Error

Derivation

1. Initial program 0.2

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
2. Using strategy rm

3. Applied unpow20.2

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

4. Applied fma-def0.2

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

5. Simplified0.2

   \[\leadsto (\color{blue}{\left((b \cdot b + \left(a \cdot a\right))_*\right)} \cdot \left(a \cdot a + b \cdot b\right) + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right))_* - 1\]
6. Using strategy rm

7. Applied fma-def0.2

   \[\leadsto (\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \color{blue}{\left((a \cdot a + \left(b \cdot b\right))_*\right)} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right))_* - 1\]
8. Using strategy rm

9. Applied add-cube-cbrt0.2

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

10. Final simplification0.2

    \[\leadsto (\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \left((a \cdot a + \left(b \cdot b\right))_*\right) + \left(4 \cdot \left(\left(3 + a\right) \cdot \left(b \cdot b\right) + \sqrt[3]{\left(a \cdot a\right) \cdot \left(1 - a\right)} \cdot \left(\sqrt[3]{\left(a \cdot a\right) \cdot \left(1 - a\right)} \cdot \sqrt[3]{\left(a \cdot a\right) \cdot \left(1 - a\right)}\right)\right)\right))_* - 1\]

Runtime

Time bar (total: 28.8s)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))