Average Error: 61.7 → 2.1
Time: 11.6s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\frac{\log \left(\left(\frac{4}{\pi \cdot f} + \left(\pi \cdot f\right) \cdot 0.08333333333333333\right) - {\left(\pi \cdot f\right)}^{3} \cdot 0.00034722222222222224\right) \cdot -4}{\pi}\]
-\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(\left(\frac{4}{\pi \cdot f} + \left(\pi \cdot f\right) \cdot 0.08333333333333333\right) - {\left(\pi \cdot f\right)}^{3} \cdot 0.00034722222222222224\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
    (-
     (+ (/ 4.0 (* PI f)) (* (* PI f) 0.08333333333333333))
     (* (pow (* PI f) 3.0) 0.00034722222222222224)))
   -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(((4.0 / (((double) M_PI) * f)) + ((((double) M_PI) * f) * 0.08333333333333333)) - (pow((((double) M_PI) * f), 3.0) * 0.00034722222222222224)) * -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 61.7

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

  2. Simplified61.7

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

  3. Taylor expanded around 0 2.2

    \[\leadsto \log \color{blue}{\left(\left(4 \cdot \frac{1}{\pi \cdot f} + 0.08333333333333333 \cdot \left(f \cdot \pi\right)\right) - 0.00034722222222222224 \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right)} \cdot \frac{-4}{\pi}\]

  4. Simplified2.2

    \[\leadsto \log \color{blue}{\left(\left(\frac{4}{\pi \cdot f} + \left(\pi \cdot f\right) \cdot 0.08333333333333333\right) - {\left(\pi \cdot f\right)}^{3} \cdot 0.00034722222222222224\right)} \cdot \frac{-4}{\pi}\]
  5. Using strategy rm

  6. Applied associate-*r/_binary642.1

    \[\leadsto \color{blue}{\frac{\log \left(\left(\frac{4}{\pi \cdot f} + \left(\pi \cdot f\right) \cdot 0.08333333333333333\right) - {\left(\pi \cdot f\right)}^{3} \cdot 0.00034722222222222224\right) \cdot -4}{\pi}}\]

  7. Final simplification2.1

    \[\leadsto \frac{\log \left(\left(\frac{4}{\pi \cdot f} + \left(\pi \cdot f\right) \cdot 0.08333333333333333\right) - {\left(\pi \cdot f\right)}^{3} \cdot 0.00034722222222222224\right) \cdot -4}{\pi}\]

Reproduce

herbie shell --seed 2020219 
(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)))))))))