Average Error: 61.4 → 2.0
Time: 16.2s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\mathsf{fma}\left(4, \log \left({f}^{\left(\frac{1}{\pi}\right)}\right) - \frac{\log \left(\frac{4}{\pi}\right)}{\pi}, \left(\pi \cdot \left(f \cdot f\right)\right) \cdot -0.08333333333333333\right) \]
-\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)
↓
\mathsf{fma}\left(4, \log \left({f}^{\left(\frac{1}{\pi}\right)}\right) - \frac{\log \left(\frac{4}{\pi}\right)}{\pi}, \left(\pi \cdot \left(f \cdot f\right)\right) \cdot -0.08333333333333333\right)

(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
 (fma
  4.0
  (- (log (pow f (/ 1.0 PI))) (/ (log (/ 4.0 PI)) PI))
  (* (* PI (* f f)) -0.08333333333333333)))
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 fma(4.0, (log(pow(f, (1.0 / ((double) M_PI)))) - (log(4.0 / ((double) M_PI)) / ((double) M_PI))), ((((double) M_PI) * (f * f)) * -0.08333333333333333));
}

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

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

  4. Simplified2.2

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

  5. Taylor expanded in f around 0 2.1

    \[\leadsto \color{blue}{4 \cdot \frac{\log f}{\pi} - \left(0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)} \]

  6. Simplified2.1

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

  7. Applied add-log-exp_binary642.1

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

  8. Simplified2.0

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

  9. Final simplification2.0

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

Reproduce

herbie shell --seed 2022039 
(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)))))))))