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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. Using strategy rm

3. Applied associate-*l/0.5

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

5. Applied add-sqr-sqrt0.5

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

6. Applied associate-/r*0.5

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

8. Applied pow10.5

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

9. Applied pow10.5

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

10. Applied pow-prod-down0.5

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

11. Simplified0.5

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

12. Final simplification0.5

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018346 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))