Average Error: 59.5 → 2.5
Time: 8.0m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[(\left(\left(\pi \cdot \frac{7}{5760}\right) \cdot \pi\right) \cdot \left(\pi \cdot {\left({\left(e^{4}\right)}^{\left(\sqrt[3]{\log f} \cdot \sqrt[3]{\log f}\right)}\right)}^{\left(\sqrt[3]{\log f}\right)}\right) + \left(\left(\frac{4}{\pi} \cdot \log f - \left(f \cdot f\right) \cdot \left(\frac{1}{12} \cdot \pi\right)\right) - \frac{4}{\pi} \cdot \log \left(\frac{4}{\pi}\right)\right))_*\]

Error

Bits error versus f

Derivation

  1. Initial program 59.5

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

  2. Taylor expanded around 0 2.5

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

  3. Applied simplify2.5

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

  4. Taylor expanded around -inf 63.6

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

  5. Applied simplify2.5

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

  7. Applied add-cube-cbrt2.5

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

  8. Applied pow-unpow2.5

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

Runtime

Time bar (total: 8.0m)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' +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)))))))))