Average Error: 59.7 → 2.3
Time: 2.8m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\frac{-4}{\pi} \cdot \log \left(\frac{e^{\frac{f \cdot \pi}{4}} + e^{\frac{-\pi}{\frac{4}{f}}}}{\sqrt{\frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right) + \left(\left(f \cdot \frac{1}{2}\right) \cdot \pi + \left(\left(f \cdot \pi\right) \cdot \left(f \cdot \pi\right)\right) \cdot \left(\frac{1}{192} \cdot \left(f \cdot \pi\right)\right)\right)}}\right) + \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\frac{1}{\sqrt{\left(\frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right) + \left({\pi}^{3} \cdot {f}^{3}\right) \cdot \frac{1}{192}\right) + \frac{1}{2} \cdot \left(f \cdot \pi\right)}}\right)\right) \cdot \left(-\sqrt{\frac{4}{\pi}}\right)\]

Error

Bits error versus f

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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 2.4

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

  4. Applied add-sqr-sqrt2.7

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

  5. Applied associate-*l*2.4

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

  6. Simplified2.4

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

  8. Applied add-sqr-sqrt2.4

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

  9. Applied *-un-lft-identity2.4

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

  10. Applied *-un-lft-identity2.4

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

  11. Applied distribute-lft-out2.4

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

  12. Applied times-frac2.4

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

  13. Applied log-prod2.4

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

  14. Applied distribute-rgt-in2.3

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

  15. Applied distribute-rgt-in2.4

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

  16. Simplified2.3

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

  17. Final simplification2.3

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

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018255 
(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)))))))))