Average Error: 59.6 → 2.6
Time: 2.3m
Precision: 64
Internal Precision: 1408
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\log \left(\left(\left(f \cdot \frac{1}{12}\right) \cdot \pi + \frac{\frac{4}{\pi}}{f}\right) - \left(\left(\frac{1}{2880} \cdot \pi\right) \cdot {f}^{3}\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(-\frac{4}{\pi}\right)\]

Error

Bits error versus f

Derivation

  1. Initial program 59.6

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

  2. Taylor expanded around 0 2.6

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\left(\frac{1}{12} \cdot \left(\pi \cdot f\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({\pi}^{3} \cdot {f}^{3}\right)\right)}\]

  3. Applied simplify2.6

    \[\leadsto \color{blue}{\log \left(\left(\left(f \cdot \frac{1}{12}\right) \cdot \pi + \frac{\frac{4}{\pi}}{f}\right) - \left(\left(\frac{1}{2880} \cdot \pi\right) \cdot {f}^{3}\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(-\frac{4}{\pi}\right)}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(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)))))))))