Average Error: 61.7 → 2.4
Time: 17.4s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[-\frac{1}{\frac{\pi}{4}} \cdot \left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{\left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot {\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2} \cdot \sqrt[3]{\sqrt{\pi}}}\right)}^{8}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)\]

Error

Bits error versus f

Derivation

  1. Initial program 61.7

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

  2. Taylor expanded around 0 2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{{\pi}^{4} \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{{\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)}}^{4} \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)\]

  5. Applied unpow-prod-down2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{\color{blue}{\left({\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)}^{4} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)} \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)\]

  6. Simplified2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{\left(\color{blue}{{\left({\left(\sqrt[3]{\pi}\right)}^{2}\right)}^{4}} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{\left({\left({\left(\sqrt[3]{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right)}^{2}\right)}^{4} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)\]

  9. Applied cbrt-prod2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{\left({\left({\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}}^{2}\right)}^{4} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)\]

  10. Applied unpow-prod-down2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{\left({\color{blue}{\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}\right)}}^{4} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)\]

  11. Applied unpow-prod-down2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{\left(\color{blue}{\left({\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}\right)}^{4} \cdot {\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}\right)}^{4}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)\]

  12. Simplified2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{\left(\left(\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8}} \cdot {\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}\right)}^{4}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)\]

  13. Simplified2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{\left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8}}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)\]
  14. Using strategy rm

  15. Applied add-cube-cbrt2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{\left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot {\left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}}}\right)}^{8}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)\]

  16. Simplified2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{\left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot {\left(\sqrt[3]{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}} \cdot \sqrt[3]{\sqrt{\pi}}}\right)}^{8}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)\]

  17. Final simplification2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{\left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot {\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2} \cdot \sqrt[3]{\sqrt{\pi}}}\right)}^{8}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right) \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020174 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary64
  (neg (* (/ 1.0 (/ PI 4.0)) (log (/ (+ (exp (* (/ PI 4.0) f)) (exp (neg (* (/ PI 4.0) f)))) (- (exp (* (/ PI 4.0) f)) (exp (neg (* (/ PI 4.0) f)))))))))