Average Error: 61.5 → 2.3
Time: 14.5s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\frac{-4 \cdot \log \left(\mathsf{fma}\left({\left(f \cdot \pi\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(0.08333333333333333, f \cdot \pi, \frac{4}{f \cdot \pi}\right)\right)\right)}{\pi} \]
-\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({\left(f \cdot \pi\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(0.08333333333333333, f \cdot \pi, \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
     (pow (* f PI) 3.0)
     -0.00034722222222222224
     (fma 0.08333333333333333 (* f PI) (/ 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(pow((f * ((double) M_PI)), 3.0), -0.00034722222222222224, fma(0.08333333333333333, (f * ((double) M_PI)), (4.0 / (f * ((double) M_PI))))))) / ((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

    \[\leadsto \log \color{blue}{\left(\left(4 \cdot \frac{1}{f \cdot \pi} + 0.08333333333333333 \cdot \left(f \cdot \pi\right)\right) - 0.00034722222222222224 \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right)} \cdot \frac{-4}{\pi} \]

  4. Simplified2.4

    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left({\left(f \cdot \pi\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)\right)} \cdot \frac{-4}{\pi} \]

  5. Applied add-sqr-sqrt_binary643.2

    \[\leadsto \log \left(\mathsf{fma}\left({\left(f \cdot \pi\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)\right) \cdot \frac{-4}{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}} \]

  6. Applied *-un-lft-identity_binary643.2

    \[\leadsto \log \left(\mathsf{fma}\left({\left(f \cdot \pi\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)\right) \cdot \frac{\color{blue}{1 \cdot -4}}{\sqrt{\pi} \cdot \sqrt{\pi}} \]

  7. Applied times-frac_binary642.7

    \[\leadsto \log \left(\mathsf{fma}\left({\left(f \cdot \pi\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{\pi}} \cdot \frac{-4}{\sqrt{\pi}}\right)} \]

  8. Applied associate-*r*_binary642.3

    \[\leadsto \color{blue}{\left(\log \left(\mathsf{fma}\left({\left(f \cdot \pi\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{-4}{\sqrt{\pi}}} \]

  9. Applied un-div-inv_binary642.5

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{fma}\left({\left(f \cdot \pi\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)\right)}{\sqrt{\pi}}} \cdot \frac{-4}{\sqrt{\pi}} \]

  10. Applied frac-times_binary643.0

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{fma}\left({\left(f \cdot \pi\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)\right) \cdot -4}{\sqrt{\pi} \cdot \sqrt{\pi}}} \]

  11. Simplified3.0

    \[\leadsto \frac{\color{blue}{-4 \cdot \log \left(\mathsf{fma}\left({\left(\pi \cdot f\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(0.08333333333333333, \pi \cdot f, \frac{4}{\pi \cdot f}\right)\right)\right)}}{\sqrt{\pi} \cdot \sqrt{\pi}} \]

  12. Simplified2.3

    \[\leadsto \frac{-4 \cdot \log \left(\mathsf{fma}\left({\left(\pi \cdot f\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(0.08333333333333333, \pi \cdot f, \frac{4}{\pi \cdot f}\right)\right)\right)}{\color{blue}{\pi}} \]

  13. Taylor expanded in f around 0 2.3

    \[\leadsto \frac{-4 \cdot \log \left(\mathsf{fma}\left({\color{blue}{\left(f \cdot \pi\right)}}^{3}, -0.00034722222222222224, \mathsf{fma}\left(0.08333333333333333, \pi \cdot f, \frac{4}{\pi \cdot f}\right)\right)\right)}{\pi} \]

  14. Final simplification2.3

    \[\leadsto \frac{-4 \cdot \log \left(\mathsf{fma}\left({\left(f \cdot \pi\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(0.08333333333333333, f \cdot \pi, \frac{4}{f \cdot \pi}\right)\right)\right)}{\pi} \]

Reproduce

herbie shell --seed 2021280 
(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)))))))))