NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.3% → 99.0%

Time: 25.4s

Alternatives: 16

Speedup: 1.1×

• 38.2% 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

Specification

Math

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

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))

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;
}

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

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;
}

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

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

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

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]

Initial Program: 72.3% accurate, 1.0× speedup

Math

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

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))

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;
}

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

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;
}

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

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

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

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]

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{{e}^{\left(x \cdot eps\_m\right)}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2e-89)
   (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (pow E (* x eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2e-89) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / pow(((double) M_E), (x * eps_m)))) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2e-89) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 / Math.pow(Math.E, (x * eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 2e-89:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 / math.pow(math.e, (x * eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 2e-89)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / (exp(1) ^ Float64(x * eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 2e-89)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / (2.71828182845904523536 ^ (x * eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 2e-89], N[(N[(N[(eps$95$m * N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[E, N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.00000000000000008e-89

   1. Initial program 59.3%

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

   3. Simplified 51.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 27.6%

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

      1. associate-+r+ 68.6%

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

      2. mul-1-neg 68.6%

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

      3. sub-neg 68.6%

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

      4. +-inverses 68.6%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. associate-*r* 68.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + \color{blue}{\left(2 \cdot x\right) \cdot e^{-1 \cdot x}}\right)}{\varepsilon}}{2} \]

      6. distribute-rgt-out 68.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(e^{-1 \cdot x} \cdot \left(2 + 2 \cdot x\right)\right)}}{\varepsilon}}{2} \]

      7. mul-1-neg 68.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(e^{\color{blue}{-x}} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 68.6%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(e^{-x} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}}{2} \]

   if 2.00000000000000008e-89 < eps

   1. Initial program 93.6%

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

   3. Simplified 85.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. exp-1-e 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \frac{1}{\color{blue}{{e}^{\left(x \cdot \varepsilon\right)}}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{{e}^{\left(x \cdot \varepsilon\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1 + e^{x \cdot eps\_m}}{2}\\ t_1 := \frac{\frac{eps\_m \cdot \left(x \cdot eps\_m + 2\right) - x}{eps\_m}}{2}\\ t_2 := \frac{1 + e^{x}}{2}\\ t_3 := \frac{eps\_m \cdot \left(x + \frac{2}{eps\_m}\right)}{2}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(1 - eps\_m\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.0125:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+40}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+66}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+75}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.92 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+81}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+106}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+107}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+108}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+110}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+129}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+131}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+142}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+143}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+156}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+160}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+167}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+172}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+173}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+185}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+187}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+189}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+196}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+200}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+206}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+211}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+229}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+229}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+235}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+236}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+240}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+240}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+265} \lor \neg \left(x \leq 9.5 \cdot 10^{+276}\right) \land \left(x \leq 1.6 \cdot 10^{+281} \lor \neg \left(x \leq 1.15 \cdot 10^{+284}\right) \land \left(x \leq 1.95 \cdot 10^{+293} \lor \neg \left(x \leq 4.1 \cdot 10^{+294}\right) \land x \leq 5 \cdot 10^{+301}\right)\right):\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (exp (* x eps_m))) 2.0))
        (t_1 (/ (/ (- (* eps_m (+ (* x eps_m) 2.0)) x) eps_m) 2.0))
        (t_2 (/ (+ 1.0 (exp x)) 2.0))
        (t_3 (/ (* eps_m (+ x (/ 2.0 eps_m))) 2.0)))
   (if (<= x -1e-280)
     (/ (+ 1.0 (pow E (* x (- 1.0 eps_m)))) 2.0)
     (if (<= x 8.5e-5)
       t_0
       (if (<= x 9e-5)
         1.0
         (if (<= x 0.0125)
           t_3
           (if (<= x 1.52e+35)
             t_0
             (if (<= x 1.65e+40)
               0.0
               (if (<= x 2e+66)
                 t_2
                 (if (<= x 1.5e+75)
                   0.0
                   (if (<= x 1.92e+77)
                     t_1
                     (if (<= x 1.3e+81)
                       0.0
                       (if (<= x 7e+106)
                         t_2
                         (if (<= x 7.5e+107)
                           0.0
                           (if (<= x 1.4e+108)
                             t_3
                             (if (<= x 4.7e+110)
                               0.0
                               (if (<= x 2.4e+129)
                                 t_2
                                 (if (<= x 2e+131)
                                   0.0
                                   (if (<= x 4e+142)
                                     t_2
                                     (if (<= x 3.4e+143)
                                       0.0
                                       (if (<= x 6e+156)
                                         t_2
                                         (if (<= x 4.2e+160)
                                           0.0
                                           (if (<= x 6.6e+167)
                                             t_1
                                             (if (<= x 2.7e+172)
                                               0.0
                                               (if (<= x 9.6e+173)
                                                 t_3
                                                 (if (<= x 1.65e+185)
                                                   0.0
                                                   (if (<= x 1.55e+187)
                                                     t_3
                                                     (if (<= x 6.5e+189)
                                                       0.0
                                                       (if (<= x 4.2e+196)
                                                         t_3
                                                         (if (<= x 9.6e+200)
                                                           0.0
                                                           (if (<= x 1.25e+206)
                                                             t_3
                                                             (if (<=
                                                                  x
                                                                  1.15e+211)
                                                               0.0
                                                               (if (<=
                                                                    x
                                                                    1.45e+229)
                                                                 t_3
                                                                 (if (<=
                                                                      x
                                                                      2e+229)
                                                                   0.0
                                                                   (if (<=
                                                                        x
                                                                        9.5e+235)
                                                                     t_2
                                                                     (if (<=
                                                                          x
                                                                          2e+236)
                                                                       0.0
                                                                       (if (<=
                                                                            x
                                                                            2e+240)
                                                                         t_2
                                                                         (if (<=
                                                                              x
                                                                              4.4e+240)
                                                                           0.0
                                                                           (if (or (<=
                                                                                    x
                                                                                    1.65e+265)
                                                                                   (and (not
                                                                                         (<=
                                                                                          x
                                                                                          9.5e+276))
                                                                                        (or (<=
                                                                                             x
                                                                                             1.6e+281)
                                                                                            (and (not
                                                                                                  (<=
                                                                                                   x
                                                                                                   1.15e+284))
                                                                                                 (or (<=
                                                                                                      x
                                                                                                      1.95e+293)
                                                                                                     (and (not
                                                                                                           (<=
                                                                                                            x
                                                                                                            4.1e+294))
                                                                                                          (<=
                                                                                                           x
                                                                                                           5e+301)))))))
                                                                             t_3
                                                                             0.0)))))))))))))))))))))))))))))))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 + exp((x * eps_m))) / 2.0;
	double t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_2 = (1.0 + exp(x)) / 2.0;
	double t_3 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double tmp;
	if (x <= -1e-280) {
		tmp = (1.0 + pow(((double) M_E), (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 8.5e-5) {
		tmp = t_0;
	} else if (x <= 9e-5) {
		tmp = 1.0;
	} else if (x <= 0.0125) {
		tmp = t_3;
	} else if (x <= 1.52e+35) {
		tmp = t_0;
	} else if (x <= 1.65e+40) {
		tmp = 0.0;
	} else if (x <= 2e+66) {
		tmp = t_2;
	} else if (x <= 1.5e+75) {
		tmp = 0.0;
	} else if (x <= 1.92e+77) {
		tmp = t_1;
	} else if (x <= 1.3e+81) {
		tmp = 0.0;
	} else if (x <= 7e+106) {
		tmp = t_2;
	} else if (x <= 7.5e+107) {
		tmp = 0.0;
	} else if (x <= 1.4e+108) {
		tmp = t_3;
	} else if (x <= 4.7e+110) {
		tmp = 0.0;
	} else if (x <= 2.4e+129) {
		tmp = t_2;
	} else if (x <= 2e+131) {
		tmp = 0.0;
	} else if (x <= 4e+142) {
		tmp = t_2;
	} else if (x <= 3.4e+143) {
		tmp = 0.0;
	} else if (x <= 6e+156) {
		tmp = t_2;
	} else if (x <= 4.2e+160) {
		tmp = 0.0;
	} else if (x <= 6.6e+167) {
		tmp = t_1;
	} else if (x <= 2.7e+172) {
		tmp = 0.0;
	} else if (x <= 9.6e+173) {
		tmp = t_3;
	} else if (x <= 1.65e+185) {
		tmp = 0.0;
	} else if (x <= 1.55e+187) {
		tmp = t_3;
	} else if (x <= 6.5e+189) {
		tmp = 0.0;
	} else if (x <= 4.2e+196) {
		tmp = t_3;
	} else if (x <= 9.6e+200) {
		tmp = 0.0;
	} else if (x <= 1.25e+206) {
		tmp = t_3;
	} else if (x <= 1.15e+211) {
		tmp = 0.0;
	} else if (x <= 1.45e+229) {
		tmp = t_3;
	} else if (x <= 2e+229) {
		tmp = 0.0;
	} else if (x <= 9.5e+235) {
		tmp = t_2;
	} else if (x <= 2e+236) {
		tmp = 0.0;
	} else if (x <= 2e+240) {
		tmp = t_2;
	} else if (x <= 4.4e+240) {
		tmp = 0.0;
	} else if ((x <= 1.65e+265) || (!(x <= 9.5e+276) && ((x <= 1.6e+281) || (!(x <= 1.15e+284) && ((x <= 1.95e+293) || (!(x <= 4.1e+294) && (x <= 5e+301))))))) {
		tmp = t_3;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (1.0 + Math.exp((x * eps_m))) / 2.0;
	double t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_2 = (1.0 + Math.exp(x)) / 2.0;
	double t_3 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double tmp;
	if (x <= -1e-280) {
		tmp = (1.0 + Math.pow(Math.E, (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 8.5e-5) {
		tmp = t_0;
	} else if (x <= 9e-5) {
		tmp = 1.0;
	} else if (x <= 0.0125) {
		tmp = t_3;
	} else if (x <= 1.52e+35) {
		tmp = t_0;
	} else if (x <= 1.65e+40) {
		tmp = 0.0;
	} else if (x <= 2e+66) {
		tmp = t_2;
	} else if (x <= 1.5e+75) {
		tmp = 0.0;
	} else if (x <= 1.92e+77) {
		tmp = t_1;
	} else if (x <= 1.3e+81) {
		tmp = 0.0;
	} else if (x <= 7e+106) {
		tmp = t_2;
	} else if (x <= 7.5e+107) {
		tmp = 0.0;
	} else if (x <= 1.4e+108) {
		tmp = t_3;
	} else if (x <= 4.7e+110) {
		tmp = 0.0;
	} else if (x <= 2.4e+129) {
		tmp = t_2;
	} else if (x <= 2e+131) {
		tmp = 0.0;
	} else if (x <= 4e+142) {
		tmp = t_2;
	} else if (x <= 3.4e+143) {
		tmp = 0.0;
	} else if (x <= 6e+156) {
		tmp = t_2;
	} else if (x <= 4.2e+160) {
		tmp = 0.0;
	} else if (x <= 6.6e+167) {
		tmp = t_1;
	} else if (x <= 2.7e+172) {
		tmp = 0.0;
	} else if (x <= 9.6e+173) {
		tmp = t_3;
	} else if (x <= 1.65e+185) {
		tmp = 0.0;
	} else if (x <= 1.55e+187) {
		tmp = t_3;
	} else if (x <= 6.5e+189) {
		tmp = 0.0;
	} else if (x <= 4.2e+196) {
		tmp = t_3;
	} else if (x <= 9.6e+200) {
		tmp = 0.0;
	} else if (x <= 1.25e+206) {
		tmp = t_3;
	} else if (x <= 1.15e+211) {
		tmp = 0.0;
	} else if (x <= 1.45e+229) {
		tmp = t_3;
	} else if (x <= 2e+229) {
		tmp = 0.0;
	} else if (x <= 9.5e+235) {
		tmp = t_2;
	} else if (x <= 2e+236) {
		tmp = 0.0;
	} else if (x <= 2e+240) {
		tmp = t_2;
	} else if (x <= 4.4e+240) {
		tmp = 0.0;
	} else if ((x <= 1.65e+265) || (!(x <= 9.5e+276) && ((x <= 1.6e+281) || (!(x <= 1.15e+284) && ((x <= 1.95e+293) || (!(x <= 4.1e+294) && (x <= 5e+301))))))) {
		tmp = t_3;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (1.0 + math.exp((x * eps_m))) / 2.0
	t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0
	t_2 = (1.0 + math.exp(x)) / 2.0
	t_3 = (eps_m * (x + (2.0 / eps_m))) / 2.0
	tmp = 0
	if x <= -1e-280:
		tmp = (1.0 + math.pow(math.e, (x * (1.0 - eps_m)))) / 2.0
	elif x <= 8.5e-5:
		tmp = t_0
	elif x <= 9e-5:
		tmp = 1.0
	elif x <= 0.0125:
		tmp = t_3
	elif x <= 1.52e+35:
		tmp = t_0
	elif x <= 1.65e+40:
		tmp = 0.0
	elif x <= 2e+66:
		tmp = t_2
	elif x <= 1.5e+75:
		tmp = 0.0
	elif x <= 1.92e+77:
		tmp = t_1
	elif x <= 1.3e+81:
		tmp = 0.0
	elif x <= 7e+106:
		tmp = t_2
	elif x <= 7.5e+107:
		tmp = 0.0
	elif x <= 1.4e+108:
		tmp = t_3
	elif x <= 4.7e+110:
		tmp = 0.0
	elif x <= 2.4e+129:
		tmp = t_2
	elif x <= 2e+131:
		tmp = 0.0
	elif x <= 4e+142:
		tmp = t_2
	elif x <= 3.4e+143:
		tmp = 0.0
	elif x <= 6e+156:
		tmp = t_2
	elif x <= 4.2e+160:
		tmp = 0.0
	elif x <= 6.6e+167:
		tmp = t_1
	elif x <= 2.7e+172:
		tmp = 0.0
	elif x <= 9.6e+173:
		tmp = t_3
	elif x <= 1.65e+185:
		tmp = 0.0
	elif x <= 1.55e+187:
		tmp = t_3
	elif x <= 6.5e+189:
		tmp = 0.0
	elif x <= 4.2e+196:
		tmp = t_3
	elif x <= 9.6e+200:
		tmp = 0.0
	elif x <= 1.25e+206:
		tmp = t_3
	elif x <= 1.15e+211:
		tmp = 0.0
	elif x <= 1.45e+229:
		tmp = t_3
	elif x <= 2e+229:
		tmp = 0.0
	elif x <= 9.5e+235:
		tmp = t_2
	elif x <= 2e+236:
		tmp = 0.0
	elif x <= 2e+240:
		tmp = t_2
	elif x <= 4.4e+240:
		tmp = 0.0
	elif (x <= 1.65e+265) or (not (x <= 9.5e+276) and ((x <= 1.6e+281) or (not (x <= 1.15e+284) and ((x <= 1.95e+293) or (not (x <= 4.1e+294) and (x <= 5e+301)))))):
		tmp = t_3
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0)
	t_1 = Float64(Float64(Float64(Float64(eps_m * Float64(Float64(x * eps_m) + 2.0)) - x) / eps_m) / 2.0)
	t_2 = Float64(Float64(1.0 + exp(x)) / 2.0)
	t_3 = Float64(Float64(eps_m * Float64(x + Float64(2.0 / eps_m))) / 2.0)
	tmp = 0.0
	if (x <= -1e-280)
		tmp = Float64(Float64(1.0 + (exp(1) ^ Float64(x * Float64(1.0 - eps_m)))) / 2.0);
	elseif (x <= 8.5e-5)
		tmp = t_0;
	elseif (x <= 9e-5)
		tmp = 1.0;
	elseif (x <= 0.0125)
		tmp = t_3;
	elseif (x <= 1.52e+35)
		tmp = t_0;
	elseif (x <= 1.65e+40)
		tmp = 0.0;
	elseif (x <= 2e+66)
		tmp = t_2;
	elseif (x <= 1.5e+75)
		tmp = 0.0;
	elseif (x <= 1.92e+77)
		tmp = t_1;
	elseif (x <= 1.3e+81)
		tmp = 0.0;
	elseif (x <= 7e+106)
		tmp = t_2;
	elseif (x <= 7.5e+107)
		tmp = 0.0;
	elseif (x <= 1.4e+108)
		tmp = t_3;
	elseif (x <= 4.7e+110)
		tmp = 0.0;
	elseif (x <= 2.4e+129)
		tmp = t_2;
	elseif (x <= 2e+131)
		tmp = 0.0;
	elseif (x <= 4e+142)
		tmp = t_2;
	elseif (x <= 3.4e+143)
		tmp = 0.0;
	elseif (x <= 6e+156)
		tmp = t_2;
	elseif (x <= 4.2e+160)
		tmp = 0.0;
	elseif (x <= 6.6e+167)
		tmp = t_1;
	elseif (x <= 2.7e+172)
		tmp = 0.0;
	elseif (x <= 9.6e+173)
		tmp = t_3;
	elseif (x <= 1.65e+185)
		tmp = 0.0;
	elseif (x <= 1.55e+187)
		tmp = t_3;
	elseif (x <= 6.5e+189)
		tmp = 0.0;
	elseif (x <= 4.2e+196)
		tmp = t_3;
	elseif (x <= 9.6e+200)
		tmp = 0.0;
	elseif (x <= 1.25e+206)
		tmp = t_3;
	elseif (x <= 1.15e+211)
		tmp = 0.0;
	elseif (x <= 1.45e+229)
		tmp = t_3;
	elseif (x <= 2e+229)
		tmp = 0.0;
	elseif (x <= 9.5e+235)
		tmp = t_2;
	elseif (x <= 2e+236)
		tmp = 0.0;
	elseif (x <= 2e+240)
		tmp = t_2;
	elseif (x <= 4.4e+240)
		tmp = 0.0;
	elseif ((x <= 1.65e+265) || (!(x <= 9.5e+276) && ((x <= 1.6e+281) || (!(x <= 1.15e+284) && ((x <= 1.95e+293) || (!(x <= 4.1e+294) && (x <= 5e+301)))))))
		tmp = t_3;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (1.0 + exp((x * eps_m))) / 2.0;
	t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	t_2 = (1.0 + exp(x)) / 2.0;
	t_3 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	tmp = 0.0;
	if (x <= -1e-280)
		tmp = (1.0 + (2.71828182845904523536 ^ (x * (1.0 - eps_m)))) / 2.0;
	elseif (x <= 8.5e-5)
		tmp = t_0;
	elseif (x <= 9e-5)
		tmp = 1.0;
	elseif (x <= 0.0125)
		tmp = t_3;
	elseif (x <= 1.52e+35)
		tmp = t_0;
	elseif (x <= 1.65e+40)
		tmp = 0.0;
	elseif (x <= 2e+66)
		tmp = t_2;
	elseif (x <= 1.5e+75)
		tmp = 0.0;
	elseif (x <= 1.92e+77)
		tmp = t_1;
	elseif (x <= 1.3e+81)
		tmp = 0.0;
	elseif (x <= 7e+106)
		tmp = t_2;
	elseif (x <= 7.5e+107)
		tmp = 0.0;
	elseif (x <= 1.4e+108)
		tmp = t_3;
	elseif (x <= 4.7e+110)
		tmp = 0.0;
	elseif (x <= 2.4e+129)
		tmp = t_2;
	elseif (x <= 2e+131)
		tmp = 0.0;
	elseif (x <= 4e+142)
		tmp = t_2;
	elseif (x <= 3.4e+143)
		tmp = 0.0;
	elseif (x <= 6e+156)
		tmp = t_2;
	elseif (x <= 4.2e+160)
		tmp = 0.0;
	elseif (x <= 6.6e+167)
		tmp = t_1;
	elseif (x <= 2.7e+172)
		tmp = 0.0;
	elseif (x <= 9.6e+173)
		tmp = t_3;
	elseif (x <= 1.65e+185)
		tmp = 0.0;
	elseif (x <= 1.55e+187)
		tmp = t_3;
	elseif (x <= 6.5e+189)
		tmp = 0.0;
	elseif (x <= 4.2e+196)
		tmp = t_3;
	elseif (x <= 9.6e+200)
		tmp = 0.0;
	elseif (x <= 1.25e+206)
		tmp = t_3;
	elseif (x <= 1.15e+211)
		tmp = 0.0;
	elseif (x <= 1.45e+229)
		tmp = t_3;
	elseif (x <= 2e+229)
		tmp = 0.0;
	elseif (x <= 9.5e+235)
		tmp = t_2;
	elseif (x <= 2e+236)
		tmp = 0.0;
	elseif (x <= 2e+240)
		tmp = t_2;
	elseif (x <= 4.4e+240)
		tmp = 0.0;
	elseif ((x <= 1.65e+265) || (~((x <= 9.5e+276)) && ((x <= 1.6e+281) || (~((x <= 1.15e+284)) && ((x <= 1.95e+293) || (~((x <= 4.1e+294)) && (x <= 5e+301)))))))
		tmp = t_3;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(eps$95$m * N[(N[(x * eps$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(eps$95$m * N[(x + N[(2.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1e-280], N[(N[(1.0 + N[Power[E, N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.5e-5], t$95$0, If[LessEqual[x, 9e-5], 1.0, If[LessEqual[x, 0.0125], t$95$3, If[LessEqual[x, 1.52e+35], t$95$0, If[LessEqual[x, 1.65e+40], 0.0, If[LessEqual[x, 2e+66], t$95$2, If[LessEqual[x, 1.5e+75], 0.0, If[LessEqual[x, 1.92e+77], t$95$1, If[LessEqual[x, 1.3e+81], 0.0, If[LessEqual[x, 7e+106], t$95$2, If[LessEqual[x, 7.5e+107], 0.0, If[LessEqual[x, 1.4e+108], t$95$3, If[LessEqual[x, 4.7e+110], 0.0, If[LessEqual[x, 2.4e+129], t$95$2, If[LessEqual[x, 2e+131], 0.0, If[LessEqual[x, 4e+142], t$95$2, If[LessEqual[x, 3.4e+143], 0.0, If[LessEqual[x, 6e+156], t$95$2, If[LessEqual[x, 4.2e+160], 0.0, If[LessEqual[x, 6.6e+167], t$95$1, If[LessEqual[x, 2.7e+172], 0.0, If[LessEqual[x, 9.6e+173], t$95$3, If[LessEqual[x, 1.65e+185], 0.0, If[LessEqual[x, 1.55e+187], t$95$3, If[LessEqual[x, 6.5e+189], 0.0, If[LessEqual[x, 4.2e+196], t$95$3, If[LessEqual[x, 9.6e+200], 0.0, If[LessEqual[x, 1.25e+206], t$95$3, If[LessEqual[x, 1.15e+211], 0.0, If[LessEqual[x, 1.45e+229], t$95$3, If[LessEqual[x, 2e+229], 0.0, If[LessEqual[x, 9.5e+235], t$95$2, If[LessEqual[x, 2e+236], 0.0, If[LessEqual[x, 2e+240], t$95$2, If[LessEqual[x, 4.4e+240], 0.0, If[Or[LessEqual[x, 1.65e+265], And[N[Not[LessEqual[x, 9.5e+276]], $MachinePrecision], Or[LessEqual[x, 1.6e+281], And[N[Not[LessEqual[x, 1.15e+284]], $MachinePrecision], Or[LessEqual[x, 1.95e+293], And[N[Not[LessEqual[x, 4.1e+294]], $MachinePrecision], LessEqual[x, 5e+301]]]]]]], t$95$3, 0.0]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if x < -9.9999999999999996e-281

   1. Initial program 71.6%

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

   3. Simplified 60.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 99.8%

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

   6. Taylor expanded in x around 0 68.9%

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

      1. sub-neg 68.9%

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

      2. metadata-eval 68.9%

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

      3. distribute-rgt-in 68.9%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 73.5%

         \[\leadsto \frac{e^{\varepsilon \cdot \color{blue}{\sqrt{x \cdot x}} + -1 \cdot x} + 1}{2} \]

      6. sqr-neg 73.5%

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

      7. sqrt-unprod 77.4%

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

      8. add-sqr-sqrt 77.4%

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

      9. neg-mul-1 77.4%

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

      10. *-un-lft-identity 77.4%

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

      11. distribute-rgt-in 77.4%

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

      12. +-commutative 77.4%

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

      13. *-commutative 77.4%

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

      14. distribute-rgt-neg-out 77.4%

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

      15. neg-sub0 77.4%

          \[\leadsto \frac{e^{\color{blue}{0 - \left(1 + \varepsilon\right) \cdot x}} + 1}{2} \]

      16. add-sqr-sqrt 0.0%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} + 1}{2} \]

      17. sqrt-unprod 66.9%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\sqrt{x \cdot x}}} + 1}{2} \]

      18. sqr-neg 66.9%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}} + 1}{2} \]

      19. sqrt-unprod 68.8%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}} + 1}{2} \]

      20. add-sqr-sqrt 68.8%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(-x\right)}} + 1}{2} \]

      21. *-commutative 68.8%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(-x\right) \cdot \left(1 + \varepsilon\right)}} + 1}{2} \]

      22. +-commutative 68.8%

          \[\leadsto \frac{e^{0 - \left(-x\right) \cdot \color{blue}{\left(\varepsilon + 1\right)}} + 1}{2} \]

      23. distribute-rgt-in 68.8%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(\varepsilon \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} + 1}{2} \]

   8. Applied egg-rr 77.4%

      \[\leadsto \frac{e^{\color{blue}{0 - x \cdot \left(\varepsilon + -1\right)}} + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 77.4%

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

      2. distribute-rgt-neg-in 77.4%

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

      3. +-commutative 77.4%

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

      4. distribute-neg-in 77.4%

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

      5. metadata-eval 77.4%

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

      6. sub-neg 77.4%

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

   10. Simplified 77.4%

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

       1. *-commutative 77.4%

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

       2. *-un-lft-identity 77.4%

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

       3. exp-prod 77.4%

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

       4. e-exp-1 77.4%

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

       5. *-commutative 77.4%

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

   12. Applied egg-rr 77.4%

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

   if -9.9999999999999996e-281 < x < 8.500000000000001e-5 or 0.012500000000000001 < x < 1.5200000000000001e35

   1. Initial program 55.0%

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

   3. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 99.5%

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

   6. Taylor expanded in x around 0 88.4%

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

   7. Taylor expanded in eps around inf 88.9%

      \[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot x}} + 1}{2} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

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

   9. Simplified 88.9%

      \[\leadsto \frac{e^{\color{blue}{x \cdot \varepsilon}} + 1}{2} \]

   if 8.500000000000001e-5 < x < 9.00000000000000057e-5

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around 0 60.9%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 9.00000000000000057e-5 < x < 0.012500000000000001 or 7.4999999999999996e107 < x < 1.3999999999999999e108 or 2.7e172 < x < 9.5999999999999997e173 or 1.65000000000000006e185 < x < 1.55000000000000006e187 or 6.50000000000000027e189 < x < 4.20000000000000029e196 or 9.6000000000000003e200 < x < 1.25e206 or 1.15000000000000005e211 < x < 1.44999999999999991e229 or 4.4000000000000003e240 < x < 1.6499999999999999e265 or 9.50000000000000013e276 < x < 1.6000000000000001e281 or 1.14999999999999992e284 < x < 1.95000000000000014e293 or 4.0999999999999998e294 < x < 5.0000000000000004e301

   1. Initial program 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in x around 0 69.8%

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

   6. Taylor expanded in x around 0 69.1%

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

      1. associate-*r* 69.1%

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

      2. neg-mul-1 69.1%

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

      3. *-commutative 69.1%

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

   8. Simplified 69.1%

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

   9. Taylor expanded in eps around inf 71.2%

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

       1. associate-*r/ 71.2%

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

       2. metadata-eval 71.2%

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

   11. Simplified 71.2%

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

   if 1.5200000000000001e35 < x < 1.6499999999999999e40 or 1.99999999999999989e66 < x < 1.5e75 or 1.92e77 < x < 1.29999999999999996e81 or 6.99999999999999962e106 < x < 7.4999999999999996e107 or 1.3999999999999999e108 < x < 4.6999999999999998e110 or 2.3999999999999999e129 < x < 1.9999999999999998e131 or 4.0000000000000002e142 < x < 3.39999999999999982e143 or 5.9999999999999999e156 < x < 4.19999999999999993e160 or 6.60000000000000036e167 < x < 2.7e172 or 9.5999999999999997e173 < x < 1.65000000000000006e185 or 1.55000000000000006e187 < x < 6.50000000000000027e189 or 4.20000000000000029e196 < x < 9.6000000000000003e200 or 1.25e206 < x < 1.15000000000000005e211 or 1.44999999999999991e229 < x < 2e229 or 9.49999999999999966e235 < x < 2.00000000000000011e236 or 2.00000000000000003e240 < x < 4.4000000000000003e240 or 1.6499999999999999e265 < x < 9.50000000000000013e276 or 1.6000000000000001e281 < x < 1.14999999999999992e284 or 1.95000000000000014e293 < x < 4.0999999999999998e294 or 5.0000000000000004e301 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 100.0%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}}}{2} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. rec-exp 100.0%

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

      4. sub-neg 100.0%

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

      5. div-sub 100.0%

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

      6. mul-1-neg 100.0%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 100.0%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 100.0%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 1.6499999999999999e40 < x < 1.99999999999999989e66 or 1.29999999999999996e81 < x < 6.99999999999999962e106 or 4.6999999999999998e110 < x < 2.3999999999999999e129 or 1.9999999999999998e131 < x < 4.0000000000000002e142 or 3.39999999999999982e143 < x < 5.9999999999999999e156 or 2e229 < x < 9.49999999999999966e235 or 2.00000000000000011e236 < x < 2.00000000000000003e240

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 57.9%

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

      1. sub-neg 57.9%

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

      2. metadata-eval 57.9%

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

      3. distribute-rgt-in 57.9%

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

      4. add-sqr-sqrt 57.9%

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

      5. sqrt-unprod 57.9%

         \[\leadsto \frac{e^{\varepsilon \cdot \color{blue}{\sqrt{x \cdot x}} + -1 \cdot x} + 1}{2} \]

      6. sqr-neg 57.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 45.2%

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

      9. neg-mul-1 45.2%

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

      10. *-un-lft-identity 45.2%

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

      11. distribute-rgt-in 45.2%

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

      12. +-commutative 45.2%

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

      13. *-commutative 45.2%

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

      14. distribute-rgt-neg-out 45.2%

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

      15. neg-sub0 45.2%

          \[\leadsto \frac{e^{\color{blue}{0 - \left(1 + \varepsilon\right) \cdot x}} + 1}{2} \]

      16. add-sqr-sqrt 45.2%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} + 1}{2} \]

      17. sqrt-unprod 45.2%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\sqrt{x \cdot x}}} + 1}{2} \]

      18. sqr-neg 45.2%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}} + 1}{2} \]

      19. sqrt-unprod 0.0%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}} + 1}{2} \]

      20. add-sqr-sqrt 57.9%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(-x\right)}} + 1}{2} \]

      21. *-commutative 57.9%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(-x\right) \cdot \left(1 + \varepsilon\right)}} + 1}{2} \]

      22. +-commutative 57.9%

          \[\leadsto \frac{e^{0 - \left(-x\right) \cdot \color{blue}{\left(\varepsilon + 1\right)}} + 1}{2} \]

      23. distribute-rgt-in 57.9%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(\varepsilon \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} + 1}{2} \]

   8. Applied egg-rr 45.2%

      \[\leadsto \frac{e^{\color{blue}{0 - x \cdot \left(\varepsilon + -1\right)}} + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 45.2%

