Average Error: 0.2 → 0.0
Time: 20.9s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\left(b \cdot b\right) \cdot 4 + \left(\left(2 \cdot \left(\sqrt[3]{{b}^{2}} \cdot \left(\left(\sqrt[3]{{b}^{2}} \cdot \sqrt[3]{{b}^{2}}\right) \cdot {a}^{2}\right)\right) + {a}^{4}\right) + {b}^{4}\right)\right) - 1\]

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Taylor expanded around -inf 0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied associate-*r*0.0

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

  6. Final simplification0.0

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

Runtime

Time bar (total: 20.9s)Debug logProfile

herbie shell --seed 2018304 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))