Average Error: 61.7 → 2.0
Time: 1.8min
Precision: binary64
Cost: 3520
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[-\frac{\log \left(\left(\sqrt{\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}} \cdot \sqrt{e^{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}}\right) \cdot 0.5\right)}{\frac{\pi}{4}}\]
-\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)
↓
-\frac{\log \left(\left(\sqrt{\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}} \cdot \sqrt{e^{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}}\right) \cdot 0.5\right)}{\frac{\pi}{4}}
(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
 (-
  (/
   (log
    (*
     (*
      (sqrt
       (/
        (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f))))
        (sinh (* (/ PI 4.0) f))))
      (sqrt
       (exp
        (log
         (/
          (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f))))
          (sinh (* (/ PI 4.0) f)))))))
     0.5))
   (/ PI 4.0))))
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 -(log((sqrt((exp((((double) M_PI) / 4.0) * f) + exp(-((((double) M_PI) / 4.0) * f))) / sinh((((double) M_PI) / 4.0) * f)) * sqrt(exp(log((exp((((double) M_PI) / 4.0) * f) + exp(-((((double) M_PI) / 4.0) * f))) / sinh((((double) M_PI) / 4.0) * f))))) * 0.5) / (((double) M_PI) / 4.0));
}

Error

Bits error versus f

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs
Alternative 1
Accuracy2.0
Cost3392
\[-\frac{\log \left(\left(\sqrt{\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}} \cdot \sqrt{\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right) \cdot 0.5\right)}{\frac{\pi}{4}}\]
Alternative 2
Accuracy2.0
Cost4992
\[-\frac{\log \left(\left(\left(\sqrt{\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}} \cdot \left|\sqrt[3]{\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right|\right) \cdot \sqrt{\sqrt[3]{\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}}}\right) \cdot 0.5\right)}{\frac{\pi}{4}}\]

Derivation

  1. Initial program 61.7

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

  3. Applied sinh-undef_binary642.0

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

  4. Applied *-un-lft-identity_binary642.0

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

  5. Applied times-frac_binary642.1

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

  6. Simplified2.1

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\color{blue}{0.5} \cdot \frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)\]
  7. Using strategy rm

  8. Applied associate-*l/_binary642.0

    \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(0.5 \cdot \frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)}{\frac{\pi}{4}}}\]

  9. Simplified2.0

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

  11. Applied add-sqr-sqrt_binary642.0

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

  13. Applied add-exp-log_binary642.0

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

  14. Applied add-exp-log_binary642.0

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

  15. Applied div-exp_binary642.0

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

  16. Simplified2.0

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

  17. Final simplification2.0

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

Reproduce

herbie shell --seed 2020322 
(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)))))))))