Average Error: 61.4 → 2.3
Time: 16.9s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\frac{\log \left(\mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{\sqrt{f} \cdot \left(\pi \cdot \sqrt{f}\right)}\right)\right) \cdot -4}{\pi} \]
-\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)
↓
\frac{\log \left(\mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{\sqrt{f} \cdot \left(\pi \cdot \sqrt{f}\right)}\right)\right) \cdot -4}{\pi}

(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
 (/
  (*
   (log
    (fma (* f PI) 0.08333333333333333 (/ 4.0 (* (sqrt f) (* PI (sqrt f))))))
   -4.0)
  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) {
	return (log(fma((f * ((double) M_PI)), 0.08333333333333333, (4.0 / (sqrt(f) * (((double) M_PI) * sqrt(f)))))) * -4.0) / ((double) M_PI);
}

Error

Bits error versus f

Derivation

  1. Initial program 61.4

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

  2. Simplified61.4

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

  3. Taylor expanded in f around 0 2.4

    \[\leadsto \log \color{blue}{\left(4 \cdot \frac{1}{f \cdot \pi} + 0.08333333333333333 \cdot \left(f \cdot \pi\right)\right)} \cdot \frac{-4}{\pi} \]

  4. Simplified2.4

    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)} \cdot \frac{-4}{\pi} \]

  5. Applied associate-*r/_binary642.3

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right) \cdot -4}{\pi}} \]

  6. Applied add-sqr-sqrt_binary642.3

    \[\leadsto \frac{\log \left(\mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{\color{blue}{\left(\sqrt{f} \cdot \sqrt{f}\right)} \cdot \pi}\right)\right) \cdot -4}{\pi} \]

  7. Applied associate-*l*_binary642.3

    \[\leadsto \frac{\log \left(\mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{\color{blue}{\sqrt{f} \cdot \left(\sqrt{f} \cdot \pi\right)}}\right)\right) \cdot -4}{\pi} \]

  8. Final simplification2.3

    \[\leadsto \frac{\log \left(\mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{\sqrt{f} \cdot \left(\pi \cdot \sqrt{f}\right)}\right)\right) \cdot -4}{\pi} \]

Reproduce

herbie shell --seed 2022068 
(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)))))))))