NMSE Section 6.1 mentioned, A

Average Error: 29.4 → 0.1

Time: 18.1s

Precision: binary64

Cost: 41284

Math

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

↓

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

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x))))) < 2

   1. Initial program 29.9

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

   2. Simplified 29.9

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

      [Start] 29.9	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Taylor expanded in eps around 0 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      rational_best.json-simplify-1 [=>] 0.0	\[ \frac{\color{blue}{\left(e^{-1 \cdot x} + e^{-1 \cdot x} \cdot x\right)} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      rational_best.json-simplify-2 [=>] 0.0	\[ \frac{\left(e^{\color{blue}{x \cdot -1}} + e^{-1 \cdot x} \cdot x\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      rational_best.json-simplify-12 [=>] 0.0	\[ \frac{\left(e^{\color{blue}{-x}} + e^{-1 \cdot x} \cdot x\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      rational_best.json-simplify-2 [=>] 0.0	\[ \frac{\left(e^{-x} + \color{blue}{x \cdot e^{-1 \cdot x}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      rational_best.json-simplify-2 [=>] 0.0	\[ \frac{\left(e^{-x} + x \cdot e^{\color{blue}{x \cdot -1}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      rational_best.json-simplify-12 [=>] 0.0	\[ \frac{\left(e^{-x} + x \cdot e^{\color{blue}{-x}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      rational_best.json-simplify-44 [=>] 0.0	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - \left(\color{blue}{e^{-1 \cdot x} \cdot \left(-1 \cdot x\right)} + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      rational_best.json-simplify-2 [=>] 0.0	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - \left(e^{-1 \cdot x} \cdot \left(-1 \cdot x\right) + \color{blue}{e^{-1 \cdot x} \cdot -1}\right)}{2} \]
      rational_best.json-simplify-47 [=>] 0.0	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - \color{blue}{e^{-1 \cdot x} \cdot \left(-1 + -1 \cdot x\right)}}{2} \]
      rational_best.json-simplify-2 [=>] 0.0	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - e^{\color{blue}{x \cdot -1}} \cdot \left(-1 + -1 \cdot x\right)}{2} \]
      rational_best.json-simplify-12 [=>] 0.0	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - e^{\color{blue}{-x}} \cdot \left(-1 + -1 \cdot x\right)}{2} \]
      rational_best.json-simplify-2 [=>] 0.0	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - e^{-x} \cdot \left(-1 + \color{blue}{x \cdot -1}\right)}{2} \]
      rational_best.json-simplify-12 [=>] 0.0	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - e^{-x} \cdot \left(-1 + \color{blue}{\left(-x\right)}\right)}{2} \]

   5. Taylor expanded in x around inf 0.0

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

   6. Simplified 0.0

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

      [Start] 0.0	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot \left(e^{-x} \cdot x\right) + -1 \cdot e^{-x}\right)}{2} \]
      rational_best.json-simplify-2 [=>] 0.0	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot \color{blue}{\left(x \cdot e^{-x}\right)} + -1 \cdot e^{-x}\right)}{2} \]
      rational_best.json-simplify-47 [=>] 0.0	\[ \frac{\left(e^{-x} + x \cdot e^{-x}\right) - \color{blue}{-1 \cdot \left(e^{-x} + x \cdot e^{-x}\right)}}{2} \]

   if 2 < (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x)))))

   1. Initial program 3.3

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

   2. Simplified 3.3

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

      [Start] 3.3	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Taylor expanded in eps around inf 2.9

