NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 100.0%

Time: 17.5s

Alternatives: 13

Speedup: 1.1×

• 38.6% 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: 73.8% 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: 100.0% accurate, 1.1× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.0)
   (/ (* 2.0 (* (exp (- x)) (+ x 1.0))) 2.0)
   (/ (+ (exp (* 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 <= 1.0) {
		tmp = (2.0 * (exp(-x) * (x + 1.0))) / 2.0;
	} else {
		tmp = (exp((x * eps_m)) + exp((-x * eps_m))) / 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) :: tmp
    if (eps_m <= 1.0d0) then
        tmp = (2.0d0 * (exp(-x) * (x + 1.0d0))) / 2.0d0
    else
        tmp = (exp((x * eps_m)) + exp((-x * eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = (2.0 * (Math.exp(-x) * (x + 1.0))) / 2.0;
	} else {
		tmp = (Math.exp((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 <= 1.0:
		tmp = (2.0 * (math.exp(-x) * (x + 1.0))) / 2.0
	else:
		tmp = (math.exp((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 <= 1.0)
		tmp = Float64(Float64(2.0 * Float64(exp(Float64(-x)) * Float64(x + 1.0))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + exp(Float64(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 <= 1.0)
		tmp = (2.0 * (exp(-x) * (x + 1.0))) / 2.0;
	else
		tmp = (exp((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, 1.0], N[(N[(2.0 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $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 < 1

   1. Initial program 61.5%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in eps around 0 71.9%

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

      1. distribute-lft-out 71.9%

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

      2. cancel-sign-sub-inv 71.9%

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

      3. metadata-eval 71.9%

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

      4. distribute-rgt1-in 71.9%

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

      5. metadata-eval 71.9%

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

      6. distribute-rgt1-in 71.9%

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

      7. *-commutative 71.9%

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

      8. mul-1-neg 71.9%

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

   7. Simplified 71.9%

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

   if 1 < 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 eps around inf 100.0%

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

      1. exp-prod 100.0%

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

      2. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. exp-prod 100.0%

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

      7. associate-*r* 100.0%

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

      8. mul-1-neg 100.0%

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

      9. neg-mul-1 100.0%

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

      10. sub-neg 100.0%

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

      11. mul-1-neg 100.0%

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

      12. associate-*r* 100.0%

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

      13. remove-double-neg 100.0%

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

      14. sub-neg 100.0%

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

      15. neg-mul-1 100.0%

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

      16. associate-*r* 100.0%

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

      17. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in eps around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. *-commutative 100.0%

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

       3. distribute-lft-neg-in 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 80.1%

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

Alternative 2: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (exp((x * (eps_m + -1.0))) + exp((x * (-1.0 - 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 = (exp((x * (eps_m + (-1.0d0)))) + exp((x * ((-1.0d0) - 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))) + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (math.exp((x * (eps_m + -1.0))) + math.exp((x * (-1.0 - 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))) + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (exp((x * (eps_m + -1.0))) + exp((x * (-1.0 - 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[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 72.8%

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

3. Simplified 72.8%

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

5. Taylor expanded in eps around inf 99.8%

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

   1. exp-prod 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. sub-neg 99.8%

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

   5. neg-mul-1 99.8%

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

   6. exp-prod 99.8%

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

   7. associate-*r* 99.8%

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

   8. mul-1-neg 99.8%

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

   9. neg-mul-1 99.8%

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

   10. sub-neg 99.8%

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

   11. mul-1-neg 99.8%

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

   12. associate-*r* 99.8%

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

   13. remove-double-neg 99.8%

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

   14. sub-neg 99.8%

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

   15. neg-mul-1 99.8%

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

   16. associate-*r* 99.8%

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

   17. associate-*r* 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 65.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := -1 + \frac{-1}{eps\_m}\\ t_1 := eps\_m \cdot t\_0\\ t_2 := \frac{1 + eps\_m}{eps\_m}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+67}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{eps\_m} - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-191}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - eps\_m\right) \cdot t\_0 + \frac{\left(1 + eps\_m\right) \cdot \left(1 + eps\_m\right) + t\_2 \cdot \frac{-1 - eps\_m}{eps\_m}}{\left(-1 - eps\_m\right) - t\_2}\right)}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-217}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-124}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{eps\_m} + \frac{1 - t\_1 \cdot t\_1}{t\_0 + t\_1}\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+23} \lor \neg \left(x \leq 6 \cdot 10^{+140}\right) \land x \leq 10^{+166}:\\ \;\;\;\;\frac{2 \cdot \left(\left(x + 1\right) \cdot e^{x}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ -1.0 eps_m)))
        (t_1 (* eps_m t_0))
        (t_2 (/ (+ 1.0 eps_m) eps_m)))
   (if (<= x -1.35e+67)
     (/ (+ 2.0 (* x (- (/ -1.0 eps_m) eps_m))) 2.0)
     (if (<= x -2.8e-191)
       (/
        (+
         2.0
         (*
          x
          (+
           (* (- 1.0 eps_m) t_0)
           (/
            (+
             (* (+ 1.0 eps_m) (+ 1.0 eps_m))
             (* t_2 (/ (- -1.0 eps_m) eps_m)))
            (- (- -1.0 eps_m) t_2)))))
        2.0)
       (if (<= x 7.5e-217)
         1.0
         (if (<= x 7e-124)
           (/
            (+ 2.0 (* x (+ (/ 1.0 eps_m) (/ (- 1.0 (* t_1 t_1)) (+ t_0 t_1)))))
            2.0)
           (if (or (<= x 1e+23) (and (not (<= x 6e+140)) (<= x 1e+166)))
             (/ (* 2.0 (* (+ x 1.0) (exp x))) 2.0)
             (/ (* 2.0 (/ x (exp x))) 2.0))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = -1.0 + (-1.0 / eps_m);
	double t_1 = eps_m * t_0;
	double t_2 = (1.0 + eps_m) / eps_m;
	double tmp;
	if (x <= -1.35e+67) {
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0;
	} else if (x <= -2.8e-191) {
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_2 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_2))))) / 2.0;
	} else if (x <= 7.5e-217) {
		tmp = 1.0;
	} else if (x <= 7e-124) {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	} else if ((x <= 1e+23) || (!(x <= 6e+140) && (x <= 1e+166))) {
		tmp = (2.0 * ((x + 1.0) * exp(x))) / 2.0;
	} else {
		tmp = (2.0 * (x / exp(x))) / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (-1.0d0) + ((-1.0d0) / eps_m)
    t_1 = eps_m * t_0
    t_2 = (1.0d0 + eps_m) / eps_m
    if (x <= (-1.35d+67)) then
        tmp = (2.0d0 + (x * (((-1.0d0) / eps_m) - eps_m))) / 2.0d0
    else if (x <= (-2.8d-191)) then
        tmp = (2.0d0 + (x * (((1.0d0 - eps_m) * t_0) + ((((1.0d0 + eps_m) * (1.0d0 + eps_m)) + (t_2 * (((-1.0d0) - eps_m) / eps_m))) / (((-1.0d0) - eps_m) - t_2))))) / 2.0d0
    else if (x <= 7.5d-217) then
        tmp = 1.0d0
    else if (x <= 7d-124) then
        tmp = (2.0d0 + (x * ((1.0d0 / eps_m) + ((1.0d0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0d0
    else if ((x <= 1d+23) .or. (.not. (x <= 6d+140)) .and. (x <= 1d+166)) then
        tmp = (2.0d0 * ((x + 1.0d0) * exp(x))) / 2.0d0
    else
        tmp = (2.0d0 * (x / exp(x))) / 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 = -1.0 + (-1.0 / eps_m);
	double t_1 = eps_m * t_0;
	double t_2 = (1.0 + eps_m) / eps_m;
	double tmp;
	if (x <= -1.35e+67) {
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0;
	} else if (x <= -2.8e-191) {
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_2 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_2))))) / 2.0;
	} else if (x <= 7.5e-217) {
		tmp = 1.0;
	} else if (x <= 7e-124) {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	} else if ((x <= 1e+23) || (!(x <= 6e+140) && (x <= 1e+166))) {
		tmp = (2.0 * ((x + 1.0) * Math.exp(x))) / 2.0;
	} else {
		tmp = (2.0 * (x / Math.exp(x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = -1.0 + (-1.0 / eps_m)
	t_1 = eps_m * t_0
	t_2 = (1.0 + eps_m) / eps_m
	tmp = 0
	if x <= -1.35e+67:
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0
	elif x <= -2.8e-191:
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_2 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_2))))) / 2.0
	elif x <= 7.5e-217:
		tmp = 1.0
	elif x <= 7e-124:
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0
	elif (x <= 1e+23) or (not (x <= 6e+140) and (x <= 1e+166)):
		tmp = (2.0 * ((x + 1.0) * math.exp(x))) / 2.0
	else:
		tmp = (2.0 * (x / math.exp(x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(-1.0 + Float64(-1.0 / eps_m))
	t_1 = Float64(eps_m * t_0)
	t_2 = Float64(Float64(1.0 + eps_m) / eps_m)
	tmp = 0.0
	if (x <= -1.35e+67)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 / eps_m) - eps_m))) / 2.0);
	elseif (x <= -2.8e-191)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 - eps_m) * t_0) + Float64(Float64(Float64(Float64(1.0 + eps_m) * Float64(1.0 + eps_m)) + Float64(t_2 * Float64(Float64(-1.0 - eps_m) / eps_m))) / Float64(Float64(-1.0 - eps_m) - t_2))))) / 2.0);
	elseif (x <= 7.5e-217)
		tmp = 1.0;
	elseif (x <= 7e-124)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(t_0 + t_1))))) / 2.0);
	elseif ((x <= 1e+23) || (!(x <= 6e+140) && (x <= 1e+166)))
		tmp = Float64(Float64(2.0 * Float64(Float64(x + 1.0) * exp(x))) / 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(x / exp(x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = -1.0 + (-1.0 / eps_m);
	t_1 = eps_m * t_0;
	t_2 = (1.0 + eps_m) / eps_m;
	tmp = 0.0;
	if (x <= -1.35e+67)
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0;
	elseif (x <= -2.8e-191)
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_2 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_2))))) / 2.0;
	elseif (x <= 7.5e-217)
		tmp = 1.0;
	elseif (x <= 7e-124)
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	elseif ((x <= 1e+23) || (~((x <= 6e+140)) && (x <= 1e+166)))
		tmp = (2.0 * ((x + 1.0) * exp(x))) / 2.0;
	else
		tmp = (2.0 * (x / exp(x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps$95$m * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]}, If[LessEqual[x, -1.35e+67], N[(N[(2.0 + N[(x * N[(N[(-1.0 / eps$95$m), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -2.8e-191], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 - eps$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(N[(-1.0 - eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - eps$95$m), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7.5e-217], 1.0, If[LessEqual[x, 7e-124], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1e+23], And[N[Not[LessEqual[x, 6e+140]], $MachinePrecision], LessEqual[x, 1e+166]]], N[(N[(2.0 * N[(N[(x + 1.0), $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.35e67

