Average Error: 59.7 → 2.1
Time: 1.7m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[-\left(\left(\frac{4}{\pi} \cdot \log \left(\sqrt[3]{\frac{4}{\pi}} \cdot \sqrt[3]{\frac{4}{\pi}}\right) + \frac{4}{\pi} \cdot \left(\log \left(\sqrt[3]{\frac{4}{\pi}}\right) - \log f\right)\right) - \left(\left(\pi \cdot \left(\pi \cdot \frac{7}{5760}\right)\right) \cdot \left({f}^{4} \cdot \pi\right) - \left(\frac{1}{12} \cdot \pi\right) \cdot \left(f \cdot f\right)\right)\right)\]

Error

Bits error versus f

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 59.7

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

  2. Taylor expanded around 0 2.1

    \[\leadsto -\color{blue}{\left(\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \frac{1}{12} \cdot \left({f}^{2} \cdot \pi\right)\right) - \left(4 \cdot \frac{\log f}{\pi} + \frac{7}{5760} \cdot \left({f}^{4} \cdot {\pi}^{3}\right)\right)\right)}\]

  3. Simplified2.1

    \[\leadsto -\color{blue}{\left(\frac{4}{\pi} \cdot \left(\log \left(\frac{4}{\pi}\right) - \log f\right) - \left(\left(\pi \cdot \left(\frac{7}{5760} \cdot \pi\right)\right) \cdot \left({f}^{4} \cdot \pi\right) - \left(\frac{1}{12} \cdot \pi\right) \cdot \left(f \cdot f\right)\right)\right)}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt2.1

    \[\leadsto -\left(\frac{4}{\pi} \cdot \left(\log \color{blue}{\left(\left(\sqrt[3]{\frac{4}{\pi}} \cdot \sqrt[3]{\frac{4}{\pi}}\right) \cdot \sqrt[3]{\frac{4}{\pi}}\right)} - \log f\right) - \left(\left(\pi \cdot \left(\frac{7}{5760} \cdot \pi\right)\right) \cdot \left({f}^{4} \cdot \pi\right) - \left(\frac{1}{12} \cdot \pi\right) \cdot \left(f \cdot f\right)\right)\right)\]

  6. Applied log-prod2.1

    \[\leadsto -\left(\frac{4}{\pi} \cdot \left(\color{blue}{\left(\log \left(\sqrt[3]{\frac{4}{\pi}} \cdot \sqrt[3]{\frac{4}{\pi}}\right) + \log \left(\sqrt[3]{\frac{4}{\pi}}\right)\right)} - \log f\right) - \left(\left(\pi \cdot \left(\frac{7}{5760} \cdot \pi\right)\right) \cdot \left({f}^{4} \cdot \pi\right) - \left(\frac{1}{12} \cdot \pi\right) \cdot \left(f \cdot f\right)\right)\right)\]

  7. Applied associate--l+2.1

    \[\leadsto -\left(\frac{4}{\pi} \cdot \color{blue}{\left(\log \left(\sqrt[3]{\frac{4}{\pi}} \cdot \sqrt[3]{\frac{4}{\pi}}\right) + \left(\log \left(\sqrt[3]{\frac{4}{\pi}}\right) - \log f\right)\right)} - \left(\left(\pi \cdot \left(\frac{7}{5760} \cdot \pi\right)\right) \cdot \left({f}^{4} \cdot \pi\right) - \left(\frac{1}{12} \cdot \pi\right) \cdot \left(f \cdot f\right)\right)\right)\]

  8. Applied distribute-lft-in2.1

    \[\leadsto -\left(\color{blue}{\left(\frac{4}{\pi} \cdot \log \left(\sqrt[3]{\frac{4}{\pi}} \cdot \sqrt[3]{\frac{4}{\pi}}\right) + \frac{4}{\pi} \cdot \left(\log \left(\sqrt[3]{\frac{4}{\pi}}\right) - \log f\right)\right)} - \left(\left(\pi \cdot \left(\frac{7}{5760} \cdot \pi\right)\right) \cdot \left({f}^{4} \cdot \pi\right) - \left(\frac{1}{12} \cdot \pi\right) \cdot \left(f \cdot f\right)\right)\right)\]

  9. Final simplification2.1

    \[\leadsto -\left(\left(\frac{4}{\pi} \cdot \log \left(\sqrt[3]{\frac{4}{\pi}} \cdot \sqrt[3]{\frac{4}{\pi}}\right) + \frac{4}{\pi} \cdot \left(\log \left(\sqrt[3]{\frac{4}{\pi}}\right) - \log f\right)\right) - \left(\left(\pi \cdot \left(\pi \cdot \frac{7}{5760}\right)\right) \cdot \left({f}^{4} \cdot \pi\right) - \left(\frac{1}{12} \cdot \pi\right) \cdot \left(f \cdot f\right)\right)\right)\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018216 +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)))))))))