Average Error: 29.5 → 0.9
Time: 6.8s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 448.5223692902876:\\ \;\;\;\;\frac{\left({x}^{3} \cdot 0.6666666666666666 + 2\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{\varepsilon + -1}\right)}^{x} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}} \cdot \left(\sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{\varepsilon + -1}\right)}^{x} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}} \cdot \sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{\varepsilon + -1}\right)}^{x} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}}\right)}{2}\\ \end{array}\]
\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 448.5223692902876:\\
\;\;\;\;\frac{\left({x}^{3} \cdot 0.6666666666666666 + 2\right) - x \cdot x}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{\varepsilon + -1}\right)}^{x} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}} \cdot \left(\sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{\varepsilon + -1}\right)}^{x} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}} \cdot \sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{\varepsilon + -1}\right)}^{x} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}}\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 448.5223692902876)
   (/ (- (+ (* (pow x 3.0) 0.6666666666666666) 2.0) (* x x)) 2.0)
   (/
    (*
     (cbrt
      (+
       (* (+ 1.0 (/ 1.0 eps)) (pow (exp (+ eps -1.0)) x))
       (/ (- 1.0 (/ 1.0 eps)) (exp (* x (+ 1.0 eps))))))
     (*
      (cbrt
       (+
        (* (+ 1.0 (/ 1.0 eps)) (pow (exp (+ eps -1.0)) x))
        (/ (- 1.0 (/ 1.0 eps)) (exp (* x (+ 1.0 eps))))))
      (cbrt
       (+
        (* (+ 1.0 (/ 1.0 eps)) (pow (exp (+ eps -1.0)) x))
        (/ (- 1.0 (/ 1.0 eps)) (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) {
	double tmp;
	if (x <= 448.5223692902876) {
		tmp = (((pow(x, 3.0) * 0.6666666666666666) + 2.0) - (x * x)) / 2.0;
	} else {
		tmp = (cbrt(((1.0 + (1.0 / eps)) * pow(exp(eps + -1.0), x)) + ((1.0 - (1.0 / eps)) / exp(x * (1.0 + eps)))) * (cbrt(((1.0 + (1.0 / eps)) * pow(exp(eps + -1.0), x)) + ((1.0 - (1.0 / eps)) / exp(x * (1.0 + eps)))) * cbrt(((1.0 + (1.0 / eps)) * pow(exp(eps + -1.0), x)) + ((1.0 - (1.0 / eps)) / exp(x * (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 < 448.5223692902876

    1. Initial program 39.1

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

    2. Taylor expanded around 0 1.2

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666666 \cdot {x}^{3} + 2\right) - {x}^{2}}}{2}\]

    3. Simplified1.2

      \[\leadsto \frac{\color{blue}{\left({x}^{3} \cdot 0.6666666666666666 + 2\right) - x \cdot x}}{2}\]

    if 448.5223692902876 < 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 add-cube-cbrt_binary64_4540.1

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

    4. Simplified0.1

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

    5. Simplified0.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 448.5223692902876:\\ \;\;\;\;\frac{\left({x}^{3} \cdot 0.6666666666666666 + 2\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{\varepsilon + -1}\right)}^{x} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}} \cdot \left(\sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{\varepsilon + -1}\right)}^{x} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}} \cdot \sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{\varepsilon + -1}\right)}^{x} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020355 
(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))