\[\left({\left({a}^2 + {b}^2\right)}^2 + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
Test:
Bouland and Aaronson, Equation (25)
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 1.7 m
Input Error: 0.5
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\left(\left({b}^2 \cdot 4\right) \cdot \left(1 - 3 \cdot a\right) + \left(a \cdot 4 + 4\right) \cdot {a}^2\right) - \left(1 - \left({a}^{4} + {b}^{4}\right)\right)\)
  1. Started with
    \[\left({\left({a}^2 + {b}^2\right)}^2 + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
    0.5
  2. Applied taylor to get
    \[\left({\left({a}^2 + {b}^2\right)}^2 + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leadsto \left(\left({b}^{4} + \left(2 \cdot \left({b}^2 \cdot {a}^2\right) + {a}^{4}\right)\right) + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
    0.3
  3. Taylor expanded around inf to get
    \[\left(\color{red}{\left({b}^{4} + \left(2 \cdot \left({b}^2 \cdot {a}^2\right) + {a}^{4}\right)\right)} + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leadsto \left(\color{blue}{\left({b}^{4} + \left(2 \cdot \left({b}^2 \cdot {a}^2\right) + {a}^{4}\right)\right)} + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
    0.3
  4. Applied taylor to get
    \[\left(\left({b}^{4} + \left(2 \cdot \left({b}^2 \cdot {a}^2\right) + {a}^{4}\right)\right) + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leadsto \left(\left({b}^{4} + \left(2 \cdot 0 + {a}^{4}\right)\right) + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
    0.0
  5. Taylor expanded around 0 to get
    \[\left(\left({b}^{4} + \left(2 \cdot \color{red}{0} + {a}^{4}\right)\right) + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leadsto \left(\left({b}^{4} + \left(2 \cdot \color{blue}{0} + {a}^{4}\right)\right) + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
    0.0
  6. Applied simplify to get
    \[\color{red}{\left(\left({b}^{4} + \left(2 \cdot 0 + {a}^{4}\right)\right) + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1} \leadsto \color{blue}{\left(\left({b}^2 \cdot 4\right) \cdot \left(1 - 3 \cdot a\right) + \left(a \cdot 4 + 4\right) \cdot {a}^2\right) - \left(1 - \left({a}^{4} + {b}^{4}\right)\right)}\]
    0.0

Original test:


(lambda ((a default) (b default))
  #:name "Bouland and Aaronson, Equation (25)"
  (- (+ (sqr (+ (sqr a) (sqr b))) (* 4 (+ (* (sqr a) (+ 1 a)) (* (sqr b) (- 1 (* 3 a)))))) 1))