NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.2% → 99.9%

Time: 25.3s

Alternatives: 14

Speedup: 1.8×

• 38.8% 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: 73.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2e-14)
   (/ (+ (/ (+ x 1.0) (exp x)) (* (+ x 1.0) (exp (- x)))) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2e-14) {
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + 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 <= 2d-14) then
        tmp = (((x + 1.0d0) / exp(x)) + ((x + 1.0d0) * exp(-x))) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + 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 <= 2e-14) {
		tmp = (((x + 1.0) / Math.exp(x)) + ((x + 1.0) * Math.exp(-x))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + 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 <= 2e-14:
		tmp = (((x + 1.0) / math.exp(x)) + ((x + 1.0) * math.exp(-x))) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + 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 <= 2e-14)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / exp(x)) + Float64(Float64(x + 1.0) * exp(Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 2e-14)
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + 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, 2e-14], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2e-14

   1. Initial program 60.3%

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

      1. fma-neg 60.2%

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

      2. /-rgt-identity 60.2%

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

      3. fma-neg 60.3%

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

      4. /-rgt-identity 60.3%

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

      5. distribute-rgt-neg-in 60.3%

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

      6. sub-neg 60.3%

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

      7. metadata-eval 60.3%

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

      8. distribute-rgt-neg-in 60.3%

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

   3. Simplified 60.3%

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

   5. Taylor expanded in eps around 0 68.8%

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

   7. Simplified 69.3%

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

      1. rec-exp 69.4%

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

      2. un-div-inv 69.4%

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

   9. Applied egg-rr 69.4%

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

   if 2e-14 < eps

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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. Taylor expanded in eps around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. 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. neg-mul-1 100.0%

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

      8. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-rgt-neg-in 100.0%

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

       3. neg-sub0 100.0%

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

       4. associate--r- 100.0%

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

       5. metadata-eval 100.0%

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

       6. +-commutative 100.0%

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

       7. +-commutative 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-259}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 410:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \left(1 + x \cdot \left(-1 - eps\_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{t\_0 + 2 \cdot \frac{x}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+162}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+179}:\\ \;\;\;\;\frac{2 \cdot t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot {x}^{2}}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -3.5e-259)
     (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
     (if (<= x 410.0)
       (/ (+ (exp (* x (+ eps_m -1.0))) (+ 1.0 (* x (- -1.0 eps_m)))) 2.0)
       (if (<= x 6.8e+93)
         (/ (+ t_0 (* 2.0 (/ x (exp x)))) 2.0)
         (if (<= x 4.2e+162)
           (/ (* x eps_m) 2.0)
           (if (<= x 6e+179)
             (/ (* 2.0 t_0) 2.0)
             (/ (* 0.25 (pow x 2.0)) eps_m))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -3.5e-259) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if (x <= 410.0) {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 6.8e+93) {
		tmp = (t_0 + (2.0 * (x / exp(x)))) / 2.0;
	} else if (x <= 4.2e+162) {
		tmp = (x * eps_m) / 2.0;
	} else if (x <= 6e+179) {
		tmp = (2.0 * t_0) / 2.0;
	} else {
		tmp = (0.25 * pow(x, 2.0)) / eps_m;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-3.5d-259)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if (x <= 410.0d0) then
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 + (x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 6.8d+93) then
        tmp = (t_0 + (2.0d0 * (x / exp(x)))) / 2.0d0
    else if (x <= 4.2d+162) then
        tmp = (x * eps_m) / 2.0d0
    else if (x <= 6d+179) then
        tmp = (2.0d0 * t_0) / 2.0d0
    else
        tmp = (0.25d0 * (x ** 2.0d0)) / eps_m
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (x <= -3.5e-259) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if (x <= 410.0) {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 6.8e+93) {
		tmp = (t_0 + (2.0 * (x / Math.exp(x)))) / 2.0;
	} else if (x <= 4.2e+162) {
		tmp = (x * eps_m) / 2.0;
	} else if (x <= 6e+179) {
		tmp = (2.0 * t_0) / 2.0;
	} else {
		tmp = (0.25 * Math.pow(x, 2.0)) / eps_m;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -3.5e-259:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif x <= 410.0:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0
	elif x <= 6.8e+93:
		tmp = (t_0 + (2.0 * (x / math.exp(x)))) / 2.0
	elif x <= 4.2e+162:
		tmp = (x * eps_m) / 2.0
	elif x <= 6e+179:
		tmp = (2.0 * t_0) / 2.0
	else:
		tmp = (0.25 * math.pow(x, 2.0)) / eps_m
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -3.5e-259)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 410.0)
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 + Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 6.8e+93)
		tmp = Float64(Float64(t_0 + Float64(2.0 * Float64(x / exp(x)))) / 2.0);
	elseif (x <= 4.2e+162)
		tmp = Float64(Float64(x * eps_m) / 2.0);
	elseif (x <= 6e+179)
		tmp = Float64(Float64(2.0 * t_0) / 2.0);
	else
		tmp = Float64(Float64(0.25 * (x ^ 2.0)) / eps_m);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (x <= -3.5e-259)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif (x <= 410.0)
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 6.8e+93)
		tmp = (t_0 + (2.0 * (x / exp(x)))) / 2.0;
	elseif (x <= 4.2e+162)
		tmp = (x * eps_m) / 2.0;
	elseif (x <= 6e+179)
		tmp = (2.0 * t_0) / 2.0;
	else
		tmp = (0.25 * (x ^ 2.0)) / eps_m;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -3.5e-259], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 410.0], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.8e+93], N[(N[(t$95$0 + N[(2.0 * N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.2e+162], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6e+179], N[(N[(2.0 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -3.5000000000000002e-259

   1. Initial program 69.8%

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

      1. fma-neg 69.7%

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

      2. /-rgt-identity 69.7%

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

      3. fma-neg 69.8%

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

      4. /-rgt-identity 69.8%

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

      5. distribute-rgt-neg-in 69.8%

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

      6. sub-neg 69.8%

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

      7. metadata-eval 69.8%

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

      8. distribute-rgt-neg-in 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in eps around inf 98.0%

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

   6. Taylor expanded in eps around -inf 98.0%

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

      1. associate-*r* 98.0%

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

      2. neg-mul-1 98.0%

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

      3. mul-1-neg 98.0%

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

      4. sub-neg 98.0%

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

      5. mul-1-neg 98.0%

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

      6. associate-*r* 98.0%

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

      7. neg-mul-1 98.0%

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

      8. mul-1-neg 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in eps around inf 98.0%

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

       1. associate-*r* 98.0%

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

       2. neg-mul-1 98.0%

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

   11. Simplified 98.0%

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

   12. Taylor expanded in x around 0 76.4%

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

   if -3.5000000000000002e-259 < x < 410

   1. Initial program 56.2%

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

      1. fma-neg 56.2%

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

      2. /-rgt-identity 56.2%

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

      3. fma-neg 56.2%

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

      4. /-rgt-identity 56.2%

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

      5. distribute-rgt-neg-in 56.2%

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

      6. sub-neg 56.2%

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

      7. metadata-eval 56.2%

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

      8. distribute-rgt-neg-in 56.2%

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

   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 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. Taylor expanded in eps around -inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

      3. mul-1-neg 99.8%

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

      4. sub-neg 99.8%

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

      5. mul-1-neg 99.8%

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

      6. 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. neg-mul-1 99.8%

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

      8. mul-1-neg 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 89.0%

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

       1. mul-1-neg 89.0%

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

       2. distribute-rgt-neg-in 89.0%

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

       3. distribute-neg-in 89.0%

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

       4. metadata-eval 89.0%

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

   11. Simplified 89.0%

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

   if 410 < x < 6.8000000000000001e93

   1. Initial program 95.9%

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

      1. fma-neg 95.9%

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

      2. /-rgt-identity 95.9%

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

      3. fma-neg 95.9%

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

      4. /-rgt-identity 95.9%

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

      5. distribute-rgt-neg-in 95.9%

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

      6. sub-neg 95.9%

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

      7. metadata-eval 95.9%

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

      8. distribute-rgt-neg-in 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in eps around 0 76.8%

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

   7. Simplified 76.8%

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

   8. Taylor expanded in x around inf 72.9%

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

      1. exp-neg 72.9%

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

      2. associate-*r/ 72.9%

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

      3. *-rgt-identity 72.9%

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

   10. Simplified 72.9%

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

   11. Taylor expanded in x around inf 72.9%

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

       1. rec-exp 72.9%

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

       2. cancel-sign-sub-inv 72.9%

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

       3. metadata-eval 72.9%

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

       4. *-lft-identity 72.9%

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

       5. neg-mul-1 72.9%

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

       6. rec-exp 72.9%

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

       7. rec-exp 72.9%

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

       8. distribute-lft-in 72.9%

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

       9. neg-mul-1 72.9%

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

       10. count-2 72.9%

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

       11. rem-exp-log 72.9%

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

       12. neg-mul-1 72.9%

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

       13. exp-sum 72.9%

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

       14. sub-neg 72.9%

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

       15. exp-diff 72.9%

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

   13. Simplified 72.9%

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

   if 6.8000000000000001e93 < x < 4.2000000000000001e162

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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. Taylor expanded in x around 0 46.1%

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

      1. mul-1-neg 46.1%

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

      2. distribute-rgt-neg-in 46.1%

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

      3. sub-neg 46.1%

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

      4. mul-1-neg 46.1%

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

      5. distribute-neg-in 46.1%

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

      6. metadata-eval 46.1%

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

      7. mul-1-neg 46.1%

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

      8. remove-double-neg 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in eps around inf 37.1%

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

       1. *-commutative 37.1%

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

   11. Simplified 37.1%

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

   if 4.2000000000000001e162 < x < 5.9999999999999996e179

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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. Taylor expanded in eps around 0 80.3%

