NMSE Section 6.1 mentioned, A

Average Accuracy: 31.0% → 56.3%

Time: 22.7s

Precision: binary64

Cost: 8516

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 430:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\varepsilon + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right) + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right), x, 2\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

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 430.0)
   (/
    (+
     (fma
      (+
       (* (+ eps 1.0) (+ -1.0 (/ 1.0 eps)))
       (* (- 1.0 eps) (+ -1.0 (/ -1.0 eps))))
      x
      2.0)
     (* eps (* eps (* x x))))
    2.0)
   0.0))

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 <= 430.0) {
		tmp = (fma((((eps + 1.0) * (-1.0 + (1.0 / eps))) + ((1.0 - eps) * (-1.0 + (-1.0 / eps)))), x, 2.0) + (eps * (eps * (x * x)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

↓

function code(x, eps)
	tmp = 0.0
	if (x <= 430.0)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(eps + 1.0) * Float64(-1.0 + Float64(1.0 / eps))) + Float64(Float64(1.0 - eps) * Float64(-1.0 + Float64(-1.0 / eps)))), x, 2.0) + Float64(eps * Float64(eps * Float64(x * x)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

↓

code[x_, eps_] := If[LessEqual[x, 430.0], N[(N[(N[(N[(N[(N[(eps + 1.0), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - eps), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] + N[(eps * N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 430

   1. Initial program 23.0%

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

   2. Simplified 23.0%

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

      [Start] 23.0	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [=>] 23.0	\[ \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]
      +-rgt-identity [<=] 23.0	\[ \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [<=] 23.0	\[ \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Taylor expanded in x around 0 55.8%

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

   4. Simplified 55.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\varepsilon + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right), x, 2\right) + \left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(1 - \varepsilon\right)}^{2} \cdot \left(1 + \frac{1}{\varepsilon}\right) - {\left(\varepsilon + 1\right)}^{2} \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)\right)}}{2} \]
      Step-by-step derivation

      [Start] 55.8

      \[ \frac{\left(-1 \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(1 - \varepsilon\right)\right) - -1 \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 + \varepsilon\right)\right)\right) \cdot x + \left(2 + \left(0.5 \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot {\left(1 - \varepsilon\right)}^{2}\right) - 0.5 \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) \cdot {x}^{2}\right)}{2} \]

      associate-+r+ [=>] 55.8

      \[ \frac{\color{blue}{\left(\left(-1 \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(1 - \varepsilon\right)\right) - -1 \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 + \varepsilon\right)\right)\right) \cdot x + 2\right) + \left(0.5 \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot {\left(1 - \varepsilon\right)}^{2}\right) - 0.5 \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) \cdot {x}^{2}}}{2} \]

   5. Taylor expanded in eps around inf 55.8%

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

   6. Simplified 61.5%

      \[\leadsto \frac{\mathsf{fma}\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\varepsilon + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right), x, 2\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)}}{2} \]
      Step-by-step derivation

      [Start] 55.8	\[ \frac{\mathsf{fma}\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\varepsilon + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right), x, 2\right) + {\varepsilon}^{2} \cdot {x}^{2}}{2} \]
      unpow2 [=>] 55.8	\[ \frac{\mathsf{fma}\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\varepsilon + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right), x, 2\right) + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {x}^{2}}{2} \]
      associate-*l* [=>] 61.5	\[ \frac{\mathsf{fma}\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\varepsilon + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right), x, 2\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot {x}^{2}\right)}}{2} \]
      unpow2 [=>] 61.5	\[ \frac{\mathsf{fma}\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\varepsilon + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right), x, 2\right) + \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(x \cdot x\right)}\right)}{2} \]

   if 430 < x

   1. Initial program 45.5%

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

   2. Simplified 45.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
      Step-by-step derivation

      [Start] 45.5	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Taylor expanded in eps around 0 46.3%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]

   4. Simplified 46.3%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
      Step-by-step derivation

      [Start] 46.3	\[ \frac{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}{2} \]
      div-sub [=>] 46.3	\[ \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]
      rec-exp [=>] 46.3	\[ \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2} \]
      mul-1-neg [<=] 46.3	\[ \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]
      +-inverses [=>] 46.3	\[ \frac{\color{blue}{0}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 430:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\varepsilon + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right) + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right), x, 2\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	56.5%
Cost	20096

\[\begin{array}{l} t_0 := \frac{x}{e^{x}}\\ \frac{t_0 + \left(t_0 + 2 \cdot e^{-x}\right)}{2} \end{array} \]

Alternative 2
Accuracy	56.5%
Cost	13760

\[\begin{array}{l} t_0 := e^{-x} \cdot \left(x + 1\right)\\ \frac{t_0 + t_0}{2} \end{array} \]

Alternative 3
Accuracy	55.6%
Cost	13632

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

Alternative 4
Accuracy	56.5%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 1000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Accuracy	16.2%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023157 
(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))