NMSE Section 6.1 mentioned, A

Average Error: 30.1 → 1.1

Time: 12.1s

Precision: 64

• could not determine a ground truth for program body

  1. x = -1.3277474740950274e+60
  2. eps = 5.872367900798126e-198

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1.9121598632801717:\\ \;\;\;\;\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

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 <= 1.9121598632801717)) {
		temp = (((0.33333333333333337 * pow(x, 3.0)) + 1.0) - (0.5 * pow(x, 2.0)));
	} else {
		temp = (((((1.0 + (1.0 / eps)) / exp(((1.0 - eps) * x))) / 2.0) - (((1.0 / eps) / exp(((1.0 + eps) * x))) / 2.0)) + ((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 < 1.9121598632801717

   1. Initial program 39.5

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

   2. Simplified 39.5

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

   3. Taylor expanded around 0 1.3

      \[\leadsto \color{blue}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\]

   if 1.9121598632801717 < 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

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

   4. Applied div-sub 0.5

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

   5. Applied div-sub 0.5

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

   6. Applied associate--r- 0.5

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.9121598632801717:\\ \;\;\;\;\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(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))