Average Error: 0.0 → 0.0
Time: 9.2s
Precision: 64
Internal Precision: 128
\[x \cdot \left(x \cdot x\right) + x \cdot x\]
↓
\[x \cdot \left(x \cdot x + x\right)\]

Error

Bits error versus x

Target

Original0.0
Target0.0
Herbie0.0
\[\left(\left(1.0 + x\right) \cdot x\right) \cdot x\]

Derivation

  1. Initial program 0.0

    \[x \cdot \left(x \cdot x\right) + x \cdot x\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{{x}^{3} + {x}^{2}}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(x + x \cdot x\right) \cdot x}\]

  4. Final simplification0.0

    \[\leadsto x \cdot \left(x \cdot x + x\right)\]

Reproduce

herbie shell --seed 2019051 
(FPCore (x)
  :name "Expression 3, p15"
  :pre (<= 0 x 2)

  :herbie-target
  (* (* (+ 1.0 x) x) x)

  (+ (* x (* x x)) (* x x)))