Average Error: 61.5 → 2.3
Time: 16.1s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\log \left({\left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)}^{\left(\sqrt{\frac{1}{\pi}}\right)}\right) \cdot \frac{-4}{\sqrt{\pi}} \]
-\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)
↓
\log \left({\left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)}^{\left(\sqrt{\frac{1}{\pi}}\right)}\right) \cdot \frac{-4}{\sqrt{\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
   (pow (fma f (* PI 0.08333333333333333) (/ 4.0 (* f PI))) (sqrt (/ 1.0 PI))))
  (/ -4.0 (sqrt 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(pow(fma(f, (((double) M_PI) * 0.08333333333333333), (4.0 / (f * ((double) M_PI)))), sqrt(1.0 / ((double) M_PI)))) * (-4.0 / sqrt((double) M_PI));
}

Error

Bits error versus f

Derivation

  1. Initial program 61.5

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

  2. Simplified61.5

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

  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 add-sqr-sqrt_binary643.3

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

  6. Applied *-un-lft-identity_binary643.3

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

  7. Applied times-frac_binary642.7

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

  8. Applied associate-*r*_binary642.4

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

  9. Applied add-log-exp_binary642.4

    \[\leadsto \color{blue}{\log \left(e^{\log \left(\mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right) \cdot \frac{1}{\sqrt{\pi}}}\right)} \cdot \frac{-4}{\sqrt{\pi}} \]

  10. Simplified2.3

    \[\leadsto \log \color{blue}{\left({\left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)}^{\left(\sqrt{\frac{1}{\pi}}\right)}\right)} \cdot \frac{-4}{\sqrt{\pi}} \]

  11. Final simplification2.3

    \[\leadsto \log \left({\left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)}^{\left(\sqrt{\frac{1}{\pi}}\right)}\right) \cdot \frac{-4}{\sqrt{\pi}} \]

Reproduce

herbie shell --seed 2022020 
(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)))))))))