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 add-sqr-sqrt0.5

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

5. Applied associate-/r*0.5

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

7. Applied add-cube-cbrt0.5

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

8. Applied sqrt-prod0.6

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

9. Applied div-inv0.5

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

10. Applied times-frac0.5

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

11. Applied associate-*l*0.4

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

12. Simplified0.4

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

13. Taylor expanded around inf 0.6

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

14. Simplified0.4

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

16. Applied add-sqr-sqrt0.4

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

17. Applied associate-*r*0.4

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

18. Simplified0.4

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

19. Final simplification0.4

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018256 +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))))