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

      1. cancel-sign-sub-inv 80.3%

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

      2. metadata-eval 80.3%

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

      3. distribute-rgt1-in 80.3%

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

      4. metadata-eval 80.3%

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

      5. neg-mul-1 80.3%

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

   8. Simplified 80.3%

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

   if 5.9999999999999996e179 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 36.1%

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

   6. Taylor expanded in x around 0 1.6%

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

      1. neg-mul-1 1.6%

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

      2. unsub-neg 1.6%

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

   8. Simplified 1.6%

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

   9. Taylor expanded in x around 0 22.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   10. Step-by-step derivation

       1. associate-*r/ 22.2%

          \[\leadsto \color{blue}{\frac{0.25 \cdot {x}^{2}}{\varepsilon}} \]

   11. Simplified 22.2%

       \[\leadsto \color{blue}{\frac{0.25 \cdot {x}^{2}}{\varepsilon}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-259}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 410:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{e^{-x} + 2 \cdot \frac{x}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+162}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+179}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot {x}^{2}}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-254}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 410:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \left(1 + x \cdot \left(-1 - eps\_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{2 \cdot e^{\log x - x}}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+156}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+179}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot {x}^{2}}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-254)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (<= x 410.0)
     (/ (+ (exp (* x (+ eps_m -1.0))) (+ 1.0 (* x (- -1.0 eps_m)))) 2.0)
     (if (<= x 5.2e+93)
       (/ (* 2.0 (exp (- (log x) x))) 2.0)
       (if (<= x 4.6e+156)
         (/ (* x eps_m) 2.0)
         (if (<= x 5e+179)
           (/ (* 2.0 (exp (- x))) 2.0)
           (/ (* 0.25 (pow x 2.0)) eps_m)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-254) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if (x <= 410.0) {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 5.2e+93) {
		tmp = (2.0 * exp((log(x) - x))) / 2.0;
	} else if (x <= 4.6e+156) {
		tmp = (x * eps_m) / 2.0;
	} else if (x <= 5e+179) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (0.25 * pow(x, 2.0)) / eps_m;
	}
	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 <= (-5d-254)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if (x <= 410.0d0) then
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 + (x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 5.2d+93) then
        tmp = (2.0d0 * exp((log(x) - x))) / 2.0d0
    else if (x <= 4.6d+156) then
        tmp = (x * eps_m) / 2.0d0
    else if (x <= 5d+179) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (0.25d0 * (x ** 2.0d0)) / eps_m
    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 <= -5e-254) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if (x <= 410.0) {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 5.2e+93) {
		tmp = (2.0 * Math.exp((Math.log(x) - x))) / 2.0;
	} else if (x <= 4.6e+156) {
		tmp = (x * eps_m) / 2.0;
	} else if (x <= 5e+179) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (0.25 * Math.pow(x, 2.0)) / eps_m;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -5e-254:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif x <= 410.0:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0
	elif x <= 5.2e+93:
		tmp = (2.0 * math.exp((math.log(x) - x))) / 2.0
	elif x <= 4.6e+156:
		tmp = (x * eps_m) / 2.0
	elif x <= 5e+179:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (0.25 * math.pow(x, 2.0)) / eps_m
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -5e-254)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 410.0)
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 + Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 5.2e+93)
		tmp = Float64(Float64(2.0 * exp(Float64(log(x) - x))) / 2.0);
	elseif (x <= 4.6e+156)
		tmp = Float64(Float64(x * eps_m) / 2.0);
	elseif (x <= 5e+179)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(0.25 * (x ^ 2.0)) / eps_m);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -5e-254)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif (x <= 410.0)
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 5.2e+93)
		tmp = (2.0 * exp((log(x) - x))) / 2.0;
	elseif (x <= 4.6e+156)
		tmp = (x * eps_m) / 2.0;
	elseif (x <= 5e+179)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (0.25 * (x ^ 2.0)) / eps_m;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5e-254], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 410.0], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.2e+93], N[(N[(2.0 * N[Exp[N[(N[Log[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.6e+156], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+179], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -5.0000000000000003e-254

   1. Initial program 69.8%

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

      1. fma-neg 69.7%

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

      2. /-rgt-identity 69.7%

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

      3. fma-neg 69.8%

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

      4. /-rgt-identity 69.8%

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

      5. distribute-rgt-neg-in 69.8%

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

      6. sub-neg 69.8%

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

      7. metadata-eval 69.8%

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

      8. distribute-rgt-neg-in 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in eps around inf 98.0%

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

   6. Taylor expanded in eps around -inf 98.0%

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

      1. associate-*r* 98.0%

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

      2. neg-mul-1 98.0%

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

      3. mul-1-neg 98.0%

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

      4. sub-neg 98.0%

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

      5. mul-1-neg 98.0%

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

      6. associate-*r* 98.0%

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

      7. neg-mul-1 98.0%

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

      8. mul-1-neg 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in eps around inf 98.0%

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

       1. associate-*r* 98.0%

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

       2. neg-mul-1 98.0%

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

   11. Simplified 98.0%

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

   12. Taylor expanded in x around 0 76.4%

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

   if -5.0000000000000003e-254 < x < 410

   1. Initial program 56.2%

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

      1. fma-neg 56.2%

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

      2. /-rgt-identity 56.2%

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

      3. fma-neg 56.2%

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

      4. /-rgt-identity 56.2%

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

      5. distribute-rgt-neg-in 56.2%

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

      6. sub-neg 56.2%

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

      7. metadata-eval 56.2%

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

      8. distribute-rgt-neg-in 56.2%

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

   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 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. Taylor expanded in eps around -inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

      3. mul-1-neg 99.8%

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

      4. sub-neg 99.8%

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

      5. mul-1-neg 99.8%

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

      6. 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. neg-mul-1 99.8%

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

      8. mul-1-neg 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 89.0%

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

       1. mul-1-neg 89.0%

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

       2. distribute-rgt-neg-in 89.0%

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

       3. distribute-neg-in 89.0%

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

       4. metadata-eval 89.0%

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

   11. Simplified 89.0%

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

   if 410 < x < 5.19999999999999999e93

   1. Initial program 95.9%

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

      1. fma-neg 95.9%

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

      2. /-rgt-identity 95.9%

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

      3. fma-neg 95.9%

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

      4. /-rgt-identity 95.9%

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

      5. distribute-rgt-neg-in 95.9%

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

      6. sub-neg 95.9%

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

      7. metadata-eval 95.9%

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

      8. distribute-rgt-neg-in 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in eps around 0 76.8%

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

   7. Simplified 76.8%

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

   8. Taylor expanded in x around inf 72.9%

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

      1. exp-neg 72.9%

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

      2. associate-*r/ 72.9%

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

      3. *-rgt-identity 72.9%

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

   10. Simplified 72.9%

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

   11. Taylor expanded in x around inf 72.8%

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

       1. rec-exp 72.8%

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

       2. cancel-sign-sub-inv 72.8%

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

       3. metadata-eval 72.8%

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

       4. *-lft-identity 72.8%

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

       5. neg-mul-1 72.8%

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

       6. rec-exp 72.8%

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

       7. rec-exp 72.8%

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

       8. distribute-lft-in 72.8%

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

       9. neg-mul-1 72.8%

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

       10. count-2 72.8%

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

       11. rem-exp-log 72.8%

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

       12. neg-mul-1 72.8%

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

       13. exp-sum 72.8%

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

       14. sub-neg 72.8%

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

       15. exp-diff 72.8%

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

       16. rem-exp-log 72.8%

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

   13. Simplified 72.8%

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

       1. add-exp-log 72.8%

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

       2. div-exp 72.8%

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

   15. Applied egg-rr 72.8%

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

   if 5.19999999999999999e93 < x < 4.5999999999999998e156

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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. Taylor expanded in x around 0 46.1%

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

      1. mul-1-neg 46.1%

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

      2. distribute-rgt-neg-in 46.1%

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

      3. sub-neg 46.1%

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

      4. mul-1-neg 46.1%

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

      5. distribute-neg-in 46.1%

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

      6. metadata-eval 46.1%

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

      7. mul-1-neg 46.1%

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

      8. remove-double-neg 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in eps around inf 37.1%

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

       1. *-commutative 37.1%

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

   11. Simplified 37.1%

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

   if 4.5999999999999998e156 < x < 5e179

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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. Taylor expanded in eps around 0 80.3%

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

      1. cancel-sign-sub-inv 80.3%

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

      2. metadata-eval 80.3%

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

      3. distribute-rgt1-in 80.3%

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

      4. metadata-eval 80.3%

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

      5. neg-mul-1 80.3%

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

   8. Simplified 80.3%

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

   if 5e179 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 36.1%

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

   6. Taylor expanded in x around 0 1.6%

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

      1. neg-mul-1 1.6%

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

      2. unsub-neg 1.6%

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

   8. Simplified 1.6%

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

   9. Taylor expanded in x around 0 22.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   10. Step-by-step derivation

       1. associate-*r/ 22.2%

          \[\leadsto \color{blue}{\frac{0.25 \cdot {x}^{2}}{\varepsilon}} \]

   11. Simplified 22.2%

       \[\leadsto \color{blue}{\frac{0.25 \cdot {x}^{2}}{\varepsilon}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-254}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 410:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{2 \cdot e^{\log x - x}}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+156}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+179}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot {x}^{2}}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2.25 \cdot 10^{-14}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2.25e-14)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2.25e-14) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + 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 <= 2.25d-14) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + 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 <= 2.25e-14) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + 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 <= 2.25e-14:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + 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 <= 2.25e-14)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(x * Float64(-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 <= 2.25e-14)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + 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, 2.25e-14], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.2499999999999999e-14

