NMSE Section 6.1 mentioned, A

Average Error: 29.4 → 4.5

Time: 7.2s

Precision: 64

• could not determine a ground truth for program body

  1. x = -8.809186652858097e+51
  2. eps = -2.0278525706399495e-199

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 5.98061178638290693 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\log \left(e^{\sqrt[3]{x}}\right) \cdot \log \left(e^{\sqrt[3]{x}}\right)\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\right)\right)\\ \end{array}\]

C

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 <= 5.980611786382907e-20)) {
		VAR = fma(1.3877787807814457e-17, (pow((log(exp(cbrt(x))) * log(exp(cbrt(x)))), 3.0) / (eps / x)), (1.0 - (0.5 * pow(x, 2.0))));
	} else {
		VAR = expm1(log1p(fma((exp(-((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

Derivation

1. Split input into 2 regimes

2. if x < 5.980611786382907e-20

   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. Simplified 38.8

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

   3. Taylor expanded around 0 6.5

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

   4. Simplified 6.5

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

   6. Applied add-cube-cbrt 6.5

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

   7. Applied unpow-prod-down 6.5

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

   8. Applied associate-/l* 6.5

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

   9. Simplified 6.5

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

   11. Applied add-log-exp 4.6

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

   13. Applied add-log-exp 4.6

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

   if 5.980611786382907e-20 < x

   1. Initial program 4.1

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

   2. Simplified 4.1

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

   4. Applied expm1-log1p-u 4.2

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 5.98061178638290693 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\log \left(e^{\sqrt[3]{x}}\right) \cdot \log \left(e^{\sqrt[3]{x}}\right)\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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))