Average Error: 61.5 → 2.3
Time: 16.1s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\log \left(\left(\left(0.001953125 \cdot \left(\frac{f}{\pi \cdot 0.125} \cdot {\pi}^{2}\right) + \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)} + \left(0.0625 \cdot \left(f \cdot \pi\right) + \left(0.8333333333333334 \cdot \left(\left(f \cdot \pi\right) \cdot -0.0625\right) + \left(0.001953125 \cdot \left(\frac{f}{\pi \cdot 0.125} \cdot {\pi}^{2}\right) + 0.75\right)\right)\right)\right)\right) + \left(\pi \cdot 0.125\right) \cdot \left(f \cdot 0.25 + f \cdot 0.5\right)\right) + \left(-0.5 - \left(\pi \cdot \frac{f \cdot 0.03125}{-1} + \left(\left(f \cdot \pi\right) \cdot 0.052083333333333336 + \left(0.25 + \left(\frac{f}{\pi \cdot 0.125} \cdot {\pi}^{2}\right) \cdot 0.00390625\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]
-\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(\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(f \cdot \pi\right) + \left(0.8333333333333334 \cdot \left(\left(f \cdot \pi\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(f \cdot 0.25 + f \cdot 0.5\right)\right) + \left(-0.5 - \left(\pi \cdot \frac{f \cdot 0.03125}{-1} + \left(\left(f \cdot \pi\right) \cdot 0.052083333333333336 + \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}
(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
   (+
    (+
     (+
      (* 0.001953125 (* (/ f (* PI 0.125)) (pow PI 2.0)))
      (+
       (/ 2.0 (* f (* PI 0.5)))
       (+
        (* 0.0625 (* f PI))
        (+
         (* 0.8333333333333334 (* (* f PI) -0.0625))
         (+ (* 0.001953125 (* (/ f (* PI 0.125)) (pow PI 2.0))) 0.75)))))
     (* (* PI 0.125) (+ (* f 0.25) (* f 0.5))))
    (-
     -0.5
     (+
      (* PI (/ (* f 0.03125) -1.0))
      (+
       (* (* f PI) 0.052083333333333336)
       (+ 0.25 (* (* (/ f (* PI 0.125)) (pow PI 2.0)) 0.00390625)))))))
  (/ -4.0 PI)))
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((((0.001953125 * ((f / (((double) M_PI) * 0.125)) * pow(((double) M_PI), 2.0))) + ((2.0 / (f * (((double) M_PI) * 0.5))) + ((0.0625 * (f * ((double) M_PI))) + ((0.8333333333333334 * ((f * ((double) M_PI)) * -0.0625)) + ((0.001953125 * ((f / (((double) M_PI) * 0.125)) * pow(((double) M_PI), 2.0))) + 0.75))))) + ((((double) M_PI) * 0.125) * ((f * 0.25) + (f * 0.5)))) + (-0.5 - ((((double) M_PI) * ((f * 0.03125) / -1.0)) + (((f * ((double) M_PI)) * 0.052083333333333336) + (0.25 + (((f / (((double) M_PI) * 0.125)) * pow(((double) M_PI), 2.0)) * 0.00390625)))))) * (-4.0 / ((double) M_PI));
}

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. Simplified61.5

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

  3. Taylor expanded around 0 2.3

    \[\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. Simplified2.3

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

  5. Simplified2.3

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

  6. Final simplification2.3

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

Reproduce

herbie shell --seed 2020295 
(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)))))))))