Average Error: 59.8 → 2.0
Time: 10.0m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\frac{\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + e^{f \cdot \frac{\pi}{4}}}{(\pi \cdot \left((\left(\frac{1}{192} \cdot \left(f \cdot f\right)\right) \cdot \left(f \cdot \left(\pi \cdot \pi\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left({\pi}^{5} \cdot \left(\frac{1}{61440} \cdot {f}^{5}\right)\right))_*}\right) \cdot \left(-4\right)}{\pi}\]

Error

Bits error versus f

Derivation

  1. Initial program 59.8

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

  2. Taylor expanded around 0 2.1

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

  3. Applied simplify2.1

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

  5. Applied associate-*r/2.0

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

Runtime

Time bar (total: 10.0m)Debug logProfile

herbie shell --seed 2020178 +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)))))))))