Average Error: 61.4 → 2.1
Time: 17.2s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\frac{-4 \cdot \log \left(\mathsf{fma}\left({\left(\pi \cdot f\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(2.066798941798942 \cdot 10^{-6}, {\pi}^{5} \cdot {f}^{5}, \mathsf{fma}\left(0.08333333333333333, \pi \cdot f, \frac{4}{\pi \cdot f}\right)\right)\right)\right)}{\pi} \]
-\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(\pi \cdot f\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(2.066798941798942 \cdot 10^{-6}, {\pi}^{5} \cdot {f}^{5}, \mathsf{fma}\left(0.08333333333333333, \pi \cdot f, \frac{4}{\pi \cdot f}\right)\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 (* PI f) 3.0)
     -0.00034722222222222224
     (fma
      2.066798941798942e-6
      (* (pow PI 5.0) (pow f 5.0))
      (fma 0.08333333333333333 (* PI f) (/ 4.0 (* PI f)))))))
  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((((double) M_PI) * f), 3.0), -0.00034722222222222224, fma(2.066798941798942e-6, (pow(((double) M_PI), 5.0) * pow(f, 5.0)), fma(0.08333333333333333, (((double) M_PI) * f), (4.0 / (((double) M_PI) * f))))))) / ((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 2.2

    \[\leadsto \log \color{blue}{\left(\left(2.066798941798942 \cdot 10^{-6} \cdot \left({f}^{5} \cdot {\pi}^{5}\right) + \left(4 \cdot \frac{1}{f \cdot \pi} + 0.08333333333333333 \cdot \left(f \cdot \pi\right)\right)\right) - 0.00034722222222222224 \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right)} \cdot \frac{-4}{\pi} \]

  4. Simplified2.2

    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left({\left(f \cdot \pi\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left({f}^{5} \cdot {\pi}^{5}, 2.066798941798942 \cdot 10^{-6}, \mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]

  5. Applied associate-*r/_binary642.1

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{fma}\left({\left(f \cdot \pi\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left({f}^{5} \cdot {\pi}^{5}, 2.066798941798942 \cdot 10^{-6}, \mathsf{fma}\left(f \cdot \pi, 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)\right)\right) \cdot -4}{\pi}} \]

  6. Simplified2.1

    \[\leadsto \frac{\color{blue}{-4 \cdot \log \left(\mathsf{fma}\left({\left(\pi \cdot f\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(2.066798941798942 \cdot 10^{-6}, {\pi}^{5} \cdot {f}^{5}, \mathsf{fma}\left(0.08333333333333333, \pi \cdot f, \frac{4}{\pi \cdot f}\right)\right)\right)\right)}}{\pi} \]

  7. Final simplification2.1

    \[\leadsto \frac{-4 \cdot \log \left(\mathsf{fma}\left({\left(\pi \cdot f\right)}^{3}, -0.00034722222222222224, \mathsf{fma}\left(2.066798941798942 \cdot 10^{-6}, {\pi}^{5} \cdot {f}^{5}, \mathsf{fma}\left(0.08333333333333333, \pi \cdot f, \frac{4}{\pi \cdot f}\right)\right)\right)\right)}{\pi} \]

Reproduce

herbie shell --seed 2022094 
(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)))))))))