Average Error: 59.5 → 2.2
Time: 2.0m
Precision: 64
Internal Precision: 1408
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\log \left(\sqrt[3]{(f \cdot \left((\left(\pi \cdot \frac{1}{192}\right) \cdot \left(\left(\pi \cdot f\right) \cdot \left(\pi \cdot f\right)\right) + \left(\frac{1}{2} \cdot \pi\right))_*\right) + \left({\pi}^{5} \cdot \left(\frac{1}{61440} \cdot {f}^{5}\right)\right))_*}\right) \cdot \left(\frac{4}{\pi} + \frac{4}{\pi}\right) + \frac{-4}{\pi} \cdot \log \left(\frac{e^{\frac{-f}{\frac{4}{\pi}}} + e^{\frac{\pi}{4} \cdot f}}{\sqrt[3]{(f \cdot \left((\left(\pi \cdot \frac{1}{192}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(f \cdot f\right)\right) + \left(\frac{1}{2} \cdot \pi\right))_*\right) + \left(\frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right)\right))_*}}\right)\]

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

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

  3. Applied simplify2.2

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

  5. Applied add-cube-cbrt2.2

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

  6. Applied *-un-lft-identity2.2

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

  7. Applied times-frac2.2

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

  8. Applied log-prod2.2

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

  9. Applied distribute-lft-in2.2

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

  10. Applied simplify2.2

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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)))))))))