Average Error: 29.5 → 0.9
Time: 4.4s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 226.99331786434493:\\ \;\;\;\;\frac{\left(\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot {\left(\sqrt[3]{x}\right)}^{3} + 2\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{\sqrt[3]{x \cdot \left(-1 - \varepsilon\right)} \cdot \sqrt[3]{x \cdot \left(-1 - \varepsilon\right)}}\right)}^{\left(\sqrt[3]{x \cdot \left(-1 - \varepsilon\right)}\right)}}{2}\\ \end{array}\]
\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 226.99331786434493:\\
\;\;\;\;\frac{\left(\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot {\left(\sqrt[3]{x}\right)}^{3} + 2\right) - x \cdot x}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{\sqrt[3]{x \cdot \left(-1 - \varepsilon\right)} \cdot \sqrt[3]{x \cdot \left(-1 - \varepsilon\right)}}\right)}^{\left(\sqrt[3]{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 226.99331786434493)
   (/
    (- (+ (* (* 0.6666666666666666 (* x x)) (pow (cbrt x) 3.0)) 2.0) (* x x))
    2.0)
   (/
    (-
     (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
     (*
      (- (/ 1.0 eps) 1.0)
      (pow
       (exp (* (cbrt (* x (- -1.0 eps))) (cbrt (* x (- -1.0 eps)))))
       (cbrt (* 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 <= 226.99331786434493) {
		tmp = ((((0.6666666666666666 * (x * x)) * pow(cbrt(x), 3.0)) + 2.0) - (x * x)) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * exp(x * (eps + -1.0))) - (((1.0 / eps) - 1.0) * pow(exp(cbrt(x * (-1.0 - eps)) * cbrt(x * (-1.0 - eps))), cbrt(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 < 226.99331786434493

    1. Initial program 39.3

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

    2. Taylor expanded around 0 1.2

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

    3. Simplified1.2

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

    5. Applied add-cube-cbrt_binary641.2

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

    6. Applied unpow-prod-down_binary641.2

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

    7. Applied associate-*r*_binary641.2

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

    8. Simplified1.2

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

    if 226.99331786434493 < 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_binary640.1

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

    4. Applied exp-prod_binary640.1

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

    5. Simplified0.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 226.99331786434493:\\ \;\;\;\;\frac{\left(\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot {\left(\sqrt[3]{x}\right)}^{3} + 2\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{\sqrt[3]{x \cdot \left(-1 - \varepsilon\right)} \cdot \sqrt[3]{x \cdot \left(-1 - \varepsilon\right)}}\right)}^{\left(\sqrt[3]{x \cdot \left(-1 - \varepsilon\right)}\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(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))