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. Simplified0.5

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

4. Applied associate-*l/0.5

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

6. Applied add-sqr-sqrt0.5

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

7. Applied associate-*r*0.5

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

8. Simplified0.5

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

10. Applied add-cube-cbrt0.5

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

11. Applied associate-/r*0.5

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

12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019004 +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: 44.3s)Debug log