision: 576
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\left(\frac{\sqrt{a1^2 + a2^2}^*}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\sqrt{a1^2 + a2^2}^*}{\sqrt[3]{\sqrt{2}}}\right) \cdot \cos th\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

  2. Initial simplification0.5

    \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot (a1 \cdot a1 + \left(a2 \cdot a2\right))_*\]
  3. Using strategy rm

  4. Applied associate-*l/0.5

    \[\leadsto \color{blue}{\frac{\cos th \cdot (a1 \cdot a1 + \left(a2 \cdot a2\right))_*}{\sqrt{2}}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\cos th \cdot (a1 \cdot a1 + \left(a2 \cdot a2\right))_*}{\color{blue}{1 \cdot \sqrt{2}}}\]

  7. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{\cos th}{1} \cdot \frac{(a1 \cdot a1 + \left(a2 \cdot a2\right))_*}{\sqrt{2}}}\]

  8. Simplified0.5

    \[\leadsto \color{blue}{\cos th} \cdot \frac{(a1 \cdot a1 + \left(a2 \cdot a2\right))_*}{\sqrt{2}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.5

    \[\leadsto \cos th \cdot \frac{(a1 \cdot a1 + \left(a2 \cdot a2\right))_*}{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}\]

  11. Applied add-sqr-sqrt0.5

    \[\leadsto \cos th \cdot \frac{\color{blue}{\sqrt{(a1 \cdot a1 + \left(a2 \cdot a2\right))_*} \cdot \sqrt{(a1 \cdot a1 + \left(a2 \cdot a2\right))_*}}}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}\]

  12. Applied times-frac0.5

    \[\leadsto \cos th \cdot \color{blue}{\left(\frac{\sqrt{(a1 \cdot a1 + \left(a2 \cdot a2\right))_*}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\sqrt{(a1 \cdot a1 + \left(a2 \cdot a2\right))_*}}{\sqrt[3]{\sqrt{2}}}\right)}\]

  13. Simplified0.5

    \[\leadsto \cos th \cdot \left(\color{blue}{\frac{\sqrt{a1^2 + a2^2}^*}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\sqrt{(a1 \cdot a1 + \left(a2 \cdot a2\right))_*}}{\sqrt[3]{\sqrt{2}}}\right)\]

  14. Simplified0.4

    \[\leadsto \cos th \cdot \left(\frac{\sqrt{a1^2 + a2^2}^*}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \color{blue}{\frac{\sqrt{a1^2 + a2^2}^*}{\sqrt[3]{\sqrt{2}}}}\right)\]

  15. Final simplification0.4

    \[\leadsto \left(\frac{\sqrt{a1^2 + a2^2}^*}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\sqrt{a1^2 + a2^2}^*}{\sqrt[3]{\sqrt{2}}}\right) \cdot \cos th\]

Runtime

Time bar (total: 40.4s)Debug logProfile

herbie shell --seed 2018277 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))