NMSE Section 6.1 mentioned, A

Average Error: 29.4 → 0.9

Time: 12.0s

Precision: binary64

Cost: 13636

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 \cdot x}{2}\\ \end{array} \]

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 360.0)
   (/ (+ (exp (* eps x)) (exp (* x (- -1.0 eps)))) 2.0)
   (/ (* 0.0 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 (x <= 360.0) {
		tmp = (exp((eps * x)) + exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (0.0 * 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 (x <= 360.0d0) then
        tmp = (exp((eps * x)) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else
        tmp = (0.0d0 * 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 (x <= 360.0) {
		tmp = (Math.exp((eps * x)) + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (0.0 * 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 x <= 360.0:
		tmp = (math.exp((eps * x)) + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = (0.0 * 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 (x <= 360.0)
		tmp = Float64(Float64(exp(Float64(eps * x)) + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	else
		tmp = Float64(Float64(0.0 * 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 (x <= 360.0)
		tmp = (exp((eps * x)) + exp((x * (-1.0 - eps)))) / 2.0;
	else
		tmp = (0.0 * 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[LessEqual[x, 360.0], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.0 * x), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 360

   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

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

      [Start] 38.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      rational_best-simplify-52 [=>] 38.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{x \cdot \left(-\left(1 - \varepsilon\right)\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      rational_best-simplify-64 [=>] 38.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \color{blue}{\left(\varepsilon - 1\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      rational_best-simplify-25 [=>] 38.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \color{blue}{\left(\varepsilon + -1\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      rational_best-simplify-25 [=>] 38.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + -1\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      rational_best-simplify-52 [=>] 38.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{x \cdot \left(-\left(1 + \varepsilon\right)\right)}}}{2} \]
      rational_best-simplify-27 [=>] 38.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{x \cdot \left(-\color{blue}{\left(\varepsilon - -1\right)}\right)}}{2} \]
      rational_best-simplify-64 [=>] 38.8	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{x \cdot \color{blue}{\left(-1 - \varepsilon\right)}}}{2} \]

   3. Taylor expanded in eps around inf 1.3

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

   4. Simplified 1.3

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

      [Start] 1.3	\[ \frac{e^{\left(\varepsilon - 1\right) \cdot x} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      rational_best-simplify-61 [=>] 1.3	\[ \frac{\color{blue}{e^{\left(\varepsilon - 1\right) \cdot x} + \left(--1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]
      rational_best-simplify-3 [=>] 1.3	\[ \frac{e^{\color{blue}{x \cdot \left(\varepsilon - 1\right)}} + \left(--1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}{2} \]
      rational_best-simplify-52 [=>] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \color{blue}{e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)} \cdot \left(--1\right)}}{2} \]
      metadata-eval [=>] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)} \cdot \color{blue}{1}}{2} \]
      rational_best-simplify-3 [=>] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \color{blue}{1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}}{2} \]
      rational_best-simplify-3 [=>] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + 1 \cdot e^{\color{blue}{\left(\left(\varepsilon + 1\right) \cdot x\right) \cdot -1}}}{2} \]
      rational_best-simplify-1 [=>] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + 1 \cdot e^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right) \cdot -1}}{2} \]
      rational_best-simplify-17 [=>] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + 1 \cdot e^{\color{blue}{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]
      rational_best-simplify-53 [<=] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + 1 \cdot e^{\color{blue}{x \cdot \left(-\left(1 + \varepsilon\right)\right)}}}{2} \]
      rational_best-simplify-27 [=>] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + 1 \cdot e^{x \cdot \left(-\color{blue}{\left(\varepsilon - -1\right)}\right)}}{2} \]
      rational_best-simplify-65 [<=] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + 1 \cdot e^{x \cdot \color{blue}{\left(-1 - \varepsilon\right)}}}{2} \]
      rational_best-simplify-51 [<=] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + 1 \cdot \color{blue}{\left(-\left(-e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)}}{2} \]
      rational_best-simplify-56 [=>] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \color{blue}{\left(-e^{x \cdot \left(-1 - \varepsilon\right)}\right) \cdot \left(-1\right)}}{2} \]
      metadata-eval [=>] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \left(-e^{x \cdot \left(-1 - \varepsilon\right)}\right) \cdot \color{blue}{-1}}{2} \]
      rational_best-simplify-18 [<=] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \color{blue}{\left(-\left(-e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)}}{2} \]
      rational_best-simplify-51 [=>] 1.3	\[ \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \color{blue}{e^{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

