Initial program 61.3
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
Taylor expanded around 0 2.4
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{0.5 \cdot \left(f \cdot \pi\right) + \left(0.00520833333333333304 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + 1.62760416666666664 \cdot 10^{-5} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right)\]
Taylor expanded around 0 2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right) + \left(0.00347222222222222246 \cdot \frac{{\pi}^{4} \cdot {f}^{4}}{{4}^{2}} + \log f\right)\right)\right)}\]
- Using strategy
rm Applied add-cube-cbrt2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right) + \left(0.00347222222222222246 \cdot \frac{{\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)}}^{4} \cdot {f}^{4}}{{4}^{2}} + \log f\right)\right)\right)\]
Applied unpow-prod-down2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right) + \left(0.00347222222222222246 \cdot \frac{\color{blue}{\left({\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)}^{4} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)} \cdot {f}^{4}}{{4}^{2}} + \log f\right)\right)\right)\]
Simplified2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right) + \left(0.00347222222222222246 \cdot \frac{\left(\color{blue}{{\left({\left(\sqrt[3]{\pi}\right)}^{2}\right)}^{4}} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + \log f\right)\right)\right)\]
- Using strategy
rm Applied add-sqr-sqrt2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right) + \left(0.00347222222222222246 \cdot \frac{\left({\left({\left(\sqrt[3]{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right)}^{2}\right)}^{4} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + \log f\right)\right)\right)\]
Applied cbrt-prod2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right) + \left(0.00347222222222222246 \cdot \frac{\left({\left({\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}}^{2}\right)}^{4} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + \log f\right)\right)\right)\]
Applied unpow-prod-down2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right) + \left(0.00347222222222222246 \cdot \frac{\left({\color{blue}{\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}\right)}}^{4} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + \log f\right)\right)\right)\]
Applied unpow-prod-down2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right) + \left(0.00347222222222222246 \cdot \frac{\left(\color{blue}{\left({\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}\right)}^{4} \cdot {\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}\right)}^{4}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + \log f\right)\right)\right)\]
Simplified2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right) + \left(0.00347222222222222246 \cdot \frac{\left(\left(\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8}} \cdot {\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}\right)}^{4}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + \log f\right)\right)\right)\]
Simplified2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right) + \left(0.00347222222222222246 \cdot \frac{\left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8}}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + \log f\right)\right)\right)\]
Final simplification2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + 0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right) + \left(0.00347222222222222246 \cdot \frac{\left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + \log f\right)\right)\right)\]
