Average Error: 30.0 → 2.1
Time: 6.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.020502406150930446:\\ \;\;\;\;\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{x}}\right) + \log \left(\sqrt[3]{e^{x}}\right)\right)}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\\ \end{array}\]
\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 \le 0.020502406150930446:\\
\;\;\;\;\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{x}}\right) + \log \left(\sqrt[3]{e^{x}}\right)\right)}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{{\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\\

\end{array}
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 VAR;
	if ((x <= 0.020502406150930446)) {
		VAR = fma(1.3877787807814457e-17, ((cbrt(pow(x, 3.0)) * ((2.0 * log(cbrt(exp(x)))) + log(cbrt(exp(x))))) / (eps / x)), (1.0 - (0.5 * pow(x, 2.0))));
	} else {
		VAR = fma((pow(exp(-1.0), ((1.0 + eps) * x)) / 2.0), (1.0 - (1.0 / eps)), ((1.0 + (1.0 / eps)) / (2.0 * exp(((1.0 - eps) * x)))));
	}
	return VAR;
}

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 < 0.020502406150930446

    1. Initial program 39.9

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

    2. Simplified39.9

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

    3. Taylor expanded around 0 7.2

      \[\leadsto \color{blue}{\left(1.38778 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon} + 1\right) - 0.5 \cdot {x}^{2}}\]

    4. Simplified7.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt7.2

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\color{blue}{\left(\sqrt[3]{{x}^{3}} \cdot \sqrt[3]{{x}^{3}}\right) \cdot \sqrt[3]{{x}^{3}}}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)\]

    7. Applied associate-/l*7.2

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \color{blue}{\frac{\sqrt[3]{{x}^{3}} \cdot \sqrt[3]{{x}^{3}}}{\frac{\varepsilon}{\sqrt[3]{{x}^{3}}}}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    8. Simplified7.2

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \sqrt[3]{{x}^{3}}}{\color{blue}{\frac{\varepsilon}{x}}}, 1 - 0.5 \cdot {x}^{2}\right)\]
    9. Using strategy rm

    10. Applied add-log-exp2.6

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \color{blue}{\log \left(e^{\sqrt[3]{{x}^{3}}}\right)}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    11. Simplified2.6

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \log \color{blue}{\left(e^{x}\right)}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]
    12. Using strategy rm

    13. Applied add-cube-cbrt2.5

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \log \color{blue}{\left(\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \sqrt[3]{e^{x}}\right)}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    14. Applied log-prod2.5

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \color{blue}{\left(\log \left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) + \log \left(\sqrt[3]{e^{x}}\right)\right)}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    15. Simplified2.5

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{x}}\right)} + \log \left(\sqrt[3]{e^{x}}\right)\right)}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    if 0.020502406150930446 < x

    1. Initial program 0.9

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

    2. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)}\]
    3. Using strategy rm

    4. Applied neg-mul-10.9

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

    5. Applied exp-prod0.9

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.020502406150930446:\\ \;\;\;\;\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{x}}\right) + \log \left(\sqrt[3]{e^{x}}\right)\right)}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))