Average Error: 61.3 → 2.3
Time: 16.7s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[1 \cdot \left(4 \cdot \left(\left(\log \left(0.005208333333333333 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.6276041666666666 \cdot 10^{-05} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \pi \cdot \left(f \cdot 0.5\right)\right)\right) - \log \left({\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)} + {\left(e^{0.25}\right)}^{\left(f \cdot \pi\right)}\right)\right) \cdot \frac{1}{\pi}\right)\right)\]
-\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(4 \cdot \left(\left(\log \left(0.005208333333333333 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.6276041666666666 \cdot 10^{-05} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \pi \cdot \left(f \cdot 0.5\right)\right)\right) - \log \left({\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)} + {\left(e^{0.25}\right)}^{\left(f \cdot \pi\right)}\right)\right) \cdot \frac{1}{\pi}\right)\right)
double code(double f) {
	return ((double) -(((double) (((double) (1.0 / ((double) (((double) M_PI) / 4.0)))) * ((double) log(((double) (((double) (((double) exp(((double) (((double) (((double) M_PI) / 4.0)) * f)))) + ((double) exp(((double) -(((double) (((double) (((double) M_PI) / 4.0)) * f)))))))) / ((double) (((double) exp(((double) (((double) (((double) M_PI) / 4.0)) * f)))) - ((double) exp(((double) -(((double) (((double) (((double) M_PI) / 4.0)) * f))))))))))))))));
}
↓
double code(double f) {
	return ((double) (1.0 * ((double) (4.0 * ((double) (((double) (((double) log(((double) (((double) (0.005208333333333333 * ((double) (((double) pow(f, 3.0)) * ((double) pow(((double) M_PI), 3.0)))))) + ((double) (((double) (1.6276041666666666e-05 * ((double) (((double) pow(f, 5.0)) * ((double) pow(((double) M_PI), 5.0)))))) + ((double) (((double) M_PI) * ((double) (f * 0.5)))))))))) - ((double) log(((double) (((double) pow(((double) exp(-0.25)), ((double) (f * ((double) M_PI))))) + ((double) pow(((double) exp(0.25)), ((double) (f * ((double) M_PI))))))))))) * ((double) (1.0 / ((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. Simplified61.4

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

  3. Taylor expanded around 0 2.2

    \[\leadsto 1 \cdot \left(4 \cdot \frac{\log \color{blue}{\left(0.005208333333333333 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.6276041666666666 \cdot 10^{-05} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + 0.5 \cdot \left(f \cdot \pi\right)\right)\right)} - \log \left({\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}\right)}{\pi}\right)\]

  4. Simplified2.2

    \[\leadsto 1 \cdot \left(4 \cdot \frac{\log \color{blue}{\left(0.005208333333333333 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.6276041666666666 \cdot 10^{-05} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \pi \cdot \left(f \cdot 0.5\right)\right)\right)} - \log \left({\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}\right)}{\pi}\right)\]
  5. Using strategy rm

  6. Applied div-inv2.3

    \[\leadsto 1 \cdot \left(4 \cdot \color{blue}{\left(\left(\log \left(0.005208333333333333 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.6276041666666666 \cdot 10^{-05} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \pi \cdot \left(f \cdot 0.5\right)\right)\right) - \log \left({\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}\right)\right) \cdot \frac{1}{\pi}\right)}\right)\]

  7. Taylor expanded around inf 2.3

    \[\leadsto 1 \cdot \left(4 \cdot \left(\left(\log \left(0.005208333333333333 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.6276041666666666 \cdot 10^{-05} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \pi \cdot \left(f \cdot 0.5\right)\right)\right) - \color{blue}{\log \left(e^{-0.25 \cdot \left(\pi \cdot f\right)} + e^{0.25 \cdot \left(\pi \cdot f\right)}\right)}\right) \cdot \frac{1}{\pi}\right)\right)\]

  8. Simplified2.3

    \[\leadsto 1 \cdot \left(4 \cdot \left(\left(\log \left(0.005208333333333333 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.6276041666666666 \cdot 10^{-05} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \pi \cdot \left(f \cdot 0.5\right)\right)\right) - \color{blue}{\log \left({\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)} + {\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)}\right)}\right) \cdot \frac{1}{\pi}\right)\right)\]

  9. Final simplification2.3

    \[\leadsto 1 \cdot \left(4 \cdot \left(\left(\log \left(0.005208333333333333 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \left(1.6276041666666666 \cdot 10^{-05} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \pi \cdot \left(f \cdot 0.5\right)\right)\right) - \log \left({\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)} + {\left(e^{0.25}\right)}^{\left(f \cdot \pi\right)}\right)\right) \cdot \frac{1}{\pi}\right)\right)\]

Reproduce

herbie shell --seed 2020184 
(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)))))))))