   1. Initial program 60.3%

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

      1. fma-neg 60.2%

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

      2. /-rgt-identity 60.2%

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

      3. fma-neg 60.3%

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

      4. /-rgt-identity 60.3%

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

      5. distribute-rgt-neg-in 60.3%

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

      6. sub-neg 60.3%

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

      7. metadata-eval 60.3%

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

      8. distribute-rgt-neg-in 60.3%

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

   3. Simplified 60.3%

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

   5. Taylor expanded in eps around inf 98.4%

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

   6. Taylor expanded in eps around 0 75.1%

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

      1. cancel-sign-sub-inv 75.1%

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

      2. metadata-eval 75.1%

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

      3. distribute-rgt1-in 75.1%

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

      4. metadata-eval 75.1%

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

      5. neg-mul-1 75.1%

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

   8. Simplified 75.1%

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

   if 2.2499999999999999e-14 < eps

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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. Taylor expanded in eps around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. 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. neg-mul-1 100.0%

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

      8. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-rgt-neg-in 100.0%

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

       3. neg-sub0 100.0%

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

       4. associate--r- 100.0%

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

       5. metadata-eval 100.0%

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

       6. +-commutative 100.0%

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

       7. +-commutative 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.25 \cdot 10^{-14}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% 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%

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

   1. fma-neg 72.8%

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

   2. /-rgt-identity 72.8%

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

   3. fma-neg 72.8%

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

   4. /-rgt-identity 72.8%

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

   5. distribute-rgt-neg-in 72.8%

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

   6. sub-neg 72.8%

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

   7. metadata-eval 72.8%

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

   8. distribute-rgt-neg-in 72.8%

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

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 98.9%

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

6. Taylor expanded in eps around -inf 98.9%

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

   1. associate-*r* 98.9%

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

   2. neg-mul-1 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. mul-1-neg 98.9%

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

   6. associate-*r* 98.9%

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

   7. neg-mul-1 98.9%

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

   8. mul-1-neg 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 6: 62.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 410:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+93}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+156}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+179}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot {x}^{2}}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (* (* x eps_m) -0.5)
   (if (<= x 410.0)
     1.0
     (if (<= x 2.1e+93)
       0.0
       (if (<= x 1.45e+156)
         (/ (* x eps_m) 2.0)
         (if (<= x 4.9e+179) 0.0 (/ (* 0.25 (pow x 2.0)) eps_m)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 410.0) {
		tmp = 1.0;
	} else if (x <= 2.1e+93) {
		tmp = 0.0;
	} else if (x <= 1.45e+156) {
		tmp = (x * eps_m) / 2.0;
	} else if (x <= 4.9e+179) {
		tmp = 0.0;
	} else {
		tmp = (0.25 * pow(x, 2.0)) / eps_m;
	}
	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 <= (-1.0d0)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 410.0d0) then
        tmp = 1.0d0
    else if (x <= 2.1d+93) then
        tmp = 0.0d0
    else if (x <= 1.45d+156) then
        tmp = (x * eps_m) / 2.0d0
    else if (x <= 4.9d+179) then
        tmp = 0.0d0
    else
        tmp = (0.25d0 * (x ** 2.0d0)) / eps_m
    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 <= -1.0) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 410.0) {
		tmp = 1.0;
	} else if (x <= 2.1e+93) {
		tmp = 0.0;
	} else if (x <= 1.45e+156) {
		tmp = (x * eps_m) / 2.0;
	} else if (x <= 4.9e+179) {
		tmp = 0.0;
	} else {
		tmp = (0.25 * Math.pow(x, 2.0)) / eps_m;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (x * eps_m) * -0.5
	elif x <= 410.0:
		tmp = 1.0
	elif x <= 2.1e+93:
		tmp = 0.0
	elif x <= 1.45e+156:
		tmp = (x * eps_m) / 2.0
	elif x <= 4.9e+179:
		tmp = 0.0
	else:
		tmp = (0.25 * math.pow(x, 2.0)) / eps_m
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 410.0)
		tmp = 1.0;
	elseif (x <= 2.1e+93)
		tmp = 0.0;
	elseif (x <= 1.45e+156)
		tmp = Float64(Float64(x * eps_m) / 2.0);
	elseif (x <= 4.9e+179)
		tmp = 0.0;
	else
		tmp = Float64(Float64(0.25 * (x ^ 2.0)) / eps_m);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 410.0)
		tmp = 1.0;
	elseif (x <= 2.1e+93)
		tmp = 0.0;
	elseif (x <= 1.45e+156)
		tmp = (x * eps_m) / 2.0;
	elseif (x <= 4.9e+179)
		tmp = 0.0;
	else
		tmp = (0.25 * (x ^ 2.0)) / eps_m;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 410.0], 1.0, If[LessEqual[x, 2.1e+93], 0.0, If[LessEqual[x, 1.45e+156], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.9e+179], 0.0, N[(N[(0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1

   1. Initial program 97.3%

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

      1. fma-neg 97.3%

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

      2. /-rgt-identity 97.3%

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

      3. fma-neg 97.3%

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

      4. /-rgt-identity 97.3%

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

      5. distribute-rgt-neg-in 97.3%

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

      6. sub-neg 97.3%

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

      7. metadata-eval 97.3%

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

      8. distribute-rgt-neg-in 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in eps around inf 97.3%

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

   6. Taylor expanded in x around 0 28.4%

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

      1. mul-1-neg 28.4%

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

      2. distribute-rgt-neg-in 28.4%

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

      3. sub-neg 28.4%

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

      4. mul-1-neg 28.4%

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

      5. distribute-neg-in 28.4%

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

      6. metadata-eval 28.4%

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

      7. mul-1-neg 28.4%

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

      8. remove-double-neg 28.4%

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

   8. Simplified 28.4%

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

   9. Taylor expanded in eps around inf 15.0%

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

       1. *-commutative 15.0%

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

   11. Simplified 15.0%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
   12. Step-by-step derivation

       1. frac-2neg 15.0%

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

       2. mul-1-neg 15.0%

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

       3. div-inv 15.0%

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

       4. associate-*r* 15.0%

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

       5. neg-mul-1 15.0%

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

       6. add-sqr-sqrt 15.0%

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

       7. sqrt-unprod 20.0%

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

       8. sqr-neg 20.0%

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

       9. sqrt-unprod 0.0%

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

       10. add-sqr-sqrt 49.6%

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

       11. metadata-eval 49.6%

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

       12. metadata-eval 49.6%

           \[\leadsto \left(x \cdot \varepsilon\right) \cdot \color{blue}{-0.5} \]

   13. Applied egg-rr 49.6%

       \[\leadsto \color{blue}{\left(x \cdot \varepsilon\right) \cdot -0.5} \]

   if -1 < x < 410

   1. Initial program 54.5%

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

      1. fma-neg 54.4%

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

      2. /-rgt-identity 54.4%

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

      3. fma-neg 54.5%

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

      4. /-rgt-identity 54.5%

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

      5. distribute-rgt-neg-in 54.5%

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

      6. sub-neg 54.5%

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

      7. metadata-eval 54.5%

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

      8. distribute-rgt-neg-in 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in x around 0 72.0%

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

   if 410 < x < 2.0999999999999998e93 or 1.45000000000000005e156 < x < 4.8999999999999999e179

   1. Initial program 96.8%

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

      1. fma-neg 96.8%

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

      2. /-rgt-identity 96.8%

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

      3. fma-neg 96.8%

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

      4. /-rgt-identity 96.8%

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

      5. distribute-rgt-neg-in 96.8%

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

      6. sub-neg 96.8%

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

      7. metadata-eval 96.8%

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

      8. distribute-rgt-neg-in 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in eps around inf 96.1%

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

   6. Taylor expanded in x around 0 16.1%

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

      1. mul-1-neg 16.1%

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

      2. distribute-rgt-neg-in 16.1%

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

      3. sub-neg 16.1%

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

      4. mul-1-neg 16.1%

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

      5. distribute-neg-in 16.1%

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

      6. metadata-eval 16.1%

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

      7. mul-1-neg 16.1%

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

      8. remove-double-neg 16.1%

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

   8. Simplified 16.1%

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

   9. Taylor expanded in x around 0 2.4%

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

       1. +-commutative 2.4%

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

       2. associate--l+ 2.4%

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

       3. +-commutative 2.4%

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

       4. add-sqr-sqrt 2.2%

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

       5. sqrt-unprod 2.1%

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

       6. mul-1-neg 2.1%

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

       7. mul-1-neg 2.1%

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

       8. sqr-neg 2.1%

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

       9. sqrt-unprod 0.0%

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

       10. add-sqr-sqrt 73.3%

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

       11. +-commutative 73.3%

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

   11. Applied egg-rr 73.3%

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

       1. associate-+r- 73.3%

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

       2. fma-udef 73.3%

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

       3. +-inverses 73.3%

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

   13. Simplified 73.3%

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

   if 2.0999999999999998e93 < x < 1.45000000000000005e156

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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. Taylor expanded in x around 0 46.1%

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

      1. mul-1-neg 46.1%

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

      2. distribute-rgt-neg-in 46.1%

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

      3. sub-neg 46.1%

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

      4. mul-1-neg 46.1%

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

      5. distribute-neg-in 46.1%

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

      6. metadata-eval 46.1%

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

      7. mul-1-neg 46.1%

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

      8. remove-double-neg 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in eps around inf 37.1%

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

       1. *-commutative 37.1%

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

   11. Simplified 37.1%

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

   if 4.8999999999999999e179 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 36.1%

