Average Error: 59.2 → 0.1
Time: 16.0s
Precision: 64
Internal Precision: 1408
\[e^{x} - 1\]
↓
\[\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + x\]

Error

Bits error versus x

Derivation

  1. Initial program 59.2

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\]

  3. Applied simplify0.1

    \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + x}\]
  4. Removed slow pow expressions.

Runtime

Time bar (total: 16.0s)Debug log

herbie shell --seed '#(151349756 408087815 228312487 2538703040 1980610373 1250971417)' 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)
  (- (exp x) 1))