Average Error: 59.8 → 0.7
Time: 2.7m
Precision: 64
Internal Precision: 128
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\left(\log \left(\frac{1}{\sqrt{\left(\frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right) + \left({\pi}^{3} \cdot {f}^{3}\right) \cdot \frac{1}{192}\right) + \frac{1}{2} \cdot \left(f \cdot \pi\right)}}\right) \cdot \left(-\sqrt{\frac{1}{\frac{\pi}{4}}}\right)\right) \cdot \sqrt{\frac{4}{\pi}} + \left(-\frac{\log \left(\frac{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{f \cdot \pi}{4}}}{\sqrt{(\pi \cdot \left((\left(\pi \cdot \pi\right) \cdot \left(\left(f \cdot f\right) \cdot \left(\frac{1}{192} \cdot f\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left(\left(\frac{1}{61440} \cdot {f}^{5}\right) \cdot {\pi}^{5}\right))_*}}\right)}{\frac{\pi}{4}}\right)\]

Error

Bits error versus f

Derivation

  1. Initial program 59.8

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

  2. Taylor expanded around 0 0.8

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

  4. Applied add-sqr-sqrt0.8

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

  5. Applied *-un-lft-identity0.8

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

  6. Applied times-frac0.8

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

  7. Applied log-prod0.9

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

  8. Applied distribute-rgt-in0.9

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

  9. Simplified0.8

    \[\leadsto -\left(\log \left(\frac{1}{\sqrt{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right) \cdot \frac{1}{\frac{\pi}{4}} + \color{blue}{\frac{\log \left(\frac{e^{\frac{f \cdot \pi}{4}} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\sqrt{(\pi \cdot \left((\left(\pi \cdot \pi\right) \cdot \left(\left(\frac{1}{192} \cdot f\right) \cdot \left(f \cdot f\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left(\left({f}^{5} \cdot \frac{1}{61440}\right) \cdot {\pi}^{5}\right))_*}}\right)}{\frac{\pi}{4}}}\right)\]
  10. Using strategy rm

  11. Applied add-sqr-sqrt0.7

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

  12. Applied associate-*r*0.7

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

  13. Simplified0.7

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

  14. Final simplification0.7

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018277 +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)))))))))