   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 3.2%

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

      1. mul-1-neg 3.2%

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

      2. distribute-lft-neg-in 3.2%

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

      3. cancel-sign-sub-inv 3.2%

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

      4. mul-1-neg 3.2%

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

      5. distribute-rgt-neg-in 3.2%

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

      6. sub-neg 3.2%

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

      7. distribute-neg-in 3.2%

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

      8. metadata-eval 3.2%

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

      9. remove-double-neg 3.2%

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

      10. remove-double-neg 3.2%

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

      11. +-commutative 3.2%

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

      12. sub-neg 3.2%

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

      13. metadata-eval 3.2%

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

      14. +-commutative 3.2%

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

   7. Simplified 3.2%

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

   8. Taylor expanded in eps around 0 35.8%

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

   9. Taylor expanded in eps around inf 35.8%

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

       1. neg-mul-1 35.8%

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

   11. Simplified 35.8%

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

   if -1.35e67 < x < -2.80000000000000012e-191

   1. Initial program 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in x around 0 55.6%

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

      1. mul-1-neg 55.6%

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

      2. distribute-lft-neg-in 55.6%

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

      3. cancel-sign-sub-inv 55.6%

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

      4. mul-1-neg 55.6%

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

      5. distribute-rgt-neg-in 55.6%

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

      6. sub-neg 55.6%

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

      7. distribute-neg-in 55.6%

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

      8. metadata-eval 55.6%

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

      9. remove-double-neg 55.6%

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

      10. remove-double-neg 55.6%

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

      11. +-commutative 55.6%

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

      12. sub-neg 55.6%

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

      13. metadata-eval 55.6%

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

      14. +-commutative 55.6%

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

   7. Simplified 55.6%

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

      1. distribute-lft-in 55.6%

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

      2. flip-+ 56.8%

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

   9. Applied egg-rr 56.8%

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

   if -2.80000000000000012e-191 < x < 7.50000000000000031e-217

   1. Initial program 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in x around 0 98.6%

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

   if 7.50000000000000031e-217 < x < 6.9999999999999997e-124

   1. Initial program 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in x around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. distribute-lft-neg-in 66.5%

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

      3. cancel-sign-sub-inv 66.5%

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

      4. mul-1-neg 66.5%

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

      5. distribute-rgt-neg-in 66.5%

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

      6. sub-neg 66.5%

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

      7. distribute-neg-in 66.5%

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

      8. metadata-eval 66.5%

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

      9. remove-double-neg 66.5%

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

      10. remove-double-neg 66.5%

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

      11. +-commutative 66.5%

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

      12. sub-neg 66.5%

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

      13. metadata-eval 66.5%

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

      14. +-commutative 66.5%

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

   7. Simplified 66.5%

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

   8. Taylor expanded in eps around 0 66.2%

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

      1. distribute-lft-in 66.2%

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

      2. flip-+ 58.3%

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

   10. Applied egg-rr 58.3%

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

   11. Taylor expanded in eps around inf 66.2%

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

   if 6.9999999999999997e-124 < x < 9.9999999999999992e22 or 5.99999999999999993e140 < x < 9.9999999999999994e165

   1. Initial program 56.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in eps around 0 53.5%

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

      1. distribute-lft-out 53.5%

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

      2. cancel-sign-sub-inv 53.5%

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

      3. metadata-eval 53.5%

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

      4. distribute-rgt1-in 53.5%

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

      5. metadata-eval 53.5%

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

      6. distribute-rgt1-in 53.5%

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

      7. *-commutative 53.5%

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

      8. mul-1-neg 53.5%

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

   7. Simplified 53.5%

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

      1. +-commutative 53.5%

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

      2. distribute-rgt-in 53.5%

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

      3. *-un-lft-identity 53.5%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 73.7%

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

      6. sqr-neg 73.7%

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

      7. sqrt-unprod 73.7%

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

      8. add-sqr-sqrt 73.7%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 73.7%

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

      11. sqr-neg 73.7%

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

      12. sqrt-unprod 73.7%

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

      13. add-sqr-sqrt 73.7%

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

   9. Applied egg-rr 73.7%

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

       1. distribute-rgt1-in 73.7%

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

       2. +-commutative 73.7%

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

   11. Simplified 73.7%

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

   if 9.9999999999999992e22 < x < 5.99999999999999993e140 or 9.9999999999999994e165 < 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%

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

   5. Taylor expanded in eps around 0 63.1%

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

      1. distribute-lft-out 63.1%

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

      2. cancel-sign-sub-inv 63.1%

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

      3. metadata-eval 63.1%

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

      4. distribute-rgt1-in 63.1%

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

      5. metadata-eval 63.1%

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

      6. distribute-rgt1-in 63.1%

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

      7. *-commutative 63.1%

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

      8. mul-1-neg 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in x around inf 63.1%

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

      1. exp-neg 63.1%

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

      2. associate-*r/ 63.1%

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

      3. *-rgt-identity 63.1%

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

   10. Simplified 63.1%

       \[\leadsto \frac{2 \cdot \color{blue}{\frac{x}{e^{x}}}}{2} \]
3. Recombined 6 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+67}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{\varepsilon} - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-191}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right) + \frac{\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right) + \frac{1 + \varepsilon}{\varepsilon} \cdot \frac{-1 - \varepsilon}{\varepsilon}}{\left(-1 - \varepsilon\right) - \frac{1 + \varepsilon}{\varepsilon}}\right)}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-217}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-124}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \frac{1 - \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right) \cdot \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{\left(-1 + \frac{-1}{\varepsilon}\right) + \varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+23} \lor \neg \left(x \leq 6 \cdot 10^{+140}\right) \land x \leq 10^{+166}:\\ \;\;\;\;\frac{2 \cdot \left(\left(x + 1\right) \cdot e^{x}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{e^{x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := -1 + \frac{-1}{eps\_m}\\ t_1 := eps\_m \cdot t\_0\\ t_2 := \frac{1 + eps\_m}{eps\_m}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+67}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{eps\_m} - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-193}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - eps\_m\right) \cdot t\_0 + \frac{\left(1 + eps\_m\right) \cdot \left(1 + eps\_m\right) + t\_2 \cdot \frac{-1 - eps\_m}{eps\_m}}{\left(-1 - eps\_m\right) - t\_2}\right)}{2}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-216}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-124}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{eps\_m} + \frac{1 - t\_1 \cdot t\_1}{t\_0 + t\_1}\right)}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+17}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ -1.0 eps_m)))
        (t_1 (* eps_m t_0))
        (t_2 (/ (+ 1.0 eps_m) eps_m)))
   (if (<= x -1.35e+67)
     (/ (+ 2.0 (* x (- (/ -1.0 eps_m) eps_m))) 2.0)
     (if (<= x -3e-193)
       (/
        (+
         2.0
         (*
          x
          (+
           (* (- 1.0 eps_m) t_0)
           (/
            (+
             (* (+ 1.0 eps_m) (+ 1.0 eps_m))
             (* t_2 (/ (- -1.0 eps_m) eps_m)))
            (- (- -1.0 eps_m) t_2)))))
        2.0)
       (if (<= x 2.65e-216)
         1.0
         (if (<= x 5e-124)
           (/
            (+ 2.0 (* x (+ (/ 1.0 eps_m) (/ (- 1.0 (* t_1 t_1)) (+ t_0 t_1)))))
            2.0)
           (if (<= x 1.12e+17)
             (/ (+ 2.0 (* x eps_m)) 2.0)
             (/ (* 2.0 (/ x (exp x))) 2.0))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = -1.0 + (-1.0 / eps_m);
	double t_1 = eps_m * t_0;
	double t_2 = (1.0 + eps_m) / eps_m;
	double tmp;
	if (x <= -1.35e+67) {
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0;
	} else if (x <= -3e-193) {
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_2 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_2))))) / 2.0;
	} else if (x <= 2.65e-216) {
		tmp = 1.0;
	} else if (x <= 5e-124) {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	} else if (x <= 1.12e+17) {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	} else {
		tmp = (2.0 * (x / exp(x))) / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (-1.0d0) + ((-1.0d0) / eps_m)
    t_1 = eps_m * t_0
    t_2 = (1.0d0 + eps_m) / eps_m
    if (x <= (-1.35d+67)) then
        tmp = (2.0d0 + (x * (((-1.0d0) / eps_m) - eps_m))) / 2.0d0
    else if (x <= (-3d-193)) then
        tmp = (2.0d0 + (x * (((1.0d0 - eps_m) * t_0) + ((((1.0d0 + eps_m) * (1.0d0 + eps_m)) + (t_2 * (((-1.0d0) - eps_m) / eps_m))) / (((-1.0d0) - eps_m) - t_2))))) / 2.0d0
    else if (x <= 2.65d-216) then
        tmp = 1.0d0
    else if (x <= 5d-124) then
        tmp = (2.0d0 + (x * ((1.0d0 / eps_m) + ((1.0d0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0d0
    else if (x <= 1.12d+17) then
        tmp = (2.0d0 + (x * eps_m)) / 2.0d0
    else
        tmp = (2.0d0 * (x / exp(x))) / 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 = -1.0 + (-1.0 / eps_m);
	double t_1 = eps_m * t_0;
	double t_2 = (1.0 + eps_m) / eps_m;
	double tmp;
	if (x <= -1.35e+67) {
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0;
	} else if (x <= -3e-193) {
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_2 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_2))))) / 2.0;
	} else if (x <= 2.65e-216) {
		tmp = 1.0;
	} else if (x <= 5e-124) {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	} else if (x <= 1.12e+17) {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	} else {
		tmp = (2.0 * (x / Math.exp(x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = -1.0 + (-1.0 / eps_m)
	t_1 = eps_m * t_0
	t_2 = (1.0 + eps_m) / eps_m
	tmp = 0
	if x <= -1.35e+67:
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0
	elif x <= -3e-193:
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_2 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_2))))) / 2.0
	elif x <= 2.65e-216:
		tmp = 1.0
	elif x <= 5e-124:
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0
	elif x <= 1.12e+17:
		tmp = (2.0 + (x * eps_m)) / 2.0
	else:
		tmp = (2.0 * (x / math.exp(x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(-1.0 + Float64(-1.0 / eps_m))
	t_1 = Float64(eps_m * t_0)
	t_2 = Float64(Float64(1.0 + eps_m) / eps_m)
	tmp = 0.0
	if (x <= -1.35e+67)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 / eps_m) - eps_m))) / 2.0);
	elseif (x <= -3e-193)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 - eps_m) * t_0) + Float64(Float64(Float64(Float64(1.0 + eps_m) * Float64(1.0 + eps_m)) + Float64(t_2 * Float64(Float64(-1.0 - eps_m) / eps_m))) / Float64(Float64(-1.0 - eps_m) - t_2))))) / 2.0);
	elseif (x <= 2.65e-216)
		tmp = 1.0;
	elseif (x <= 5e-124)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(t_0 + t_1))))) / 2.0);
	elseif (x <= 1.12e+17)
		tmp = Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(x / exp(x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = -1.0 + (-1.0 / eps_m);
	t_1 = eps_m * t_0;
	t_2 = (1.0 + eps_m) / eps_m;
	tmp = 0.0;
	if (x <= -1.35e+67)
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0;
	elseif (x <= -3e-193)
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_2 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_2))))) / 2.0;
	elseif (x <= 2.65e-216)
		tmp = 1.0;
	elseif (x <= 5e-124)
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	elseif (x <= 1.12e+17)
		tmp = (2.0 + (x * eps_m)) / 2.0;
	else
		tmp = (2.0 * (x / exp(x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps$95$m * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]}, If[LessEqual[x, -1.35e+67], N[(N[(2.0 + N[(x * N[(N[(-1.0 / eps$95$m), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -3e-193], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 - eps$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(N[(-1.0 - eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - eps$95$m), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.65e-216], 1.0, If[LessEqual[x, 5e-124], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.12e+17], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.35e67

