Average Error: 61.7 → 1.8
Time: 16.5s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{\frac{4}{\pi}}\\ t_1 := f \cdot \frac{\pi}{4}\\ -t_0 \cdot \left(\log \left(\frac{\cosh t_1}{\sinh t_1}\right) \cdot t_0\right) \end{array} \]
-\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 := \sqrt{\frac{4}{\pi}}\\
t_1 := f \cdot \frac{\pi}{4}\\
-t_0 \cdot \left(\log \left(\frac{\cosh t_1}{\sinh t_1}\right) \cdot t_0\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 (sqrt (/ 4.0 PI))) (t_1 (* f (/ PI 4.0))))
   (- (* t_0 (* (log (/ (cosh t_1) (sinh t_1))) t_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 = sqrt(4.0 / ((double) M_PI));
	double t_1 = f * (((double) M_PI) / 4.0);
	return -(t_0 * (log(cosh(t_1) / sinh(t_1)) * t_0));
}

Error

Bits error versus f

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 61.7

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

  3. Applied sinh-undef_binary641.9

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

  4. Applied associate-/r*_binary641.9

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

  5. Simplified1.9

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\color{blue}{\cosh \left(f \cdot \frac{\pi}{4}\right)}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right) \]
  6. Using strategy rm

  7. Applied add-sqr-sqrt_binary642.2

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

  8. Applied associate-*l*_binary641.8

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

  9. Simplified1.8

    \[\leadsto -\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \color{blue}{\left(\log \left(\frac{\cosh \left(f \cdot \frac{\pi}{4}\right)}{\sinh \left(f \cdot \frac{\pi}{4}\right)}\right) \cdot \sqrt{\frac{4}{\pi}}\right)} \]
  10. Using strategy rm

  11. Applied associate-*r*_binary641.8

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

  12. Simplified1.8

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

  13. Final simplification1.8

    \[\leadsto -\sqrt{\frac{4}{\pi}} \cdot \left(\log \left(\frac{\cosh \left(f \cdot \frac{\pi}{4}\right)}{\sinh \left(f \cdot \frac{\pi}{4}\right)}\right) \cdot \sqrt{\frac{4}{\pi}}\right) \]

Reproduce

herbie shell --seed 2021202 
(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)))))))))