Average Error: 59.3 → 0.1
Time: 16.6s
Precision: 64
Internal precision: 128
\[e^{x} - 1\]
⬇
\[x + {x}^2 \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)\]

Error

Bits error versus x

Target

Original59.3
Comparison0.1
Herbie0.1
\[ x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{{x}^2}{6}\right) \]

Derivation

  1. Initial program 59.3

    \[e^{x} - 1\]

  2. Applied taylor 0.1

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

  3. Taylor expanded around 0 0.1

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

  4. Applied simplify 0.1

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

Runtime

Time bar (total: 16.6s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(2902752189 169865760 36032119 3038042540 4030741990 3025584327)'
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))