   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 3.2%

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

      1. mul-1-neg 3.2%

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

      2. distribute-lft-neg-in 3.2%

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

      3. cancel-sign-sub-inv 3.2%

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

      4. mul-1-neg 3.2%

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

      5. distribute-rgt-neg-in 3.2%

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

      6. sub-neg 3.2%

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

      7. distribute-neg-in 3.2%

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

      8. metadata-eval 3.2%

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

      9. remove-double-neg 3.2%

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

      10. remove-double-neg 3.2%

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

      11. +-commutative 3.2%

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

      12. sub-neg 3.2%

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

      13. metadata-eval 3.2%

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

      14. +-commutative 3.2%

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

   7. Simplified 3.2%

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

   8. Taylor expanded in eps around 0 35.8%

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

   9. Taylor expanded in eps around inf 35.8%

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

       1. neg-mul-1 35.8%

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

   11. Simplified 35.8%

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

   if -1.35e67 < x < -2.9999999999999999e-193

   1. Initial program 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in x around 0 55.6%

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

      1. mul-1-neg 55.6%

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

      2. distribute-lft-neg-in 55.6%

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

      3. cancel-sign-sub-inv 55.6%

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

      4. mul-1-neg 55.6%

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

      5. distribute-rgt-neg-in 55.6%

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

      6. sub-neg 55.6%

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

      7. distribute-neg-in 55.6%

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

      8. metadata-eval 55.6%

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

      9. remove-double-neg 55.6%

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

      10. remove-double-neg 55.6%

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

      11. +-commutative 55.6%

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

      12. sub-neg 55.6%

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

      13. metadata-eval 55.6%

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

      14. +-commutative 55.6%

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

   7. Simplified 55.6%

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

      1. distribute-lft-in 55.6%

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

      2. flip-+ 56.8%

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

   9. Applied egg-rr 56.8%

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

   if -2.9999999999999999e-193 < x < 2.64999999999999989e-216

   1. Initial program 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in x around 0 98.6%

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

   if 2.64999999999999989e-216 < x < 5.0000000000000003e-124

   1. Initial program 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in x around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. distribute-lft-neg-in 66.5%

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

      3. cancel-sign-sub-inv 66.5%

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

      4. mul-1-neg 66.5%

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

      5. distribute-rgt-neg-in 66.5%

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

      6. sub-neg 66.5%

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

      7. distribute-neg-in 66.5%

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

      8. metadata-eval 66.5%

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

      9. remove-double-neg 66.5%

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

      10. remove-double-neg 66.5%

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

      11. +-commutative 66.5%

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

      12. sub-neg 66.5%

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

      13. metadata-eval 66.5%

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

      14. +-commutative 66.5%

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

   7. Simplified 66.5%

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

   8. Taylor expanded in eps around 0 66.2%

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

      1. distribute-lft-in 66.2%

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

      2. flip-+ 58.3%

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

   10. Applied egg-rr 58.3%

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

   11. Taylor expanded in eps around inf 66.2%

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

   if 5.0000000000000003e-124 < x < 1.12e17

   1. Initial program 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in x around 0 62.3%

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

      1. mul-1-neg 62.3%

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

      2. distribute-lft-neg-in 62.3%

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

      3. cancel-sign-sub-inv 62.3%

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

      4. mul-1-neg 62.3%

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

      5. distribute-rgt-neg-in 62.3%

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

      6. sub-neg 62.3%

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

      7. distribute-neg-in 62.3%

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

      8. metadata-eval 62.3%

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

      9. remove-double-neg 62.3%

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

      10. remove-double-neg 62.3%

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

      11. +-commutative 62.3%

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

      12. sub-neg 62.3%

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

      13. metadata-eval 62.3%

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

      14. +-commutative 62.3%

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

   7. Simplified 62.3%

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

   8. Taylor expanded in eps around 0 61.9%

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

   9. Taylor expanded in eps around 0 61.9%

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

   if 1.12e17 < 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%

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

   5. Taylor expanded in eps around 0 56.3%

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

      1. distribute-lft-out 56.3%

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

      2. cancel-sign-sub-inv 56.3%

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

      3. metadata-eval 56.3%

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

      4. distribute-rgt1-in 56.3%

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

      5. metadata-eval 56.3%

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

      6. distribute-rgt1-in 56.3%

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

      7. *-commutative 56.3%

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

      8. mul-1-neg 56.3%

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

   7. Simplified 56.3%

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

   8. Taylor expanded in x around inf 56.3%

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

      1. exp-neg 56.3%

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

      2. associate-*r/ 56.3%

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

      3. *-rgt-identity 56.3%

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

   10. Simplified 56.3%

       \[\leadsto \frac{2 \cdot \color{blue}{\frac{x}{e^{x}}}}{2} \]
3. Recombined 6 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+67}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{\varepsilon} - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-193}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right) + \frac{\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right) + \frac{1 + \varepsilon}{\varepsilon} \cdot \frac{-1 - \varepsilon}{\varepsilon}}{\left(-1 - \varepsilon\right) - \frac{1 + \varepsilon}{\varepsilon}}\right)}{2}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-216}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-124}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \frac{1 - \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right) \cdot \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{\left(-1 + \frac{-1}{\varepsilon}\right) + \varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}\right)}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+17}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{e^{x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := -1 + \frac{-1}{eps\_m}\\ t_1 := eps\_m \cdot t\_0\\ \mathbf{if}\;eps\_m \leq 1:\\ \;\;\;\;\frac{2 \cdot \left(e^{-x} \cdot \left(x + 1\right)\right)}{2}\\ \mathbf{elif}\;eps\_m \leq 9.5 \cdot 10^{+205}:\\ \;\;\;\;\frac{2 \cdot \left(\left(x + 1\right) \cdot e^{x}\right)}{2}\\ \mathbf{elif}\;eps\_m \leq 4.8 \cdot 10^{+249} \lor \neg \left(eps\_m \leq 8.5 \cdot 10^{+295}\right) \land eps\_m \leq 3.5 \cdot 10^{+304}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - eps\_m\right) \cdot t\_0 + \frac{-1 + \left(1 + eps\_m\right) \cdot \left(1 + eps\_m\right)}{\left(-1 - eps\_m\right) - \frac{1 + eps\_m}{eps\_m}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{eps\_m} + \frac{1 - t\_1 \cdot t\_1}{t\_0 + t\_1}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ -1.0 eps_m))) (t_1 (* eps_m t_0)))
   (if (<= eps_m 1.0)
     (/ (* 2.0 (* (exp (- x)) (+ x 1.0))) 2.0)
     (if (<= eps_m 9.5e+205)
       (/ (* 2.0 (* (+ x 1.0) (exp x))) 2.0)
       (if (or (<= eps_m 4.8e+249)
               (and (not (<= eps_m 8.5e+295)) (<= eps_m 3.5e+304)))
         (/
          (+
           2.0
           (*
            x
            (+
             (* (- 1.0 eps_m) t_0)
             (/
              (+ -1.0 (* (+ 1.0 eps_m) (+ 1.0 eps_m)))
              (- (- -1.0 eps_m) (/ (+ 1.0 eps_m) eps_m))))))
          2.0)
         (/
          (+ 2.0 (* x (+ (/ 1.0 eps_m) (/ (- 1.0 (* t_1 t_1)) (+ t_0 t_1)))))
          2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = -1.0 + (-1.0 / eps_m);
	double t_1 = eps_m * t_0;
	double tmp;
	if (eps_m <= 1.0) {
		tmp = (2.0 * (exp(-x) * (x + 1.0))) / 2.0;
	} else if (eps_m <= 9.5e+205) {
		tmp = (2.0 * ((x + 1.0) * exp(x))) / 2.0;
	} else if ((eps_m <= 4.8e+249) || (!(eps_m <= 8.5e+295) && (eps_m <= 3.5e+304))) {
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((-1.0 + ((1.0 + eps_m) * (1.0 + eps_m))) / ((-1.0 - eps_m) - ((1.0 + eps_m) / eps_m)))))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 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) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) + ((-1.0d0) / eps_m)
    t_1 = eps_m * t_0
    if (eps_m <= 1.0d0) then
        tmp = (2.0d0 * (exp(-x) * (x + 1.0d0))) / 2.0d0
    else if (eps_m <= 9.5d+205) then
        tmp = (2.0d0 * ((x + 1.0d0) * exp(x))) / 2.0d0
    else if ((eps_m <= 4.8d+249) .or. (.not. (eps_m <= 8.5d+295)) .and. (eps_m <= 3.5d+304)) then
        tmp = (2.0d0 + (x * (((1.0d0 - eps_m) * t_0) + (((-1.0d0) + ((1.0d0 + eps_m) * (1.0d0 + eps_m))) / (((-1.0d0) - eps_m) - ((1.0d0 + eps_m) / eps_m)))))) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((1.0d0 / eps_m) + ((1.0d0 - (t_1 * t_1)) / (t_0 + t_1))))) / 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 = -1.0 + (-1.0 / eps_m);
	double t_1 = eps_m * t_0;
	double tmp;
	if (eps_m <= 1.0) {
		tmp = (2.0 * (Math.exp(-x) * (x + 1.0))) / 2.0;
	} else if (eps_m <= 9.5e+205) {
		tmp = (2.0 * ((x + 1.0) * Math.exp(x))) / 2.0;
	} else if ((eps_m <= 4.8e+249) || (!(eps_m <= 8.5e+295) && (eps_m <= 3.5e+304))) {
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((-1.0 + ((1.0 + eps_m) * (1.0 + eps_m))) / ((-1.0 - eps_m) - ((1.0 + eps_m) / eps_m)))))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = -1.0 + (-1.0 / eps_m)
	t_1 = eps_m * t_0
	tmp = 0
	if eps_m <= 1.0:
		tmp = (2.0 * (math.exp(-x) * (x + 1.0))) / 2.0
	elif eps_m <= 9.5e+205:
		tmp = (2.0 * ((x + 1.0) * math.exp(x))) / 2.0
	elif (eps_m <= 4.8e+249) or (not (eps_m <= 8.5e+295) and (eps_m <= 3.5e+304)):
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((-1.0 + ((1.0 + eps_m) * (1.0 + eps_m))) / ((-1.0 - eps_m) - ((1.0 + eps_m) / eps_m)))))) / 2.0
	else:
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(-1.0 + Float64(-1.0 / eps_m))
	t_1 = Float64(eps_m * t_0)
	tmp = 0.0
	if (eps_m <= 1.0)
		tmp = Float64(Float64(2.0 * Float64(exp(Float64(-x)) * Float64(x + 1.0))) / 2.0);
	elseif (eps_m <= 9.5e+205)
		tmp = Float64(Float64(2.0 * Float64(Float64(x + 1.0) * exp(x))) / 2.0);
	elseif ((eps_m <= 4.8e+249) || (!(eps_m <= 8.5e+295) && (eps_m <= 3.5e+304)))
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 - eps_m) * t_0) + Float64(Float64(-1.0 + Float64(Float64(1.0 + eps_m) * Float64(1.0 + eps_m))) / Float64(Float64(-1.0 - eps_m) - Float64(Float64(1.0 + eps_m) / eps_m)))))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(t_0 + t_1))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = -1.0 + (-1.0 / eps_m);
	t_1 = eps_m * t_0;
	tmp = 0.0;
	if (eps_m <= 1.0)
		tmp = (2.0 * (exp(-x) * (x + 1.0))) / 2.0;
	elseif (eps_m <= 9.5e+205)
		tmp = (2.0 * ((x + 1.0) * exp(x))) / 2.0;
	elseif ((eps_m <= 4.8e+249) || (~((eps_m <= 8.5e+295)) && (eps_m <= 3.5e+304)))
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((-1.0 + ((1.0 + eps_m) * (1.0 + eps_m))) / ((-1.0 - eps_m) - ((1.0 + eps_m) / eps_m)))))) / 2.0;
	else
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps$95$m * t$95$0), $MachinePrecision]}, If[LessEqual[eps$95$m, 1.0], N[(N[(2.0 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 9.5e+205], N[(N[(2.0 * N[(N[(x + 1.0), $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[eps$95$m, 4.8e+249], And[N[Not[LessEqual[eps$95$m, 8.5e+295]], $MachinePrecision], LessEqual[eps$95$m, 3.5e+304]]], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 - eps$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(-1.0 + N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - eps$95$m), $MachinePrecision] - N[(N[(1.0 + eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 1

