NMSE Section 6.1 mentioned, A

Average Error: 29.3 → 0.5

Time: 9.6s

Precision: binary64

Cost: 13953

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1009973366871088 \cdot 10^{-16}:\\ \;\;\;\;\frac{2 \cdot \frac{x \cdot x - 1}{e^{x} \cdot \left(x - 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{x \cdot \varepsilon - x}}{2}\\ \end{array}\]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

↓

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.1009973366871088e-16)
   (/ (* 2.0 (/ (- (* x x) 1.0) (* (exp x) (- x 1.0)))) 2.0)
   (/ (+ (exp (* x (- -1.0 eps))) (exp (- (* x eps) x))) 2.0)))

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 tmp;
	if (x <= -1.1009973366871088e-16) {
		tmp = (2.0 * (((x * x) - 1.0) / (exp(x) * (x - 1.0)))) / 2.0;
	} else {
		tmp = (exp(x * (-1.0 - eps)) + exp((x * eps) - x)) / 2.0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Alternatives

Alternative 1
Error	0.6
Cost	7040

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

Alternative 2
Error	1.1
Cost	13632

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

Alternative 3
Error	29.3
Cost	14400

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

Alternative 4
Error	29.3
Cost	27264

\[\frac{\sqrt[3]{{\left(\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}\right)}^{3}}}{2}\]

Alternative 5
Error	29.9
Cost	34368

\[\frac{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{1 + \frac{1}{\varepsilon}}\right) \cdot \left(e^{x \cdot \left(-1 + \varepsilon\right)} \cdot \sqrt[3]{1 + \frac{1}{\varepsilon}}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(-1 + \frac{1}{\varepsilon}\right)}{2}\]

Alternative 6
Error	29.9
Cost	47936

\[\frac{\sqrt[3]{\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}}} \cdot \left(\sqrt[3]{\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}}} \cdot \sqrt[3]{\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}}}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(-1 + \frac{1}{\varepsilon}\right)}{2}\]

Alternative 7
Error	29.7
Cost	62272

\[\frac{\sqrt[3]{\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}} \cdot \left(\sqrt[3]{\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}} \cdot \sqrt[3]{\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}}\right)}{2}\]

Alternative 8
Error	46.7
Cost	62464

\[\frac{\frac{{\left(\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}}\right)}^{3} - {\left(\frac{-1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}\right)}^{3}}{\frac{-1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}} \cdot \left(\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}} + \frac{-1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}\right) + {\left(\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}}\right)}^{2}}}{2}\]

Alternative 9
Error	46.7
Cost	62848

\[\frac{\frac{{\left(\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}}\right)}^{3} - {\left(\frac{-1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}\right)}^{3}}{\frac{-1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}} \cdot \left(\frac{1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}} + \frac{-1 + \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}\right) + \left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{x \cdot \left(1 - \varepsilon\right)}\right)}^{-2}\right)}}{2}\]

Derivation

1. Split input into 2 regimes

2. if x < -1.10099733668710882e-16

   1. Initial program 57.4

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

   2. Taylor expanded around 0 2.2

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

   3. Simplified 2.1

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

   5. Applied neg-sub0_binary64_414 2.1

      \[\leadsto \frac{2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{0 - x}}\right)}{2}\]

   6. Applied exp-diff_binary64_467 2.0

      \[\leadsto \frac{2 \cdot \left(\left(x + 1\right) \cdot \color{blue}{\frac{e^{0}}{e^{x}}}\right)}{2}\]

   7. Applied flip-+_binary64_393 2.3

      \[\leadsto \frac{2 \cdot \left(\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}} \cdot \frac{e^{0}}{e^{x}}\right)}{2}\]

   8. Applied frac-times_binary64_429 2.2

      \[\leadsto \frac{2 \cdot \color{blue}{\frac{\left(x \cdot x - 1 \cdot 1\right) \cdot e^{0}}{\left(x - 1\right) \cdot e^{x}}}}{2}\]

   9. Simplified 2.2

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

   10. Simplified 2.2

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

   11. Simplified 2.2

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

   if -1.10099733668710882e-16 < x

   1. Initial program 28.8

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

   2. Taylor expanded around inf 0.5

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

   3. Simplified 0.5

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

   4. Simplified 0.5

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1009973366871088 \cdot 10^{-16}:\\ \;\;\;\;\frac{2 \cdot \frac{x \cdot x - 1}{e^{x} \cdot \left(x - 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{x \cdot \varepsilon - x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))