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 0 0.0

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

3. Simplified0.2

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

5. Applied pow10.2

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

6. Applied pow-plus0.2

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

7. Applied pow10.2

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

8. Applied pow10.2

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

9. Applied pow-prod-up0.2

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

10. Applied pow-prod-up0.1

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

11. Simplified0.1

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

13. Applied associate-*r*0.1

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

14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019100 
(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))