NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.8%

Time: 19.5s

Precision: binary64

Cost: 14024

• 38.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.99999999999999997e77
  2. eps = -2e-231

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + x\right) \cdot e^{-x}\\ t_1 := e^{\varepsilon \cdot \left(-x\right)}\\ \mathbf{if}\;\varepsilon \leq -1:\\ \;\;\;\;\frac{t_1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + t_1}{2}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (+ 1.0 x) (exp (- x)))) (t_1 (exp (* eps (- x)))))
   (if (<= eps -1.0)
     (/ (+ t_1 (exp (* eps x))) 2.0)
     (if (<= eps 5e-29)
       (/ (+ t_0 t_0) 2.0)
       (/ (+ (exp (* x (+ -1.0 eps))) t_1) 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 t_0 = (1.0 + x) * exp(-x);
	double t_1 = exp((eps * -x));
	double tmp;
	if (eps <= -1.0) {
		tmp = (t_1 + exp((eps * x))) / 2.0;
	} else if (eps <= 5e-29) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps))) + t_1) / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 + x) * exp(-x)
    t_1 = exp((eps * -x))
    if (eps <= (-1.0d0)) then
        tmp = (t_1 + exp((eps * x))) / 2.0d0
    else if (eps <= 5d-29) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps))) + t_1) / 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 t_0 = (1.0 + x) * Math.exp(-x);
	double t_1 = Math.exp((eps * -x));
	double tmp;
	if (eps <= -1.0) {
		tmp = (t_1 + Math.exp((eps * x))) / 2.0;
	} else if (eps <= 5e-29) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps))) + t_1) / 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):
	t_0 = (1.0 + x) * math.exp(-x)
	t_1 = math.exp((eps * -x))
	tmp = 0
	if eps <= -1.0:
		tmp = (t_1 + math.exp((eps * x))) / 2.0
	elif eps <= 5e-29:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps))) + t_1) / 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)
	t_0 = Float64(Float64(1.0 + x) * exp(Float64(-x)))
	t_1 = exp(Float64(eps * Float64(-x)))
	tmp = 0.0
	if (eps <= -1.0)
		tmp = Float64(Float64(t_1 + exp(Float64(eps * x))) / 2.0);
	elseif (eps <= 5e-29)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps))) + t_1) / 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)
	t_0 = (1.0 + x) * exp(-x);
	t_1 = exp((eps * -x));
	tmp = 0.0;
	if (eps <= -1.0)
		tmp = (t_1 + exp((eps * x))) / 2.0;
	elseif (eps <= 5e-29)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps))) + t_1) / 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_] := Block[{t$95$0 = N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eps, -1.0], N[(N[(t$95$1 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 5e-29], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1

   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. Taylor expanded in eps around inf 100.0%

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

   5. Simplified 100.0%

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

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

   6. Taylor expanded in eps around inf 100.0%

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

   7. Simplified 100.0%

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

      [Start] 100.0%	\[ \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
      cancel-sign-sub-inv [=>] 100.0%	\[ \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2} \]
      mul-1-neg [=>] 100.0%	\[ \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
      *-commutative [<=] 100.0%	\[ \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
      metadata-eval [=>] 100.0%	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{1} \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
      mul-1-neg [=>] 100.0%	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{-\varepsilon \cdot x}}}{2} \]
      *-commutative [=>] 100.0%	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{-\color{blue}{x \cdot \varepsilon}}}{2} \]
      distribute-lft-neg-out [<=] 100.0%	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{\left(-x\right) \cdot \varepsilon}}}{2} \]
      *-lft-identity [=>] 100.0%	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{e^{\left(-x\right) \cdot \varepsilon}}}{2} \]
      +-commutative [=>] 100.0%	\[ \frac{\color{blue}{e^{\left(-x\right) \cdot \varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}}}{2} \]
      *-commutative [=>] 100.0%	\[ \frac{e^{\color{blue}{\varepsilon \cdot \left(-x\right)}} + e^{-x \cdot \left(1 - \varepsilon\right)}}{2} \]
      distribute-lft-neg-in [=>] 100.0%	\[ \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2} \]

   8. Taylor expanded in eps around inf 100.0%

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

   if -1 < eps < 4.99999999999999986e-29

   1. Initial program 38.8%

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

   2. Simplified 38.8%

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

      [Start] 38.8%	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [=>] 38.8%	\[ \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]
      +-rgt-identity [<=] 38.8%	\[ \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [<=] 38.8%	\[ \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Taylor expanded in eps around 0 98.9%

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

   4. Simplified 100.0%

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

      [Start] 98.9%	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      *-commutative [=>] 98.9%	\[ \frac{\left(\color{blue}{x \cdot e^{-1 \cdot x}} + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      distribute-lft1-in [=>] 98.9%	\[ \frac{\color{blue}{\left(x + 1\right) \cdot e^{-1 \cdot x}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      neg-mul-1 [<=] 98.9%	\[ \frac{\left(x + 1\right) \cdot e^{\color{blue}{-x}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      distribute-lft-out [=>] 98.9%	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{-1 \cdot \left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)}}{2} \]
      mul-1-neg [=>] 98.9%	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{\left(-\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)\right)}}{2} \]
      *-commutative [=>] 98.9%	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(\color{blue}{x \cdot e^{-1 \cdot x}} + e^{-1 \cdot x}\right)\right)}{2} \]
      distribute-lft1-in [=>] 100.0%	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\color{blue}{\left(x + 1\right) \cdot e^{-1 \cdot x}}\right)}{2} \]
      neg-mul-1 [<=] 100.0%	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)}{2} \]

   if 4.99999999999999986e-29 < eps

   1. Initial program 97.5%

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

   2. Simplified 97.5%

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

      [Start] 97.5%	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [=>] 97.5%	\[ \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]
      +-rgt-identity [<=] 97.5%	\[ \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [<=] 97.5%	\[ \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   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. Taylor expanded in eps around inf 100.0%

