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

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9989605912940498:\\ \;\;\;\;\frac{e^{x}}{\left(\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} - 1}\right) \cdot \sqrt[3]{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Original	40.3
Target	39.9
Herbie	0.6

Derivation

Split input into 2 regimes
if (exp x) < 0.9989605912940498
1. Initial program 0.0
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Using strategy rm
3. Applied add-cube-cbrt0.0
  \[\leadsto \frac{e^{x}}{\color{blue}{\left(\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} - 1}\right) \cdot \sqrt[3]{e^{x} - 1}}}\]
if 0.9989605912940498 < (exp x)
1. Initial program 60.3
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Taylor expanded around 0 1.0
  \[\leadsto \frac{e^{x}}{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}\]
3. Taylor expanded around 0 0.9
  \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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

Error

Target

Derivation

`if (exp x) < 0.9989605912940498`

`if 0.9989605912940498 < (exp x)`

Runtime