NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.2% → 99.3%

Time: 18.1s

Precision: binary64

Cost: 13705

• 38.3% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.25 \cdot 10^{+14} \lor \neg \left(\varepsilon \leq 1.9 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{e^{x}} \cdot \left(1 + x\right)}{2}\\ \end{array} \]

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 (or (<= eps -1.25e+14) (not (<= eps 1.9e-11)))
   (/ (+ (exp (* eps x)) (exp (* eps (- x)))) 2.0)
   (/ (* (/ 2.0 (exp x)) (+ 1.0 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 ((eps <= -1.25e+14) || !(eps <= 1.9e-11)) {
		tmp = (exp((eps * x)) + exp((eps * -x))) / 2.0;
	} else {
		tmp = ((2.0 / exp(x)) * (1.0 + x)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.25d+14)) .or. (.not. (eps <= 1.9d-11))) then
        tmp = (exp((eps * x)) + exp((eps * -x))) / 2.0d0
    else
        tmp = ((2.0d0 / exp(x)) * (1.0d0 + x)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0;
}

↓

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.25e+14) || !(eps <= 1.9e-11)) {
		tmp = (Math.exp((eps * x)) + Math.exp((eps * -x))) / 2.0;
	} else {
		tmp = ((2.0 / Math.exp(x)) * (1.0 + x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

↓

def code(x, eps):
	tmp = 0
	if (eps <= -1.25e+14) or not (eps <= 1.9e-11):
		tmp = (math.exp((eps * x)) + math.exp((eps * -x))) / 2.0
	else:
		tmp = ((2.0 / math.exp(x)) * (1.0 + x)) / 2.0
	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 ((eps <= -1.25e+14) || !(eps <= 1.9e-11))
		tmp = Float64(Float64(exp(Float64(eps * x)) + exp(Float64(eps * Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 / exp(x)) * Float64(1.0 + x)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

↓

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.25e+14) || ~((eps <= 1.9e-11)))
		tmp = (exp((eps * x)) + exp((eps * -x))) / 2.0;
	else
		tmp = ((2.0 / exp(x)) * (1.0 + x)) / 2.0;
	end
	tmp_2 = 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[Or[LessEqual[eps, -1.25e+14], N[Not[LessEqual[eps, 1.9e-11]], $MachinePrecision]], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.25e14 or 1.8999999999999999e-11 < eps

   1. Initial program 100.0%

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

   2. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [=>] 100.0	\[ \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]
      +-rgt-identity [<=] 100.0	\[ \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [<=] 100.0	\[ \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Taylor expanded in eps around inf 100.0%

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

   4. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      neg-mul-1 [<=] 100.0	\[ \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      distribute-rgt-neg-in [=>] 100.0	\[ \frac{e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      mul-1-neg [=>] 100.0	\[ \frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]
      mul-1-neg [=>] 100.0	\[ \frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(-e^{\color{blue}{-\left(\varepsilon + 1\right) \cdot x}}\right)}{2} \]
      +-commutative [<=] 100.0	\[ \frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(-e^{-\color{blue}{\left(1 + \varepsilon\right)} \cdot x}\right)}{2} \]
      *-commutative [=>] 100.0	\[ \frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(-e^{-\color{blue}{x \cdot \left(1 + \varepsilon\right)}}\right)}{2} \]
      +-commutative [=>] 100.0	\[ \frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(-e^{-x \cdot \color{blue}{\left(\varepsilon + 1\right)}}\right)}{2} \]

   5. Taylor expanded in eps around inf 100.0%

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

   6. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{e^{\varepsilon \cdot x} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}{2} \]
      *-commutative [=>] 100.0	\[ \frac{e^{\color{blue}{x \cdot \varepsilon}} - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)}{2} \]

   7. Taylor expanded in eps around inf 100.0%

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

   if -1.25e14 < eps < 1.8999999999999999e-11

   1. Initial program 41.6%

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

   2. Simplified 38.0%

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

      [Start] 41.6	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Taylor expanded in eps around 0 41.9%

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

   4. Simplified 99.1%

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

      [Start] 41.9	\[ \frac{\left(\frac{e^{-1 \cdot x}}{\varepsilon} + \left(e^{-1 \cdot x} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right)\right) - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)}{2} \]
      associate--r+ [=>] 42.2	\[ \frac{\color{blue}{\left(\left(\frac{e^{-1 \cdot x}}{\varepsilon} + \left(e^{-1 \cdot x} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right)\right) - \frac{1}{\varepsilon \cdot e^{x}}\right) - -1 \cdot \frac{x}{e^{x}}}}{2} \]
      sub-neg [=>] 42.2	\[ \frac{\color{blue}{\left(\left(\frac{e^{-1 \cdot x}}{\varepsilon} + \left(e^{-1 \cdot x} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right)\right) - \frac{1}{\varepsilon \cdot e^{x}}\right) + \left(--1 \cdot \frac{x}{e^{x}}\right)}}{2} \]

