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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 *-un-lft-identity0.5

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

9. Applied sqrt-prod0.5

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

10. Applied div-inv0.5

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

11. Applied times-frac0.5

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

12. Applied associate-*l*0.5

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

13. Simplified0.5

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

14. Final simplification0.5

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

Reproduce

herbie shell --seed 2018365 +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))))

Details

Time bar (total: 43.1s)Debug log