Average Error: 0.2 → 0.0
Time: 43.0s
Precision: 64
Internal Precision: 576
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left((\left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right) \cdot 2 + \left({a}^{4} + {b}^{4}\right))_* + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(3 + a\right) + \sqrt[3]{1 - a} \cdot \left(\left(\sqrt[3]{1 - a} \cdot \sqrt[3]{1 - a}\right) \cdot \left(a \cdot a\right)\right)\right)\right) - 1\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Taylor expanded around -inf 0.0

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

  3. Simplified0.0

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

  5. Applied add-cube-cbrt0.0

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

  6. Applied associate-*r*0.0

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

  7. Final simplification0.0

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

Runtime

Time bar (total: 43.0s)Debug logProfile

herbie shell --seed 2018256 +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))