\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]

↓

\[\left(4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) + (\left(2 \cdot {a}^{2}\right) \cdot \left({b}^{2}\right) + \left({a}^{4} + {b}^{4}\right))_*\right) - 1\]

Derivation

Initial program 0.2
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
Using strategy rm
Applied expm1-log1p-u2.1
\[\leadsto \left(\color{blue}{(e^{\log_* (1 + {\left(a \cdot a + b \cdot b\right)}^{2})} - 1)^*} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
Taylor expanded around 0 0.0
\[\leadsto \left(\color{blue}{\left({b}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right)\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
Simplified0.0
\[\leadsto \left(\color{blue}{(\left(2 \cdot {a}^{2}\right) \cdot \left({b}^{2}\right) + \left({a}^{4} + {b}^{4}\right))_*} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
Final simplification0.0
\[\leadsto \left(4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) + (\left(2 \cdot {a}^{2}\right) \cdot \left({b}^{2}\right) + \left({a}^{4} + {b}^{4}\right))_*\right) - 1\]

Runtime

herbie shell --seed 2018221 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))

Error

Derivation

Runtime