NMSE Section 6.1 mentioned, A

Average Error: 29.7 → 1.0

Time: 5.8s

Precision: 64

• could not determine a ground truth for program body

  1. x = -1.5126695650221918e+290
  2. eps = 1.0699143029134086e-283

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 199.934473320751266:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{2}, 0.66666666666666674 \cdot x - 1, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - 1 \cdot \left(\frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon} - \frac{1}{e^{\mathsf{fma}\left(x, \varepsilon, 1 \cdot x\right)}}\right)}{2}\\ \end{array}\]

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 temp;
	if ((x <= 199.93447332075127)) {
		temp = (fma(pow(x, 2.0), ((0.6666666666666667 * x) - 1.0), 2.0) / 2.0);
	} else {
		temp = ((((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (1.0 * ((exp(-((x * eps) + (1.0 * x))) / eps) - (1.0 / exp(fma(x, eps, (1.0 * x))))))) / 2.0);
	}
	return temp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if x < 199.93447332075127

   1. Initial program 39.4

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

   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

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

   if 199.93447332075127 < 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. Taylor expanded around inf 0.1

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

   3. Simplified 0.1

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 199.934473320751266:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{2}, 0.66666666666666674 \cdot x - 1, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - 1 \cdot \left(\frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon} - \frac{1}{e^{\mathsf{fma}\left(x, \varepsilon, 1 \cdot x\right)}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +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))