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

      2. distribute-rgt-neg-in 45.2%

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

      3. +-commutative 45.2%

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

      4. distribute-neg-in 45.2%

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

      5. metadata-eval 45.2%

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

      6. sub-neg 45.2%

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

   10. Simplified 45.2%

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

   11. Taylor expanded in eps around 0 100.0%

       \[\leadsto \frac{\color{blue}{e^{x}} + 1}{2} \]

   if 1.5e75 < x < 1.92e77 or 4.19999999999999993e160 < x < 6.60000000000000036e167

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around 0 0.1%

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

      1. associate-*r* 0.1%

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

      2. neg-mul-1 0.1%

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

      3. *-commutative 0.1%

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

   8. Simplified 0.1%

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

   9. Taylor expanded in eps around 0 0.0%

      \[\leadsto \frac{\color{blue}{\frac{-1 \cdot x + \varepsilon \cdot \left(2 + \varepsilon \cdot x\right)}{\varepsilon}}}{2} \]
3. Recombined 7 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(1 - \varepsilon\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.0125:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{+35}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+40}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+66}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+75}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.92 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+81}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+106}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+107}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+108}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+110}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+131}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+142}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+143}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+156}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+160}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+172}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+173}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+185}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+187}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+189}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+196}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+200}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+206}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+211}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+229}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+229}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+235}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+236}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+240}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+265} \lor \neg \left(x \leq 9.5 \cdot 10^{+276}\right) \land \left(x \leq 1.6 \cdot 10^{+281} \lor \neg \left(x \leq 1.15 \cdot 10^{+284}\right) \land \left(x \leq 1.95 \cdot 10^{+293} \lor \neg \left(x \leq 4.1 \cdot 10^{+294}\right) \land x \leq 5 \cdot 10^{+301}\right)\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1 + e^{x \cdot eps\_m}}{2}\\ t_1 := \frac{\frac{eps\_m \cdot \left(x \cdot eps\_m + 2\right) - x}{eps\_m}}{2}\\ t_2 := \frac{1 + e^{x}}{2}\\ t_3 := \frac{eps\_m \cdot \left(x + \frac{2}{eps\_m}\right)}{2}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.0125:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+39}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+65}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+74}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.58 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+80}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+107}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+107}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+107}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 10^{+110}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+129}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+131}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+142}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+150}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+157}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+162}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+172}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+174}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+183}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+186}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+189}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+198}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+198}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+205}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+207}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+228}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+229}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+236}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+236}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+240}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+240}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+266} \lor \neg \left(x \leq 2.3 \cdot 10^{+274}\right) \land \left(x \leq 5.6 \cdot 10^{+280} \lor \neg \left(x \leq 1.3 \cdot 10^{+284}\right) \land \left(x \leq 2.55 \cdot 10^{+294} \lor \neg \left(x \leq 2.7 \cdot 10^{+295}\right) \land x \leq 5 \cdot 10^{+301}\right)\right):\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (exp (* x eps_m))) 2.0))
        (t_1 (/ (/ (- (* eps_m (+ (* x eps_m) 2.0)) x) eps_m) 2.0))
        (t_2 (/ (+ 1.0 (exp x)) 2.0))
        (t_3 (/ (* eps_m (+ x (/ 2.0 eps_m))) 2.0)))
   (if (<= x -1e-280)
     (/ (+ 1.0 (exp (* x (- 1.0 eps_m)))) 2.0)
     (if (<= x 8.5e-5)
       t_0
       (if (<= x 9e-5)
         1.0
         (if (<= x 0.0125)
           t_3
           (if (<= x 3.9e+36)
             t_0
             (if (<= x 5e+39)
               0.0
               (if (<= x 2e+65)
                 t_2
                 (if (<= x 5e+74)
                   0.0
                   (if (<= x 1.58e+78)
                     t_1
                     (if (<= x 5e+80)
                       0.0
                       (if (<= x 1.45e+107)
                         t_2
                         (if (<= x 5e+107)
                           0.0
                           (if (<= x 8e+107)
                             t_3
                             (if (<= x 1e+110)
                               0.0
                               (if (<= x 4.5e+129)
                                 t_2
                                 (if (<= x 2e+131)
                                   0.0
                                   (if (<= x 6e+142)
                                     t_2
                                     (if (<= x 2.25e+150)
                                       0.0
                                       (if (<= x 8.5e+157)
                                         t_2
                                         (if (<= x 2e+162)
                                           0.0
                                           (if (<= x 6e+168)
                                             t_1
                                             (if (<= x 4.2e+172)
                                               0.0
                                               (if (<= x 4e+174)
                                                 t_3
                                                 (if (<= x 3.4e+183)
                                                   0.0
                                                   (if (<= x 6e+186)
                                                     t_3
                                                     (if (<= x 4.6e+189)
                                                       0.0
                                                       (if (<= x 3.3e+198)
                                                         t_3
                                                         (if (<= x 8e+198)
                                                           0.0
                                                           (if (<= x 7e+205)
                                                             t_3
                                                             (if (<=
                                                                  x
                                                                  7.5e+207)
                                                               0.0
                                                               (if (<=
                                                                    x
                                                                    6.5e+228)
                                                                 t_3
                                                                 (if (<=
                                                                      x
                                                                      2e+229)
                                                                   0.0
                                                                   (if (<=
                                                                        x
                                                                        1.45e+236)
                                                                     t_2
                                                                     (if (<=
                                                                          x
                                                                          2e+236)
                                                                       0.0
                                                                       (if (<=
                                                                            x
                                                                            3e+240)
                                                                         t_2
                                                                         (if (<=
                                                                              x
                                                                              6.8e+240)
                                                                           0.0
                                                                           (if (or (<=
                                                                                    x
                                                                                    3.6e+266)
                                                                                   (and (not
                                                                                         (<=
                                                                                          x
                                                                                          2.3e+274))
                                                                                        (or (<=
                                                                                             x
                                                                                             5.6e+280)
                                                                                            (and (not
                                                                                                  (<=
                                                                                                   x
                                                                                                   1.3e+284))
                                                                                                 (or (<=
                                                                                                      x
                                                                                                      2.55e+294)
                                                                                                     (and (not
                                                                                                           (<=
                                                                                                            x
                                                                                                            2.7e+295))
                                                                                                          (<=
                                                                                                           x
                                                                                                           5e+301)))))))
                                                                             t_3
                                                                             0.0)))))))))))))))))))))))))))))))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 + exp((x * eps_m))) / 2.0;
	double t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_2 = (1.0 + exp(x)) / 2.0;
	double t_3 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double tmp;
	if (x <= -1e-280) {
		tmp = (1.0 + exp((x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 8.5e-5) {
		tmp = t_0;
	} else if (x <= 9e-5) {
		tmp = 1.0;
	} else if (x <= 0.0125) {
		tmp = t_3;
	} else if (x <= 3.9e+36) {
		tmp = t_0;
	} else if (x <= 5e+39) {
		tmp = 0.0;
	} else if (x <= 2e+65) {
		tmp = t_2;
	} else if (x <= 5e+74) {
		tmp = 0.0;
	} else if (x <= 1.58e+78) {
		tmp = t_1;
	} else if (x <= 5e+80) {
		tmp = 0.0;
	} else if (x <= 1.45e+107) {
		tmp = t_2;
	} else if (x <= 5e+107) {
		tmp = 0.0;
	} else if (x <= 8e+107) {
		tmp = t_3;
	} else if (x <= 1e+110) {
		tmp = 0.0;
	} else if (x <= 4.5e+129) {
		tmp = t_2;
	} else if (x <= 2e+131) {
		tmp = 0.0;
	} else if (x <= 6e+142) {
		tmp = t_2;
	} else if (x <= 2.25e+150) {
		tmp = 0.0;
	} else if (x <= 8.5e+157) {
		tmp = t_2;
	} else if (x <= 2e+162) {
		tmp = 0.0;
	} else if (x <= 6e+168) {
		tmp = t_1;
	} else if (x <= 4.2e+172) {
		tmp = 0.0;
	} else if (x <= 4e+174) {
		tmp = t_3;
	} else if (x <= 3.4e+183) {
		tmp = 0.0;
	} else if (x <= 6e+186) {
		tmp = t_3;
	} else if (x <= 4.6e+189) {
		tmp = 0.0;
	} else if (x <= 3.3e+198) {
		tmp = t_3;
	} else if (x <= 8e+198) {
		tmp = 0.0;
	} else if (x <= 7e+205) {
		tmp = t_3;
	} else if (x <= 7.5e+207) {
		tmp = 0.0;
	} else if (x <= 6.5e+228) {
		tmp = t_3;
	} else if (x <= 2e+229) {
		tmp = 0.0;
	} else if (x <= 1.45e+236) {
		tmp = t_2;
	} else if (x <= 2e+236) {
		tmp = 0.0;
	} else if (x <= 3e+240) {
		tmp = t_2;
	} else if (x <= 6.8e+240) {
		tmp = 0.0;
	} else if ((x <= 3.6e+266) || (!(x <= 2.3e+274) && ((x <= 5.6e+280) || (!(x <= 1.3e+284) && ((x <= 2.55e+294) || (!(x <= 2.7e+295) && (x <= 5e+301))))))) {
		tmp = t_3;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (1.0d0 + exp((x * eps_m))) / 2.0d0
    t_1 = (((eps_m * ((x * eps_m) + 2.0d0)) - x) / eps_m) / 2.0d0
    t_2 = (1.0d0 + exp(x)) / 2.0d0
    t_3 = (eps_m * (x + (2.0d0 / eps_m))) / 2.0d0
    if (x <= (-1d-280)) then
        tmp = (1.0d0 + exp((x * (1.0d0 - eps_m)))) / 2.0d0
    else if (x <= 8.5d-5) then
        tmp = t_0
    else if (x <= 9d-5) then
        tmp = 1.0d0
    else if (x <= 0.0125d0) then
        tmp = t_3
    else if (x <= 3.9d+36) then
        tmp = t_0
    else if (x <= 5d+39) then
        tmp = 0.0d0
    else if (x <= 2d+65) then
        tmp = t_2
    else if (x <= 5d+74) then
        tmp = 0.0d0
    else if (x <= 1.58d+78) then
        tmp = t_1
    else if (x <= 5d+80) then
        tmp = 0.0d0
    else if (x <= 1.45d+107) then
        tmp = t_2
    else if (x <= 5d+107) then
        tmp = 0.0d0
    else if (x <= 8d+107) then
        tmp = t_3
    else if (x <= 1d+110) then
        tmp = 0.0d0
    else if (x <= 4.5d+129) then
        tmp = t_2
    else if (x <= 2d+131) then
        tmp = 0.0d0
    else if (x <= 6d+142) then
        tmp = t_2
    else if (x <= 2.25d+150) then
        tmp = 0.0d0
    else if (x <= 8.5d+157) then
        tmp = t_2
    else if (x <= 2d+162) then
        tmp = 0.0d0
    else if (x <= 6d+168) then
        tmp = t_1
    else if (x <= 4.2d+172) then
        tmp = 0.0d0
    else if (x <= 4d+174) then
        tmp = t_3
    else if (x <= 3.4d+183) then
        tmp = 0.0d0
    else if (x <= 6d+186) then
        tmp = t_3
    else if (x <= 4.6d+189) then
        tmp = 0.0d0
    else if (x <= 3.3d+198) then
        tmp = t_3
    else if (x <= 8d+198) then
        tmp = 0.0d0
    else if (x <= 7d+205) then
        tmp = t_3
    else if (x <= 7.5d+207) then
        tmp = 0.0d0
    else if (x <= 6.5d+228) then
        tmp = t_3
    else if (x <= 2d+229) then
        tmp = 0.0d0
    else if (x <= 1.45d+236) then
        tmp = t_2
    else if (x <= 2d+236) then
        tmp = 0.0d0
    else if (x <= 3d+240) then
        tmp = t_2
    else if (x <= 6.8d+240) then
        tmp = 0.0d0
    else if ((x <= 3.6d+266) .or. (.not. (x <= 2.3d+274)) .and. (x <= 5.6d+280) .or. (.not. (x <= 1.3d+284)) .and. (x <= 2.55d+294) .or. (.not. (x <= 2.7d+295)) .and. (x <= 5d+301)) then
        tmp = t_3
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (1.0 + Math.exp((x * eps_m))) / 2.0;
	double t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_2 = (1.0 + Math.exp(x)) / 2.0;
	double t_3 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double tmp;
	if (x <= -1e-280) {
		tmp = (1.0 + Math.exp((x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 8.5e-5) {
		tmp = t_0;
	} else if (x <= 9e-5) {
		tmp = 1.0;
	} else if (x <= 0.0125) {
		tmp = t_3;
	} else if (x <= 3.9e+36) {
		tmp = t_0;
	} else if (x <= 5e+39) {
		tmp = 0.0;
	} else if (x <= 2e+65) {
		tmp = t_2;
	} else if (x <= 5e+74) {
		tmp = 0.0;
	} else if (x <= 1.58e+78) {
		tmp = t_1;
	} else if (x <= 5e+80) {
		tmp = 0.0;
	} else if (x <= 1.45e+107) {
		tmp = t_2;
	} else if (x <= 5e+107) {
		tmp = 0.0;
	} else if (x <= 8e+107) {
		tmp = t_3;
	} else if (x <= 1e+110) {
		tmp = 0.0;
	} else if (x <= 4.5e+129) {
		tmp = t_2;
	} else if (x <= 2e+131) {
		tmp = 0.0;
	} else if (x <= 6e+142) {
		tmp = t_2;
	} else if (x <= 2.25e+150) {
		tmp = 0.0;
	} else if (x <= 8.5e+157) {
		tmp = t_2;
	} else if (x <= 2e+162) {
		tmp = 0.0;
	} else if (x <= 6e+168) {
		tmp = t_1;
	} else if (x <= 4.2e+172) {
		tmp = 0.0;
	} else if (x <= 4e+174) {
		tmp = t_3;
	} else if (x <= 3.4e+183) {
		tmp = 0.0;
	} else if (x <= 6e+186) {
		tmp = t_3;
	} else if (x <= 4.6e+189) {
		tmp = 0.0;
	} else if (x <= 3.3e+198) {
		tmp = t_3;
	} else if (x <= 8e+198) {
		tmp = 0.0;
	} else if (x <= 7e+205) {
		tmp = t_3;
	} else if (x <= 7.5e+207) {
		tmp = 0.0;
	} else if (x <= 6.5e+228) {
		tmp = t_3;
	} else if (x <= 2e+229) {
		tmp = 0.0;
	} else if (x <= 1.45e+236) {
		tmp = t_2;
	} else if (x <= 2e+236) {
		tmp = 0.0;
	} else if (x <= 3e+240) {
		tmp = t_2;
	} else if (x <= 6.8e+240) {
		tmp = 0.0;
	} else if ((x <= 3.6e+266) || (!(x <= 2.3e+274) && ((x <= 5.6e+280) || (!(x <= 1.3e+284) && ((x <= 2.55e+294) || (!(x <= 2.7e+295) && (x <= 5e+301))))))) {
		tmp = t_3;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (1.0 + math.exp((x * eps_m))) / 2.0
	t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0
	t_2 = (1.0 + math.exp(x)) / 2.0
	t_3 = (eps_m * (x + (2.0 / eps_m))) / 2.0
	tmp = 0
	if x <= -1e-280:
		tmp = (1.0 + math.exp((x * (1.0 - eps_m)))) / 2.0
	elif x <= 8.5e-5:
		tmp = t_0
	elif x <= 9e-5:
		tmp = 1.0
	elif x <= 0.0125:
		tmp = t_3
	elif x <= 3.9e+36:
		tmp = t_0
	elif x <= 5e+39:
		tmp = 0.0
	elif x <= 2e+65:
		tmp = t_2
	elif x <= 5e+74:
		tmp = 0.0
	elif x <= 1.58e+78:
		tmp = t_1
	elif x <= 5e+80:
		tmp = 0.0
	elif x <= 1.45e+107:
		tmp = t_2
	elif x <= 5e+107:
		tmp = 0.0
	elif x <= 8e+107:
		tmp = t_3
	elif x <= 1e+110:
		tmp = 0.0
	elif x <= 4.5e+129:
		tmp = t_2
	elif x <= 2e+131:
		tmp = 0.0
	elif x <= 6e+142:
		tmp = t_2
	elif x <= 2.25e+150:
		tmp = 0.0
	elif x <= 8.5e+157:
		tmp = t_2
	elif x <= 2e+162:
		tmp = 0.0
	elif x <= 6e+168:
		tmp = t_1
	elif x <= 4.2e+172:
		tmp = 0.0
	elif x <= 4e+174:
		tmp = t_3
	elif x <= 3.4e+183:
		tmp = 0.0
	elif x <= 6e+186:
		tmp = t_3
	elif x <= 4.6e+189:
		tmp = 0.0
	elif x <= 3.3e+198:
		tmp = t_3
	elif x <= 8e+198:
		tmp = 0.0
	elif x <= 7e+205:
		tmp = t_3
	elif x <= 7.5e+207:
		tmp = 0.0
	elif x <= 6.5e+228:
		tmp = t_3
	elif x <= 2e+229:
		tmp = 0.0
	elif x <= 1.45e+236:
		tmp = t_2
	elif x <= 2e+236:
		tmp = 0.0
	elif x <= 3e+240:
		tmp = t_2
	elif x <= 6.8e+240:
		tmp = 0.0
	elif (x <= 3.6e+266) or (not (x <= 2.3e+274) and ((x <= 5.6e+280) or (not (x <= 1.3e+284) and ((x <= 2.55e+294) or (not (x <= 2.7e+295) and (x <= 5e+301)))))):
		tmp = t_3
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0)
	t_1 = Float64(Float64(Float64(Float64(eps_m * Float64(Float64(x * eps_m) + 2.0)) - x) / eps_m) / 2.0)
	t_2 = Float64(Float64(1.0 + exp(x)) / 2.0)
	t_3 = Float64(Float64(eps_m * Float64(x + Float64(2.0 / eps_m))) / 2.0)
	tmp = 0.0
	if (x <= -1e-280)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(1.0 - eps_m)))) / 2.0);
	elseif (x <= 8.5e-5)
		tmp = t_0;
	elseif (x <= 9e-5)
		tmp = 1.0;
	elseif (x <= 0.0125)
		tmp = t_3;
	elseif (x <= 3.9e+36)
		tmp = t_0;
	elseif (x <= 5e+39)
		tmp = 0.0;
	elseif (x <= 2e+65)
		tmp = t_2;
	elseif (x <= 5e+74)
		tmp = 0.0;
	elseif (x <= 1.58e+78)
		tmp = t_1;
	elseif (x <= 5e+80)
		tmp = 0.0;
	elseif (x <= 1.45e+107)
		tmp = t_2;
	elseif (x <= 5e+107)
		tmp = 0.0;
	elseif (x <= 8e+107)
		tmp = t_3;
	elseif (x <= 1e+110)
		tmp = 0.0;
	elseif (x <= 4.5e+129)
		tmp = t_2;
	elseif (x <= 2e+131)
		tmp = 0.0;
	elseif (x <= 6e+142)
		tmp = t_2;
	elseif (x <= 2.25e+150)
		tmp = 0.0;
	elseif (x <= 8.5e+157)
		tmp = t_2;
	elseif (x <= 2e+162)
		tmp = 0.0;
	elseif (x <= 6e+168)
		tmp = t_1;
	elseif (x <= 4.2e+172)
		tmp = 0.0;
	elseif (x <= 4e+174)
		tmp = t_3;
	elseif (x <= 3.4e+183)
		tmp = 0.0;
	elseif (x <= 6e+186)
		tmp = t_3;
	elseif (x <= 4.6e+189)
		tmp = 0.0;
	elseif (x <= 3.3e+198)
		tmp = t_3;
	elseif (x <= 8e+198)
		tmp = 0.0;
	elseif (x <= 7e+205)
		tmp = t_3;
	elseif (x <= 7.5e+207)
		tmp = 0.0;
	elseif (x <= 6.5e+228)
		tmp = t_3;
	elseif (x <= 2e+229)
		tmp = 0.0;
	elseif (x <= 1.45e+236)
		tmp = t_2;
	elseif (x <= 2e+236)
		tmp = 0.0;
	elseif (x <= 3e+240)
		tmp = t_2;
	elseif (x <= 6.8e+240)
		tmp = 0.0;
	elseif ((x <= 3.6e+266) || (!(x <= 2.3e+274) && ((x <= 5.6e+280) || (!(x <= 1.3e+284) && ((x <= 2.55e+294) || (!(x <= 2.7e+295) && (x <= 5e+301)))))))
		tmp = t_3;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (1.0 + exp((x * eps_m))) / 2.0;
	t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	t_2 = (1.0 + exp(x)) / 2.0;
	t_3 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	tmp = 0.0;
	if (x <= -1e-280)
		tmp = (1.0 + exp((x * (1.0 - eps_m)))) / 2.0;
	elseif (x <= 8.5e-5)
		tmp = t_0;
	elseif (x <= 9e-5)
		tmp = 1.0;
	elseif (x <= 0.0125)
		tmp = t_3;
	elseif (x <= 3.9e+36)
		tmp = t_0;
	elseif (x <= 5e+39)
		tmp = 0.0;
	elseif (x <= 2e+65)
		tmp = t_2;
	elseif (x <= 5e+74)
		tmp = 0.0;
	elseif (x <= 1.58e+78)
		tmp = t_1;
	elseif (x <= 5e+80)
		tmp = 0.0;
	elseif (x <= 1.45e+107)
		tmp = t_2;
	elseif (x <= 5e+107)
		tmp = 0.0;
	elseif (x <= 8e+107)
		tmp = t_3;
	elseif (x <= 1e+110)
		tmp = 0.0;
	elseif (x <= 4.5e+129)
		tmp = t_2;
	elseif (x <= 2e+131)
		tmp = 0.0;
	elseif (x <= 6e+142)
		tmp = t_2;
	elseif (x <= 2.25e+150)
		tmp = 0.0;
	elseif (x <= 8.5e+157)
		tmp = t_2;
	elseif (x <= 2e+162)
		tmp = 0.0;
	elseif (x <= 6e+168)
		tmp = t_1;
	elseif (x <= 4.2e+172)
		tmp = 0.0;
	elseif (x <= 4e+174)
		tmp = t_3;
	elseif (x <= 3.4e+183)
		tmp = 0.0;
	elseif (x <= 6e+186)
		tmp = t_3;
	elseif (x <= 4.6e+189)
		tmp = 0.0;
	elseif (x <= 3.3e+198)
		tmp = t_3;
	elseif (x <= 8e+198)
		tmp = 0.0;
	elseif (x <= 7e+205)
		tmp = t_3;
	elseif (x <= 7.5e+207)
		tmp = 0.0;
	elseif (x <= 6.5e+228)
		tmp = t_3;
	elseif (x <= 2e+229)
		tmp = 0.0;
	elseif (x <= 1.45e+236)
		tmp = t_2;
	elseif (x <= 2e+236)
		tmp = 0.0;
	elseif (x <= 3e+240)
		tmp = t_2;
	elseif (x <= 6.8e+240)
		tmp = 0.0;
	elseif ((x <= 3.6e+266) || (~((x <= 2.3e+274)) && ((x <= 5.6e+280) || (~((x <= 1.3e+284)) && ((x <= 2.55e+294) || (~((x <= 2.7e+295)) && (x <= 5e+301)))))))
		tmp = t_3;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(eps$95$m * N[(N[(x * eps$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(eps$95$m * N[(x + N[(2.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1e-280], N[(N[(1.0 + N[Exp[N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.5e-5], t$95$0, If[LessEqual[x, 9e-5], 1.0, If[LessEqual[x, 0.0125], t$95$3, If[LessEqual[x, 3.9e+36], t$95$0, If[LessEqual[x, 5e+39], 0.0, If[LessEqual[x, 2e+65], t$95$2, If[LessEqual[x, 5e+74], 0.0, If[LessEqual[x, 1.58e+78], t$95$1, If[LessEqual[x, 5e+80], 0.0, If[LessEqual[x, 1.45e+107], t$95$2, If[LessEqual[x, 5e+107], 0.0, If[LessEqual[x, 8e+107], t$95$3, If[LessEqual[x, 1e+110], 0.0, If[LessEqual[x, 4.5e+129], t$95$2, If[LessEqual[x, 2e+131], 0.0, If[LessEqual[x, 6e+142], t$95$2, If[LessEqual[x, 2.25e+150], 0.0, If[LessEqual[x, 8.5e+157], t$95$2, If[LessEqual[x, 2e+162], 0.0, If[LessEqual[x, 6e+168], t$95$1, If[LessEqual[x, 4.2e+172], 0.0, If[LessEqual[x, 4e+174], t$95$3, If[LessEqual[x, 3.4e+183], 0.0, If[LessEqual[x, 6e+186], t$95$3, If[LessEqual[x, 4.6e+189], 0.0, If[LessEqual[x, 3.3e+198], t$95$3, If[LessEqual[x, 8e+198], 0.0, If[LessEqual[x, 7e+205], t$95$3, If[LessEqual[x, 7.5e+207], 0.0, If[LessEqual[x, 6.5e+228], t$95$3, If[LessEqual[x, 2e+229], 0.0, If[LessEqual[x, 1.45e+236], t$95$2, If[LessEqual[x, 2e+236], 0.0, If[LessEqual[x, 3e+240], t$95$2, If[LessEqual[x, 6.8e+240], 0.0, If[Or[LessEqual[x, 3.6e+266], And[N[Not[LessEqual[x, 2.3e+274]], $MachinePrecision], Or[LessEqual[x, 5.6e+280], And[N[Not[LessEqual[x, 1.3e+284]], $MachinePrecision], Or[LessEqual[x, 2.55e+294], And[N[Not[LessEqual[x, 2.7e+295]], $MachinePrecision], LessEqual[x, 5e+301]]]]]]], t$95$3, 0.0]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if x < -9.9999999999999996e-281

   1. Initial program 71.6%

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

   3. Simplified 60.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 99.8%

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

   6. Taylor expanded in x around 0 68.9%

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

      1. sub-neg 68.9%

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

      2. metadata-eval 68.9%

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

      3. distribute-rgt-in 68.9%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 73.5%

         \[\leadsto \frac{e^{\varepsilon \cdot \color{blue}{\sqrt{x \cdot x}} + -1 \cdot x} + 1}{2} \]

      6. sqr-neg 73.5%

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

      7. sqrt-unprod 77.4%

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

      8. add-sqr-sqrt 77.4%

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

      9. neg-mul-1 77.4%

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

      10. *-un-lft-identity 77.4%

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

      11. distribute-rgt-in 77.4%

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

      12. +-commutative 77.4%

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

      13. *-commutative 77.4%

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

      14. distribute-rgt-neg-out 77.4%

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

      15. neg-sub0 77.4%

          \[\leadsto \frac{e^{\color{blue}{0 - \left(1 + \varepsilon\right) \cdot x}} + 1}{2} \]

      16. add-sqr-sqrt 0.0%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} + 1}{2} \]

      17. sqrt-unprod 66.9%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\sqrt{x \cdot x}}} + 1}{2} \]

      18. sqr-neg 66.9%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}} + 1}{2} \]

      19. sqrt-unprod 68.8%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}} + 1}{2} \]

      20. add-sqr-sqrt 68.8%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(-x\right)}} + 1}{2} \]

      21. *-commutative 68.8%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(-x\right) \cdot \left(1 + \varepsilon\right)}} + 1}{2} \]

      22. +-commutative 68.8%

          \[\leadsto \frac{e^{0 - \left(-x\right) \cdot \color{blue}{\left(\varepsilon + 1\right)}} + 1}{2} \]

      23. distribute-rgt-in 68.8%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(\varepsilon \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} + 1}{2} \]

   8. Applied egg-rr 77.4%

      \[\leadsto \frac{e^{\color{blue}{0 - x \cdot \left(\varepsilon + -1\right)}} + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 77.4%

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

      2. distribute-rgt-neg-in 77.4%

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

      3. +-commutative 77.4%

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

      4. distribute-neg-in 77.4%

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

      5. metadata-eval 77.4%

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

      6. sub-neg 77.4%

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

   10. Simplified 77.4%

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

   if -9.9999999999999996e-281 < x < 8.500000000000001e-5 or 0.012500000000000001 < x < 3.90000000000000021e36

   1. Initial program 55.0%

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

   3. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 99.5%

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

   6. Taylor expanded in x around 0 88.4%

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

   7. Taylor expanded in eps around inf 88.9%

      \[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot x}} + 1}{2} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

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

   9. Simplified 88.9%

      \[\leadsto \frac{e^{\color{blue}{x \cdot \varepsilon}} + 1}{2} \]

   if 8.500000000000001e-5 < x < 9.00000000000000057e-5

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around 0 60.9%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 9.00000000000000057e-5 < x < 0.012500000000000001 or 5.0000000000000002e107 < x < 7.9999999999999998e107 or 4.2000000000000003e172 < x < 4.00000000000000028e174 or 3.4e183 < x < 5.99999999999999964e186 or 4.6e189 < x < 3.29999999999999994e198 or 8.00000000000000014e198 < x < 6.9999999999999996e205 or 7.49999999999999986e207 < x < 6.5e228 or 6.80000000000000017e240 < x < 3.59999999999999988e266 or 2.30000000000000003e274 < x < 5.60000000000000002e280 or 1.2999999999999999e284 < x < 2.55e294 or 2.7000000000000002e295 < x < 5.0000000000000004e301

   1. Initial program 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in x around 0 69.8%

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

   6. Taylor expanded in x around 0 69.1%

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

      1. associate-*r* 69.1%

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

      2. neg-mul-1 69.1%

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

      3. *-commutative 69.1%

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

   8. Simplified 69.1%

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

   9. Taylor expanded in eps around inf 71.2%

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

       1. associate-*r/ 71.2%

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

       2. metadata-eval 71.2%

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

   11. Simplified 71.2%

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

   if 3.90000000000000021e36 < x < 5.00000000000000015e39 or 2e65 < x < 4.99999999999999963e74 or 1.58000000000000004e78 < x < 4.99999999999999961e80 or 1.44999999999999994e107 < x < 5.0000000000000002e107 or 7.9999999999999998e107 < x < 1e110 or 4.5000000000000001e129 < x < 1.9999999999999998e131 or 5.99999999999999949e142 < x < 2.25e150 or 8.4999999999999998e157 < x < 1.9999999999999999e162 or 5.9999999999999996e168 < x < 4.2000000000000003e172 or 4.00000000000000028e174 < x < 3.4e183 or 5.99999999999999964e186 < x < 4.6e189 or 3.29999999999999994e198 < x < 8.00000000000000014e198 or 6.9999999999999996e205 < x < 7.49999999999999986e207 or 6.5e228 < x < 2e229 or 1.45e236 < x < 2.00000000000000011e236 or 2.9999999999999999e240 < x < 6.80000000000000017e240 or 3.59999999999999988e266 < x < 2.30000000000000003e274 or 5.60000000000000002e280 < x < 1.2999999999999999e284 or 2.55e294 < x < 2.7000000000000002e295 or 5.0000000000000004e301 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 100.0%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}}}{2} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. rec-exp 100.0%

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

      4. sub-neg 100.0%

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

      5. div-sub 100.0%

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

      6. mul-1-neg 100.0%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 100.0%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 100.0%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 5.00000000000000015e39 < x < 2e65 or 4.99999999999999961e80 < x < 1.44999999999999994e107 or 1e110 < x < 4.5000000000000001e129 or 1.9999999999999998e131 < x < 5.99999999999999949e142 or 2.25e150 < x < 8.4999999999999998e157 or 2e229 < x < 1.45e236 or 2.00000000000000011e236 < x < 2.9999999999999999e240

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 57.9%

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

      1. sub-neg 57.9%

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

      2. metadata-eval 57.9%

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

      3. distribute-rgt-in 57.9%

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

      4. add-sqr-sqrt 57.9%

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

      5. sqrt-unprod 57.9%

         \[\leadsto \frac{e^{\varepsilon \cdot \color{blue}{\sqrt{x \cdot x}} + -1 \cdot x} + 1}{2} \]

      6. sqr-neg 57.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 45.2%

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

      9. neg-mul-1 45.2%

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

      10. *-un-lft-identity 45.2%

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

      11. distribute-rgt-in 45.2%

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

      12. +-commutative 45.2%

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

      13. *-commutative 45.2%

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

      14. distribute-rgt-neg-out 45.2%

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

      15. neg-sub0 45.2%

          \[\leadsto \frac{e^{\color{blue}{0 - \left(1 + \varepsilon\right) \cdot x}} + 1}{2} \]

      16. add-sqr-sqrt 45.2%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} + 1}{2} \]

      17. sqrt-unprod 45.2%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\sqrt{x \cdot x}}} + 1}{2} \]

      18. sqr-neg 45.2%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}} + 1}{2} \]

      19. sqrt-unprod 0.0%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}} + 1}{2} \]

      20. add-sqr-sqrt 57.9%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(-x\right)}} + 1}{2} \]

      21. *-commutative 57.9%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(-x\right) \cdot \left(1 + \varepsilon\right)}} + 1}{2} \]

      22. +-commutative 57.9%

          \[\leadsto \frac{e^{0 - \left(-x\right) \cdot \color{blue}{\left(\varepsilon + 1\right)}} + 1}{2} \]

      23. distribute-rgt-in 57.9%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(\varepsilon \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} + 1}{2} \]

   8. Applied egg-rr 45.2%

      \[\leadsto \frac{e^{\color{blue}{0 - x \cdot \left(\varepsilon + -1\right)}} + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 45.2%

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

      2. distribute-rgt-neg-in 45.2%

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

      3. +-commutative 45.2%

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

      4. distribute-neg-in 45.2%

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

      5. metadata-eval 45.2%

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

      6. sub-neg 45.2%

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

   10. Simplified 45.2%

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

   11. Taylor expanded in eps around 0 100.0%

       \[\leadsto \frac{\color{blue}{e^{x}} + 1}{2} \]

   if 4.99999999999999963e74 < x < 1.58000000000000004e78 or 1.9999999999999999e162 < x < 5.9999999999999996e168

