Average Error: 0.2 → 0.0
Time: 40.6s
Precision: 64
Internal Precision: 128
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \frac{\left(b \cdot b\right) \cdot \left(9 - a \cdot a\right)}{3 - a}\right) + \left(\left({a}^{4} + \left({b}^{2} \cdot {a}^{2}\right) \cdot 2\right) + {b}^{4}\right)\right) - 1\]

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

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

  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. Using strategy rm

  4. Applied flip-+0.0

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

  5. Applied associate-*r/0.0

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

  6. Final simplification0.0

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \frac{\left(b \cdot b\right) \cdot \left(9 - a \cdot a\right)}{3 - a}\right) + \left(\left({a}^{4} + \left({b}^{2} \cdot {a}^{2}\right) \cdot 2\right) + {b}^{4}\right)\right) - 1\]

Runtime

Time bar (total: 40.6s)Debug logProfile

herbie shell --seed 2018346 
(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))