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

   6. Taylor expanded in x around 0 1.6%

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

      1. neg-mul-1 1.6%

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

      2. unsub-neg 1.6%

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

   8. Simplified 1.6%

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

   9. Taylor expanded in x around 0 22.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   10. Step-by-step derivation

       1. associate-*r/ 22.2%

          \[\leadsto \color{blue}{\frac{0.25 \cdot {x}^{2}}{\varepsilon}} \]

   11. Simplified 22.2%

       \[\leadsto \color{blue}{\frac{0.25 \cdot {x}^{2}}{\varepsilon}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 410:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+93}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+156}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+179}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot {x}^{2}}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-258}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 410:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \left(1 + x \cdot \left(-1 - eps\_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+161}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+179}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot {x}^{2}}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-258)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (<= x 410.0)
     (/ (+ (exp (* x (+ eps_m -1.0))) (+ 1.0 (* x (- -1.0 eps_m)))) 2.0)
     (if (<= x 7.5e+89)
       (/ (* 2.0 (/ x (exp x))) 2.0)
       (if (<= x 1.6e+161)
         (/ (* x eps_m) 2.0)
         (if (<= x 4.8e+179)
           (/ (* 2.0 (exp (- x))) 2.0)
           (/ (* 0.25 (pow x 2.0)) eps_m)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-258) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if (x <= 410.0) {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 7.5e+89) {
		tmp = (2.0 * (x / exp(x))) / 2.0;
	} else if (x <= 1.6e+161) {
		tmp = (x * eps_m) / 2.0;
	} else if (x <= 4.8e+179) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (0.25 * pow(x, 2.0)) / eps_m;
	}
	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 <= (-1d-258)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if (x <= 410.0d0) then
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 + (x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 7.5d+89) then
        tmp = (2.0d0 * (x / exp(x))) / 2.0d0
    else if (x <= 1.6d+161) then
        tmp = (x * eps_m) / 2.0d0
    else if (x <= 4.8d+179) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (0.25d0 * (x ** 2.0d0)) / eps_m
    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 <= -1e-258) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if (x <= 410.0) {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 7.5e+89) {
		tmp = (2.0 * (x / Math.exp(x))) / 2.0;
	} else if (x <= 1.6e+161) {
		tmp = (x * eps_m) / 2.0;
	} else if (x <= 4.8e+179) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (0.25 * Math.pow(x, 2.0)) / eps_m;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e-258:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif x <= 410.0:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0
	elif x <= 7.5e+89:
		tmp = (2.0 * (x / math.exp(x))) / 2.0
	elif x <= 1.6e+161:
		tmp = (x * eps_m) / 2.0
	elif x <= 4.8e+179:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (0.25 * math.pow(x, 2.0)) / eps_m
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-258)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 410.0)
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 + Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 7.5e+89)
		tmp = Float64(Float64(2.0 * Float64(x / exp(x))) / 2.0);
	elseif (x <= 1.6e+161)
		tmp = Float64(Float64(x * eps_m) / 2.0);
	elseif (x <= 4.8e+179)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(0.25 * (x ^ 2.0)) / eps_m);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1e-258)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif (x <= 410.0)
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 7.5e+89)
		tmp = (2.0 * (x / exp(x))) / 2.0;
	elseif (x <= 1.6e+161)
		tmp = (x * eps_m) / 2.0;
	elseif (x <= 4.8e+179)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (0.25 * (x ^ 2.0)) / eps_m;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1e-258], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 410.0], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7.5e+89], N[(N[(2.0 * N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.6e+161], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.8e+179], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -9.99999999999999954e-259

   1. Initial program 69.8%

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

      1. fma-neg 69.7%

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

      2. /-rgt-identity 69.7%

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

      3. fma-neg 69.8%

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

      4. /-rgt-identity 69.8%

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

      5. distribute-rgt-neg-in 69.8%

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

      6. sub-neg 69.8%

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

      7. metadata-eval 69.8%

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

      8. distribute-rgt-neg-in 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in eps around inf 98.0%

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

   6. Taylor expanded in eps around -inf 98.0%

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

      1. associate-*r* 98.0%

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

      2. neg-mul-1 98.0%

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

      3. mul-1-neg 98.0%

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

      4. sub-neg 98.0%

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

      5. mul-1-neg 98.0%

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

      6. associate-*r* 98.0%

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

      7. neg-mul-1 98.0%

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

      8. mul-1-neg 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in eps around inf 98.0%

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

       1. associate-*r* 98.0%

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

       2. neg-mul-1 98.0%

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

   11. Simplified 98.0%

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

   12. Taylor expanded in x around 0 76.4%

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

   if -9.99999999999999954e-259 < x < 410

   1. Initial program 56.2%

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

      1. fma-neg 56.2%

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

      2. /-rgt-identity 56.2%

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

      3. fma-neg 56.2%

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

      4. /-rgt-identity 56.2%

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

      5. distribute-rgt-neg-in 56.2%

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

      6. sub-neg 56.2%

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

      7. metadata-eval 56.2%

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

      8. distribute-rgt-neg-in 56.2%

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

   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 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. Taylor expanded in eps around -inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

      3. mul-1-neg 99.8%

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

      4. sub-neg 99.8%

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

      5. mul-1-neg 99.8%

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

      6. 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. neg-mul-1 99.8%

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

      8. mul-1-neg 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 89.0%

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

       1. mul-1-neg 89.0%

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

       2. distribute-rgt-neg-in 89.0%

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

       3. distribute-neg-in 89.0%

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

       4. metadata-eval 89.0%

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

   11. Simplified 89.0%

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

   if 410 < x < 7.49999999999999947e89

   1. Initial program 95.9%

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

      1. fma-neg 95.9%

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

      2. /-rgt-identity 95.9%

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

      3. fma-neg 95.9%

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

      4. /-rgt-identity 95.9%

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

      5. distribute-rgt-neg-in 95.9%

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

      6. sub-neg 95.9%

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

      7. metadata-eval 95.9%

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

      8. distribute-rgt-neg-in 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in eps around 0 76.8%

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

   7. Simplified 76.8%

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

   8. Taylor expanded in x around inf 72.9%

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

      1. exp-neg 72.9%

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

      2. associate-*r/ 72.9%

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

      3. *-rgt-identity 72.9%

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

   10. Simplified 72.9%

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

   11. Taylor expanded in x around inf 72.8%

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

       1. rec-exp 72.8%

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

       2. cancel-sign-sub-inv 72.8%

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

       3. metadata-eval 72.8%

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

       4. *-lft-identity 72.8%

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

       5. neg-mul-1 72.8%

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

       6. rec-exp 72.8%

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

       7. rec-exp 72.8%

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

       8. distribute-lft-in 72.8%

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

       9. neg-mul-1 72.8%

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

       10. count-2 72.8%

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

       11. rem-exp-log 72.8%

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

       12. neg-mul-1 72.8%

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

       13. exp-sum 72.8%

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

       14. sub-neg 72.8%

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

       15. exp-diff 72.8%

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

       16. rem-exp-log 72.8%

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

   13. Simplified 72.8%

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

   if 7.49999999999999947e89 < x < 1.60000000000000001e161

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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. Taylor expanded in x around 0 46.1%

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

      1. mul-1-neg 46.1%

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

      2. distribute-rgt-neg-in 46.1%

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

      3. sub-neg 46.1%

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

      4. mul-1-neg 46.1%

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

      5. distribute-neg-in 46.1%

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

      6. metadata-eval 46.1%

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

      7. mul-1-neg 46.1%

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

      8. remove-double-neg 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in eps around inf 37.1%

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

       1. *-commutative 37.1%

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

   11. Simplified 37.1%

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

   if 1.60000000000000001e161 < x < 4.80000000000000025e179

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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. Taylor expanded in eps around 0 80.3%

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

      1. cancel-sign-sub-inv 80.3%

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

      2. metadata-eval 80.3%

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

      3. distribute-rgt1-in 80.3%

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

      4. metadata-eval 80.3%

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

      5. neg-mul-1 80.3%

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

   8. Simplified 80.3%

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

   if 4.80000000000000025e179 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 36.1%

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

   6. Taylor expanded in x around 0 1.6%

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

      1. neg-mul-1 1.6%

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

      2. unsub-neg 1.6%

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

   8. Simplified 1.6%

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

   9. Taylor expanded in x around 0 22.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   10. Step-by-step derivation

       1. associate-*r/ 22.2%

          \[\leadsto \color{blue}{\frac{0.25 \cdot {x}^{2}}{\varepsilon}} \]

   11. Simplified 22.2%

       \[\leadsto \color{blue}{\frac{0.25 \cdot {x}^{2}}{\varepsilon}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-258}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 410:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+161}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+179}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot {x}^{2}}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+179}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot {x}^{2}}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.4)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (<= x 6.8e+93)
     (/ (* 2.0 (/ x (exp x))) 2.0)
     (if (<= x 2.8e+155)
       (/ (* x eps_m) 2.0)
       (if (<= x 5e+179)
         (/ (* 2.0 (exp (- x))) 2.0)
         (/ (* 0.25 (pow x 2.0)) eps_m))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.4) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if (x <= 6.8e+93) {
		tmp = (2.0 * (x / exp(x))) / 2.0;
	} else if (x <= 2.8e+155) {
		tmp = (x * eps_m) / 2.0;
	} else if (x <= 5e+179) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (0.25 * pow(x, 2.0)) / eps_m;
	}
	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 <= 1.4d0) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if (x <= 6.8d+93) then
        tmp = (2.0d0 * (x / exp(x))) / 2.0d0
    else if (x <= 2.8d+155) then
        tmp = (x * eps_m) / 2.0d0
    else if (x <= 5d+179) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (0.25d0 * (x ** 2.0d0)) / eps_m
    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 <= 1.4) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if (x <= 6.8e+93) {
		tmp = (2.0 * (x / Math.exp(x))) / 2.0;
	} else if (x <= 2.8e+155) {
		tmp = (x * eps_m) / 2.0;
	} else if (x <= 5e+179) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (0.25 * Math.pow(x, 2.0)) / eps_m;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.4:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif x <= 6.8e+93:
		tmp = (2.0 * (x / math.exp(x))) / 2.0
	elif x <= 2.8e+155:
		tmp = (x * eps_m) / 2.0
	elif x <= 5e+179:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (0.25 * math.pow(x, 2.0)) / eps_m
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 6.8e+93)
		tmp = Float64(Float64(2.0 * Float64(x / exp(x))) / 2.0);
	elseif (x <= 2.8e+155)
		tmp = Float64(Float64(x * eps_m) / 2.0);
	elseif (x <= 5e+179)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(0.25 * (x ^ 2.0)) / eps_m);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif (x <= 6.8e+93)
		tmp = (2.0 * (x / exp(x))) / 2.0;
	elseif (x <= 2.8e+155)
		tmp = (x * eps_m) / 2.0;
	elseif (x <= 5e+179)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (0.25 * (x ^ 2.0)) / eps_m;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.4], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.8e+93], N[(N[(2.0 * N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.8e+155], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+179], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.3999999999999999

