Average Error: 61.5 → 2.0
Time: 15.0s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[1 \cdot \frac{2 \cdot \left(-\log \left({\left(\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\pi \cdot \frac{f}{4}\right)}\right)}^{0.3333333333333333}\right)\right)}{\frac{\pi}{4}} - 1 \cdot \frac{\log \left(\sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\pi \cdot \frac{f}{4}\right)}}\right)}{\frac{\pi}{4}}\]
-\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)
↓
1 \cdot \frac{2 \cdot \left(-\log \left({\left(\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\pi \cdot \frac{f}{4}\right)}\right)}^{0.3333333333333333}\right)\right)}{\frac{\pi}{4}} - 1 \cdot \frac{\log \left(\sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\pi \cdot \frac{f}{4}\right)}}\right)}{\frac{\pi}{4}}
double code(double f) {
	return ((double) -(((double) ((1.0 / (((double) M_PI) / 4.0)) * ((double) log((((double) (((double) exp(((double) ((((double) M_PI) / 4.0) * f)))) + ((double) exp(((double) -(((double) ((((double) M_PI) / 4.0) * f)))))))) / ((double) (((double) exp(((double) ((((double) M_PI) / 4.0) * f)))) - ((double) exp(((double) -(((double) ((((double) M_PI) / 4.0) * f)))))))))))))));
}
↓
double code(double f) {
	return ((double) (((double) (1.0 * (((double) (2.0 * ((double) -(((double) log(((double) pow((((double) cosh(((double) (((double) M_PI) * (f / 4.0))))) / ((double) sinh(((double) (((double) M_PI) * (f / 4.0)))))), 0.3333333333333333)))))))) / (((double) M_PI) / 4.0)))) - ((double) (1.0 * (((double) log(((double) cbrt((((double) cosh(((double) (((double) M_PI) * (f / 4.0))))) / ((double) sinh(((double) (((double) M_PI) * (f / 4.0)))))))))) / (((double) M_PI) / 4.0))))));
}

Error

Bits error versus f

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 61.5

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

  3. Applied sinh-undef2.2

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

  4. Applied associate-/r*2.2

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

  5. Simplified2.2

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\color{blue}{\cosh \left(\pi \cdot \frac{f}{4}\right)}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)\]
  6. Using strategy rm

  7. Applied add-cube-cbrt2.2

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\left(\sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}} \cdot \sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right) \cdot \sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right)}\]

  8. Applied log-prod2.2

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}} \cdot \sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right) + \log \left(\sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right)\right)}\]

  9. Applied distribute-lft-in2.2

    \[\leadsto -\color{blue}{\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}} \cdot \sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right) + \frac{1}{\frac{\pi}{4}} \cdot \log \left(\sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right)\right)}\]

  10. Simplified2.2

    \[\leadsto -\left(\color{blue}{1 \cdot \frac{2 \cdot \log \left(\sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\pi \cdot \frac{f}{4}\right)}}\right)}{\frac{\pi}{4}}} + \frac{1}{\frac{\pi}{4}} \cdot \log \left(\sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right)\right)\]

  11. Simplified2.2

    \[\leadsto -\left(1 \cdot \frac{2 \cdot \log \left(\sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\pi \cdot \frac{f}{4}\right)}}\right)}{\frac{\pi}{4}} + \color{blue}{1 \cdot \frac{\log \left(\sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\pi \cdot \frac{f}{4}\right)}}\right)}{\frac{\pi}{4}}}\right)\]
  12. Using strategy rm

  13. Applied pow1/32.0

    \[\leadsto -\left(1 \cdot \frac{2 \cdot \log \color{blue}{\left({\left(\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\pi \cdot \frac{f}{4}\right)}\right)}^{0.3333333333333333}\right)}}{\frac{\pi}{4}} + 1 \cdot \frac{\log \left(\sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\pi \cdot \frac{f}{4}\right)}}\right)}{\frac{\pi}{4}}\right)\]

  14. Final simplification2.0

    \[\leadsto 1 \cdot \frac{2 \cdot \left(-\log \left({\left(\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\pi \cdot \frac{f}{4}\right)}\right)}^{0.3333333333333333}\right)\right)}{\frac{\pi}{4}} - 1 \cdot \frac{\log \left(\sqrt[3]{\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\pi \cdot \frac{f}{4}\right)}}\right)}{\frac{\pi}{4}}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary64
  (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))) (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))))))