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 *-un-lft-identity0.6

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

10. Applied times-frac0.5

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

11. Simplified0.5

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

13. Applied *-un-lft-identity0.5

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

14. Applied div-inv0.5

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

15. Applied times-frac0.5

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

16. Applied associate-*r*0.5

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

17. Final simplification0.5

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018339 +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))))