Error

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Target

Derivation

1. Split input into 2 regimes

2. if x < -0.001988566952708495

   1. Initial program 0.0

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

   3. Applied add-log-exp0.0

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

   if -0.001988566952708495 < x

   1. Initial program 59.9

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

   2. Taylor expanded around 0 1.0

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

   4. Applied add-cube-cbrt1.0

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

   6. Applied pow1/332.4

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

   7. Applied pow1/332.4

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

   8. Applied pow-prod-down1.0

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

   9. Simplified1.0

      \[\leadsto {\color{blue}{\left(x \cdot \left(\frac{1}{144} \cdot x\right)\right)}}^{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{12} \cdot x} + \left(\frac{1}{x} + \frac{1}{2}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.001988566952708495:\\ \;\;\;\;\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot \left(\frac{1}{144} \cdot x\right)\right)}^{\frac{1}{3}} \cdot \sqrt[3]{x \cdot \frac{1}{12}} + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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

Details

Time bar (total: 10.4s)Debug log