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around 0 0.1%

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

      1. associate-*r* 0.1%

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

      2. neg-mul-1 0.1%

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

      3. *-commutative 0.1%

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

   8. Simplified 0.1%

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

   9. Taylor expanded in eps around 0 0.0%

      \[\leadsto \frac{\color{blue}{\frac{-1 \cdot x + \varepsilon \cdot \left(2 + \varepsilon \cdot x\right)}{\varepsilon}}}{2} \]
3. Recombined 7 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.0125:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+36}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+39}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+74}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.58 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+80}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+107}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+107}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+110}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+129}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+131}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+142}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+150}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+162}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+172}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+174}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+183}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+186}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+189}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+198}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+198}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+205}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+207}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+228}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+229}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+236}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+236}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+240}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+240}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+266} \lor \neg \left(x \leq 2.3 \cdot 10^{+274}\right) \land \left(x \leq 5.6 \cdot 10^{+280} \lor \neg \left(x \leq 1.3 \cdot 10^{+284}\right) \land \left(x \leq 2.55 \cdot 10^{+294} \lor \neg \left(x \leq 2.7 \cdot 10^{+295}\right) \land x \leq 5 \cdot 10^{+301}\right)\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1 + e^{x \cdot eps\_m}}{2}\\ t_1 := \frac{\frac{eps\_m \cdot \left(x \cdot eps\_m + 2\right) - x}{eps\_m}}{2}\\ t_2 := \frac{eps\_m \cdot \left(x + \frac{2}{eps\_m}\right)}{2}\\ t_3 := \frac{1 + e^{x}}{2}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-282}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(1 - eps\_m\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.19:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+39}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+61}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 10^{+76}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+103}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+107}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+107}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 10^{+110}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+128}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+135}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+142}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 10^{+143}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+158}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+160}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+173}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+174}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+186}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+187}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+191}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+195}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+201}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+206}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+206}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+228}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+229}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+235}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+236}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+240}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+240}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+265} \lor \neg \left(x \leq 2.1 \cdot 10^{+274}\right) \land \left(x \leq 1.5 \cdot 10^{+281} \lor \neg \left(x \leq 1.25 \cdot 10^{+289}\right) \land \left(x \leq 7.8 \cdot 10^{+293} \lor \neg \left(x \leq 4.4 \cdot 10^{+294}\right) \land x \leq 5.3 \cdot 10^{+301}\right)\right):\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (exp (* x eps_m))) 2.0))
        (t_1 (/ (/ (- (* eps_m (+ (* x eps_m) 2.0)) x) eps_m) 2.0))
        (t_2 (/ (* eps_m (+ x (/ 2.0 eps_m))) 2.0))
        (t_3 (/ (+ 1.0 (exp x)) 2.0)))
   (if (<= x -2e-282)
     (/ (+ 1.0 (pow E (* x (- 1.0 eps_m)))) 2.0)
     (if (<= x 8.5e-5)
       t_0
       (if (<= x 0.19)
         (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
         (if (<= x 8.5e+38)
           t_0
           (if (<= x 7.2e+39)
             0.0
             (if (<= x 4e+61)
               t_3
               (if (<= x 1e+76)
                 0.0
                 (if (<= x 2.25e+77)
                   t_1
                   (if (<= x 1.5e+80)
                     0.0
                     (if (<= x 1.35e+103)
                       t_3
                       (if (<= x 7.5e+107)
                         0.0
                         (if (<= x 9e+107)
                           t_2
                           (if (<= x 1e+110)
                             0.0
                             (if (<= x 5e+128)
                               t_3
                               (if (<= x 3e+135)
                                 0.0
                                 (if (<= x 2.2e+142)
                                   t_3
                                   (if (<= x 1e+143)
                                     0.0
                                     (if (<= x 1e+158)
                                       t_3
                                       (if (<= x 1.35e+160)
                                         0.0
                                         (if (<= x 2.3e+169)
                                           t_1
                                           (if (<= x 1.7e+173)
                                             0.0
                                             (if (<= x 1.3e+174)
                                               t_2
                                               (if (<= x 3.5e+186)
                                                 0.0
                                                 (if (<= x 2.6e+187)
                                                   t_2
                                                   (if (<= x 7e+191)
                                                     0.0
                                                     (if (<= x 3.8e+195)
                                                       t_2
                                                       (if (<= x 3.4e+201)
                                                         0.0
                                                         (if (<= x 1.3e+206)
                                                           t_2
                                                           (if (<= x 3.2e+206)
                                                             0.0
                                                             (if (<=
                                                                  x
                                                                  5.8e+228)
                                                               t_2
                                                               (if (<=
                                                                    x
                                                                    2.15e+229)
                                                                 0.0
                                                                 (if (<=
                                                                      x
                                                                      1.1e+235)
                                                                   t_3
                                                                   (if (<=
                                                                        x
                                                                        1.55e+236)
                                                                     0.0
                                                                     (if (<=
                                                                          x
                                                                          2e+240)
                                                                       t_3
                                                                       (if (<=
                                                                            x
                                                                            4.3e+240)
                                                                         0.0
                                                                         (if (or (<=
                                                                                  x
                                                                                  2.55e+265)
                                                                                 (and (not
                                                                                       (<=
                                                                                        x
                                                                                        2.1e+274))
                                                                                      (or (<=
                                                                                           x
                                                                                           1.5e+281)
                                                                                          (and (not
                                                                                                (<=
                                                                                                 x
                                                                                                 1.25e+289))
                                                                                               (or (<=
                                                                                                    x
                                                                                                    7.8e+293)
                                                                                                   (and (not
                                                                                                         (<=
                                                                                                          x
                                                                                                          4.4e+294))
                                                                                                        (<=
                                                                                                         x
                                                                                                         5.3e+301)))))))
                                                                           t_2
                                                                           0.0))))))))))))))))))))))))))))))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 + exp((x * eps_m))) / 2.0;
	double t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_2 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double t_3 = (1.0 + exp(x)) / 2.0;
	double tmp;
	if (x <= -2e-282) {
		tmp = (1.0 + pow(((double) M_E), (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 8.5e-5) {
		tmp = t_0;
	} else if (x <= 0.19) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else if (x <= 8.5e+38) {
		tmp = t_0;
	} else if (x <= 7.2e+39) {
		tmp = 0.0;
	} else if (x <= 4e+61) {
		tmp = t_3;
	} else if (x <= 1e+76) {
		tmp = 0.0;
	} else if (x <= 2.25e+77) {
		tmp = t_1;
	} else if (x <= 1.5e+80) {
		tmp = 0.0;
	} else if (x <= 1.35e+103) {
		tmp = t_3;
	} else if (x <= 7.5e+107) {
		tmp = 0.0;
	} else if (x <= 9e+107) {
		tmp = t_2;
	} else if (x <= 1e+110) {
		tmp = 0.0;
	} else if (x <= 5e+128) {
		tmp = t_3;
	} else if (x <= 3e+135) {
		tmp = 0.0;
	} else if (x <= 2.2e+142) {
		tmp = t_3;
	} else if (x <= 1e+143) {
		tmp = 0.0;
	} else if (x <= 1e+158) {
		tmp = t_3;
	} else if (x <= 1.35e+160) {
		tmp = 0.0;
	} else if (x <= 2.3e+169) {
		tmp = t_1;
	} else if (x <= 1.7e+173) {
		tmp = 0.0;
	} else if (x <= 1.3e+174) {
		tmp = t_2;
	} else if (x <= 3.5e+186) {
		tmp = 0.0;
	} else if (x <= 2.6e+187) {
		tmp = t_2;
	} else if (x <= 7e+191) {
		tmp = 0.0;
	} else if (x <= 3.8e+195) {
		tmp = t_2;
	} else if (x <= 3.4e+201) {
		tmp = 0.0;
	} else if (x <= 1.3e+206) {
		tmp = t_2;
	} else if (x <= 3.2e+206) {
		tmp = 0.0;
	} else if (x <= 5.8e+228) {
		tmp = t_2;
	} else if (x <= 2.15e+229) {
		tmp = 0.0;
	} else if (x <= 1.1e+235) {
		tmp = t_3;
	} else if (x <= 1.55e+236) {
		tmp = 0.0;
	} else if (x <= 2e+240) {
		tmp = t_3;
	} else if (x <= 4.3e+240) {
		tmp = 0.0;
	} else if ((x <= 2.55e+265) || (!(x <= 2.1e+274) && ((x <= 1.5e+281) || (!(x <= 1.25e+289) && ((x <= 7.8e+293) || (!(x <= 4.4e+294) && (x <= 5.3e+301))))))) {
		tmp = t_2;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (1.0 + Math.exp((x * eps_m))) / 2.0;
	double t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_2 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double t_3 = (1.0 + Math.exp(x)) / 2.0;
	double tmp;
	if (x <= -2e-282) {
		tmp = (1.0 + Math.pow(Math.E, (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 8.5e-5) {
		tmp = t_0;
	} else if (x <= 0.19) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else if (x <= 8.5e+38) {
		tmp = t_0;
	} else if (x <= 7.2e+39) {
		tmp = 0.0;
	} else if (x <= 4e+61) {
		tmp = t_3;
	} else if (x <= 1e+76) {
		tmp = 0.0;
	} else if (x <= 2.25e+77) {
		tmp = t_1;
	} else if (x <= 1.5e+80) {
		tmp = 0.0;
	} else if (x <= 1.35e+103) {
		tmp = t_3;
	} else if (x <= 7.5e+107) {
		tmp = 0.0;
	} else if (x <= 9e+107) {
		tmp = t_2;
	} else if (x <= 1e+110) {
		tmp = 0.0;
	} else if (x <= 5e+128) {
		tmp = t_3;
	} else if (x <= 3e+135) {
		tmp = 0.0;
	} else if (x <= 2.2e+142) {
		tmp = t_3;
	} else if (x <= 1e+143) {
		tmp = 0.0;
	} else if (x <= 1e+158) {
		tmp = t_3;
	} else if (x <= 1.35e+160) {
		tmp = 0.0;
	} else if (x <= 2.3e+169) {
		tmp = t_1;
	} else if (x <= 1.7e+173) {
		tmp = 0.0;
	} else if (x <= 1.3e+174) {
		tmp = t_2;
	} else if (x <= 3.5e+186) {
		tmp = 0.0;
	} else if (x <= 2.6e+187) {
		tmp = t_2;
	} else if (x <= 7e+191) {
		tmp = 0.0;
	} else if (x <= 3.8e+195) {
		tmp = t_2;
	} else if (x <= 3.4e+201) {
		tmp = 0.0;
	} else if (x <= 1.3e+206) {
		tmp = t_2;
	} else if (x <= 3.2e+206) {
		tmp = 0.0;
	} else if (x <= 5.8e+228) {
		tmp = t_2;
	} else if (x <= 2.15e+229) {
		tmp = 0.0;
	} else if (x <= 1.1e+235) {
		tmp = t_3;
	} else if (x <= 1.55e+236) {
		tmp = 0.0;
	} else if (x <= 2e+240) {
		tmp = t_3;
	} else if (x <= 4.3e+240) {
		tmp = 0.0;
	} else if ((x <= 2.55e+265) || (!(x <= 2.1e+274) && ((x <= 1.5e+281) || (!(x <= 1.25e+289) && ((x <= 7.8e+293) || (!(x <= 4.4e+294) && (x <= 5.3e+301))))))) {
		tmp = t_2;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (1.0 + math.exp((x * eps_m))) / 2.0
	t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0
	t_2 = (eps_m * (x + (2.0 / eps_m))) / 2.0
	t_3 = (1.0 + math.exp(x)) / 2.0
	tmp = 0
	if x <= -2e-282:
		tmp = (1.0 + math.pow(math.e, (x * (1.0 - eps_m)))) / 2.0
	elif x <= 8.5e-5:
		tmp = t_0
	elif x <= 0.19:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	elif x <= 8.5e+38:
		tmp = t_0
	elif x <= 7.2e+39:
		tmp = 0.0
	elif x <= 4e+61:
		tmp = t_3
	elif x <= 1e+76:
		tmp = 0.0
	elif x <= 2.25e+77:
		tmp = t_1
	elif x <= 1.5e+80:
		tmp = 0.0
	elif x <= 1.35e+103:
		tmp = t_3
	elif x <= 7.5e+107:
		tmp = 0.0
	elif x <= 9e+107:
		tmp = t_2
	elif x <= 1e+110:
		tmp = 0.0
	elif x <= 5e+128:
		tmp = t_3
	elif x <= 3e+135:
		tmp = 0.0
	elif x <= 2.2e+142:
		tmp = t_3
	elif x <= 1e+143:
		tmp = 0.0
	elif x <= 1e+158:
		tmp = t_3
	elif x <= 1.35e+160:
		tmp = 0.0
	elif x <= 2.3e+169:
		tmp = t_1
	elif x <= 1.7e+173:
		tmp = 0.0
	elif x <= 1.3e+174:
		tmp = t_2
	elif x <= 3.5e+186:
		tmp = 0.0
	elif x <= 2.6e+187:
		tmp = t_2
	elif x <= 7e+191:
		tmp = 0.0
	elif x <= 3.8e+195:
		tmp = t_2
	elif x <= 3.4e+201:
		tmp = 0.0
	elif x <= 1.3e+206:
		tmp = t_2
	elif x <= 3.2e+206:
		tmp = 0.0
	elif x <= 5.8e+228:
		tmp = t_2
	elif x <= 2.15e+229:
		tmp = 0.0
	elif x <= 1.1e+235:
		tmp = t_3
	elif x <= 1.55e+236:
		tmp = 0.0
	elif x <= 2e+240:
		tmp = t_3
	elif x <= 4.3e+240:
		tmp = 0.0
	elif (x <= 2.55e+265) or (not (x <= 2.1e+274) and ((x <= 1.5e+281) or (not (x <= 1.25e+289) and ((x <= 7.8e+293) or (not (x <= 4.4e+294) and (x <= 5.3e+301)))))):
		tmp = t_2
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0)
	t_1 = Float64(Float64(Float64(Float64(eps_m * Float64(Float64(x * eps_m) + 2.0)) - x) / eps_m) / 2.0)
	t_2 = Float64(Float64(eps_m * Float64(x + Float64(2.0 / eps_m))) / 2.0)
	t_3 = Float64(Float64(1.0 + exp(x)) / 2.0)
	tmp = 0.0
	if (x <= -2e-282)
		tmp = Float64(Float64(1.0 + (exp(1) ^ Float64(x * Float64(1.0 - eps_m)))) / 2.0);
	elseif (x <= 8.5e-5)
		tmp = t_0;
	elseif (x <= 0.19)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	elseif (x <= 8.5e+38)
		tmp = t_0;
	elseif (x <= 7.2e+39)
		tmp = 0.0;
	elseif (x <= 4e+61)
		tmp = t_3;
	elseif (x <= 1e+76)
		tmp = 0.0;
	elseif (x <= 2.25e+77)
		tmp = t_1;
	elseif (x <= 1.5e+80)
		tmp = 0.0;
	elseif (x <= 1.35e+103)
		tmp = t_3;
	elseif (x <= 7.5e+107)
		tmp = 0.0;
	elseif (x <= 9e+107)
		tmp = t_2;
	elseif (x <= 1e+110)
		tmp = 0.0;
	elseif (x <= 5e+128)
		tmp = t_3;
	elseif (x <= 3e+135)
		tmp = 0.0;
	elseif (x <= 2.2e+142)
		tmp = t_3;
	elseif (x <= 1e+143)
		tmp = 0.0;
	elseif (x <= 1e+158)
		tmp = t_3;
	elseif (x <= 1.35e+160)
		tmp = 0.0;
	elseif (x <= 2.3e+169)
		tmp = t_1;
	elseif (x <= 1.7e+173)
		tmp = 0.0;
	elseif (x <= 1.3e+174)
		tmp = t_2;
	elseif (x <= 3.5e+186)
		tmp = 0.0;
	elseif (x <= 2.6e+187)
		tmp = t_2;
	elseif (x <= 7e+191)
		tmp = 0.0;
	elseif (x <= 3.8e+195)
		tmp = t_2;
	elseif (x <= 3.4e+201)
		tmp = 0.0;
	elseif (x <= 1.3e+206)
		tmp = t_2;
	elseif (x <= 3.2e+206)
		tmp = 0.0;
	elseif (x <= 5.8e+228)
		tmp = t_2;
	elseif (x <= 2.15e+229)
		tmp = 0.0;
	elseif (x <= 1.1e+235)
		tmp = t_3;
	elseif (x <= 1.55e+236)
		tmp = 0.0;
	elseif (x <= 2e+240)
		tmp = t_3;
	elseif (x <= 4.3e+240)
		tmp = 0.0;
	elseif ((x <= 2.55e+265) || (!(x <= 2.1e+274) && ((x <= 1.5e+281) || (!(x <= 1.25e+289) && ((x <= 7.8e+293) || (!(x <= 4.4e+294) && (x <= 5.3e+301)))))))
		tmp = t_2;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (1.0 + exp((x * eps_m))) / 2.0;
	t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	t_2 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	t_3 = (1.0 + exp(x)) / 2.0;
	tmp = 0.0;
	if (x <= -2e-282)
		tmp = (1.0 + (2.71828182845904523536 ^ (x * (1.0 - eps_m)))) / 2.0;
	elseif (x <= 8.5e-5)
		tmp = t_0;
	elseif (x <= 0.19)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	elseif (x <= 8.5e+38)
		tmp = t_0;
	elseif (x <= 7.2e+39)
		tmp = 0.0;
	elseif (x <= 4e+61)
		tmp = t_3;
	elseif (x <= 1e+76)
		tmp = 0.0;
	elseif (x <= 2.25e+77)
		tmp = t_1;
	elseif (x <= 1.5e+80)
		tmp = 0.0;
	elseif (x <= 1.35e+103)
		tmp = t_3;
	elseif (x <= 7.5e+107)
		tmp = 0.0;
	elseif (x <= 9e+107)
		tmp = t_2;
	elseif (x <= 1e+110)
		tmp = 0.0;
	elseif (x <= 5e+128)
		tmp = t_3;
	elseif (x <= 3e+135)
		tmp = 0.0;
	elseif (x <= 2.2e+142)
		tmp = t_3;
	elseif (x <= 1e+143)
		tmp = 0.0;
	elseif (x <= 1e+158)
		tmp = t_3;
	elseif (x <= 1.35e+160)
		tmp = 0.0;
	elseif (x <= 2.3e+169)
		tmp = t_1;
	elseif (x <= 1.7e+173)
		tmp = 0.0;
	elseif (x <= 1.3e+174)
		tmp = t_2;
	elseif (x <= 3.5e+186)
		tmp = 0.0;
	elseif (x <= 2.6e+187)
		tmp = t_2;
	elseif (x <= 7e+191)
		tmp = 0.0;
	elseif (x <= 3.8e+195)
		tmp = t_2;
	elseif (x <= 3.4e+201)
		tmp = 0.0;
	elseif (x <= 1.3e+206)
		tmp = t_2;
	elseif (x <= 3.2e+206)
		tmp = 0.0;
	elseif (x <= 5.8e+228)
		tmp = t_2;
	elseif (x <= 2.15e+229)
		tmp = 0.0;
	elseif (x <= 1.1e+235)
		tmp = t_3;
	elseif (x <= 1.55e+236)
		tmp = 0.0;
	elseif (x <= 2e+240)
		tmp = t_3;
	elseif (x <= 4.3e+240)
		tmp = 0.0;
	elseif ((x <= 2.55e+265) || (~((x <= 2.1e+274)) && ((x <= 1.5e+281) || (~((x <= 1.25e+289)) && ((x <= 7.8e+293) || (~((x <= 4.4e+294)) && (x <= 5.3e+301)))))))
		tmp = t_2;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(eps$95$m * N[(N[(x * eps$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(eps$95$m * N[(x + N[(2.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2e-282], N[(N[(1.0 + N[Power[E, N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.5e-5], t$95$0, If[LessEqual[x, 0.19], N[(N[(N[(eps$95$m * N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.5e+38], t$95$0, If[LessEqual[x, 7.2e+39], 0.0, If[LessEqual[x, 4e+61], t$95$3, If[LessEqual[x, 1e+76], 0.0, If[LessEqual[x, 2.25e+77], t$95$1, If[LessEqual[x, 1.5e+80], 0.0, If[LessEqual[x, 1.35e+103], t$95$3, If[LessEqual[x, 7.5e+107], 0.0, If[LessEqual[x, 9e+107], t$95$2, If[LessEqual[x, 1e+110], 0.0, If[LessEqual[x, 5e+128], t$95$3, If[LessEqual[x, 3e+135], 0.0, If[LessEqual[x, 2.2e+142], t$95$3, If[LessEqual[x, 1e+143], 0.0, If[LessEqual[x, 1e+158], t$95$3, If[LessEqual[x, 1.35e+160], 0.0, If[LessEqual[x, 2.3e+169], t$95$1, If[LessEqual[x, 1.7e+173], 0.0, If[LessEqual[x, 1.3e+174], t$95$2, If[LessEqual[x, 3.5e+186], 0.0, If[LessEqual[x, 2.6e+187], t$95$2, If[LessEqual[x, 7e+191], 0.0, If[LessEqual[x, 3.8e+195], t$95$2, If[LessEqual[x, 3.4e+201], 0.0, If[LessEqual[x, 1.3e+206], t$95$2, If[LessEqual[x, 3.2e+206], 0.0, If[LessEqual[x, 5.8e+228], t$95$2, If[LessEqual[x, 2.15e+229], 0.0, If[LessEqual[x, 1.1e+235], t$95$3, If[LessEqual[x, 1.55e+236], 0.0, If[LessEqual[x, 2e+240], t$95$3, If[LessEqual[x, 4.3e+240], 0.0, If[Or[LessEqual[x, 2.55e+265], And[N[Not[LessEqual[x, 2.1e+274]], $MachinePrecision], Or[LessEqual[x, 1.5e+281], And[N[Not[LessEqual[x, 1.25e+289]], $MachinePrecision], Or[LessEqual[x, 7.8e+293], And[N[Not[LessEqual[x, 4.4e+294]], $MachinePrecision], LessEqual[x, 5.3e+301]]]]]]], t$95$2, 0.0]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if x < -2e-282

   1. Initial program 71.6%

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

   3. Simplified 60.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 99.8%

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

   6. Taylor expanded in x around 0 68.9%

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

      1. sub-neg 68.9%

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

      2. metadata-eval 68.9%

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

      3. distribute-rgt-in 68.9%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 73.5%

         \[\leadsto \frac{e^{\varepsilon \cdot \color{blue}{\sqrt{x \cdot x}} + -1 \cdot x} + 1}{2} \]

      6. sqr-neg 73.5%

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

      7. sqrt-unprod 77.4%

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

      8. add-sqr-sqrt 77.4%

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

      9. neg-mul-1 77.4%

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

      10. *-un-lft-identity 77.4%

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

      11. distribute-rgt-in 77.4%

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

      12. +-commutative 77.4%

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

      13. *-commutative 77.4%

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

      14. distribute-rgt-neg-out 77.4%

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

      15. neg-sub0 77.4%

          \[\leadsto \frac{e^{\color{blue}{0 - \left(1 + \varepsilon\right) \cdot x}} + 1}{2} \]

      16. add-sqr-sqrt 0.0%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} + 1}{2} \]

      17. sqrt-unprod 66.9%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\sqrt{x \cdot x}}} + 1}{2} \]

      18. sqr-neg 66.9%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}} + 1}{2} \]

      19. sqrt-unprod 68.8%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}} + 1}{2} \]

      20. add-sqr-sqrt 68.8%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(-x\right)}} + 1}{2} \]

      21. *-commutative 68.8%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(-x\right) \cdot \left(1 + \varepsilon\right)}} + 1}{2} \]

      22. +-commutative 68.8%

          \[\leadsto \frac{e^{0 - \left(-x\right) \cdot \color{blue}{\left(\varepsilon + 1\right)}} + 1}{2} \]

      23. distribute-rgt-in 68.8%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(\varepsilon \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} + 1}{2} \]

   8. Applied egg-rr 77.4%

      \[\leadsto \frac{e^{\color{blue}{0 - x \cdot \left(\varepsilon + -1\right)}} + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 77.4%

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

      2. distribute-rgt-neg-in 77.4%

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

      3. +-commutative 77.4%

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

      4. distribute-neg-in 77.4%

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

      5. metadata-eval 77.4%

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

      6. sub-neg 77.4%

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

   10. Simplified 77.4%

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

       1. *-commutative 77.4%

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

       2. *-un-lft-identity 77.4%

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

       3. exp-prod 77.4%

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

       4. e-exp-1 77.4%

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

       5. *-commutative 77.4%

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

   12. Applied egg-rr 77.4%

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

   if -2e-282 < x < 8.500000000000001e-5 or 0.19 < x < 8.4999999999999997e38

   1. Initial program 55.0%

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

   3. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 99.5%

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

   6. Taylor expanded in x around 0 88.4%

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

   7. Taylor expanded in eps around inf 88.9%

      \[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot x}} + 1}{2} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

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

   9. Simplified 88.9%

      \[\leadsto \frac{e^{\color{blue}{x \cdot \varepsilon}} + 1}{2} \]

   if 8.500000000000001e-5 < x < 0.19

   1. Initial program 3.1%

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

   3. Simplified 3.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 3.1%

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

      1. associate-+r+ 99.2%

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

      2. mul-1-neg 99.2%

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

      3. sub-neg 99.2%

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

      4. +-inverses 99.2%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. associate-*r* 99.2%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + \color{blue}{\left(2 \cdot x\right) \cdot e^{-1 \cdot x}}\right)}{\varepsilon}}{2} \]

      6. distribute-rgt-out 99.2%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(e^{-1 \cdot x} \cdot \left(2 + 2 \cdot x\right)\right)}}{\varepsilon}}{2} \]

      7. mul-1-neg 99.2%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(e^{\color{blue}{-x}} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 99.2%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(e^{-x} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}}{2} \]

   if 8.4999999999999997e38 < x < 7.19999999999999969e39 or 3.9999999999999998e61 < x < 1e76 or 2.25000000000000012e77 < x < 1.49999999999999993e80 or 1.34999999999999996e103 < x < 7.4999999999999996e107 or 9e107 < x < 1e110 or 5e128 < x < 3e135 or 2.19999999999999987e142 < x < 1e143 or 9.99999999999999953e157 < x < 1.35e160 or 2.2999999999999999e169 < x < 1.70000000000000011e173 or 1.2999999999999999e174 < x < 3.49999999999999987e186 or 2.5999999999999999e187 < x < 6.9999999999999994e191 or 3.8e195 < x < 3.4e201 or 1.29999999999999994e206 < x < 3.20000000000000005e206 or 5.80000000000000003e228 < x < 2.14999999999999996e229 or 1.1e235 < x < 1.55e236 or 2.00000000000000003e240 < x < 4.3e240 or 2.55000000000000011e265 < x < 2.10000000000000016e274 or 1.50000000000000005e281 < x < 1.25000000000000008e289 or 7.80000000000000055e293 < x < 4.4000000000000002e294 or 5.3000000000000001e301 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 100.0%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}}}{2} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. rec-exp 100.0%

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

      4. sub-neg 100.0%

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

      5. div-sub 100.0%

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

      6. mul-1-neg 100.0%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 100.0%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 100.0%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 7.19999999999999969e39 < x < 3.9999999999999998e61 or 1.49999999999999993e80 < x < 1.34999999999999996e103 or 1e110 < x < 5e128 or 3e135 < x < 2.19999999999999987e142 or 1e143 < x < 9.99999999999999953e157 or 2.14999999999999996e229 < x < 1.1e235 or 1.55e236 < x < 2.00000000000000003e240

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 57.9%

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

      1. sub-neg 57.9%

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

      2. metadata-eval 57.9%

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

      3. distribute-rgt-in 57.9%

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

      4. add-sqr-sqrt 57.9%

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

      5. sqrt-unprod 57.9%

         \[\leadsto \frac{e^{\varepsilon \cdot \color{blue}{\sqrt{x \cdot x}} + -1 \cdot x} + 1}{2} \]

      6. sqr-neg 57.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 45.2%

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

      9. neg-mul-1 45.2%

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

      10. *-un-lft-identity 45.2%

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

      11. distribute-rgt-in 45.2%

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

      12. +-commutative 45.2%

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

      13. *-commutative 45.2%

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

      14. distribute-rgt-neg-out 45.2%

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

      15. neg-sub0 45.2%

          \[\leadsto \frac{e^{\color{blue}{0 - \left(1 + \varepsilon\right) \cdot x}} + 1}{2} \]

      16. add-sqr-sqrt 45.2%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} + 1}{2} \]

      17. sqrt-unprod 45.2%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\sqrt{x \cdot x}}} + 1}{2} \]

      18. sqr-neg 45.2%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}} + 1}{2} \]

      19. sqrt-unprod 0.0%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}} + 1}{2} \]

      20. add-sqr-sqrt 57.9%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(-x\right)}} + 1}{2} \]

      21. *-commutative 57.9%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(-x\right) \cdot \left(1 + \varepsilon\right)}} + 1}{2} \]

      22. +-commutative 57.9%

          \[\leadsto \frac{e^{0 - \left(-x\right) \cdot \color{blue}{\left(\varepsilon + 1\right)}} + 1}{2} \]

      23. distribute-rgt-in 57.9%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(\varepsilon \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} + 1}{2} \]

   8. Applied egg-rr 45.2%

      \[\leadsto \frac{e^{\color{blue}{0 - x \cdot \left(\varepsilon + -1\right)}} + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 45.2%

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

      2. distribute-rgt-neg-in 45.2%

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

      3. +-commutative 45.2%

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

      4. distribute-neg-in 45.2%

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

      5. metadata-eval 45.2%

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

      6. sub-neg 45.2%

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

   10. Simplified 45.2%

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

   11. Taylor expanded in eps around 0 100.0%

       \[\leadsto \frac{\color{blue}{e^{x}} + 1}{2} \]

   if 1e76 < x < 2.25000000000000012e77 or 1.35e160 < x < 2.2999999999999999e169

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around 0 0.1%

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

      1. associate-*r* 0.1%

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

      2. neg-mul-1 0.1%

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

      3. *-commutative 0.1%

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

   8. Simplified 0.1%

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

   9. Taylor expanded in eps around 0 0.0%

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

   if 7.4999999999999996e107 < x < 9e107 or 1.70000000000000011e173 < x < 1.2999999999999999e174 or 3.49999999999999987e186 < x < 2.5999999999999999e187 or 6.9999999999999994e191 < x < 3.8e195 or 3.4e201 < x < 1.29999999999999994e206 or 3.20000000000000005e206 < x < 5.80000000000000003e228 or 4.3e240 < x < 2.55000000000000011e265 or 2.10000000000000016e274 < x < 1.50000000000000005e281 or 1.25000000000000008e289 < x < 7.80000000000000055e293 or 4.4000000000000002e294 < x < 5.3000000000000001e301

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 74.2%

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

   6. Taylor expanded in x around 0 73.3%

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

      1. associate-*r* 73.3%

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

      2. neg-mul-1 73.3%

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

      3. *-commutative 73.3%

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

   8. Simplified 73.3%

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

   9. Taylor expanded in eps around inf 73.3%

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

       1. associate-*r/ 73.3%

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

       2. metadata-eval 73.3%

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

   11. Simplified 73.3%

       \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}}{2} \]
3. Recombined 7 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-282}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(1 - \varepsilon\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 0.19:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+39}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+61}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 10^{+76}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+103}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+107}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+107}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+110}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+128}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+135}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+142}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 10^{+143}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+158}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+160}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+173}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+174}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+186}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+187}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+191}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+195}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+201}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+206}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+206}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+228}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+229}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+235}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+236}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+240}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+265} \lor \neg \left(x \leq 2.1 \cdot 10^{+274}\right) \land \left(x \leq 1.5 \cdot 10^{+281} \lor \neg \left(x \leq 1.25 \cdot 10^{+289}\right) \land \left(x \leq 7.8 \cdot 10^{+293} \lor \neg \left(x \leq 4.4 \cdot 10^{+294}\right) \land x \leq 5.3 \cdot 10^{+301}\right)\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 7 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{e}^{\left(x \cdot eps\_m\right)}} + e^{x \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 7e-15)
   (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   (/ (+ (/ 1.0 (pow E (* x eps_m))) (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 7e-15) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = ((1.0 / pow(((double) M_E), (x * eps_m))) + exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 7e-15) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = ((1.0 / Math.pow(Math.E, (x * eps_m))) + Math.exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 7e-15:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	else:
		tmp = ((1.0 / math.pow(math.e, (x * eps_m))) + math.exp((x * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 7e-15)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 / (exp(1) ^ Float64(x * eps_m))) + exp(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 7e-15)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	else
		tmp = ((1.0 / (2.71828182845904523536 ^ (x * eps_m))) + exp((x * eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 7e-15], N[(N[(N[(eps$95$m * N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 / N[Power[E, N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 7.0000000000000001e-15

   1. Initial program 58.4%

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

   3. Simplified 51.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 29.2%

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

      1. associate-+r+ 71.1%

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

      2. mul-1-neg 71.1%

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

      3. sub-neg 71.1%

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

      4. +-inverses 71.1%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. associate-*r* 71.1%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + \color{blue}{\left(2 \cdot x\right) \cdot e^{-1 \cdot x}}\right)}{\varepsilon}}{2} \]

      6. distribute-rgt-out 71.1%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(e^{-1 \cdot x} \cdot \left(2 + 2 \cdot x\right)\right)}}{\varepsilon}}{2} \]

      7. mul-1-neg 71.1%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(e^{\color{blue}{-x}} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 71.1%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(e^{-x} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}}{2} \]

   if 7.0000000000000001e-15 < eps

   1. Initial program 100.0%

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

   3. Simplified 90.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. exp-1-e 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   15. Simplified 100.0%

       \[\leadsto \frac{e^{\color{blue}{x \cdot \varepsilon}} + \frac{1}{{e}^{\left(x \cdot \varepsilon\right)}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 7 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{e}^{\left(x \cdot \varepsilon\right)}} + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot eps\_m}\\ \mathbf{if}\;eps\_m \leq 7 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + \frac{1}{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x eps_m))))
   (if (<= eps_m 7e-15)
     (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
     (/ (+ t_0 (/ 1.0 t_0)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * eps_m));
	double tmp;
	if (eps_m <= 7e-15) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (t_0 + (1.0 / t_0)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * eps_m))
    if (eps_m <= 7d-15) then
        tmp = ((eps_m * (exp(-x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else
        tmp = (t_0 + (1.0d0 / t_0)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * eps_m));
	double tmp;
	if (eps_m <= 7e-15) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (t_0 + (1.0 / t_0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * eps_m))
	tmp = 0
	if eps_m <= 7e-15:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	else:
		tmp = (t_0 + (1.0 / t_0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * eps_m))
	tmp = 0.0
	if (eps_m <= 7e-15)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(t_0 + Float64(1.0 / t_0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * eps_m));
	tmp = 0.0;
	if (eps_m <= 7e-15)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	else
		tmp = (t_0 + (1.0 / t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eps$95$m, 7e-15], N[(N[(N[(eps$95$m * N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 7.0000000000000001e-15