   5. Taylor expanded in eps around inf 1.2

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

   if 360 < 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. Simplified 0.1

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

      [Start] 0.1	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      rational_best-simplify-52 [=>] 0.1	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{x \cdot \left(-\left(1 - \varepsilon\right)\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      rational_best-simplify-64 [=>] 0.1	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \color{blue}{\left(\varepsilon - 1\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      rational_best-simplify-25 [=>] 0.1	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \color{blue}{\left(\varepsilon + -1\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      rational_best-simplify-25 [=>] 0.1	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + -1\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      rational_best-simplify-52 [=>] 0.1	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{x \cdot \left(-\left(1 + \varepsilon\right)\right)}}}{2} \]
      rational_best-simplify-27 [=>] 0.1	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{x \cdot \left(-\color{blue}{\left(\varepsilon - -1\right)}\right)}}{2} \]
      rational_best-simplify-64 [=>] 0.1	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{x \cdot \color{blue}{\left(-1 - \varepsilon\right)}}}{2} \]

   3. Taylor expanded in x around 0 63.4

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

   4. Simplified 63.4

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

      [Start] 63.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\left(\frac{1}{\varepsilon} + -1 \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)\right)\right) - 1\right)}{2} \]
      rational_best-simplify-25 [=>] 63.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \color{blue}{\left(\left(\frac{1}{\varepsilon} + -1 \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)\right)\right) + -1\right)}}{2} \]
      rational_best-simplify-1 [=>] 63.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \color{blue}{\left(-1 + \left(\frac{1}{\varepsilon} + -1 \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)\right)\right)\right)}}{2} \]
      rational_best-simplify-113 [=>] 63.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(-1 + \left(\frac{1}{\varepsilon} + \color{blue}{\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(-1 \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)\right)}\right)\right)}{2} \]
      rational_best-simplify-3 [=>] 63.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(-1 + \left(\frac{1}{\varepsilon} + \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(\left(\left(1 + \varepsilon\right) \cdot x\right) \cdot -1\right)}\right)\right)}{2} \]
      rational_best-simplify-113 [=>] 63.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(-1 + \left(\frac{1}{\varepsilon} + \color{blue}{\left(\left(1 + \varepsilon\right) \cdot x\right) \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot -1\right)}\right)\right)}{2} \]
      rational_best-simplify-3 [=>] 63.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(-1 + \left(\frac{1}{\varepsilon} + \color{blue}{\left(x \cdot \left(1 + \varepsilon\right)\right)} \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot -1\right)\right)\right)}{2} \]
      rational_best-simplify-1 [<=] 63.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(-1 + \left(\frac{1}{\varepsilon} + \left(x \cdot \color{blue}{\left(\varepsilon + 1\right)}\right) \cdot \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot -1\right)\right)\right)}{2} \]
      rational_best-simplify-18 [<=] 63.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(-1 + \left(\frac{1}{\varepsilon} + \left(x \cdot \left(\varepsilon + 1\right)\right) \cdot \color{blue}{\left(-\left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right)}{2} \]
      rational_best-simplify-65 [<=] 63.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(-1 + \left(\frac{1}{\varepsilon} + \left(x \cdot \left(\varepsilon + 1\right)\right) \cdot \color{blue}{\left(1 - \frac{1}{\varepsilon}\right)}\right)\right)}{2} \]

   5. Taylor expanded in x around 0 35.6

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

   6. Taylor expanded in x around inf 1.4

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

   7. Taylor expanded in eps around 0 0.1

      \[\leadsto \frac{\color{blue}{0} \cdot x}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 \cdot x}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	35204

\[\begin{array}{l} t_0 := e^{x \cdot \left(-1 - \varepsilon\right)}\\ t_1 := t_0 \cdot \left(1 - \frac{1}{\varepsilon}\right)\\ t_2 := \left(\frac{1}{\varepsilon} + -1\right) \cdot t_0\\ \frac{\begin{array}{l} \mathbf{if}\;t_2 \ne 0:\\ \;\;\;\;\frac{t_1 \cdot \left(\left(-e^{x \cdot \varepsilon - x}\right) - t_0\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\varepsilon \cdot x - x} + t_1\\ \end{array}}{2} \end{array} \]

Alternative 2
Error	1.0
Cost	13632

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

Alternative 3
Error	1.3
Cost	7364

\[\begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{\left(1 + x \cdot \varepsilon\right) + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 \cdot x}{2}\\ \end{array} \]

Alternative 4
Error	1.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{0 \cdot x}{2}\\ \end{array} \]

Alternative 5
Error	16.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023104 
(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))