Average Error: 45.3 → 0.1
Time: 29.5s
Precision: 64
Internal precision: 1408
\[\frac{e^{x}}{e^{x} - 1}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -3.859884253300168 \cdot 10^{-07}:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x + x} - 1}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original45.3
Comparison45.3
Herbie0.1
\[ \frac{1}{1 - e^{-x}} \]

Derivation

  1. Split input into 2 regimes.

  2. if x < -3.859884253300168e-07

    1. Initial program 0.2

      \[\frac{e^{x}}{e^{x} - 1}\]
    2. Using strategy rm

    3. Applied flip-- 0.2

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}\]

    4. Applied simplify 0.2

      \[\leadsto \frac{e^{x}}{\frac{\color{blue}{e^{x + x} - 1}}{e^{x} + 1}}\]

    if -3.859884253300168e-07 < x

    1. Initial program 60.7

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

    2. 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: 29.5s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(2329929097 3210370195 3111198779 2406363002 3511342718 2136436390)'
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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