   5. Taylor expanded in x around inf 99.1%

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

   6. Simplified 100.0%

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

      [Start] 99.1	\[ \frac{2 \cdot \frac{x}{e^{x}} + 2 \cdot e^{-x}}{2} \]
      +-commutative [=>] 99.1	\[ \frac{\color{blue}{2 \cdot e^{-x} + 2 \cdot \frac{x}{e^{x}}}}{2} \]
      neg-mul-1 [=>] 99.1	\[ \frac{2 \cdot e^{\color{blue}{-1 \cdot x}} + 2 \cdot \frac{x}{e^{x}}}{2} \]
      *-lft-identity [<=] 99.1	\[ \frac{\color{blue}{1 \cdot \left(2 \cdot e^{-1 \cdot x}\right)} + 2 \cdot \frac{x}{e^{x}}}{2} \]
      associate-*r/ [=>] 99.1	\[ \frac{1 \cdot \left(2 \cdot e^{-1 \cdot x}\right) + \color{blue}{\frac{2 \cdot x}{e^{x}}}}{2} \]
      associate-*l/ [<=] 99.1	\[ \frac{1 \cdot \left(2 \cdot e^{-1 \cdot x}\right) + \color{blue}{\frac{2}{e^{x}} \cdot x}}{2} \]
      metadata-eval [<=] 99.1	\[ \frac{1 \cdot \left(2 \cdot e^{-1 \cdot x}\right) + \frac{\color{blue}{2 \cdot 1}}{e^{x}} \cdot x}{2} \]
      associate-*r/ [<=] 99.1	\[ \frac{1 \cdot \left(2 \cdot e^{-1 \cdot x}\right) + \color{blue}{\left(2 \cdot \frac{1}{e^{x}}\right)} \cdot x}{2} \]
      exp-neg [<=] 99.1	\[ \frac{1 \cdot \left(2 \cdot e^{-1 \cdot x}\right) + \left(2 \cdot \color{blue}{e^{-x}}\right) \cdot x}{2} \]
      neg-mul-1 [=>] 99.1	\[ \frac{1 \cdot \left(2 \cdot e^{-1 \cdot x}\right) + \left(2 \cdot e^{\color{blue}{-1 \cdot x}}\right) \cdot x}{2} \]
      *-commutative [=>] 99.1	\[ \frac{1 \cdot \left(2 \cdot e^{-1 \cdot x}\right) + \color{blue}{x \cdot \left(2 \cdot e^{-1 \cdot x}\right)}}{2} \]
      distribute-rgt-out [=>] 100.0	\[ \frac{\color{blue}{\left(2 \cdot e^{-1 \cdot x}\right) \cdot \left(1 + x\right)}}{2} \]
      neg-mul-1 [<=] 100.0	\[ \frac{\left(2 \cdot e^{\color{blue}{-x}}\right) \cdot \left(1 + x\right)}{2} \]
      exp-neg [=>] 100.0	\[ \frac{\left(2 \cdot \color{blue}{\frac{1}{e^{x}}}\right) \cdot \left(1 + x\right)}{2} \]
      associate-*r/ [=>] 100.0	\[ \frac{\color{blue}{\frac{2 \cdot 1}{e^{x}}} \cdot \left(1 + x\right)}{2} \]
      metadata-eval [=>] 100.0	\[ \frac{\frac{\color{blue}{2}}{e^{x}} \cdot \left(1 + x\right)}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.25 \cdot 10^{+14} \lor \neg \left(\varepsilon \leq 1.9 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{e^{x}} \cdot \left(1 + x\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.9%
Cost	13632

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

Alternative 2
Accuracy	76.6%
Cost	8033

\[\begin{array}{l} t_0 := \frac{\frac{2}{e^{x}} \cdot \left(1 + x\right)}{2}\\ t_1 := \frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-47}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} - -1}{2}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+201}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot \left(2 + x \cdot \left(x + -2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+224} \lor \neg \left(x \leq 8.5 \cdot 10^{+260}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)}{2}\\ \end{array} \]

Alternative 3
Accuracy	76.7%
Cost	8033

\[\begin{array}{l} t_0 := \frac{\frac{2}{e^{x}} \cdot \left(1 + x\right)}{2}\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} - -1}{2}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+34}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 + x \cdot \left(\varepsilon + -1\right)\right)}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+201}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot \left(2 + x \cdot \left(x + -2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+224} \lor \neg \left(x \leq 1.5 \cdot 10^{+261}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)}{2}\\ \end{array} \]

Alternative 4
Accuracy	77.7%
Cost	7769

\[\begin{array}{l} t_0 := \frac{\frac{2}{e^{x}} \cdot \left(1 + x\right)}{2}\\ \mathbf{if}\;x \leq -2700000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 400000000000:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} - -1}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+201}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot \left(2 + x \cdot \left(x + -2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+224} \lor \neg \left(x \leq 1.5 \cdot 10^{+261}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)}{2}\\ \end{array} \]

Alternative 5
Accuracy	77.4%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -2700000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2200000000000:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} - -1}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+108}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot \left(2 + x \cdot \left(x + -2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+225}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+287}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Accuracy	70.0%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 11500000000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+108}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+201}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot \left(2 + x \cdot \left(x + -2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+224}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+287}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Accuracy	63.7%
Cost	1496

\[\begin{array}{l} t_0 := \frac{2 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)}{2}\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 11500000000:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(1 - \varepsilon\right) \cdot \left(-1 - \frac{1}{\varepsilon}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+108}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+202}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot \left(2 + x \cdot \left(x + -2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+224}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+286}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Accuracy	63.2%
Cost	1364

\[\begin{array}{l} t_0 := \frac{2 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)}{2}\\ \mathbf{if}\;x \leq 11500000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+108}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+202}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot \left(2 + x \cdot \left(x + -2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+224}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+287}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Accuracy	63.2%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;x \leq 11500000000 \lor \neg \left(x \leq 10^{+224}\right) \land x \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Accuracy	60.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -0.385:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Accuracy	60.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -260:\\ \;\;\;\;\frac{\frac{0.5}{\varepsilon} \cdot \left(x \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Accuracy	59.8%
Cost	516

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+136}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{elif}\;x \leq 11500000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13
Accuracy	57.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14
Accuracy	57.1%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 11500000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15
Accuracy	16.3%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023160 
(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))