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

   5. Simplified 100.0%

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

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

   6. Taylor expanded in eps around inf 100.0%

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

   7. Simplified 100.0%

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

      [Start] 100.0%	\[ \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
      cancel-sign-sub-inv [=>] 100.0%	\[ \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}}{2} \]
      mul-1-neg [=>] 100.0%	\[ \frac{e^{\color{blue}{-\left(1 - \varepsilon\right) \cdot x}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
      *-commutative [<=] 100.0%	\[ \frac{e^{-\color{blue}{x \cdot \left(1 - \varepsilon\right)}} + \left(--1\right) \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
      metadata-eval [=>] 100.0%	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{1} \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
      mul-1-neg [=>] 100.0%	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{-\varepsilon \cdot x}}}{2} \]
      *-commutative [=>] 100.0%	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{-\color{blue}{x \cdot \varepsilon}}}{2} \]
      distribute-lft-neg-out [<=] 100.0%	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + 1 \cdot e^{\color{blue}{\left(-x\right) \cdot \varepsilon}}}{2} \]
      *-lft-identity [=>] 100.0%	\[ \frac{e^{-x \cdot \left(1 - \varepsilon\right)} + \color{blue}{e^{\left(-x\right) \cdot \varepsilon}}}{2} \]
      +-commutative [=>] 100.0%	\[ \frac{\color{blue}{e^{\left(-x\right) \cdot \varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}}}{2} \]
      *-commutative [=>] 100.0%	\[ \frac{e^{\color{blue}{\varepsilon \cdot \left(-x\right)}} + e^{-x \cdot \left(1 - \varepsilon\right)}}{2} \]
      distribute-lft-neg-in [=>] 100.0%	\[ \frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\color{blue}{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{2} \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1:\\ \;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot e^{-x} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	14024

\[\begin{array}{l} t_0 := \left(1 + x\right) \cdot e^{-x}\\ t_1 := e^{\varepsilon \cdot \left(-x\right)}\\ \mathbf{if}\;\varepsilon \leq -1:\\ \;\;\;\;\frac{t_1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + t_1}{2}\\ \end{array} \]

Alternative 2
Accuracy	98.8%
Cost	13832

\[\begin{array}{l} t_0 := e^{x \cdot \left(-1 + \varepsilon\right)}\\ t_1 := e^{\varepsilon \cdot \left(-x\right)}\\ \mathbf{if}\;\varepsilon \leq -1:\\ \;\;\;\;\frac{t_1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{t_0 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + t_1}{2}\\ \end{array} \]

Alternative 3
Accuracy	98.8%
Cost	13632

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

Alternative 4
Accuracy	92.1%
Cost	13572

\[\begin{array}{l} \mathbf{if}\;x \leq 135:\\ \;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]

Alternative 5
Accuracy	82.1%
Cost	8524

\[\begin{array}{l} t_0 := e^{x \cdot \left(-1 - \varepsilon\right)}\\ t_1 := \frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\ \mathbf{if}\;x \leq -4900:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(1 - \left(1 - \varepsilon\right) \cdot x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + t_0 \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2}\\ \end{array} \]

Alternative 6
Accuracy	82.0%
Cost	8012

\[\begin{array}{l} t_0 := \frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\ t_1 := e^{x \cdot \left(-1 - \varepsilon\right)}\\ \mathbf{if}\;x \leq -4900:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(1 + \varepsilon \cdot x\right) + t_1 \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{2}\\ \end{array} \]

Alternative 7
Accuracy	82.1%
Cost	8012

\[\begin{array}{l} t_0 := e^{x \cdot \left(-1 - \varepsilon\right)}\\ t_1 := \frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\ \mathbf{if}\;x \leq -4900:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right) \cdot t_0}{2}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2}\\ \end{array} \]

Alternative 8
Accuracy	83.1%
Cost	7761

\[\begin{array}{l} \mathbf{if}\;x \leq -4900:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-77} \lor \neg \left(x \leq 3.5 \cdot 10^{-57}\right) \land x \leq 6.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]

Alternative 9
Accuracy	82.9%
Cost	7377

\[\begin{array}{l} \mathbf{if}\;x \leq -4900:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-77} \lor \neg \left(x \leq 2 \cdot 10^{-56}\right) \land x \leq 6.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]

Alternative 10
Accuracy	77.1%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -4900:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 520000000000:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Accuracy	70.3%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 13000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Accuracy	60.4%
Cost	1732

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \left(1 - \varepsilon\right) \cdot x\right) + \left(1 + \left(\left(x - x\right) - \varepsilon \cdot x\right)\right)}{2}\\ \mathbf{elif}\;x \leq 13000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13
Accuracy	60.4%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \left(\left(x - x\right) - \varepsilon \cdot x\right)\right)}{2}\\ \mathbf{elif}\;x \leq 13000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14
Accuracy	59.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 - \left(1 - \varepsilon\right) \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15
Accuracy	60.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+160}:\\ \;\;\;\;\frac{-0.25 \cdot \left(x \cdot x\right)}{\varepsilon}\\ \mathbf{elif}\;x \leq 13000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16
Accuracy	60.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 13000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 17
Accuracy	57.3%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 13000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 18
Accuracy	15.9%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023167 
(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))