Average Error: 59.5 → 2.3
Time: 1.9m
Precision: 64
Internal Precision: 1408
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\left(-\sqrt{\frac{4}{\pi}}\right) \cdot \left(\sqrt{\frac{4}{\pi}} \cdot \log \left(\frac{e^{\frac{-f}{\frac{4}{\pi}}} + e^{\frac{\pi}{4} \cdot f}}{(\pi \cdot \left((\left(\left(f \cdot f\right) \cdot \left(f \cdot \frac{1}{192}\right)\right) \cdot \left(\pi \cdot \pi\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left(\frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right)\right))_*}\right)\right)\]

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 2.4

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

  3. Applied simplify2.4

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

  5. Applied add-sqr-sqrt2.6

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

  6. Applied distribute-lft-neg-in2.6

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

  7. Applied associate-*l*2.3

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +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)))))))))