Average Error: 30.5 → 1.1
Time: 6.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 210.292076175756:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, 0.0625, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1}{\sqrt{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}} \cdot \frac{1 + \frac{1}{\varepsilon}}{\sqrt{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}}\right)\\ \end{array}\]
\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 210.292076175756:\\
\;\;\;\;\mathsf{fma}\left({x}^{4}, 0.0625, 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}{\sqrt{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}} \cdot \frac{1 + \frac{1}{\varepsilon}}{\sqrt{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}}\right)\\

\end{array}
double code(double x, double eps) {
	return ((double) (((double) (((double) (((double) (1.0 + ((double) (1.0 / eps)))) * ((double) exp(((double) -(((double) (((double) (1.0 - eps)) * x)))))))) - ((double) (((double) (((double) (1.0 / eps)) - 1.0)) * ((double) exp(((double) -(((double) (((double) (1.0 + eps)) * x)))))))))) / 2.0));
}
↓
double code(double x, double eps) {
	double VAR;
	if ((x <= 210.29207617575594)) {
		VAR = ((double) fma(((double) pow(x, 4.0)), 0.0625, ((double) (1.0 - ((double) (0.5 * ((double) pow(x, 2.0))))))));
	} else {
		VAR = ((double) fma(((double) (((double) exp(((double) -(((double) (((double) (1.0 + eps)) * x)))))) / 2.0)), ((double) (1.0 - ((double) (1.0 / eps)))), ((double) (((double) (1.0 / ((double) sqrt(((double) (2.0 * ((double) exp(((double) (((double) (1.0 - eps)) * x)))))))))) * ((double) (((double) (1.0 + ((double) (1.0 / eps)))) / ((double) sqrt(((double) (2.0 * ((double) exp(((double) (((double) (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 < 210.29207617575594

    1. Initial program 40.1

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

    2. Simplified40.1

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

    3. Taylor expanded around 0 7.8

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

    4. Simplified7.8

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

    6. Applied expm1-log1p-u7.8

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)\right)\right)}\]

    7. Taylor expanded around 0 1.5

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log 2 - 0.25 \cdot {x}^{2}}\right)\]

    8. Taylor expanded around 0 1.5

      \[\leadsto \color{blue}{\left(0.0625 \cdot {x}^{4} + 1\right) - 0.5 \cdot {x}^{2}}\]

    9. Simplified1.5

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.0625, 1 - 0.5 \cdot {x}^{2}\right)}\]

    if 210.29207617575594 < x

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied *-un-lft-identity0.1

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

    6. Applied times-frac0.1

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 210.292076175756:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, 0.0625, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1}{\sqrt{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}} \cdot \frac{1 + \frac{1}{\varepsilon}}{\sqrt{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +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))