   1. Initial program 63.0%

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

      1. fma-neg 63.0%

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

      2. /-rgt-identity 63.0%

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

      3. fma-neg 63.0%

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

      4. /-rgt-identity 63.0%

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

      5. distribute-rgt-neg-in 63.0%

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

      6. sub-neg 63.0%

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

      7. metadata-eval 63.0%

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

      8. distribute-rgt-neg-in 63.0%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in eps around -inf 98.9%

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

      1. associate-*r* 98.9%

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

      2. neg-mul-1 98.9%

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

      3. mul-1-neg 98.9%

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

      4. sub-neg 98.9%

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

      5. mul-1-neg 98.9%

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

      6. associate-*r* 98.9%

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

      7. neg-mul-1 98.9%

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

      8. mul-1-neg 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in eps around inf 98.9%

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

       1. associate-*r* 98.9%

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

       2. neg-mul-1 98.9%

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

   11. Simplified 98.9%

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

   12. Taylor expanded in x around 0 81.0%

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

   if 1.3999999999999999 < x < 6.8000000000000001e93

   1. Initial program 95.9%

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

      1. fma-neg 95.9%

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

      2. /-rgt-identity 95.9%

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

      3. fma-neg 95.9%

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

      4. /-rgt-identity 95.9%

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

      5. distribute-rgt-neg-in 95.9%

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

      6. sub-neg 95.9%

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

      7. metadata-eval 95.9%

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

      8. distribute-rgt-neg-in 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in eps around 0 76.8%

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

   7. Simplified 76.8%

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

   8. Taylor expanded in x around inf 72.9%

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

      1. exp-neg 72.9%

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

      2. associate-*r/ 72.9%

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

      3. *-rgt-identity 72.9%

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

   10. Simplified 72.9%

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

   11. Taylor expanded in x around inf 72.8%

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

       1. rec-exp 72.8%

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

       2. cancel-sign-sub-inv 72.8%

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

       3. metadata-eval 72.8%

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

       4. *-lft-identity 72.8%

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

       5. neg-mul-1 72.8%

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

       6. rec-exp 72.8%

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

       7. rec-exp 72.8%

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

       8. distribute-lft-in 72.8%

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

       9. neg-mul-1 72.8%

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

       10. count-2 72.8%

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

       11. rem-exp-log 72.8%

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

       12. neg-mul-1 72.8%

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

       13. exp-sum 72.8%

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

       14. sub-neg 72.8%

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

       15. exp-diff 72.8%

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

       16. rem-exp-log 72.8%

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

   13. Simplified 72.8%

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

   if 6.8000000000000001e93 < x < 2.80000000000000016e155

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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. Taylor expanded in x around 0 46.1%

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

      1. mul-1-neg 46.1%

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

      2. distribute-rgt-neg-in 46.1%

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

      3. sub-neg 46.1%

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

      4. mul-1-neg 46.1%

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

      5. distribute-neg-in 46.1%

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

      6. metadata-eval 46.1%

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

      7. mul-1-neg 46.1%

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

      8. remove-double-neg 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in eps around inf 37.1%

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

       1. *-commutative 37.1%

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

   11. Simplified 37.1%

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

   if 2.80000000000000016e155 < x < 5e179

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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. Taylor expanded in eps around 0 80.3%

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

      1. cancel-sign-sub-inv 80.3%

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

      2. metadata-eval 80.3%

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

      3. distribute-rgt1-in 80.3%

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

      4. metadata-eval 80.3%

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

      5. neg-mul-1 80.3%

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

   8. Simplified 80.3%

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

   if 5e179 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 36.1%

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

   6. Taylor expanded in x around 0 1.6%

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

      1. neg-mul-1 1.6%

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

      2. unsub-neg 1.6%

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

   8. Simplified 1.6%

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

   9. Taylor expanded in x around 0 22.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   10. Step-by-step derivation

       1. associate-*r/ 22.2%

          \[\leadsto \color{blue}{\frac{0.25 \cdot {x}^{2}}{\varepsilon}} \]

   11. Simplified 22.2%

       \[\leadsto \color{blue}{\frac{0.25 \cdot {x}^{2}}{\varepsilon}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+179}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot {x}^{2}}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(-eps\_m\right)}\\ \mathbf{if}\;eps\_m \leq 0.26:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;eps\_m \leq 5.4 \cdot 10^{+287}:\\ \;\;\;\;\frac{\left(1 + x \cdot \left(eps\_m + -1\right)\right) + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (- eps_m)))))
   (if (<= eps_m 0.26)
     (/ (* 2.0 (exp (- x))) 2.0)
     (if (<= eps_m 5.4e+287)
       (/ (+ (+ 1.0 (* x (+ eps_m -1.0))) t_0) 2.0)
       (/ (+ 1.0 t_0) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * -eps_m));
	double tmp;
	if (eps_m <= 0.26) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else if (eps_m <= 5.4e+287) {
		tmp = ((1.0 + (x * (eps_m + -1.0))) + t_0) / 2.0;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * -eps_m))
    if (eps_m <= 0.26d0) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else if (eps_m <= 5.4d+287) then
        tmp = ((1.0d0 + (x * (eps_m + (-1.0d0)))) + t_0) / 2.0d0
    else
        tmp = (1.0d0 + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * -eps_m));
	double tmp;
	if (eps_m <= 0.26) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else if (eps_m <= 5.4e+287) {
		tmp = ((1.0 + (x * (eps_m + -1.0))) + t_0) / 2.0;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * -eps_m))
	tmp = 0
	if eps_m <= 0.26:
		tmp = (2.0 * math.exp(-x)) / 2.0
	elif eps_m <= 5.4e+287:
		tmp = ((1.0 + (x * (eps_m + -1.0))) + t_0) / 2.0
	else:
		tmp = (1.0 + t_0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(-eps_m)))
	tmp = 0.0
	if (eps_m <= 0.26)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	elseif (eps_m <= 5.4e+287)
		tmp = Float64(Float64(Float64(1.0 + Float64(x * Float64(eps_m + -1.0))) + t_0) / 2.0);
	else
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * -eps_m));
	tmp = 0.0;
	if (eps_m <= 0.26)
		tmp = (2.0 * exp(-x)) / 2.0;
	elseif (eps_m <= 5.4e+287)
		tmp = ((1.0 + (x * (eps_m + -1.0))) + t_0) / 2.0;
	else
		tmp = (1.0 + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eps$95$m, 0.26], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 5.4e+287], N[(N[(N[(1.0 + N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 0.26000000000000001

   1. Initial program 60.5%

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

      1. fma-neg 60.5%

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

      2. /-rgt-identity 60.5%

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

      3. fma-neg 60.5%

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

      4. /-rgt-identity 60.5%

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

      5. distribute-rgt-neg-in 60.5%

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

      6. sub-neg 60.5%

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

      7. metadata-eval 60.5%

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

      8. distribute-rgt-neg-in 60.5%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in eps around inf 98.4%

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

   6. Taylor expanded in eps around 0 75.3%

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

      1. cancel-sign-sub-inv 75.3%

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

      2. metadata-eval 75.3%

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

      3. distribute-rgt1-in 75.3%

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

      4. metadata-eval 75.3%

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

      5. neg-mul-1 75.3%

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

   8. Simplified 75.3%

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

   if 0.26000000000000001 < eps < 5.3999999999999998e287

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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. Taylor expanded in eps around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. 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. neg-mul-1 100.0%

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

      8. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0 61.3%

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

       1. mul-1-neg 61.3%

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

       2. distribute-lft-neg-out 61.3%

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

       3. *-commutative 61.3%

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

   14. Simplified 61.3%

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

   if 5.3999999999999998e287 < eps

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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. Taylor expanded in eps around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. 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. neg-mul-1 100.0%

