Average Error: 59.7 → 2.2
Time: 48.8s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\frac{-4 \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, e^{\log \left(\frac{4}{f \cdot \pi}\right)}\right)\right)}{\pi} \]
-\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{-4 \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, e^{\log \left(\frac{4}{f \cdot \pi}\right)}\right)\right)}{\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
 (/
  (*
   -4.0
   (log (fma f (* PI 0.08333333333333333) (exp (log (/ 4.0 (* f PI)))))))
  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 (-4.0 * log(fma(f, (((double) M_PI) * 0.08333333333333333), exp(log(4.0 / (f * ((double) M_PI))))))) / ((double) M_PI);
}

Error

Bits error versus f

Derivation

  1. Initial program 59.7

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

  2. Simplified59.7

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

  3. Taylor expanded in f around 0 2.3

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

  4. Simplified2.3

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

  5. Applied add-sqr-sqrt_binary643.1

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

  6. Applied *-un-lft-identity_binary643.1

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

  7. Applied times-frac_binary642.5

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

  8. Applied associate-*r*_binary642.2

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

  9. Applied un-div-inv_binary642.3

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

  10. Applied frac-times_binary642.9

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

  11. Simplified2.9

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

  12. Simplified2.2

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

  13. Applied add-exp-log_binary642.2

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

  14. Applied add-exp-log_binary642.2

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

  15. Applied prod-exp_binary642.2

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

  16. Applied add-exp-log_binary642.2

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

  17. Applied div-exp_binary642.2

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

  18. Simplified2.2

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

  19. Final simplification2.2

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

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)))))))))