Average Error: 59.6 → 0.7
Time: 5.5m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\frac{1}{\frac{\frac{\pi}{4}}{\log \left(\tanh \left(\frac{\pi}{4} \cdot f\right)\right)}}\]

Error

Bits error versus f

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 59.6

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

  2. Initial simplification0.7

    \[\leadsto \frac{\log \left(\tanh \left(\frac{\pi}{4} \cdot f\right)\right)}{\frac{\pi}{4}}\]
  3. Using strategy rm

  4. Applied clear-num0.7

    \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\pi}{4}}{\log \left(\tanh \left(\frac{\pi}{4} \cdot f\right)\right)}}}\]

  5. Final simplification0.7

    \[\leadsto \frac{1}{\frac{\frac{\pi}{4}}{\log \left(\tanh \left(\frac{\pi}{4} \cdot f\right)\right)}}\]

Runtime

Time bar (total: 5.5m)Debug logProfile

herbie shell --seed 2018214 +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)))))))))