Average Error: 59.7 → 2.1
Time: 4.7m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\frac{-1}{\frac{\pi}{4}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + \left({f}^{2} \cdot {\pi}^{2}\right) \cdot \frac{1}{48}\right) - \left(\log f + \left({f}^{4} \cdot {\pi}^{4}\right) \cdot \frac{7}{23040}\right)\right)\]

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 2.1

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

  3. Taylor expanded around 0 2.1

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\left(\log \left(\frac{4}{\pi}\right) + \frac{1}{48} \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(\frac{7}{23040} \cdot \left({f}^{4} \cdot {\pi}^{4}\right) + \log f\right)\right)}\]

  4. Final simplification2.1

    \[\leadsto \frac{-1}{\frac{\pi}{4}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + \left({f}^{2} \cdot {\pi}^{2}\right) \cdot \frac{1}{48}\right) - \left(\log f + \left({f}^{4} \cdot {\pi}^{4}\right) \cdot \frac{7}{23040}\right)\right)\]

Runtime

Time bar (total: 4.7m)Debug logProfile

herbie shell --seed 2018216 
(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)))))))))