Average Error: 0.0 → 0.0
Time: 37.8s
Precision: 64
Internal Precision: 384
\[\left(\left(\left(1.0 + 0.5 \cdot x\right) - \left(0.125 \cdot x\right) \cdot x\right) + \left(\left(0.0625 \cdot x\right) \cdot x\right) \cdot x\right) - \left(\left(\left(0.0390625 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\]
↓
\[\left(\left(1.0 + x \cdot 0.5\right) - 0.0390625 \cdot {x}^{4}\right) - \left(0.125 - 0.0625 \cdot x\right) \cdot \left(x \cdot x\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\left(\left(\left(1.0 + 0.5 \cdot x\right) - \left(0.125 \cdot x\right) \cdot x\right) + \left(\left(0.0625 \cdot x\right) \cdot x\right) \cdot x\right) - \left(\left(\left(0.0390625 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \left(\left(\left(1.0 + 0.5 \cdot x\right) - \left(0.125 \cdot x\right) \cdot x\right) + \left(\left(0.0625 \cdot x\right) \cdot x\right) \cdot x\right) - \color{blue}{0.0390625 \cdot {x}^{4}}\]

  3. Applied simplify0.0

    \[\leadsto \color{blue}{\left(\left(1.0 + x \cdot 0.5\right) - 0.0390625 \cdot {x}^{4}\right) - \left(0.125 - 0.0625 \cdot x\right) \cdot \left(x \cdot x\right)}\]

Runtime

Time bar (total: 37.8s)Debug log

herbie shell --seed '#(1743936871 1855164119 3668777427 1254258049 132811564 1366975197)' 
(FPCore (x)
  :name "sqroot"
  :pre (<= 0 x 1)
  (- (+ (- (+ 1.0 (* 0.5 x)) (* (* 0.125 x) x)) (* (* (* 0.0625 x) x) x)) (* (* (* (* 0.0390625 x) x) x) x)))