Average Error: 0.2 → 0.2
Time: 43.4s
Precision: 64
Internal Precision: 320
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \sqrt[3]{a + 3} \cdot \left(\left(\sqrt[3]{a + 3} \cdot \sqrt[3]{a + 3}\right) \cdot \left(b \cdot b\right)\right)\right) \cdot 4\right) - 1\]

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

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

  3. Applied add-cube-cbrt0.2

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

  4. Applied associate-*r*0.2

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

  5. Final simplification0.2

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

Runtime

Time bar (total: 43.4s)Debug logProfile

herbie shell --seed 2018242 
(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))