Average Error: 0.2 → 0.0
Time: 45.2s
Precision: 64
Internal Precision: 320
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\left({b}^{4} + {a}^{4}\right) - 1\right) + \left(4 + a \cdot \left(a \cdot 2\right)\right) \cdot \left(b \cdot b\right)\]

Error

Bits error versus a

Bits error versus b

Try it out

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. Using strategy rm

  3. Applied add-cube-cbrt0.7

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

  4. Applied unpow-prod-down0.7

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

  5. Applied simplify0.5

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

  6. Taylor expanded around 0 0.0

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

  7. Applied simplify0.0

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

Runtime

Time bar (total: 45.2s)Debug logProfile

herbie shell --seed 2018199 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))