Average Error: 59.8 → 2.2
Time: 5.0m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\left(\log \left(\frac{1}{\left(f \cdot \pi\right) \cdot \frac{1}{2} + \left(\left({\pi}^{5} \cdot {f}^{5}\right) \cdot \frac{1}{61440} + \frac{1}{192} \cdot \left({\pi}^{3} \cdot {f}^{3}\right)\right)}\right) + \log \left(e^{\frac{\pi}{4} \cdot \left(-f\right)} + e^{\frac{\pi}{4} \cdot f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\]

Error

Bits error versus f

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 59.8

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

  2. Taylor expanded around 0 2.2

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

  4. Applied div-inv2.2

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

  5. Applied log-prod2.2

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

  6. Final simplification2.2

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

Runtime

Time bar (total: 5.0m)Debug logProfile

herbie shell --seed 2018220 
(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)))))))))