NMSE Section 6.1 mentioned, A

Average Error: 29.7 → 2.4

Time: 6.1s

Precision: 64

• could not determine a ground truth for program body

  1. x = -4.284609575541583e+261
  2. eps = 1.0138034241742799e-181

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1.8493523655100802 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\left(2 \cdot \log \left(\sqrt[3]{e^{x}}\right) + \log \left(\sqrt[3]{e^{x}}\right)\right) \cdot \sqrt[3]{{x}^{3}}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

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 temp;
	if ((x <= 1.8493523655100802e-05)) {
		temp = fma(1.3877787807814457e-17, ((((2.0 * log(cbrt(exp(x)))) + log(cbrt(exp(x)))) * cbrt(pow(x, 3.0))) / (eps / x)), (1.0 - (0.5 * pow(x, 2.0))));
	} else {
		temp = ((((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0);
	}
	return temp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if x < 1.8493523655100802e-05

   1. Initial program 39.2

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

   2. Simplified 39.3

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

   3. Taylor expanded around 0 7.1

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

   4. Simplified 7.1

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

   6. Applied add-cube-cbrt 7.1

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

   7. Applied associate-/l* 7.1

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

   8. Simplified 7.1

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

   10. Applied add-log-exp 2.8

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

   11. Simplified 2.8

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

   13. Applied add-cube-cbrt 2.6

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

   14. Applied log-prod 2.6

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

   15. Simplified 2.6

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

   if 1.8493523655100802e-05 < x

   1. Initial program 1.6

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.8493523655100802 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\left(2 \cdot \log \left(\sqrt[3]{e^{x}}\right) + \log \left(\sqrt[3]{e^{x}}\right)\right) \cdot \sqrt[3]{{x}^{3}}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 +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))