   1. Initial program 58.4%

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

   3. Simplified 51.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 29.2%

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

      1. associate-+r+ 71.1%

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

      2. mul-1-neg 71.1%

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

      3. sub-neg 71.1%

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

      4. +-inverses 71.1%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. associate-*r* 71.1%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + \color{blue}{\left(2 \cdot x\right) \cdot e^{-1 \cdot x}}\right)}{\varepsilon}}{2} \]

      6. distribute-rgt-out 71.1%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(e^{-1 \cdot x} \cdot \left(2 + 2 \cdot x\right)\right)}}{\varepsilon}}{2} \]

      7. mul-1-neg 71.1%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(e^{\color{blue}{-x}} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 71.1%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(e^{-x} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}}{2} \]

   if 7.0000000000000001e-15 < eps

   1. Initial program 100.0%

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

   3. Simplified 90.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

      \[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot x}} + \frac{1}{e^{x \cdot \varepsilon}}}{2} \]
   10. Step-by-step derivation

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \frac{e^{\color{blue}{x \cdot \varepsilon}} + \frac{1}{e^{x \cdot \varepsilon}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 7 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \frac{1}{e^{x \cdot \varepsilon}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{e^{x + x \cdot eps\_m}}}{2} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (/ (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (exp (+ x (* x eps_m))))) 2.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (exp((x * (eps_m + -1.0))) + (1.0 / exp((x + (x * eps_m))))) / 2.0;
}

Fortran

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

Java

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

Python

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

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / exp(Float64(x + Float64(x * eps_m))))) / 2.0)
end

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[N[(x + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

3. Simplified 65.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
4. Add Preprocessing

5. Taylor expanded in eps around inf 99.2%

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

6. Final simplification 99.2%

   \[\leadsto \frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{e^{x + x \cdot \varepsilon}}}{2} \]
7. Add Preprocessing

Alternative 8: 59.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(eps\_m + \frac{-1}{eps\_m}\right)\\ t_1 := \frac{2 - x \cdot eps\_m}{2}\\ t_2 := \frac{2 + t\_0}{2}\\ t_3 := \frac{1 + e^{x}}{2}\\ t_4 := \frac{eps\_m \cdot \left(x + \frac{2}{eps\_m}\right)}{2}\\ t_5 := \frac{\frac{eps\_m \cdot \left(x \cdot eps\_m + 2\right) - x}{eps\_m}}{2}\\ t_6 := \frac{2 + \frac{x + eps\_m \cdot \left(x \cdot 0 - x \cdot eps\_m\right)}{eps\_m}}{2}\\ \mathbf{if}\;eps\_m \leq 1.9 \cdot 10^{+107}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eps\_m \leq 2.1 \cdot 10^{+179}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;eps\_m \leq 1.32 \cdot 10^{+181}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 3.4 \cdot 10^{+181}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;eps\_m \leq 1.55 \cdot 10^{+184}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot \left(1 + x \cdot \left(eps\_m + -1\right)\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{elif}\;eps\_m \leq 1.1 \cdot 10^{+187}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 3.5 \cdot 10^{+192}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;eps\_m \leq 9.6 \cdot 10^{+197}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 1.65 \cdot 10^{+203}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;eps\_m \leq 2.65 \cdot 10^{+228}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 2.7 \cdot 10^{+228}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 6.2 \cdot 10^{+230}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 9.2 \cdot 10^{+232}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;eps\_m \leq 1.4 \cdot 10^{+236}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 1.45 \cdot 10^{+236}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 2.05 \cdot 10^{+237}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 3.3 \cdot 10^{+237}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 5.9 \cdot 10^{+237}:\\ \;\;\;\;\frac{2 - t\_0}{2}\\ \mathbf{elif}\;eps\_m \leq 6 \cdot 10^{+241}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 1.5 \cdot 10^{+243}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 3.2 \cdot 10^{+248}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eps\_m \leq 3.25 \cdot 10^{+248}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 1.5 \cdot 10^{+254}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 5.2 \cdot 10^{+265}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;eps\_m \leq 1.3 \cdot 10^{+268}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;eps\_m \leq 5 \cdot 10^{+278}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 9.5 \cdot 10^{+279}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;eps\_m \leq 2.3 \cdot 10^{+283}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 2 \cdot 10^{+295} \lor \neg \left(eps\_m \leq 2.7 \cdot 10^{+299}\right):\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ eps_m (/ -1.0 eps_m))))
        (t_1 (/ (- 2.0 (* x eps_m)) 2.0))
        (t_2 (/ (+ 2.0 t_0) 2.0))
        (t_3 (/ (+ 1.0 (exp x)) 2.0))
        (t_4 (/ (* eps_m (+ x (/ 2.0 eps_m))) 2.0))
        (t_5 (/ (/ (- (* eps_m (+ (* x eps_m) 2.0)) x) eps_m) 2.0))
        (t_6
         (/ (+ 2.0 (/ (+ x (* eps_m (- (* x 0.0) (* x eps_m)))) eps_m)) 2.0)))
   (if (<= eps_m 1.9e+107)
     t_3
     (if (<= eps_m 2.1e+179)
       t_6
       (if (<= eps_m 1.32e+181)
         t_5
         (if (<= eps_m 3.4e+181)
           t_6
           (if (<= eps_m 1.55e+184)
             (/
              (+
               (* (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (* x (+ eps_m -1.0))))
               (+ 1.0 (/ -1.0 eps_m)))
              2.0)
             (if (<= eps_m 1.1e+187)
               t_5
               (if (<= eps_m 3.5e+192)
                 t_6
                 (if (<= eps_m 9.6e+197)
                   t_5
                   (if (<= eps_m 1.65e+203)
                     t_6
                     (if (<= eps_m 2.65e+228)
                       t_5
                       (if (<= eps_m 2.7e+228)
                         t_1
                         (if (<= eps_m 6.2e+230)
                           t_4
                           (if (<= eps_m 9.2e+232)
                             t_6
                             (if (<= eps_m 1.4e+236)
                               t_5
                               (if (<= eps_m 1.45e+236)
                                 t_1
                                 (if (<= eps_m 2.05e+237)
                                   t_5
                                   (if (<= eps_m 3.3e+237)
                                     t_4
                                     (if (<= eps_m 5.9e+237)
                                       (/ (- 2.0 t_0) 2.0)
                                       (if (<= eps_m 6e+241)
                                         t_2
                                         (if (<= eps_m 1.5e+243)
                                           t_5
                                           (if (<= eps_m 3.2e+248)
                                             t_3
                                             (if (<= eps_m 3.25e+248)
                                               t_1
                                               (if (<= eps_m 1.5e+254)
                                                 t_2
                                                 (if (<= eps_m 5.2e+265)
                                                   t_6
                                                   (if (<= eps_m 1.3e+268)
                                                     t_5
                                                     (if (<= eps_m 2e+272)
                                                       t_6
                                                       (if (<= eps_m 5e+278)
                                                         t_5
                                                         (if (<=
                                                              eps_m
                                                              9.5e+279)
                                                           t_6
                                                           (if (<=
                                                                eps_m
                                                                2.3e+283)
                                                             t_4
                                                             (if (or (<=
                                                                      eps_m
                                                                      2e+295)
                                                                     (not
                                                                      (<=
                                                                       eps_m
                                                                       2.7e+299)))
                                                               t_6
                                                               t_5))))))))))))))))))))))))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (eps_m + (-1.0 / eps_m));
	double t_1 = (2.0 - (x * eps_m)) / 2.0;
	double t_2 = (2.0 + t_0) / 2.0;
	double t_3 = (1.0 + exp(x)) / 2.0;
	double t_4 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double t_5 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_6 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	double tmp;
	if (eps_m <= 1.9e+107) {
		tmp = t_3;
	} else if (eps_m <= 2.1e+179) {
		tmp = t_6;
	} else if (eps_m <= 1.32e+181) {
		tmp = t_5;
	} else if (eps_m <= 3.4e+181) {
		tmp = t_6;
	} else if (eps_m <= 1.55e+184) {
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else if (eps_m <= 1.1e+187) {
		tmp = t_5;
	} else if (eps_m <= 3.5e+192) {
		tmp = t_6;
	} else if (eps_m <= 9.6e+197) {
		tmp = t_5;
	} else if (eps_m <= 1.65e+203) {
		tmp = t_6;
	} else if (eps_m <= 2.65e+228) {
		tmp = t_5;
	} else if (eps_m <= 2.7e+228) {
		tmp = t_1;
	} else if (eps_m <= 6.2e+230) {
		tmp = t_4;
	} else if (eps_m <= 9.2e+232) {
		tmp = t_6;
	} else if (eps_m <= 1.4e+236) {
		tmp = t_5;
	} else if (eps_m <= 1.45e+236) {
		tmp = t_1;
	} else if (eps_m <= 2.05e+237) {
		tmp = t_5;
	} else if (eps_m <= 3.3e+237) {
		tmp = t_4;
	} else if (eps_m <= 5.9e+237) {
		tmp = (2.0 - t_0) / 2.0;
	} else if (eps_m <= 6e+241) {
		tmp = t_2;
	} else if (eps_m <= 1.5e+243) {
		tmp = t_5;
	} else if (eps_m <= 3.2e+248) {
		tmp = t_3;
	} else if (eps_m <= 3.25e+248) {
		tmp = t_1;
	} else if (eps_m <= 1.5e+254) {
		tmp = t_2;
	} else if (eps_m <= 5.2e+265) {
		tmp = t_6;
	} else if (eps_m <= 1.3e+268) {
		tmp = t_5;
	} else if (eps_m <= 2e+272) {
		tmp = t_6;
	} else if (eps_m <= 5e+278) {
		tmp = t_5;
	} else if (eps_m <= 9.5e+279) {
		tmp = t_6;
	} else if (eps_m <= 2.3e+283) {
		tmp = t_4;
	} else if ((eps_m <= 2e+295) || !(eps_m <= 2.7e+299)) {
		tmp = t_6;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_0 = x * (eps_m + ((-1.0d0) / eps_m))
    t_1 = (2.0d0 - (x * eps_m)) / 2.0d0
    t_2 = (2.0d0 + t_0) / 2.0d0
    t_3 = (1.0d0 + exp(x)) / 2.0d0
    t_4 = (eps_m * (x + (2.0d0 / eps_m))) / 2.0d0
    t_5 = (((eps_m * ((x * eps_m) + 2.0d0)) - x) / eps_m) / 2.0d0
    t_6 = (2.0d0 + ((x + (eps_m * ((x * 0.0d0) - (x * eps_m)))) / eps_m)) / 2.0d0
    if (eps_m <= 1.9d+107) then
        tmp = t_3
    else if (eps_m <= 2.1d+179) then
        tmp = t_6
    else if (eps_m <= 1.32d+181) then
        tmp = t_5
    else if (eps_m <= 3.4d+181) then
        tmp = t_6
    else if (eps_m <= 1.55d+184) then
        tmp = (((1.0d0 + (1.0d0 / eps_m)) * (1.0d0 + (x * (eps_m + (-1.0d0))))) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else if (eps_m <= 1.1d+187) then
        tmp = t_5
    else if (eps_m <= 3.5d+192) then
        tmp = t_6
    else if (eps_m <= 9.6d+197) then
        tmp = t_5
    else if (eps_m <= 1.65d+203) then
        tmp = t_6
    else if (eps_m <= 2.65d+228) then
        tmp = t_5
    else if (eps_m <= 2.7d+228) then
        tmp = t_1
    else if (eps_m <= 6.2d+230) then
        tmp = t_4
    else if (eps_m <= 9.2d+232) then
        tmp = t_6
    else if (eps_m <= 1.4d+236) then
        tmp = t_5
    else if (eps_m <= 1.45d+236) then
        tmp = t_1
    else if (eps_m <= 2.05d+237) then
        tmp = t_5
    else if (eps_m <= 3.3d+237) then
        tmp = t_4
    else if (eps_m <= 5.9d+237) then
        tmp = (2.0d0 - t_0) / 2.0d0
    else if (eps_m <= 6d+241) then
        tmp = t_2
    else if (eps_m <= 1.5d+243) then
        tmp = t_5
    else if (eps_m <= 3.2d+248) then
        tmp = t_3
    else if (eps_m <= 3.25d+248) then
        tmp = t_1
    else if (eps_m <= 1.5d+254) then
        tmp = t_2
    else if (eps_m <= 5.2d+265) then
        tmp = t_6
    else if (eps_m <= 1.3d+268) then
        tmp = t_5
    else if (eps_m <= 2d+272) then
        tmp = t_6
    else if (eps_m <= 5d+278) then
        tmp = t_5
    else if (eps_m <= 9.5d+279) then
        tmp = t_6
    else if (eps_m <= 2.3d+283) then
        tmp = t_4
    else if ((eps_m <= 2d+295) .or. (.not. (eps_m <= 2.7d+299))) then
        tmp = t_6
    else
        tmp = t_5
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (eps_m + (-1.0 / eps_m));
	double t_1 = (2.0 - (x * eps_m)) / 2.0;
	double t_2 = (2.0 + t_0) / 2.0;
	double t_3 = (1.0 + Math.exp(x)) / 2.0;
	double t_4 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double t_5 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_6 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	double tmp;
	if (eps_m <= 1.9e+107) {
		tmp = t_3;
	} else if (eps_m <= 2.1e+179) {
		tmp = t_6;
	} else if (eps_m <= 1.32e+181) {
		tmp = t_5;
	} else if (eps_m <= 3.4e+181) {
		tmp = t_6;
	} else if (eps_m <= 1.55e+184) {
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else if (eps_m <= 1.1e+187) {
		tmp = t_5;
	} else if (eps_m <= 3.5e+192) {
		tmp = t_6;
	} else if (eps_m <= 9.6e+197) {
		tmp = t_5;
	} else if (eps_m <= 1.65e+203) {
		tmp = t_6;
	} else if (eps_m <= 2.65e+228) {
		tmp = t_5;
	} else if (eps_m <= 2.7e+228) {
		tmp = t_1;
	} else if (eps_m <= 6.2e+230) {
		tmp = t_4;
	} else if (eps_m <= 9.2e+232) {
		tmp = t_6;
	} else if (eps_m <= 1.4e+236) {
		tmp = t_5;
	} else if (eps_m <= 1.45e+236) {
		tmp = t_1;
	} else if (eps_m <= 2.05e+237) {
		tmp = t_5;
	} else if (eps_m <= 3.3e+237) {
		tmp = t_4;
	} else if (eps_m <= 5.9e+237) {
		tmp = (2.0 - t_0) / 2.0;
	} else if (eps_m <= 6e+241) {
		tmp = t_2;
	} else if (eps_m <= 1.5e+243) {
		tmp = t_5;
	} else if (eps_m <= 3.2e+248) {
		tmp = t_3;
	} else if (eps_m <= 3.25e+248) {
		tmp = t_1;
	} else if (eps_m <= 1.5e+254) {
		tmp = t_2;
	} else if (eps_m <= 5.2e+265) {
		tmp = t_6;
	} else if (eps_m <= 1.3e+268) {
		tmp = t_5;
	} else if (eps_m <= 2e+272) {
		tmp = t_6;
	} else if (eps_m <= 5e+278) {
		tmp = t_5;
	} else if (eps_m <= 9.5e+279) {
		tmp = t_6;
	} else if (eps_m <= 2.3e+283) {
		tmp = t_4;
	} else if ((eps_m <= 2e+295) || !(eps_m <= 2.7e+299)) {
		tmp = t_6;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (eps_m + (-1.0 / eps_m))
	t_1 = (2.0 - (x * eps_m)) / 2.0
	t_2 = (2.0 + t_0) / 2.0
	t_3 = (1.0 + math.exp(x)) / 2.0
	t_4 = (eps_m * (x + (2.0 / eps_m))) / 2.0
	t_5 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0
	t_6 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0
	tmp = 0
	if eps_m <= 1.9e+107:
		tmp = t_3
	elif eps_m <= 2.1e+179:
		tmp = t_6
	elif eps_m <= 1.32e+181:
		tmp = t_5
	elif eps_m <= 3.4e+181:
		tmp = t_6
	elif eps_m <= 1.55e+184:
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + (1.0 + (-1.0 / eps_m))) / 2.0
	elif eps_m <= 1.1e+187:
		tmp = t_5
	elif eps_m <= 3.5e+192:
		tmp = t_6
	elif eps_m <= 9.6e+197:
		tmp = t_5
	elif eps_m <= 1.65e+203:
		tmp = t_6
	elif eps_m <= 2.65e+228:
		tmp = t_5
	elif eps_m <= 2.7e+228:
		tmp = t_1
	elif eps_m <= 6.2e+230:
		tmp = t_4
	elif eps_m <= 9.2e+232:
		tmp = t_6
	elif eps_m <= 1.4e+236:
		tmp = t_5
	elif eps_m <= 1.45e+236:
		tmp = t_1
	elif eps_m <= 2.05e+237:
		tmp = t_5
	elif eps_m <= 3.3e+237:
		tmp = t_4
	elif eps_m <= 5.9e+237:
		tmp = (2.0 - t_0) / 2.0
	elif eps_m <= 6e+241:
		tmp = t_2
	elif eps_m <= 1.5e+243:
		tmp = t_5
	elif eps_m <= 3.2e+248:
		tmp = t_3
	elif eps_m <= 3.25e+248:
		tmp = t_1
	elif eps_m <= 1.5e+254:
		tmp = t_2
	elif eps_m <= 5.2e+265:
		tmp = t_6
	elif eps_m <= 1.3e+268:
		tmp = t_5
	elif eps_m <= 2e+272:
		tmp = t_6
	elif eps_m <= 5e+278:
		tmp = t_5
	elif eps_m <= 9.5e+279:
		tmp = t_6
	elif eps_m <= 2.3e+283:
		tmp = t_4
	elif (eps_m <= 2e+295) or not (eps_m <= 2.7e+299):
		tmp = t_6
	else:
		tmp = t_5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(eps_m + Float64(-1.0 / eps_m)))
	t_1 = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0)
	t_2 = Float64(Float64(2.0 + t_0) / 2.0)
	t_3 = Float64(Float64(1.0 + exp(x)) / 2.0)
	t_4 = Float64(Float64(eps_m * Float64(x + Float64(2.0 / eps_m))) / 2.0)
	t_5 = Float64(Float64(Float64(Float64(eps_m * Float64(Float64(x * eps_m) + 2.0)) - x) / eps_m) / 2.0)
	t_6 = Float64(Float64(2.0 + Float64(Float64(x + Float64(eps_m * Float64(Float64(x * 0.0) - Float64(x * eps_m)))) / eps_m)) / 2.0)
	tmp = 0.0
	if (eps_m <= 1.9e+107)
		tmp = t_3;
	elseif (eps_m <= 2.1e+179)
		tmp = t_6;
	elseif (eps_m <= 1.32e+181)
		tmp = t_5;
	elseif (eps_m <= 3.4e+181)
		tmp = t_6;
	elseif (eps_m <= 1.55e+184)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * Float64(1.0 + Float64(x * Float64(eps_m + -1.0)))) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	elseif (eps_m <= 1.1e+187)
		tmp = t_5;
	elseif (eps_m <= 3.5e+192)
		tmp = t_6;
	elseif (eps_m <= 9.6e+197)
		tmp = t_5;
	elseif (eps_m <= 1.65e+203)
		tmp = t_6;
	elseif (eps_m <= 2.65e+228)
		tmp = t_5;
	elseif (eps_m <= 2.7e+228)
		tmp = t_1;
	elseif (eps_m <= 6.2e+230)
		tmp = t_4;
	elseif (eps_m <= 9.2e+232)
		tmp = t_6;
	elseif (eps_m <= 1.4e+236)
		tmp = t_5;
	elseif (eps_m <= 1.45e+236)
		tmp = t_1;
	elseif (eps_m <= 2.05e+237)
		tmp = t_5;
	elseif (eps_m <= 3.3e+237)
		tmp = t_4;
	elseif (eps_m <= 5.9e+237)
		tmp = Float64(Float64(2.0 - t_0) / 2.0);
	elseif (eps_m <= 6e+241)
		tmp = t_2;
	elseif (eps_m <= 1.5e+243)
		tmp = t_5;
	elseif (eps_m <= 3.2e+248)
		tmp = t_3;
	elseif (eps_m <= 3.25e+248)
		tmp = t_1;
	elseif (eps_m <= 1.5e+254)
		tmp = t_2;
	elseif (eps_m <= 5.2e+265)
		tmp = t_6;
	elseif (eps_m <= 1.3e+268)
		tmp = t_5;
	elseif (eps_m <= 2e+272)
		tmp = t_6;
	elseif (eps_m <= 5e+278)
		tmp = t_5;
	elseif (eps_m <= 9.5e+279)
		tmp = t_6;
	elseif (eps_m <= 2.3e+283)
		tmp = t_4;
	elseif ((eps_m <= 2e+295) || !(eps_m <= 2.7e+299))
		tmp = t_6;
	else
		tmp = t_5;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (eps_m + (-1.0 / eps_m));
	t_1 = (2.0 - (x * eps_m)) / 2.0;
	t_2 = (2.0 + t_0) / 2.0;
	t_3 = (1.0 + exp(x)) / 2.0;
	t_4 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	t_5 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	t_6 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	tmp = 0.0;
	if (eps_m <= 1.9e+107)
		tmp = t_3;
	elseif (eps_m <= 2.1e+179)
		tmp = t_6;
	elseif (eps_m <= 1.32e+181)
		tmp = t_5;
	elseif (eps_m <= 3.4e+181)
		tmp = t_6;
	elseif (eps_m <= 1.55e+184)
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + (1.0 + (-1.0 / eps_m))) / 2.0;
	elseif (eps_m <= 1.1e+187)
		tmp = t_5;
	elseif (eps_m <= 3.5e+192)
		tmp = t_6;
	elseif (eps_m <= 9.6e+197)
		tmp = t_5;
	elseif (eps_m <= 1.65e+203)
		tmp = t_6;
	elseif (eps_m <= 2.65e+228)
		tmp = t_5;
	elseif (eps_m <= 2.7e+228)
		tmp = t_1;
	elseif (eps_m <= 6.2e+230)
		tmp = t_4;
	elseif (eps_m <= 9.2e+232)
		tmp = t_6;
	elseif (eps_m <= 1.4e+236)
		tmp = t_5;
	elseif (eps_m <= 1.45e+236)
		tmp = t_1;
	elseif (eps_m <= 2.05e+237)
		tmp = t_5;
	elseif (eps_m <= 3.3e+237)
		tmp = t_4;
	elseif (eps_m <= 5.9e+237)
		tmp = (2.0 - t_0) / 2.0;
	elseif (eps_m <= 6e+241)
		tmp = t_2;
	elseif (eps_m <= 1.5e+243)
		tmp = t_5;
	elseif (eps_m <= 3.2e+248)
		tmp = t_3;
	elseif (eps_m <= 3.25e+248)
		tmp = t_1;
	elseif (eps_m <= 1.5e+254)
		tmp = t_2;
	elseif (eps_m <= 5.2e+265)
		tmp = t_6;
	elseif (eps_m <= 1.3e+268)
		tmp = t_5;
	elseif (eps_m <= 2e+272)
		tmp = t_6;
	elseif (eps_m <= 5e+278)
		tmp = t_5;
	elseif (eps_m <= 9.5e+279)
		tmp = t_6;
	elseif (eps_m <= 2.3e+283)
		tmp = t_4;
	elseif ((eps_m <= 2e+295) || ~((eps_m <= 2.7e+299)))
		tmp = t_6;
	else
		tmp = t_5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(eps$95$m + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(eps$95$m * N[(x + N[(2.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(eps$95$m * N[(N[(x * eps$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$6 = N[(N[(2.0 + N[(N[(x + N[(eps$95$m * N[(N[(x * 0.0), $MachinePrecision] - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[eps$95$m, 1.9e+107], t$95$3, If[LessEqual[eps$95$m, 2.1e+179], t$95$6, If[LessEqual[eps$95$m, 1.32e+181], t$95$5, If[LessEqual[eps$95$m, 3.4e+181], t$95$6, If[LessEqual[eps$95$m, 1.55e+184], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 1.1e+187], t$95$5, If[LessEqual[eps$95$m, 3.5e+192], t$95$6, If[LessEqual[eps$95$m, 9.6e+197], t$95$5, If[LessEqual[eps$95$m, 1.65e+203], t$95$6, If[LessEqual[eps$95$m, 2.65e+228], t$95$5, If[LessEqual[eps$95$m, 2.7e+228], t$95$1, If[LessEqual[eps$95$m, 6.2e+230], t$95$4, If[LessEqual[eps$95$m, 9.2e+232], t$95$6, If[LessEqual[eps$95$m, 1.4e+236], t$95$5, If[LessEqual[eps$95$m, 1.45e+236], t$95$1, If[LessEqual[eps$95$m, 2.05e+237], t$95$5, If[LessEqual[eps$95$m, 3.3e+237], t$95$4, If[LessEqual[eps$95$m, 5.9e+237], N[(N[(2.0 - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 6e+241], t$95$2, If[LessEqual[eps$95$m, 1.5e+243], t$95$5, If[LessEqual[eps$95$m, 3.2e+248], t$95$3, If[LessEqual[eps$95$m, 3.25e+248], t$95$1, If[LessEqual[eps$95$m, 1.5e+254], t$95$2, If[LessEqual[eps$95$m, 5.2e+265], t$95$6, If[LessEqual[eps$95$m, 1.3e+268], t$95$5, If[LessEqual[eps$95$m, 2e+272], t$95$6, If[LessEqual[eps$95$m, 5e+278], t$95$5, If[LessEqual[eps$95$m, 9.5e+279], t$95$6, If[LessEqual[eps$95$m, 2.3e+283], t$95$4, If[Or[LessEqual[eps$95$m, 2e+295], N[Not[LessEqual[eps$95$m, 2.7e+299]], $MachinePrecision]], t$95$6, t$95$5]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if eps < 1.8999999999999999e107 or 1.49999999999999992e243 < eps < 3.19999999999999985e248

   1. Initial program 65.9%

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

   3. Simplified 59.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 99.0%

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

   6. Taylor expanded in x around 0 69.2%

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

      1. sub-neg 69.2%

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

      2. metadata-eval 69.2%

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

      3. distribute-rgt-in 69.2%

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

      4. add-sqr-sqrt 34.9%

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

      5. sqrt-unprod 60.4%

         \[\leadsto \frac{e^{\varepsilon \cdot \color{blue}{\sqrt{x \cdot x}} + -1 \cdot x} + 1}{2} \]

      6. sqr-neg 60.4%

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

      7. sqrt-unprod 25.4%

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

      8. add-sqr-sqrt 65.7%

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

      9. neg-mul-1 65.7%

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

      10. *-un-lft-identity 65.7%

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

      11. distribute-rgt-in 65.7%

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

      12. +-commutative 65.7%

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

      13. *-commutative 65.7%

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

      14. distribute-rgt-neg-out 65.7%

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

      15. neg-sub0 65.7%

          \[\leadsto \frac{e^{\color{blue}{0 - \left(1 + \varepsilon\right) \cdot x}} + 1}{2} \]

      16. add-sqr-sqrt 40.2%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} + 1}{2} \]

      17. sqrt-unprod 73.2%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\sqrt{x \cdot x}}} + 1}{2} \]

      18. sqr-neg 73.2%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}} + 1}{2} \]

      19. sqrt-unprod 34.3%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}} + 1}{2} \]

      20. add-sqr-sqrt 69.0%

          \[\leadsto \frac{e^{0 - \left(1 + \varepsilon\right) \cdot \color{blue}{\left(-x\right)}} + 1}{2} \]

      21. *-commutative 69.0%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(-x\right) \cdot \left(1 + \varepsilon\right)}} + 1}{2} \]

      22. +-commutative 69.0%

          \[\leadsto \frac{e^{0 - \left(-x\right) \cdot \color{blue}{\left(\varepsilon + 1\right)}} + 1}{2} \]

      23. distribute-rgt-in 69.0%

          \[\leadsto \frac{e^{0 - \color{blue}{\left(\varepsilon \cdot \left(-x\right) + 1 \cdot \left(-x\right)\right)}} + 1}{2} \]

   8. Applied egg-rr 65.5%

      \[\leadsto \frac{e^{\color{blue}{0 - x \cdot \left(\varepsilon + -1\right)}} + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 65.5%

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

      2. distribute-rgt-neg-in 65.5%

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

      3. +-commutative 65.5%

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

      4. distribute-neg-in 65.5%

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

      5. metadata-eval 65.5%

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

      6. sub-neg 65.5%

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

   10. Simplified 65.5%

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

   11. Taylor expanded in eps around 0 65.8%

       \[\leadsto \frac{\color{blue}{e^{x}} + 1}{2} \]

   if 1.8999999999999999e107 < eps < 2.0999999999999999e179 or 1.31999999999999996e181 < eps < 3.40000000000000031e181 or 1.0999999999999999e187 < eps < 3.49999999999999983e192 or 9.5999999999999995e197 < eps < 1.64999999999999995e203 or 6.19999999999999963e230 < eps < 9.20000000000000024e232 or 1.50000000000000003e254 < eps < 5.2000000000000003e265 or 1.29999999999999997e268 < eps < 2.0000000000000001e272 or 5.00000000000000029e278 < eps < 9.4999999999999991e279 or 2.3000000000000002e283 < eps < 2e295 or 2.7e299 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 33.6%

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

   6. Taylor expanded in x around 0 17.9%

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

      1. associate-*r* 17.9%

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

      2. neg-mul-1 17.9%

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

      3. *-commutative 17.9%

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

   8. Simplified 17.9%

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

      1. distribute-lft-in 17.9%

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

      2. *-rgt-identity 17.9%

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

      3. distribute-rgt-in 17.9%

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

      4. add-sqr-sqrt 8.7%

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

      5. sqrt-unprod 17.5%

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

      6. sqr-neg 17.5%

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

      7. sqrt-unprod 8.6%

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

      8. add-sqr-sqrt 50.8%

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

      9. un-div-inv 50.8%

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

      10. add-sqr-sqrt 42.2%

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

      11. sqrt-unprod 24.6%

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

      12. sqr-neg 24.6%

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

      13. sqrt-unprod 8.6%

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

      14. add-sqr-sqrt 50.8%

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

   10. Applied egg-rr 50.8%

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

   11. Taylor expanded in eps around 0 80.3%

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

       1. associate-+r+ 80.3%

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

       2. distribute-rgt1-in 80.3%

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

       3. metadata-eval 80.3%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(\color{blue}{0} \cdot x + -1 \cdot \left(\varepsilon \cdot x\right)\right)}{\varepsilon}}{2} \]

       4. associate-*r* 80.3%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(0 \cdot x + \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot x}\right)}{\varepsilon}}{2} \]

       5. neg-mul-1 80.3%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(0 \cdot x + \color{blue}{\left(-\varepsilon\right)} \cdot x\right)}{\varepsilon}}{2} \]

       6. *-commutative 80.3%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(0 \cdot x + \color{blue}{x \cdot \left(-\varepsilon\right)}\right)}{\varepsilon}}{2} \]

   13. Simplified 80.3%

       \[\leadsto \frac{2 + \color{blue}{\frac{x + \varepsilon \cdot \left(0 \cdot x + x \cdot \left(-\varepsilon\right)\right)}{\varepsilon}}}{2} \]

   if 2.0999999999999999e179 < eps < 1.31999999999999996e181 or 1.5499999999999999e184 < eps < 1.0999999999999999e187 or 3.49999999999999983e192 < eps < 9.5999999999999995e197 or 1.64999999999999995e203 < eps < 2.65e228 or 9.20000000000000024e232 < eps < 1.39999999999999996e236 or 1.45e236 < eps < 2.05000000000000001e237 or 6.00000000000000031e241 < eps < 1.49999999999999992e243 or 5.2000000000000003e265 < eps < 1.29999999999999997e268 or 2.0000000000000001e272 < eps < 5.00000000000000029e278 or 2e295 < eps < 2.7e299

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in x around 0 63.8%

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

      1. associate-*r* 63.8%

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

      2. neg-mul-1 63.8%

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

      3. *-commutative 63.8%

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

   8. Simplified 63.8%

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

   9. Taylor expanded in eps around 0 100.0%

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

   if 3.40000000000000031e181 < eps < 1.5499999999999999e184

   1. Initial program 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in x around 0 40.1%

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

   6. Taylor expanded in x around 0 24.0%

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

   if 2.65e228 < eps < 2.7000000000000002e228 or 1.39999999999999996e236 < eps < 1.45e236 or 3.19999999999999985e248 < eps < 3.25000000000000024e248

