Average Error: 61.5 → 2.1
Time: 15.0s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[1 \cdot \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0003472222222222224\right) + \left(4 \cdot \frac{\log f}{\pi} + \left(0.01388888888888889 \cdot \left(\frac{{\left({\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right)}^{\left(\sqrt{3}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) - \left(\pi \cdot \left(0.08333333333333334 \cdot \left(f \cdot f\right)\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)\right)\right)\]
-\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)
↓
1 \cdot \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0003472222222222224\right) + \left(4 \cdot \frac{\log f}{\pi} + \left(0.01388888888888889 \cdot \left(\frac{{\left({\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right)}^{\left(\sqrt{3}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) - \left(\pi \cdot \left(0.08333333333333334 \cdot \left(f \cdot f\right)\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)\right)\right)
(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
 (*
  1.0
  (+
   (* (pow PI 3.0) (* (pow f 4.0) 0.0003472222222222224))
   (+
    (* 4.0 (/ (log f) PI))
    (-
     (*
      0.01388888888888889
      (*
       (/
        (pow
         (*
          (pow
           (*
            (* (cbrt (sqrt PI)) (cbrt (sqrt PI)))
            (* (cbrt (sqrt PI)) (cbrt (sqrt PI))))
           (sqrt 3.0))
          (pow (cbrt PI) (sqrt 3.0)))
         (sqrt 3.0))
        4.0)
       (/ (pow f 4.0) 4.0)))
     (+
      (* PI (* 0.08333333333333334 (* f f)))
      (* 4.0 (/ (log (/ 4.0 PI)) PI))))))))
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) (1.0 * ((double) (((double) (((double) pow(((double) M_PI), 3.0)) * ((double) (((double) pow(f, 4.0)) * 0.0003472222222222224)))) + ((double) (((double) (4.0 * (((double) log(f)) / ((double) M_PI)))) + ((double) (((double) (0.01388888888888889 * ((double) ((((double) pow(((double) (((double) pow(((double) (((double) (((double) cbrt(((double) sqrt(((double) M_PI))))) * ((double) cbrt(((double) sqrt(((double) M_PI))))))) * ((double) (((double) cbrt(((double) sqrt(((double) M_PI))))) * ((double) cbrt(((double) sqrt(((double) M_PI))))))))), ((double) sqrt(3.0)))) * ((double) pow(((double) cbrt(((double) M_PI))), ((double) sqrt(3.0)))))), ((double) sqrt(3.0)))) / 4.0) * (((double) pow(f, 4.0)) / 4.0))))) - ((double) (((double) (((double) M_PI) * ((double) (0.08333333333333334 * ((double) (f * f)))))) + ((double) (4.0 * (((double) log((4.0 / ((double) M_PI)))) / ((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 Error: 61.5 bits

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

  2. SimplifiedError: 61.5 bits

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

  3. Taylor expanded around 0 Error: 2.1 bits

    \[\leadsto 1 \cdot \color{blue}{\left(\left(0.0003472222222222224 \cdot \left({\pi}^{3} \cdot {f}^{4}\right) + \left(4 \cdot \frac{\log f}{\pi} + 0.01388888888888889 \cdot \frac{{\pi}^{3} \cdot {f}^{4}}{{4}^{2}}\right)\right) - \left(0.08333333333333334 \cdot \left({f}^{2} \cdot \pi\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)}\]

  4. SimplifiedError: 2.1 bits

    \[\leadsto 1 \cdot \color{blue}{\left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0003472222222222224\right) + \left(4 \cdot \frac{\log f}{\pi} + \left(0.01388888888888889 \cdot \left(\frac{{\pi}^{3}}{4} \cdot \frac{{f}^{4}}{4}\right) - \left(\pi \cdot \left(0.08333333333333334 \cdot \left(f \cdot f\right)\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)\right)\right)}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrtError: 2.1 bits

    \[\leadsto 1 \cdot \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0003472222222222224\right) + \left(4 \cdot \frac{\log f}{\pi} + \left(0.01388888888888889 \cdot \left(\frac{{\pi}^{\color{blue}{\left(\sqrt{3} \cdot \sqrt{3}\right)}}}{4} \cdot \frac{{f}^{4}}{4}\right) - \left(\pi \cdot \left(0.08333333333333334 \cdot \left(f \cdot f\right)\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)\right)\right)\]

