Average Error: 59.5 → 2.2
Time: 1.8m
Precision: 64
Internal Precision: 1408
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[-(\pi \cdot \left(\left(f \cdot \frac{1}{12}\right) \cdot f - \left(\frac{7}{5760} \cdot {f}^{4}\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(\frac{4}{\pi} \cdot \log \left(\frac{4}{\pi}\right) - \frac{4}{\pi} \cdot \log f\right))_*\]

Error

Bits error versus f

Derivation

  1. Initial program 59.5

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

  2. Taylor expanded around 0 2.2

    \[\leadsto -\color{blue}{\left(\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \frac{1}{12} \cdot \left(\pi \cdot {f}^{2}\right)\right) - \left(\frac{7}{5760} \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + 4 \cdot \frac{\log f}{\pi}\right)\right)}\]

  3. Applied simplify2.2

    \[\leadsto \color{blue}{-(\pi \cdot \left(\left(f \cdot \frac{1}{12}\right) \cdot f - \left(\frac{7}{5760} \cdot {f}^{4}\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(\frac{4}{\pi} \cdot \log \left(\frac{4}{\pi}\right) - \frac{4}{\pi} \cdot \log f\right))_*}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +o rules:numerics
(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)))))))))