Average Error: 58.8 → 0.4
Time: 1.5m
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 58.8

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.4

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

  3. Applied simplify0.4

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

Runtime

Time bar (total: 1.5m)Debug log

herbie shell --seed '#(2094665535 3654809497 1731717781 3082199462 566033875 575777438)' 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)
  (- (exp x) 1))