   1. Initial program 61.5%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in eps around 0 71.9%

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

      1. distribute-lft-out 71.9%

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

      2. cancel-sign-sub-inv 71.9%

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

      3. metadata-eval 71.9%

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

      4. distribute-rgt1-in 71.9%

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

      5. metadata-eval 71.9%

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

      6. distribute-rgt1-in 71.9%

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

      7. *-commutative 71.9%

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

      8. mul-1-neg 71.9%

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

   7. Simplified 71.9%

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

   if 1 < eps < 9.4999999999999997e205

   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 eps around 0 38.6%

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

      1. distribute-lft-out 38.6%

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

      2. cancel-sign-sub-inv 38.6%

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

      3. metadata-eval 38.6%

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

      4. distribute-rgt1-in 38.6%

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

      5. metadata-eval 38.6%

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

      6. distribute-rgt1-in 38.6%

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

      7. *-commutative 38.6%

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

      8. mul-1-neg 38.6%

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

   7. Simplified 38.6%

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

      1. +-commutative 38.6%

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

      2. distribute-rgt-in 38.6%

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

      3. *-un-lft-identity 38.6%

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

      4. add-sqr-sqrt 24.2%

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

      5. sqrt-unprod 58.3%

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

      6. sqr-neg 58.3%

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

      7. sqrt-unprod 34.1%

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

      8. add-sqr-sqrt 58.3%

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

      9. add-sqr-sqrt 24.3%

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

      10. sqrt-unprod 58.3%

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

      11. sqr-neg 58.3%

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

      12. sqrt-unprod 34.1%

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

      13. add-sqr-sqrt 58.8%

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

   9. Applied egg-rr 58.8%

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

       1. distribute-rgt1-in 58.8%

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

       2. +-commutative 58.8%

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

   11. Simplified 58.8%

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

   if 9.4999999999999997e205 < eps < 4.8e249 or 8.5000000000000003e295 < eps < 3.4999999999999998e304

   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 16.0%

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

      1. mul-1-neg 16.0%

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

      2. distribute-lft-neg-in 16.0%

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

      3. cancel-sign-sub-inv 16.0%

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

      4. mul-1-neg 16.0%

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

      5. distribute-rgt-neg-in 16.0%

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

      6. sub-neg 16.0%

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

      7. distribute-neg-in 16.0%

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

      8. metadata-eval 16.0%

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

      9. remove-double-neg 16.0%

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

      10. remove-double-neg 16.0%

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

      11. +-commutative 16.0%

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

      12. sub-neg 16.0%

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

      13. metadata-eval 16.0%

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

      14. +-commutative 16.0%

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

   7. Simplified 16.0%

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

      1. distribute-lft-in 16.0%

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

      2. flip-+ 60.3%

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

   9. Applied egg-rr 60.3%

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

   10. Taylor expanded in eps around inf 60.3%

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

   if 4.8e249 < eps < 8.5000000000000003e295 or 3.4999999999999998e304 < 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%

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

      1. mul-1-neg 12.8%

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

      2. distribute-lft-neg-in 12.8%

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

      3. cancel-sign-sub-inv 12.8%

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

      4. mul-1-neg 12.8%

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

      5. distribute-rgt-neg-in 12.8%

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

      6. sub-neg 12.8%

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

      7. distribute-neg-in 12.8%

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

      8. metadata-eval 12.8%

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

      9. remove-double-neg 12.8%

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

      10. remove-double-neg 12.8%

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

      11. +-commutative 12.8%

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

      12. sub-neg 12.8%

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

      13. metadata-eval 12.8%

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

      14. +-commutative 12.8%

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

   7. Simplified 12.8%

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

   8. Taylor expanded in eps around 0 38.3%

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

      1. distribute-lft-in 38.3%

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

      2. flip-+ 70.3%

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

   10. Applied egg-rr 70.3%

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

   11. Taylor expanded in eps around inf 70.3%

       \[\leadsto \frac{2 + x \cdot \left(\frac{\color{blue}{1} - \left(\varepsilon \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{\left(-1 + \frac{-1}{\varepsilon}\right) - \varepsilon \cdot \left(1 + \frac{1}{\varepsilon}\right)} + \frac{1}{\varepsilon}\right)}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{2 \cdot \left(e^{-x} \cdot \left(x + 1\right)\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{+205}:\\ \;\;\;\;\frac{2 \cdot \left(\left(x + 1\right) \cdot e^{x}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.8 \cdot 10^{+249} \lor \neg \left(\varepsilon \leq 8.5 \cdot 10^{+295}\right) \land \varepsilon \leq 3.5 \cdot 10^{+304}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right) + \frac{-1 + \left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)}{\left(-1 - \varepsilon\right) - \frac{1 + \varepsilon}{\varepsilon}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \frac{1 - \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right) \cdot \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{\left(-1 + \frac{-1}{\varepsilon}\right) + \varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := -1 + \frac{-1}{eps\_m}\\ t_1 := eps\_m \cdot t\_0\\ t_2 := t\_0 + t\_1\\ t_3 := \frac{1 + eps\_m}{eps\_m}\\ t_4 := t\_1 \cdot t\_1\\ t_5 := 1 + \frac{1}{eps\_m}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+67}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{eps\_m} - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-191}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - eps\_m\right) \cdot t\_0 + \frac{\left(1 + eps\_m\right) \cdot \left(1 + eps\_m\right) + t\_3 \cdot \frac{-1 - eps\_m}{eps\_m}}{\left(-1 - eps\_m\right) - t\_3}\right)}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-217}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-124}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{eps\_m} + \frac{1 - t\_4}{t\_2}\right)}{2}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{eps\_m} + \frac{t\_5 \cdot t\_5 - t\_4}{t\_2}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ -1.0 eps_m)))
        (t_1 (* eps_m t_0))
        (t_2 (+ t_0 t_1))
        (t_3 (/ (+ 1.0 eps_m) eps_m))
        (t_4 (* t_1 t_1))
        (t_5 (+ 1.0 (/ 1.0 eps_m))))
   (if (<= x -1.35e+67)
     (/ (+ 2.0 (* x (- (/ -1.0 eps_m) eps_m))) 2.0)
     (if (<= x -5.6e-191)
       (/
        (+
         2.0
         (*
          x
          (+
           (* (- 1.0 eps_m) t_0)
           (/
            (+
             (* (+ 1.0 eps_m) (+ 1.0 eps_m))
             (* t_3 (/ (- -1.0 eps_m) eps_m)))
            (- (- -1.0 eps_m) t_3)))))
        2.0)
       (if (<= x 6.8e-217)
         1.0
         (if (<= x 1.45e-124)
           (/ (+ 2.0 (* x (+ (/ 1.0 eps_m) (/ (- 1.0 t_4) t_2)))) 2.0)
           (if (<= x 2.9e-8)
             (/ (+ 2.0 (* x eps_m)) 2.0)
             (/
              (+ 2.0 (* x (+ (/ 1.0 eps_m) (/ (- (* t_5 t_5) t_4) t_2))))
              2.0))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = -1.0 + (-1.0 / eps_m);
	double t_1 = eps_m * t_0;
	double t_2 = t_0 + t_1;
	double t_3 = (1.0 + eps_m) / eps_m;
	double t_4 = t_1 * t_1;
	double t_5 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -1.35e+67) {
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0;
	} else if (x <= -5.6e-191) {
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_3 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_3))))) / 2.0;
	} else if (x <= 6.8e-217) {
		tmp = 1.0;
	} else if (x <= 1.45e-124) {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - t_4) / t_2)))) / 2.0;
	} else if (x <= 2.9e-8) {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps_m) + (((t_5 * t_5) - t_4) / t_2)))) / 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (-1.0d0) + ((-1.0d0) / eps_m)
    t_1 = eps_m * t_0
    t_2 = t_0 + t_1
    t_3 = (1.0d0 + eps_m) / eps_m
    t_4 = t_1 * t_1
    t_5 = 1.0d0 + (1.0d0 / eps_m)
    if (x <= (-1.35d+67)) then
        tmp = (2.0d0 + (x * (((-1.0d0) / eps_m) - eps_m))) / 2.0d0
    else if (x <= (-5.6d-191)) then
        tmp = (2.0d0 + (x * (((1.0d0 - eps_m) * t_0) + ((((1.0d0 + eps_m) * (1.0d0 + eps_m)) + (t_3 * (((-1.0d0) - eps_m) / eps_m))) / (((-1.0d0) - eps_m) - t_3))))) / 2.0d0
    else if (x <= 6.8d-217) then
        tmp = 1.0d0
    else if (x <= 1.45d-124) then
        tmp = (2.0d0 + (x * ((1.0d0 / eps_m) + ((1.0d0 - t_4) / t_2)))) / 2.0d0
    else if (x <= 2.9d-8) then
        tmp = (2.0d0 + (x * eps_m)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((1.0d0 / eps_m) + (((t_5 * t_5) - t_4) / t_2)))) / 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 = -1.0 + (-1.0 / eps_m);
	double t_1 = eps_m * t_0;
	double t_2 = t_0 + t_1;
	double t_3 = (1.0 + eps_m) / eps_m;
	double t_4 = t_1 * t_1;
	double t_5 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -1.35e+67) {
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0;
	} else if (x <= -5.6e-191) {
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_3 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_3))))) / 2.0;
	} else if (x <= 6.8e-217) {
		tmp = 1.0;
	} else if (x <= 1.45e-124) {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - t_4) / t_2)))) / 2.0;
	} else if (x <= 2.9e-8) {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps_m) + (((t_5 * t_5) - t_4) / t_2)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = -1.0 + (-1.0 / eps_m)
	t_1 = eps_m * t_0
	t_2 = t_0 + t_1
	t_3 = (1.0 + eps_m) / eps_m
	t_4 = t_1 * t_1
	t_5 = 1.0 + (1.0 / eps_m)
	tmp = 0
	if x <= -1.35e+67:
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0
	elif x <= -5.6e-191:
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_3 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_3))))) / 2.0
	elif x <= 6.8e-217:
		tmp = 1.0
	elif x <= 1.45e-124:
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - t_4) / t_2)))) / 2.0
	elif x <= 2.9e-8:
		tmp = (2.0 + (x * eps_m)) / 2.0
	else:
		tmp = (2.0 + (x * ((1.0 / eps_m) + (((t_5 * t_5) - t_4) / t_2)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(-1.0 + Float64(-1.0 / eps_m))
	t_1 = Float64(eps_m * t_0)
	t_2 = Float64(t_0 + t_1)
	t_3 = Float64(Float64(1.0 + eps_m) / eps_m)
	t_4 = Float64(t_1 * t_1)
	t_5 = Float64(1.0 + Float64(1.0 / eps_m))
	tmp = 0.0
	if (x <= -1.35e+67)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 / eps_m) - eps_m))) / 2.0);
	elseif (x <= -5.6e-191)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 - eps_m) * t_0) + Float64(Float64(Float64(Float64(1.0 + eps_m) * Float64(1.0 + eps_m)) + Float64(t_3 * Float64(Float64(-1.0 - eps_m) / eps_m))) / Float64(Float64(-1.0 - eps_m) - t_3))))) / 2.0);
	elseif (x <= 6.8e-217)
		tmp = 1.0;
	elseif (x <= 1.45e-124)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 - t_4) / t_2)))) / 2.0);
	elseif (x <= 2.9e-8)
		tmp = Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps_m) + Float64(Float64(Float64(t_5 * t_5) - t_4) / t_2)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = -1.0 + (-1.0 / eps_m);
	t_1 = eps_m * t_0;
	t_2 = t_0 + t_1;
	t_3 = (1.0 + eps_m) / eps_m;
	t_4 = t_1 * t_1;
	t_5 = 1.0 + (1.0 / eps_m);
	tmp = 0.0;
	if (x <= -1.35e+67)
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0;
	elseif (x <= -5.6e-191)
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_3 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_3))))) / 2.0;
	elseif (x <= 6.8e-217)
		tmp = 1.0;
	elseif (x <= 1.45e-124)
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - t_4) / t_2)))) / 2.0;
	elseif (x <= 2.9e-8)
		tmp = (2.0 + (x * eps_m)) / 2.0;
	else
		tmp = (2.0 + (x * ((1.0 / eps_m) + (((t_5 * t_5) - t_4) / t_2)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps$95$m * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 + eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+67], N[(N[(2.0 + N[(x * N[(N[(-1.0 / eps$95$m), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -5.6e-191], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 - eps$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(N[(-1.0 - eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - eps$95$m), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.8e-217], 1.0, If[LessEqual[x, 1.45e-124], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - t$95$4), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.9e-8], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(N[(t$95$5 * t$95$5), $MachinePrecision] - t$95$4), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.35e67

