Average Error: 45.6 → 0.1
Time: 6.0s
Precision: 64
Internal precision: 1408
\[\frac{e^{x}}{e^{x} - 1}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -4.010384934750878 \cdot 10^{-06}:\\ \;\;\;\;\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original45.6
Comparison45.5
Herbie0.1
\[ \frac{1}{1 - e^{-x}} \]

Derivation

  1. Split input into 2 regimes.

  2. if x < -4.010384934750878e-06

    1. Initial program 0.1

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

    3. Applied add-log-exp 0.1

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

    if -4.010384934750878e-06 < x

    1. Initial program 60.7

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

    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: 6.0s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1066785882 2324371342 4059510649 1466361199 2701357084 1216585281)'
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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