Average Error: 59.5 → 2.5
Time: 2.1m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\log \left(\sqrt{(\left(\pi \cdot f\right) \cdot \frac{1}{12} + \left((\left(\left(\pi \cdot f\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(-\frac{1}{2880}\right) \cdot \left(f \cdot f\right)\right) + \left(\frac{\frac{4}{\pi}}{f}\right))_*\right))_*}\right) \cdot \left(-\frac{4}{\pi}\right) + \left(\log \left(\sqrt{\left(\frac{1}{\pi \cdot f} \cdot 4 + \frac{1}{12} \cdot \left(\pi \cdot f\right)\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)}\right) \cdot \sqrt{\frac{1}{\frac{\pi}{4}}}\right) \cdot \left(-\sqrt{\frac{4}{\pi}}\right)\]

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

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt2.8

    \[\leadsto -\color{blue}{\left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \sqrt{\frac{1}{\frac{\pi}{4}}}\right)} \cdot \log \left(\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right)\]

  5. Applied associate-*l*2.5

    \[\leadsto -\color{blue}{\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right)\right)}\]

  6. Simplified2.5

    \[\leadsto -\color{blue}{\sqrt{\frac{4}{\pi}}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right)\right)\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt2.5

    \[\leadsto -\sqrt{\frac{4}{\pi}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \color{blue}{\left(\sqrt{\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)} \cdot \sqrt{\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)}\right)}\right)\]

  9. Applied log-prod2.5

    \[\leadsto -\sqrt{\frac{4}{\pi}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \color{blue}{\left(\log \left(\sqrt{\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)}\right) + \log \left(\sqrt{\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)}\right)\right)}\right)\]

  10. Applied distribute-lft-in2.5

    \[\leadsto -\sqrt{\frac{4}{\pi}} \cdot \color{blue}{\left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\sqrt{\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)}\right) + \sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\sqrt{\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)}\right)\right)}\]

  11. Applied distribute-rgt-in2.5

    \[\leadsto -\color{blue}{\left(\left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\sqrt{\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)}\right)\right) \cdot \sqrt{\frac{4}{\pi}} + \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\sqrt{\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)}\right)\right) \cdot \sqrt{\frac{4}{\pi}}\right)}\]

  12. Simplified2.5

    \[\leadsto -\left(\left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\sqrt{\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)}\right)\right) \cdot \sqrt{\frac{4}{\pi}} + \color{blue}{\log \left(\sqrt{(\left(\pi \cdot f\right) \cdot \frac{1}{12} + \left((\left(\left(\pi \cdot f\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(f \cdot f\right) \cdot \left(-\frac{1}{2880}\right)\right) + \left(\frac{\frac{4}{\pi}}{f}\right))_*\right))_*}\right) \cdot \frac{4}{\pi}}\right)\]

  13. Final simplification2.5

    \[\leadsto \log \left(\sqrt{(\left(\pi \cdot f\right) \cdot \frac{1}{12} + \left((\left(\left(\pi \cdot f\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(-\frac{1}{2880}\right) \cdot \left(f \cdot f\right)\right) + \left(\frac{\frac{4}{\pi}}{f}\right))_*\right))_*}\right) \cdot \left(-\frac{4}{\pi}\right) + \left(\log \left(\sqrt{\left(\frac{1}{\pi \cdot f} \cdot 4 + \frac{1}{12} \cdot \left(\pi \cdot f\right)\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)}\right) \cdot \sqrt{\frac{1}{\frac{\pi}{4}}}\right) \cdot \left(-\sqrt{\frac{4}{\pi}}\right)\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018234 +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)))))))))