Average Error: 29.6 → 1.0
Time: 10.3s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 10.306908003704974:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\log \left(e^{0.6666666666666666 \cdot {x}^{3}}\right) + 2\right) - x \cdot x\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {e}^{\left(x \cdot \left(-1 - \varepsilon\right)\right)}}{2}\\ \end{array}\]
\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 10.306908003704974:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\left(\log \left(e^{0.6666666666666666 \cdot {x}^{3}}\right) + 2\right) - x \cdot x\right)}^{3}}}{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 {e}^{\left(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 10.306908003704974)
   (/
    (cbrt
     (pow
      (- (+ (log (exp (* 0.6666666666666666 (pow x 3.0)))) 2.0) (* x x))
      3.0))
    2.0)
   (/
    (-
     (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
     (* (- (/ 1.0 eps) 1.0) (pow E (* 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 <= 10.306908003704974) {
		tmp = cbrt(pow(((log(exp(0.6666666666666666 * pow(x, 3.0))) + 2.0) - (x * x)), 3.0)) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * exp(x * (eps + -1.0))) - (((1.0 / eps) - 1.0) * pow(((double) M_E), (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 < 10.3069080037049741

    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-cbrt-cube_binary64_1101.2

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

    6. Simplified1.2

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

    8. Applied add-log-exp_binary64_1131.2

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

    if 10.3069080037049741 < 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 *-un-lft-identity_binary64_770.3

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

    4. Applied exp-prod_binary64_1260.3

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

    5. Simplified0.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10.306908003704974:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\log \left(e^{0.6666666666666666 \cdot {x}^{3}}\right) + 2\right) - x \cdot x\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {e}^{\left(x \cdot \left(-1 - \varepsilon\right)\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020303 
(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))