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

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9994820352830056:\\ \;\;\;\;\frac{e^{x}}{\sqrt[3]{{\left(e^{x} - 1\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\\ \end{array}\]

Original	40.1
Target	39.8
Herbie	0.6

Derivation

Split input into 2 regimes
if (exp x) < 0.9994820352830056
1. Initial program 0.0
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Using strategy rm
3. Applied add-cbrt-cube0.0
  \[\leadsto \frac{e^{x}}{\color{blue}{\sqrt[3]{\left(\left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)\right) \cdot \left(e^{x} - 1\right)}}}\]
4. Applied simplify0.0
  \[\leadsto \frac{e^{x}}{\sqrt[3]{\color{blue}{{\left(e^{x} - 1\right)}^{3}}}}\]
if 0.9994820352830056 < (exp x)
1. Initial program 60.3
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Taylor expanded around 0 0.8
  \[\leadsto \frac{e^{x}}{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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

Error

Target

Derivation

`if (exp x) < 0.9994820352830056`

`if 0.9994820352830056 < (exp x)`

Runtime