Error

Target

Derivation

1. Split input into 3 regimes

2. if (cbrt (pow (/ (exp x) (expm1 x)) 3)) < -68.58172063229419

   1. Initial program 62.1

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

   2. Applied simplify0.0

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

   3. Taylor expanded around 0 0.1

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

   4. Applied simplify0.1

      \[\leadsto \color{blue}{\frac{1}{x} + (\frac{1}{12} \cdot x + \frac{1}{2})_*}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt0.1

      \[\leadsto \frac{1}{x} + \color{blue}{\sqrt{(\frac{1}{12} \cdot x + \frac{1}{2})_*} \cdot \sqrt{(\frac{1}{12} \cdot x + \frac{1}{2})_*}}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt0.1

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

   9. Applied associate-*l*0.1

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

   11. Applied add-cube-cbrt0.1

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

   if -68.58172063229419 < (cbrt (pow (/ (exp x) (expm1 x)) 3)) < 1711.090160547773

   1. Initial program 0.0

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

   2. Applied simplify0.0

      \[\leadsto \color{blue}{\frac{e^{x}}{(e^{x} - 1)^*}}\]
   3. Using strategy rm

   4. Applied add-cbrt-cube0.1

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

   5. Applied add-cbrt-cube0.2

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

   6. Applied cbrt-undiv0.2

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

   7. Applied simplify0.1

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

   if 1711.090160547773 < (cbrt (pow (/ (exp x) (expm1 x)) 3))

   1. Initial program 59.4

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

   2. Applied simplify1.3

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

   3. Taylor expanded around 0 1.1

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

   4. Applied simplify1.1

      \[\leadsto \color{blue}{\frac{1}{x} + (\frac{1}{12} \cdot x + \frac{1}{2})_*}\]
   5. Using strategy rm

   6. Applied add-exp-log1.1

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

Runtime

Time bar (total: 19.7s)Debug logProfile

herbie shell --seed 2018199 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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