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. 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 associate-*r*0.5

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

7. Applied add-cube-cbrt0.5

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

8. Applied associate-/r*0.5

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

10. Applied associate-*l/0.5

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

11. Final simplification0.5

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

Reproduce

herbie shell --seed 2019091 +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))))