Average Error: 61.6 → 2.0
Time: 13.6s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[1 \cdot \frac{-\log \left(\sqrt{\cosh \left(\pi \cdot \frac{f}{4}\right)}\right)}{\frac{\pi}{4}} - 1 \cdot \frac{\log \left(\frac{\sqrt{\cosh \left(\pi \cdot \frac{f}{4}\right)}}{\sinh \left(\pi \cdot \frac{f}{4}\right)}\right)}{\frac{\pi}{4}}\]
-\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{-\log \left(\sqrt{\cosh \left(\pi \cdot \frac{f}{4}\right)}\right)}{\frac{\pi}{4}} - 1 \cdot \frac{\log \left(\frac{\sqrt{\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) -(((double) log(((double) sqrt(((double) cosh(((double) (((double) M_PI) * (f / 4.0))))))))))) / (((double) M_PI) / 4.0)))) - ((double) (1.0 * (((double) log((((double) sqrt(((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 61.6

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

  3. Applied sinh-undef2.0

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

  4. Applied associate-/r*2.0

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

  5. Simplified2.0

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

  7. Applied *-un-lft-identity2.0

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

  8. Applied add-sqr-sqrt2.1

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

  9. Applied times-frac2.1

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

  10. Applied log-prod2.1

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

  11. Applied distribute-lft-in2.1

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

  12. Simplified2.1

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

  13. Simplified2.0

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

  14. Final simplification2.0

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

Reproduce

herbie shell --seed 2020198 
(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)))))))))