   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 3.2%

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

      1. mul-1-neg 3.2%

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

      2. distribute-lft-neg-in 3.2%

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

      3. cancel-sign-sub-inv 3.2%

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

      4. mul-1-neg 3.2%

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

      5. distribute-rgt-neg-in 3.2%

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

      6. sub-neg 3.2%

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

      7. distribute-neg-in 3.2%

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

      8. metadata-eval 3.2%

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

      9. remove-double-neg 3.2%

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

      10. remove-double-neg 3.2%

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

      11. +-commutative 3.2%

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

      12. sub-neg 3.2%

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

      13. metadata-eval 3.2%

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

      14. +-commutative 3.2%

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

   7. Simplified 3.2%

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

   8. Taylor expanded in eps around 0 35.8%

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

   9. Taylor expanded in eps around inf 35.8%

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

       1. neg-mul-1 35.8%

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

   11. Simplified 35.8%

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

   if -1.35e67 < x < -5.60000000000000023e-191

   1. Initial program 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in x around 0 55.6%

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

      1. mul-1-neg 55.6%

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

      2. distribute-lft-neg-in 55.6%

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

      3. cancel-sign-sub-inv 55.6%

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

      4. mul-1-neg 55.6%

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

      5. distribute-rgt-neg-in 55.6%

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

      6. sub-neg 55.6%

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

      7. distribute-neg-in 55.6%

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

      8. metadata-eval 55.6%

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

      9. remove-double-neg 55.6%

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

      10. remove-double-neg 55.6%

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

      11. +-commutative 55.6%

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

      12. sub-neg 55.6%

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

      13. metadata-eval 55.6%

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

      14. +-commutative 55.6%

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

   7. Simplified 55.6%

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

      1. distribute-lft-in 55.6%

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

      2. flip-+ 56.8%

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

   9. Applied egg-rr 56.8%

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

   if -5.60000000000000023e-191 < x < 6.80000000000000032e-217

   1. Initial program 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in x around 0 98.6%

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

   if 6.80000000000000032e-217 < x < 1.4500000000000001e-124

   1. Initial program 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in x around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. distribute-lft-neg-in 66.5%

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

      3. cancel-sign-sub-inv 66.5%

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

      4. mul-1-neg 66.5%

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

      5. distribute-rgt-neg-in 66.5%

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

      6. sub-neg 66.5%

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

      7. distribute-neg-in 66.5%

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

      8. metadata-eval 66.5%

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

      9. remove-double-neg 66.5%

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

      10. remove-double-neg 66.5%

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

      11. +-commutative 66.5%

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

      12. sub-neg 66.5%

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

      13. metadata-eval 66.5%

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

      14. +-commutative 66.5%

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

   7. Simplified 66.5%

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

   8. Taylor expanded in eps around 0 66.2%

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

      1. distribute-lft-in 66.2%

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

      2. flip-+ 58.3%

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

   10. Applied egg-rr 58.3%

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

   11. Taylor expanded in eps around inf 66.2%

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

   if 1.4500000000000001e-124 < x < 2.9000000000000002e-8

   1. Initial program 42.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in x around 0 69.7%

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

      1. mul-1-neg 69.7%

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

      2. distribute-lft-neg-in 69.7%

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

      3. cancel-sign-sub-inv 69.7%

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

      4. mul-1-neg 69.7%

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

      5. distribute-rgt-neg-in 69.7%

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

      6. sub-neg 69.7%

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

      7. distribute-neg-in 69.7%

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

      8. metadata-eval 69.7%

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

      9. remove-double-neg 69.7%

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

      10. remove-double-neg 69.7%

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

      11. +-commutative 69.7%

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

      12. sub-neg 69.7%

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

      13. metadata-eval 69.7%

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

      14. +-commutative 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in eps around 0 69.4%

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

   9. Taylor expanded in eps around 0 69.4%

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

   if 2.9000000000000002e-8 < 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%

