Average Error: 0.2 → 0.0

Time: 2.2m

Precision: 64

Internal Precision: 320

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

↓

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

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Initial program 0.2
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
Initial simplification0.2
\[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(\left(b \cdot b\right) \cdot 4 - 1\right)\]
Taylor expanded around -inf 0.0
\[\leadsto \color{blue}{\left({b}^{4} + \left({a}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right)} + \left(\left(b \cdot b\right) \cdot 4 - 1\right)\]
Using strategy rm
Applied pow-prod-down0.0
\[\leadsto \left({b}^{4} + \left({a}^{4} + 2 \cdot \color{blue}{{\left(a \cdot b\right)}^{2}}\right)\right) + \left(\left(b \cdot b\right) \cdot 4 - 1\right)\]
Final simplification0.0
\[\leadsto \left(\left({\left(a \cdot b\right)}^{2} \cdot 2 + {a}^{4}\right) + {b}^{4}\right) + \left(\left(b \cdot b\right) \cdot 4 - 1\right)\]

herbie shell --seed 2018230 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))