Average Error: 58.6 → 0.5
Time: 5.6s
Precision: 64
Internal Precision: 128
\[e^{x} - 1\]
↓
\[x + \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.5
Herbie0.5
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.6

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019026 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1 (/ x 2)) (/ (* x x) 6)))

  (- (exp x) 1))