Average Error: 61.4 → 1.9
Time: 12.2s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[-\frac{\log \left(\frac{\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sqrt[3]{\sinh \left(\frac{\pi}{4} \cdot f\right)} \cdot \sqrt[3]{\sinh \left(\frac{\pi}{4} \cdot f\right)}}}{\sqrt[3]{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right)}{\frac{\pi}{4}}\]
-\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)
↓
-\frac{\log \left(\frac{\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sqrt[3]{\sinh \left(\frac{\pi}{4} \cdot f\right)} \cdot \sqrt[3]{\sinh \left(\frac{\pi}{4} \cdot f\right)}}}{\sqrt[3]{\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
 (-
  (/
   (log
    (/
     (/
      (cosh (* (/ PI 4.0) f))
      (* (cbrt (sinh (* (/ PI 4.0) f))) (cbrt (sinh (* (/ PI 4.0) f)))))
     (cbrt (sinh (* (/ 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((cosh((((double) M_PI) / 4.0) * f) / (cbrt(sinh((((double) M_PI) / 4.0) * f)) * cbrt(sinh((((double) M_PI) / 4.0) * f)))) / cbrt(sinh((((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 61.4

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

  3. Applied sinh-undef_binary642.0

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

  4. Applied associate-/r*_binary642.0

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

  5. Simplified2.0

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

  7. Applied associate-*l/_binary641.9

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

  8. Simplified1.9

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

  10. Applied add-cube-cbrt_binary641.9

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

  11. Applied associate-/r*_binary641.9

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

  12. Final simplification1.9

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

Reproduce

herbie shell --seed 2020224 
(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)))))))))