\[\left({\left({a}^2 + {b}^2\right)}^2 + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
Test:
Bouland and Aaronson, Equation (25)
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 26.6 s
Input Error: 0.2
Output Error: 0.1
Log: ⚲
Profile: 🕒
\((\left((\left(1 - a \cdot 3\right) * \left({b}^2\right) + \left((a * a + a)_* \cdot a\right))_*\right) * 4 + \left((\left(\left(b \cdot 2\right) \cdot b\right) * \left(a \cdot a\right) + \left({a}^{4} + {b}^{4}\right))_*\right))_* - 1\)
  1. Started with
    \[\left({\left({a}^2 + {b}^2\right)}^2 + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
    0.2
  2. Applied taylor to get
    \[\left({\left({a}^2 + {b}^2\right)}^2 + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leadsto \left(\left({b}^{4} + \left(2 \cdot \left({b}^2 \cdot {a}^2\right) + {a}^{4}\right)\right) + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
    0.1
  3. Taylor expanded around inf to get
    \[\left(\color{red}{\left({b}^{4} + \left(2 \cdot \left({b}^2 \cdot {a}^2\right) + {a}^{4}\right)\right)} + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leadsto \left(\color{blue}{\left({b}^{4} + \left(2 \cdot \left({b}^2 \cdot {a}^2\right) + {a}^{4}\right)\right)} + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
    0.1
  4. Applied simplify to get
    \[\color{red}{\left(\left({b}^{4} + \left(2 \cdot \left({b}^2 \cdot {a}^2\right) + {a}^{4}\right)\right) + 4 \cdot \left({a}^2 \cdot \left(1 + a\right) + {b}^2 \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1} \leadsto \color{blue}{(\left((\left(1 - a \cdot 3\right) * \left({b}^2\right) + \left((a * a + a)_* \cdot a\right))_*\right) * 4 + \left((\left(\left(b \cdot 2\right) \cdot b\right) * \left(a \cdot a\right) + \left({a}^{4} + {b}^{4}\right))_*\right))_* - 1}\]
    0.1

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))