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

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

5. Applied associate-*r*0.5

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

6. Simplified0.5

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

8. Applied pow10.5

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

9. Applied pow10.5

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

10. Applied pow-prod-down0.5

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

11. Simplified0.5

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

12. Final simplification0.5

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

Runtime

Time bar (total: 30.2s)Debug logProfile

herbie shell --seed 2018353 +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))))