Average Error: 30.0 → 1.0
Time: 5.2s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 12.271515233532067:\\ \;\;\;\;\frac{\sqrt[3]{8 + \left(x \cdot x\right) \cdot \left(x \cdot 8 - 12\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt[3]{{\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}} \cdot \sqrt[3]{{\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}}\right)\right) \cdot \sqrt[3]{e^{x \cdot \left(\varepsilon + -1\right)}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array}\]
\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 12.271515233532067:\\
\;\;\;\;\frac{\sqrt[3]{8 + \left(x \cdot x\right) \cdot \left(x \cdot 8 - 12\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt[3]{{\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}} \cdot \sqrt[3]{{\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}}\right)\right) \cdot \sqrt[3]{e^{x \cdot \left(\varepsilon + -1\right)}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot 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 12.271515233532067)
   (/ (cbrt (+ 8.0 (* (* x x) (- (* x 8.0) 12.0)))) 2.0)
   (/
    (-
     (*
      (*
       (+ 1.0 (/ 1.0 eps))
       (* (cbrt (pow (exp x) (+ eps -1.0))) (cbrt (pow (exp x) (+ eps -1.0)))))
      (cbrt (exp (* x (+ eps -1.0)))))
     (* (+ (/ 1.0 eps) -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) {
	double tmp;
	if (x <= 12.271515233532067) {
		tmp = cbrt(8.0 + ((x * x) * ((x * 8.0) - 12.0))) / 2.0;
	} else {
		tmp = ((((1.0 + (1.0 / eps)) * (cbrt(pow(exp(x), (eps + -1.0))) * cbrt(pow(exp(x), (eps + -1.0))))) * cbrt(exp(x * (eps + -1.0)))) - (((1.0 / eps) + -1.0) * 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 < 12.271515233532067

    1. Initial program 39.5

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

    2. Taylor expanded around 0 1.1

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

    3. Simplified1.1

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

    5. Applied add-cbrt-cube_binary64_4471.2

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

    6. Simplified1.2

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

    7. Taylor expanded around 0 1.2

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(8 \cdot {x}^{3} + 8\right) - 12 \cdot {x}^{2}}}}{2}\]

    8. Simplified1.2

      \[\leadsto \frac{\sqrt[3]{\color{blue}{8 + \left(x \cdot x\right) \cdot \left(x \cdot 8 - 12\right)}}}{2}\]

    if 12.271515233532067 < x

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt_binary64_4460.3

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

    4. Applied associate-*r*_binary64_3560.3

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

    5. Simplified0.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 12.271515233532067:\\ \;\;\;\;\frac{\sqrt[3]{8 + \left(x \cdot x\right) \cdot \left(x \cdot 8 - 12\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt[3]{{\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}} \cdot \sqrt[3]{{\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}}\right)\right) \cdot \sqrt[3]{e^{x \cdot \left(\varepsilon + -1\right)}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(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))