Average Error: 30.1 → 30.1
Time: 11.1s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}}{2}\]
\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
↓
\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}}{2}
(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))
↓
(FPCore (x eps)
 :precision binary64
 (/
  (+
   (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
   (/ (- 1.0 (/ 1.0 eps)) (exp (* x (+ 1.0 eps)))))
  2.0))
double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}
↓
double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(x * (eps + -1.0))) + ((1.0 - (1.0 / eps)) / exp(x * (1.0 + eps)))) / 2.0;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.1

    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
  2. Using strategy rm

  3. Applied sub-neg_binary6430.1

    \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2}\]

  4. Simplified30.1

    \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}}}{2}\]

  5. Final simplification30.1

    \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}}{2}\]

Reproduce

herbie shell --seed 2020268 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))