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

   5. Taylor expanded in x around 0 3.1%

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

      1. mul-1-neg 3.1%

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

      2. distribute-lft-neg-in 3.1%

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

      3. cancel-sign-sub-inv 3.1%

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

      4. mul-1-neg 3.1%

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

      5. distribute-rgt-neg-in 3.1%

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

      6. sub-neg 3.1%

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

      7. distribute-neg-in 3.1%

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

      8. metadata-eval 3.1%

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

      9. remove-double-neg 3.1%

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

      10. remove-double-neg 3.1%

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

      11. +-commutative 3.1%

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

      12. sub-neg 3.1%

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

      13. metadata-eval 3.1%

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

      14. +-commutative 3.1%

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

   7. Simplified 3.1%

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

   8. Taylor expanded in eps around 0 16.0%

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

      1. distribute-lft-in 16.0%

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

      2. flip-+ 18.8%

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

   10. Applied egg-rr 18.8%

       \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\frac{\left(-1 + \frac{-1}{\varepsilon}\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right) - \left(\varepsilon \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(\varepsilon \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{\left(-1 + \frac{-1}{\varepsilon}\right) - \varepsilon \cdot \left(1 + \frac{1}{\varepsilon}\right)}} + \frac{1}{\varepsilon}\right)}{2} \]
3. Recombined 6 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+67}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{\varepsilon} - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-191}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right) + \frac{\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right) + \frac{1 + \varepsilon}{\varepsilon} \cdot \frac{-1 - \varepsilon}{\varepsilon}}{\left(-1 - \varepsilon\right) - \frac{1 + \varepsilon}{\varepsilon}}\right)}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-217}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-124}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \frac{1 - \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right) \cdot \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{\left(-1 + \frac{-1}{\varepsilon}\right) + \varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}\right)}{2}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right) \cdot \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{\left(-1 + \frac{-1}{\varepsilon}\right) + \varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := -1 + \frac{-1}{eps\_m}\\ t_1 := eps\_m \cdot t\_0\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-123}:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-190}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - eps\_m\right) \cdot t\_0 + \frac{-1 + \left(1 + eps\_m\right) \cdot \left(1 + eps\_m\right)}{\left(-1 - eps\_m\right) - \frac{1 + eps\_m}{eps\_m}}\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-216}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-124} \lor \neg \left(x \leq 4.3 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{eps\_m} + \frac{1 - t\_1 \cdot t\_1}{t\_0 + t\_1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ -1.0 eps_m))) (t_1 (* eps_m t_0)))
   (if (<= x -1.1e-123)
     (/ (- 2.0 (* x eps_m)) 2.0)
     (if (<= x -1e-190)
       (/
        (+
         2.0
         (*
          x
          (+
           (* (- 1.0 eps_m) t_0)
           (/
            (+ -1.0 (* (+ 1.0 eps_m) (+ 1.0 eps_m)))
            (- (- -1.0 eps_m) (/ (+ 1.0 eps_m) eps_m))))))
        2.0)
       (if (<= x 1.6e-216)
         1.0
         (if (or (<= x 3.35e-124) (not (<= x 4.3e-7)))
           (/
            (+ 2.0 (* x (+ (/ 1.0 eps_m) (/ (- 1.0 (* t_1 t_1)) (+ t_0 t_1)))))
            2.0)
           (/ (+ 2.0 (* x eps_m)) 2.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = -1.0 + (-1.0 / eps_m);
	double t_1 = eps_m * t_0;
	double tmp;
	if (x <= -1.1e-123) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= -1e-190) {
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((-1.0 + ((1.0 + eps_m) * (1.0 + eps_m))) / ((-1.0 - eps_m) - ((1.0 + eps_m) / eps_m)))))) / 2.0;
	} else if (x <= 1.6e-216) {
		tmp = 1.0;
	} else if ((x <= 3.35e-124) || !(x <= 4.3e-7)) {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 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) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) + ((-1.0d0) / eps_m)
    t_1 = eps_m * t_0
    if (x <= (-1.1d-123)) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else if (x <= (-1d-190)) then
        tmp = (2.0d0 + (x * (((1.0d0 - eps_m) * t_0) + (((-1.0d0) + ((1.0d0 + eps_m) * (1.0d0 + eps_m))) / (((-1.0d0) - eps_m) - ((1.0d0 + eps_m) / eps_m)))))) / 2.0d0
    else if (x <= 1.6d-216) then
        tmp = 1.0d0
    else if ((x <= 3.35d-124) .or. (.not. (x <= 4.3d-7))) then
        tmp = (2.0d0 + (x * ((1.0d0 / eps_m) + ((1.0d0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0d0
    else
        tmp = (2.0d0 + (x * eps_m)) / 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 = -1.0 + (-1.0 / eps_m);
	double t_1 = eps_m * t_0;
	double tmp;
	if (x <= -1.1e-123) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= -1e-190) {
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((-1.0 + ((1.0 + eps_m) * (1.0 + eps_m))) / ((-1.0 - eps_m) - ((1.0 + eps_m) / eps_m)))))) / 2.0;
	} else if (x <= 1.6e-216) {
		tmp = 1.0;
	} else if ((x <= 3.35e-124) || !(x <= 4.3e-7)) {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = -1.0 + (-1.0 / eps_m)
	t_1 = eps_m * t_0
	tmp = 0
	if x <= -1.1e-123:
		tmp = (2.0 - (x * eps_m)) / 2.0
	elif x <= -1e-190:
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((-1.0 + ((1.0 + eps_m) * (1.0 + eps_m))) / ((-1.0 - eps_m) - ((1.0 + eps_m) / eps_m)))))) / 2.0
	elif x <= 1.6e-216:
		tmp = 1.0
	elif (x <= 3.35e-124) or not (x <= 4.3e-7):
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0
	else:
		tmp = (2.0 + (x * eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(-1.0 + Float64(-1.0 / eps_m))
	t_1 = Float64(eps_m * t_0)
	tmp = 0.0
	if (x <= -1.1e-123)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	elseif (x <= -1e-190)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 - eps_m) * t_0) + Float64(Float64(-1.0 + Float64(Float64(1.0 + eps_m) * Float64(1.0 + eps_m))) / Float64(Float64(-1.0 - eps_m) - Float64(Float64(1.0 + eps_m) / eps_m)))))) / 2.0);
	elseif (x <= 1.6e-216)
		tmp = 1.0;
	elseif ((x <= 3.35e-124) || !(x <= 4.3e-7))
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(t_0 + t_1))))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = -1.0 + (-1.0 / eps_m);
	t_1 = eps_m * t_0;
	tmp = 0.0;
	if (x <= -1.1e-123)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	elseif (x <= -1e-190)
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((-1.0 + ((1.0 + eps_m) * (1.0 + eps_m))) / ((-1.0 - eps_m) - ((1.0 + eps_m) / eps_m)))))) / 2.0;
	elseif (x <= 1.6e-216)
		tmp = 1.0;
	elseif ((x <= 3.35e-124) || ~((x <= 4.3e-7)))
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	else
		tmp = (2.0 + (x * eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps$95$m * t$95$0), $MachinePrecision]}, If[LessEqual[x, -1.1e-123], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1e-190], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 - eps$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(-1.0 + N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - eps$95$m), $MachinePrecision] - N[(N[(1.0 + eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.6e-216], 1.0, If[Or[LessEqual[x, 3.35e-124], N[Not[LessEqual[x, 4.3e-7]], $MachinePrecision]], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.10000000000000003e-123

   1. Initial program 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in x around 0 27.7%

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

      1. mul-1-neg 27.7%

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

      2. distribute-lft-neg-in 27.7%

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

      3. cancel-sign-sub-inv 27.7%

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

      4. mul-1-neg 27.7%

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

      5. distribute-rgt-neg-in 27.7%

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

      6. sub-neg 27.7%

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

      7. distribute-neg-in 27.7%

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

      8. metadata-eval 27.7%

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

      9. remove-double-neg 27.7%

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

      10. remove-double-neg 27.7%

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

      11. +-commutative 27.7%

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

      12. sub-neg 27.7%

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

      13. metadata-eval 27.7%

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

      14. +-commutative 27.7%

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

   7. Simplified 27.7%

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

   8. Taylor expanded in eps around 0 44.8%

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

   9. Taylor expanded in eps around 0 44.8%

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

       1. neg-mul-1 44.8%

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

   11. Simplified 44.8%

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

   if -1.10000000000000003e-123 < x < -1e-190

   1. Initial program 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in x around 0 57.6%

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

      1. mul-1-neg 57.6%

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

      2. distribute-lft-neg-in 57.6%

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

      3. cancel-sign-sub-inv 57.6%

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

      4. mul-1-neg 57.6%

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

      5. distribute-rgt-neg-in 57.6%

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

      6. sub-neg 57.6%

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

      7. distribute-neg-in 57.6%

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

      8. metadata-eval 57.6%

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

      9. remove-double-neg 57.6%

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

      10. remove-double-neg 57.6%

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

      11. +-commutative 57.6%

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

      12. sub-neg 57.6%

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

      13. metadata-eval 57.6%

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

      14. +-commutative 57.6%

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

   7. Simplified 57.6%

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

      1. distribute-lft-in 57.6%

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

      2. flip-+ 68.8%

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

   9. Applied egg-rr 68.8%

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

   10. Taylor expanded in eps around inf 68.8%

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

   if -1e-190 < x < 1.60000000000000013e-216

   1. Initial program 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in x around 0 98.6%

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

   if 1.60000000000000013e-216 < x < 3.35e-124 or 4.3000000000000001e-7 < x

   1. Initial program 87.3%

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

   3. Simplified 87.3%

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

   5. Taylor expanded in x around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. distribute-lft-neg-in 22.7%

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

      3. cancel-sign-sub-inv 22.7%

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

      4. mul-1-neg 22.7%

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

      5. distribute-rgt-neg-in 22.7%

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

      6. sub-neg 22.7%

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

      7. distribute-neg-in 22.7%

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

      8. metadata-eval 22.7%

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

      9. remove-double-neg 22.7%

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

      10. remove-double-neg 22.7%

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

      11. +-commutative 22.7%

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

      12. sub-neg 22.7%

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

      13. metadata-eval 22.7%

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

      14. +-commutative 22.7%

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

   7. Simplified 22.7%

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

   8. Taylor expanded in eps around 0 31.5%

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

      1. distribute-lft-in 31.5%

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

      2. flip-+ 31.0%

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

   10. Applied egg-rr 31.0%

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

   11. Taylor expanded in eps around inf 33.3%

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

   if 3.35e-124 < x < 4.3000000000000001e-7

   1. Initial program 42.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in x around 0 69.7%

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

      1. mul-1-neg 69.7%

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

      2. distribute-lft-neg-in 69.7%

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

      3. cancel-sign-sub-inv 69.7%

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

      4. mul-1-neg 69.7%

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

      5. distribute-rgt-neg-in 69.7%

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

      6. sub-neg 69.7%

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

      7. distribute-neg-in 69.7%

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

      8. metadata-eval 69.7%

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

      9. remove-double-neg 69.7%

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

      10. remove-double-neg 69.7%

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

      11. +-commutative 69.7%

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

      12. sub-neg 69.7%

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

      13. metadata-eval 69.7%

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

      14. +-commutative 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in eps around 0 69.4%