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

   4. Simplified 2.9

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

      [Start] 2.9	\[ \frac{e^{\left(\varepsilon - 1\right) \cdot x} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      rational_best.json-simplify-2 [=>] 2.9	\[ \frac{e^{\left(\varepsilon - 1\right) \cdot x} - \color{blue}{e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)} \cdot -1}}{2} \]
      rational_best.json-simplify-12 [=>] 2.9	\[ \frac{e^{\left(\varepsilon - 1\right) \cdot x} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]
      rational_best.json-simplify-11 [=>] 2.9	\[ \frac{e^{\left(\varepsilon - 1\right) \cdot x} - \color{blue}{\left(0 - e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]
      rational_best.json-simplify-46 [=>] 2.9	\[ \frac{\color{blue}{e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)} + \left(e^{\left(\varepsilon - 1\right) \cdot x} - 0\right)}}{2} \]
      rational_best.json-simplify-1 [=>] 2.9	\[ \frac{e^{-1 \cdot \left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)} + \left(e^{\left(\varepsilon - 1\right) \cdot x} - 0\right)}{2} \]
      rational_best.json-simplify-2 [=>] 2.9	\[ \frac{e^{-1 \cdot \color{blue}{\left(x \cdot \left(1 + \varepsilon\right)\right)}} + \left(e^{\left(\varepsilon - 1\right) \cdot x} - 0\right)}{2} \]
      rational_best.json-simplify-44 [=>] 2.9	\[ \frac{e^{\color{blue}{x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)}} + \left(e^{\left(\varepsilon - 1\right) \cdot x} - 0\right)}{2} \]
      rational_best.json-simplify-2 [=>] 2.9	\[ \frac{e^{x \cdot \color{blue}{\left(\left(1 + \varepsilon\right) \cdot -1\right)}} + \left(e^{\left(\varepsilon - 1\right) \cdot x} - 0\right)}{2} \]
      rational_best.json-simplify-12 [=>] 2.9	\[ \frac{e^{x \cdot \color{blue}{\left(-\left(1 + \varepsilon\right)\right)}} + \left(e^{\left(\varepsilon - 1\right) \cdot x} - 0\right)}{2} \]
      rational_best.json-simplify-11 [=>] 2.9	\[ \frac{e^{x \cdot \color{blue}{\left(0 - \left(1 + \varepsilon\right)\right)}} + \left(e^{\left(\varepsilon - 1\right) \cdot x} - 0\right)}{2} \]
      rational_best.json-simplify-1 [<=] 2.9	\[ \frac{e^{x \cdot \left(0 - \color{blue}{\left(\varepsilon + 1\right)}\right)} + \left(e^{\left(\varepsilon - 1\right) \cdot x} - 0\right)}{2} \]
      rational_best.json-simplify-16 [=>] 2.9	\[ \frac{e^{x \cdot \left(0 - \color{blue}{\left(\varepsilon - -1\right)}\right)} + \left(e^{\left(\varepsilon - 1\right) \cdot x} - 0\right)}{2} \]
      rational_best.json-simplify-46 [=>] 2.9	\[ \frac{e^{x \cdot \color{blue}{\left(-1 + \left(0 - \varepsilon\right)\right)}} + \left(e^{\left(\varepsilon - 1\right) \cdot x} - 0\right)}{2} \]
      rational_best.json-simplify-11 [<=] 2.9	\[ \frac{e^{x \cdot \left(-1 + \color{blue}{\left(-\varepsilon\right)}\right)} + \left(e^{\left(\varepsilon - 1\right) \cdot x} - 0\right)}{2} \]
      rational_best.json-simplify-6 [=>] 2.9	\[ \frac{e^{x \cdot \left(-1 + \left(-\varepsilon\right)\right)} + \color{blue}{e^{\left(\varepsilon - 1\right) \cdot x}}}{2} \]
      rational_best.json-simplify-2 [=>] 2.9	\[ \frac{e^{x \cdot \left(-1 + \left(-\varepsilon\right)\right)} + e^{\color{blue}{x \cdot \left(\varepsilon - 1\right)}}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	34756

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

Alternative 2
Error	0.1
Cost	28228

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

Alternative 3
Error	1.0
Cost	13696

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

Alternative 4
Error	1.7
Cost	13440

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

Alternative 5
Error	1.5
Cost	13248

\[\begin{array}{l} t_0 := e^{-x}\\ \frac{t_0 + t_0}{2} \end{array} \]

Alternative 6
Error	1.9
Cost	576

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

Alternative 7
Error	16.5
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023090 
(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))