Average Error: 0.0 → 0.0
Time: 5.7s
Precision: 64
Internal Precision: 320
\[x \cdot \left(x \cdot x\right) + x \cdot x\]
↓
\[(x \cdot x + \left({\left(\frac{1}{x}\right)}^{-3}\right))_*\]

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. Initial simplification0.0

    \[\leadsto (\left(x \cdot x\right) \cdot x + \left(x \cdot x\right))_*\]
  3. Using strategy rm

  4. Applied add-cbrt-cube20.4

    \[\leadsto \color{blue}{\sqrt[3]{\left((\left(x \cdot x\right) \cdot x + \left(x \cdot x\right))_* \cdot (\left(x \cdot x\right) \cdot x + \left(x \cdot x\right))_*\right) \cdot (\left(x \cdot x\right) \cdot x + \left(x \cdot x\right))_*}}\]

  5. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{{x}^{2} + {\left(\frac{1}{x}\right)}^{-3}}\]

  6. Simplified0.0

    \[\leadsto \color{blue}{(x \cdot x + \left({\left(\frac{1}{x}\right)}^{-3}\right))_*}\]

  7. Final simplification0.0

    \[\leadsto (x \cdot x + \left({\left(\frac{1}{x}\right)}^{-3}\right))_*\]

Runtime

Time bar (total: 5.7s)Debug logProfile

herbie shell --seed 2018249 +o rules:numerics
(FPCore (x)
  :name "Expression 3, p15"
  :pre (<= 0 x 2)

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

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