Average Error: 61.3 → 2.4
Time: 14.6s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[{\pi}^{3} \cdot \left({f}^{4} \cdot 0.0012152777777777778\right) + \left(4 \cdot \frac{\log f}{\pi} - \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\pi \cdot \left(f \cdot f\right)\right) \cdot 0.08333333333333333\right)\right)\]
-\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)
↓
{\pi}^{3} \cdot \left({f}^{4} \cdot 0.0012152777777777778\right) + \left(4 \cdot \frac{\log f}{\pi} - \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\pi \cdot \left(f \cdot f\right)\right) \cdot 0.08333333333333333\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
 (+
  (* (pow PI 3.0) (* (pow f 4.0) 0.0012152777777777778))
  (-
   (* 4.0 (/ (log f) PI))
   (+
    (* 4.0 (/ (log (/ 4.0 PI)) PI))
    (* (* PI (* f f)) 0.08333333333333333)))))
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 (pow(((double) M_PI), 3.0) * (pow(f, 4.0) * 0.0012152777777777778)) + ((4.0 * (log(f) / ((double) M_PI))) - ((4.0 * (log(4.0 / ((double) M_PI)) / ((double) M_PI))) + ((((double) M_PI) * (f * f)) * 0.08333333333333333)));
}

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. Simplified61.3

    \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{e^{\frac{\pi}{4} \cdot f} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right) \cdot \frac{-4}{\pi}}\]

  3. Taylor expanded around 0 2.4

    \[\leadsto \color{blue}{\left(\left(0.020833333333333332 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot {\pi}^{4}\right)\right)\right)} \cdot \frac{-4}{\pi}\]

  4. Simplified2.4

    \[\leadsto \color{blue}{\left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot {\pi}^{4}\right)\right)\right)} \cdot \frac{-4}{\pi}\]

  5. Taylor expanded around 0 2.4

    \[\leadsto \color{blue}{\left(0.0012152777777777778 \cdot \left({f}^{4} \cdot {\pi}^{3}\right) + 4 \cdot \frac{\log f}{\pi}\right) - \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right)}\]

  6. Simplified2.4

    \[\leadsto \color{blue}{{\pi}^{3} \cdot \left({f}^{4} \cdot 0.0012152777777777778\right) + \left(4 \cdot \frac{\log f}{\pi} - \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\pi \cdot \left(f \cdot f\right)\right) \cdot 0.08333333333333333\right)\right)}\]

  7. Final simplification2.4

    \[\leadsto {\pi}^{3} \cdot \left({f}^{4} \cdot 0.0012152777777777778\right) + \left(4 \cdot \frac{\log f}{\pi} - \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(\pi \cdot \left(f \cdot f\right)\right) \cdot 0.08333333333333333\right)\right)\]

Reproduce

herbie shell --seed 2020274 
(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)))))))))