Average Error: 61.3 → 2.4
Time: 15.2s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\frac{-4}{\pi} \cdot \left(\left(2 \cdot \log \left(\sqrt[3]{{\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)} + e^{f \cdot \frac{\pi}{4}}}\right) - \log f\right) + \log \left(\frac{{2}^{1.3333333333333333}}{\pi}\right)\right) \]
-\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)
↓
\frac{-4}{\pi} \cdot \left(\left(2 \cdot \log \left(\sqrt[3]{{\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)} + e^{f \cdot \frac{\pi}{4}}}\right) - \log f\right) + \log \left(\frac{{2}^{1.3333333333333333}}{\pi}\right)\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
 (*
  (/ -4.0 PI)
  (+
   (-
    (* 2.0 (log (cbrt (+ (pow (exp -0.25) (* PI f)) (exp (* f (/ PI 4.0)))))))
    (log f))
   (log (/ (pow 2.0 1.3333333333333333) 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 / ((double) M_PI)) * (((2.0 * log(cbrt(pow(exp(-0.25), (((double) M_PI) * f)) + exp(f * (((double) M_PI) / 4.0))))) - log(f)) + log(pow(2.0, 1.3333333333333333) / ((double) M_PI)));
}

Error

Bits error versus f

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 61.3

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

  2. Simplified61.3

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

  3. Taylor expanded in f around 0 2.8

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

  4. Simplified2.8

    \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{f \cdot \left(\pi \cdot 0.5\right)}\right) \cdot \frac{-4}{\pi}} \]
  5. Using strategy rm

  6. Applied add-cube-cbrt_binary642.8

    \[\leadsto \log \left(\frac{\color{blue}{\left(\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right) \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}}}{f \cdot \left(\pi \cdot 0.5\right)}\right) \cdot \frac{-4}{\pi} \]

  7. Applied times-frac_binary642.8

    \[\leadsto \log \color{blue}{\left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}}{f} \cdot \frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}}{\pi \cdot 0.5}\right)} \cdot \frac{-4}{\pi} \]

  8. Applied log-prod_binary642.7

    \[\leadsto \color{blue}{\left(\log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}}{f}\right) + \log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}}{\pi \cdot 0.5}\right)\right)} \cdot \frac{-4}{\pi} \]

  9. Simplified2.7

    \[\leadsto \left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{{\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)} + e^{f \cdot \frac{\pi}{4}}}\right) - \log f\right)} + \log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}}{\pi \cdot 0.5}\right)\right) \cdot \frac{-4}{\pi} \]

  10. Simplified2.7

    \[\leadsto \left(\left(2 \cdot \log \left(\sqrt[3]{{\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)} + e^{f \cdot \frac{\pi}{4}}}\right) - \log f\right) + \color{blue}{\log \left(\frac{\sqrt[3]{{\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)} + e^{f \cdot \frac{\pi}{4}}}}{\pi \cdot 0.5}\right)}\right) \cdot \frac{-4}{\pi} \]

  11. Taylor expanded in f around 0 2.4

    \[\leadsto \left(\left(2 \cdot \log \left(\sqrt[3]{{\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)} + e^{f \cdot \frac{\pi}{4}}}\right) - \log f\right) + \color{blue}{\log \left(2 \cdot \left({2}^{0.3333333333333333} \cdot \frac{1}{\pi}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

  12. Simplified2.4

    \[\leadsto \color{blue}{\frac{-4}{\pi} \cdot \left(\left(2 \cdot \log \left(\sqrt[3]{{\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)} + e^{f \cdot \frac{\pi}{4}}}\right) - \log f\right) + \log \left(\frac{{2}^{1.3333333333333333}}{\pi}\right)\right)} \]

  13. Final simplification2.4

    \[\leadsto \frac{-4}{\pi} \cdot \left(\left(2 \cdot \log \left(\sqrt[3]{{\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)} + e^{f \cdot \frac{\pi}{4}}}\right) - \log f\right) + \log \left(\frac{{2}^{1.3333333333333333}}{\pi}\right)\right) \]

Reproduce

herbie shell --seed 2021197 
(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)))))))))