Average Error: 59.6 → 3.3
Time: 7.6m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\left(-\frac{4}{\pi}\right) \cdot \log \left(\sqrt[3]{\frac{4}{\pi}} \cdot \sqrt[3]{\frac{4}{\pi}}\right) + (\left(-\frac{4}{\pi}\right) \cdot \left(\log \left(\sqrt[3]{\frac{4}{\pi}}\right)\right) + \left((\left(-f\right) \cdot \left((\left(\pi \cdot f\right) \cdot \frac{7}{96} + \frac{3}{2})_*\right) + \left(\frac{\log f}{\frac{\pi}{4}}\right))_*\right))_*\]

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 3.3

    \[\leadsto -\color{blue}{\left(\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\frac{3}{2} \cdot f + \frac{7}{96} \cdot \left(\pi \cdot {f}^{2}\right)\right)\right) - 4 \cdot \frac{\log f}{\pi}\right)}\]

  3. Applied simplify3.3

    \[\leadsto \color{blue}{\left(-\frac{4}{\pi}\right) \cdot \log \left(\frac{4}{\pi}\right) - \left((\left(\frac{7}{96} \cdot f\right) \cdot \left(f \cdot \pi\right) + \left(f \cdot \frac{3}{2}\right))_* - \frac{\log f}{\frac{\pi}{4}}\right)}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt3.3

    \[\leadsto \left(-\frac{4}{\pi}\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{\frac{4}{\pi}} \cdot \sqrt[3]{\frac{4}{\pi}}\right) \cdot \sqrt[3]{\frac{4}{\pi}}\right)} - \left((\left(\frac{7}{96} \cdot f\right) \cdot \left(f \cdot \pi\right) + \left(f \cdot \frac{3}{2}\right))_* - \frac{\log f}{\frac{\pi}{4}}\right)\]

  6. Applied log-prod3.3

    \[\leadsto \left(-\frac{4}{\pi}\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{\frac{4}{\pi}} \cdot \sqrt[3]{\frac{4}{\pi}}\right) + \log \left(\sqrt[3]{\frac{4}{\pi}}\right)\right)} - \left((\left(\frac{7}{96} \cdot f\right) \cdot \left(f \cdot \pi\right) + \left(f \cdot \frac{3}{2}\right))_* - \frac{\log f}{\frac{\pi}{4}}\right)\]

  7. Applied distribute-lft-in3.3

    \[\leadsto \color{blue}{\left(\left(-\frac{4}{\pi}\right) \cdot \log \left(\sqrt[3]{\frac{4}{\pi}} \cdot \sqrt[3]{\frac{4}{\pi}}\right) + \left(-\frac{4}{\pi}\right) \cdot \log \left(\sqrt[3]{\frac{4}{\pi}}\right)\right)} - \left((\left(\frac{7}{96} \cdot f\right) \cdot \left(f \cdot \pi\right) + \left(f \cdot \frac{3}{2}\right))_* - \frac{\log f}{\frac{\pi}{4}}\right)\]

  8. Applied associate--l+3.3

    \[\leadsto \color{blue}{\left(-\frac{4}{\pi}\right) \cdot \log \left(\sqrt[3]{\frac{4}{\pi}} \cdot \sqrt[3]{\frac{4}{\pi}}\right) + \left(\left(-\frac{4}{\pi}\right) \cdot \log \left(\sqrt[3]{\frac{4}{\pi}}\right) - \left((\left(\frac{7}{96} \cdot f\right) \cdot \left(f \cdot \pi\right) + \left(f \cdot \frac{3}{2}\right))_* - \frac{\log f}{\frac{\pi}{4}}\right)\right)}\]

  9. Applied simplify3.3

    \[\leadsto \left(-\frac{4}{\pi}\right) \cdot \log \left(\sqrt[3]{\frac{4}{\pi}} \cdot \sqrt[3]{\frac{4}{\pi}}\right) + \color{blue}{(\left(-\frac{4}{\pi}\right) \cdot \left(\log \left(\sqrt[3]{\frac{4}{\pi}}\right)\right) + \left((\left(-f\right) \cdot \left((\left(\pi \cdot f\right) \cdot \frac{7}{96} + \frac{3}{2})_*\right) + \left(\frac{\log f}{\frac{\pi}{4}}\right))_*\right))_*}\]

Runtime

Time bar (total: 7.6m)Debug logProfile

herbie shell --seed 2018195 +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)))))))))