Average Error: 0.2 → 0.0
Time: 32.1s
Precision: 64
Internal Precision: 320
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[(\left(b \cdot b\right) \cdot \left((\left(a \cdot 2\right) \cdot a + 4)_*\right) + \left({b}^{4}\right))_* + \left({a}^{4} - 1\right)\]

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  4. Applied unpow-prod-down0.2

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

  5. Applied simplify0.2

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

  6. Applied simplify0.2

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

  7. Taylor expanded around 0 0.0

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

  8. Applied simplify0.0

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

Runtime

Time bar (total: 32.1s)Debug logProfile

herbie shell --seed 2018199 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))