NMSE Section 6.1 mentioned, A

Average Error: 30.1 → 1.0

Time: 5.4s

Precision: 64

• could not determine a ground truth for program body

  1. x = -5.82365347619112e+126
  2. eps = 1.1430041436012933e-89

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 330.46423576118536:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{2}, x \cdot 0.66666666666666674 - 1, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 - \varepsilon\right) \cdot x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}{2}\\ \end{array}\]

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 VAR;
	if ((x <= 330.46423576118536)) {
		VAR = (fma(pow(x, 2.0), ((x * 0.6666666666666667) - 1.0), 2.0) / 2.0);
	} else {
		VAR = ((((1.0 + (1.0 / eps)) * pow(exp(-1.0), ((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * pow(exp(-1.0), ((1.0 + eps) * x)))) / 2.0);
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if x < 330.46423576118536

   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. 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

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

   if 330.46423576118536 < 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. Using strategy rm

   3. Applied neg-mul-1 0.1

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

   4. Applied exp-prod 0.1

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

   6. Applied neg-mul-1 0.1

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

   7. Applied exp-prod 0.1

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 330.46423576118536:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{2}, x \cdot 0.66666666666666674 - 1, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 - \varepsilon\right) \cdot x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +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))