Average Error: 61.5 → 2.1
Time: 15.7s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[1 \cdot \left(4 \cdot \frac{\log \left(\frac{0.00520833333333333304 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.62760416666666664 \cdot 10^{-5} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + f \cdot \left(\pi \cdot 0.5\right)\right)}{\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}}\right) - \log \left(\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}\right)}{\pi}\right)\]

Error

Bits error versus f

Derivation

  1. Initial program 61.5

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

  2. Simplified61.5

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

  3. Taylor expanded around 0 2.1

    \[\leadsto 1 \cdot \left(4 \cdot \frac{\log \left(\frac{\color{blue}{0.00520833333333333304 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.62760416666666664 \cdot 10^{-5} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + 0.5 \cdot \left(f \cdot \pi\right)\right)}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}\right)}{\pi}\right)\]

  4. Simplified2.1

    \[\leadsto 1 \cdot \left(4 \cdot \frac{\log \left(\frac{\color{blue}{0.00520833333333333304 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.62760416666666664 \cdot 10^{-5} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \pi \cdot \left(f \cdot 0.5\right)\right)}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}\right)}{\pi}\right)\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt2.1

    \[\leadsto 1 \cdot \left(4 \cdot \frac{\log \left(\frac{0.00520833333333333304 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.62760416666666664 \cdot 10^{-5} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \pi \cdot \left(f \cdot 0.5\right)\right)}{\color{blue}{\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}} \cdot \sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}}}\right)}{\pi}\right)\]

  7. Applied *-un-lft-identity2.1

    \[\leadsto 1 \cdot \left(4 \cdot \frac{\log \left(\frac{\color{blue}{1 \cdot \left(0.00520833333333333304 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.62760416666666664 \cdot 10^{-5} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \pi \cdot \left(f \cdot 0.5\right)\right)\right)}}{\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}} \cdot \sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}}\right)}{\pi}\right)\]

  8. Applied times-frac2.1

    \[\leadsto 1 \cdot \left(4 \cdot \frac{\log \color{blue}{\left(\frac{1}{\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}} \cdot \frac{0.00520833333333333304 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.62760416666666664 \cdot 10^{-5} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \pi \cdot \left(f \cdot 0.5\right)\right)}{\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}}\right)}}{\pi}\right)\]

  9. Applied log-prod2.1

    \[\leadsto 1 \cdot \left(4 \cdot \frac{\color{blue}{\log \left(\frac{1}{\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}}\right) + \log \left(\frac{0.00520833333333333304 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.62760416666666664 \cdot 10^{-5} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \pi \cdot \left(f \cdot 0.5\right)\right)}{\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}}\right)}}{\pi}\right)\]

  10. Simplified2.1

    \[\leadsto 1 \cdot \left(4 \cdot \frac{\color{blue}{\left(-\log \left(\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}\right)\right)} + \log \left(\frac{0.00520833333333333304 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.62760416666666664 \cdot 10^{-5} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \pi \cdot \left(f \cdot 0.5\right)\right)}{\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}}\right)}{\pi}\right)\]

  11. Simplified2.1

    \[\leadsto 1 \cdot \left(4 \cdot \frac{\left(-\log \left(\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}\right)\right) + \color{blue}{\log \left(\frac{0.00520833333333333304 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.62760416666666664 \cdot 10^{-5} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + f \cdot \left(\pi \cdot 0.5\right)\right)}{\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}}\right)}}{\pi}\right)\]

  12. Final simplification2.1

    \[\leadsto 1 \cdot \left(4 \cdot \frac{\log \left(\frac{0.00520833333333333304 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.62760416666666664 \cdot 10^{-5} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + f \cdot \left(\pi \cdot 0.5\right)\right)}{\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}}\right) - \log \left(\sqrt{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}\right)}{\pi}\right)\]

Reproduce

herbie shell --seed 2020184 
(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)))))))))