Average Error: 59.7 → 2.2
Time: 3.5m
Precision: 64
Internal Precision: 1408
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[-\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{\frac{-\pi}{\frac{4}{f}}}}\right) + \log \left(\frac{\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right) + \left(\frac{1}{2} \cdot \left(\pi \cdot f\right) + \frac{1}{192} \cdot \left({\pi}^{3} \cdot {f}^{3}\right)\right)}\right)\right)\]

Error

Bits error versus f

Derivation

  1. Initial program 59.7

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

  2. Taylor expanded around 0 2.3

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

  4. Applied *-un-lft-identity2.3

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

  5. Applied add-sqr-sqrt2.3

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

  6. Applied times-frac2.3

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

  7. Applied log-prod2.2

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

  8. Applied simplify2.2

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

Runtime

Time bar (total: 3.5m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(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)))))))))