Average Error: 61.4 → 2.4
Time: 14.6s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[{\pi}^{3} \cdot \left(0.0008680555555555555 \cdot {f}^{4}\right) + \left(4 \cdot \left(\frac{\log f}{\pi} - \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right) + \left(f \cdot \left(\pi \cdot f\right)\right) \cdot -0.08333333333333333\right)\]
-\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(0.0008680555555555555 \cdot {f}^{4}\right) + \left(4 \cdot \left(\frac{\log f}{\pi} - \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right) + \left(f \cdot \left(\pi \cdot f\right)\right) \cdot -0.08333333333333333\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) (* 0.0008680555555555555 (pow f 4.0)))
  (+
   (* 4.0 (- (/ (log f) PI) (/ (log (/ 4.0 PI)) PI)))
   (* (* f (* PI 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) * (0.0008680555555555555 * pow(f, 4.0))) + ((4.0 * ((log(f) / ((double) M_PI)) - (log(4.0 / ((double) M_PI)) / ((double) M_PI)))) + ((f * (((double) M_PI) * 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 61.4

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

  2. Simplified61.4

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

  3. Taylor expanded around 0 2.4

    \[\leadsto \log \color{blue}{\left(\left(\frac{\log \left(e^{-0.25}\right) \cdot \pi}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)} + \left(0.125 \cdot \frac{{\pi}^{3} \cdot \left({\log \left(e^{-0.25}\right)}^{2} \cdot f\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \left(0.5 \cdot \frac{{\pi}^{2} \cdot \left({\log \left(e^{-0.25}\right)}^{2} \cdot f\right)}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)} + \left(0.001953125 \cdot \frac{{\pi}^{4} \cdot f}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(2 \cdot \frac{1}{f \cdot \left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)} + \left(0.03125 \cdot \frac{{\pi}^{2} \cdot f}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)} + \left(0.8333333333333334 \cdot \frac{{\pi}^{3} \cdot \left({\log \left(e^{-0.25}\right)}^{3} \cdot f\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \left(0.5 \cdot \frac{{\pi}^{4} \cdot \left({\log \left(e^{-0.25}\right)}^{4} \cdot f\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)} + \frac{{\pi}^{2} \cdot {\log \left(e^{-0.25}\right)}^{2}}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}}\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(0.03125 \cdot \frac{{\pi}^{3} \cdot \left(\log \left(e^{-0.25}\right) \cdot f\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + \left(0.0625 \cdot \frac{{\pi}^{4} \cdot \left({\log \left(e^{-0.25}\right)}^{2} \cdot f\right)}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{3}} + \left(0.013020833333333334 \cdot \frac{{\pi}^{3} \cdot f}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}} + 0.0625 \cdot \frac{{\pi}^{2}}{{\left(0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)\right)}^{2}}\right)\right)\right)\right)} \cdot \frac{-4}{\pi}\]

  4. Simplified2.4

    \[\leadsto \log \color{blue}{\left(\left(\frac{\pi}{\frac{\pi}{-0.5}} + \left(0.125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot 0.0625\right)}{{\pi}^{2} \cdot 0.25} + \left(0.5 \cdot \left(\frac{{\pi}^{2}}{\frac{\pi}{0.125}} \cdot f\right) + \left(0.001953125 \cdot \frac{f \cdot {\pi}^{4}}{{\pi}^{3} \cdot 0.125} + \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \left(0.03125 \cdot \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5} + \left(0.8333333333333334 \cdot \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\pi}^{2} \cdot 0.25} + \left(0.5 \cdot \left(\frac{f \cdot {\pi}^{4}}{{\pi}^{3} \cdot 0.125} \cdot 0.00390625\right) + \left(\frac{\pi \cdot 0.5}{\pi} + \frac{{\pi}^{2} \cdot 0.0625}{{\pi}^{2} \cdot 0.25}\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(0.03125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot -0.25\right)}{{\pi}^{2} \cdot 0.25} + \left(0.013020833333333334 \cdot \frac{f \cdot {\pi}^{3}}{{\pi}^{2} \cdot 0.25} + 0.0625 \cdot \left(\frac{{\pi}^{2}}{{\pi}^{2} \cdot 0.25} + \frac{f \cdot {\pi}^{4}}{{\pi}^{3} \cdot 0.125} \cdot 0.0625\right)\right)\right)\right)} \cdot \frac{-4}{\pi}\]

  5. Simplified2.4

    \[\leadsto \color{blue}{\log \left(\left(\left(0.001953125 \cdot \left(\frac{f}{\pi \cdot 0.125} \cdot {\pi}^{2}\right) + \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \left(0.0625 \cdot \left(\pi \cdot f\right) + \left(0.8333333333333334 \cdot \left(\left(\pi \cdot f\right) \cdot -0.0625\right) + \left(0.001953125 \cdot \left(\frac{f}{\pi \cdot 0.125} \cdot {\pi}^{2}\right) + 0.75\right)\right)\right)\right)\right) + \left(\pi \cdot 0.125\right) \cdot \left(0.25 \cdot f + f \cdot 0.5\right)\right) + \left(-0.5 - \left(\pi \cdot \frac{f \cdot 0.03125}{-1} + \left(0.052083333333333336 \cdot \left(\pi \cdot f\right) + \left(0.25 + \left(\frac{f}{\pi \cdot 0.125} \cdot {\pi}^{2}\right) \cdot 0.00390625\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi}}\]

  6. Taylor expanded around 0 2.4

    \[\leadsto \color{blue}{\left(0.0008680555555555555 \cdot \left({f}^{4} \cdot {\pi}^{3}\right) + 4 \cdot \frac{\log f}{\pi}\right) - \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right)}\]

  7. Simplified2.4

    \[\leadsto \color{blue}{{\pi}^{3} \cdot \left(0.0008680555555555555 \cdot {f}^{4}\right) + \left(4 \cdot \left(\frac{\log f}{\pi} - \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right) + \left(f \cdot \left(f \cdot \pi\right)\right) \cdot -0.08333333333333333\right)}\]

  8. Final simplification2.4

    \[\leadsto {\pi}^{3} \cdot \left(0.0008680555555555555 \cdot {f}^{4}\right) + \left(4 \cdot \left(\frac{\log f}{\pi} - \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right) + \left(f \cdot \left(\pi \cdot f\right)\right) \cdot -0.08333333333333333\right)\]

Reproduce

herbie shell --seed 2020281 
(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)))))))))