Average Error: 61.5 → 2.5
Time: 11.9s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0012152777777777778\right) + -4 \cdot \frac{1}{\frac{\pi}{\log \left(\frac{4}{\pi}\right) - \log f}}\right) + \left(\pi \cdot \left(f \cdot f\right)\right) \cdot -0.08333333333333333\]
-\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({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0012152777777777778\right) + -4 \cdot \frac{1}{\frac{\pi}{\log \left(\frac{4}{\pi}\right) - \log f}}\right) + \left(\pi \cdot \left(f \cdot f\right)\right) \cdot -0.08333333333333333
(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 (/ 1.0 (/ PI (- (log (/ 4.0 PI)) (log f))))))
  (* (* PI (* f f)) -0.08333333333333333)))
double code(double f) {
	return ((double) -(((double) ((1.0 / (((double) M_PI) / 4.0)) * ((double) log((((double) (((double) exp(((double) ((((double) M_PI) / 4.0) * f)))) + ((double) exp(((double) -(((double) ((((double) M_PI) / 4.0) * f)))))))) / ((double) (((double) exp(((double) ((((double) M_PI) / 4.0) * f)))) - ((double) exp(((double) -(((double) ((((double) M_PI) / 4.0) * f)))))))))))))));
}
↓
double code(double f) {
	return ((double) (((double) (((double) (((double) pow(((double) M_PI), 3.0)) * ((double) (((double) pow(f, 4.0)) * 0.0012152777777777778)))) + ((double) (-4.0 * (1.0 / (((double) M_PI) / ((double) (((double) log((4.0 / ((double) M_PI)))) - ((double) log(f)))))))))) + ((double) (((double) (((double) M_PI) * ((double) (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 61.5

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

  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 2.4

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

  4. Simplified2.4

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

  5. Taylor expanded around inf 2.5

    \[\leadsto \color{blue}{0.0012152777777777778 \cdot \left({f}^{4} \cdot {\pi}^{3}\right) - \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \left(4 \cdot \frac{\log \left(\frac{1}{f}\right)}{\pi} + 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right)\right)}\]

  6. Simplified2.4

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

  8. Applied frac-sub_binary642.5

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

  9. Simplified2.5

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

  11. Applied clear-num_binary642.5

    \[\leadsto \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0012152777777777778\right) + -4 \cdot \color{blue}{\frac{1}{\frac{\pi \cdot \pi}{\pi \cdot \left(\log \left(\frac{4}{\pi}\right) - \log f\right)}}}\right) + \left(\pi \cdot \left(f \cdot f\right)\right) \cdot -0.08333333333333333\]

  12. Simplified2.5

    \[\leadsto \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0012152777777777778\right) + -4 \cdot \frac{1}{\color{blue}{\frac{\pi}{\log \left(\frac{4}{\pi}\right) - \log f}}}\right) + \left(\pi \cdot \left(f \cdot f\right)\right) \cdot -0.08333333333333333\]

  13. Final simplification2.5

    \[\leadsto \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0012152777777777778\right) + -4 \cdot \frac{1}{\frac{\pi}{\log \left(\frac{4}{\pi}\right) - \log f}}\right) + \left(\pi \cdot \left(f \cdot f\right)\right) \cdot -0.08333333333333333\]

Reproduce

herbie shell --seed 2020210 
(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)))))))))