Average Error: 28.9 → 1.0
Time: 6.5s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 1.9184223926600088:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(0.6666666666666666 \cdot {x}^{3} + 2\right) - x \cdot x\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right) \cdot \frac{\sqrt[3]{\frac{1}{\varepsilon} - 1}}{e^{x \cdot \left(1 + \varepsilon\right)}}}{2}\\ \end{array}\]
\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 1.9184223926600088:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\left(0.6666666666666666 \cdot {x}^{3} + 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(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right) \cdot \frac{\sqrt[3]{\frac{1}{\varepsilon} - 1}}{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 1.9184223926600088)
   (/
    (cbrt (pow (- (+ (* 0.6666666666666666 (pow x 3.0)) 2.0) (* x x)) 3.0))
    2.0)
   (/
    (-
     (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
     (*
      (* (cbrt (- (/ 1.0 eps) 1.0)) (cbrt (- (/ 1.0 eps) 1.0)))
      (/ (cbrt (- (/ 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 <= 1.9184223926600088) {
		tmp = cbrt(pow((((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))) - ((cbrt((1.0 / eps) - 1.0) * cbrt((1.0 / eps) - 1.0)) * (cbrt((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 < 1.9184223926600088

    1. Initial program 38.8

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

    2. Taylor expanded around 0 1.2

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

    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_7961.2

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

    6. Simplified1.2

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

    if 1.9184223926600088 < x

    1. Initial program 0.5

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

    3. Applied add-cube-cbrt_binary64_7950.5

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

    4. Applied associate-*l*_binary64_7010.5

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

    5. Simplified0.5

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9184223926600088:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(0.6666666666666666 \cdot {x}^{3} + 2\right) - x \cdot x\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right) \cdot \frac{\sqrt[3]{\frac{1}{\varepsilon} - 1}}{e^{x \cdot \left(1 + \varepsilon\right)}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020343 
(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))