NMSE Section 6.1 mentioned, A

Average Error: 29.4 → 1.1

Time: 5.6s

Precision: 64

• could not determine a ground truth for program body

  1. x = -2.9980302389965735e+156
  2. eps = 0.06406876111163662

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 2.5494520086503014:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\mathsf{fma}\left({x}^{2}, 0.66666666666666674 \cdot x - 1, 2\right)\right)}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\\ \end{array}\]

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 <= 2.5494520086503014)) {
		VAR = (log1p(log(exp(expm1(fma(pow(x, 2.0), ((0.6666666666666667 * x) - 1.0), 2.0))))) / 2.0);
	} else {
		VAR = 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 < 2.5494520086503014

   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. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

   3. Simplified 1.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}}{2}\]

   4. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

   5. Simplified 1.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, 0.66666666666666674 \cdot x - 1, 2\right)}}{2}\]
   6. Using strategy rm

   7. Applied log1p-expm1-u 1.3

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left({x}^{2}, 0.66666666666666674 \cdot x - 1, 2\right)\right)\right)}}{2}\]
   8. Using strategy rm

   9. Applied add-log-exp 1.3

      \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\log \left(e^{\mathsf{expm1}\left(\mathsf{fma}\left({x}^{2}, 0.66666666666666674 \cdot x - 1, 2\right)\right)}\right)}\right)}{2}\]

   if 2.5494520086503014 < x

   1. Initial program 0.5

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

   2. Simplified 0.5

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.5494520086503014:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\mathsf{fma}\left({x}^{2}, 0.66666666666666674 \cdot x - 1, 2\right)\right)}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +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))