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

      8. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0 100.0%

       \[\leadsto \frac{\color{blue}{1} - \left(-e^{\left(-\varepsilon\right) \cdot x}\right)}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.26:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 5.4 \cdot 10^{+287}:\\ \;\;\;\;\frac{\left(1 + x \cdot \left(\varepsilon + -1\right)\right) + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{2 \cdot e^{-x}}{2}\\ \mathbf{if}\;x \leq 6.8 \cdot 10^{+93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+160}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+179}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot {x}^{2}}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 (exp (- x))) 2.0)))
   (if (<= x 6.8e+93)
     t_0
     (if (<= x 1.8e+160)
       (/ (* x eps_m) 2.0)
       (if (<= x 4.9e+179) t_0 (/ (* 0.25 (pow x 2.0)) eps_m))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (2.0 * exp(-x)) / 2.0;
	double tmp;
	if (x <= 6.8e+93) {
		tmp = t_0;
	} else if (x <= 1.8e+160) {
		tmp = (x * eps_m) / 2.0;
	} else if (x <= 4.9e+179) {
		tmp = t_0;
	} else {
		tmp = (0.25 * pow(x, 2.0)) / eps_m;
	}
	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 = (2.0d0 * exp(-x)) / 2.0d0
    if (x <= 6.8d+93) then
        tmp = t_0
    else if (x <= 1.8d+160) then
        tmp = (x * eps_m) / 2.0d0
    else if (x <= 4.9d+179) then
        tmp = t_0
    else
        tmp = (0.25d0 * (x ** 2.0d0)) / eps_m
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (2.0 * Math.exp(-x)) / 2.0;
	double tmp;
	if (x <= 6.8e+93) {
		tmp = t_0;
	} else if (x <= 1.8e+160) {
		tmp = (x * eps_m) / 2.0;
	} else if (x <= 4.9e+179) {
		tmp = t_0;
	} else {
		tmp = (0.25 * Math.pow(x, 2.0)) / eps_m;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (2.0 * math.exp(-x)) / 2.0
	tmp = 0
	if x <= 6.8e+93:
		tmp = t_0
	elif x <= 1.8e+160:
		tmp = (x * eps_m) / 2.0
	elif x <= 4.9e+179:
		tmp = t_0
	else:
		tmp = (0.25 * math.pow(x, 2.0)) / eps_m
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0)
	tmp = 0.0
	if (x <= 6.8e+93)
		tmp = t_0;
	elseif (x <= 1.8e+160)
		tmp = Float64(Float64(x * eps_m) / 2.0);
	elseif (x <= 4.9e+179)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.25 * (x ^ 2.0)) / eps_m);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (2.0 * exp(-x)) / 2.0;
	tmp = 0.0;
	if (x <= 6.8e+93)
		tmp = t_0;
	elseif (x <= 1.8e+160)
		tmp = (x * eps_m) / 2.0;
	elseif (x <= 4.9e+179)
		tmp = t_0;
	else
		tmp = (0.25 * (x ^ 2.0)) / eps_m;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, 6.8e+93], t$95$0, If[LessEqual[x, 1.8e+160], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.9e+179], t$95$0, N[(N[(0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.8000000000000001e93 or 1.80000000000000011e160 < x < 4.8999999999999999e179

   1. Initial program 66.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 66.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 66.5%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 66.6%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 66.6%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 66.6%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 66.6%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 66.6%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 66.6%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Simplified 66.6%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 98.6%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]

   6. Taylor expanded in eps around 0 76.4%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}}}{2} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv 76.4%

         \[\leadsto \frac{\color{blue}{e^{-1 \cdot x} + \left(--1\right) \cdot e^{-1 \cdot x}}}{2} \]

      2. metadata-eval 76.4%

         \[\leadsto \frac{e^{-1 \cdot x} + \color{blue}{1} \cdot e^{-1 \cdot x}}{2} \]

      3. distribute-rgt1-in 76.4%

         \[\leadsto \frac{\color{blue}{\left(1 + 1\right) \cdot e^{-1 \cdot x}}}{2} \]

      4. metadata-eval 76.4%

         \[\leadsto \frac{\color{blue}{2} \cdot e^{-1 \cdot x}}{2} \]

      5. neg-mul-1 76.4%

         \[\leadsto \frac{2 \cdot e^{\color{blue}{-x}}}{2} \]

   8. Simplified 76.4%

      \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]

   if 6.8000000000000001e93 < x < 1.80000000000000011e160

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 100.0%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   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. Taylor expanded in x around 0 46.1%

      \[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 46.1%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-x \cdot \left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      2. distribute-rgt-neg-in 46.1%

         \[\leadsto \frac{\left(1 + \color{blue}{x \cdot \left(-\left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      3. sub-neg 46.1%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      4. mul-1-neg 46.1%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      5. distribute-neg-in 46.1%

         \[\leadsto \frac{\left(1 + x \cdot \color{blue}{\left(\left(-1\right) + \left(--1 \cdot \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      6. metadata-eval 46.1%

         \[\leadsto \frac{\left(1 + x \cdot \left(\color{blue}{-1} + \left(--1 \cdot \varepsilon\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      7. mul-1-neg 46.1%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      8. remove-double-neg 46.1%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \color{blue}{\varepsilon}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   8. Simplified 46.1%

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(-1 + \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   9. Taylor expanded in eps around inf 37.1%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   10. Step-by-step derivation

       1. *-commutative 37.1%

          \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   11. Simplified 37.1%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   if 4.8999999999999999e179 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - \frac{-1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 36.1%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]

   6. Taylor expanded in x around 0 1.6%

      \[\leadsto \frac{\frac{e^{-1 \cdot x} - \color{blue}{\left(1 + -1 \cdot x\right)}}{\varepsilon}}{2} \]
   7. Step-by-step derivation

      1. neg-mul-1 1.6%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} - \left(1 + \color{blue}{\left(-x\right)}\right)}{\varepsilon}}{2} \]

      2. unsub-neg 1.6%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} - \color{blue}{\left(1 - x\right)}}{\varepsilon}}{2} \]

   8. Simplified 1.6%

      \[\leadsto \frac{\frac{e^{-1 \cdot x} - \color{blue}{\left(1 - x\right)}}{\varepsilon}}{2} \]

   9. Taylor expanded in x around 0 22.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   10. Step-by-step derivation

       1. associate-*r/ 22.2%

          \[\leadsto \color{blue}{\frac{0.25 \cdot {x}^{2}}{\varepsilon}} \]

   11. Simplified 22.2%

       \[\leadsto \color{blue}{\frac{0.25 \cdot {x}^{2}}{\varepsilon}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+160}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+179}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot {x}^{2}}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.2% accurate, 11.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 410:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+93}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.95)
   (* (* x eps_m) -0.5)
   (if (<= x 410.0) 1.0 (if (<= x 3.8e+93) 0.0 (/ (* x eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.95) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 410.0) {
		tmp = 1.0;
	} else if (x <= 3.8e+93) {
		tmp = 0.0;
	} else {
		tmp = (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 <= (-0.95d0)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 410.0d0) then
        tmp = 1.0d0
    else if (x <= 3.8d+93) then
        tmp = 0.0d0
    else
        tmp = (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 <= -0.95) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 410.0) {
		tmp = 1.0;
	} else if (x <= 3.8e+93) {
		tmp = 0.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.95:
		tmp = (x * eps_m) * -0.5
	elif x <= 410.0:
		tmp = 1.0
	elif x <= 3.8e+93:
		tmp = 0.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.95)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 410.0)
		tmp = 1.0;
	elseif (x <= 3.8e+93)
		tmp = 0.0;
	else
		tmp = 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 (x <= -0.95)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 410.0)
		tmp = 1.0;
	elseif (x <= 3.8e+93)
		tmp = 0.0;
	else
		tmp = (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, -0.95], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 410.0], 1.0, If[LessEqual[x, 3.8e+93], 0.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.94999999999999996

   1. Initial program 97.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 97.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 97.3%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 97.3%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 97.3%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 97.3%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 97.3%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 97.3%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 97.3%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Simplified 97.3%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 97.3%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]

   6. Taylor expanded in x around 0 28.4%

      \[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 28.4%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-x \cdot \left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      2. distribute-rgt-neg-in 28.4%

         \[\leadsto \frac{\left(1 + \color{blue}{x \cdot \left(-\left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      3. sub-neg 28.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      4. mul-1-neg 28.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      5. distribute-neg-in 28.4%

         \[\leadsto \frac{\left(1 + x \cdot \color{blue}{\left(\left(-1\right) + \left(--1 \cdot \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      6. metadata-eval 28.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(\color{blue}{-1} + \left(--1 \cdot \varepsilon\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      7. mul-1-neg 28.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      8. remove-double-neg 28.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \color{blue}{\varepsilon}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   8. Simplified 28.4%

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(-1 + \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   9. Taylor expanded in eps around inf 15.0%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   10. Step-by-step derivation

       1. *-commutative 15.0%

          \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   11. Simplified 15.0%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
   12. Step-by-step derivation

       1. frac-2neg 15.0%

          \[\leadsto \color{blue}{\frac{-x \cdot \varepsilon}{-2}} \]

       2. mul-1-neg 15.0%

          \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \varepsilon\right)}}{-2} \]

       3. div-inv 15.0%

          \[\leadsto \color{blue}{\left(-1 \cdot \left(x \cdot \varepsilon\right)\right) \cdot \frac{1}{-2}} \]

       4. associate-*r* 15.0%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot x\right) \cdot \varepsilon\right)} \cdot \frac{1}{-2} \]

       5. neg-mul-1 15.0%

          \[\leadsto \left(\color{blue}{\left(-x\right)} \cdot \varepsilon\right) \cdot \frac{1}{-2} \]

       6. add-sqr-sqrt 15.0%

          \[\leadsto \left(\color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \varepsilon\right) \cdot \frac{1}{-2} \]

       7. sqrt-unprod 20.0%

          \[\leadsto \left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \cdot \varepsilon\right) \cdot \frac{1}{-2} \]

       8. sqr-neg 20.0%

          \[\leadsto \left(\sqrt{\color{blue}{x \cdot x}} \cdot \varepsilon\right) \cdot \frac{1}{-2} \]

       9. sqrt-unprod 0.0%

          \[\leadsto \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \varepsilon\right) \cdot \frac{1}{-2} \]

       10. add-sqr-sqrt 49.6%

           \[\leadsto \left(\color{blue}{x} \cdot \varepsilon\right) \cdot \frac{1}{-2} \]

