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

↓

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

Error

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

Derivation

Initial program 0.5
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
Applied simplify0.5
\[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)}\]
Using strategy rm
Applied add-sqr-sqrt0.5
\[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(\sqrt{a2 \cdot a2 + a1 \cdot a1} \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}\right)}\]
Applied associate-*r*0.5
\[\leadsto \color{blue}{\left(\frac{\cos th}{\sqrt{2}} \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}\right) \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}}\]

Runtime

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))