Average Error: 29.9 → 0.9
Time: 5.2s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 523.0649211994058:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot \left(x \cdot 0.6666666666666667\right) + \left(2 - 1 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \sqrt{{\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)}}\right) \cdot \sqrt{e^{x \cdot \left(\varepsilon - 1\right)}} + e^{x \cdot \left(\left(-\varepsilon\right) - 1\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array}\]
\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 523.0649211994058:\\
\;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot \left(x \cdot 0.6666666666666667\right) + \left(2 - 1 \cdot \left(x \cdot x\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \sqrt{{\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)}}\right) \cdot \sqrt{e^{x \cdot \left(\varepsilon - 1\right)}} + e^{x \cdot \left(\left(-\varepsilon\right) - 1\right)} \cdot \left(1 - \frac{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 523.0649211994058)
   (/
    (+
     (* (pow (* (cbrt x) (cbrt x)) 3.0) (* x 0.6666666666666667))
     (- 2.0 (* 1.0 (* x x))))
    2.0)
   (/
    (+
     (*
      (* (+ 1.0 (/ 1.0 eps)) (sqrt (pow (exp (- 1.0 eps)) (- x))))
      (sqrt (exp (* x (- eps 1.0)))))
     (* (exp (* x (- (- eps) 1.0))) (- 1.0 (/ 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 <= 523.0649211994058)) {
		tmp = (((double) (((double) (((double) pow(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), 3.0)) * ((double) (x * 0.6666666666666667)))) + ((double) (2.0 - ((double) (1.0 * ((double) (x * x)))))))) / 2.0);
	} else {
		tmp = (((double) (((double) (((double) (((double) (1.0 + (1.0 / eps))) * ((double) sqrt(((double) pow(((double) exp(((double) (1.0 - eps)))), ((double) -(x)))))))) * ((double) sqrt(((double) exp(((double) (x * ((double) (eps - 1.0)))))))))) + ((double) (((double) exp(((double) (x * ((double) (((double) -(eps)) - 1.0)))))) * ((double) (1.0 - (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 < 523.064921199405831

    1. Initial program Error: 39.8 bits

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

    2. Taylor expanded around 0 Error: 1.2 bits

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

    3. SimplifiedError: 1.2 bits

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

    5. Applied add-cube-cbrtError: 1.2 bits

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

    6. Applied unpow-prod-downError: 1.2 bits

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

    7. Applied associate-*l*Error: 1.2 bits

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

    8. SimplifiedError: 1.2 bits

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

    if 523.064921199405831 < x

    1. Initial program Error: 0.1 bits

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

    3. Applied add-sqr-sqrtError: 0.1 bits

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

    4. Applied associate-*r*Error: 0.1 bits

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

    5. SimplifiedError: 0.1 bits

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

  4. Final simplificationError: 0.9 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 523.0649211994058:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot \left(x \cdot 0.6666666666666667\right) + \left(2 - 1 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \sqrt{{\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)}}\right) \cdot \sqrt{e^{x \cdot \left(\varepsilon - 1\right)}} + e^{x \cdot \left(\left(-\varepsilon\right) - 1\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array}\]

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))