Average Error: 0.2 → 0.0
Time: 18.5s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + \left(\left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right) \cdot 2 + \left({b}^{4} + {a}^{4}\right)\right)\right) - 1\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019093 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))