Average Error: 29.1 → 5.5
Time: 8.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.00987646901126133 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1}{\frac{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}{1 + \frac{1}{\varepsilon}}}\right)\\ \end{array}\]
\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 1.00987646901126133 \cdot 10^{-4}:\\
\;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1}{\frac{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}{1 + \frac{1}{\varepsilon}}}\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 <= 0.00010098764690112613)) {
		VAR = ((cbrt(fma(1.3877787807814457e-17, (pow(x, 3.0) / eps), (1.0 - (0.5 * pow(x, 2.0))))) * (pow(1.0, 0.3333333333333333) - (0.16666666666666666 * (pow(x, 2.0) * pow(1.0, 0.3333333333333333))))) * cbrt(fma(1.3877787807814457e-17, (pow((cbrt(x) * cbrt(x)), 3.0) / (eps / x)), (1.0 - (0.5 * pow(x, 2.0))))));
	} else {
		VAR = fma((exp(-((1.0 + eps) * x)) / 2.0), (1.0 - (1.0 / eps)), (1.0 / ((2.0 * exp(((1.0 - eps) * x))) / (1.0 + (1.0 / eps)))));
	}
	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 < 0.00010098764690112613

    1. Initial program 38.7

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

    2. Simplified38.7

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

    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

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

    7. Taylor expanded around 0 7.0

      \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \color{blue}{\left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt7.0

      \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\]

    10. Applied unpow-prod-down7.0

      \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot {\left(\sqrt[3]{x}\right)}^{3}}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\]

    11. Applied associate-/l*7.0

      \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \color{blue}{\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{{\left(\sqrt[3]{x}\right)}^{3}}}}, 1 - 0.5 \cdot {x}^{2}\right)}\]

    12. Simplified7.0

      \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\color{blue}{\frac{\varepsilon}{x}}}, 1 - 0.5 \cdot {x}^{2}\right)}\]

    if 0.00010098764690112613 < x

    1. Initial program 1.2

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

    2. Simplified1.2

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

    4. Applied clear-num1.2

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.00987646901126133 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1}{\frac{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}{1 + \frac{1}{\varepsilon}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +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))