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

   9. Taylor expanded in eps around 0 69.4%

      \[\leadsto \frac{2 + x \cdot \color{blue}{\varepsilon}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-123}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-190}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right) + \frac{-1 + \left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)}{\left(-1 - \varepsilon\right) - \frac{1 + \varepsilon}{\varepsilon}}\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-216}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-124} \lor \neg \left(x \leq 4.3 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \frac{1 - \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right) \cdot \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{\left(-1 + \frac{-1}{\varepsilon}\right) + \varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := -1 + \frac{-1}{eps\_m}\\ t_1 := eps\_m \cdot t\_0\\ t_2 := \frac{1 + eps\_m}{eps\_m}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+67}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{eps\_m} - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-193}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - eps\_m\right) \cdot t\_0 + \frac{\left(1 + eps\_m\right) \cdot \left(1 + eps\_m\right) + t\_2 \cdot \frac{-1 - eps\_m}{eps\_m}}{\left(-1 - eps\_m\right) - t\_2}\right)}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-217}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-125} \lor \neg \left(x \leq 5.4 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{eps\_m} + \frac{1 - t\_1 \cdot t\_1}{t\_0 + t\_1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ -1.0 eps_m)))
        (t_1 (* eps_m t_0))
        (t_2 (/ (+ 1.0 eps_m) eps_m)))
   (if (<= x -1.35e+67)
     (/ (+ 2.0 (* x (- (/ -1.0 eps_m) eps_m))) 2.0)
     (if (<= x -3.9e-193)
       (/
        (+
         2.0
         (*
          x
          (+
           (* (- 1.0 eps_m) t_0)
           (/
            (+
             (* (+ 1.0 eps_m) (+ 1.0 eps_m))
             (* t_2 (/ (- -1.0 eps_m) eps_m)))
            (- (- -1.0 eps_m) t_2)))))
        2.0)
       (if (<= x 3.5e-217)
         1.0
         (if (or (<= x 3.9e-125) (not (<= x 5.4e-6)))
           (/
            (+ 2.0 (* x (+ (/ 1.0 eps_m) (/ (- 1.0 (* t_1 t_1)) (+ t_0 t_1)))))
            2.0)
           (/ (+ 2.0 (* x eps_m)) 2.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = -1.0 + (-1.0 / eps_m);
	double t_1 = eps_m * t_0;
	double t_2 = (1.0 + eps_m) / eps_m;
	double tmp;
	if (x <= -1.35e+67) {
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0;
	} else if (x <= -3.9e-193) {
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_2 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_2))))) / 2.0;
	} else if (x <= 3.5e-217) {
		tmp = 1.0;
	} else if ((x <= 3.9e-125) || !(x <= 5.4e-6)) {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (-1.0d0) + ((-1.0d0) / eps_m)
    t_1 = eps_m * t_0
    t_2 = (1.0d0 + eps_m) / eps_m
    if (x <= (-1.35d+67)) then
        tmp = (2.0d0 + (x * (((-1.0d0) / eps_m) - eps_m))) / 2.0d0
    else if (x <= (-3.9d-193)) then
        tmp = (2.0d0 + (x * (((1.0d0 - eps_m) * t_0) + ((((1.0d0 + eps_m) * (1.0d0 + eps_m)) + (t_2 * (((-1.0d0) - eps_m) / eps_m))) / (((-1.0d0) - eps_m) - t_2))))) / 2.0d0
    else if (x <= 3.5d-217) then
        tmp = 1.0d0
    else if ((x <= 3.9d-125) .or. (.not. (x <= 5.4d-6))) then
        tmp = (2.0d0 + (x * ((1.0d0 / eps_m) + ((1.0d0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0d0
    else
        tmp = (2.0d0 + (x * eps_m)) / 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 = -1.0 + (-1.0 / eps_m);
	double t_1 = eps_m * t_0;
	double t_2 = (1.0 + eps_m) / eps_m;
	double tmp;
	if (x <= -1.35e+67) {
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0;
	} else if (x <= -3.9e-193) {
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_2 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_2))))) / 2.0;
	} else if (x <= 3.5e-217) {
		tmp = 1.0;
	} else if ((x <= 3.9e-125) || !(x <= 5.4e-6)) {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = -1.0 + (-1.0 / eps_m)
	t_1 = eps_m * t_0
	t_2 = (1.0 + eps_m) / eps_m
	tmp = 0
	if x <= -1.35e+67:
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0
	elif x <= -3.9e-193:
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_2 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_2))))) / 2.0
	elif x <= 3.5e-217:
		tmp = 1.0
	elif (x <= 3.9e-125) or not (x <= 5.4e-6):
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0
	else:
		tmp = (2.0 + (x * eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(-1.0 + Float64(-1.0 / eps_m))
	t_1 = Float64(eps_m * t_0)
	t_2 = Float64(Float64(1.0 + eps_m) / eps_m)
	tmp = 0.0
	if (x <= -1.35e+67)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 / eps_m) - eps_m))) / 2.0);
	elseif (x <= -3.9e-193)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 - eps_m) * t_0) + Float64(Float64(Float64(Float64(1.0 + eps_m) * Float64(1.0 + eps_m)) + Float64(t_2 * Float64(Float64(-1.0 - eps_m) / eps_m))) / Float64(Float64(-1.0 - eps_m) - t_2))))) / 2.0);
	elseif (x <= 3.5e-217)
		tmp = 1.0;
	elseif ((x <= 3.9e-125) || !(x <= 5.4e-6))
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(t_0 + t_1))))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = -1.0 + (-1.0 / eps_m);
	t_1 = eps_m * t_0;
	t_2 = (1.0 + eps_m) / eps_m;
	tmp = 0.0;
	if (x <= -1.35e+67)
		tmp = (2.0 + (x * ((-1.0 / eps_m) - eps_m))) / 2.0;
	elseif (x <= -3.9e-193)
		tmp = (2.0 + (x * (((1.0 - eps_m) * t_0) + ((((1.0 + eps_m) * (1.0 + eps_m)) + (t_2 * ((-1.0 - eps_m) / eps_m))) / ((-1.0 - eps_m) - t_2))))) / 2.0;
	elseif (x <= 3.5e-217)
		tmp = 1.0;
	elseif ((x <= 3.9e-125) || ~((x <= 5.4e-6)))
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - (t_1 * t_1)) / (t_0 + t_1))))) / 2.0;
	else
		tmp = (2.0 + (x * eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps$95$m * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]}, If[LessEqual[x, -1.35e+67], N[(N[(2.0 + N[(x * N[(N[(-1.0 / eps$95$m), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -3.9e-193], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 - eps$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(N[(-1.0 - eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - eps$95$m), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.5e-217], 1.0, If[Or[LessEqual[x, 3.9e-125], N[Not[LessEqual[x, 5.4e-6]], $MachinePrecision]], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.35e67

   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 3.2%

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

      1. mul-1-neg 3.2%

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

      2. distribute-lft-neg-in 3.2%

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

      3. cancel-sign-sub-inv 3.2%

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

      4. mul-1-neg 3.2%

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

      5. distribute-rgt-neg-in 3.2%

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

      6. sub-neg 3.2%

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

      7. distribute-neg-in 3.2%

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

      8. metadata-eval 3.2%

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

      9. remove-double-neg 3.2%

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

      10. remove-double-neg 3.2%

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

      11. +-commutative 3.2%

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

      12. sub-neg 3.2%

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

      13. metadata-eval 3.2%

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

      14. +-commutative 3.2%

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

   7. Simplified 3.2%

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

   8. Taylor expanded in eps around 0 35.8%

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

   9. Taylor expanded in eps around inf 35.8%

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

       1. neg-mul-1 35.8%

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

   11. Simplified 35.8%

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

   if -1.35e67 < x < -3.8999999999999999e-193

   1. Initial program 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in x around 0 55.6%

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

      1. mul-1-neg 55.6%

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

      2. distribute-lft-neg-in 55.6%

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

      3. cancel-sign-sub-inv 55.6%

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

      4. mul-1-neg 55.6%

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

      5. distribute-rgt-neg-in 55.6%

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

      6. sub-neg 55.6%

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

      7. distribute-neg-in 55.6%

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

      8. metadata-eval 55.6%

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

      9. remove-double-neg 55.6%

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

      10. remove-double-neg 55.6%

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

      11. +-commutative 55.6%

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

      12. sub-neg 55.6%

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

      13. metadata-eval 55.6%

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

      14. +-commutative 55.6%

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

   7. Simplified 55.6%

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

      1. distribute-lft-in 55.6%

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

      2. flip-+ 56.8%

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

   9. Applied egg-rr 56.8%

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

   if -3.8999999999999999e-193 < x < 3.5e-217

   1. Initial program 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in x around 0 98.6%

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

   if 3.5e-217 < x < 3.89999999999999982e-125 or 5.39999999999999997e-6 < x

   1. Initial program 87.3%

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

   3. Simplified 87.3%

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

   5. Taylor expanded in x around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. distribute-lft-neg-in 22.7%

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

      3. cancel-sign-sub-inv 22.7%

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

      4. mul-1-neg 22.7%

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

      5. distribute-rgt-neg-in 22.7%

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

      6. sub-neg 22.7%

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

      7. distribute-neg-in 22.7%

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

      8. metadata-eval 22.7%

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

      9. remove-double-neg 22.7%

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

      10. remove-double-neg 22.7%

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

      11. +-commutative 22.7%

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

      12. sub-neg 22.7%

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

      13. metadata-eval 22.7%

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

      14. +-commutative 22.7%

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

   7. Simplified 22.7%

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

   8. Taylor expanded in eps around 0 31.5%

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

      1. distribute-lft-in 31.5%

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

      2. flip-+ 31.0%

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

   10. Applied egg-rr 31.0%

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

   11. Taylor expanded in eps around inf 33.3%

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

   if 3.89999999999999982e-125 < x < 5.39999999999999997e-6

   1. Initial program 42.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in x around 0 69.7%

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

      1. mul-1-neg 69.7%

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

      2. distribute-lft-neg-in 69.7%

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

      3. cancel-sign-sub-inv 69.7%

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

      4. mul-1-neg 69.7%

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

      5. distribute-rgt-neg-in 69.7%

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

      6. sub-neg 69.7%

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

      7. distribute-neg-in 69.7%

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

      8. metadata-eval 69.7%

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

      9. remove-double-neg 69.7%

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

      10. remove-double-neg 69.7%

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

      11. +-commutative 69.7%

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

      12. sub-neg 69.7%

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

      13. metadata-eval 69.7%

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

      14. +-commutative 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in eps around 0 69.4%

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

   9. Taylor expanded in eps around 0 69.4%

      \[\leadsto \frac{2 + x \cdot \color{blue}{\varepsilon}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+67}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{\varepsilon} - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-193}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right) + \frac{\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right) + \frac{1 + \varepsilon}{\varepsilon} \cdot \frac{-1 - \varepsilon}{\varepsilon}}{\left(-1 - \varepsilon\right) - \frac{1 + \varepsilon}{\varepsilon}}\right)}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-217}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-125} \lor \neg \left(x \leq 5.4 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \frac{1 - \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right) \cdot \left(\varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{\left(-1 + \frac{-1}{\varepsilon}\right) + \varepsilon \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(1 - eps\_m\right) \cdot \left(-1 + \frac{-1}{eps\_m}\right)\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-194}:\\ \;\;\;\;\frac{2 + x \cdot \left(t\_0 + \frac{-1 + \left(1 + eps\_m\right) \cdot \left(1 + eps\_m\right)}{\left(-1 - eps\_m\right) - \frac{1 + eps\_m}{eps\_m}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{eps\_m} + t\_0\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (- 1.0 eps_m) (+ -1.0 (/ -1.0 eps_m)))))
   (if (<= x -1.6e-123)
     (/ (- 2.0 (* x eps_m)) 2.0)
     (if (<= x -1.05e-194)
       (/
        (+
         2.0
         (*
          x
          (+
           t_0
           (/
            (+ -1.0 (* (+ 1.0 eps_m) (+ 1.0 eps_m)))
            (- (- -1.0 eps_m) (/ (+ 1.0 eps_m) eps_m))))))
        2.0)
       (/ (+ 2.0 (* x (+ (/ 1.0 eps_m) t_0))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 - eps_m) * (-1.0 + (-1.0 / eps_m));
	double tmp;
	if (x <= -1.6e-123) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= -1.05e-194) {
		tmp = (2.0 + (x * (t_0 + ((-1.0 + ((1.0 + eps_m) * (1.0 + eps_m))) / ((-1.0 - eps_m) - ((1.0 + eps_m) / eps_m)))))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps_m) + 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 = (1.0d0 - eps_m) * ((-1.0d0) + ((-1.0d0) / eps_m))
    if (x <= (-1.6d-123)) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else if (x <= (-1.05d-194)) then
        tmp = (2.0d0 + (x * (t_0 + (((-1.0d0) + ((1.0d0 + eps_m) * (1.0d0 + eps_m))) / (((-1.0d0) - eps_m) - ((1.0d0 + eps_m) / eps_m)))))) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((1.0d0 / eps_m) + 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 = (1.0 - eps_m) * (-1.0 + (-1.0 / eps_m));
	double tmp;
	if (x <= -1.6e-123) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= -1.05e-194) {
		tmp = (2.0 + (x * (t_0 + ((-1.0 + ((1.0 + eps_m) * (1.0 + eps_m))) / ((-1.0 - eps_m) - ((1.0 + eps_m) / eps_m)))))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps_m) + t_0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (1.0 - eps_m) * (-1.0 + (-1.0 / eps_m))
	tmp = 0
	if x <= -1.6e-123:
		tmp = (2.0 - (x * eps_m)) / 2.0
	elif x <= -1.05e-194:
		tmp = (2.0 + (x * (t_0 + ((-1.0 + ((1.0 + eps_m) * (1.0 + eps_m))) / ((-1.0 - eps_m) - ((1.0 + eps_m) / eps_m)))))) / 2.0
	else:
		tmp = (2.0 + (x * ((1.0 / eps_m) + t_0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 - eps_m) * Float64(-1.0 + Float64(-1.0 / eps_m)))
	tmp = 0.0
	if (x <= -1.6e-123)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	elseif (x <= -1.05e-194)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(t_0 + Float64(Float64(-1.0 + Float64(Float64(1.0 + eps_m) * Float64(1.0 + eps_m))) / Float64(Float64(-1.0 - eps_m) - Float64(Float64(1.0 + eps_m) / eps_m)))))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps_m) + t_0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (1.0 - eps_m) * (-1.0 + (-1.0 / eps_m));
	tmp = 0.0;
	if (x <= -1.6e-123)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	elseif (x <= -1.05e-194)
		tmp = (2.0 + (x * (t_0 + ((-1.0 + ((1.0 + eps_m) * (1.0 + eps_m))) / ((-1.0 - eps_m) - ((1.0 + eps_m) / eps_m)))))) / 2.0;
	else
		tmp = (2.0 + (x * ((1.0 / eps_m) + 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[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6e-123], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.05e-194], N[(N[(2.0 + N[(x * N[(t$95$0 + N[(N[(-1.0 + N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - eps$95$m), $MachinePrecision] - N[(N[(1.0 + eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.59999999999999989e-123

