NMSE Section 6.1 mentioned, A

Average Error: 29.4 → 0.1

Time: 12.3s

Precision: binary64

Cost: 28228

Math

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

↓

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

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
 (let* ((t_0 (exp (- x))))
   (if (<=
        (-
         (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
         (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
        2.0)
     (/ (+ (* t_0 (+ x 1.0)) (* (+ x 1.0) t_0)) 2.0)
     (/ (- (exp (- (* x eps) x)) (- (exp (- (* (- eps) x) 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 t_0 = exp(-x);
	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_0 * (x + 1.0)) + ((x + 1.0) * t_0)) / 2.0;
	} else {
		tmp = (exp(((x * eps) - x)) - -exp(((-eps * x) - 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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    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_0 * (x + 1.0d0)) + ((x + 1.0d0) * t_0)) / 2.0d0
    else
        tmp = (exp(((x * eps) - x)) - -exp(((-eps * x) - 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 t_0 = Math.exp(-x);
	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_0 * (x + 1.0)) + ((x + 1.0) * t_0)) / 2.0;
	} else {
		tmp = (Math.exp(((x * eps) - x)) - -Math.exp(((-eps * x) - 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):
	t_0 = math.exp(-x)
	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_0 * (x + 1.0)) + ((x + 1.0) * t_0)) / 2.0
	else:
		tmp = (math.exp(((x * eps) - x)) - -math.exp(((-eps * x) - 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)
	t_0 = exp(Float64(-x))
	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(Float64(t_0 * Float64(x + 1.0)) + Float64(Float64(x + 1.0) * t_0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(Float64(x * eps) - x)) - Float64(-exp(Float64(Float64(Float64(-eps) * x) - 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)
	t_0 = exp(-x);
	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_0 * (x + 1.0)) + ((x + 1.0) * t_0)) / 2.0;
	else
		tmp = (exp(((x * eps) - x)) - -exp(((-eps * x) - 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_] := Block[{t$95$0 = N[Exp[(-x)], $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[(N[(t$95$0 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] - (-N[Exp[N[(N[((-eps) * x), $MachinePrecision] - x), $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

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

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

   4. Simplified 25.1

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

      [Start] 29.9	\[ \frac{\left(\frac{e^{-x}}{\varepsilon} + \left(e^{-x} + e^{-x} \cdot x\right)\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} + -1 \cdot e^{-1 \cdot x}\right)\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 25.1	\[ \frac{\color{blue}{\left(e^{-x} + e^{-x} \cdot x\right) + \left(\frac{e^{-x}}{\varepsilon} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} + -1 \cdot e^{-1 \cdot x}\right)\right)\right)}}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 25.1	\[ \frac{\color{blue}{\left(e^{-x} \cdot x + e^{-x}\right)} + \left(\frac{e^{-x}}{\varepsilon} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} + -1 \cdot e^{-1 \cdot x}\right)\right)\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [=>] 25.1	\[ \frac{\left(e^{-x} \cdot x + e^{\color{blue}{x \cdot -1}}\right) + \left(\frac{e^{-x}}{\varepsilon} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} + -1 \cdot e^{-1 \cdot x}\right)\right)\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 25.1	\[ \frac{\left(e^{-x} \cdot x + e^{\color{blue}{-1 \cdot x}}\right) + \left(\frac{e^{-x}}{\varepsilon} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} + -1 \cdot e^{-1 \cdot x}\right)\right)\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 25.1	\[ \frac{\left(\color{blue}{x \cdot e^{-x}} + e^{-1 \cdot x}\right) + \left(\frac{e^{-x}}{\varepsilon} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} + -1 \cdot e^{-1 \cdot x}\right)\right)\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [=>] 25.1	\[ \frac{\left(x \cdot e^{\color{blue}{x \cdot -1}} + e^{-1 \cdot x}\right) + \left(\frac{e^{-x}}{\varepsilon} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} + -1 \cdot e^{-1 \cdot x}\right)\right)\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 25.1	\[ \frac{\left(x \cdot e^{\color{blue}{-1 \cdot x}} + e^{-1 \cdot x}\right) + \left(\frac{e^{-x}}{\varepsilon} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} + -1 \cdot e^{-1 \cdot x}\right)\right)\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [<=] 25.1	\[ \frac{\left(x \cdot e^{-1 \cdot x} + \color{blue}{e^{-1 \cdot x} \cdot 1}\right) + \left(\frac{e^{-x}}{\varepsilon} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} + -1 \cdot e^{-1 \cdot x}\right)\right)\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 25.1	\[ \frac{\color{blue}{e^{-1 \cdot x} \cdot \left(x + 1\right)} + \left(\frac{e^{-x}}{\varepsilon} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} + -1 \cdot e^{-1 \cdot x}\right)\right)\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 25.1	\[ \frac{e^{\color{blue}{x \cdot -1}} \cdot \left(x + 1\right) + \left(\frac{e^{-x}}{\varepsilon} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} + -1 \cdot e^{-1 \cdot x}\right)\right)\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [<=] 25.1	\[ \frac{e^{\color{blue}{-x}} \cdot \left(x + 1\right) + \left(\frac{e^{-x}}{\varepsilon} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} + -1 \cdot e^{-1 \cdot x}\right)\right)\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 25.1	\[ \frac{e^{-x} \cdot \left(x + 1\right) + \left(\frac{e^{-x}}{\varepsilon} - \color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} + \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)\right)}\right)}{2} \]

   5. Applied egg-rr 0.0

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

   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

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

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

   4. Simplified 2.9

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

      [Start] 2.9	\[ \frac{e^{\varepsilon \cdot x - x} - -1 \cdot e^{\left(-1 \cdot \varepsilon - 1\right) \cdot x}}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 2.9	\[ \frac{e^{\color{blue}{x \cdot \varepsilon} - x} - -1 \cdot e^{\left(-1 \cdot \varepsilon - 1\right) \cdot x}}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 2.9	\[ \frac{e^{x \cdot \varepsilon - x} - \color{blue}{e^{\left(-1 \cdot \varepsilon - 1\right) \cdot x} \cdot -1}}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 2.9	\[ \frac{e^{x \cdot \varepsilon - x} - \color{blue}{\left(-e^{\left(-1 \cdot \varepsilon - 1\right) \cdot x}\right)}}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 2.9	\[ \frac{e^{x \cdot \varepsilon - x} - \left(-e^{\color{blue}{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-13 [=>] 2.9	\[ \frac{e^{x \cdot \varepsilon - x} - \left(-e^{\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot x - x \cdot 1}}\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 2.9	\[ \frac{e^{x \cdot \varepsilon - x} - \left(-e^{\color{blue}{\left(\varepsilon \cdot -1\right)} \cdot x - x \cdot 1}\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 2.9	\[ \frac{e^{x \cdot \varepsilon - x} - \left(-e^{\color{blue}{\left(-\varepsilon\right)} \cdot x - x \cdot 1}\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 2.9	\[ \frac{e^{x \cdot \varepsilon - x} - \left(-e^{\left(-\varepsilon\right) \cdot x - \color{blue}{x}}\right)}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.6
Cost	13760

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

Alternative 2
Error	0.9
Cost	13636

\[\begin{array}{l} \mathbf{if}\;x \leq 3:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot x}{2}\\ \end{array} \]

Alternative 3
Error	1.7
Cost	13504

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

Alternative 4
Error	1.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot x}{2}\\ \end{array} \]

Alternative 5
Error	1.2
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot x}{2}\\ \end{array} \]

Alternative 6
Error	1.2
Cost	452

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

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))