Average Error: 29.4 → 1.0
Time: 4.6s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 312.8937051629594:\\ \;\;\;\;\frac{2 + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666667 - 1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon - 1\right)} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}}{2}\\ \end{array}\]
\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}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 312.8937051629594:\\
\;\;\;\;\frac{2 + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666667 - 1\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\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}\\

\end{array}
(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
 (if (<= x 312.8937051629594)
   (/ (+ 2.0 (* x (* x (- (* x 0.6666666666666667) 1.0)))) 2.0)
   (/
    (+
     (* (+ 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 (((double) (((double) (((double) (1.0 + (1.0 / eps))) * ((double) exp(((double) -(((double) (((double) (1.0 - eps)) * x)))))))) - ((double) (((double) ((1.0 / eps) - 1.0)) * ((double) exp(((double) -(((double) (((double) (1.0 + eps)) * x)))))))))) / 2.0);
}
↓
double code(double x, double eps) {
	double tmp;
	if ((x <= 312.8937051629594)) {
		tmp = (((double) (2.0 + ((double) (x * ((double) (x * ((double) (((double) (x * 0.6666666666666667)) - 1.0)))))))) / 2.0);
	} else {
		tmp = (((double) (((double) (((double) (1.0 + (1.0 / eps))) * ((double) exp(((double) (x * ((double) (eps - 1.0)))))))) + (((double) (1.0 - (1.0 / eps))) / ((double) exp(((double) (x * ((double) (1.0 + eps))))))))) / 2.0);
	}
	return tmp;
}

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. Split input into 2 regimes

  2. if x < 312.893705162959407

    1. Initial program 38.9

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

    2. Taylor expanded around 0 1.2

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.2

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667 \cdot {x}^{3} + 2\right) - 1 \cdot \left(x \cdot x\right)}}{2}\]

    4. Taylor expanded around 0 1.2

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    5. Simplified1.2

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(x \cdot \left(0.6666666666666667 \cdot x - 1\right)\right)}}{2}\]

    if 312.893705162959407 < x

    1. Initial program 0.1

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

    3. Applied sub-neg_binary640.1

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

    4. Simplified0.1

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 312.8937051629594:\\ \;\;\;\;\frac{2 + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666667 - 1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon - 1\right)} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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))