\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.00845879631570754:\\ \;\;\;\;\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\right)}}\right)\\ \mathbf{elif}\;x \le 0.008115788255173338:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}} \cdot \sqrt[3]{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}}\right)\right)\\ \end{array}\]

Derivation

Split input into 3 regimes
if x < -0.00845879631570754
1. Initial program 0.0
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. Initial simplification0.0
  \[\leadsto \frac{2}{1 + e^{-2 \cdot x}} - 1\]
3. Using strategy rm
4. Applied add-cube-cbrt0.0
  \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\]
5. Using strategy rm
6. Applied add-cbrt-cube0.0
  \[\leadsto \left(\sqrt[3]{\color{blue}{\sqrt[3]{\left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}}} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\]
if -0.00845879631570754 < x < 0.008115788255173338
1. Initial program 59.0
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. Initial simplification59.0
  \[\leadsto \frac{2}{1 + e^{-2 \cdot x}} - 1\]
3. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]
if 0.008115788255173338 < x
1. Initial program 0.0
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. Initial simplification0.0
  \[\leadsto \frac{2}{1 + e^{-2 \cdot x}} - 1\]
3. Using strategy rm
4. Applied add-cube-cbrt0.0
  \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\]
5. Using strategy rm
6. Applied add-cube-cbrt0.0
  \[\leadsto \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}} \cdot \sqrt[3]{\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)}\]
Recombined 3 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00845879631570754:\\ \;\;\;\;\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\right)}}\right)\\ \mathbf{elif}\;x \le 0.008115788255173338:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}} \cdot \sqrt[3]{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}}\right)\right)\\ \end{array}\]

Error

Try it out

Derivation

`if x < -0.00845879631570754`

`if -0.00845879631570754 < x < 0.008115788255173338`

`if 0.008115788255173338 < x`

Runtime

In
Out