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Derivation

1. Initial program 0.6

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

2. Applied simplify0.6

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

4. Applied div-inv0.6

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

5. Applied associate-*l*0.6

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

6. Applied simplify0.5

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

8. Applied add-cube-cbrt0.5

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

9. Applied add-sqr-sqrt0.5

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

10. Applied times-frac0.5

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))