Average Error: 61.5 → 2.7
Time: 15.2s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[-4 \cdot \frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\pi} \]
-\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)
↓
-4 \cdot \frac{\log \left(\frac{4}{f \cdot \pi}\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 (/ 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(4.0 / (f * ((double) M_PI))) / ((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 61.5

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

  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 around 0 2.8

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

  4. Simplified2.8

    \[\leadsto \log \color{blue}{\left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)} \cdot \frac{-4}{\pi} \]

  5. Taylor expanded around 0 2.7

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

  6. Simplified2.7

    \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}} \]
  7. Using strategy rm

  8. Applied *-commutative_binary642.7

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

  9. Final simplification2.7

    \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\pi} \]

Reproduce

herbie shell --seed 2021198 
(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)))))))))