Average Error: 61.3 → 2.5
Time: 14.7s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\left(\left(2 \cdot \log \left(\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right) - \log \pi\right) + \log \left(\frac{\sqrt[3]{{2}^{0.6666666666666666}} \cdot \sqrt[3]{\sqrt[3]{2}}}{f \cdot 0.5}\right)\right) \cdot \frac{-4}{\pi}\]
-\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(2 \cdot \log \left(\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right) - \log \pi\right) + \log \left(\frac{\sqrt[3]{{2}^{0.6666666666666666}} \cdot \sqrt[3]{\sqrt[3]{2}}}{f \cdot 0.5}\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
 (*
  (+
   (-
    (* 2.0 (log (cbrt (+ (exp (* (/ PI 4.0) f)) (pow (exp -0.25) (* PI f))))))
    (log PI))
   (log
    (/ (* (cbrt (pow 2.0 0.6666666666666666)) (cbrt (cbrt 2.0))) (* f 0.5))))
  (/ -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 (((2.0 * log(cbrt(exp((((double) M_PI) / 4.0) * f) + pow(exp(-0.25), (((double) M_PI) * f))))) - log((double) M_PI)) + log((cbrt(pow(2.0, 0.6666666666666666)) * cbrt(cbrt(2.0))) / (f * 0.5))) * (-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 2.9

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

  4. Simplified2.9

    \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\color{blue}{\pi \cdot \left(0.5 \cdot f\right)}}\right) \cdot \frac{-4}{\pi}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt_binary642.9

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

  7. Applied times-frac_binary642.9

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

  8. Applied log-prod_binary642.8

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

  9. Simplified2.8

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

  10. Simplified2.8

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

  11. Taylor expanded around 0 2.5

    \[\leadsto \left(\left(2 \cdot \log \left(\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right) - \log \pi\right) + \log \left(\frac{\color{blue}{{2}^{0.3333333333333333}}}{f \cdot 0.5}\right)\right) \cdot \frac{-4}{\pi}\]

  12. Simplified2.5

    \[\leadsto \left(\left(2 \cdot \log \left(\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right) - \log \pi\right) + \log \left(\frac{\color{blue}{\sqrt[3]{2}}}{f \cdot 0.5}\right)\right) \cdot \frac{-4}{\pi}\]
  13. Using strategy rm

  14. Applied add-cube-cbrt_binary642.5

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

  15. Applied cbrt-prod_binary642.5

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

  16. Simplified2.5

    \[\leadsto \left(\left(2 \cdot \log \left(\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right) - \log \pi\right) + \log \left(\frac{\color{blue}{\sqrt[3]{{2}^{0.6666666666666666}}} \cdot \sqrt[3]{\sqrt[3]{2}}}{f \cdot 0.5}\right)\right) \cdot \frac{-4}{\pi}\]

  17. Final simplification2.5

    \[\leadsto \left(\left(2 \cdot \log \left(\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right) - \log \pi\right) + \log \left(\frac{\sqrt[3]{{2}^{0.6666666666666666}} \cdot \sqrt[3]{\sqrt[3]{2}}}{f \cdot 0.5}\right)\right) \cdot \frac{-4}{\pi}\]

Reproduce

herbie shell --seed 2020344 
(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)))))))))