Error

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 *-un-lft-identity0.8

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

5. Applied log-prod0.8

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

6. Applied distribute-lft-in0.8

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

7. Simplified0.7

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

8. Final simplification0.7

   \[\leadsto \frac{-1}{\frac{\pi}{4}} \cdot 0 + \left(-\frac{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\left(-\pi\right) \cdot \frac{f}{4}}}{(\pi \cdot \left((\left(\pi \cdot \pi\right) \cdot \left(\left(f \cdot f\right) \cdot \left(f \cdot \frac{1}{192}\right)\right) + \left(\frac{1}{2} \cdot f\right))_*\right) + \left({f}^{5} \cdot \left({\pi}^{5} \cdot \frac{1}{61440}\right)\right))_*}\right)}{\frac{\pi}{4}}\right)\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(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: 59.8s)Debug log