Average Error: 0.2 → 0.0
Time: 25.7s
Precision: 64
Internal Precision: 576
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\left(2 \cdot \left({b}^{2} \cdot {a}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right) - 1\right) + \left(\left(\left(b \cdot b\right) \cdot 4\right) \cdot \left(1 - 3 \cdot a\right) + \left(a \cdot 4\right) \cdot \left(a \cdot a + a\right)\right)\]

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

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

  2. Initial simplification0.2

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

  3. Taylor expanded around 0 0.0

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

  4. Final simplification0.0

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

Runtime

Time bar (total: 25.7s)Debug logProfile

herbie shell --seed 2018215 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))