Average Error: 58.7 → 0.4
Time: 51.1s
Precision: 64
Internal Precision: 1408
\[e^{x} - 1\]
↓
\[x + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  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

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

Runtime

Time bar (total: 51.1s)Debug logProfile

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)
  (- (exp x) 1))