Error

Derivation

1. Initial program 59.5

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

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-cube-cbrt0.8

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

5. Applied log-prod0.9

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

6. Applied distribute-rgt-in0.9

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

7. Simplified0.8

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

8. Final simplification0.8

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018354 +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)))))))))