Average Error: 59.6 → 2.2
Time: 3.3m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\left(\frac{-1}{48} \cdot \left(\left(f \cdot f\right) \cdot \left(\pi \cdot \pi\right)\right) - \left(\left(\log \left(\frac{4}{\pi}\right) - \log f\right) - \left(\frac{7}{23040} \cdot {f}^{4}\right) \cdot {\pi}^{4}\right)\right) \cdot \frac{4}{\pi}\]

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 2.2

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\left(\frac{1}{48} \cdot \left({\pi}^{2} \cdot {f}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \frac{7}{23040} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)}\]

  3. Applied simplify2.2

    \[\leadsto \color{blue}{\left(\frac{-1}{48} \cdot \left(\left(f \cdot f\right) \cdot \left(\pi \cdot \pi\right)\right) - \left(\left(\log \left(\frac{4}{\pi}\right) - \log f\right) - \left(\frac{7}{23040} \cdot {f}^{4}\right) \cdot {\pi}^{4}\right)\right) \cdot \frac{4}{\pi}}\]

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(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)))))))))