Average Error: 61.3 → 2.5
Time: 14.2s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\left(\left(f \cdot \left(f \cdot {\pi}^{2}\right)\right) \cdot 0.0078125 + \left(\left(\log \left(\frac{4}{\pi}\right) + \left(0.5 \cdot \left(\pi \cdot \left(f \cdot -0.25\right)\right) + \left(f \cdot \pi\right) \cdot 0.125\right)\right) - \left(\left(f \cdot \left(f \cdot {\pi}^{2}\right)\right) \cdot 0.0026041666666666665 + \left(\log f + 0.0625 \cdot \left(\left(f \cdot \left(f \cdot {\pi}^{2}\right)\right) \cdot -0.25\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]
-\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)
↓
\left(\left(f \cdot \left(f \cdot {\pi}^{2}\right)\right) \cdot 0.0078125 + \left(\left(\log \left(\frac{4}{\pi}\right) + \left(0.5 \cdot \left(\pi \cdot \left(f \cdot -0.25\right)\right) + \left(f \cdot \pi\right) \cdot 0.125\right)\right) - \left(\left(f \cdot \left(f \cdot {\pi}^{2}\right)\right) \cdot 0.0026041666666666665 + \left(\log f + 0.0625 \cdot \left(\left(f \cdot \left(f \cdot {\pi}^{2}\right)\right) \cdot -0.25\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
 (*
  (+
   (* (* f (* f (pow PI 2.0))) 0.0078125)
   (-
    (+ (log (/ 4.0 PI)) (+ (* 0.5 (* PI (* f -0.25))) (* (* f PI) 0.125)))
    (+
     (* (* f (* f (pow PI 2.0))) 0.0026041666666666665)
     (+ (log f) (* 0.0625 (* (* f (* f (pow PI 2.0))) -0.25))))))
  (/ -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 (((f * (f * pow(((double) M_PI), 2.0))) * 0.0078125) + ((log(4.0 / ((double) M_PI)) + ((0.5 * (((double) M_PI) * (f * -0.25))) + ((f * ((double) M_PI)) * 0.125))) - (((f * (f * pow(((double) M_PI), 2.0))) * 0.0026041666666666665) + (log(f) + (0.0625 * ((f * (f * pow(((double) M_PI), 2.0))) * -0.25)))))) * (-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 2.6

    \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\color{blue}{\left(0.03125 \cdot \left({\pi}^{2} \cdot {f}^{2}\right) + \left(0.0026041666666666665 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + 0.25 \cdot \left(\pi \cdot f\right)\right)\right) - \left(0.16666666666666666 \cdot \left({f}^{3} \cdot \left({\log \left(e^{-0.25}\right)}^{3} \cdot {\pi}^{3}\right)\right) + \left(f \cdot \left(\log \left(e^{-0.25}\right) \cdot \pi\right) + 0.5 \cdot \left({\pi}^{2} \cdot \left({\log \left(e^{-0.25}\right)}^{2} \cdot {f}^{2}\right)\right)\right)\right)}}\right) \cdot \frac{-4}{\pi}\]

  4. Simplified2.6

    \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\color{blue}{\left(0.0026041666666666665 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + f \cdot \left(\pi \cdot 0.25 + 0.03125 \cdot \left(f \cdot {\pi}^{2}\right)\right)\right) + \left(\left(\pi \cdot \left(f \cdot 0.25\right) - 0.16666666666666666 \cdot \left({f}^{3} \cdot \left({\pi}^{3} \cdot -0.015625\right)\right)\right) - 0.5 \cdot \left({\pi}^{2} \cdot \left(\left(f \cdot f\right) \cdot 0.0625\right)\right)\right)}}\right) \cdot \frac{-4}{\pi}\]

  5. Taylor expanded around 0 2.5

    \[\leadsto \color{blue}{\left(\left(0.125 \cdot \left({\pi}^{2} \cdot \left({\log \left(e^{-0.25}\right)}^{2} \cdot {f}^{2}\right)\right) + \left(\log \left(\frac{4}{\pi}\right) + \left(0.5 \cdot \left(\pi \cdot \left(\log \left(e^{-0.25}\right) \cdot f\right)\right) + 0.125 \cdot \left(\pi \cdot f\right)\right)\right)\right) - \left(0.0026041666666666665 \cdot \left({\pi}^{2} \cdot {f}^{2}\right) + \left(\log f + 0.0625 \cdot \left({\pi}^{2} \cdot \left(\log \left(e^{-0.25}\right) \cdot {f}^{2}\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi}\]

  6. Simplified2.5

    \[\leadsto \color{blue}{\left(\left(f \cdot \left(f \cdot {\pi}^{2}\right)\right) \cdot 0.0078125 + \left(\left(\log \left(\frac{4}{\pi}\right) + \left(0.5 \cdot \left(\pi \cdot \left(f \cdot -0.25\right)\right) + \left(\pi \cdot f\right) \cdot 0.125\right)\right) - \left(0.0026041666666666665 \cdot \left(f \cdot \left(f \cdot {\pi}^{2}\right)\right) + \left(\log f + 0.0625 \cdot \left(-0.25 \cdot \left(f \cdot \left(f \cdot {\pi}^{2}\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi}\]

  7. Final simplification2.5

    \[\leadsto \left(\left(f \cdot \left(f \cdot {\pi}^{2}\right)\right) \cdot 0.0078125 + \left(\left(\log \left(\frac{4}{\pi}\right) + \left(0.5 \cdot \left(\pi \cdot \left(f \cdot -0.25\right)\right) + \left(f \cdot \pi\right) \cdot 0.125\right)\right) - \left(\left(f \cdot \left(f \cdot {\pi}^{2}\right)\right) \cdot 0.0026041666666666665 + \left(\log f + 0.0625 \cdot \left(\left(f \cdot \left(f \cdot {\pi}^{2}\right)\right) \cdot -0.25\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

Reproduce

herbie shell --seed 2020280 
(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)))))))))