Average Error: 60.0 → 1.6
Time: 1.4m
Precision: 64
Internal precision: 128
\[-\frac1{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
\[\left(-\frac1{\frac{\pi}{4}}\right) \cdot \log \left(\frac{e^{\frac{-f}{\frac{4}{\pi}}} + e^{\frac{f}{\frac{4}{\pi}}}}{\left({f}^3 \cdot {\pi}^3\right) \cdot \frac{1}{192} + \left(\frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \frac{1}{2} \cdot \left(\pi \cdot f\right)\right)}\right)\]

Error

Bits error versus f

Derivation

  1. Initial program 60.0

    \[-\frac1{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
  2. Applied taylor 1.6

    \[\leadsto -\frac1{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{2} \cdot \left(\pi \cdot f\right) + \left(\frac{1}{192} \cdot \left({\pi}^{3} \cdot {f}^{3}\right) + \frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right)\right)}\right)\]
  3. Taylor expanded around 0 1.6

    \[\leadsto -\frac1{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\frac{1}{2} \cdot \left(\pi \cdot f\right) + \left(\frac{1}{192} \cdot \left({\pi}^{3} \cdot {f}^{3}\right) + \frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right)\right)}}\right)\]
  4. Applied simplify 1.6

    \[\leadsto \color{blue}{\left(-\frac1{\frac{\pi}{4}}\right) \cdot \log \left(\frac{e^{\frac{-f}{\frac{4}{\pi}}} + e^{\frac{f}{\frac{4}{\pi}}}}{\left({f}^3 \cdot {\pi}^3\right) \cdot \frac{1}{192} + \left(\frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \frac{1}{2} \cdot \left(\pi \cdot f\right)\right)}\right)}\]
  5. Removed slow pow expressions

Runtime

Time bar (total: 1.4m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(3637839332 532107118 3790453781 2338559806 115313347 1588793255)'
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  (- (* (/ (/ PI 4)) (log (/ (+ (exp (* (/ PI 4) f)) (exp (- (* (/ PI 4) f)))) (- (exp (* (/ PI 4) f)) (exp (- (* (/ PI 4) f)))))))))