Average Error: 0.2 → 0.2
Time: 19.7s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[(\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \left((a \cdot a + \left(b \cdot b\right))_*\right) + \left((\left(b \cdot 4\right) \cdot b + -1)_*\right))_*\]

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. Initial simplification0.2

    \[\leadsto (\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \left((b \cdot b + \left(a \cdot a\right))_*\right) + \left((\left(4 \cdot b\right) \cdot b + -1)_*\right))_*\]

  3. Taylor expanded around -inf 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

    \[\leadsto (\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \left((a \cdot a + \left(b \cdot b\right))_*\right) + \left((\left(b \cdot 4\right) \cdot b + -1)_*\right))_*\]

Runtime

Time bar (total: 19.7s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.20.20.00.20%
herbie shell --seed 2018355 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))