Average Error: 59.7 → 2.2
Time: 4.3m
Precision: 64
Internal Precision: 1408
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[-\left(\left(\frac{4 + 4}{\pi} \cdot \log \left(\sqrt[3]{\frac{4}{\pi}}\right) + \frac{4}{\pi} \cdot \left(\log \left(\sqrt[3]{\frac{4}{\pi}}\right) - \log f\right)\right) + \left(\left(f \cdot f\right) \cdot \left(\frac{1}{12} \cdot \pi\right) - \left(\left(\frac{7}{5760} \cdot \pi\right) \cdot {f}^{4}\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\]

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 2.2

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

  3. Applied simplify2.2

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

  5. Applied add-cube-cbrt2.2

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

  6. Applied log-prod2.2

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

  7. Applied associate--l+2.2

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

  8. Applied distribute-lft-in2.2

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

  9. Applied simplify2.2

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

  10. Applied simplify2.2

    \[\leadsto -\left(\left(\color{blue}{\frac{4 + 4}{\pi}} \cdot \log \left(\sqrt[3]{\frac{4}{\pi}}\right) + \frac{4}{\pi} \cdot \left(\log \left(\sqrt[3]{\frac{4}{\pi}}\right) - \log f\right)\right) + \left(\left(f \cdot f\right) \cdot \left(\frac{1}{12} \cdot \pi\right) - \left(\left(\frac{7}{5760} \cdot \pi\right) \cdot {f}^{4}\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\]
  11. Removed slow pow expressions.

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(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)))))))))