Average Error: 61.5 → 1.4
Time: 13.8s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}} \leq \infty:\\ \;\;\;\;\frac{\log \left(\frac{e^{0.25 \cdot \left(\pi \cdot f\right)} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\left(\pi \cdot f\right) \cdot 0.5}\right) \cdot -4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
-\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)
↓
\begin{array}{l}
\mathbf{if}\;\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}} \leq \infty:\\
\;\;\;\;\frac{\log \left(\frac{e^{0.25 \cdot \left(\pi \cdot f\right)} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\left(\pi \cdot f\right) \cdot 0.5}\right) \cdot -4}{\pi}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
(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
 (if (<=
      (/
       (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f))))
       (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))
      INFINITY)
   (/
    (*
     (log
      (/
       (+ (exp (* 0.25 (* PI f))) (pow (exp -0.25) (* PI f)))
       (* (* PI f) 0.5)))
     -4.0)
    PI)
   0.0))
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) {
	double tmp;
	if (((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) INFINITY)) {
		tmp = (log((exp(0.25 * (((double) M_PI) * f)) + pow(exp(-0.25), (((double) M_PI) * f))) / ((((double) M_PI) * f) * 0.5)) * -4.0) / ((double) M_PI);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Error

Bits error versus f

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 PI.f64 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 PI.f64 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 PI.f64 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 PI.f64 4) f))))) < +inf.0

    1. Initial program 61.5

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

    2. Simplified61.5

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

    3. Taylor expanded around 0 1.5

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

    4. Simplified1.5

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

    6. Applied associate-*r/_binary641.4

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

    7. Simplified1.4

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

    8. Taylor expanded around 0 1.4

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

    if +inf.0 < (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 PI.f64 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 PI.f64 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 PI.f64 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 PI.f64 4) f)))))

    1. Initial program 0

      \[0\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}} \leq \infty:\\ \;\;\;\;\frac{\log \left(\frac{e^{0.25 \cdot \left(\pi \cdot f\right)} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\left(\pi \cdot f\right) \cdot 0.5}\right) \cdot -4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2021126 
(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)))))))))