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. neg-mul-1 0.0%

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

      3. *-commutative 0.0%

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

   8. Simplified 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.0%

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

      3. sqr-neg 0.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-un-lft-identity 100.0%

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

      8. distribute-rgt-in 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. *-lft-identity 100.0%

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

       3. distribute-rgt-out 100.0%

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

       4. +-commutative 100.0%

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

       5. associate-+r- 100.0%

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

       6. div-sub 100.0%

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

       7. *-inverses 100.0%

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

       8. associate-+l- 100.0%

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

       9. metadata-eval 100.0%

          \[\leadsto \frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} - \color{blue}{0}\right) - \varepsilon\right)}{2} \]

   12. Simplified 100.0%

       \[\leadsto \frac{2 + \color{blue}{x \cdot \left(\left(\frac{1}{\varepsilon} - 0\right) - \varepsilon\right)}}{2} \]

   13. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

       3. *-commutative 100.0%

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

   15. Simplified 100.0%

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

   if 2.7000000000000002e228 < eps < 6.19999999999999963e230 or 2.05000000000000001e237 < eps < 3.3000000000000001e237 or 9.4999999999999991e279 < eps < 2.3000000000000002e283

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in x around 0 67.8%

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

      1. associate-*r* 67.8%

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

      2. neg-mul-1 67.8%

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

      3. *-commutative 67.8%

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

   8. Simplified 67.8%

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

   9. Taylor expanded in eps around inf 67.8%

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

       1. associate-*r/ 67.8%

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

       2. metadata-eval 67.8%

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

   11. Simplified 67.8%

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

   if 3.3000000000000001e237 < eps < 5.8999999999999997e237

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around 0 0.5%

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

      1. associate-*r* 0.5%

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

      2. neg-mul-1 0.5%

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

      3. *-commutative 0.5%

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

   8. Simplified 0.5%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 3.1%

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

      3. sqr-neg 3.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 3.7%

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

      6. distribute-rgt-in 3.7%

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

      7. *-un-lft-identity 3.7%

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

      8. distribute-rgt-in 3.7%

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

   10. Applied egg-rr 3.7%

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

       1. associate-*l/ 3.7%

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

       2. *-lft-identity 3.7%

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

       3. distribute-rgt-out 3.7%

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

       4. +-commutative 3.7%

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

       5. associate-+r- 3.7%

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

       6. div-sub 3.7%

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

       7. *-inverses 3.7%

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

       8. associate-+l- 3.7%

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

       9. metadata-eval 3.7%

          \[\leadsto \frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} - \color{blue}{0}\right) - \varepsilon\right)}{2} \]

   12. Simplified 3.7%

       \[\leadsto \frac{2 + \color{blue}{x \cdot \left(\left(\frac{1}{\varepsilon} - 0\right) - \varepsilon\right)}}{2} \]

   if 5.8999999999999997e237 < eps < 6.00000000000000031e241 or 3.25000000000000024e248 < eps < 1.50000000000000003e254

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in x around 0 67.7%

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

      1. mul-1-neg 67.7%

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

      2. distribute-rgt-neg-in 67.7%

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

      3. mul-1-neg 67.7%

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

      4. mul-1-neg 67.7%

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

      5. distribute-rgt-neg-in 67.7%

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

      6. neg-sub0 67.7%

         \[\leadsto \frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(0 - \left(1 - \varepsilon\right)\right)}\right)}{2} \]

      7. associate--r- 67.7%

         \[\leadsto \frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(\left(0 - 1\right) + \varepsilon\right)}\right)}{2} \]

      8. metadata-eval 67.7%

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

      9. +-commutative 67.7%

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

      10. +-commutative 67.7%

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

      11. distribute-rgt1-in 67.7%

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

      12. +-commutative 67.7%

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

      13. distribute-rgt-in 67.7%

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

      14. neg-mul-1 67.7%

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

      15. rgt-mult-inverse 67.7%

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

      16. associate-+l+ 67.7%

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

      17. +-commutative 67.7%

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

      18. associate-+r+ 67.7%

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

      19. metadata-eval 67.7%

          \[\leadsto \frac{2 + x \cdot \left(\varepsilon + \left(\color{blue}{0} + \left(-\frac{1}{\varepsilon}\right)\right)\right)}{2} \]

      20. sub-neg 67.7%

          \[\leadsto \frac{2 + x \cdot \left(\varepsilon + \color{blue}{\left(0 - \frac{1}{\varepsilon}\right)}\right)}{2} \]

      21. neg-sub0 67.7%

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

      22. distribute-neg-frac 67.7%

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

      23. metadata-eval 67.7%

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

   8. Simplified 67.7%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}}{2} \]
3. Recombined 8 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.9 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{+179}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.32 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.4 \cdot 10^{+181}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.55 \cdot 10^{+184}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + x \cdot \left(\varepsilon + -1\right)\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.1 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{+192}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 9.6 \cdot 10^{+197}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.65 \cdot 10^{+203}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.65 \cdot 10^{+228}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.7 \cdot 10^{+228}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{+230}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 9.2 \cdot 10^{+232}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{+236}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.45 \cdot 10^{+236}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.05 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{+237}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 5.9 \cdot 10^{+237}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{+241}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.5 \cdot 10^{+243}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{+248}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.25 \cdot 10^{+248}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.5 \cdot 10^{+254}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{+265}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{+268}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{+278}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{+279}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{+283}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+295} \lor \neg \left(\varepsilon \leq 2.7 \cdot 10^{+299}\right):\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{\frac{eps\_m \cdot \left(x \cdot eps\_m + 2\right) - x}{eps\_m}}{2}\\ t_1 := \frac{2 + \frac{x + eps\_m \cdot \left(x \cdot 0 - x \cdot eps\_m\right)}{eps\_m}}{2}\\ t_2 := \frac{2 - x \cdot eps\_m}{2}\\ t_3 := x \cdot \left(eps\_m + \frac{-1}{eps\_m}\right)\\ t_4 := \frac{2 + t\_3}{2}\\ t_5 := \frac{eps\_m \cdot \left(x + \frac{2}{eps\_m}\right)}{2}\\ \mathbf{if}\;eps\_m \leq 1.28 \cdot 10^{+64}:\\ \;\;\;\;1\\ \mathbf{elif}\;eps\_m \leq 8.8 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 5 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 2.3 \cdot 10^{+165}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 2 \cdot 10^{+179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 3.35 \cdot 10^{+180}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 8.2 \cdot 10^{+183}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 1.55 \cdot 10^{+184}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot \left(1 + x \cdot \left(eps\_m + -1\right)\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{elif}\;eps\_m \leq 8.5 \cdot 10^{+186}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 2 \cdot 10^{+191}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 1.9 \cdot 10^{+198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 2.8 \cdot 10^{+203}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 2.65 \cdot 10^{+228}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 3.2 \cdot 10^{+228}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 2 \cdot 10^{+230}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 2.9 \cdot 10^{+232}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 1.35 \cdot 10^{+236}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 1.45 \cdot 10^{+236}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 2.05 \cdot 10^{+237}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 5.8 \cdot 10^{+237}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 6.3 \cdot 10^{+237}:\\ \;\;\;\;\frac{2 - t\_3}{2}\\ \mathbf{elif}\;eps\_m \leq 6 \cdot 10^{+241}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 6.1 \cdot 10^{+246}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 6.2 \cdot 10^{+246}:\\ \;\;\;\;1\\ \mathbf{elif}\;eps\_m \leq 3.25 \cdot 10^{+248}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 4.9 \cdot 10^{+253}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 3.2 \cdot 10^{+265}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 1.3 \cdot 10^{+268}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 4.8 \cdot 10^{+272}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 1.85 \cdot 10^{+278}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 1.4 \cdot 10^{+280}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 1.7 \cdot 10^{+283}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 3.6 \cdot 10^{+295} \lor \neg \left(eps\_m \leq 1.45 \cdot 10^{+299}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (/ (- (* eps_m (+ (* x eps_m) 2.0)) x) eps_m) 2.0))
        (t_1
         (/ (+ 2.0 (/ (+ x (* eps_m (- (* x 0.0) (* x eps_m)))) eps_m)) 2.0))
        (t_2 (/ (- 2.0 (* x eps_m)) 2.0))
        (t_3 (* x (+ eps_m (/ -1.0 eps_m))))
        (t_4 (/ (+ 2.0 t_3) 2.0))
        (t_5 (/ (* eps_m (+ x (/ 2.0 eps_m))) 2.0)))
   (if (<= eps_m 1.28e+64)
     1.0
     (if (<= eps_m 8.8e+144)
       t_1
       (if (<= eps_m 5e+153)
         t_0
         (if (<= eps_m 2.3e+165)
           t_4
           (if (<= eps_m 2e+179)
             t_1
             (if (<= eps_m 3.35e+180)
               t_0
               (if (<= eps_m 8.2e+183)
                 t_1
                 (if (<= eps_m 1.55e+184)
                   (/
                    (+
                     (* (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (* x (+ eps_m -1.0))))
                     (+ 1.0 (/ -1.0 eps_m)))
                    2.0)
                   (if (<= eps_m 8.5e+186)
                     t_0
                     (if (<= eps_m 2e+191)
                       t_1
                       (if (<= eps_m 1.9e+198)
                         t_0
                         (if (<= eps_m 2.8e+203)
                           t_1
                           (if (<= eps_m 2.65e+228)
                             t_0
                             (if (<= eps_m 3.2e+228)
                               t_2
                               (if (<= eps_m 2e+230)
                                 t_5
                                 (if (<= eps_m 2.9e+232)
                                   t_1
                                   (if (<= eps_m 1.35e+236)
                                     t_0
                                     (if (<= eps_m 1.45e+236)
                                       t_2
                                       (if (<= eps_m 2.05e+237)
                                         t_0
                                         (if (<= eps_m 5.8e+237)
                                           t_5
                                           (if (<= eps_m 6.3e+237)
                                             (/ (- 2.0 t_3) 2.0)
                                             (if (<= eps_m 6e+241)
                                               t_4
                                               (if (<= eps_m 6.1e+246)
                                                 t_0
                                                 (if (<= eps_m 6.2e+246)
                                                   1.0
                                                   (if (<= eps_m 3.25e+248)
                                                     t_2
                                                     (if (<= eps_m 4.9e+253)
                                                       t_4
                                                       (if (<= eps_m 3.2e+265)
                                                         t_1
                                                         (if (<=
                                                              eps_m
                                                              1.3e+268)
                                                           t_0
                                                           (if (<=
                                                                eps_m
                                                                4.8e+272)
                                                             t_1
                                                             (if (<=
                                                                  eps_m
                                                                  1.85e+278)
                                                               t_0
                                                               (if (<=
                                                                    eps_m
                                                                    1.4e+280)
                                                                 t_1
                                                                 (if (<=
                                                                      eps_m
                                                                      1.7e+283)
                                                                   t_5
                                                                   (if (or (<=
                                                                            eps_m
                                                                            3.6e+295)
                                                                           (not
                                                                            (<=
                                                                             eps_m
                                                                             1.45e+299)))
                                                                     t_1
                                                                     t_0)))))))))))))))))))))))))))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_1 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	double t_2 = (2.0 - (x * eps_m)) / 2.0;
	double t_3 = x * (eps_m + (-1.0 / eps_m));
	double t_4 = (2.0 + t_3) / 2.0;
	double t_5 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double tmp;
	if (eps_m <= 1.28e+64) {
		tmp = 1.0;
	} else if (eps_m <= 8.8e+144) {
		tmp = t_1;
	} else if (eps_m <= 5e+153) {
		tmp = t_0;
	} else if (eps_m <= 2.3e+165) {
		tmp = t_4;
	} else if (eps_m <= 2e+179) {
		tmp = t_1;
	} else if (eps_m <= 3.35e+180) {
		tmp = t_0;
	} else if (eps_m <= 8.2e+183) {
		tmp = t_1;
	} else if (eps_m <= 1.55e+184) {
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else if (eps_m <= 8.5e+186) {
		tmp = t_0;
	} else if (eps_m <= 2e+191) {
		tmp = t_1;
	} else if (eps_m <= 1.9e+198) {
		tmp = t_0;
	} else if (eps_m <= 2.8e+203) {
		tmp = t_1;
	} else if (eps_m <= 2.65e+228) {
		tmp = t_0;
	} else if (eps_m <= 3.2e+228) {
		tmp = t_2;
	} else if (eps_m <= 2e+230) {
		tmp = t_5;
	} else if (eps_m <= 2.9e+232) {
		tmp = t_1;
	} else if (eps_m <= 1.35e+236) {
		tmp = t_0;
	} else if (eps_m <= 1.45e+236) {
		tmp = t_2;
	} else if (eps_m <= 2.05e+237) {
		tmp = t_0;
	} else if (eps_m <= 5.8e+237) {
		tmp = t_5;
	} else if (eps_m <= 6.3e+237) {
		tmp = (2.0 - t_3) / 2.0;
	} else if (eps_m <= 6e+241) {
		tmp = t_4;
	} else if (eps_m <= 6.1e+246) {
		tmp = t_0;
	} else if (eps_m <= 6.2e+246) {
		tmp = 1.0;
	} else if (eps_m <= 3.25e+248) {
		tmp = t_2;
	} else if (eps_m <= 4.9e+253) {
		tmp = t_4;
	} else if (eps_m <= 3.2e+265) {
		tmp = t_1;
	} else if (eps_m <= 1.3e+268) {
		tmp = t_0;
	} else if (eps_m <= 4.8e+272) {
		tmp = t_1;
	} else if (eps_m <= 1.85e+278) {
		tmp = t_0;
	} else if (eps_m <= 1.4e+280) {
		tmp = t_1;
	} else if (eps_m <= 1.7e+283) {
		tmp = t_5;
	} else if ((eps_m <= 3.6e+295) || !(eps_m <= 1.45e+299)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (((eps_m * ((x * eps_m) + 2.0d0)) - x) / eps_m) / 2.0d0
    t_1 = (2.0d0 + ((x + (eps_m * ((x * 0.0d0) - (x * eps_m)))) / eps_m)) / 2.0d0
    t_2 = (2.0d0 - (x * eps_m)) / 2.0d0
    t_3 = x * (eps_m + ((-1.0d0) / eps_m))
    t_4 = (2.0d0 + t_3) / 2.0d0
    t_5 = (eps_m * (x + (2.0d0 / eps_m))) / 2.0d0
    if (eps_m <= 1.28d+64) then
        tmp = 1.0d0
    else if (eps_m <= 8.8d+144) then
        tmp = t_1
    else if (eps_m <= 5d+153) then
        tmp = t_0
    else if (eps_m <= 2.3d+165) then
        tmp = t_4
    else if (eps_m <= 2d+179) then
        tmp = t_1
    else if (eps_m <= 3.35d+180) then
        tmp = t_0
    else if (eps_m <= 8.2d+183) then
        tmp = t_1
    else if (eps_m <= 1.55d+184) then
        tmp = (((1.0d0 + (1.0d0 / eps_m)) * (1.0d0 + (x * (eps_m + (-1.0d0))))) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else if (eps_m <= 8.5d+186) then
        tmp = t_0
    else if (eps_m <= 2d+191) then
        tmp = t_1
    else if (eps_m <= 1.9d+198) then
        tmp = t_0
    else if (eps_m <= 2.8d+203) then
        tmp = t_1
    else if (eps_m <= 2.65d+228) then
        tmp = t_0
    else if (eps_m <= 3.2d+228) then
        tmp = t_2
    else if (eps_m <= 2d+230) then
        tmp = t_5
    else if (eps_m <= 2.9d+232) then
        tmp = t_1
    else if (eps_m <= 1.35d+236) then
        tmp = t_0
    else if (eps_m <= 1.45d+236) then
        tmp = t_2
    else if (eps_m <= 2.05d+237) then
        tmp = t_0
    else if (eps_m <= 5.8d+237) then
        tmp = t_5
    else if (eps_m <= 6.3d+237) then
        tmp = (2.0d0 - t_3) / 2.0d0
    else if (eps_m <= 6d+241) then
        tmp = t_4
    else if (eps_m <= 6.1d+246) then
        tmp = t_0
    else if (eps_m <= 6.2d+246) then
        tmp = 1.0d0
    else if (eps_m <= 3.25d+248) then
        tmp = t_2
    else if (eps_m <= 4.9d+253) then
        tmp = t_4
    else if (eps_m <= 3.2d+265) then
        tmp = t_1
    else if (eps_m <= 1.3d+268) then
        tmp = t_0
    else if (eps_m <= 4.8d+272) then
        tmp = t_1
    else if (eps_m <= 1.85d+278) then
        tmp = t_0
    else if (eps_m <= 1.4d+280) then
        tmp = t_1
    else if (eps_m <= 1.7d+283) then
        tmp = t_5
    else if ((eps_m <= 3.6d+295) .or. (.not. (eps_m <= 1.45d+299))) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_1 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	double t_2 = (2.0 - (x * eps_m)) / 2.0;
	double t_3 = x * (eps_m + (-1.0 / eps_m));
	double t_4 = (2.0 + t_3) / 2.0;
	double t_5 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double tmp;
	if (eps_m <= 1.28e+64) {
		tmp = 1.0;
	} else if (eps_m <= 8.8e+144) {
		tmp = t_1;
	} else if (eps_m <= 5e+153) {
		tmp = t_0;
	} else if (eps_m <= 2.3e+165) {
		tmp = t_4;
	} else if (eps_m <= 2e+179) {
		tmp = t_1;
	} else if (eps_m <= 3.35e+180) {
		tmp = t_0;
	} else if (eps_m <= 8.2e+183) {
		tmp = t_1;
	} else if (eps_m <= 1.55e+184) {
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else if (eps_m <= 8.5e+186) {
		tmp = t_0;
	} else if (eps_m <= 2e+191) {
		tmp = t_1;
	} else if (eps_m <= 1.9e+198) {
		tmp = t_0;
	} else if (eps_m <= 2.8e+203) {
		tmp = t_1;
	} else if (eps_m <= 2.65e+228) {
		tmp = t_0;
	} else if (eps_m <= 3.2e+228) {
		tmp = t_2;
	} else if (eps_m <= 2e+230) {
		tmp = t_5;
	} else if (eps_m <= 2.9e+232) {
		tmp = t_1;
	} else if (eps_m <= 1.35e+236) {
		tmp = t_0;
	} else if (eps_m <= 1.45e+236) {
		tmp = t_2;
	} else if (eps_m <= 2.05e+237) {
		tmp = t_0;
	} else if (eps_m <= 5.8e+237) {
		tmp = t_5;
	} else if (eps_m <= 6.3e+237) {
		tmp = (2.0 - t_3) / 2.0;
	} else if (eps_m <= 6e+241) {
		tmp = t_4;
	} else if (eps_m <= 6.1e+246) {
		tmp = t_0;
	} else if (eps_m <= 6.2e+246) {
		tmp = 1.0;
	} else if (eps_m <= 3.25e+248) {
		tmp = t_2;
	} else if (eps_m <= 4.9e+253) {
		tmp = t_4;
	} else if (eps_m <= 3.2e+265) {
		tmp = t_1;
	} else if (eps_m <= 1.3e+268) {
		tmp = t_0;
	} else if (eps_m <= 4.8e+272) {
		tmp = t_1;
	} else if (eps_m <= 1.85e+278) {
		tmp = t_0;
	} else if (eps_m <= 1.4e+280) {
		tmp = t_1;
	} else if (eps_m <= 1.7e+283) {
		tmp = t_5;
	} else if ((eps_m <= 3.6e+295) || !(eps_m <= 1.45e+299)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0
	t_1 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0
	t_2 = (2.0 - (x * eps_m)) / 2.0
	t_3 = x * (eps_m + (-1.0 / eps_m))
	t_4 = (2.0 + t_3) / 2.0
	t_5 = (eps_m * (x + (2.0 / eps_m))) / 2.0
	tmp = 0
	if eps_m <= 1.28e+64:
		tmp = 1.0
	elif eps_m <= 8.8e+144:
		tmp = t_1
	elif eps_m <= 5e+153:
		tmp = t_0
	elif eps_m <= 2.3e+165:
		tmp = t_4
	elif eps_m <= 2e+179:
		tmp = t_1
	elif eps_m <= 3.35e+180:
		tmp = t_0
	elif eps_m <= 8.2e+183:
		tmp = t_1
	elif eps_m <= 1.55e+184:
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + (1.0 + (-1.0 / eps_m))) / 2.0
	elif eps_m <= 8.5e+186:
		tmp = t_0
	elif eps_m <= 2e+191:
		tmp = t_1
	elif eps_m <= 1.9e+198:
		tmp = t_0
	elif eps_m <= 2.8e+203:
		tmp = t_1
	elif eps_m <= 2.65e+228:
		tmp = t_0
	elif eps_m <= 3.2e+228:
		tmp = t_2
	elif eps_m <= 2e+230:
		tmp = t_5
	elif eps_m <= 2.9e+232:
		tmp = t_1
	elif eps_m <= 1.35e+236:
		tmp = t_0
	elif eps_m <= 1.45e+236:
		tmp = t_2
	elif eps_m <= 2.05e+237:
		tmp = t_0
	elif eps_m <= 5.8e+237:
		tmp = t_5
	elif eps_m <= 6.3e+237:
		tmp = (2.0 - t_3) / 2.0
	elif eps_m <= 6e+241:
		tmp = t_4
	elif eps_m <= 6.1e+246:
		tmp = t_0
	elif eps_m <= 6.2e+246:
		tmp = 1.0
	elif eps_m <= 3.25e+248:
		tmp = t_2
	elif eps_m <= 4.9e+253:
		tmp = t_4
	elif eps_m <= 3.2e+265:
		tmp = t_1
	elif eps_m <= 1.3e+268:
		tmp = t_0
	elif eps_m <= 4.8e+272:
		tmp = t_1
	elif eps_m <= 1.85e+278:
		tmp = t_0
	elif eps_m <= 1.4e+280:
		tmp = t_1
	elif eps_m <= 1.7e+283:
		tmp = t_5
	elif (eps_m <= 3.6e+295) or not (eps_m <= 1.45e+299):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(Float64(eps_m * Float64(Float64(x * eps_m) + 2.0)) - x) / eps_m) / 2.0)
	t_1 = Float64(Float64(2.0 + Float64(Float64(x + Float64(eps_m * Float64(Float64(x * 0.0) - Float64(x * eps_m)))) / eps_m)) / 2.0)
	t_2 = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0)
	t_3 = Float64(x * Float64(eps_m + Float64(-1.0 / eps_m)))
	t_4 = Float64(Float64(2.0 + t_3) / 2.0)
	t_5 = Float64(Float64(eps_m * Float64(x + Float64(2.0 / eps_m))) / 2.0)
	tmp = 0.0
	if (eps_m <= 1.28e+64)
		tmp = 1.0;
	elseif (eps_m <= 8.8e+144)
		tmp = t_1;
	elseif (eps_m <= 5e+153)
		tmp = t_0;
	elseif (eps_m <= 2.3e+165)
		tmp = t_4;
	elseif (eps_m <= 2e+179)
		tmp = t_1;
	elseif (eps_m <= 3.35e+180)
		tmp = t_0;
	elseif (eps_m <= 8.2e+183)
		tmp = t_1;
	elseif (eps_m <= 1.55e+184)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * Float64(1.0 + Float64(x * Float64(eps_m + -1.0)))) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	elseif (eps_m <= 8.5e+186)
		tmp = t_0;
	elseif (eps_m <= 2e+191)
		tmp = t_1;
	elseif (eps_m <= 1.9e+198)
		tmp = t_0;
	elseif (eps_m <= 2.8e+203)
		tmp = t_1;
	elseif (eps_m <= 2.65e+228)
		tmp = t_0;
	elseif (eps_m <= 3.2e+228)
		tmp = t_2;
	elseif (eps_m <= 2e+230)
		tmp = t_5;
	elseif (eps_m <= 2.9e+232)
		tmp = t_1;
	elseif (eps_m <= 1.35e+236)
		tmp = t_0;
	elseif (eps_m <= 1.45e+236)
		tmp = t_2;
	elseif (eps_m <= 2.05e+237)
		tmp = t_0;
	elseif (eps_m <= 5.8e+237)
		tmp = t_5;
	elseif (eps_m <= 6.3e+237)
		tmp = Float64(Float64(2.0 - t_3) / 2.0);
	elseif (eps_m <= 6e+241)
		tmp = t_4;
	elseif (eps_m <= 6.1e+246)
		tmp = t_0;
	elseif (eps_m <= 6.2e+246)
		tmp = 1.0;
	elseif (eps_m <= 3.25e+248)
		tmp = t_2;
	elseif (eps_m <= 4.9e+253)
		tmp = t_4;
	elseif (eps_m <= 3.2e+265)
		tmp = t_1;
	elseif (eps_m <= 1.3e+268)
		tmp = t_0;
	elseif (eps_m <= 4.8e+272)
		tmp = t_1;
	elseif (eps_m <= 1.85e+278)
		tmp = t_0;
	elseif (eps_m <= 1.4e+280)
		tmp = t_1;
	elseif (eps_m <= 1.7e+283)
		tmp = t_5;
	elseif ((eps_m <= 3.6e+295) || !(eps_m <= 1.45e+299))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	t_1 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	t_2 = (2.0 - (x * eps_m)) / 2.0;
	t_3 = x * (eps_m + (-1.0 / eps_m));
	t_4 = (2.0 + t_3) / 2.0;
	t_5 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	tmp = 0.0;
	if (eps_m <= 1.28e+64)
		tmp = 1.0;
	elseif (eps_m <= 8.8e+144)
		tmp = t_1;
	elseif (eps_m <= 5e+153)
		tmp = t_0;
	elseif (eps_m <= 2.3e+165)
		tmp = t_4;
	elseif (eps_m <= 2e+179)
		tmp = t_1;
	elseif (eps_m <= 3.35e+180)
		tmp = t_0;
	elseif (eps_m <= 8.2e+183)
		tmp = t_1;
	elseif (eps_m <= 1.55e+184)
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + (1.0 + (-1.0 / eps_m))) / 2.0;
	elseif (eps_m <= 8.5e+186)
		tmp = t_0;
	elseif (eps_m <= 2e+191)
		tmp = t_1;
	elseif (eps_m <= 1.9e+198)
		tmp = t_0;
	elseif (eps_m <= 2.8e+203)
		tmp = t_1;
	elseif (eps_m <= 2.65e+228)
		tmp = t_0;
	elseif (eps_m <= 3.2e+228)
		tmp = t_2;
	elseif (eps_m <= 2e+230)
		tmp = t_5;
	elseif (eps_m <= 2.9e+232)
		tmp = t_1;
	elseif (eps_m <= 1.35e+236)
		tmp = t_0;
	elseif (eps_m <= 1.45e+236)
		tmp = t_2;
	elseif (eps_m <= 2.05e+237)
		tmp = t_0;
	elseif (eps_m <= 5.8e+237)
		tmp = t_5;
	elseif (eps_m <= 6.3e+237)
		tmp = (2.0 - t_3) / 2.0;
	elseif (eps_m <= 6e+241)
		tmp = t_4;
	elseif (eps_m <= 6.1e+246)
		tmp = t_0;
	elseif (eps_m <= 6.2e+246)
		tmp = 1.0;
	elseif (eps_m <= 3.25e+248)
		tmp = t_2;
	elseif (eps_m <= 4.9e+253)
		tmp = t_4;
	elseif (eps_m <= 3.2e+265)
		tmp = t_1;
	elseif (eps_m <= 1.3e+268)
		tmp = t_0;
	elseif (eps_m <= 4.8e+272)
		tmp = t_1;
	elseif (eps_m <= 1.85e+278)
		tmp = t_0;
	elseif (eps_m <= 1.4e+280)
		tmp = t_1;
	elseif (eps_m <= 1.7e+283)
		tmp = t_5;
	elseif ((eps_m <= 3.6e+295) || ~((eps_m <= 1.45e+299)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(eps$95$m * N[(N[(x * eps$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(x + N[(eps$95$m * N[(N[(x * 0.0), $MachinePrecision] - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(x * N[(eps$95$m + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(2.0 + t$95$3), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(eps$95$m * N[(x + N[(2.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[eps$95$m, 1.28e+64], 1.0, If[LessEqual[eps$95$m, 8.8e+144], t$95$1, If[LessEqual[eps$95$m, 5e+153], t$95$0, If[LessEqual[eps$95$m, 2.3e+165], t$95$4, If[LessEqual[eps$95$m, 2e+179], t$95$1, If[LessEqual[eps$95$m, 3.35e+180], t$95$0, If[LessEqual[eps$95$m, 8.2e+183], t$95$1, If[LessEqual[eps$95$m, 1.55e+184], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 8.5e+186], t$95$0, If[LessEqual[eps$95$m, 2e+191], t$95$1, If[LessEqual[eps$95$m, 1.9e+198], t$95$0, If[LessEqual[eps$95$m, 2.8e+203], t$95$1, If[LessEqual[eps$95$m, 2.65e+228], t$95$0, If[LessEqual[eps$95$m, 3.2e+228], t$95$2, If[LessEqual[eps$95$m, 2e+230], t$95$5, If[LessEqual[eps$95$m, 2.9e+232], t$95$1, If[LessEqual[eps$95$m, 1.35e+236], t$95$0, If[LessEqual[eps$95$m, 1.45e+236], t$95$2, If[LessEqual[eps$95$m, 2.05e+237], t$95$0, If[LessEqual[eps$95$m, 5.8e+237], t$95$5, If[LessEqual[eps$95$m, 6.3e+237], N[(N[(2.0 - t$95$3), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 6e+241], t$95$4, If[LessEqual[eps$95$m, 6.1e+246], t$95$0, If[LessEqual[eps$95$m, 6.2e+246], 1.0, If[LessEqual[eps$95$m, 3.25e+248], t$95$2, If[LessEqual[eps$95$m, 4.9e+253], t$95$4, If[LessEqual[eps$95$m, 3.2e+265], t$95$1, If[LessEqual[eps$95$m, 1.3e+268], t$95$0, If[LessEqual[eps$95$m, 4.8e+272], t$95$1, If[LessEqual[eps$95$m, 1.85e+278], t$95$0, If[LessEqual[eps$95$m, 1.4e+280], t$95$1, If[LessEqual[eps$95$m, 1.7e+283], t$95$5, If[Or[LessEqual[eps$95$m, 3.6e+295], N[Not[LessEqual[eps$95$m, 1.45e+299]], $MachinePrecision]], t$95$1, t$95$0]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if eps < 1.28000000000000004e64 or 6.10000000000000028e246 < eps < 6.19999999999999977e246

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 54.2%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 1.28000000000000004e64 < eps < 8.79999999999999952e144 or 2.30000000000000016e165 < eps < 1.99999999999999996e179 or 3.3499999999999999e180 < eps < 8.20000000000000029e183 or 8.4999999999999999e186 < eps < 2.00000000000000015e191 or 1.89999999999999994e198 < eps < 2.7999999999999999e203 or 2.0000000000000002e230 < eps < 2.90000000000000023e232 or 4.9000000000000001e253 < eps < 3.20000000000000014e265 or 1.29999999999999997e268 < eps < 4.8000000000000001e272 or 1.85000000000000011e278 < eps < 1.40000000000000001e280 or 1.7000000000000001e283 < eps < 3.59999999999999984e295 or 1.44999999999999996e299 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 36.9%

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

   6. Taylor expanded in x around 0 17.2%

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

      1. associate-*r* 17.2%

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

      2. neg-mul-1 17.2%

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

      3. *-commutative 17.2%

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

   8. Simplified 17.2%

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

      1. distribute-lft-in 17.2%

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

      2. *-rgt-identity 17.2%

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

      3. distribute-rgt-in 17.2%

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

      4. add-sqr-sqrt 2.4%

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

      5. sqrt-unprod 14.3%

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

      6. sqr-neg 14.3%

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

      7. sqrt-unprod 11.7%

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

      8. add-sqr-sqrt 41.8%

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

      9. un-div-inv 41.8%

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

      10. add-sqr-sqrt 30.1%

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

      11. sqrt-unprod 20.5%

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

      12. sqr-neg 20.5%

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

      13. sqrt-unprod 11.7%

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

      14. add-sqr-sqrt 41.8%

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

   10. Applied egg-rr 41.8%

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

   11. Taylor expanded in eps around 0 68.0%

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

       1. associate-+r+ 68.0%

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

       2. distribute-rgt1-in 68.0%

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

       3. metadata-eval 68.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(\color{blue}{0} \cdot x + -1 \cdot \left(\varepsilon \cdot x\right)\right)}{\varepsilon}}{2} \]

       4. associate-*r* 68.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(0 \cdot x + \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot x}\right)}{\varepsilon}}{2} \]

       5. neg-mul-1 68.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(0 \cdot x + \color{blue}{\left(-\varepsilon\right)} \cdot x\right)}{\varepsilon}}{2} \]

       6. *-commutative 68.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(0 \cdot x + \color{blue}{x \cdot \left(-\varepsilon\right)}\right)}{\varepsilon}}{2} \]

