Average Error: 59.7 → 2.2
Time: 2.4m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\frac{-1}{\left(\frac{1}{4} \cdot \frac{\pi}{\log \left(\frac{4}{\pi}\right) - \log f} + \left(\frac{7}{92160} \cdot \frac{{f}^{4} \cdot {\pi}^{5}}{{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)}^{2}} + \frac{{f}^{4} \cdot {\pi}^{5}}{{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)}^{3}} \cdot \frac{1}{9216}\right)\right) - \frac{{\pi}^{3} \cdot {f}^{2}}{{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)}^{2}} \cdot \frac{1}{192}}\]

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 2.2

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

  3. Simplified2.2

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left((\frac{1}{48} \cdot \left(\left(f \cdot \pi\right) \cdot \left(f \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right)\right))_* - (\left({f}^{4} \cdot \frac{7}{23040}\right) \cdot \left({\pi}^{4}\right) + \left(\log f\right))_*\right)}\]
  4. Using strategy rm

  5. Applied associate-*l/2.1

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

  6. Simplified2.1

    \[\leadsto -\frac{\color{blue}{(\left(\left(f \cdot \pi\right) \cdot \left(f \cdot \pi\right)\right) \cdot \frac{1}{48} + \left(\log \left(\frac{4}{\pi}\right)\right))_* - (\left({\pi}^{4}\right) \cdot \left({f}^{4} \cdot \frac{7}{23040}\right) + \left(\log f\right))_*}}{\frac{\pi}{4}}\]
  7. Using strategy rm

  8. Applied clear-num2.2

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

  9. Taylor expanded around 0 2.2

    \[\leadsto -\frac{1}{\color{blue}{\left(\frac{1}{4} \cdot \frac{\pi}{\log \left(\frac{4}{\pi}\right) - \log f} + \left(\frac{7}{92160} \cdot \frac{{\pi}^{5} \cdot {f}^{4}}{{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)}^{2}} + \frac{1}{9216} \cdot \frac{{\pi}^{5} \cdot {f}^{4}}{{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)}^{3}}\right)\right) - \frac{1}{192} \cdot \frac{{f}^{2} \cdot {\pi}^{3}}{{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)}^{2}}}}\]

  10. Final simplification2.2

    \[\leadsto \frac{-1}{\left(\frac{1}{4} \cdot \frac{\pi}{\log \left(\frac{4}{\pi}\right) - \log f} + \left(\frac{7}{92160} \cdot \frac{{f}^{4} \cdot {\pi}^{5}}{{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)}^{2}} + \frac{{f}^{4} \cdot {\pi}^{5}}{{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)}^{3}} \cdot \frac{1}{9216}\right)\right) - \frac{{\pi}^{3} \cdot {f}^{2}}{{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)}^{2}} \cdot \frac{1}{192}}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018249 +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)))))))))