Average Error: 61.5 → 2.0
Time: 12.0s
Precision: binary64
Cost: 20288
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}}\]
-\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)
↓
\log \left(\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \frac{-1}{\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)) (sinh (* (/ PI 4.0) f))))
  (/ -1.0 (/ 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) / sinh((((double) M_PI) / 4.0) * f)) * (-1.0 / (((double) M_PI) / 4.0));
}

Error

Bits error versus f

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error2.3
Cost7424
\[-\frac{\log \left(\frac{4}{\pi \cdot f} + \left(\pi \cdot f\right) \cdot 0.08333333333333333\right)}{\frac{\pi}{4}}\]
Alternative 2
Error2.7
Cost6976
\[\frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi} \cdot -4\]
Alternative 3
Error55.2
Cost64
\[-1\]
Alternative 4
Error60.8
Cost64
\[0\]
Alternative 5
Error63.0
Cost64
\[1\]

Error

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-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 cosh-undef_binary642.0

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

  5. Applied times-frac_binary642.0

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

  6. Simplified2.0

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

  7. Simplified2.0

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

  8. Final simplification2.0

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

Reproduce

herbie shell --seed 2021044 
(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)))))))))