Average Error: 29.4 → 1.0
Time: 6.5s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
↓
\[\frac{e^{-\left(x + \varepsilon \cdot x\right)} + \frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}}}{2} \]
\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{e^{-\left(x + \varepsilon \cdot x\right)} + \frac{1}{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
 (/ (+ (exp (- (+ x (* eps x)))) (/ 1.0 (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 (exp(-(x + (eps * x))) + (1.0 / 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 29.4

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

  2. Applied egg-rr29.4

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

  3. Taylor expanded in eps around inf 1.0

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

  4. Final simplification1.0

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

Reproduce

herbie shell --seed 2022130 
(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))