Average Error: 29.0 → 1.0
Time: 6.2s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 368.7430660701782:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left({x}^{3} \cdot 0.6666666666666666 + 2\right) - x \cdot x\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array}\]
\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 368.7430660701782:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\left({x}^{3} \cdot 0.6666666666666666 + 2\right) - x \cdot x\right)}^{3}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\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 368.7430660701782)
   (/
    (cbrt (pow (- (+ (* (pow x 3.0) 0.6666666666666666) 2.0) (* x x)) 3.0))
    2.0)
   (/
    (-
     (/ (+ 1.0 (/ 1.0 eps)) (exp (* x (- 1.0 eps))))
     (* (- (/ 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 <= 368.7430660701782) {
		tmp = cbrt(pow((((pow(x, 3.0) * 0.6666666666666666) + 2.0) - (x * x)), 3.0)) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps)) / exp(x * (1.0 - eps))) - (((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 < 368.7430660701782

    1. Initial program 38.6

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

    2. Taylor expanded around 0 1.3

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

    3. Simplified1.3

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

    5. Applied add-cbrt-cube_binary64_1151.3

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

    6. Simplified1.3

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

    if 368.7430660701782 < x

    1. Initial program 0.0

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

    3. Applied exp-neg_binary64_1260.0

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

    4. Applied un-div-inv_binary64_770.0

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 368.7430660701782:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left({x}^{3} \cdot 0.6666666666666666 + 2\right) - x \cdot x\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020277 
(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))