Average Error: 59.7 → 2.2
Time: 3.5m
Precision: 64
Internal Precision: 1408
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\left(-4\right) \cdot \frac{(\left(\pi \cdot \left(\frac{1}{48} \cdot \pi\right)\right) \cdot \left(f \cdot f\right) + \left(\log \left(\frac{4}{\pi}\right)\right))_* - (\left({\pi}^{4} \cdot \frac{7}{23040}\right) \cdot \left({f}^{4}\right) + \left(\log f\right))_*}{\pi}\]

Error

Bits error versus f

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 2.3

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

  3. Applied simplify2.3

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

  4. Taylor expanded around 0 2.2

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

  5. Applied simplify2.2

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

  7. Applied div-inv2.2

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

  8. Applied distribute-lft-neg-in2.2

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

  9. Applied associate-*l*2.2

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

  10. Applied simplify2.2

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

Runtime

Time bar (total: 3.5m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +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)))))))))