Average Error: 44.9 → 0.2
Time: 3.4s
Precision: 64
Internal precision: 1408
\[\frac{e^{x}}{e^{x} - 1}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -5.075430944100181 \cdot 10^{-08}:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original44.9
Comparison44.9
Herbie0.2
\[ \frac{1}{1 - e^{-x}} \]

Derivation

  1. Split input into 2 regimes.

  2. if x < -5.075430944100181e-08

    1. Initial program 0.2

      \[\frac{e^{x}}{e^{x} - 1}\]

    if -5.075430944100181e-08 < x

    1. Initial program 60.8

      \[\frac{e^{x}}{e^{x} - 1}\]

    2. Applied taylor 0.1

      \[\leadsto \frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\]

    3. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 3.4s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1066372953 114334025 411438303 1288252006 2962405338 2829794477)'
(FPCore (x)
  :name "expq2 (section 3.11)"

  :target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))