Average Error: 61.5 → 1.6
Time: 17.1s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\frac{\log \sinh \left(\frac{\pi}{4} \cdot f\right)}{\frac{\pi}{4}} - \frac{\log \cosh \left(\frac{\pi}{4} \cdot f\right)}{\frac{\pi}{4}}\]
-\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)
↓
\frac{\log \sinh \left(\frac{\pi}{4} \cdot f\right)}{\frac{\pi}{4}} - \frac{\log \cosh \left(\frac{\pi}{4} \cdot f\right)}{\frac{\pi}{4}}
(FPCore (f)
 :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)))))))))
↓
(FPCore (f)
 :precision binary64
 (-
  (/ (log (sinh (* (/ PI 4.0) f))) (/ PI 4.0))
  (/ (log (cosh (* (/ PI 4.0) f))) (/ PI 4.0))))
double code(double f) {
	return -((1.0 / (((double) M_PI) / 4.0)) * log((exp((((double) M_PI) / 4.0) * f) + exp(-((((double) M_PI) / 4.0) * f))) / (exp((((double) M_PI) / 4.0) * f) - exp(-((((double) M_PI) / 4.0) * f)))));
}
↓
double code(double f) {
	return (log(sinh((((double) M_PI) / 4.0) * f)) / (((double) M_PI) / 4.0)) - (log(cosh((((double) M_PI) / 4.0) * f)) / (((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 2sinh-undef_binary641.7

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

  5. Applied associate-*l/_binary641.6

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

  6. Simplified1.6

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

  8. Applied log-div_binary641.6

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

  9. Applied div-sub_binary641.6

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

  10. Final simplification1.6

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

Reproduce

herbie shell --seed 2020289 
(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)))))))))