Average Error: 0.2 → 0.0
Time: 26.8s
Precision: 64
Internal precision: 128
\[\left({\left({a}^2 + {b}^2\right)}^2 + 4 \cdot {b}^2\right) - 1\]
⬇
\[\left(4 + \left(a + a\right) \cdot a\right) \cdot {b}^2 + \left(\left({b}^{4} - 1\right) + {a}^{4}\right)\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Applied taylor 0.0

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

  3. Taylor expanded around inf 0.0

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

  4. Applied simplify 0.0

    \[\leadsto \color{blue}{\left(4 + \left(a + a\right) \cdot a\right) \cdot {b}^2 + \left(\left({b}^{4} - 1\right) + {a}^{4}\right)}\]
  5. Removed slow pow expressions

Runtime

Time bar (total: 26.8s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(4087086052 3684086861 2629883594 1179437848 2959370164 1609658556)'
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (sqr (+ (sqr a) (sqr b))) (* 4 (sqr b))) 1))