Average Error: 29.1 → 1.0
Time: 6.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 357.6465136238462:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 \le 357.6465136238462:\\
\;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\

\end{array}
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) {
	double VAR;
	if ((x <= 357.6465136238462)) {
		VAR = (fma(pow(x, 3.0), 0.6666666666666667, (2.0 - (1.0 * pow(x, 2.0)))) / 2.0);
	} else {
		VAR = ((((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) / exp(((1.0 + eps) * x)))) / 2.0);
	}
	return VAR;
}

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 < 357.6465136238462

    1. Initial program 38.6

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

    2. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}}{2}\]

    if 357.6465136238462 < x

    1. Initial program 0.0

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

    3. Applied exp-neg0.1

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

    4. Applied un-div-inv0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 357.6465136238462:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))