Average Error: 29.0 → 1.0
Time: 3.9s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 166.76597640670573:\\ \;\;\;\;\frac{{x}^{3} \cdot 0.6666666666666667 + \left(2 - 1 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon - 1\right)} + e^{x \cdot \left(\left(-\varepsilon\right) - 1\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array}\]
\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 166.76597640670573:\\
\;\;\;\;\frac{{x}^{3} \cdot 0.6666666666666667 + \left(2 - 1 \cdot \left(x \cdot x\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon - 1\right)} + e^{x \cdot \left(\left(-\varepsilon\right) - 1\right)} \cdot \left(1 - \frac{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 166.76597640670573)
   (/ (+ (* (pow x 3.0) 0.6666666666666667) (- 2.0 (* 1.0 (* x x)))) 2.0)
   (/
    (+
     (* (+ 1.0 (/ 1.0 eps)) (exp (* x (- eps 1.0))))
     (* (exp (* x (- (- eps) 1.0))) (- 1.0 (/ 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 <= 166.76597640670573)) {
		tmp = (((double) (((double) (((double) pow(x, 3.0)) * 0.6666666666666667)) + ((double) (2.0 - ((double) (1.0 * ((double) (x * x)))))))) / 2.0);
	} else {
		tmp = (((double) (((double) (((double) (1.0 + (1.0 / eps))) * ((double) exp(((double) (x * ((double) (eps - 1.0)))))))) + ((double) (((double) exp(((double) (x * ((double) (((double) -(eps)) - 1.0)))))) * ((double) (1.0 - (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 < 166.765976406705732

    1. Initial program Error: 38.9 bits

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

    2. Taylor expanded around 0 Error: 1.2 bits

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

    3. SimplifiedError: 1.2 bits

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

    if 166.765976406705732 < x

    1. Initial program Error: 0.2 bits

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

  4. Final simplificationError: 1.0 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 166.76597640670573:\\ \;\;\;\;\frac{{x}^{3} \cdot 0.6666666666666667 + \left(2 - 1 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon - 1\right)} + e^{x \cdot \left(\left(-\varepsilon\right) - 1\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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))