Average Error: 59.2 → 0.1
Time: 4.5s
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

Target

Original59.2
Comparison0.1
Herbie0.1
\[ x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right) \]

Derivation

  1. Initial program 59.2

    \[e^{x} - 1\]

  2. Applied taylor 0.1

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

  3. Taylor expanded around 0 0.1

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

  4. Applied simplify 0.1

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

Runtime

Time bar (total: 4.5s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1068028399 4028058041 2917032441 2563479541 765645300 1132738916)'
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))