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Target

Derivation

1. Split input into 2 regimes

2. if x < -0.0018546157771360632

   1. Initial program 0.0

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

   3. Applied flip3--0.0

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

   4. Applied associate-/r/0.0

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

   5. Simplified0.0

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

   if -0.0018546157771360632 < x

   1. Initial program 60.3

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

   2. Taylor expanded around 0 0.8

      \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt0.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{12} \cdot x} \cdot \sqrt[3]{\frac{1}{12} \cdot x}\right) \cdot \sqrt[3]{\frac{1}{12} \cdot x}} + \left(\frac{1}{x} + \frac{1}{2}\right)\]
   5. Using strategy rm

   6. Applied add-exp-log0.8

      \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\frac{1}{12} \cdot x} \cdot \sqrt[3]{\frac{1}{12} \cdot x}\right)}} \cdot \sqrt[3]{\frac{1}{12} \cdot x} + \left(\frac{1}{x} + \frac{1}{2}\right)\]
   7. Using strategy rm

   8. Applied *-un-lft-identity0.8

      \[\leadsto e^{\color{blue}{1 \cdot \log \left(\sqrt[3]{\frac{1}{12} \cdot x} \cdot \sqrt[3]{\frac{1}{12} \cdot x}\right)}} \cdot \sqrt[3]{\frac{1}{12} \cdot x} + \left(\frac{1}{x} + \frac{1}{2}\right)\]

   9. Applied exp-prod0.8

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\sqrt[3]{\frac{1}{12} \cdot x} \cdot \sqrt[3]{\frac{1}{12} \cdot x}\right)\right)}} \cdot \sqrt[3]{\frac{1}{12} \cdot x} + \left(\frac{1}{x} + \frac{1}{2}\right)\]

   10. Simplified0.8

       \[\leadsto {\color{blue}{e}}^{\left(\log \left(\sqrt[3]{\frac{1}{12} \cdot x} \cdot \sqrt[3]{\frac{1}{12} \cdot x}\right)\right)} \cdot \sqrt[3]{\frac{1}{12} \cdot x} + \left(\frac{1}{x} + \frac{1}{2}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0018546157771360632:\\ \;\;\;\;\frac{e^{x}}{{\left(e^{x}\right)}^{3} - 1} \cdot \left(\left(e^{x} + 1\right) + e^{x + x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \sqrt[3]{x \cdot \frac{1}{12}} \cdot {e}^{\left(\log \left(\sqrt[3]{x \cdot \frac{1}{12}} \cdot \sqrt[3]{x \cdot \frac{1}{12}}\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 9.0s)Debug logProfile

herbie shell --seed 2018297 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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