Average Error: 61.3 → 1.9
Time: 31.0s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := \sqrt{e^{t_0} + e^{-t_0}}\\ -\frac{\log \left(\frac{t_1}{2} \cdot \frac{t_1}{\sinh t_0}\right)}{\frac{\pi}{4}} \end{array} \]
-\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 := \sqrt{e^{t_0} + e^{-t_0}}\\
-\frac{\log \left(\frac{t_1}{2} \cdot \frac{t_1}{\sinh t_0}\right)}{\frac{\pi}{4}}
\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 (sqrt (+ (exp t_0) (exp (- t_0))))))
   (- (/ (log (* (/ t_1 2.0) (/ t_1 (sinh t_0)))) (/ PI 4.0)))))
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 = (((double) M_PI) / 4.0) * f;
	double t_1 = sqrt(exp(t_0) + exp(-t_0));
	return -(log((t_1 / 2.0) * (t_1 / sinh(t_0))) / (((double) M_PI) / 4.0));
}

Error

Bits error versus f

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 61.3

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

  2. Applied sinh-undef_binary642.0

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

  3. Applied add-sqr-sqrt_binary642.0

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

  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 associate-*l/_binary641.9

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

  6. Final simplification1.9

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

Reproduce

herbie shell --seed 2022067 
(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)))))))))