Average Error: 0.0 → 0.5
Time: 7.2s
Precision: 64
Internal Precision: 320
\[x \cdot \left(x \cdot x\right) + x \cdot x\]
↓
\[\sqrt{(\left(x \cdot x\right) \cdot x + \left(x \cdot x\right))_*} \cdot (\left(x \cdot x\right) \cdot \left((x \cdot \frac{-1}{8} + \frac{1}{2})_*\right) + x)_*\]

Error

Bits error versus x

Target

Original0.0
Target0.0
Herbie0.5
\[\left(\left(1.0 + x\right) \cdot x\right) \cdot x\]

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-sqr-sqrt0.0

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

  5. Taylor expanded around 0 0.5

    \[\leadsto \sqrt{(\left(x \cdot x\right) \cdot x + \left(x \cdot x\right))_*} \cdot \color{blue}{\left(\left(x + \frac{1}{2} \cdot {x}^{2}\right) - \frac{1}{8} \cdot {x}^{3}\right)}\]

  6. Simplified0.5

    \[\leadsto \sqrt{(\left(x \cdot x\right) \cdot x + \left(x \cdot x\right))_*} \cdot \color{blue}{(\left(x \cdot x\right) \cdot \left((x \cdot \frac{-1}{8} + \frac{1}{2})_*\right) + x)_*}\]

  7. Final simplification0.5

    \[\leadsto \sqrt{(\left(x \cdot x\right) \cdot x + \left(x \cdot x\right))_*} \cdot (\left(x \cdot x\right) \cdot \left((x \cdot \frac{-1}{8} + \frac{1}{2})_*\right) + x)_*\]

Runtime

Time bar (total: 7.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.50.50.00.40%
herbie shell --seed 2018286 +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)))