Average Error: 0.2 → 0.2
Time: 25.6s
Precision: 64
Internal Precision: 320
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[(4 \cdot \left((\left(1 - 3 \cdot a\right) \cdot \left(b \cdot b\right) + \left((\left(a \cdot a\right) \cdot a + \left(a \cdot a\right))_*\right))_*\right) + \left((\left((b \cdot b + \left(a \cdot a\right))_*\right) \cdot \left((b \cdot b + \left(a \cdot a\right))_*\right) + \left(-1\right))_*\right))_*\]

Error

Bits error versus a

Bits error versus b

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

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

  3. Final simplification0.2

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

Runtime

Time bar (total: 25.6s)Debug logProfile

herbie shell --seed 2018221 +o rules:numerics
(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))