   13. Simplified 68.0%

       \[\leadsto \frac{2 + \color{blue}{\frac{x + \varepsilon \cdot \left(0 \cdot x + x \cdot \left(-\varepsilon\right)\right)}{\varepsilon}}}{2} \]

   if 8.79999999999999952e144 < eps < 5.00000000000000018e153 or 1.99999999999999996e179 < eps < 3.3499999999999999e180 or 1.5499999999999999e184 < eps < 8.4999999999999999e186 or 2.00000000000000015e191 < eps < 1.89999999999999994e198 or 2.7999999999999999e203 < eps < 2.65e228 or 2.90000000000000023e232 < eps < 1.3500000000000001e236 or 1.45e236 < eps < 2.05000000000000001e237 or 6.00000000000000031e241 < eps < 6.10000000000000028e246 or 3.20000000000000014e265 < eps < 1.29999999999999997e268 or 4.8000000000000001e272 < eps < 1.85000000000000011e278 or 3.59999999999999984e295 < eps < 1.44999999999999996e299

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in x around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. neg-mul-1 66.7%

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

      3. *-commutative 66.7%

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

   8. Simplified 66.7%

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

   9. Taylor expanded in eps around 0 100.0%

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

   if 5.00000000000000018e153 < eps < 2.30000000000000016e165 or 6.30000000000000008e237 < eps < 6.00000000000000031e241 or 3.25000000000000024e248 < eps < 4.9000000000000001e253

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in x around 0 83.9%

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

      1. mul-1-neg 83.9%

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

      2. distribute-rgt-neg-in 83.9%

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

      3. mul-1-neg 83.9%

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

      4. mul-1-neg 83.9%

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

      5. distribute-rgt-neg-in 83.9%

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

      6. neg-sub0 83.9%

         \[\leadsto \frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(0 - \left(1 - \varepsilon\right)\right)}\right)}{2} \]

      7. associate--r- 83.9%

         \[\leadsto \frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(\left(0 - 1\right) + \varepsilon\right)}\right)}{2} \]

      8. metadata-eval 83.9%

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

      9. +-commutative 83.9%

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

      10. +-commutative 83.9%

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

      11. distribute-rgt1-in 83.9%

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

      12. +-commutative 83.9%

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

      13. distribute-rgt-in 83.9%

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

      14. neg-mul-1 83.9%

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

      15. rgt-mult-inverse 83.9%

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

      16. associate-+l+ 83.9%

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

      17. +-commutative 83.9%

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

      18. associate-+r+ 83.9%

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

      19. metadata-eval 83.9%

          \[\leadsto \frac{2 + x \cdot \left(\varepsilon + \left(\color{blue}{0} + \left(-\frac{1}{\varepsilon}\right)\right)\right)}{2} \]

      20. sub-neg 83.9%

          \[\leadsto \frac{2 + x \cdot \left(\varepsilon + \color{blue}{\left(0 - \frac{1}{\varepsilon}\right)}\right)}{2} \]

      21. neg-sub0 83.9%

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

      22. distribute-neg-frac 83.9%

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

      23. metadata-eval 83.9%

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

   8. Simplified 83.9%

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

   if 8.20000000000000029e183 < eps < 1.5499999999999999e184

   1. Initial program 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in x around 0 40.1%

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

   6. Taylor expanded in x around 0 24.0%

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

   if 2.65e228 < eps < 3.2000000000000003e228 or 1.3500000000000001e236 < eps < 1.45e236 or 6.19999999999999977e246 < eps < 3.25000000000000024e248

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. neg-mul-1 0.0%

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

      3. *-commutative 0.0%

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

   8. Simplified 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.0%

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

      3. sqr-neg 0.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-un-lft-identity 100.0%

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

      8. distribute-rgt-in 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. *-lft-identity 100.0%

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

       3. distribute-rgt-out 100.0%

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

       4. +-commutative 100.0%

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

       5. associate-+r- 100.0%

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

       6. div-sub 100.0%

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

       7. *-inverses 100.0%

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

       8. associate-+l- 100.0%

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

       9. metadata-eval 100.0%

          \[\leadsto \frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} - \color{blue}{0}\right) - \varepsilon\right)}{2} \]

   12. Simplified 100.0%

       \[\leadsto \frac{2 + \color{blue}{x \cdot \left(\left(\frac{1}{\varepsilon} - 0\right) - \varepsilon\right)}}{2} \]

   13. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

       3. *-commutative 100.0%

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

   15. Simplified 100.0%

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

   if 3.2000000000000003e228 < eps < 2.0000000000000002e230 or 2.05000000000000001e237 < eps < 5.8000000000000002e237 or 1.40000000000000001e280 < eps < 1.7000000000000001e283

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in x around 0 67.8%

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

      1. associate-*r* 67.8%

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

      2. neg-mul-1 67.8%

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

      3. *-commutative 67.8%

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

   8. Simplified 67.8%

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

   9. Taylor expanded in eps around inf 67.8%

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

       1. associate-*r/ 67.8%

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

       2. metadata-eval 67.8%

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

   11. Simplified 67.8%

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

   if 5.8000000000000002e237 < eps < 6.30000000000000008e237

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around 0 0.5%

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

      1. associate-*r* 0.5%

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

      2. neg-mul-1 0.5%

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

      3. *-commutative 0.5%

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

   8. Simplified 0.5%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 3.1%

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

      3. sqr-neg 3.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 3.7%

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

      6. distribute-rgt-in 3.7%

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

      7. *-un-lft-identity 3.7%

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

      8. distribute-rgt-in 3.7%

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

   10. Applied egg-rr 3.7%

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

       1. associate-*l/ 3.7%

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

       2. *-lft-identity 3.7%

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

       3. distribute-rgt-out 3.7%

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

       4. +-commutative 3.7%

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

       5. associate-+r- 3.7%

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

       6. div-sub 3.7%

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

       7. *-inverses 3.7%

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

       8. associate-+l- 3.7%

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

       9. metadata-eval 3.7%

          \[\leadsto \frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} - \color{blue}{0}\right) - \varepsilon\right)}{2} \]

   12. Simplified 3.7%

       \[\leadsto \frac{2 + \color{blue}{x \cdot \left(\left(\frac{1}{\varepsilon} - 0\right) - \varepsilon\right)}}{2} \]
3. Recombined 8 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.28 \cdot 10^{+64}:\\ \;\;\;\;1\\ \mathbf{elif}\;\varepsilon \leq 8.8 \cdot 10^{+144}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{+165}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.35 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{+183}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.55 \cdot 10^{+184}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + x \cdot \left(\varepsilon + -1\right)\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 8.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+191}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{+198}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{+203}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.65 \cdot 10^{+228}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{+228}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{+232}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.35 \cdot 10^{+236}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.45 \cdot 10^{+236}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.05 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 5.8 \cdot 10^{+237}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 6.3 \cdot 10^{+237}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{+241}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 6.1 \cdot 10^{+246}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{+246}:\\ \;\;\;\;1\\ \mathbf{elif}\;\varepsilon \leq 3.25 \cdot 10^{+248}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.9 \cdot 10^{+253}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{+265}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{+268}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.8 \cdot 10^{+272}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.85 \cdot 10^{+278}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{+280}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{+283}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{+295} \lor \neg \left(\varepsilon \leq 1.45 \cdot 10^{+299}\right):\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{2 - x \cdot eps\_m}{2}\\ t_1 := x \cdot \left(eps\_m + \frac{-1}{eps\_m}\right)\\ t_2 := \frac{2 + t\_1}{2}\\ t_3 := \frac{eps\_m \cdot \left(x + \frac{2}{eps\_m}\right)}{2}\\ t_4 := \frac{\frac{eps\_m \cdot \left(x \cdot eps\_m + 2\right) - x}{eps\_m}}{2}\\ t_5 := \frac{2 + \frac{x + eps\_m \cdot \left(x \cdot 0 - x \cdot eps\_m\right)}{eps\_m}}{2}\\ \mathbf{if}\;eps\_m \leq 2.7 \cdot 10^{+63}:\\ \;\;\;\;1\\ \mathbf{elif}\;eps\_m \leq 1.2 \cdot 10^{+145}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 1.9 \cdot 10^{+153}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 3 \cdot 10^{+165}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 2 \cdot 10^{+179}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 1.55 \cdot 10^{+181}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 2.5 \cdot 10^{+181}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 7.4 \cdot 10^{+186}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 4.2 \cdot 10^{+191}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 3.2 \cdot 10^{+197}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 1.9 \cdot 10^{+203}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 2.65 \cdot 10^{+228}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 2.7 \cdot 10^{+228}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 6.1 \cdot 10^{+230}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eps\_m \leq 4.1 \cdot 10^{+232}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 1.1 \cdot 10^{+236}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 1.45 \cdot 10^{+236}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 1.55 \cdot 10^{+237}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 3.3 \cdot 10^{+237}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eps\_m \leq 5.9 \cdot 10^{+237}:\\ \;\;\;\;\frac{2 - t\_1}{2}\\ \mathbf{elif}\;eps\_m \leq 6 \cdot 10^{+241}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 6.1 \cdot 10^{+246}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 6.8 \cdot 10^{+246}:\\ \;\;\;\;1\\ \mathbf{elif}\;eps\_m \leq 5.2 \cdot 10^{+248}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 1.5 \cdot 10^{+254}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 4 \cdot 10^{+264}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 1.42 \cdot 10^{+268}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 2.45 \cdot 10^{+272}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 5 \cdot 10^{+275}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 1.45 \cdot 10^{+280}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 1.45 \cdot 10^{+283}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eps\_m \leq 3.7 \cdot 10^{+295} \lor \neg \left(eps\_m \leq 2 \cdot 10^{+299}\right):\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (- 2.0 (* x eps_m)) 2.0))
        (t_1 (* x (+ eps_m (/ -1.0 eps_m))))
        (t_2 (/ (+ 2.0 t_1) 2.0))
        (t_3 (/ (* eps_m (+ x (/ 2.0 eps_m))) 2.0))
        (t_4 (/ (/ (- (* eps_m (+ (* x eps_m) 2.0)) x) eps_m) 2.0))
        (t_5
         (/ (+ 2.0 (/ (+ x (* eps_m (- (* x 0.0) (* x eps_m)))) eps_m)) 2.0)))
   (if (<= eps_m 2.7e+63)
     1.0
     (if (<= eps_m 1.2e+145)
       t_5
       (if (<= eps_m 1.9e+153)
         t_4
         (if (<= eps_m 3e+165)
           t_2
           (if (<= eps_m 2e+179)
             t_5
             (if (<= eps_m 1.55e+181)
               t_4
               (if (<= eps_m 2.5e+181)
                 t_5
                 (if (<= eps_m 7.4e+186)
                   t_4
                   (if (<= eps_m 4.2e+191)
                     t_5
                     (if (<= eps_m 3.2e+197)
                       t_4
                       (if (<= eps_m 1.9e+203)
                         t_5
                         (if (<= eps_m 2.65e+228)
                           t_4
                           (if (<= eps_m 2.7e+228)
                             t_0
                             (if (<= eps_m 6.1e+230)
                               t_3
                               (if (<= eps_m 4.1e+232)
                                 t_5
                                 (if (<= eps_m 1.1e+236)
                                   t_4
                                   (if (<= eps_m 1.45e+236)
                                     t_0
                                     (if (<= eps_m 1.55e+237)
                                       t_4
                                       (if (<= eps_m 3.3e+237)
                                         t_3
                                         (if (<= eps_m 5.9e+237)
                                           (/ (- 2.0 t_1) 2.0)
                                           (if (<= eps_m 6e+241)
                                             t_2
                                             (if (<= eps_m 6.1e+246)
                                               t_4
                                               (if (<= eps_m 6.8e+246)
                                                 1.0
                                                 (if (<= eps_m 5.2e+248)
                                                   t_0
                                                   (if (<= eps_m 1.5e+254)
                                                     t_2
                                                     (if (<= eps_m 4e+264)
                                                       t_5
                                                       (if (<= eps_m 1.42e+268)
                                                         t_4
                                                         (if (<=
                                                              eps_m
                                                              2.45e+272)
                                                           t_5
                                                           (if (<=
                                                                eps_m
                                                                5e+275)
                                                             t_4
                                                             (if (<=
                                                                  eps_m
                                                                  1.45e+280)
                                                               t_5
                                                               (if (<=
                                                                    eps_m
                                                                    1.45e+283)
                                                                 t_3
                                                                 (if (or (<=
                                                                          eps_m
                                                                          3.7e+295)
                                                                         (not
                                                                          (<=
                                                                           eps_m
                                                                           2e+299)))
                                                                   t_5
                                                                   t_4))))))))))))))))))))))))))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (2.0 - (x * eps_m)) / 2.0;
	double t_1 = x * (eps_m + (-1.0 / eps_m));
	double t_2 = (2.0 + t_1) / 2.0;
	double t_3 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double t_4 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_5 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	double tmp;
	if (eps_m <= 2.7e+63) {
		tmp = 1.0;
	} else if (eps_m <= 1.2e+145) {
		tmp = t_5;
	} else if (eps_m <= 1.9e+153) {
		tmp = t_4;
	} else if (eps_m <= 3e+165) {
		tmp = t_2;
	} else if (eps_m <= 2e+179) {
		tmp = t_5;
	} else if (eps_m <= 1.55e+181) {
		tmp = t_4;
	} else if (eps_m <= 2.5e+181) {
		tmp = t_5;
	} else if (eps_m <= 7.4e+186) {
		tmp = t_4;
	} else if (eps_m <= 4.2e+191) {
		tmp = t_5;
	} else if (eps_m <= 3.2e+197) {
		tmp = t_4;
	} else if (eps_m <= 1.9e+203) {
		tmp = t_5;
	} else if (eps_m <= 2.65e+228) {
		tmp = t_4;
	} else if (eps_m <= 2.7e+228) {
		tmp = t_0;
	} else if (eps_m <= 6.1e+230) {
		tmp = t_3;
	} else if (eps_m <= 4.1e+232) {
		tmp = t_5;
	} else if (eps_m <= 1.1e+236) {
		tmp = t_4;
	} else if (eps_m <= 1.45e+236) {
		tmp = t_0;
	} else if (eps_m <= 1.55e+237) {
		tmp = t_4;
	} else if (eps_m <= 3.3e+237) {
		tmp = t_3;
	} else if (eps_m <= 5.9e+237) {
		tmp = (2.0 - t_1) / 2.0;
	} else if (eps_m <= 6e+241) {
		tmp = t_2;
	} else if (eps_m <= 6.1e+246) {
		tmp = t_4;
	} else if (eps_m <= 6.8e+246) {
		tmp = 1.0;
	} else if (eps_m <= 5.2e+248) {
		tmp = t_0;
	} else if (eps_m <= 1.5e+254) {
		tmp = t_2;
	} else if (eps_m <= 4e+264) {
		tmp = t_5;
	} else if (eps_m <= 1.42e+268) {
		tmp = t_4;
	} else if (eps_m <= 2.45e+272) {
		tmp = t_5;
	} else if (eps_m <= 5e+275) {
		tmp = t_4;
	} else if (eps_m <= 1.45e+280) {
		tmp = t_5;
	} else if (eps_m <= 1.45e+283) {
		tmp = t_3;
	} else if ((eps_m <= 3.7e+295) || !(eps_m <= 2e+299)) {
		tmp = t_5;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (2.0d0 - (x * eps_m)) / 2.0d0
    t_1 = x * (eps_m + ((-1.0d0) / eps_m))
    t_2 = (2.0d0 + t_1) / 2.0d0
    t_3 = (eps_m * (x + (2.0d0 / eps_m))) / 2.0d0
    t_4 = (((eps_m * ((x * eps_m) + 2.0d0)) - x) / eps_m) / 2.0d0
    t_5 = (2.0d0 + ((x + (eps_m * ((x * 0.0d0) - (x * eps_m)))) / eps_m)) / 2.0d0
    if (eps_m <= 2.7d+63) then
        tmp = 1.0d0
    else if (eps_m <= 1.2d+145) then
        tmp = t_5
    else if (eps_m <= 1.9d+153) then
        tmp = t_4
    else if (eps_m <= 3d+165) then
        tmp = t_2
    else if (eps_m <= 2d+179) then
        tmp = t_5
    else if (eps_m <= 1.55d+181) then
        tmp = t_4
    else if (eps_m <= 2.5d+181) then
        tmp = t_5
    else if (eps_m <= 7.4d+186) then
        tmp = t_4
    else if (eps_m <= 4.2d+191) then
        tmp = t_5
    else if (eps_m <= 3.2d+197) then
        tmp = t_4
    else if (eps_m <= 1.9d+203) then
        tmp = t_5
    else if (eps_m <= 2.65d+228) then
        tmp = t_4
    else if (eps_m <= 2.7d+228) then
        tmp = t_0
    else if (eps_m <= 6.1d+230) then
        tmp = t_3
    else if (eps_m <= 4.1d+232) then
        tmp = t_5
    else if (eps_m <= 1.1d+236) then
        tmp = t_4
    else if (eps_m <= 1.45d+236) then
        tmp = t_0
    else if (eps_m <= 1.55d+237) then
        tmp = t_4
    else if (eps_m <= 3.3d+237) then
        tmp = t_3
    else if (eps_m <= 5.9d+237) then
        tmp = (2.0d0 - t_1) / 2.0d0
    else if (eps_m <= 6d+241) then
        tmp = t_2
    else if (eps_m <= 6.1d+246) then
        tmp = t_4
    else if (eps_m <= 6.8d+246) then
        tmp = 1.0d0
    else if (eps_m <= 5.2d+248) then
        tmp = t_0
    else if (eps_m <= 1.5d+254) then
        tmp = t_2
    else if (eps_m <= 4d+264) then
        tmp = t_5
    else if (eps_m <= 1.42d+268) then
        tmp = t_4
    else if (eps_m <= 2.45d+272) then
        tmp = t_5
    else if (eps_m <= 5d+275) then
        tmp = t_4
    else if (eps_m <= 1.45d+280) then
        tmp = t_5
    else if (eps_m <= 1.45d+283) then
        tmp = t_3
    else if ((eps_m <= 3.7d+295) .or. (.not. (eps_m <= 2d+299))) then
        tmp = t_5
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (2.0 - (x * eps_m)) / 2.0;
	double t_1 = x * (eps_m + (-1.0 / eps_m));
	double t_2 = (2.0 + t_1) / 2.0;
	double t_3 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double t_4 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_5 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	double tmp;
	if (eps_m <= 2.7e+63) {
		tmp = 1.0;
	} else if (eps_m <= 1.2e+145) {
		tmp = t_5;
	} else if (eps_m <= 1.9e+153) {
		tmp = t_4;
	} else if (eps_m <= 3e+165) {
		tmp = t_2;
	} else if (eps_m <= 2e+179) {
		tmp = t_5;
	} else if (eps_m <= 1.55e+181) {
		tmp = t_4;
	} else if (eps_m <= 2.5e+181) {
		tmp = t_5;
	} else if (eps_m <= 7.4e+186) {
		tmp = t_4;
	} else if (eps_m <= 4.2e+191) {
		tmp = t_5;
	} else if (eps_m <= 3.2e+197) {
		tmp = t_4;
	} else if (eps_m <= 1.9e+203) {
		tmp = t_5;
	} else if (eps_m <= 2.65e+228) {
		tmp = t_4;
	} else if (eps_m <= 2.7e+228) {
		tmp = t_0;
	} else if (eps_m <= 6.1e+230) {
		tmp = t_3;
	} else if (eps_m <= 4.1e+232) {
		tmp = t_5;
	} else if (eps_m <= 1.1e+236) {
		tmp = t_4;
	} else if (eps_m <= 1.45e+236) {
		tmp = t_0;
	} else if (eps_m <= 1.55e+237) {
		tmp = t_4;
	} else if (eps_m <= 3.3e+237) {
		tmp = t_3;
	} else if (eps_m <= 5.9e+237) {
		tmp = (2.0 - t_1) / 2.0;
	} else if (eps_m <= 6e+241) {
		tmp = t_2;
	} else if (eps_m <= 6.1e+246) {
		tmp = t_4;
	} else if (eps_m <= 6.8e+246) {
		tmp = 1.0;
	} else if (eps_m <= 5.2e+248) {
		tmp = t_0;
	} else if (eps_m <= 1.5e+254) {
		tmp = t_2;
	} else if (eps_m <= 4e+264) {
		tmp = t_5;
	} else if (eps_m <= 1.42e+268) {
		tmp = t_4;
	} else if (eps_m <= 2.45e+272) {
		tmp = t_5;
	} else if (eps_m <= 5e+275) {
		tmp = t_4;
	} else if (eps_m <= 1.45e+280) {
		tmp = t_5;
	} else if (eps_m <= 1.45e+283) {
		tmp = t_3;
	} else if ((eps_m <= 3.7e+295) || !(eps_m <= 2e+299)) {
		tmp = t_5;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (2.0 - (x * eps_m)) / 2.0
	t_1 = x * (eps_m + (-1.0 / eps_m))
	t_2 = (2.0 + t_1) / 2.0
	t_3 = (eps_m * (x + (2.0 / eps_m))) / 2.0
	t_4 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0
	t_5 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0
	tmp = 0
	if eps_m <= 2.7e+63:
		tmp = 1.0
	elif eps_m <= 1.2e+145:
		tmp = t_5
	elif eps_m <= 1.9e+153:
		tmp = t_4
	elif eps_m <= 3e+165:
		tmp = t_2
	elif eps_m <= 2e+179:
		tmp = t_5
	elif eps_m <= 1.55e+181:
		tmp = t_4
	elif eps_m <= 2.5e+181:
		tmp = t_5
	elif eps_m <= 7.4e+186:
		tmp = t_4
	elif eps_m <= 4.2e+191:
		tmp = t_5
	elif eps_m <= 3.2e+197:
		tmp = t_4
	elif eps_m <= 1.9e+203:
		tmp = t_5
	elif eps_m <= 2.65e+228:
		tmp = t_4
	elif eps_m <= 2.7e+228:
		tmp = t_0
	elif eps_m <= 6.1e+230:
		tmp = t_3
	elif eps_m <= 4.1e+232:
		tmp = t_5
	elif eps_m <= 1.1e+236:
		tmp = t_4
	elif eps_m <= 1.45e+236:
		tmp = t_0
	elif eps_m <= 1.55e+237:
		tmp = t_4
	elif eps_m <= 3.3e+237:
		tmp = t_3
	elif eps_m <= 5.9e+237:
		tmp = (2.0 - t_1) / 2.0
	elif eps_m <= 6e+241:
		tmp = t_2
	elif eps_m <= 6.1e+246:
		tmp = t_4
	elif eps_m <= 6.8e+246:
		tmp = 1.0
	elif eps_m <= 5.2e+248:
		tmp = t_0
	elif eps_m <= 1.5e+254:
		tmp = t_2
	elif eps_m <= 4e+264:
		tmp = t_5
	elif eps_m <= 1.42e+268:
		tmp = t_4
	elif eps_m <= 2.45e+272:
		tmp = t_5
	elif eps_m <= 5e+275:
		tmp = t_4
	elif eps_m <= 1.45e+280:
		tmp = t_5
	elif eps_m <= 1.45e+283:
		tmp = t_3
	elif (eps_m <= 3.7e+295) or not (eps_m <= 2e+299):
		tmp = t_5
	else:
		tmp = t_4
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0)
	t_1 = Float64(x * Float64(eps_m + Float64(-1.0 / eps_m)))
	t_2 = Float64(Float64(2.0 + t_1) / 2.0)
	t_3 = Float64(Float64(eps_m * Float64(x + Float64(2.0 / eps_m))) / 2.0)
	t_4 = Float64(Float64(Float64(Float64(eps_m * Float64(Float64(x * eps_m) + 2.0)) - x) / eps_m) / 2.0)
	t_5 = Float64(Float64(2.0 + Float64(Float64(x + Float64(eps_m * Float64(Float64(x * 0.0) - Float64(x * eps_m)))) / eps_m)) / 2.0)
	tmp = 0.0
	if (eps_m <= 2.7e+63)
		tmp = 1.0;
	elseif (eps_m <= 1.2e+145)
		tmp = t_5;
	elseif (eps_m <= 1.9e+153)
		tmp = t_4;
	elseif (eps_m <= 3e+165)
		tmp = t_2;
	elseif (eps_m <= 2e+179)
		tmp = t_5;
	elseif (eps_m <= 1.55e+181)
		tmp = t_4;
	elseif (eps_m <= 2.5e+181)
		tmp = t_5;
	elseif (eps_m <= 7.4e+186)
		tmp = t_4;
	elseif (eps_m <= 4.2e+191)
		tmp = t_5;
	elseif (eps_m <= 3.2e+197)
		tmp = t_4;
	elseif (eps_m <= 1.9e+203)
		tmp = t_5;
	elseif (eps_m <= 2.65e+228)
		tmp = t_4;
	elseif (eps_m <= 2.7e+228)
		tmp = t_0;
	elseif (eps_m <= 6.1e+230)
		tmp = t_3;
	elseif (eps_m <= 4.1e+232)
		tmp = t_5;
	elseif (eps_m <= 1.1e+236)
		tmp = t_4;
	elseif (eps_m <= 1.45e+236)
		tmp = t_0;
	elseif (eps_m <= 1.55e+237)
		tmp = t_4;
	elseif (eps_m <= 3.3e+237)
		tmp = t_3;
	elseif (eps_m <= 5.9e+237)
		tmp = Float64(Float64(2.0 - t_1) / 2.0);
	elseif (eps_m <= 6e+241)
		tmp = t_2;
	elseif (eps_m <= 6.1e+246)
		tmp = t_4;
	elseif (eps_m <= 6.8e+246)
		tmp = 1.0;
	elseif (eps_m <= 5.2e+248)
		tmp = t_0;
	elseif (eps_m <= 1.5e+254)
		tmp = t_2;
	elseif (eps_m <= 4e+264)
		tmp = t_5;
	elseif (eps_m <= 1.42e+268)
		tmp = t_4;
	elseif (eps_m <= 2.45e+272)
		tmp = t_5;
	elseif (eps_m <= 5e+275)
		tmp = t_4;
	elseif (eps_m <= 1.45e+280)
		tmp = t_5;
	elseif (eps_m <= 1.45e+283)
		tmp = t_3;
	elseif ((eps_m <= 3.7e+295) || !(eps_m <= 2e+299))
		tmp = t_5;
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (2.0 - (x * eps_m)) / 2.0;
	t_1 = x * (eps_m + (-1.0 / eps_m));
	t_2 = (2.0 + t_1) / 2.0;
	t_3 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	t_4 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	t_5 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	tmp = 0.0;
	if (eps_m <= 2.7e+63)
		tmp = 1.0;
	elseif (eps_m <= 1.2e+145)
		tmp = t_5;
	elseif (eps_m <= 1.9e+153)
		tmp = t_4;
	elseif (eps_m <= 3e+165)
		tmp = t_2;
	elseif (eps_m <= 2e+179)
		tmp = t_5;
	elseif (eps_m <= 1.55e+181)
		tmp = t_4;
	elseif (eps_m <= 2.5e+181)
		tmp = t_5;
	elseif (eps_m <= 7.4e+186)
		tmp = t_4;
	elseif (eps_m <= 4.2e+191)
		tmp = t_5;
	elseif (eps_m <= 3.2e+197)
		tmp = t_4;
	elseif (eps_m <= 1.9e+203)
		tmp = t_5;
	elseif (eps_m <= 2.65e+228)
		tmp = t_4;
	elseif (eps_m <= 2.7e+228)
		tmp = t_0;
	elseif (eps_m <= 6.1e+230)
		tmp = t_3;
	elseif (eps_m <= 4.1e+232)
		tmp = t_5;
	elseif (eps_m <= 1.1e+236)
		tmp = t_4;
	elseif (eps_m <= 1.45e+236)
		tmp = t_0;
	elseif (eps_m <= 1.55e+237)
		tmp = t_4;
	elseif (eps_m <= 3.3e+237)
		tmp = t_3;
	elseif (eps_m <= 5.9e+237)
		tmp = (2.0 - t_1) / 2.0;
	elseif (eps_m <= 6e+241)
		tmp = t_2;
	elseif (eps_m <= 6.1e+246)
		tmp = t_4;
	elseif (eps_m <= 6.8e+246)
		tmp = 1.0;
	elseif (eps_m <= 5.2e+248)
		tmp = t_0;
	elseif (eps_m <= 1.5e+254)
		tmp = t_2;
	elseif (eps_m <= 4e+264)
		tmp = t_5;
	elseif (eps_m <= 1.42e+268)
		tmp = t_4;
	elseif (eps_m <= 2.45e+272)
		tmp = t_5;
	elseif (eps_m <= 5e+275)
		tmp = t_4;
	elseif (eps_m <= 1.45e+280)
		tmp = t_5;
	elseif (eps_m <= 1.45e+283)
		tmp = t_3;
	elseif ((eps_m <= 3.7e+295) || ~((eps_m <= 2e+299)))
		tmp = t_5;
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(eps$95$m + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(eps$95$m * N[(x + N[(2.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(eps$95$m * N[(N[(x * eps$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(2.0 + N[(N[(x + N[(eps$95$m * N[(N[(x * 0.0), $MachinePrecision] - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[eps$95$m, 2.7e+63], 1.0, If[LessEqual[eps$95$m, 1.2e+145], t$95$5, If[LessEqual[eps$95$m, 1.9e+153], t$95$4, If[LessEqual[eps$95$m, 3e+165], t$95$2, If[LessEqual[eps$95$m, 2e+179], t$95$5, If[LessEqual[eps$95$m, 1.55e+181], t$95$4, If[LessEqual[eps$95$m, 2.5e+181], t$95$5, If[LessEqual[eps$95$m, 7.4e+186], t$95$4, If[LessEqual[eps$95$m, 4.2e+191], t$95$5, If[LessEqual[eps$95$m, 3.2e+197], t$95$4, If[LessEqual[eps$95$m, 1.9e+203], t$95$5, If[LessEqual[eps$95$m, 2.65e+228], t$95$4, If[LessEqual[eps$95$m, 2.7e+228], t$95$0, If[LessEqual[eps$95$m, 6.1e+230], t$95$3, If[LessEqual[eps$95$m, 4.1e+232], t$95$5, If[LessEqual[eps$95$m, 1.1e+236], t$95$4, If[LessEqual[eps$95$m, 1.45e+236], t$95$0, If[LessEqual[eps$95$m, 1.55e+237], t$95$4, If[LessEqual[eps$95$m, 3.3e+237], t$95$3, If[LessEqual[eps$95$m, 5.9e+237], N[(N[(2.0 - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 6e+241], t$95$2, If[LessEqual[eps$95$m, 6.1e+246], t$95$4, If[LessEqual[eps$95$m, 6.8e+246], 1.0, If[LessEqual[eps$95$m, 5.2e+248], t$95$0, If[LessEqual[eps$95$m, 1.5e+254], t$95$2, If[LessEqual[eps$95$m, 4e+264], t$95$5, If[LessEqual[eps$95$m, 1.42e+268], t$95$4, If[LessEqual[eps$95$m, 2.45e+272], t$95$5, If[LessEqual[eps$95$m, 5e+275], t$95$4, If[LessEqual[eps$95$m, 1.45e+280], t$95$5, If[LessEqual[eps$95$m, 1.45e+283], t$95$3, If[Or[LessEqual[eps$95$m, 3.7e+295], N[Not[LessEqual[eps$95$m, 2e+299]], $MachinePrecision]], t$95$5, t$95$4]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if eps < 2.70000000000000017e63 or 6.10000000000000028e246 < eps < 6.79999999999999977e246

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 54.2%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 2.70000000000000017e63 < eps < 1.19999999999999996e145 or 2.9999999999999999e165 < eps < 1.99999999999999996e179 or 1.54999999999999995e181 < eps < 2.5000000000000002e181 or 7.4e186 < eps < 4.2000000000000001e191 or 3.1999999999999998e197 < eps < 1.90000000000000012e203 or 6.0999999999999999e230 < eps < 4.10000000000000002e232 or 1.50000000000000003e254 < eps < 4.00000000000000018e264 or 1.4200000000000001e268 < eps < 2.4500000000000001e272 or 5.0000000000000003e275 < eps < 1.44999999999999993e280 or 1.4499999999999999e283 < eps < 3.7000000000000002e295 or 2.0000000000000001e299 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 36.9%

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

   6. Taylor expanded in x around 0 17.2%

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

      1. associate-*r* 17.2%

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

      2. neg-mul-1 17.2%

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

      3. *-commutative 17.2%

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

   8. Simplified 17.2%

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

      1. distribute-lft-in 17.2%

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

      2. *-rgt-identity 17.2%

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

      3. distribute-rgt-in 17.2%

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

      4. add-sqr-sqrt 2.4%

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

      5. sqrt-unprod 14.3%

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

      6. sqr-neg 14.3%

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

      7. sqrt-unprod 11.7%

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

      8. add-sqr-sqrt 41.8%

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

      9. un-div-inv 41.8%

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

      10. add-sqr-sqrt 30.1%

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

      11. sqrt-unprod 20.5%

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

      12. sqr-neg 20.5%

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

      13. sqrt-unprod 11.7%

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

      14. add-sqr-sqrt 41.8%

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

   10. Applied egg-rr 41.8%

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

   11. Taylor expanded in eps around 0 68.0%

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

       1. associate-+r+ 68.0%

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

       2. distribute-rgt1-in 68.0%

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

       3. metadata-eval 68.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(\color{blue}{0} \cdot x + -1 \cdot \left(\varepsilon \cdot x\right)\right)}{\varepsilon}}{2} \]

       4. associate-*r* 68.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(0 \cdot x + \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot x}\right)}{\varepsilon}}{2} \]

