Average Error: 61.3 → 1.7
Time: 13.6s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[-1 \cdot \frac{\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}{\frac{\pi}{4}}\]
-\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(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\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
 (-
  (*
   1.0
   (/ (log (/ (cosh (* (/ PI 4.0) f)) (sinh (* (/ PI 4.0) f)))) (/ PI 4.0)))))
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) log((((double) cosh(((double) ((((double) M_PI) / 4.0) * f)))) / ((double) sinh(((double) ((((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 61.3

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

  3. Applied sinh-undef_binary641.8

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

  4. Applied associate-/r*_binary641.8

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

  5. Simplified1.8

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

  7. Applied div-inv_binary641.8

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

  8. Applied associate-*l*_binary641.8

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

  9. Simplified1.7

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

  10. Final simplification1.7

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

Reproduce

herbie shell --seed 2020204 
(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)))))))))