Average Error: 61.3 → 2.3
Time: 21.5s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\log \left(\left(\frac{\pi \cdot -0.25}{\pi \cdot 0.5} + \left(0.125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot 0.0625\right)}{{\left(\pi \cdot 0.5\right)}^{2}} + \left(0.5 \cdot \frac{\left(f \cdot 0.0625\right) \cdot {\pi}^{2}}{\pi \cdot 0.5} + \left(0.001953125 \cdot \frac{f \cdot {\pi}^{4}}{0.125 \cdot {\pi}^{3}} + \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f} + \left(0.03125 \cdot \frac{f \cdot {\pi}^{2}}{\pi \cdot 0.5} + \left(0.8333333333333334 \cdot \frac{f \cdot \left({\pi}^{3} \cdot -0.015625\right)}{{\left(\pi \cdot 0.5\right)}^{2}} + \left(0.5 \cdot \frac{{\pi}^{4} \cdot \left(f \cdot 0.00390625\right)}{0.125 \cdot {\pi}^{3}} + \left(0.25 \cdot \frac{\pi}{\pi \cdot 0.5} + \frac{0.0625 \cdot {\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(0.03125 \cdot \frac{{\pi}^{3} \cdot \left(-0.25 \cdot f\right)}{{\left(\pi \cdot 0.5\right)}^{2}} + \left(0.0625 \cdot \frac{0.0625 \cdot \left(f \cdot {\pi}^{4}\right)}{0.125 \cdot {\pi}^{3}} + \left(\frac{0.0625 \cdot {\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}} + 0.013020833333333334 \cdot \frac{{\pi}^{3} \cdot f}{{\left(\pi \cdot 0.5\right)}^{2}}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]
-\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(\frac{\pi \cdot -0.25}{\pi \cdot 0.5} + \left(0.125 \cdot \frac{{\pi}^{3} \cdot \left(f \cdot 0.0625\right)}{{\left(\pi \cdot 0.5\right)}^{2}} + \left(0.5 \cdot \frac{\left(f \cdot 0.0625\right) \cdot {\pi}^{2}}{\pi \cdot 0.5} + \left(0.001953125 \cdot \frac{f \cdot {\pi}^{4}}{0.125 \cdot {\pi}^{3}} + \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f} + \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)}{{\left(\pi \cdot 0.5\right)}^{2}} + \left(0.5 \cdot \frac{{\pi}^{4} \cdot \left(f \cdot 0.00390625\right)}{0.125 \cdot {\pi}^{3}} + \left(0.25 \cdot \frac{\pi}{\pi \cdot 0.5} + \frac{0.0625 \cdot {\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}}\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(0.03125 \cdot \frac{{\pi}^{3} \cdot \left(-0.25 \cdot f\right)}{{\left(\pi \cdot 0.5\right)}^{2}} + \left(0.0625 \cdot \frac{0.0625 \cdot \left(f \cdot {\pi}^{4}\right)}{0.125 \cdot {\pi}^{3}} + \left(\frac{0.0625 \cdot {\pi}^{2}}{{\left(\pi \cdot 0.5\right)}^{2}} + 0.013020833333333334 \cdot \frac{{\pi}^{3} \cdot f}{{\left(\pi \cdot 0.5\right)}^{2}}\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
   (-
    (+
     (/ (* PI -0.25) (* PI 0.5))
     (+
      (* 0.125 (/ (* (pow PI 3.0) (* f 0.0625)) (pow (* PI 0.5) 2.0)))
      (+
       (* 0.5 (/ (* (* f 0.0625) (pow PI 2.0)) (* PI 0.5)))
       (+
        (* 0.001953125 (/ (* f (pow PI 4.0)) (* 0.125 (pow PI 3.0))))
        (+
         (/ 2.0 (* (* PI 0.5) f))
         (+
          (* 0.03125 (/ (* f (pow PI 2.0)) (* PI 0.5)))
          (+
           (*
            0.8333333333333334
            (/ (* f (* (pow PI 3.0) -0.015625)) (pow (* PI 0.5) 2.0)))
           (+
            (*
             0.5
             (/ (* (pow PI 4.0) (* f 0.00390625)) (* 0.125 (pow PI 3.0))))
            (+
             (* 0.25 (/ PI (* PI 0.5)))
             (/ (* 0.0625 (pow PI 2.0)) (pow (* PI 0.5) 2.0)))))))))))
    (+
     (* 0.03125 (/ (* (pow PI 3.0) (* -0.25 f)) (pow (* PI 0.5) 2.0)))
     (+
      (* 0.0625 (/ (* 0.0625 (* f (pow PI 4.0))) (* 0.125 (pow PI 3.0))))
      (+
       (/ (* 0.0625 (pow PI 2.0)) (pow (* PI 0.5) 2.0))
       (*
        0.013020833333333334
        (/ (* (pow PI 3.0) f) (pow (* PI 0.5) 2.0))))))))
  (/ -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((((((double) M_PI) * -0.25) / (((double) M_PI) * 0.5)) + ((0.125 * ((pow(((double) M_PI), 3.0) * (f * 0.0625)) / pow((((double) M_PI) * 0.5), 2.0))) + ((0.5 * (((f * 0.0625) * pow(((double) M_PI), 2.0)) / (((double) M_PI) * 0.5))) + ((0.001953125 * ((f * pow(((double) M_PI), 4.0)) / (0.125 * pow(((double) M_PI), 3.0)))) + ((2.0 / ((((double) M_PI) * 0.5) * f)) + ((0.03125 * ((f * pow(((double) M_PI), 2.0)) / (((double) M_PI) * 0.5))) + ((0.8333333333333334 * ((f * (pow(((double) M_PI), 3.0) * -0.015625)) / pow((((double) M_PI) * 0.5), 2.0))) + ((0.5 * ((pow(((double) M_PI), 4.0) * (f * 0.00390625)) / (0.125 * pow(((double) M_PI), 3.0)))) + ((0.25 * (((double) M_PI) / (((double) M_PI) * 0.5))) + ((0.0625 * pow(((double) M_PI), 2.0)) / pow((((double) M_PI) * 0.5), 2.0))))))))))) - ((0.03125 * ((pow(((double) M_PI), 3.0) * (-0.25 * f)) / pow((((double) M_PI) * 0.5), 2.0))) + ((0.0625 * ((0.0625 * (f * pow(((double) M_PI), 4.0))) / (0.125 * pow(((double) M_PI), 3.0)))) + (((0.0625 * pow(((double) M_PI), 2.0)) / pow((((double) M_PI) * 0.5), 2.0)) + (0.013020833333333334 * ((pow(((double) M_PI), 3.0) * f) / pow((((double) M_PI) * 0.5), 2.0))))))) * (-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 61.3

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

  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 2.3

    \[\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

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

  5. Final simplification2.3

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

Reproduce

herbie shell --seed 2020353 
(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)))))))))