\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

↓

\[(\left((a \cdot a + \left(b \cdot b\right))_*\right) \cdot \left((a \cdot a + \left(b \cdot b\right))_*\right) + \left((\left(b \cdot b\right) \cdot 4 + -1)_*\right))_*\]

Error

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

Derivation

Initial program 0.2
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
Simplified0.2
\[\leadsto \color{blue}{(\left((a \cdot a + \left(b \cdot b\right))_*\right) \cdot \left((a \cdot a + \left(b \cdot b\right))_*\right) + \left((\left(b \cdot b\right) \cdot 4 + -1)_*\right))_*}\]
Final simplification0.2
\[\leadsto (\left((a \cdot a + \left(b \cdot b\right))_*\right) \cdot \left((a \cdot a + \left(b \cdot b\right))_*\right) + \left((\left(b \cdot b\right) \cdot 4 + -1)_*\right))_*\]

Reproduce

herbie shell --seed 2019090 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))