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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `a1`

Derivation

Initial program 0.5
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
Initial simplification0.5
\[\leadsto \frac{\cos th}{\sqrt{2}} \cdot (a1 \cdot a1 + \left(a2 \cdot a2\right))_*\]
Using strategy rm
Applied add-sqr-sqrt0.5
\[\leadsto \frac{\cos th}{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}} \cdot (a1 \cdot a1 + \left(a2 \cdot a2\right))_*\]
Applied associate-/r*0.5
\[\leadsto \color{blue}{\frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}} \cdot (a1 \cdot a1 + \left(a2 \cdot a2\right))_*\]
Final simplification0.5
\[\leadsto (a1 \cdot a1 + \left(a2 \cdot a2\right))_* \cdot \frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}\]

Runtime

herbie shell --seed 2018215 +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))))