   1. Initial program 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in x around 0 27.7%

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

      1. mul-1-neg 27.7%

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

      2. distribute-lft-neg-in 27.7%

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

      3. cancel-sign-sub-inv 27.7%

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

      4. mul-1-neg 27.7%

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

      5. distribute-rgt-neg-in 27.7%

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

      6. sub-neg 27.7%

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

      7. distribute-neg-in 27.7%

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

      8. metadata-eval 27.7%

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

      9. remove-double-neg 27.7%

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

      10. remove-double-neg 27.7%

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

      11. +-commutative 27.7%

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

      12. sub-neg 27.7%

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

      13. metadata-eval 27.7%

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

      14. +-commutative 27.7%

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

   7. Simplified 27.7%

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

   8. Taylor expanded in eps around 0 44.8%

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

   9. Taylor expanded in eps around 0 44.8%

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

       1. neg-mul-1 44.8%

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

   11. Simplified 44.8%

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

   if -1.59999999999999989e-123 < x < -1.05e-194

   1. Initial program 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in x around 0 57.6%

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

      1. mul-1-neg 57.6%

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

      2. distribute-lft-neg-in 57.6%

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

      3. cancel-sign-sub-inv 57.6%

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

      4. mul-1-neg 57.6%

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

      5. distribute-rgt-neg-in 57.6%

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

      6. sub-neg 57.6%

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

      7. distribute-neg-in 57.6%

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

      8. metadata-eval 57.6%

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

      9. remove-double-neg 57.6%

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

      10. remove-double-neg 57.6%

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

      11. +-commutative 57.6%

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

      12. sub-neg 57.6%

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

      13. metadata-eval 57.6%

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

      14. +-commutative 57.6%

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

   7. Simplified 57.6%

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

      1. distribute-lft-in 57.6%

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

      2. flip-+ 68.8%

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

   9. Applied egg-rr 68.8%

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

   10. Taylor expanded in eps around inf 68.8%

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

   if -1.05e-194 < x

   1. Initial program 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in x around 0 56.4%

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

      1. mul-1-neg 56.4%

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

      2. distribute-lft-neg-in 56.4%

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

      3. cancel-sign-sub-inv 56.4%

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

      4. mul-1-neg 56.4%

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

      5. distribute-rgt-neg-in 56.4%

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

      6. sub-neg 56.4%

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

      7. distribute-neg-in 56.4%

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

      8. metadata-eval 56.4%

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

      9. remove-double-neg 56.4%

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

      10. remove-double-neg 56.4%

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

      11. +-commutative 56.4%

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

      12. sub-neg 56.4%

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

      13. metadata-eval 56.4%

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

      14. +-commutative 56.4%

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

   7. Simplified 56.4%

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

   8. Taylor expanded in eps around 0 60.1%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-194}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right) + \frac{-1 + \left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)}{\left(-1 - \varepsilon\right) - \frac{1 + \varepsilon}{\varepsilon}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.9% accurate, 9.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{eps\_m} + \left(1 - eps\_m\right) \cdot \left(-1 + \frac{-1}{eps\_m}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.5e-242)
   (/ (- 2.0 (* x eps_m)) 2.0)
   (/
    (+ 2.0 (* x (+ (/ 1.0 eps_m) (* (- 1.0 eps_m) (+ -1.0 (/ -1.0 eps_m))))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.5e-242) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 + (-1.0 / eps_m)))))) / 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) :: tmp
    if (x <= (-1.5d-242)) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((1.0d0 / eps_m) + ((1.0d0 - eps_m) * ((-1.0d0) + ((-1.0d0) / eps_m)))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.5e-242) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 + (-1.0 / eps_m)))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.5e-242:
		tmp = (2.0 - (x * eps_m)) / 2.0
	else:
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 + (-1.0 / eps_m)))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.5e-242)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 - eps_m) * Float64(-1.0 + Float64(-1.0 / eps_m)))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.5e-242)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	else
		tmp = (2.0 + (x * ((1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 + (-1.0 / eps_m)))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.5e-242], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e-242

   1. Initial program 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in x around 0 44.0%

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

      1. mul-1-neg 44.0%

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

      2. distribute-lft-neg-in 44.0%

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

      3. cancel-sign-sub-inv 44.0%

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

      4. mul-1-neg 44.0%

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

      5. distribute-rgt-neg-in 44.0%

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

      6. sub-neg 44.0%

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

      7. distribute-neg-in 44.0%

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

      8. metadata-eval 44.0%

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

      9. remove-double-neg 44.0%

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

      10. remove-double-neg 44.0%

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

      11. +-commutative 44.0%

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

      12. sub-neg 44.0%

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

      13. metadata-eval 44.0%

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

      14. +-commutative 44.0%

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

   7. Simplified 44.0%

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

   8. Taylor expanded in eps around 0 54.9%

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

   9. Taylor expanded in eps around 0 54.9%

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

       1. neg-mul-1 54.9%

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

   11. Simplified 54.9%

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

   if -1.5e-242 < x

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in x around 0 52.2%

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

      1. mul-1-neg 52.2%

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

      2. distribute-lft-neg-in 52.2%

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

      3. cancel-sign-sub-inv 52.2%

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

      4. mul-1-neg 52.2%

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

      5. distribute-rgt-neg-in 52.2%

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

      6. sub-neg 52.2%

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

      7. distribute-neg-in 52.2%

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

      8. metadata-eval 52.2%

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

      9. remove-double-neg 52.2%

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

      10. remove-double-neg 52.2%

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

      11. +-commutative 52.2%

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

      12. sub-neg 52.2%

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

      13. metadata-eval 52.2%

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

      14. +-commutative 52.2%

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

   7. Simplified 52.2%

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

   8. Taylor expanded in eps around 0 56.5%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.9% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.5e-242) (/ (- 2.0 (* x eps_m)) 2.0) (/ (+ 2.0 (* x eps_m)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.5e-242) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 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) :: tmp
    if (x <= (-1.5d-242)) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else
        tmp = (2.0d0 + (x * eps_m)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.5e-242) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.5e-242:
		tmp = (2.0 - (x * eps_m)) / 2.0
	else:
		tmp = (2.0 + (x * eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.5e-242)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + 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 (x <= -1.5e-242)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	else
		tmp = (2.0 + (x * eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.5e-242], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e-242

   1. Initial program 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in x around 0 44.0%

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

      1. mul-1-neg 44.0%

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

      2. distribute-lft-neg-in 44.0%

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

      3. cancel-sign-sub-inv 44.0%

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

      4. mul-1-neg 44.0%

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

      5. distribute-rgt-neg-in 44.0%

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

      6. sub-neg 44.0%

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

      7. distribute-neg-in 44.0%

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

      8. metadata-eval 44.0%

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

      9. remove-double-neg 44.0%

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

      10. remove-double-neg 44.0%

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

      11. +-commutative 44.0%

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

      12. sub-neg 44.0%

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

      13. metadata-eval 44.0%

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

      14. +-commutative 44.0%

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

   7. Simplified 44.0%

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

   8. Taylor expanded in eps around 0 54.9%

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

   9. Taylor expanded in eps around 0 54.9%

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

       1. neg-mul-1 54.9%

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

   11. Simplified 54.9%

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

   if -1.5e-242 < x

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in x around 0 52.2%

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

      1. mul-1-neg 52.2%

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

      2. distribute-lft-neg-in 52.2%

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

      3. cancel-sign-sub-inv 52.2%

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

      4. mul-1-neg 52.2%

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

      5. distribute-rgt-neg-in 52.2%

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

      6. sub-neg 52.2%

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

      7. distribute-neg-in 52.2%

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

      8. metadata-eval 52.2%

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

      9. remove-double-neg 52.2%

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

      10. remove-double-neg 52.2%

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

      11. +-commutative 52.2%

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

      12. sub-neg 52.2%

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

      13. metadata-eval 52.2%

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

      14. +-commutative 52.2%

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

   7. Simplified 52.2%

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

   8. Taylor expanded in eps around 0 56.5%

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

   9. Taylor expanded in eps around 0 56.4%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.0% 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 72.8%

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

3. Simplified 72.8%

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

5. Taylor expanded in x around 0 49.0%

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

   1. mul-1-neg 49.0%

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

   2. distribute-lft-neg-in 49.0%

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

   3. cancel-sign-sub-inv 49.0%

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

   4. mul-1-neg 49.0%

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

   5. distribute-rgt-neg-in 49.0%

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

   6. sub-neg 49.0%

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

   7. distribute-neg-in 49.0%

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

   8. metadata-eval 49.0%

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

   9. remove-double-neg 49.0%

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

   10. remove-double-neg 49.0%

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

   11. +-commutative 49.0%

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

   12. sub-neg 49.0%

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

   13. metadata-eval 49.0%

       \[\leadsto \frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 + \varepsilon\right) + \left(\varepsilon + 1\right) \cdot \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right)\right)}{2} \]

   14. +-commutative 49.0%

       \[\leadsto \frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 + \varepsilon\right) + \left(\varepsilon + 1\right) \cdot \color{blue}{\left(-1 + \frac{1}{\varepsilon}\right)}\right)}{2} \]

7. Simplified 49.0%

   \[\leadsto \frac{\color{blue}{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 + \varepsilon\right) + \left(\varepsilon + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

8. Taylor expanded in eps around 0 55.4%

   \[\leadsto \frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 + \varepsilon\right) + \color{blue}{\frac{1}{\varepsilon}}\right)}{2} \]

9. Taylor expanded in eps around 0 55.3%

   \[\leadsto \frac{2 + x \cdot \color{blue}{\varepsilon}}{2} \]

10. Final simplification 55.3%

    \[\leadsto \frac{2 + x \cdot \varepsilon}{2} \]
11. Add Preprocessing

Alternative 13: 43.8% 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 72.8%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

3. Simplified 72.8%

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 49.0%

   \[\leadsto \frac{\color{blue}{2}}{2} \]

6. Final simplification 49.0%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024048 
(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))