Input Error: 45.0b
Output Error: 0.1b
Time: 9.4s
Precision: 64b
Ground Truth: 128b
\[\frac{e^{x}}{e^{x} - 1}\]
⬇
\[\begin{cases} \frac{e^{x}}{\sqrt[3]{{\left(e^{x} - 1\right)}^3}} & \text{when } x \le -2.110554417076555 \cdot 10^{-07} \\ \frac{1}{x} + \left(\frac{1}{2} + \frac{1}{12} \cdot x\right) & \text{otherwise} \end{cases}\]

Error

Bits error versus x

Derivation

    if x < -2.110554417076555e-07

    1. Initial program 0.2b

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

    3. Applied add-cbrt-cube 0.2b

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

    if -2.110554417076555e-07 < x

    1. Initial program 60.8b

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

    2. Applied taylor 0.1b

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

    3. Taylor expanded around 0 0.1b

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

Runtime

Total time: 9.4s Debug log

herbie --seed '#(1077686180 1792156481 1245604953 765346327 4231839880 732746889)'
(FPCore (x)
  :name "NMSE section 3.11"
  
  :target
  (/ 1 (- 1 (exp (- x))))(/ (exp x) (- (exp x) 1)))