Error

Derivation

1. Initial program 58.6

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]

2. Simplified58.6

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

3. Taylor expanded around 0 0.2

   \[\leadsto \color{blue}{\left(2 \cdot x + \left(\frac{2}{3} \cdot {x}^{3} + \frac{2}{5} \cdot {x}^{5}\right)\right)} \cdot \frac{1}{2}\]

4. Simplified0.2

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

6. Applied flip3-+0.2

   \[\leadsto \left(x \cdot \color{blue}{\frac{{\left(x \cdot \left(\frac{2}{3} \cdot x\right)\right)}^{3} + {2}^{3}}{\left(x \cdot \left(\frac{2}{3} \cdot x\right)\right) \cdot \left(x \cdot \left(\frac{2}{3} \cdot x\right)\right) + \left(2 \cdot 2 - \left(x \cdot \left(\frac{2}{3} \cdot x\right)\right) \cdot 2\right)}} + {x}^{5} \cdot \frac{2}{5}\right) \cdot \frac{1}{2}\]

7. Applied associate-*r/0.2

   \[\leadsto \left(\color{blue}{\frac{x \cdot \left({\left(x \cdot \left(\frac{2}{3} \cdot x\right)\right)}^{3} + {2}^{3}\right)}{\left(x \cdot \left(\frac{2}{3} \cdot x\right)\right) \cdot \left(x \cdot \left(\frac{2}{3} \cdot x\right)\right) + \left(2 \cdot 2 - \left(x \cdot \left(\frac{2}{3} \cdot x\right)\right) \cdot 2\right)}} + {x}^{5} \cdot \frac{2}{5}\right) \cdot \frac{1}{2}\]

8. Final simplification0.2

   \[\leadsto \left(\frac{2}{5} \cdot {x}^{5} + \frac{x \cdot \left({\left(x \cdot \left(x \cdot \frac{2}{3}\right)\right)}^{3} + 8\right)}{\left(4 - \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot 2\right) + \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right)}\right) \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2019100 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))