       5. neg-mul-1 68.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(0 \cdot x + \color{blue}{\left(-\varepsilon\right)} \cdot x\right)}{\varepsilon}}{2} \]

       6. *-commutative 68.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(0 \cdot x + \color{blue}{x \cdot \left(-\varepsilon\right)}\right)}{\varepsilon}}{2} \]

   13. Simplified 68.0%

       \[\leadsto \frac{2 + \color{blue}{\frac{x + \varepsilon \cdot \left(0 \cdot x + x \cdot \left(-\varepsilon\right)\right)}{\varepsilon}}}{2} \]

   if 1.19999999999999996e145 < eps < 1.89999999999999983e153 or 1.99999999999999996e179 < eps < 1.54999999999999995e181 or 2.5000000000000002e181 < eps < 7.4e186 or 4.2000000000000001e191 < eps < 3.1999999999999998e197 or 1.90000000000000012e203 < eps < 2.65e228 or 4.10000000000000002e232 < eps < 1.09999999999999989e236 or 1.45e236 < eps < 1.54999999999999995e237 or 6.00000000000000031e241 < eps < 6.10000000000000028e246 or 4.00000000000000018e264 < eps < 1.4200000000000001e268 or 2.4500000000000001e272 < eps < 5.0000000000000003e275 or 3.7000000000000002e295 < eps < 2.0000000000000001e299

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in x around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. neg-mul-1 66.7%

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

      3. *-commutative 66.7%

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

   8. Simplified 66.7%

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

   9. Taylor expanded in eps around 0 100.0%

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

   if 1.89999999999999983e153 < eps < 2.9999999999999999e165 or 5.8999999999999997e237 < eps < 6.00000000000000031e241 or 5.20000000000000019e248 < eps < 1.50000000000000003e254

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in x around 0 83.9%

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

      1. mul-1-neg 83.9%

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

      2. distribute-rgt-neg-in 83.9%

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

      3. mul-1-neg 83.9%

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

      4. mul-1-neg 83.9%

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

      5. distribute-rgt-neg-in 83.9%

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

      6. neg-sub0 83.9%

         \[\leadsto \frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(0 - \left(1 - \varepsilon\right)\right)}\right)}{2} \]

      7. associate--r- 83.9%

         \[\leadsto \frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(\left(0 - 1\right) + \varepsilon\right)}\right)}{2} \]

      8. metadata-eval 83.9%

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

      9. +-commutative 83.9%

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

      10. +-commutative 83.9%

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

      11. distribute-rgt1-in 83.9%

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

      12. +-commutative 83.9%

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

      13. distribute-rgt-in 83.9%

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

      14. neg-mul-1 83.9%

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

      15. rgt-mult-inverse 83.9%

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

      16. associate-+l+ 83.9%

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

      17. +-commutative 83.9%

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

      18. associate-+r+ 83.9%

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

      19. metadata-eval 83.9%

          \[\leadsto \frac{2 + x \cdot \left(\varepsilon + \left(\color{blue}{0} + \left(-\frac{1}{\varepsilon}\right)\right)\right)}{2} \]

      20. sub-neg 83.9%

          \[\leadsto \frac{2 + x \cdot \left(\varepsilon + \color{blue}{\left(0 - \frac{1}{\varepsilon}\right)}\right)}{2} \]

      21. neg-sub0 83.9%

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

      22. distribute-neg-frac 83.9%

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

      23. metadata-eval 83.9%

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

   8. Simplified 83.9%

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

   if 2.65e228 < eps < 2.7000000000000002e228 or 1.09999999999999989e236 < eps < 1.45e236 or 6.79999999999999977e246 < eps < 5.20000000000000019e248

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. neg-mul-1 0.0%

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

      3. *-commutative 0.0%

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

   8. Simplified 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.0%

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

      3. sqr-neg 0.0%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-un-lft-identity 100.0%

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

      8. distribute-rgt-in 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. *-lft-identity 100.0%

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

       3. distribute-rgt-out 100.0%

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

       4. +-commutative 100.0%

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

       5. associate-+r- 100.0%

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

       6. div-sub 100.0%

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

       7. *-inverses 100.0%

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

       8. associate-+l- 100.0%

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

       9. metadata-eval 100.0%

          \[\leadsto \frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} - \color{blue}{0}\right) - \varepsilon\right)}{2} \]

   12. Simplified 100.0%

       \[\leadsto \frac{2 + \color{blue}{x \cdot \left(\left(\frac{1}{\varepsilon} - 0\right) - \varepsilon\right)}}{2} \]

   13. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

       3. *-commutative 100.0%

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

   15. Simplified 100.0%

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

   if 2.7000000000000002e228 < eps < 6.0999999999999999e230 or 1.54999999999999995e237 < eps < 3.3000000000000001e237 or 1.44999999999999993e280 < eps < 1.4499999999999999e283

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in x around 0 67.8%

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

      1. associate-*r* 67.8%

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

      2. neg-mul-1 67.8%

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

      3. *-commutative 67.8%

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

   8. Simplified 67.8%

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

   9. Taylor expanded in eps around inf 67.8%

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

       1. associate-*r/ 67.8%

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

       2. metadata-eval 67.8%

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

   11. Simplified 67.8%

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

   if 3.3000000000000001e237 < eps < 5.8999999999999997e237

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around 0 0.5%

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

      1. associate-*r* 0.5%

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

      2. neg-mul-1 0.5%

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

      3. *-commutative 0.5%

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

   8. Simplified 0.5%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 3.1%

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

      3. sqr-neg 3.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 3.7%

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

      6. distribute-rgt-in 3.7%

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

      7. *-un-lft-identity 3.7%

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

      8. distribute-rgt-in 3.7%

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

   10. Applied egg-rr 3.7%

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

       1. associate-*l/ 3.7%

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

       2. *-lft-identity 3.7%

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

       3. distribute-rgt-out 3.7%

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

       4. +-commutative 3.7%

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

       5. associate-+r- 3.7%

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

       6. div-sub 3.7%

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

       7. *-inverses 3.7%

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

       8. associate-+l- 3.7%

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

       9. metadata-eval 3.7%

          \[\leadsto \frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} - \color{blue}{0}\right) - \varepsilon\right)}{2} \]

   12. Simplified 3.7%

       \[\leadsto \frac{2 + \color{blue}{x \cdot \left(\left(\frac{1}{\varepsilon} - 0\right) - \varepsilon\right)}}{2} \]
3. Recombined 7 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.7 \cdot 10^{+63}:\\ \;\;\;\;1\\ \mathbf{elif}\;\varepsilon \leq 1.2 \cdot 10^{+145}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{+165}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.55 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 7.4 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{+191}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{+197}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{+203}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.65 \cdot 10^{+228}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.7 \cdot 10^{+228}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 6.1 \cdot 10^{+230}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.1 \cdot 10^{+232}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.1 \cdot 10^{+236}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.45 \cdot 10^{+236}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.55 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{+237}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 5.9 \cdot 10^{+237}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{+241}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 6.1 \cdot 10^{+246}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{+246}:\\ \;\;\;\;1\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{+248}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.5 \cdot 10^{+254}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{+264}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.42 \cdot 10^{+268}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.45 \cdot 10^{+272}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.45 \cdot 10^{+280}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.45 \cdot 10^{+283}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.7 \cdot 10^{+295} \lor \neg \left(\varepsilon \leq 2 \cdot 10^{+299}\right):\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{eps\_m \cdot \left(x + \frac{2}{eps\_m}\right)}{2}\\ t_1 := \frac{\frac{eps\_m \cdot \left(x \cdot eps\_m + 2\right) - x}{eps\_m}}{2}\\ t_2 := \frac{2 - x \cdot eps\_m}{2}\\ t_3 := x \cdot \left(eps\_m + \frac{-1}{eps\_m}\right)\\ t_4 := \frac{2 - t\_3}{2}\\ t_5 := \frac{2 + t\_3}{2}\\ \mathbf{if}\;eps\_m \leq 8.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(1 - eps\_m\right) - x\right)}{2}\\ \mathbf{elif}\;eps\_m \leq 9 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 1.4 \cdot 10^{+166}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 2 \cdot 10^{+179}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 1.55 \cdot 10^{+181}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 1.58 \cdot 10^{+181}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 1.15 \cdot 10^{+187}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 2 \cdot 10^{+191}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 5 \cdot 10^{+197}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 1.65 \cdot 10^{+203}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 2.65 \cdot 10^{+228}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 4.3 \cdot 10^{+228}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 2.75 \cdot 10^{+230}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 2.9 \cdot 10^{+232}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 1.4 \cdot 10^{+236}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 1.45 \cdot 10^{+236}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 2.05 \cdot 10^{+237}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 5.8 \cdot 10^{+237}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 5.9 \cdot 10^{+237}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 6 \cdot 10^{+241}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 6.1 \cdot 10^{+246}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 8 \cdot 10^{+246}:\\ \;\;\;\;1\\ \mathbf{elif}\;eps\_m \leq 3.25 \cdot 10^{+248}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 6.5 \cdot 10^{+254}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;eps\_m \leq 4 \cdot 10^{+264}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 1.45 \cdot 10^{+268}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 2 \cdot 10^{+268}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eps\_m \leq 1.05 \cdot 10^{+272}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 1.45 \cdot 10^{+279}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 4.8 \cdot 10^{+279}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 2.5 \cdot 10^{+283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 1.4 \cdot 10^{+295}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eps\_m \leq 2.7 \cdot 10^{+299}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (* eps_m (+ x (/ 2.0 eps_m))) 2.0))
        (t_1 (/ (/ (- (* eps_m (+ (* x eps_m) 2.0)) x) eps_m) 2.0))
        (t_2 (/ (- 2.0 (* x eps_m)) 2.0))
        (t_3 (* x (+ eps_m (/ -1.0 eps_m))))
        (t_4 (/ (- 2.0 t_3) 2.0))
        (t_5 (/ (+ 2.0 t_3) 2.0)))
   (if (<= eps_m 8.2e+68)
     (/ (+ 2.0 (- (* x (- 1.0 eps_m)) x)) 2.0)
     (if (<= eps_m 9e+153)
       t_1
       (if (<= eps_m 1.4e+166)
         t_5
         (if (<= eps_m 2e+179)
           t_2
           (if (<= eps_m 1.55e+181)
             t_1
             (if (<= eps_m 1.58e+181)
               t_2
               (if (<= eps_m 1.15e+187)
                 t_1
                 (if (<= eps_m 2e+191)
                   t_2
                   (if (<= eps_m 5e+197)
                     t_1
                     (if (<= eps_m 1.65e+203)
                       t_2
                       (if (<= eps_m 2.65e+228)
                         t_1
                         (if (<= eps_m 4.3e+228)
                           t_2
                           (if (<= eps_m 2.75e+230)
                             t_0
                             (if (<= eps_m 2.9e+232)
                               t_4
                               (if (<= eps_m 1.4e+236)
                                 t_1
                                 (if (<= eps_m 1.45e+236)
                                   t_2
                                   (if (<= eps_m 2.05e+237)
                                     t_1
                                     (if (<= eps_m 5.8e+237)
                                       t_0
                                       (if (<= eps_m 5.9e+237)
                                         t_4
                                         (if (<= eps_m 6e+241)
                                           t_5
                                           (if (<= eps_m 6.1e+246)
                                             t_1
                                             (if (<= eps_m 8e+246)
                                               1.0
                                               (if (<= eps_m 3.25e+248)
                                                 t_2
                                                 (if (<= eps_m 6.5e+254)
                                                   t_5
                                                   (if (<= eps_m 4e+264)
                                                     t_2
                                                     (if (<= eps_m 1.45e+268)
                                                       t_1
                                                       (if (<= eps_m 2e+268)
                                                         t_2
                                                         (if (<=
                                                              eps_m
                                                              1.05e+272)
                                                           t_4
                                                           (if (<=
                                                                eps_m
                                                                1.45e+279)
                                                             t_1
                                                             (if (<=
                                                                  eps_m
                                                                  4.8e+279)
                                                               t_4
                                                               (if (<=
                                                                    eps_m
                                                                    2.5e+283)
                                                                 t_0
                                                                 (if (<=
                                                                      eps_m
                                                                      1.4e+295)
                                                                   t_4
                                                                   (if (<=
                                                                        eps_m
                                                                        2.7e+299)
                                                                     t_1
                                                                     t_2)))))))))))))))))))))))))))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_2 = (2.0 - (x * eps_m)) / 2.0;
	double t_3 = x * (eps_m + (-1.0 / eps_m));
	double t_4 = (2.0 - t_3) / 2.0;
	double t_5 = (2.0 + t_3) / 2.0;
	double tmp;
	if (eps_m <= 8.2e+68) {
		tmp = (2.0 + ((x * (1.0 - eps_m)) - x)) / 2.0;
	} else if (eps_m <= 9e+153) {
		tmp = t_1;
	} else if (eps_m <= 1.4e+166) {
		tmp = t_5;
	} else if (eps_m <= 2e+179) {
		tmp = t_2;
	} else if (eps_m <= 1.55e+181) {
		tmp = t_1;
	} else if (eps_m <= 1.58e+181) {
		tmp = t_2;
	} else if (eps_m <= 1.15e+187) {
		tmp = t_1;
	} else if (eps_m <= 2e+191) {
		tmp = t_2;
	} else if (eps_m <= 5e+197) {
		tmp = t_1;
	} else if (eps_m <= 1.65e+203) {
		tmp = t_2;
	} else if (eps_m <= 2.65e+228) {
		tmp = t_1;
	} else if (eps_m <= 4.3e+228) {
		tmp = t_2;
	} else if (eps_m <= 2.75e+230) {
		tmp = t_0;
	} else if (eps_m <= 2.9e+232) {
		tmp = t_4;
	} else if (eps_m <= 1.4e+236) {
		tmp = t_1;
	} else if (eps_m <= 1.45e+236) {
		tmp = t_2;
	} else if (eps_m <= 2.05e+237) {
		tmp = t_1;
	} else if (eps_m <= 5.8e+237) {
		tmp = t_0;
	} else if (eps_m <= 5.9e+237) {
		tmp = t_4;
	} else if (eps_m <= 6e+241) {
		tmp = t_5;
	} else if (eps_m <= 6.1e+246) {
		tmp = t_1;
	} else if (eps_m <= 8e+246) {
		tmp = 1.0;
	} else if (eps_m <= 3.25e+248) {
		tmp = t_2;
	} else if (eps_m <= 6.5e+254) {
		tmp = t_5;
	} else if (eps_m <= 4e+264) {
		tmp = t_2;
	} else if (eps_m <= 1.45e+268) {
		tmp = t_1;
	} else if (eps_m <= 2e+268) {
		tmp = t_2;
	} else if (eps_m <= 1.05e+272) {
		tmp = t_4;
	} else if (eps_m <= 1.45e+279) {
		tmp = t_1;
	} else if (eps_m <= 4.8e+279) {
		tmp = t_4;
	} else if (eps_m <= 2.5e+283) {
		tmp = t_0;
	} else if (eps_m <= 1.4e+295) {
		tmp = t_4;
	} else if (eps_m <= 2.7e+299) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (eps_m * (x + (2.0d0 / eps_m))) / 2.0d0
    t_1 = (((eps_m * ((x * eps_m) + 2.0d0)) - x) / eps_m) / 2.0d0
    t_2 = (2.0d0 - (x * eps_m)) / 2.0d0
    t_3 = x * (eps_m + ((-1.0d0) / eps_m))
    t_4 = (2.0d0 - t_3) / 2.0d0
    t_5 = (2.0d0 + t_3) / 2.0d0
    if (eps_m <= 8.2d+68) then
        tmp = (2.0d0 + ((x * (1.0d0 - eps_m)) - x)) / 2.0d0
    else if (eps_m <= 9d+153) then
        tmp = t_1
    else if (eps_m <= 1.4d+166) then
        tmp = t_5
    else if (eps_m <= 2d+179) then
        tmp = t_2
    else if (eps_m <= 1.55d+181) then
        tmp = t_1
    else if (eps_m <= 1.58d+181) then
        tmp = t_2
    else if (eps_m <= 1.15d+187) then
        tmp = t_1
    else if (eps_m <= 2d+191) then
        tmp = t_2
    else if (eps_m <= 5d+197) then
        tmp = t_1
    else if (eps_m <= 1.65d+203) then
        tmp = t_2
    else if (eps_m <= 2.65d+228) then
        tmp = t_1
    else if (eps_m <= 4.3d+228) then
        tmp = t_2
    else if (eps_m <= 2.75d+230) then
        tmp = t_0
    else if (eps_m <= 2.9d+232) then
        tmp = t_4
    else if (eps_m <= 1.4d+236) then
        tmp = t_1
    else if (eps_m <= 1.45d+236) then
        tmp = t_2
    else if (eps_m <= 2.05d+237) then
        tmp = t_1
    else if (eps_m <= 5.8d+237) then
        tmp = t_0
    else if (eps_m <= 5.9d+237) then
        tmp = t_4
    else if (eps_m <= 6d+241) then
        tmp = t_5
    else if (eps_m <= 6.1d+246) then
        tmp = t_1
    else if (eps_m <= 8d+246) then
        tmp = 1.0d0
    else if (eps_m <= 3.25d+248) then
        tmp = t_2
    else if (eps_m <= 6.5d+254) then
        tmp = t_5
    else if (eps_m <= 4d+264) then
        tmp = t_2
    else if (eps_m <= 1.45d+268) then
        tmp = t_1
    else if (eps_m <= 2d+268) then
        tmp = t_2
    else if (eps_m <= 1.05d+272) then
        tmp = t_4
    else if (eps_m <= 1.45d+279) then
        tmp = t_1
    else if (eps_m <= 4.8d+279) then
        tmp = t_4
    else if (eps_m <= 2.5d+283) then
        tmp = t_0
    else if (eps_m <= 1.4d+295) then
        tmp = t_4
    else if (eps_m <= 2.7d+299) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	double t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double t_2 = (2.0 - (x * eps_m)) / 2.0;
	double t_3 = x * (eps_m + (-1.0 / eps_m));
	double t_4 = (2.0 - t_3) / 2.0;
	double t_5 = (2.0 + t_3) / 2.0;
	double tmp;
	if (eps_m <= 8.2e+68) {
		tmp = (2.0 + ((x * (1.0 - eps_m)) - x)) / 2.0;
	} else if (eps_m <= 9e+153) {
		tmp = t_1;
	} else if (eps_m <= 1.4e+166) {
		tmp = t_5;
	} else if (eps_m <= 2e+179) {
		tmp = t_2;
	} else if (eps_m <= 1.55e+181) {
		tmp = t_1;
	} else if (eps_m <= 1.58e+181) {
		tmp = t_2;
	} else if (eps_m <= 1.15e+187) {
		tmp = t_1;
	} else if (eps_m <= 2e+191) {
		tmp = t_2;
	} else if (eps_m <= 5e+197) {
		tmp = t_1;
	} else if (eps_m <= 1.65e+203) {
		tmp = t_2;
	} else if (eps_m <= 2.65e+228) {
		tmp = t_1;
	} else if (eps_m <= 4.3e+228) {
		tmp = t_2;
	} else if (eps_m <= 2.75e+230) {
		tmp = t_0;
	} else if (eps_m <= 2.9e+232) {
		tmp = t_4;
	} else if (eps_m <= 1.4e+236) {
		tmp = t_1;
	} else if (eps_m <= 1.45e+236) {
		tmp = t_2;
	} else if (eps_m <= 2.05e+237) {
		tmp = t_1;
	} else if (eps_m <= 5.8e+237) {
		tmp = t_0;
	} else if (eps_m <= 5.9e+237) {
		tmp = t_4;
	} else if (eps_m <= 6e+241) {
		tmp = t_5;
	} else if (eps_m <= 6.1e+246) {
		tmp = t_1;
	} else if (eps_m <= 8e+246) {
		tmp = 1.0;
	} else if (eps_m <= 3.25e+248) {
		tmp = t_2;
	} else if (eps_m <= 6.5e+254) {
		tmp = t_5;
	} else if (eps_m <= 4e+264) {
		tmp = t_2;
	} else if (eps_m <= 1.45e+268) {
		tmp = t_1;
	} else if (eps_m <= 2e+268) {
		tmp = t_2;
	} else if (eps_m <= 1.05e+272) {
		tmp = t_4;
	} else if (eps_m <= 1.45e+279) {
		tmp = t_1;
	} else if (eps_m <= 4.8e+279) {
		tmp = t_4;
	} else if (eps_m <= 2.5e+283) {
		tmp = t_0;
	} else if (eps_m <= 1.4e+295) {
		tmp = t_4;
	} else if (eps_m <= 2.7e+299) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (eps_m * (x + (2.0 / eps_m))) / 2.0
	t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0
	t_2 = (2.0 - (x * eps_m)) / 2.0
	t_3 = x * (eps_m + (-1.0 / eps_m))
	t_4 = (2.0 - t_3) / 2.0
	t_5 = (2.0 + t_3) / 2.0
	tmp = 0
	if eps_m <= 8.2e+68:
		tmp = (2.0 + ((x * (1.0 - eps_m)) - x)) / 2.0
	elif eps_m <= 9e+153:
		tmp = t_1
	elif eps_m <= 1.4e+166:
		tmp = t_5
	elif eps_m <= 2e+179:
		tmp = t_2
	elif eps_m <= 1.55e+181:
		tmp = t_1
	elif eps_m <= 1.58e+181:
		tmp = t_2
	elif eps_m <= 1.15e+187:
		tmp = t_1
	elif eps_m <= 2e+191:
		tmp = t_2
	elif eps_m <= 5e+197:
		tmp = t_1
	elif eps_m <= 1.65e+203:
		tmp = t_2
	elif eps_m <= 2.65e+228:
		tmp = t_1
	elif eps_m <= 4.3e+228:
		tmp = t_2
	elif eps_m <= 2.75e+230:
		tmp = t_0
	elif eps_m <= 2.9e+232:
		tmp = t_4
	elif eps_m <= 1.4e+236:
		tmp = t_1
	elif eps_m <= 1.45e+236:
		tmp = t_2
	elif eps_m <= 2.05e+237:
		tmp = t_1
	elif eps_m <= 5.8e+237:
		tmp = t_0
	elif eps_m <= 5.9e+237:
		tmp = t_4
	elif eps_m <= 6e+241:
		tmp = t_5
	elif eps_m <= 6.1e+246:
		tmp = t_1
	elif eps_m <= 8e+246:
		tmp = 1.0
	elif eps_m <= 3.25e+248:
		tmp = t_2
	elif eps_m <= 6.5e+254:
		tmp = t_5
	elif eps_m <= 4e+264:
		tmp = t_2
	elif eps_m <= 1.45e+268:
		tmp = t_1
	elif eps_m <= 2e+268:
		tmp = t_2
	elif eps_m <= 1.05e+272:
		tmp = t_4
	elif eps_m <= 1.45e+279:
		tmp = t_1
	elif eps_m <= 4.8e+279:
		tmp = t_4
	elif eps_m <= 2.5e+283:
		tmp = t_0
	elif eps_m <= 1.4e+295:
		tmp = t_4
	elif eps_m <= 2.7e+299:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(eps_m * Float64(x + Float64(2.0 / eps_m))) / 2.0)
	t_1 = Float64(Float64(Float64(Float64(eps_m * Float64(Float64(x * eps_m) + 2.0)) - x) / eps_m) / 2.0)
	t_2 = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0)
	t_3 = Float64(x * Float64(eps_m + Float64(-1.0 / eps_m)))
	t_4 = Float64(Float64(2.0 - t_3) / 2.0)
	t_5 = Float64(Float64(2.0 + t_3) / 2.0)
	tmp = 0.0
	if (eps_m <= 8.2e+68)
		tmp = Float64(Float64(2.0 + Float64(Float64(x * Float64(1.0 - eps_m)) - x)) / 2.0);
	elseif (eps_m <= 9e+153)
		tmp = t_1;
	elseif (eps_m <= 1.4e+166)
		tmp = t_5;
	elseif (eps_m <= 2e+179)
		tmp = t_2;
	elseif (eps_m <= 1.55e+181)
		tmp = t_1;
	elseif (eps_m <= 1.58e+181)
		tmp = t_2;
	elseif (eps_m <= 1.15e+187)
		tmp = t_1;
	elseif (eps_m <= 2e+191)
		tmp = t_2;
	elseif (eps_m <= 5e+197)
		tmp = t_1;
	elseif (eps_m <= 1.65e+203)
		tmp = t_2;
	elseif (eps_m <= 2.65e+228)
		tmp = t_1;
	elseif (eps_m <= 4.3e+228)
		tmp = t_2;
	elseif (eps_m <= 2.75e+230)
		tmp = t_0;
	elseif (eps_m <= 2.9e+232)
		tmp = t_4;
	elseif (eps_m <= 1.4e+236)
		tmp = t_1;
	elseif (eps_m <= 1.45e+236)
		tmp = t_2;
	elseif (eps_m <= 2.05e+237)
		tmp = t_1;
	elseif (eps_m <= 5.8e+237)
		tmp = t_0;
	elseif (eps_m <= 5.9e+237)
		tmp = t_4;
	elseif (eps_m <= 6e+241)
		tmp = t_5;
	elseif (eps_m <= 6.1e+246)
		tmp = t_1;
	elseif (eps_m <= 8e+246)
		tmp = 1.0;
	elseif (eps_m <= 3.25e+248)
		tmp = t_2;
	elseif (eps_m <= 6.5e+254)
		tmp = t_5;
	elseif (eps_m <= 4e+264)
		tmp = t_2;
	elseif (eps_m <= 1.45e+268)
		tmp = t_1;
	elseif (eps_m <= 2e+268)
		tmp = t_2;
	elseif (eps_m <= 1.05e+272)
		tmp = t_4;
	elseif (eps_m <= 1.45e+279)
		tmp = t_1;
	elseif (eps_m <= 4.8e+279)
		tmp = t_4;
	elseif (eps_m <= 2.5e+283)
		tmp = t_0;
	elseif (eps_m <= 1.4e+295)
		tmp = t_4;
	elseif (eps_m <= 2.7e+299)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	t_2 = (2.0 - (x * eps_m)) / 2.0;
	t_3 = x * (eps_m + (-1.0 / eps_m));
	t_4 = (2.0 - t_3) / 2.0;
	t_5 = (2.0 + t_3) / 2.0;
	tmp = 0.0;
	if (eps_m <= 8.2e+68)
		tmp = (2.0 + ((x * (1.0 - eps_m)) - x)) / 2.0;
	elseif (eps_m <= 9e+153)
		tmp = t_1;
	elseif (eps_m <= 1.4e+166)
		tmp = t_5;
	elseif (eps_m <= 2e+179)
		tmp = t_2;
	elseif (eps_m <= 1.55e+181)
		tmp = t_1;
	elseif (eps_m <= 1.58e+181)
		tmp = t_2;
	elseif (eps_m <= 1.15e+187)
		tmp = t_1;
	elseif (eps_m <= 2e+191)
		tmp = t_2;
	elseif (eps_m <= 5e+197)
		tmp = t_1;
	elseif (eps_m <= 1.65e+203)
		tmp = t_2;
	elseif (eps_m <= 2.65e+228)
		tmp = t_1;
	elseif (eps_m <= 4.3e+228)
		tmp = t_2;
	elseif (eps_m <= 2.75e+230)
		tmp = t_0;
	elseif (eps_m <= 2.9e+232)
		tmp = t_4;
	elseif (eps_m <= 1.4e+236)
		tmp = t_1;
	elseif (eps_m <= 1.45e+236)
		tmp = t_2;
	elseif (eps_m <= 2.05e+237)
		tmp = t_1;
	elseif (eps_m <= 5.8e+237)
		tmp = t_0;
	elseif (eps_m <= 5.9e+237)
		tmp = t_4;
	elseif (eps_m <= 6e+241)
		tmp = t_5;
	elseif (eps_m <= 6.1e+246)
		tmp = t_1;
	elseif (eps_m <= 8e+246)
		tmp = 1.0;
	elseif (eps_m <= 3.25e+248)
		tmp = t_2;
	elseif (eps_m <= 6.5e+254)
		tmp = t_5;
	elseif (eps_m <= 4e+264)
		tmp = t_2;
	elseif (eps_m <= 1.45e+268)
		tmp = t_1;
	elseif (eps_m <= 2e+268)
		tmp = t_2;
	elseif (eps_m <= 1.05e+272)
		tmp = t_4;
	elseif (eps_m <= 1.45e+279)
		tmp = t_1;
	elseif (eps_m <= 4.8e+279)
		tmp = t_4;
	elseif (eps_m <= 2.5e+283)
		tmp = t_0;
	elseif (eps_m <= 1.4e+295)
		tmp = t_4;
	elseif (eps_m <= 2.7e+299)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(eps$95$m * N[(x + N[(2.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(eps$95$m * N[(N[(x * eps$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(x * N[(eps$95$m + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(2.0 - t$95$3), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(2.0 + t$95$3), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[eps$95$m, 8.2e+68], N[(N[(2.0 + N[(N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 9e+153], t$95$1, If[LessEqual[eps$95$m, 1.4e+166], t$95$5, If[LessEqual[eps$95$m, 2e+179], t$95$2, If[LessEqual[eps$95$m, 1.55e+181], t$95$1, If[LessEqual[eps$95$m, 1.58e+181], t$95$2, If[LessEqual[eps$95$m, 1.15e+187], t$95$1, If[LessEqual[eps$95$m, 2e+191], t$95$2, If[LessEqual[eps$95$m, 5e+197], t$95$1, If[LessEqual[eps$95$m, 1.65e+203], t$95$2, If[LessEqual[eps$95$m, 2.65e+228], t$95$1, If[LessEqual[eps$95$m, 4.3e+228], t$95$2, If[LessEqual[eps$95$m, 2.75e+230], t$95$0, If[LessEqual[eps$95$m, 2.9e+232], t$95$4, If[LessEqual[eps$95$m, 1.4e+236], t$95$1, If[LessEqual[eps$95$m, 1.45e+236], t$95$2, If[LessEqual[eps$95$m, 2.05e+237], t$95$1, If[LessEqual[eps$95$m, 5.8e+237], t$95$0, If[LessEqual[eps$95$m, 5.9e+237], t$95$4, If[LessEqual[eps$95$m, 6e+241], t$95$5, If[LessEqual[eps$95$m, 6.1e+246], t$95$1, If[LessEqual[eps$95$m, 8e+246], 1.0, If[LessEqual[eps$95$m, 3.25e+248], t$95$2, If[LessEqual[eps$95$m, 6.5e+254], t$95$5, If[LessEqual[eps$95$m, 4e+264], t$95$2, If[LessEqual[eps$95$m, 1.45e+268], t$95$1, If[LessEqual[eps$95$m, 2e+268], t$95$2, If[LessEqual[eps$95$m, 1.05e+272], t$95$4, If[LessEqual[eps$95$m, 1.45e+279], t$95$1, If[LessEqual[eps$95$m, 4.8e+279], t$95$4, If[LessEqual[eps$95$m, 2.5e+283], t$95$0, If[LessEqual[eps$95$m, 1.4e+295], t$95$4, If[LessEqual[eps$95$m, 2.7e+299], t$95$1, t$95$2]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if eps < 8.1999999999999998e68

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 32.5%

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

   6. Taylor expanded in x around 0 34.3%

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

      1. associate-*r* 34.3%

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

      2. neg-mul-1 34.3%

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

      3. *-commutative 34.3%

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

   8. Simplified 34.3%

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

      1. distribute-lft-in 34.3%

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

      2. *-rgt-identity 34.3%

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

      3. distribute-rgt-in 34.3%

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

      4. add-sqr-sqrt 17.8%

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

      5. sqrt-unprod 40.9%

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

      6. sqr-neg 40.9%

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

      7. sqrt-unprod 21.6%

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

      8. add-sqr-sqrt 36.8%

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

      9. un-div-inv 36.8%

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

      10. add-sqr-sqrt 15.2%

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

      11. sqrt-unprod 38.2%

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

      12. sqr-neg 38.2%

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

      13. sqrt-unprod 21.6%

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

      14. add-sqr-sqrt 36.8%

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

   10. Applied egg-rr 36.8%

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

   11. Taylor expanded in eps around inf 57.9%

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

       1. mul-1-neg 57.9%

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

   13. Simplified 57.9%

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

   if 8.1999999999999998e68 < eps < 9.0000000000000002e153 or 1.99999999999999996e179 < eps < 1.54999999999999995e181 or 1.58000000000000002e181 < eps < 1.15000000000000002e187 or 2.00000000000000015e191 < eps < 5.00000000000000009e197 or 1.64999999999999995e203 < eps < 2.65e228 or 2.90000000000000023e232 < eps < 1.39999999999999996e236 or 1.45e236 < eps < 2.05000000000000001e237 or 6.00000000000000031e241 < eps < 6.10000000000000028e246 or 4.00000000000000018e264 < eps < 1.4500000000000001e268 or 1.04999999999999998e272 < eps < 1.44999999999999988e279 or 1.4000000000000001e295 < eps < 2.7e299

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 76.4%

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

   6. Taylor expanded in x around 0 43.2%

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

      1. associate-*r* 43.2%

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

      2. neg-mul-1 43.2%

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

      3. *-commutative 43.2%

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

   8. Simplified 43.2%

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

   9. Taylor expanded in eps around 0 59.3%

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

   if 9.0000000000000002e153 < eps < 1.39999999999999998e166 or 5.8999999999999997e237 < eps < 6.00000000000000031e241 or 3.25000000000000024e248 < eps < 6.50000000000000024e254

