Average Error: 61.5 → 1.9
Time: 14.7s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}} \leq 275729.2830638932:\\ \;\;\;\;\log \left(\left(e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}\right) \cdot \frac{1}{e^{\frac{\pi}{4} \cdot f} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \end{array}\]
-\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 275729.2830638932:\\
\;\;\;\;\log \left(\left(e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}\right) \cdot \frac{1}{e^{\frac{\pi}{4} \cdot f} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right) \cdot \frac{-4}{\pi}\\

\mathbf{else}:\\
\;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\

\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)))))
      275729.2830638932)
   (*
    (log
     (*
      (+ (exp (* (/ PI 4.0) f)) (pow (exp -0.25) (* PI f)))
      (/ 1.0 (- (exp (* (/ PI 4.0) f)) (pow (exp -0.25) (* PI f))))))
    (/ -4.0 PI))
   (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) 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) {
	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)))) <= 275729.2830638932) {
		tmp = log((exp((((double) M_PI) / 4.0) * f) + pow(exp(-0.25), (((double) M_PI) * f))) * (1.0 / (exp((((double) M_PI) / 4.0) * f) - pow(exp(-0.25), (((double) M_PI) * f))))) * (-4.0 / ((double) M_PI));
	} else {
		tmp = -4.0 * ((log(4.0 / ((double) M_PI)) - log(f)) / ((double) M_PI));
	}
	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))))) < 275729.28306389321

    1. Initial program 11.7

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

    2. Simplified11.8

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

    4. Applied div-inv_binary6411.9

      \[\leadsto \log \color{blue}{\left(\left(e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}\right) \cdot \frac{1}{e^{\frac{\pi}{4} \cdot f} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right)} \cdot \frac{-4}{\pi}\]

    5. Simplified11.9

      \[\leadsto \log \left(\left(e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}\right) \cdot \color{blue}{\frac{1}{e^{f \cdot \frac{\pi}{4}} - {\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)}}}\right) \cdot \frac{-4}{\pi}\]

    if 275729.28306389321 < (/.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 62.7

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

    2. Simplified62.7

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

    3. Taylor expanded around 0 1.7

      \[\leadsto \color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - \pi \cdot \log \left(e^{-0.25}\right)}\right) - \log f\right)} \cdot \frac{-4}{\pi}\]

    4. Simplified1.7

      \[\leadsto \color{blue}{\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right)} \cdot \frac{-4}{\pi}\]

    5. Taylor expanded around 0 1.6

      \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}} \leq 275729.2830638932:\\ \;\;\;\;\log \left(\left(e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}\right) \cdot \frac{1}{e^{\frac{\pi}{4} \cdot f} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(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)))))))))