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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.006883813457963097:\\ \;\;\;\;\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right)} - 1\\ \mathbf{elif}\;x \le 0.012613869149178572:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{(\left(\frac{2}{1 - {\left(e^{x}\right)}^{\left(-2 + -2\right)}}\right) \cdot \left(1 - e^{-2 \cdot x}\right) + \left(-1\right))_*} \cdot \sqrt[3]{(\left(\frac{2}{1 - {\left(e^{x}\right)}^{\left(-2 + -2\right)}}\right) \cdot \left(1 - e^{-2 \cdot x}\right) + \left(-1\right))_*}\right) \cdot \sqrt[3]{(\left(\frac{2}{1 - {\left(e^{x}\right)}^{\left(-2 + -2\right)}}\right) \cdot \left(1 - e^{-2 \cdot x}\right) + \left(-1\right))_*}\\ \end{array}\]

Derivation

Split input into 3 regimes
if x < -0.006883813457963097
1. Initial program 0.0
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. Using strategy rm
3. Applied add-cbrt-cube0.0
  \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}}} - 1\]
if -0.006883813457963097 < x < 0.012613869149178572
1. Initial program 59.0
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. 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 0.012613869149178572 < x
1. Initial program 0.0
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. Using strategy rm
3. Applied flip-+0.0
  \[\leadsto \frac{2}{\color{blue}{\frac{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}}{1 - e^{-2 \cdot x}}}} - 1\]
4. Applied associate-/r/0.0
  \[\leadsto \color{blue}{\frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}} \cdot \left(1 - e^{-2 \cdot x}\right)} - 1\]
5. Applied fma-neg0.0
  \[\leadsto \color{blue}{(\left(\frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}}\right) \cdot \left(1 - e^{-2 \cdot x}\right) + \left(-1\right))_*}\]
6. Simplified0.0
  \[\leadsto (\color{blue}{\left(\frac{2}{1 - {\left(e^{x}\right)}^{\left(-2 + -2\right)}}\right)} \cdot \left(1 - e^{-2 \cdot x}\right) + \left(-1\right))_*\]
7. Using strategy rm
8. Applied add-cube-cbrt0.0
  \[\leadsto \color{blue}{\left(\sqrt[3]{(\left(\frac{2}{1 - {\left(e^{x}\right)}^{\left(-2 + -2\right)}}\right) \cdot \left(1 - e^{-2 \cdot x}\right) + \left(-1\right))_*} \cdot \sqrt[3]{(\left(\frac{2}{1 - {\left(e^{x}\right)}^{\left(-2 + -2\right)}}\right) \cdot \left(1 - e^{-2 \cdot x}\right) + \left(-1\right))_*}\right) \cdot \sqrt[3]{(\left(\frac{2}{1 - {\left(e^{x}\right)}^{\left(-2 + -2\right)}}\right) \cdot \left(1 - e^{-2 \cdot x}\right) + \left(-1\right))_*}}\]
Recombined 3 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006883813457963097:\\ \;\;\;\;\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right)} - 1\\ \mathbf{elif}\;x \le 0.012613869149178572:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{(\left(\frac{2}{1 - {\left(e^{x}\right)}^{\left(-2 + -2\right)}}\right) \cdot \left(1 - e^{-2 \cdot x}\right) + \left(-1\right))_*} \cdot \sqrt[3]{(\left(\frac{2}{1 - {\left(e^{x}\right)}^{\left(-2 + -2\right)}}\right) \cdot \left(1 - e^{-2 \cdot x}\right) + \left(-1\right))_*}\right) \cdot \sqrt[3]{(\left(\frac{2}{1 - {\left(e^{x}\right)}^{\left(-2 + -2\right)}}\right) \cdot \left(1 - e^{-2 \cdot x}\right) + \left(-1\right))_*}\\ \end{array}\]

Error

Derivation

`if x < -0.006883813457963097`

`if -0.006883813457963097 < x < 0.012613869149178572`

`if 0.012613869149178572 < x`

Runtime