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in x around 0 83.9%

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

      1. mul-1-neg 83.9%

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

      2. distribute-rgt-neg-in 83.9%

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

      3. mul-1-neg 83.9%

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

      4. mul-1-neg 83.9%

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

      5. distribute-rgt-neg-in 83.9%

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

      6. neg-sub0 83.9%

         \[\leadsto \frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(0 - \left(1 - \varepsilon\right)\right)}\right)}{2} \]

      7. associate--r- 83.9%

         \[\leadsto \frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(\left(0 - 1\right) + \varepsilon\right)}\right)}{2} \]

      8. metadata-eval 83.9%

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

      9. +-commutative 83.9%

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

      10. +-commutative 83.9%

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

      11. distribute-rgt1-in 83.9%

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

      12. +-commutative 83.9%

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

      13. distribute-rgt-in 83.9%

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

      14. neg-mul-1 83.9%

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

      15. rgt-mult-inverse 83.9%

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

      16. associate-+l+ 83.9%

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

      17. +-commutative 83.9%

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

      18. associate-+r+ 83.9%

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

      19. metadata-eval 83.9%

          \[\leadsto \frac{2 + x \cdot \left(\varepsilon + \left(\color{blue}{0} + \left(-\frac{1}{\varepsilon}\right)\right)\right)}{2} \]

      20. sub-neg 83.9%

          \[\leadsto \frac{2 + x \cdot \left(\varepsilon + \color{blue}{\left(0 - \frac{1}{\varepsilon}\right)}\right)}{2} \]

      21. neg-sub0 83.9%

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

      22. distribute-neg-frac 83.9%

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

      23. metadata-eval 83.9%

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

   8. Simplified 83.9%

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

   if 1.39999999999999998e166 < eps < 1.99999999999999996e179 or 1.54999999999999995e181 < eps < 1.58000000000000002e181 or 1.15000000000000002e187 < eps < 2.00000000000000015e191 or 5.00000000000000009e197 < eps < 1.64999999999999995e203 or 2.65e228 < eps < 4.30000000000000032e228 or 1.39999999999999996e236 < eps < 1.45e236 or 8.00000000000000055e246 < eps < 3.25000000000000024e248 or 6.50000000000000024e254 < eps < 4.00000000000000018e264 or 1.4500000000000001e268 < eps < 1.9999999999999999e268 or 2.7e299 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 10.0%

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

   6. Taylor expanded in x around 0 7.2%

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

      1. associate-*r* 7.2%

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

      2. neg-mul-1 7.2%

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

      3. *-commutative 7.2%

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

   8. Simplified 7.2%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 7.4%

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

      3. sqr-neg 7.4%

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

      4. sqrt-unprod 7.1%

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

      5. add-sqr-sqrt 59.8%

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

      6. distribute-rgt-in 59.8%

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

      7. *-un-lft-identity 59.8%

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

      8. distribute-rgt-in 59.8%

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

   10. Applied egg-rr 59.8%

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

       1. associate-*l/ 59.8%

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

       2. *-lft-identity 59.8%

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

       3. distribute-rgt-out 59.8%

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

       4. +-commutative 59.8%

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

       5. associate-+r- 59.8%

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

       6. div-sub 59.8%

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

       7. *-inverses 59.8%

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

       8. associate-+l- 59.8%

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

       9. metadata-eval 59.8%

          \[\leadsto \frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} - \color{blue}{0}\right) - \varepsilon\right)}{2} \]

   12. Simplified 59.8%

       \[\leadsto \frac{2 + \color{blue}{x \cdot \left(\left(\frac{1}{\varepsilon} - 0\right) - \varepsilon\right)}}{2} \]

   13. Taylor expanded in eps around inf 59.8%

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

       1. associate-*r* 59.8%

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

       2. neg-mul-1 59.8%

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

       3. *-commutative 59.8%

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

   15. Simplified 59.8%

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

   if 4.30000000000000032e228 < eps < 2.7499999999999999e230 or 2.05000000000000001e237 < eps < 5.8000000000000002e237 or 4.8000000000000001e279 < eps < 2.5000000000000002e283

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0 67.8%

      \[\leadsto \frac{\color{blue}{2 + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. associate-*r* 67.8%

         \[\leadsto \frac{2 + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}}{2} \]

      2. neg-mul-1 67.8%

         \[\leadsto \frac{2 + \color{blue}{\left(-x\right)} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}{2} \]

      3. *-commutative 67.8%

         \[\leadsto \frac{2 + \left(-x\right) \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

   8. Simplified 67.8%

      \[\leadsto \frac{\color{blue}{2 + \left(-x\right) \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

   9. Taylor expanded in eps around inf 67.8%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(x + 2 \cdot \frac{1}{\varepsilon}\right)}}{2} \]
   10. Step-by-step derivation

       1. associate-*r/ 67.8%

          \[\leadsto \frac{\varepsilon \cdot \left(x + \color{blue}{\frac{2 \cdot 1}{\varepsilon}}\right)}{2} \]

       2. metadata-eval 67.8%

          \[\leadsto \frac{\varepsilon \cdot \left(x + \frac{\color{blue}{2}}{\varepsilon}\right)}{2} \]

   11. Simplified 67.8%

       \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}}{2} \]

   if 2.7499999999999999e230 < eps < 2.90000000000000023e232 or 5.8000000000000002e237 < eps < 5.8999999999999997e237 or 1.9999999999999999e268 < eps < 1.04999999999999998e272 or 1.44999999999999988e279 < eps < 4.8000000000000001e279 or 2.5000000000000002e283 < eps < 1.4000000000000001e295

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 3.1%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0 0.3%

      \[\leadsto \frac{\color{blue}{2 + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. associate-*r* 0.3%

         \[\leadsto \frac{2 + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}}{2} \]

      2. neg-mul-1 0.3%

         \[\leadsto \frac{2 + \color{blue}{\left(-x\right)} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}{2} \]

      3. *-commutative 0.3%

         \[\leadsto \frac{2 + \left(-x\right) \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

   8. Simplified 0.3%

      \[\leadsto \frac{\color{blue}{2 + \left(-x\right) \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.3%

         \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2} \]

      2. sqrt-unprod 1.4%

         \[\leadsto \frac{2 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2} \]

      3. sqr-neg 1.4%

         \[\leadsto \frac{2 + \sqrt{\color{blue}{x \cdot x}} \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2} \]

      4. sqrt-unprod 0.0%

         \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2} \]

      5. add-sqr-sqrt 45.1%

         \[\leadsto \frac{2 + \color{blue}{x} \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2} \]

      6. distribute-rgt-in 45.1%

         \[\leadsto \frac{2 + x \cdot \color{blue}{\left(1 \cdot \left(1 - \varepsilon\right) + \frac{1}{\varepsilon} \cdot \left(1 - \varepsilon\right)\right)}}{2} \]

      7. *-un-lft-identity 45.1%

         \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(1 - \varepsilon\right)} + \frac{1}{\varepsilon} \cdot \left(1 - \varepsilon\right)\right)}{2} \]

      8. distribute-rgt-in 45.1%

         \[\leadsto \frac{2 + \color{blue}{\left(\left(1 - \varepsilon\right) \cdot x + \left(\frac{1}{\varepsilon} \cdot \left(1 - \varepsilon\right)\right) \cdot x\right)}}{2} \]

   10. Applied egg-rr 45.1%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(1 - \varepsilon\right) \cdot x + \left(\frac{1}{\varepsilon} \cdot \left(1 - \varepsilon\right)\right) \cdot x\right)}}{2} \]
   11. Step-by-step derivation

       1. associate-*l/ 45.1%

          \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot x + \color{blue}{\frac{1 \cdot \left(1 - \varepsilon\right)}{\varepsilon}} \cdot x\right)}{2} \]

       2. *-lft-identity 45.1%

          \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot x + \frac{\color{blue}{1 - \varepsilon}}{\varepsilon} \cdot x\right)}{2} \]

       3. distribute-rgt-out 45.1%

          \[\leadsto \frac{2 + \color{blue}{x \cdot \left(\left(1 - \varepsilon\right) + \frac{1 - \varepsilon}{\varepsilon}\right)}}{2} \]

       4. +-commutative 45.1%

          \[\leadsto \frac{2 + x \cdot \color{blue}{\left(\frac{1 - \varepsilon}{\varepsilon} + \left(1 - \varepsilon\right)\right)}}{2} \]

       5. associate-+r- 45.1%

          \[\leadsto \frac{2 + x \cdot \color{blue}{\left(\left(\frac{1 - \varepsilon}{\varepsilon} + 1\right) - \varepsilon\right)}}{2} \]

       6. div-sub 45.1%

          \[\leadsto \frac{2 + x \cdot \left(\left(\color{blue}{\left(\frac{1}{\varepsilon} - \frac{\varepsilon}{\varepsilon}\right)} + 1\right) - \varepsilon\right)}{2} \]

       7. *-inverses 45.1%

          \[\leadsto \frac{2 + x \cdot \left(\left(\left(\frac{1}{\varepsilon} - \color{blue}{1}\right) + 1\right) - \varepsilon\right)}{2} \]

       8. associate-+l- 45.1%

          \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(\frac{1}{\varepsilon} - \left(1 - 1\right)\right)} - \varepsilon\right)}{2} \]

       9. metadata-eval 45.1%

          \[\leadsto \frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} - \color{blue}{0}\right) - \varepsilon\right)}{2} \]

   12. Simplified 45.1%

       \[\leadsto \frac{2 + \color{blue}{x \cdot \left(\left(\frac{1}{\varepsilon} - 0\right) - \varepsilon\right)}}{2} \]

   if 6.10000000000000028e246 < eps < 8.00000000000000055e246

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{2}}{2} \]
3. Recombined 7 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(1 - \varepsilon\right) - x\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 9 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{+166}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.55 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.58 \cdot 10^{+181}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+191}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.65 \cdot 10^{+203}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.65 \cdot 10^{+228}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.3 \cdot 10^{+228}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.75 \cdot 10^{+230}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{+232}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{+236}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.45 \cdot 10^{+236}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.05 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 5.8 \cdot 10^{+237}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 5.9 \cdot 10^{+237}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{+241}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 6.1 \cdot 10^{+246}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 8 \cdot 10^{+246}:\\ \;\;\;\;1\\ \mathbf{elif}\;\varepsilon \leq 3.25 \cdot 10^{+248}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{+254}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{+264}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.45 \cdot 10^{+268}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.05 \cdot 10^{+272}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.45 \cdot 10^{+279}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.8 \cdot 10^{+279}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.5 \cdot 10^{+283}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{+295}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.7 \cdot 10^{+299}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{2 + \frac{x + eps\_m \cdot \left(x \cdot 0 - x \cdot eps\_m\right)}{eps\_m}}{2}\\ t_1 := \frac{\frac{eps\_m \cdot \left(x \cdot eps\_m + 2\right) - x}{eps\_m}}{2}\\ \mathbf{if}\;eps\_m \leq 3.1 \cdot 10^{+254}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{elif}\;eps\_m \leq 7.2 \cdot 10^{+272}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 2 \cdot 10^{+276}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eps\_m \leq 1.45 \cdot 10^{+280}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 1.7 \cdot 10^{+283}:\\ \;\;\;\;\frac{eps\_m \cdot \left(x + \frac{2}{eps\_m}\right)}{2}\\ \mathbf{elif}\;eps\_m \leq 4 \cdot 10^{+295} \lor \neg \left(eps\_m \leq 2.6 \cdot 10^{+299}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0
         (/ (+ 2.0 (/ (+ x (* eps_m (- (* x 0.0) (* x eps_m)))) eps_m)) 2.0))
        (t_1 (/ (/ (- (* eps_m (+ (* x eps_m) 2.0)) x) eps_m) 2.0)))
   (if (<= eps_m 3.1e+254)
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (if (<= eps_m 7.2e+272)
       t_0
       (if (<= eps_m 2e+276)
         t_1
         (if (<= eps_m 1.45e+280)
           t_0
           (if (<= eps_m 1.7e+283)
             (/ (* eps_m (+ x (/ 2.0 eps_m))) 2.0)
             (if (or (<= eps_m 4e+295) (not (<= eps_m 2.6e+299)))
               t_0
               t_1))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	double t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double tmp;
	if (eps_m <= 3.1e+254) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else if (eps_m <= 7.2e+272) {
		tmp = t_0;
	} else if (eps_m <= 2e+276) {
		tmp = t_1;
	} else if (eps_m <= 1.45e+280) {
		tmp = t_0;
	} else if (eps_m <= 1.7e+283) {
		tmp = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	} else if ((eps_m <= 4e+295) || !(eps_m <= 2.6e+299)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (2.0d0 + ((x + (eps_m * ((x * 0.0d0) - (x * eps_m)))) / eps_m)) / 2.0d0
    t_1 = (((eps_m * ((x * eps_m) + 2.0d0)) - x) / eps_m) / 2.0d0
    if (eps_m <= 3.1d+254) then
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    else if (eps_m <= 7.2d+272) then
        tmp = t_0
    else if (eps_m <= 2d+276) then
        tmp = t_1
    else if (eps_m <= 1.45d+280) then
        tmp = t_0
    else if (eps_m <= 1.7d+283) then
        tmp = (eps_m * (x + (2.0d0 / eps_m))) / 2.0d0
    else if ((eps_m <= 4d+295) .or. (.not. (eps_m <= 2.6d+299))) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	double t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double tmp;
	if (eps_m <= 3.1e+254) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else if (eps_m <= 7.2e+272) {
		tmp = t_0;
	} else if (eps_m <= 2e+276) {
		tmp = t_1;
	} else if (eps_m <= 1.45e+280) {
		tmp = t_0;
	} else if (eps_m <= 1.7e+283) {
		tmp = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	} else if ((eps_m <= 4e+295) || !(eps_m <= 2.6e+299)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0
	t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0
	tmp = 0
	if eps_m <= 3.1e+254:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	elif eps_m <= 7.2e+272:
		tmp = t_0
	elif eps_m <= 2e+276:
		tmp = t_1
	elif eps_m <= 1.45e+280:
		tmp = t_0
	elif eps_m <= 1.7e+283:
		tmp = (eps_m * (x + (2.0 / eps_m))) / 2.0
	elif (eps_m <= 4e+295) or not (eps_m <= 2.6e+299):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(2.0 + Float64(Float64(x + Float64(eps_m * Float64(Float64(x * 0.0) - Float64(x * eps_m)))) / eps_m)) / 2.0)
	t_1 = Float64(Float64(Float64(Float64(eps_m * Float64(Float64(x * eps_m) + 2.0)) - x) / eps_m) / 2.0)
	tmp = 0.0
	if (eps_m <= 3.1e+254)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	elseif (eps_m <= 7.2e+272)
		tmp = t_0;
	elseif (eps_m <= 2e+276)
		tmp = t_1;
	elseif (eps_m <= 1.45e+280)
		tmp = t_0;
	elseif (eps_m <= 1.7e+283)
		tmp = Float64(Float64(eps_m * Float64(x + Float64(2.0 / eps_m))) / 2.0);
	elseif ((eps_m <= 4e+295) || !(eps_m <= 2.6e+299))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	t_1 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	tmp = 0.0;
	if (eps_m <= 3.1e+254)
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	elseif (eps_m <= 7.2e+272)
		tmp = t_0;
	elseif (eps_m <= 2e+276)
		tmp = t_1;
	elseif (eps_m <= 1.45e+280)
		tmp = t_0;
	elseif (eps_m <= 1.7e+283)
		tmp = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	elseif ((eps_m <= 4e+295) || ~((eps_m <= 2.6e+299)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(2.0 + N[(N[(x + N[(eps$95$m * N[(N[(x * 0.0), $MachinePrecision] - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(eps$95$m * N[(N[(x * eps$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[eps$95$m, 3.1e+254], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 7.2e+272], t$95$0, If[LessEqual[eps$95$m, 2e+276], t$95$1, If[LessEqual[eps$95$m, 1.45e+280], t$95$0, If[LessEqual[eps$95$m, 1.7e+283], N[(N[(eps$95$m * N[(x + N[(2.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[eps$95$m, 4e+295], N[Not[LessEqual[eps$95$m, 2.6e+299]], $MachinePrecision]], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 3.1000000000000002e254

   1. Initial program 72.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 99.2%

      \[\leadsto \frac{\color{blue}{e^{x \cdot \left(\varepsilon - 1\right)} + \frac{1}{e^{x + \varepsilon \cdot x}}}}{2} \]

   6. Taylor expanded in x around 0 67.8%

      \[\leadsto \frac{e^{x \cdot \left(\varepsilon - 1\right)} + \color{blue}{1}}{2} \]

   7. Taylor expanded in eps around inf 68.1%

      \[\leadsto \frac{e^{\color{blue}{\varepsilon \cdot x}} + 1}{2} \]
   8. Step-by-step derivation

      1. *-commutative 87.1%

         \[\leadsto \frac{e^{\color{blue}{x \cdot \varepsilon}} + \frac{1}{{e}^{\left(x \cdot \varepsilon\right)}}}{2} \]

   9. Simplified 68.1%

      \[\leadsto \frac{e^{\color{blue}{x \cdot \varepsilon}} + 1}{2} \]

   if 3.1000000000000002e254 < eps < 7.1999999999999995e272 or 2.0000000000000001e276 < eps < 1.44999999999999993e280 or 1.7000000000000001e283 < eps < 3.9999999999999999e295 or 2.6e299 < eps

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 12.8%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0 0.9%

      \[\leadsto \frac{\color{blue}{2 + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. associate-*r* 0.9%

         \[\leadsto \frac{2 + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}}{2} \]

      2. neg-mul-1 0.9%

         \[\leadsto \frac{2 + \color{blue}{\left(-x\right)} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}{2} \]

      3. *-commutative 0.9%

         \[\leadsto \frac{2 + \left(-x\right) \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

   8. Simplified 0.9%

      \[\leadsto \frac{\color{blue}{2 + \left(-x\right) \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]
   9. Step-by-step derivation

      1. distribute-lft-in 0.9%

         \[\leadsto \frac{2 + \left(-x\right) \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot 1 + \left(1 - \varepsilon\right) \cdot \frac{1}{\varepsilon}\right)}}{2} \]

      2. *-rgt-identity 0.9%

         \[\leadsto \frac{2 + \left(-x\right) \cdot \left(\color{blue}{\left(1 - \varepsilon\right)} + \left(1 - \varepsilon\right) \cdot \frac{1}{\varepsilon}\right)}{2} \]

      3. distribute-rgt-in 0.9%

         \[\leadsto \frac{2 + \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(-x\right) + \left(\left(1 - \varepsilon\right) \cdot \frac{1}{\varepsilon}\right) \cdot \left(-x\right)\right)}}{2} \]

      4. add-sqr-sqrt 0.1%

         \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} + \left(\left(1 - \varepsilon\right) \cdot \frac{1}{\varepsilon}\right) \cdot \left(-x\right)\right)}{2} \]

      5. sqrt-unprod 1.0%

         \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + \left(\left(1 - \varepsilon\right) \cdot \frac{1}{\varepsilon}\right) \cdot \left(-x\right)\right)}{2} \]

      6. sqr-neg 1.0%

         \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot \sqrt{\color{blue}{x \cdot x}} + \left(\left(1 - \varepsilon\right) \cdot \frac{1}{\varepsilon}\right) \cdot \left(-x\right)\right)}{2} \]

      7. sqrt-unprod 0.0%

         \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + \left(\left(1 - \varepsilon\right) \cdot \frac{1}{\varepsilon}\right) \cdot \left(-x\right)\right)}{2} \]

      8. add-sqr-sqrt 42.7%

         \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot \color{blue}{x} + \left(\left(1 - \varepsilon\right) \cdot \frac{1}{\varepsilon}\right) \cdot \left(-x\right)\right)}{2} \]

      9. un-div-inv 42.7%

         \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot x + \color{blue}{\frac{1 - \varepsilon}{\varepsilon}} \cdot \left(-x\right)\right)}{2} \]

      10. add-sqr-sqrt 42.7%

          \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot x + \frac{1 - \varepsilon}{\varepsilon} \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}\right)}{2} \]

      11. sqrt-unprod 22.7%

          \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot x + \frac{1 - \varepsilon}{\varepsilon} \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)}{2} \]

      12. sqr-neg 22.7%

          \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot x + \frac{1 - \varepsilon}{\varepsilon} \cdot \sqrt{\color{blue}{x \cdot x}}\right)}{2} \]

      13. sqrt-unprod 0.0%

          \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot x + \frac{1 - \varepsilon}{\varepsilon} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)}{2} \]

      14. add-sqr-sqrt 42.7%

          \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot x + \frac{1 - \varepsilon}{\varepsilon} \cdot \color{blue}{x}\right)}{2} \]

   10. Applied egg-rr 42.7%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(1 - \varepsilon\right) \cdot x + \frac{1 - \varepsilon}{\varepsilon} \cdot x\right)}}{2} \]

   11. Taylor expanded in eps around 0 90.0%

       \[\leadsto \frac{2 + \color{blue}{\frac{x + \varepsilon \cdot \left(x + \left(-1 \cdot x + -1 \cdot \left(\varepsilon \cdot x\right)\right)\right)}{\varepsilon}}}{2} \]
   12. Step-by-step derivation

       1. associate-+r+ 90.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \color{blue}{\left(\left(x + -1 \cdot x\right) + -1 \cdot \left(\varepsilon \cdot x\right)\right)}}{\varepsilon}}{2} \]

       2. distribute-rgt1-in 90.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(\color{blue}{\left(-1 + 1\right) \cdot x} + -1 \cdot \left(\varepsilon \cdot x\right)\right)}{\varepsilon}}{2} \]

       3. metadata-eval 90.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(\color{blue}{0} \cdot x + -1 \cdot \left(\varepsilon \cdot x\right)\right)}{\varepsilon}}{2} \]

       4. associate-*r* 90.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(0 \cdot x + \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot x}\right)}{\varepsilon}}{2} \]

       5. neg-mul-1 90.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(0 \cdot x + \color{blue}{\left(-\varepsilon\right)} \cdot x\right)}{\varepsilon}}{2} \]

       6. *-commutative 90.0%

          \[\leadsto \frac{2 + \frac{x + \varepsilon \cdot \left(0 \cdot x + \color{blue}{x \cdot \left(-\varepsilon\right)}\right)}{\varepsilon}}{2} \]

   13. Simplified 90.0%

       \[\leadsto \frac{2 + \color{blue}{\frac{x + \varepsilon \cdot \left(0 \cdot x + x \cdot \left(-\varepsilon\right)\right)}{\varepsilon}}}{2} \]

   if 7.1999999999999995e272 < eps < 2.0000000000000001e276 or 3.9999999999999999e295 < eps < 2.6e299

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0 6.6%

      \[\leadsto \frac{\color{blue}{2 + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. associate-*r* 6.6%

         \[\leadsto \frac{2 + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}}{2} \]

      2. neg-mul-1 6.6%

         \[\leadsto \frac{2 + \color{blue}{\left(-x\right)} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}{2} \]

      3. *-commutative 6.6%

         \[\leadsto \frac{2 + \left(-x\right) \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

   8. Simplified 6.6%

      \[\leadsto \frac{\color{blue}{2 + \left(-x\right) \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

   9. Taylor expanded in eps around 0 100.0%

      \[\leadsto \frac{\color{blue}{\frac{-1 \cdot x + \varepsilon \cdot \left(2 + \varepsilon \cdot x\right)}{\varepsilon}}}{2} \]

   if 1.44999999999999993e280 < eps < 1.7000000000000001e283

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{2 + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \frac{2 + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}}{2} \]

      2. neg-mul-1 100.0%

         \[\leadsto \frac{2 + \color{blue}{\left(-x\right)} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}{2} \]

      3. *-commutative 100.0%

         \[\leadsto \frac{2 + \left(-x\right) \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

   8. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{2 + \left(-x\right) \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

   9. Taylor expanded in eps around inf 100.0%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(x + 2 \cdot \frac{1}{\varepsilon}\right)}}{2} \]
   10. Step-by-step derivation

       1. associate-*r/ 100.0%

          \[\leadsto \frac{\varepsilon \cdot \left(x + \color{blue}{\frac{2 \cdot 1}{\varepsilon}}\right)}{2} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{\varepsilon \cdot \left(x + \frac{\color{blue}{2}}{\varepsilon}\right)}{2} \]

   11. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.1 \cdot 10^{+254}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 7.2 \cdot 10^{+272}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.45 \cdot 10^{+280}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{+283}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{+295} \lor \neg \left(\varepsilon \leq 2.6 \cdot 10^{+299}\right):\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.4% accurate, 25.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{eps\_m \cdot \left(x + \frac{2}{eps\_m}\right)}{2} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (/ (* eps_m (+ x (/ 2.0 eps_m))) 2.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (eps_m * (x + (2.0 / eps_m))) / 2.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = (eps_m * (x + (2.0d0 / eps_m))) / 2.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return (eps_m * (x + (2.0 / eps_m))) / 2.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (eps_m * (x + (2.0 / eps_m))) / 2.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(eps_m * Float64(x + Float64(2.0 / eps_m))) / 2.0)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (eps_m * (x + (2.0 / eps_m))) / 2.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(eps$95$m * N[(x + N[(2.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

3. Simplified 73.5%

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 40.1%

   \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

6. Taylor expanded in x around 0 35.1%

   \[\leadsto \frac{\color{blue}{2 + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]
7. Step-by-step derivation

   1. associate-*r* 35.1%

      \[\leadsto \frac{2 + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}}{2} \]

   2. neg-mul-1 35.1%

      \[\leadsto \frac{2 + \color{blue}{\left(-x\right)} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}{2} \]

   3. *-commutative 35.1%

      \[\leadsto \frac{2 + \left(-x\right) \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

8. Simplified 35.1%

   \[\leadsto \frac{\color{blue}{2 + \left(-x\right) \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

9. Taylor expanded in eps around inf 50.1%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(x + 2 \cdot \frac{1}{\varepsilon}\right)}}{2} \]
10. Step-by-step derivation

    1. associate-*r/ 50.1%

       \[\leadsto \frac{\varepsilon \cdot \left(x + \color{blue}{\frac{2 \cdot 1}{\varepsilon}}\right)}{2} \]

    2. metadata-eval 50.1%

       \[\leadsto \frac{\varepsilon \cdot \left(x + \frac{\color{blue}{2}}{\varepsilon}\right)}{2} \]

11. Simplified 50.1%

    \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}}{2} \]
12. Add Preprocessing

Alternative 14: 50.8% accurate, 32.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{2 - x \cdot eps\_m}{2} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (/ (- 2.0 (* x eps_m)) 2.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (2.0 - (x * eps_m)) / 2.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = (2.0d0 - (x * eps_m)) / 2.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return (2.0 - (x * eps_m)) / 2.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (2.0 - (x * eps_m)) / 2.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (2.0 - (x * eps_m)) / 2.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

3. Simplified 73.5%

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 40.1%

   \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

6. Taylor expanded in x around 0 35.1%

   \[\leadsto \frac{\color{blue}{2 + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]
7. Step-by-step derivation

   1. associate-*r* 35.1%

      \[\leadsto \frac{2 + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}}{2} \]

   2. neg-mul-1 35.1%

      \[\leadsto \frac{2 + \color{blue}{\left(-x\right)} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}{2} \]

   3. *-commutative 35.1%

      \[\leadsto \frac{2 + \left(-x\right) \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

8. Simplified 35.1%

   \[\leadsto \frac{\color{blue}{2 + \left(-x\right) \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 15.2%

      \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2} \]

   2. sqrt-unprod 38.1%

      \[\leadsto \frac{2 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2} \]

   3. sqr-neg 38.1%

      \[\leadsto \frac{2 + \sqrt{\color{blue}{x \cdot x}} \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2} \]

   4. sqrt-unprod 18.2%

      \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2} \]

   5. add-sqr-sqrt 36.9%

      \[\leadsto \frac{2 + \color{blue}{x} \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2} \]

   6. distribute-rgt-in 36.9%

      \[\leadsto \frac{2 + x \cdot \color{blue}{\left(1 \cdot \left(1 - \varepsilon\right) + \frac{1}{\varepsilon} \cdot \left(1 - \varepsilon\right)\right)}}{2} \]

   7. *-un-lft-identity 36.9%

      \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(1 - \varepsilon\right)} + \frac{1}{\varepsilon} \cdot \left(1 - \varepsilon\right)\right)}{2} \]

   8. distribute-rgt-in 36.9%

      \[\leadsto \frac{2 + \color{blue}{\left(\left(1 - \varepsilon\right) \cdot x + \left(\frac{1}{\varepsilon} \cdot \left(1 - \varepsilon\right)\right) \cdot x\right)}}{2} \]

10. Applied egg-rr 36.9%

    \[\leadsto \frac{2 + \color{blue}{\left(\left(1 - \varepsilon\right) \cdot x + \left(\frac{1}{\varepsilon} \cdot \left(1 - \varepsilon\right)\right) \cdot x\right)}}{2} \]
11. Step-by-step derivation

    1. associate-*l/ 36.9%

       \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot x + \color{blue}{\frac{1 \cdot \left(1 - \varepsilon\right)}{\varepsilon}} \cdot x\right)}{2} \]

    2. *-lft-identity 36.9%

       \[\leadsto \frac{2 + \left(\left(1 - \varepsilon\right) \cdot x + \frac{\color{blue}{1 - \varepsilon}}{\varepsilon} \cdot x\right)}{2} \]

    3. distribute-rgt-out 36.9%

       \[\leadsto \frac{2 + \color{blue}{x \cdot \left(\left(1 - \varepsilon\right) + \frac{1 - \varepsilon}{\varepsilon}\right)}}{2} \]

    4. +-commutative 36.9%

       \[\leadsto \frac{2 + x \cdot \color{blue}{\left(\frac{1 - \varepsilon}{\varepsilon} + \left(1 - \varepsilon\right)\right)}}{2} \]

    5. associate-+r- 36.9%

       \[\leadsto \frac{2 + x \cdot \color{blue}{\left(\left(\frac{1 - \varepsilon}{\varepsilon} + 1\right) - \varepsilon\right)}}{2} \]

    6. div-sub 36.9%

       \[\leadsto \frac{2 + x \cdot \left(\left(\color{blue}{\left(\frac{1}{\varepsilon} - \frac{\varepsilon}{\varepsilon}\right)} + 1\right) - \varepsilon\right)}{2} \]

    7. *-inverses 36.9%

       \[\leadsto \frac{2 + x \cdot \left(\left(\left(\frac{1}{\varepsilon} - \color{blue}{1}\right) + 1\right) - \varepsilon\right)}{2} \]

    8. associate-+l- 36.9%

       \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(\frac{1}{\varepsilon} - \left(1 - 1\right)\right)} - \varepsilon\right)}{2} \]

    9. metadata-eval 36.9%

       \[\leadsto \frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} - \color{blue}{0}\right) - \varepsilon\right)}{2} \]

12. Simplified 36.9%

    \[\leadsto \frac{2 + \color{blue}{x \cdot \left(\left(\frac{1}{\varepsilon} - 0\right) - \varepsilon\right)}}{2} \]

13. Taylor expanded in eps around inf 52.1%

    \[\leadsto \frac{2 + \color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
14. Step-by-step derivation

    1. associate-*r* 52.1%

       \[\leadsto \frac{2 + \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot x}}{2} \]

    2. neg-mul-1 52.1%

       \[\leadsto \frac{2 + \color{blue}{\left(-\varepsilon\right)} \cdot x}{2} \]

    3. *-commutative 52.1%

       \[\leadsto \frac{2 + \color{blue}{x \cdot \left(-\varepsilon\right)}}{2} \]

15. Simplified 52.1%

    \[\leadsto \frac{2 + \color{blue}{x \cdot \left(-\varepsilon\right)}}{2} \]

16. Final simplification 52.1%

    \[\leadsto \frac{2 - x \cdot \varepsilon}{2} \]
17. Add Preprocessing

Alternative 15: 44.5% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 1.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 1.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 1.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 73.5%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

3. Simplified 73.5%

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 44.4%

   \[\leadsto \frac{\color{blue}{2}}{2} \]

6. Final simplification 44.4%

   \[\leadsto 1 \]
7. Add Preprocessing

Alternative 16: 15.6% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 0.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 73.5%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

3. Simplified 61.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
4. Add Preprocessing

5. Taylor expanded in eps around 0 14.0%

   \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}}}{2} \]
6. Step-by-step derivation

   1. mul-1-neg 14.0%

      \[\leadsto \frac{\frac{e^{-1 \cdot x} + \color{blue}{\left(-e^{-1 \cdot x}\right)}}{\varepsilon}}{2} \]

   2. mul-1-neg 14.0%

      \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-e^{\color{blue}{-x}}\right)}{\varepsilon}}{2} \]

   3. rec-exp 13.9%

      \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-\color{blue}{\frac{1}{e^{x}}}\right)}{\varepsilon}}{2} \]

   4. sub-neg 13.9%

      \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

   5. div-sub 13.9%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

   6. mul-1-neg 13.9%

      \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

   7. rec-exp 14.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

   8. +-inverses 14.2%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

7. Simplified 14.2%

   \[\leadsto \frac{\color{blue}{0}}{2} \]

8. Final simplification 14.2%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))