Average Error: 59.6 → 2.1
Time: 3.6m
Precision: 64
Internal Precision: 1408
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[-\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{61440} \cdot \left({\pi}^{5} \cdot {f}^{5}\right) + \left(\frac{1}{2} \cdot \left(\pi \cdot f\right) + \frac{1}{192} \cdot \left({\pi}^{3} \cdot {f}^{3}\right)\right)}\right)\right)\]

Error

Bits error versus f

Derivation

  1. Initial program 59.6

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

  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. Using strategy rm

  4. Applied add-sqr-sqrt2.4

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

  5. Applied associate-*l*2.1

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

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' 
(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)))))))))