Average Error: 61.5 → 2.4
Time: 12.6s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\left(4 \cdot \frac{\log f}{\pi} + {\pi}^{3} \cdot \left({f}^{4} \cdot 0.001215277777777778\right)\right) - \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + 0.08333333333333334 \cdot \left(\pi \cdot \left(f \cdot f\right)\right)\right)\]
-\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)
↓
\left(4 \cdot \frac{\log f}{\pi} + {\pi}^{3} \cdot \left({f}^{4} \cdot 0.001215277777777778\right)\right) - \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + 0.08333333333333334 \cdot \left(\pi \cdot \left(f \cdot f\right)\right)\right)
(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
 (-
  (+
   (* 4.0 (/ (log f) PI))
   (* (pow PI 3.0) (* (pow f 4.0) 0.001215277777777778)))
  (+ (* 4.0 (/ (log (/ 4.0 PI)) PI)) (* 0.08333333333333334 (* PI (* f f))))))
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) (((double) (4.0 * (((double) log(f)) / ((double) M_PI)))) + ((double) (((double) pow(((double) M_PI), 3.0)) * ((double) (((double) pow(f, 4.0)) * 0.001215277777777778)))))) - ((double) (((double) (4.0 * (((double) log((4.0 / ((double) M_PI)))) / ((double) M_PI)))) + ((double) (0.08333333333333334 * ((double) (((double) M_PI) * ((double) (f * f))))))))));
}

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. Taylor expanded around 0 2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\left(0.020833333333333336 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.0034722222222222225 \cdot \frac{{\pi}^{4} \cdot {f}^{4}}{{4}^{2}} + 8.68055555555556 \cdot 10^{-05} \cdot \left({\pi}^{4} \cdot {f}^{4}\right)\right)\right)\right)}\]

  3. Simplified2.4

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\left(0.020833333333333336 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(f \cdot f\right)\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left(0.0034722222222222225 \cdot \frac{{\pi}^{4} \cdot {f}^{4}}{4 \cdot 4} + \left({\pi}^{4} \cdot {f}^{4}\right) \cdot 8.68055555555556 \cdot 10^{-05}\right)\right)\right)}\]

  4. Taylor expanded around 0 2.4

    \[\leadsto -\color{blue}{\left(\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + 0.08333333333333334 \cdot \left({f}^{2} \cdot \pi\right)\right) - \left(4 \cdot \frac{\log f}{\pi} + 0.001215277777777778 \cdot \left({f}^{4} \cdot {\pi}^{3}\right)\right)\right)}\]

  5. Simplified2.4

    \[\leadsto -\color{blue}{\left(\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + 0.08333333333333334 \cdot \left(\pi \cdot \left(f \cdot f\right)\right)\right) - \left(4 \cdot \frac{\log f}{\pi} + {\pi}^{3} \cdot \left({f}^{4} \cdot 0.001215277777777778\right)\right)\right)}\]

  6. Final simplification2.4

    \[\leadsto \left(4 \cdot \frac{\log f}{\pi} + {\pi}^{3} \cdot \left({f}^{4} \cdot 0.001215277777777778\right)\right) - \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + 0.08333333333333334 \cdot \left(\pi \cdot \left(f \cdot f\right)\right)\right)\]

Reproduce

herbie shell --seed 2020210 
(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)))))))))