Average Error: 61.6 → 2.1
Time: 18.1s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot \left({\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}}\right)}^{8} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{\pi}}}\right)}^{8}\right)\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]
-\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(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot \left({\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}}\right)}^{8} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{\pi}}}\right)}^{8}\right)\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\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
 (*
  (-
   (+ (* 0.020833333333333332 (* (* f f) (pow PI 2.0))) (log (/ 4.0 PI)))
   (+
    (log f)
    (*
     0.00030381944444444445
     (*
      (pow f 4.0)
      (*
       (*
        (pow (cbrt (sqrt PI)) 8.0)
        (*
         (pow (cbrt (pow (cbrt (sqrt PI)) 2.0)) 8.0)
         (pow (cbrt (cbrt (sqrt PI))) 8.0)))
       (pow (cbrt PI) 4.0))))))
  (/ -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 (((0.020833333333333332 * ((f * f) * pow(((double) M_PI), 2.0))) + log(4.0 / ((double) M_PI))) - (log(f) + (0.00030381944444444445 * (pow(f, 4.0) * ((pow(cbrt(sqrt((double) M_PI)), 8.0) * (pow(cbrt(pow(cbrt(sqrt((double) M_PI)), 2.0)), 8.0) * pow(cbrt(cbrt(sqrt((double) M_PI))), 8.0))) * pow(cbrt((double) M_PI), 4.0)))))) * (-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 61.6

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

  2. Simplified61.6

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

  3. Taylor expanded around 0 2.1

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

  4. Simplified2.1

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

  6. Applied add-cube-cbrt_binary642.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot {\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)}}^{4}\right)\right)\right) \cdot \frac{-4}{\pi}\]

  7. Applied unpow-prod-down_binary642.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \color{blue}{\left({\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)}^{4} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)}\right)\right)\right) \cdot \frac{-4}{\pi}\]

  8. Simplified2.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left(\color{blue}{{\left({\left(\sqrt[3]{\pi}\right)}^{2}\right)}^{4}} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt_binary642.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left({\left({\left(\sqrt[3]{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right)}^{2}\right)}^{4} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

  11. Applied cbrt-prod_binary642.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left({\left({\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}}^{2}\right)}^{4} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

  12. Applied unpow-prod-down_binary642.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left({\color{blue}{\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}\right)}}^{4} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

  13. Applied unpow-prod-down_binary642.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left(\color{blue}{\left({\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}\right)}^{4} \cdot {\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}\right)}^{4}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

  14. Simplified2.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left(\left(\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8}} \cdot {\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}\right)}^{4}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

  15. Simplified2.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8}}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]
  16. Using strategy rm

  17. Applied add-cube-cbrt_binary642.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot {\left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}}}\right)}^{8}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

  18. Applied cbrt-prod_binary642.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot {\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\pi}}}\right)}}^{8}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

  19. Applied unpow-prod-down_binary642.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot \color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}}\right)}^{8} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{\pi}}}\right)}^{8}\right)}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

  20. Simplified2.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot \left(\color{blue}{{\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}}\right)}^{8}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{\pi}}}\right)}^{8}\right)\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

  21. Final simplification2.1

    \[\leadsto \left(\left(0.020833333333333332 \cdot \left(\left(f \cdot f\right) \cdot {\pi}^{2}\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left(\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{8} \cdot \left({\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{2}}\right)}^{8} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{\pi}}}\right)}^{8}\right)\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]

Reproduce

herbie shell --seed 2020231 
(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)))))))))