Error

Derivation

1. Initial program 0.2

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

2. Taylor expanded around inf 0.0

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

3. Simplified0.0

   \[\leadsto \left(\color{blue}{(\left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right) \cdot 2 + \left({b}^{4} + {a}^{4}\right))_*} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
4. Using strategy rm

5. Applied flip--0.0

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

6. Applied associate-*r/0.0

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

7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019026 +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))