Average Error: 29.9 → 0.9
Time: 4.8s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 269.31082505911274:\\ \;\;\;\;\frac{\left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot \left(x \cdot 0.6666666666666667\right) + 2\right) - 1 \cdot \left(x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{e^{x \cdot \left(\varepsilon - 1\right)}} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \sqrt{e^{x \cdot \left(\varepsilon - 1\right)}}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-x \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array}\]
\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 269.31082505911274:\\
\;\;\;\;\frac{\left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot \left(x \cdot 0.6666666666666667\right) + 2\right) - 1 \cdot \left(x \cdot x\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{e^{x \cdot \left(\varepsilon - 1\right)}} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \sqrt{e^{x \cdot \left(\varepsilon - 1\right)}}\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 269.31082505911274)
   (/
    (-
     (+ (* (pow (* (cbrt x) (cbrt x)) 3.0) (* x 0.6666666666666667)) 2.0)
     (* 1.0 (* x x)))
    2.0)
   (/
    (-
     (*
      (sqrt (exp (* x (- eps 1.0))))
      (* (+ 1.0 (/ 1.0 eps)) (sqrt (exp (* x (- eps 1.0))))))
     (* (- (/ 1.0 eps) 1.0) (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 <= 269.31082505911274)) {
		tmp = (((double) (((double) (((double) (((double) pow(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), 3.0)) * ((double) (x * 0.6666666666666667)))) + 2.0)) - ((double) (1.0 * ((double) (x * x)))))) / 2.0);
	} else {
		tmp = (((double) (((double) (((double) sqrt(((double) exp(((double) (x * ((double) (eps - 1.0)))))))) * ((double) (((double) (1.0 + (1.0 / eps))) * ((double) sqrt(((double) exp(((double) (x * ((double) (eps - 1.0)))))))))))) - ((double) (((double) ((1.0 / eps) - 1.0)) * ((double) exp(((double) -(((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 < 269.31082505911274

    1. Initial program 39.8

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

    2. Taylor expanded around 0 1.2

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667 \cdot \left(e^{-0} \cdot {x}^{3}\right) + 2 \cdot e^{-0}\right) - 1 \cdot \left(e^{-0} \cdot {x}^{2}\right)}}{2}\]

    3. Simplified1.2

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

    5. Applied add-cube-cbrt_binary641.2

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

    6. Applied unpow-prod-down_binary641.2

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

    7. Applied associate-*l*_binary641.2

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

    8. Simplified1.2

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

    if 269.31082505911274 < 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-sqr-sqrt_binary640.1

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

    4. Applied associate-*r*_binary640.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 269.31082505911274:\\ \;\;\;\;\frac{\left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot \left(x \cdot 0.6666666666666667\right) + 2\right) - 1 \cdot \left(x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{e^{x \cdot \left(\varepsilon - 1\right)}} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \sqrt{e^{x \cdot \left(\varepsilon - 1\right)}}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-x \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))