Average Error: 0.2 → 0.0
Time: 1.8m
Precision: 64
Internal Precision: 576
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right) + \left(\left({b}^{4} + {a}^{4}\right) - 1\right)\]

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-cbrt-cube14.1

    \[\leadsto \left(\color{blue}{\sqrt[3]{\left({\left(a \cdot a + b \cdot b\right)}^{2} \cdot {\left(a \cdot a + b \cdot b\right)}^{2}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{2}}} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  4. Applied simplify14.1

    \[\leadsto \left(\sqrt[3]{\color{blue}{{\left(\left(b \cdot b + a \cdot a\right) \cdot \left(b \cdot b + a \cdot a\right)\right)}^{3}}} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  5. Taylor expanded around 0 14.1

    \[\leadsto \left(\sqrt[3]{{\color{blue}{\left(2 \cdot \left({b}^{2} \cdot {a}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right)}}^{3}} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  6. Applied simplify0.0

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))