Average Error: 59.6 → 2.4
Time: 3.7m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[-\left(\left((\left(\frac{7}{5760} \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(-\pi\right) \cdot {f}^{4}\right) + \left(\left(f \cdot \frac{1}{12}\right) \cdot \left(f \cdot \pi\right)\right))_* - \left(\frac{4}{\pi} \cdot \left(\log f - \log 4\right) + \frac{4}{\pi} \cdot \log \pi\right)\right) + 0\right)\]

Error

Bits error versus f

Derivation

  1. Initial program 59.6

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

  2. Taylor expanded around 0 2.3

    \[\leadsto -\color{blue}{\left(\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \frac{1}{12} \cdot \left(\pi \cdot {f}^{2}\right)\right) - \left(\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}\right)\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt63.5

    \[\leadsto -\left(\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \frac{1}{12} \cdot \left(\pi \cdot {f}^{2}\right)\right) - \color{blue}{\sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}} \cdot \sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}}}\right)\]

  5. Applied add-sqr-sqrt63.5

    \[\leadsto -\left(\color{blue}{\sqrt{4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \frac{1}{12} \cdot \left(\pi \cdot {f}^{2}\right)} \cdot \sqrt{4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \frac{1}{12} \cdot \left(\pi \cdot {f}^{2}\right)}} - \sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}} \cdot \sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}}\right)\]

  6. Applied prod-diff63.5

    \[\leadsto -\color{blue}{\left((\left(\sqrt{4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \frac{1}{12} \cdot \left(\pi \cdot {f}^{2}\right)}\right) \cdot \left(\sqrt{4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \frac{1}{12} \cdot \left(\pi \cdot {f}^{2}\right)}\right) + \left(-\sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}} \cdot \sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}}\right))_* + (\left(-\sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}}\right) \cdot \left(\sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}}\right) + \left(\sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}} \cdot \sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}}\right))_*\right)}\]

  7. Applied simplify63.5

    \[\leadsto -\left(\color{blue}{\left((\left(\frac{7}{5760} \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(-\pi\right) \cdot {f}^{4}\right) + \left(\left(f \cdot \frac{1}{12}\right) \cdot \left(f \cdot \pi\right)\right))_* - \frac{4}{\pi} \cdot \left(\log f - \log \left(\frac{4}{\pi}\right)\right)\right)} + (\left(-\sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}}\right) \cdot \left(\sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}}\right) + \left(\sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}} \cdot \sqrt{\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}}\right))_*\right)\]

  8. Applied simplify2.3

    \[\leadsto -\left(\left((\left(\frac{7}{5760} \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(-\pi\right) \cdot {f}^{4}\right) + \left(\left(f \cdot \frac{1}{12}\right) \cdot \left(f \cdot \pi\right)\right))_* - \frac{4}{\pi} \cdot \left(\log f - \log \left(\frac{4}{\pi}\right)\right)\right) + \color{blue}{0}\right)\]
  9. Using strategy rm

  10. Applied log-div2.3

    \[\leadsto -\left(\left((\left(\frac{7}{5760} \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(-\pi\right) \cdot {f}^{4}\right) + \left(\left(f \cdot \frac{1}{12}\right) \cdot \left(f \cdot \pi\right)\right))_* - \frac{4}{\pi} \cdot \left(\log f - \color{blue}{\left(\log 4 - \log \pi\right)}\right)\right) + 0\right)\]

  11. Applied associate--r-2.4

    \[\leadsto -\left(\left((\left(\frac{7}{5760} \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(-\pi\right) \cdot {f}^{4}\right) + \left(\left(f \cdot \frac{1}{12}\right) \cdot \left(f \cdot \pi\right)\right))_* - \frac{4}{\pi} \cdot \color{blue}{\left(\left(\log f - \log 4\right) + \log \pi\right)}\right) + 0\right)\]

  12. Applied distribute-lft-in2.4

    \[\leadsto -\left(\left((\left(\frac{7}{5760} \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(-\pi\right) \cdot {f}^{4}\right) + \left(\left(f \cdot \frac{1}{12}\right) \cdot \left(f \cdot \pi\right)\right))_* - \color{blue}{\left(\frac{4}{\pi} \cdot \left(\log f - \log 4\right) + \frac{4}{\pi} \cdot \log \pi\right)}\right) + 0\right)\]

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018199 +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)))))))))