Average Error: 59.7 → 2.0
Time: 3.7m
Precision: 64
Internal Precision: 1408
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\frac{-4}{\pi} \cdot \left(\log \left(e^{\frac{-f}{\frac{4}{\pi}}} + e^{\frac{\pi}{4} \cdot f}\right) + \left(-\log \left((\left(\pi \cdot f\right) \cdot \left((\frac{1}{192} \cdot \left(\left(\pi \cdot f\right) \cdot \left(\pi \cdot f\right)\right) + \frac{1}{2})_*\right) + \left(\frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right)\right))_*\right)\right)\right)\]

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 2.0

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

  3. Applied simplify2.0

    \[\leadsto \color{blue}{\frac{-4}{\pi} \cdot \log \left(\frac{e^{\frac{-f}{\frac{4}{\pi}}} + e^{\frac{\pi}{4} \cdot f}}{(f \cdot \left((\left(\left(\pi \cdot \frac{1}{192}\right) \cdot \pi\right) \cdot \left(\left(f \cdot \pi\right) \cdot f\right) + \left(\frac{1}{2} \cdot \pi\right))_*\right) + \left(\frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right)\right))_*}\right)}\]
  4. Using strategy rm

  5. Applied div-inv2.0

    \[\leadsto \frac{-4}{\pi} \cdot \log \color{blue}{\left(\left(e^{\frac{-f}{\frac{4}{\pi}}} + e^{\frac{\pi}{4} \cdot f}\right) \cdot \frac{1}{(f \cdot \left((\left(\left(\pi \cdot \frac{1}{192}\right) \cdot \pi\right) \cdot \left(\left(f \cdot \pi\right) \cdot f\right) + \left(\frac{1}{2} \cdot \pi\right))_*\right) + \left(\frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right)\right))_*}\right)}\]

  6. Applied log-prod2.0

    \[\leadsto \frac{-4}{\pi} \cdot \color{blue}{\left(\log \left(e^{\frac{-f}{\frac{4}{\pi}}} + e^{\frac{\pi}{4} \cdot f}\right) + \log \left(\frac{1}{(f \cdot \left((\left(\left(\pi \cdot \frac{1}{192}\right) \cdot \pi\right) \cdot \left(\left(f \cdot \pi\right) \cdot f\right) + \left(\frac{1}{2} \cdot \pi\right))_*\right) + \left(\frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right)\right))_*}\right)\right)}\]

  7. Applied simplify2.0

    \[\leadsto \frac{-4}{\pi} \cdot \left(\log \left(e^{\frac{-f}{\frac{4}{\pi}}} + e^{\frac{\pi}{4} \cdot f}\right) + \color{blue}{\left(-\log \left((\left(\pi \cdot f\right) \cdot \left((\frac{1}{192} \cdot \left(\left(\pi \cdot f\right) \cdot \left(\pi \cdot f\right)\right) + \frac{1}{2})_*\right) + \left(\frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right)\right))_*\right)\right)}\right)\]

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +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)))))))))