Error

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-sqr-sqrt0.5

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

11. Applied associate-/l*0.5

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

12. Simplified0.5

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

13. Final simplification0.5

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

Runtime

Time bar (total: 41.0s)Debug logProfile

herbie shell --seed 2018273 +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))))