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Derivation

1. Initial program 59.7

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]

2. Taylor expanded around 0 0.8

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right)\]
3. Using strategy rm

4. Applied add-sqr-sqrt1.1

   \[\leadsto -\color{blue}{\left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \sqrt{\frac{1}{\frac{\pi}{4}}}\right)} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right)\]

5. Applied associate-*l*0.8

   \[\leadsto -\color{blue}{\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right)\right)}\]

6. Simplified0.8

   \[\leadsto -\color{blue}{\sqrt{\frac{4}{\pi}}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right)\right)\]
7. Using strategy rm

8. Applied add-sqr-sqrt0.8

   \[\leadsto -\sqrt{\frac{4}{\pi}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\sqrt{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)} \cdot \sqrt{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}}\right)\right)\]

9. Applied add-cube-cbrt0.8

   \[\leadsto -\sqrt{\frac{4}{\pi}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}}{\sqrt{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)} \cdot \sqrt{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right)\right)\]

10. Applied times-frac0.8

    \[\leadsto -\sqrt{\frac{4}{\pi}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \color{blue}{\left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sqrt{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}} \cdot \frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sqrt{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right)}\right)\]

11. Final simplification0.8

    \[\leadsto \left(\log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot \left(-f\right)} + e^{f \cdot \frac{\pi}{4}}}}{\sqrt{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right)}} \cdot \frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot \left(-f\right)} + e^{f \cdot \frac{\pi}{4}}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot \left(-f\right)} + e^{f \cdot \frac{\pi}{4}}}}{\sqrt{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right)}}\right) \cdot \sqrt{\frac{1}{\frac{\pi}{4}}}\right) \cdot \left(-\sqrt{\frac{4}{\pi}}\right)\]

Reproduce

herbie shell --seed 2018362 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  (- (* (/ 1 (/ PI 4)) (log (/ (+ (exp (* (/ PI 4) f)) (exp (- (* (/ PI 4) f)))) (- (exp (* (/ PI 4) f)) (exp (- (* (/ PI 4) f)))))))))

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