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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

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

5. Applied add-sqr-sqrt0.5

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

6. Applied associate-/r*0.5

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

8. Applied add-sqr-sqrt0.5

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

9. Applied associate-/r*0.5

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

11. Applied add-cube-cbrt0.5

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

12. Applied sqrt-prod0.6

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

13. Applied div-inv0.5

    \[\leadsto \frac{\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} + \frac{\color{blue}{\cos th \cdot \frac{1}{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]

14. Applied times-frac0.5

    \[\leadsto \frac{\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} + \color{blue}{\left(\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right)} \cdot \left(a2 \cdot a2\right)\]

15. Simplified0.5

    \[\leadsto \frac{\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} + \left(\color{blue}{\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a2 \cdot a2\right)\]

16. Final simplification0.5

    \[\leadsto \frac{\frac{\left(a1 \cdot a1\right) \cdot \cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} + \left(\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a2 \cdot a2\right)\]

Runtime

Time bar (total: 53.1s)Debug logProfile

herbie shell --seed 2018285 +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))))