Average Error: 45.1 → 0.1
Time: 11.1s
Precision: 64
Internal precision: 1408
\[\frac{e^{x}}{e^{x} - 1}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -4.695602164921309 \cdot 10^{-06}:\\ \;\;\;\;\frac{e^{x}}{\frac{{\left(e^{x}\right)}^3 - 1}{\left(e^{x} + 1\right) + e^{x + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]

Error

Bits error versus x

Target

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

Derivation

  1. Split input into 2 regimes.

  2. if x < -4.695602164921309e-06

    1. Initial program 0.1

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

    3. Applied flip3-- 0.1

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

    4. Applied simplify 0.1

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

    5. Applied simplify 0.1

      \[\leadsto \frac{e^{x}}{\frac{{\left(e^{x}\right)}^3 - 1}{\color{blue}{\left(e^{x} + 1\right) + e^{x + x}}}}\]

    if -4.695602164921309e-06 < x

    1. Initial program 60.6

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

    2. Applied taylor 0.1

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

    3. Taylor expanded around 0 0.1

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

Runtime

Time bar (total: 11.1s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(3283856077 3183919125 3399458751 396155847 3688194147 1862413033)'
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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