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Derivation

1. Initial program 59.6

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

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-exp-log0.8

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

5. Applied add-exp-log0.8

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

6. Applied div-exp0.8

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

8. Applied associate-*l/0.7

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

9. Simplified0.7

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

10. Final simplification0.7

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

Reproduce

herbie shell --seed 2018360 
(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)))))))))

Details

Time bar (total: 53.0s)Debug log