Average Error: 61.4 → 2.3
Time: 14.6s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\frac{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3} + \left(1.6276041666666666 \cdot 10^{-05} \cdot \left({\pi}^{5} \cdot {f}^{5}\right) + \left(\pi \cdot f\right) \cdot 0.5\right)}\right) \cdot -4}{\pi}\]
-\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)
↓
\frac{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3} + \left(1.6276041666666666 \cdot 10^{-05} \cdot \left({\pi}^{5} \cdot {f}^{5}\right) + \left(\pi \cdot f\right) \cdot 0.5\right)}\right) \cdot -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
 (/
  (*
   (log
    (/
     (+ (exp (* (/ PI 4.0) f)) (pow (exp -0.25) (* PI f)))
     (+
      (* 0.005208333333333333 (pow (* PI f) 3.0))
      (+
       (* 1.6276041666666666e-05 (* (pow PI 5.0) (pow f 5.0)))
       (* (* PI 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 (log((exp((((double) M_PI) / 4.0) * f) + pow(exp(-0.25), (((double) M_PI) * f))) / ((0.005208333333333333 * pow((((double) M_PI) * f), 3.0)) + ((1.6276041666666666e-05 * (pow(((double) M_PI), 5.0) * pow(f, 5.0))) + ((((double) M_PI) * 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 61.4

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

  2. Simplified61.4

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

  3. Taylor expanded around 0 2.4

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

  4. Simplified2.4

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

  6. Applied associate-*r/_binary642.3

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

  7. Simplified2.3

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

  8. Final simplification2.3

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

Reproduce

herbie shell --seed 2020232 
(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)))))))))