Average Error: 61.6 → 2.2
Time: 14.4s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[-\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\pi \cdot \left(f \cdot \left(f \cdot 0.083333333333333343\right)\right) - \left(0.0138888888888888899 \cdot \left(\frac{{\left({\left({\left(\sqrt[3]{\pi}\right)}^{2}\right)}^{\left(\sqrt{3}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) + \left(4 \cdot \frac{\log f}{\pi} + 3.472222222222224 \cdot 10^{-4} \cdot \left({f}^{4} \cdot {\pi}^{3}\right)\right)\right)\right)\right)\]

Error

Bits error versus f

Derivation

  1. Initial program 61.6

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

  2. Taylor expanded around 0 2.3

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\left(4 \cdot \frac{1}{\pi \cdot f} + 0.083333333333333343 \cdot \left(f \cdot \pi\right)\right) - 3.472222222222224 \cdot 10^{-4} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right)}\]

  3. Simplified2.3

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{4}{\pi \cdot f} + \left(\pi \cdot \left(f \cdot 0.083333333333333343\right) - {\left(\pi \cdot f\right)}^{3} \cdot 3.472222222222224 \cdot 10^{-4}\right)\right)}\]

  4. Taylor expanded around 0 2.2

    \[\leadsto -\color{blue}{\left(\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + 0.083333333333333343 \cdot \left({f}^{2} \cdot \pi\right)\right) - \left(0.0138888888888888899 \cdot \frac{{f}^{4} \cdot {\pi}^{3}}{{4}^{2}} + \left(4 \cdot \frac{\log f}{\pi} + 3.472222222222224 \cdot 10^{-4} \cdot \left({f}^{4} \cdot {\pi}^{3}\right)\right)\right)\right)}\]

  5. Simplified2.2

    \[\leadsto -\color{blue}{\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\pi \cdot \left(f \cdot \left(f \cdot 0.083333333333333343\right)\right) - \left(0.0138888888888888899 \cdot \left(\frac{{\pi}^{3}}{4} \cdot \frac{{f}^{4}}{4}\right) + \left(4 \cdot \frac{\log f}{\pi} + 3.472222222222224 \cdot 10^{-4} \cdot \left({\pi}^{3} \cdot {f}^{4}\right)\right)\right)\right)\right)}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt2.2

    \[\leadsto -\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\pi \cdot \left(f \cdot \left(f \cdot 0.083333333333333343\right)\right) - \left(0.0138888888888888899 \cdot \left(\frac{{\pi}^{\color{blue}{\left(\sqrt{3} \cdot \sqrt{3}\right)}}}{4} \cdot \frac{{f}^{4}}{4}\right) + \left(4 \cdot \frac{\log f}{\pi} + 3.472222222222224 \cdot 10^{-4} \cdot \left({\pi}^{3} \cdot {f}^{4}\right)\right)\right)\right)\right)\]

  8. Applied pow-unpow2.2

    \[\leadsto -\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\pi \cdot \left(f \cdot \left(f \cdot 0.083333333333333343\right)\right) - \left(0.0138888888888888899 \cdot \left(\frac{\color{blue}{{\left({\pi}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}}{4} \cdot \frac{{f}^{4}}{4}\right) + \left(4 \cdot \frac{\log f}{\pi} + 3.472222222222224 \cdot 10^{-4} \cdot \left({\pi}^{3} \cdot {f}^{4}\right)\right)\right)\right)\right)\]
  9. Using strategy rm

  10. Applied add-cube-cbrt2.2

    \[\leadsto -\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\pi \cdot \left(f \cdot \left(f \cdot 0.083333333333333343\right)\right) - \left(0.0138888888888888899 \cdot \left(\frac{{\left({\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)}}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) + \left(4 \cdot \frac{\log f}{\pi} + 3.472222222222224 \cdot 10^{-4} \cdot \left({\pi}^{3} \cdot {f}^{4}\right)\right)\right)\right)\right)\]

  11. Applied unpow-prod-down2.2

    \[\leadsto -\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\pi \cdot \left(f \cdot \left(f \cdot 0.083333333333333343\right)\right) - \left(0.0138888888888888899 \cdot \left(\frac{{\color{blue}{\left({\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) + \left(4 \cdot \frac{\log f}{\pi} + 3.472222222222224 \cdot 10^{-4} \cdot \left({\pi}^{3} \cdot {f}^{4}\right)\right)\right)\right)\right)\]

  12. Simplified2.2

    \[\leadsto -\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\pi \cdot \left(f \cdot \left(f \cdot 0.083333333333333343\right)\right) - \left(0.0138888888888888899 \cdot \left(\frac{{\left(\color{blue}{{\left({\left(\sqrt[3]{\pi}\right)}^{2}\right)}^{\left(\sqrt{3}\right)}} \cdot {\left(\sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) + \left(4 \cdot \frac{\log f}{\pi} + 3.472222222222224 \cdot 10^{-4} \cdot \left({\pi}^{3} \cdot {f}^{4}\right)\right)\right)\right)\right)\]

  13. Final simplification2.2

    \[\leadsto -\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\pi \cdot \left(f \cdot \left(f \cdot 0.083333333333333343\right)\right) - \left(0.0138888888888888899 \cdot \left(\frac{{\left({\left({\left(\sqrt[3]{\pi}\right)}^{2}\right)}^{\left(\sqrt{3}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) + \left(4 \cdot \frac{\log f}{\pi} + 3.472222222222224 \cdot 10^{-4} \cdot \left({f}^{4} \cdot {\pi}^{3}\right)\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020181 
(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)))))))))