Average Error: 61.6 → 1.9
Time: 13.6s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\log \left(\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\pi \cdot \frac{f}{4}\right)}\right) \cdot \left(-\sqrt{4 \cdot \frac{1}{\pi}}\right)\right)\]
-\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)
↓
\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\log \left(\frac{\cosh \left(\pi \cdot \frac{f}{4}\right)}{\sinh \left(\pi \cdot \frac{f}{4}\right)}\right) \cdot \left(-\sqrt{4 \cdot \frac{1}{\pi}}\right)\right)
double code(double f) {
	return ((double) -(((double) ((1.0 / (((double) M_PI) / 4.0)) * ((double) log((((double) (((double) exp(((double) ((((double) M_PI) / 4.0) * f)))) + ((double) exp(((double) -(((double) ((((double) M_PI) / 4.0) * f)))))))) / ((double) (((double) exp(((double) ((((double) M_PI) / 4.0) * f)))) - ((double) exp(((double) -(((double) ((((double) M_PI) / 4.0) * f)))))))))))))));
}
↓
double code(double f) {
	return ((double) (((double) sqrt((1.0 / (((double) M_PI) / 4.0)))) * ((double) (((double) log((((double) cosh(((double) (((double) M_PI) * (f / 4.0))))) / ((double) sinh(((double) (((double) M_PI) * (f / 4.0)))))))) * ((double) -(((double) sqrt(((double) (4.0 * (1.0 / ((double) M_PI))))))))))));
}

Error

Bits error versus f

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Results

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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. Using strategy rm

  3. Applied sinh-undef1.9

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

  4. Applied associate-/r*1.9

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

  5. Simplified1.9

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

  7. Applied add-sqr-sqrt2.2

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

  8. Applied associate-*l*1.9

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

  9. Simplified1.9

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

  10. Final simplification1.9

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

Reproduce

herbie shell --seed 2020196 
(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)))))))))