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. Simplified0.2

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

3. Taylor expanded around inf 0.0

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

4. Simplified0.0

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

6. Applied add-cube-cbrt0.0

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

7. Applied associate-*l*0.0

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

8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019068 
(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))