       11. metadata-eval 49.6%

           \[\leadsto \left(x \cdot \varepsilon\right) \cdot \frac{1}{\color{blue}{-2}} \]

       12. metadata-eval 49.6%

           \[\leadsto \left(x \cdot \varepsilon\right) \cdot \color{blue}{-0.5} \]

   13. Applied egg-rr 49.6%

       \[\leadsto \color{blue}{\left(x \cdot \varepsilon\right) \cdot -0.5} \]

   if -0.94999999999999996 < x < 410

   1. Initial program 54.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 54.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 54.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 54.5%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 54.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 54.5%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 54.5%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 54.5%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 54.5%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Simplified 54.5%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 72.0%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 410 < x < 3.7999999999999998e93

   1. Initial program 95.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 95.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 95.9%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 95.9%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 95.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 95.9%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 95.9%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 95.9%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 95.9%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 94.9%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]

   6. Taylor expanded in x around 0 14.4%

      \[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 14.4%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-x \cdot \left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      2. distribute-rgt-neg-in 14.4%

         \[\leadsto \frac{\left(1 + \color{blue}{x \cdot \left(-\left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      3. sub-neg 14.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      4. mul-1-neg 14.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      5. distribute-neg-in 14.4%

         \[\leadsto \frac{\left(1 + x \cdot \color{blue}{\left(\left(-1\right) + \left(--1 \cdot \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      6. metadata-eval 14.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(\color{blue}{-1} + \left(--1 \cdot \varepsilon\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      7. mul-1-neg 14.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      8. remove-double-neg 14.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \color{blue}{\varepsilon}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   8. Simplified 14.4%

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(-1 + \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   9. Taylor expanded in x around 0 2.5%

      \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \varepsilon\right)\right) - -1 \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}{2} \]
   10. Step-by-step derivation

       1. +-commutative 2.5%

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(-1 + \varepsilon\right) + 1\right)} - -1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}{2} \]

       2. associate--l+ 2.5%

          \[\leadsto \frac{\color{blue}{x \cdot \left(-1 + \varepsilon\right) + \left(1 - -1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}}{2} \]

       3. +-commutative 2.5%

          \[\leadsto \frac{x \cdot \color{blue}{\left(\varepsilon + -1\right)} + \left(1 - -1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}{2} \]

       4. add-sqr-sqrt 2.4%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\sqrt{-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)} \cdot \sqrt{-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}\right)}{2} \]

       5. sqrt-unprod 2.3%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\sqrt{\left(-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right) \cdot \left(-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}}\right)}{2} \]

       6. mul-1-neg 2.3%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \sqrt{\color{blue}{\left(-\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)} \cdot \left(-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}\right)}{2} \]

       7. mul-1-neg 2.3%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \sqrt{\left(-\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right) \cdot \color{blue}{\left(-\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}}\right)}{2} \]

       8. sqr-neg 2.3%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \sqrt{\color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right) \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}\right)}{2} \]

       9. sqrt-unprod 0.0%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\sqrt{1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)} \cdot \sqrt{1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)}{2} \]

       10. add-sqr-sqrt 71.4%

           \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}\right)}{2} \]

       11. +-commutative 71.4%

           \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right) + 1\right)}\right)}{2} \]

   11. Applied egg-rr 71.3%

       \[\leadsto \frac{\color{blue}{x \cdot \left(\varepsilon + -1\right) + \left(1 - \mathsf{fma}\left(x, \varepsilon + -1, 1\right)\right)}}{2} \]
   12. Step-by-step derivation

       1. associate-+r- 71.3%

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\varepsilon + -1\right) + 1\right) - \mathsf{fma}\left(x, \varepsilon + -1, 1\right)}}{2} \]

       2. fma-udef 71.3%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \varepsilon + -1, 1\right)} - \mathsf{fma}\left(x, \varepsilon + -1, 1\right)}{2} \]

       3. +-inverses 71.3%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   13. Simplified 71.3%

       \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 3.7999999999999998e93 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 100.0%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   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. Taylor expanded in x around 0 23.6%

      \[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 23.6%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-x \cdot \left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      2. distribute-rgt-neg-in 23.6%

         \[\leadsto \frac{\left(1 + \color{blue}{x \cdot \left(-\left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      3. sub-neg 23.6%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      4. mul-1-neg 23.6%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      5. distribute-neg-in 23.6%

         \[\leadsto \frac{\left(1 + x \cdot \color{blue}{\left(\left(-1\right) + \left(--1 \cdot \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      6. metadata-eval 23.6%

         \[\leadsto \frac{\left(1 + x \cdot \left(\color{blue}{-1} + \left(--1 \cdot \varepsilon\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      7. mul-1-neg 23.6%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      8. remove-double-neg 23.6%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \color{blue}{\varepsilon}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   8. Simplified 23.6%

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(-1 + \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   9. Taylor expanded in eps around inf 20.2%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   10. Step-by-step derivation

       1. *-commutative 20.2%

          \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   11. Simplified 20.2%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 410:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+93}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.8% accurate, 20.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 410:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0) (* (* x eps_m) -0.5) (if (<= x 410.0) 1.0 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 410.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 410.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 410.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (x * eps_m) * -0.5
	elif x <= 410.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 410.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 410.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 410.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 97.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 97.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 97.3%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 97.3%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 97.3%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 97.3%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 97.3%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 97.3%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 97.3%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Simplified 97.3%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 97.3%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]

   6. Taylor expanded in x around 0 28.4%

      \[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 28.4%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-x \cdot \left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      2. distribute-rgt-neg-in 28.4%

         \[\leadsto \frac{\left(1 + \color{blue}{x \cdot \left(-\left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      3. sub-neg 28.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      4. mul-1-neg 28.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      5. distribute-neg-in 28.4%

         \[\leadsto \frac{\left(1 + x \cdot \color{blue}{\left(\left(-1\right) + \left(--1 \cdot \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      6. metadata-eval 28.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(\color{blue}{-1} + \left(--1 \cdot \varepsilon\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      7. mul-1-neg 28.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      8. remove-double-neg 28.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \color{blue}{\varepsilon}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   8. Simplified 28.4%

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(-1 + \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   9. Taylor expanded in eps around inf 15.0%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   10. Step-by-step derivation

       1. *-commutative 15.0%

          \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   11. Simplified 15.0%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
   12. Step-by-step derivation

       1. frac-2neg 15.0%

          \[\leadsto \color{blue}{\frac{-x \cdot \varepsilon}{-2}} \]

       2. mul-1-neg 15.0%

          \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \varepsilon\right)}}{-2} \]

       3. div-inv 15.0%

          \[\leadsto \color{blue}{\left(-1 \cdot \left(x \cdot \varepsilon\right)\right) \cdot \frac{1}{-2}} \]

       4. associate-*r* 15.0%

          \[\leadsto \color{blue}{\left(\left(-1 \cdot x\right) \cdot \varepsilon\right)} \cdot \frac{1}{-2} \]

       5. neg-mul-1 15.0%

          \[\leadsto \left(\color{blue}{\left(-x\right)} \cdot \varepsilon\right) \cdot \frac{1}{-2} \]

       6. add-sqr-sqrt 15.0%

          \[\leadsto \left(\color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \varepsilon\right) \cdot \frac{1}{-2} \]

       7. sqrt-unprod 20.0%

          \[\leadsto \left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \cdot \varepsilon\right) \cdot \frac{1}{-2} \]

       8. sqr-neg 20.0%

          \[\leadsto \left(\sqrt{\color{blue}{x \cdot x}} \cdot \varepsilon\right) \cdot \frac{1}{-2} \]

       9. sqrt-unprod 0.0%

          \[\leadsto \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \varepsilon\right) \cdot \frac{1}{-2} \]

       10. add-sqr-sqrt 49.6%

           \[\leadsto \left(\color{blue}{x} \cdot \varepsilon\right) \cdot \frac{1}{-2} \]

       11. metadata-eval 49.6%

           \[\leadsto \left(x \cdot \varepsilon\right) \cdot \frac{1}{\color{blue}{-2}} \]

       12. metadata-eval 49.6%

           \[\leadsto \left(x \cdot \varepsilon\right) \cdot \color{blue}{-0.5} \]

   13. Applied egg-rr 49.6%

       \[\leadsto \color{blue}{\left(x \cdot \varepsilon\right) \cdot -0.5} \]

   if -1 < x < 410

   1. Initial program 54.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 54.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 54.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 54.5%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 54.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 54.5%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 54.5%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 54.5%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 54.5%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Simplified 54.5%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 72.0%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 410 < x

   1. Initial program 99.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 99.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 99.0%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 99.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 99.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 99.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 99.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 99.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 98.8%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]

   6. Taylor expanded in x around 0 21.4%

      \[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 21.4%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-x \cdot \left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      2. distribute-rgt-neg-in 21.4%

         \[\leadsto \frac{\left(1 + \color{blue}{x \cdot \left(-\left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      3. sub-neg 21.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      4. mul-1-neg 21.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      5. distribute-neg-in 21.4%

         \[\leadsto \frac{\left(1 + x \cdot \color{blue}{\left(\left(-1\right) + \left(--1 \cdot \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      6. metadata-eval 21.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(\color{blue}{-1} + \left(--1 \cdot \varepsilon\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      7. mul-1-neg 21.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      8. remove-double-neg 21.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \color{blue}{\varepsilon}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   8. Simplified 21.4%

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(-1 + \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   9. Taylor expanded in x around 0 1.3%

      \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \varepsilon\right)\right) - -1 \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}{2} \]
   10. Step-by-step derivation

       1. +-commutative 1.3%

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(-1 + \varepsilon\right) + 1\right)} - -1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}{2} \]

