Average Error: 61.6 → 1.8
Time: 24.8s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\begin{array}{l} t_0 := f \cdot \frac{\pi}{4}\\ t_1 := \sqrt{e^{t_0} + e^{-t_0}}\\ \frac{\log \left(\frac{t_1}{\sinh t_0}\right)}{\pi} \cdot -4 - \frac{4}{\pi} \cdot \log \left(\frac{t_1}{2}\right) \end{array} \]
-\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 := f \cdot \frac{\pi}{4}\\
t_1 := \sqrt{e^{t_0} + e^{-t_0}}\\
\frac{\log \left(\frac{t_1}{\sinh t_0}\right)}{\pi} \cdot -4 - \frac{4}{\pi} \cdot \log \left(\frac{t_1}{2}\right)
\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 (* f (/ PI 4.0))) (t_1 (sqrt (+ (exp t_0) (exp (- t_0))))))
   (-
    (* (/ (log (/ t_1 (sinh t_0))) PI) -4.0)
    (* (/ 4.0 PI) (log (/ t_1 2.0))))))
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 t_0 = f * (((double) M_PI) / 4.0);
	double t_1 = sqrt((exp(t_0) + exp(-t_0)));
	return ((log((t_1 / sinh(t_0))) / ((double) M_PI)) * -4.0) - ((4.0 / ((double) M_PI)) * log((t_1 / 2.0)));
}

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. Applied sinh-undef_binary641.9

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{2 \cdot \sinh \left(\frac{\pi}{4} \cdot f\right)}}\right) \]

  3. Applied add-sqr-sqrt_binary641.9

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\color{blue}{\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}} \cdot \sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}}{2 \cdot \sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \]

  4. Applied times-frac_binary642.0

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{2} \cdot \frac{\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)} \]

  5. Applied log-prod_binary642.1

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{2}\right) + \log \left(\frac{\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)\right)} \]

  6. Applied distribute-rgt-in_binary641.9

    \[\leadsto -\color{blue}{\left(\log \left(\frac{\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{2}\right) \cdot \frac{1}{\frac{\pi}{4}} + \log \left(\frac{\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \frac{1}{\frac{\pi}{4}}\right)} \]

  7. Simplified1.9

    \[\leadsto -\left(\color{blue}{\frac{4}{\pi} \cdot \log \left(\frac{\sqrt{e^{f \cdot \frac{\pi}{4}} + e^{-f \cdot \frac{\pi}{4}}}}{2}\right)} + \log \left(\frac{\sqrt{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \cdot \frac{1}{\frac{\pi}{4}}\right) \]

  8. Simplified1.9

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

  9. Applied div-inv_binary641.9

    \[\leadsto -\left(\frac{4}{\pi} \cdot \log \left(\frac{\sqrt{e^{f \cdot \frac{\pi}{4}} + e^{-f \cdot \frac{\pi}{4}}}}{2}\right) + \color{blue}{\left(4 \cdot \frac{1}{\pi}\right)} \cdot \log \left(\frac{\sqrt{e^{f \cdot \frac{\pi}{4}} + e^{-f \cdot \frac{\pi}{4}}}}{\sinh \left(f \cdot \frac{\pi}{4}\right)}\right)\right) \]

  10. Applied associate-*l*_binary641.9

    \[\leadsto -\left(\frac{4}{\pi} \cdot \log \left(\frac{\sqrt{e^{f \cdot \frac{\pi}{4}} + e^{-f \cdot \frac{\pi}{4}}}}{2}\right) + \color{blue}{4 \cdot \left(\frac{1}{\pi} \cdot \log \left(\frac{\sqrt{e^{f \cdot \frac{\pi}{4}} + e^{-f \cdot \frac{\pi}{4}}}}{\sinh \left(f \cdot \frac{\pi}{4}\right)}\right)\right)}\right) \]

  11. Simplified1.8

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

  12. Final simplification1.8

    \[\leadsto \frac{\log \left(\frac{\sqrt{e^{f \cdot \frac{\pi}{4}} + e^{-f \cdot \frac{\pi}{4}}}}{\sinh \left(f \cdot \frac{\pi}{4}\right)}\right)}{\pi} \cdot -4 - \frac{4}{\pi} \cdot \log \left(\frac{\sqrt{e^{f \cdot \frac{\pi}{4}} + e^{-f \cdot \frac{\pi}{4}}}}{2}\right) \]

Reproduce

herbie shell --seed 2022160 
(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)))))))))