Average Error: 59.5 → 2.1
Time: 3.6m
Precision: 64
Internal Precision: 1408
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\frac{-4}{\pi} \cdot \left(\log \left(\frac{\sqrt{e^{\frac{-f}{\frac{4}{\pi}}} + e^{f \cdot \frac{\pi}{4}}}}{1}\right) + \log \left(\frac{\sqrt{e^{\frac{-f}{\frac{4}{\pi}}} + e^{f \cdot \frac{\pi}{4}}}}{\frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right) + \left(\frac{1}{2} \cdot \left(\pi \cdot f\right) + \frac{1}{192} \cdot \left({\pi}^{3} \cdot {f}^{3}\right)\right)}\right)\right)\]

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. Applied simplify59.5

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

  3. Taylor expanded around 0 2.2

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

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

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

  6. Applied add-sqr-sqrt2.2

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

  7. Applied times-frac2.2

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

  8. Applied log-prod2.1

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

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(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)))))))))