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. Initial simplification0.5

   \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot (a1 \cdot a1 + \left(a2 \cdot a2\right))_*\]
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-cube-cbrt0.5

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

7. Applied associate-/r*0.5

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

8. Taylor expanded around -inf 0.6

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

9. Simplified0.4

   \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{2}} \cdot \cos th\right) \cdot (a1 \cdot a1 + \left(a2 \cdot a2\right))_*}}{\sqrt[3]{\sqrt{2}}}\]
10. Using strategy rm

11. Applied associate-*l*0.4

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

12. Final simplification0.4

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

Runtime

Time bar (total: 40.6s)Debug logProfile

herbie shell --seed 2018351 +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))))