Average Error: 61.6 → 2.3
Time: 12.0s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\left(\left(0.020833333333333332 \cdot \left(\left(\pi \cdot f\right) \cdot \left(\pi \cdot f\right)\right) + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + 0.00030381944444444445 \cdot \left({f}^{4} \cdot \left({\left(\sqrt[3]{\pi}\right)}^{8} \cdot {\left(\sqrt[3]{\pi}\right)}^{4}\right)\right)\right)\right) \cdot \frac{-4}{\pi}\]
-\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(\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 \left({\left(\sqrt[3]{\pi}\right)}^{8} \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 (* (* PI f) (* PI f))) (log (/ 4.0 PI)))
   (+
    (log f)
    (*
     0.00030381944444444445
     (* (pow f 4.0) (* (pow (cbrt 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 * ((((double) M_PI) * f) * (((double) M_PI) * f))) + log(4.0 / ((double) M_PI))) - (log(f) + (0.00030381944444444445 * (pow(f, 4.0) * (pow(cbrt((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 2.3

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

  4. Simplified2.3

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

  6. Applied add-cube-cbrt_binary642.3

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

  7. Applied unpow-prod-down_binary642.3

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

  8. Simplified2.3

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

  9. Final simplification2.3

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

Reproduce

herbie shell --seed 2020220 
(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)))))))))