  7. Applied pow-unpowError: 2.1 bits

    \[\leadsto 1 \cdot \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0003472222222222224\right) + \left(4 \cdot \frac{\log f}{\pi} + \left(0.01388888888888889 \cdot \left(\frac{\color{blue}{{\left({\pi}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}}{4} \cdot \frac{{f}^{4}}{4}\right) - \left(\pi \cdot \left(0.08333333333333334 \cdot \left(f \cdot f\right)\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)\right)\right)\]
  8. Using strategy rm

  9. Applied add-cube-cbrtError: 2.1 bits

    \[\leadsto 1 \cdot \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0003472222222222224\right) + \left(4 \cdot \frac{\log f}{\pi} + \left(0.01388888888888889 \cdot \left(\frac{{\left({\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)}}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) - \left(\pi \cdot \left(0.08333333333333334 \cdot \left(f \cdot f\right)\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)\right)\right)\]

  10. Applied unpow-prod-downError: 2.1 bits

    \[\leadsto 1 \cdot \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0003472222222222224\right) + \left(4 \cdot \frac{\log f}{\pi} + \left(0.01388888888888889 \cdot \left(\frac{{\color{blue}{\left({\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) - \left(\pi \cdot \left(0.08333333333333334 \cdot \left(f \cdot f\right)\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)\right)\right)\]
  11. Using strategy rm

  12. Applied add-sqr-sqrtError: 2.1 bits

    \[\leadsto 1 \cdot \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0003472222222222224\right) + \left(4 \cdot \frac{\log f}{\pi} + \left(0.01388888888888889 \cdot \left(\frac{{\left({\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right)}^{\left(\sqrt{3}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) - \left(\pi \cdot \left(0.08333333333333334 \cdot \left(f \cdot f\right)\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)\right)\right)\]

  13. Applied cbrt-prodError: 2.1 bits

    \[\leadsto 1 \cdot \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0003472222222222224\right) + \left(4 \cdot \frac{\log f}{\pi} + \left(0.01388888888888889 \cdot \left(\frac{{\left({\left(\sqrt[3]{\pi} \cdot \color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}\right)}^{\left(\sqrt{3}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) - \left(\pi \cdot \left(0.08333333333333334 \cdot \left(f \cdot f\right)\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)\right)\right)\]

  14. Applied add-sqr-sqrtError: 2.1 bits

    \[\leadsto 1 \cdot \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0003472222222222224\right) + \left(4 \cdot \frac{\log f}{\pi} + \left(0.01388888888888889 \cdot \left(\frac{{\left({\left(\sqrt[3]{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right)}^{\left(\sqrt{3}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) - \left(\pi \cdot \left(0.08333333333333334 \cdot \left(f \cdot f\right)\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)\right)\right)\]

  15. Applied cbrt-prodError: 2.1 bits

    \[\leadsto 1 \cdot \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0003472222222222224\right) + \left(4 \cdot \frac{\log f}{\pi} + \left(0.01388888888888889 \cdot \left(\frac{{\left({\left(\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right)}^{\left(\sqrt{3}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) - \left(\pi \cdot \left(0.08333333333333334 \cdot \left(f \cdot f\right)\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)\right)\right)\]

  16. Applied swap-sqrError: 2.1 bits

    \[\leadsto 1 \cdot \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0003472222222222224\right) + \left(4 \cdot \frac{\log f}{\pi} + \left(0.01388888888888889 \cdot \left(\frac{{\left({\color{blue}{\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right)}}^{\left(\sqrt{3}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) - \left(\pi \cdot \left(0.08333333333333334 \cdot \left(f \cdot f\right)\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)\right)\right)\]

  17. Final simplificationError: 2.1 bits

    \[\leadsto 1 \cdot \left({\pi}^{3} \cdot \left({f}^{4} \cdot 0.0003472222222222224\right) + \left(4 \cdot \frac{\log f}{\pi} + \left(0.01388888888888889 \cdot \left(\frac{{\left({\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right)}^{\left(\sqrt{3}\right)} \cdot {\left(\sqrt[3]{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}{4} \cdot \frac{{f}^{4}}{4}\right) - \left(\pi \cdot \left(0.08333333333333334 \cdot \left(f \cdot f\right)\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020203 
(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)))))))))