       2. associate--l+ 1.3%

          \[\leadsto \frac{\color{blue}{x \cdot \left(-1 + \varepsilon\right) + \left(1 - -1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}}{2} \]

       3. +-commutative 1.3%

          \[\leadsto \frac{x \cdot \color{blue}{\left(\varepsilon + -1\right)} + \left(1 - -1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}{2} \]

       4. add-sqr-sqrt 1.2%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\sqrt{-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)} \cdot \sqrt{-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}\right)}{2} \]

       5. sqrt-unprod 1.0%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\sqrt{\left(-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right) \cdot \left(-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}}\right)}{2} \]

       6. mul-1-neg 1.0%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \sqrt{\color{blue}{\left(-\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)} \cdot \left(-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}\right)}{2} \]

       7. mul-1-neg 1.0%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \sqrt{\left(-\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right) \cdot \color{blue}{\left(-\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}}\right)}{2} \]

       8. sqr-neg 1.0%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \sqrt{\color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right) \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}\right)}{2} \]

       9. sqrt-unprod 0.0%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\sqrt{1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)} \cdot \sqrt{1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)}{2} \]

       10. add-sqr-sqrt 59.0%

           \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}\right)}{2} \]

       11. +-commutative 59.0%

           \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right) + 1\right)}\right)}{2} \]

   11. Applied egg-rr 46.0%

       \[\leadsto \frac{\color{blue}{x \cdot \left(\varepsilon + -1\right) + \left(1 - \mathsf{fma}\left(x, \varepsilon + -1, 1\right)\right)}}{2} \]
   12. Step-by-step derivation

       1. associate-+r- 46.0%

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\varepsilon + -1\right) + 1\right) - \mathsf{fma}\left(x, \varepsilon + -1, 1\right)}}{2} \]

       2. fma-udef 46.0%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \varepsilon + -1, 1\right)} - \mathsf{fma}\left(x, \varepsilon + -1, 1\right)}{2} \]

       3. +-inverses 46.6%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   13. Simplified 46.6%

       \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 410:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.9% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 410:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 410.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 410.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 410.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 410.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 410.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 410.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 410.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 410.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 410

   1. Initial program 63.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 63.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 63.0%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 63.0%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 63.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 63.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 63.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 63.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 63.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Simplified 63.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 58.3%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 410 < x

   1. Initial program 99.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 99.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 99.0%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 99.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 99.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 99.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 99.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 99.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 98.8%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]

   6. Taylor expanded in x around 0 21.4%

      \[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 21.4%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-x \cdot \left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      2. distribute-rgt-neg-in 21.4%

         \[\leadsto \frac{\left(1 + \color{blue}{x \cdot \left(-\left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      3. sub-neg 21.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      4. mul-1-neg 21.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-\left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      5. distribute-neg-in 21.4%

         \[\leadsto \frac{\left(1 + x \cdot \color{blue}{\left(\left(-1\right) + \left(--1 \cdot \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      6. metadata-eval 21.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(\color{blue}{-1} + \left(--1 \cdot \varepsilon\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      7. mul-1-neg 21.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      8. remove-double-neg 21.4%

         \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \color{blue}{\varepsilon}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   8. Simplified 21.4%

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(-1 + \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   9. Taylor expanded in x around 0 1.3%

      \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \varepsilon\right)\right) - -1 \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}{2} \]
   10. Step-by-step derivation

       1. +-commutative 1.3%

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(-1 + \varepsilon\right) + 1\right)} - -1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}{2} \]

       2. associate--l+ 1.3%

          \[\leadsto \frac{\color{blue}{x \cdot \left(-1 + \varepsilon\right) + \left(1 - -1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}}{2} \]

       3. +-commutative 1.3%

          \[\leadsto \frac{x \cdot \color{blue}{\left(\varepsilon + -1\right)} + \left(1 - -1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}{2} \]

       4. add-sqr-sqrt 1.2%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\sqrt{-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)} \cdot \sqrt{-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}\right)}{2} \]

       5. sqrt-unprod 1.0%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\sqrt{\left(-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right) \cdot \left(-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}}\right)}{2} \]

       6. mul-1-neg 1.0%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \sqrt{\color{blue}{\left(-\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)} \cdot \left(-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}\right)}{2} \]

       7. mul-1-neg 1.0%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \sqrt{\left(-\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right) \cdot \color{blue}{\left(-\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}}\right)}{2} \]

       8. sqr-neg 1.0%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \sqrt{\color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right) \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}\right)}{2} \]

       9. sqrt-unprod 0.0%

          \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\sqrt{1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)} \cdot \sqrt{1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)}{2} \]

       10. add-sqr-sqrt 59.0%

           \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}\right)}{2} \]

       11. +-commutative 59.0%

           \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right) + 1\right)}\right)}{2} \]

   11. Applied egg-rr 46.0%

       \[\leadsto \frac{\color{blue}{x \cdot \left(\varepsilon + -1\right) + \left(1 - \mathsf{fma}\left(x, \varepsilon + -1, 1\right)\right)}}{2} \]
   12. Step-by-step derivation

       1. associate-+r- 46.0%

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\varepsilon + -1\right) + 1\right) - \mathsf{fma}\left(x, \varepsilon + -1, 1\right)}}{2} \]

       2. fma-udef 46.0%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \varepsilon + -1, 1\right)} - \mathsf{fma}\left(x, \varepsilon + -1, 1\right)}{2} \]

       3. +-inverses 46.6%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   13. Simplified 46.6%

       \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 410:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 16.1% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 0.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 72.8%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Step-by-step derivation

   1. fma-neg 72.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

   2. /-rgt-identity 72.8%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

   3. fma-neg 72.8%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   4. /-rgt-identity 72.8%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   5. distribute-rgt-neg-in 72.8%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   6. sub-neg 72.8%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   7. metadata-eval 72.8%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   8. distribute-rgt-neg-in 72.8%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

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 98.9%

   \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]

6. Taylor expanded in x around 0 59.7%

   \[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]
7. Step-by-step derivation

   1. mul-1-neg 59.7%

      \[\leadsto \frac{\left(1 + \color{blue}{\left(-x \cdot \left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   2. distribute-rgt-neg-in 59.7%

      \[\leadsto \frac{\left(1 + \color{blue}{x \cdot \left(-\left(1 - \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   3. sub-neg 59.7%

      \[\leadsto \frac{\left(1 + x \cdot \left(-\color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   4. mul-1-neg 59.7%

      \[\leadsto \frac{\left(1 + x \cdot \left(-\left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   5. distribute-neg-in 59.7%

      \[\leadsto \frac{\left(1 + x \cdot \color{blue}{\left(\left(-1\right) + \left(--1 \cdot \varepsilon\right)\right)}\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   6. metadata-eval 59.7%

      \[\leadsto \frac{\left(1 + x \cdot \left(\color{blue}{-1} + \left(--1 \cdot \varepsilon\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   7. mul-1-neg 59.7%

      \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

   8. remove-double-neg 59.7%

      \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \color{blue}{\varepsilon}\right)\right) - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

8. Simplified 59.7%

   \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(-1 + \varepsilon\right)\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

9. Taylor expanded in x around 0 41.9%

   \[\leadsto \frac{\left(1 + x \cdot \left(-1 + \varepsilon\right)\right) - -1 \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}{2} \]
10. Step-by-step derivation

    1. +-commutative 41.9%

       \[\leadsto \frac{\color{blue}{\left(x \cdot \left(-1 + \varepsilon\right) + 1\right)} - -1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}{2} \]

    2. associate--l+ 41.9%

       \[\leadsto \frac{\color{blue}{x \cdot \left(-1 + \varepsilon\right) + \left(1 - -1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}}{2} \]

    3. +-commutative 41.9%

       \[\leadsto \frac{x \cdot \color{blue}{\left(\varepsilon + -1\right)} + \left(1 - -1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}{2} \]

    4. add-sqr-sqrt 0.6%

       \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\sqrt{-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)} \cdot \sqrt{-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}\right)}{2} \]

    5. sqrt-unprod 2.0%

       \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\sqrt{\left(-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right) \cdot \left(-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}}\right)}{2} \]

    6. mul-1-neg 2.0%

       \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \sqrt{\color{blue}{\left(-\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)} \cdot \left(-1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}\right)}{2} \]

    7. mul-1-neg 2.0%

       \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \sqrt{\left(-\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right) \cdot \color{blue}{\left(-\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)}}\right)}{2} \]

    8. sqr-neg 2.0%

       \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \sqrt{\color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right) \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}\right)}{2} \]

    9. sqrt-unprod 1.7%

       \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\sqrt{1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)} \cdot \sqrt{1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)}{2} \]

    10. add-sqr-sqrt 20.2%

        \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}\right)}{2} \]

    11. +-commutative 20.2%

        \[\leadsto \frac{x \cdot \left(\varepsilon + -1\right) + \left(1 - \color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right) + 1\right)}\right)}{2} \]

11. Applied egg-rr 14.4%

    \[\leadsto \frac{\color{blue}{x \cdot \left(\varepsilon + -1\right) + \left(1 - \mathsf{fma}\left(x, \varepsilon + -1, 1\right)\right)}}{2} \]
12. Step-by-step derivation

    1. associate-+r- 14.2%

       \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\varepsilon + -1\right) + 1\right) - \mathsf{fma}\left(x, \varepsilon + -1, 1\right)}}{2} \]

    2. fma-udef 14.2%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \varepsilon + -1, 1\right)} - \mathsf{fma}\left(x, \varepsilon + -1, 1\right)}{2} \]

    3. +-inverses 14.5%

       \[\leadsto \frac{\color{blue}{0}}{2} \]

13. Simplified 14.5%

    \[\leadsto \frac{\color{blue}{0}}{2} \]

14. Final simplification 14.5%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))