Average Error: 28.6 → 0.6
Time: 12.8s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\frac{2 \cdot \frac{1 + x}{e^{x}}}{2}\]
\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{2 \cdot \frac{1 + x}{e^{x}}}{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 (/ (* 2.0 (/ (+ 1.0 x) (exp x))) 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 (2.0 * ((1.0 + x) / exp(x))) / 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 28.6

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Taylor expanded around inf 0.6

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

  5. Simplified0.6

    \[\leadsto \frac{2 \cdot \color{blue}{\frac{1 + x}{e^{x}}}}{2}\]

  6. Final simplification0.6

    \[\leadsto \frac{2 \cdot \frac{1 + x}{e^{x}}}{2}\]

Reproduce

herbie shell --seed 2021044 
(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))