Average Error: 61.5 → 1.9
Time: 16.4s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t_0}\\ \mathbf{if}\;\begin{array}{l} t_2 := e^{-t_0}\\ \frac{t_1 + t_2}{t_1 - t_2} \leq 181103.68332153108 \end{array}:\\ \;\;\;\;\begin{array}{l} t_3 := {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}\\ \log \left(\frac{e^{\frac{1}{\frac{4}{\pi \cdot f}}} + t_3}{t_1 - t_3}\right) \cdot \frac{-4}{\pi} \end{array}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \end{array} \]
-\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}
t_0 := \frac{\pi}{4} \cdot f\\
t_1 := e^{t_0}\\
\mathbf{if}\;\begin{array}{l}
t_2 := e^{-t_0}\\
\frac{t_1 + t_2}{t_1 - t_2} \leq 181103.68332153108
\end{array}:\\
\;\;\;\;\begin{array}{l}
t_3 := {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}\\
\log \left(\frac{e^{\frac{1}{\frac{4}{\pi \cdot f}}} + t_3}{t_1 - t_3}\right) \cdot \frac{-4}{\pi}
\end{array}\\

\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
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)))
   (if (let* ((t_2 (exp (- t_0))))
         (<= (/ (+ t_1 t_2) (- t_1 t_2)) 181103.68332153108))
     (let* ((t_3 (pow (exp -0.25) (* PI f))))
       (*
        (log (/ (+ (exp (/ 1.0 (/ 4.0 (* PI f)))) t_3) (- t_1 t_3)))
        (/ -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 t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	double tmp;
	if (((t_1 + t_2) / (t_1 - t_2)) <= 181103.68332153108) {
		double t_3_1 = pow(exp(-0.25), (((double) M_PI) * f));
		tmp = log((exp(1.0 / (4.0 / (((double) M_PI) * f))) + t_3_1) / (t_1 - t_3_1)) * (-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))))) < 181103.683321531076

    1. Initial program 14.2

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

    2. Simplified14.2

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

    4. Applied associate-*l/_binary6414.2

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

    5. Applied clear-num_binary6414.2

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

    if 181103.683321531076 < (/.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

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

    4. Simplified1.7

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

    6. Applied clear-num_binary641.7

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{f \cdot \left(\pi \cdot 0.5\right)}\right) \cdot \color{blue}{\frac{1}{\frac{\pi}{-4}}} \]

    7. Applied un-div-inv_binary641.6

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

    8. Applied associate-/r/_binary641.6

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

    9. Simplified1.6

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

    10. Taylor expanded around 0 1.5

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}} \leq 181103.68332153108:\\ \;\;\;\;\log \left(\frac{e^{\frac{1}{\frac{4}{\pi \cdot f}}} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{e^{\frac{\pi}{4} \cdot f} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(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)))))))))