Average Error: 61.7 → 2.2
Time: 33.3s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[-1 \cdot \left(\frac{\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.00347222222222222246 \cdot \frac{{\pi}^{4} \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-5} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)}{\pi} \cdot 4\right)\]

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 2.2

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

  4. Applied div-inv2.2

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

  5. Applied associate-*l*2.2

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

  6. Simplified2.2

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

  7. Final simplification2.2

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

Reproduce

herbie shell --seed 2020162 
(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)))))))))