Average Error: 61.6 → 2.7
Time: 13.5s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\frac{-4 \cdot \log \left(\frac{{\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)} + e^{f \cdot \frac{\pi}{4}}}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\pi} \]
-\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{-4 \cdot \log \left(\frac{{\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)} + e^{f \cdot \frac{\pi}{4}}}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\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
 (/
  (*
   -4.0
   (log
    (/
     (+ (pow (exp -0.25) (* PI f)) (exp (* f (/ PI 4.0))))
     (* f (* PI 0.5)))))
  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 (-4.0 * log((pow(exp(-0.25), (((double) M_PI) * f)) + exp(f * (((double) M_PI) / 4.0))) / (f * (((double) M_PI) * 0.5)))) / ((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 2.8

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

  4. Simplified2.8

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

  6. Applied associate-*r/_binary642.7

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

  7. Simplified2.7

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

  8. Final simplification2.7

    \[\leadsto \frac{-4 \cdot \log \left(\frac{{\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)} + e^{f \cdot \frac{\pi}{4}}}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\pi} \]

Reproduce

herbie shell --seed 2021196 
(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)))))))))