Average Error: 61.5 → 2.0
Time: 18.1s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sqrt{\sinh \left(\frac{\pi}{4} \cdot f\right)}}}{\sqrt{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right)\]
-\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{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sqrt{\sinh \left(\frac{\pi}{4} \cdot f\right)}}}{\sqrt{\sinh \left(\frac{\pi}{4} \cdot f\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
 (-
  (*
   (/ 1.0 (/ PI 4.0))
   (log
    (/
     (/ (cosh (* (/ PI 4.0) f)) (sqrt (sinh (* (/ PI 4.0) f))))
     (sqrt (sinh (* (/ PI 4.0) f))))))))
double code(double f) {
	return ((double) -(((double) ((1.0 / (((double) M_PI) / 4.0)) * ((double) log((((double) (((double) exp(((double) ((((double) M_PI) / 4.0) * f)))) + ((double) exp(((double) -(((double) ((((double) M_PI) / 4.0) * f)))))))) / ((double) (((double) exp(((double) ((((double) M_PI) / 4.0) * f)))) - ((double) exp(((double) -(((double) ((((double) M_PI) / 4.0) * f)))))))))))))));
}
↓
double code(double f) {
	return ((double) -(((double) ((1.0 / (((double) M_PI) / 4.0)) * ((double) log(((((double) cosh(((double) ((((double) M_PI) / 4.0) * f)))) / ((double) sqrt(((double) sinh(((double) ((((double) M_PI) / 4.0) * f))))))) / ((double) sqrt(((double) sinh(((double) ((((double) M_PI) / 4.0) * f)))))))))))));
}

Error

Bits error versus f

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program Error: 61.5 bits

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

  3. Applied sinh-undefError: 2.0 bits

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

  4. Applied associate-/r*Error: 2.0 bits

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

  5. SimplifiedError: 2.0 bits

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\color{blue}{\cosh \left(\frac{\pi}{4} \cdot f\right)}}{\sinh \left(\frac{\pi}{4} \cdot f\right)}\right)\]
  6. Using strategy rm

  7. Applied add-sqr-sqrtError: 2.0 bits

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

  8. Applied associate-/r*Error: 2.0 bits

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

  9. Final simplificationError: 2.0 bits

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\frac{\cosh \left(\frac{\pi}{4} \cdot f\right)}{\sqrt{\sinh \left(\frac{\pi}{4} \cdot f\right)}}}{\sqrt{\sinh \left(\frac{\pi}{4} \cdot f\right)}}\right)\]

Reproduce

herbie shell --seed 2020205 
(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)))))))))