Average Error: 59.7 → 1.9
Time: 4.4m
Precision: 64
Internal Precision: 1408
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \le 2.8134282826637952 \cdot 10^{-05}:\\ \;\;\;\;\left(\left(\left(f \cdot \frac{1}{48}\right) \cdot \left(f \cdot \left(\pi \cdot \pi\right)\right) - \log f\right) + \left(\log \left(\frac{4}{\pi}\right) - \frac{7}{23040} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;-\log \left({\left(\frac{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{\left(-f\right) \cdot \frac{\pi}{4}}}\right)}^{\left(\frac{4}{\pi}\right)}\right)\\ \end{array}\]

Error

Bits error versus f

Derivation

  1. Split input into 2 regimes

  2. if (* (/ PI 4) f) < 2.8134282826637952e-05

    1. Initial program 60.7

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

    2. Taylor expanded around 0 0.5

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

    3. Taylor expanded around 0 0.4

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\left(\frac{1}{48} \cdot \left({\pi}^{2} \cdot {f}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \frac{7}{23040} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)}\]

    4. Applied simplify0.4

      \[\leadsto \color{blue}{\left(\left(\left(f \cdot \frac{1}{48}\right) \cdot \left(f \cdot \left(\pi \cdot \pi\right)\right) - \log f\right) + \left(\log \left(\frac{4}{\pi}\right) - \frac{7}{23040} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right) \cdot \frac{-4}{\pi}}\]

    if 2.8134282826637952e-05 < (* (/ PI 4) f)

    1. Initial program 36.0

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

    3. Applied add-log-exp36.0

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

    4. Applied simplify36.0

      \[\leadsto -\log \color{blue}{\left({\left(\frac{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{\left(-f\right) \cdot \frac{\pi}{4}}}\right)}^{\left(\frac{4}{\pi}\right)}\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 4.4m)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' 
(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)))))))))