Error: 45.4 → 0.2
Time: 13.7s
Precision: 64
Ground Truth: 128
\[\frac{e^{x}}{e^{x} - 1}\]
⬇
\[\begin{cases} e^{x} \cdot \frac{1}{e^{x} - 1} & \text{when } x \le -1.8773586721409905 \cdot 10^{-08} \\ \frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right) & \text{otherwise} \end{cases}\]

Error

Bits error versus x

Derivation

    if x < -1.8773586721409905e-08

    1. Initial program 0.3

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

    3. Applied div-inv 0.3

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

    if -1.8773586721409905e-08 < x

    1. Initial program 60.7

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

    2. Applied taylor 0.1

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

    3. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]
  1. Removed slow pow expressions

Runtime

Total time: 13.7s Debug log

Please report a bug with the following info:

herbie --seed '#(2600893960 2918582358 1462551469 2542791771 2229689717 1439756912)'
(FPCore (x)
  :name "NMSE section 3.11"
  
  :target
  (/ 1 (- 1 (exp (- x))))(/ (exp x) (- (exp x) 1)))