NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.8% → 99.8%

Time: 19.9s

Alternatives: 17

Speedup: 1.7×

• 37.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.99999999999999997e77
  2. eps = -2e-231

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \mathbf{if}\;eps\_m \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{t\_0 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{e^{x + eps\_m \cdot x}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (+ x 1.0) (exp (- x)))))
   (if (<= eps_m 2e-40)
     (/ (+ t_0 t_0) 2.0)
     (/ (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (exp (+ x (* eps_m x))))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x + 1.0) * exp(-x);
	double tmp;
	if (eps_m <= 2e-40) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / exp((x + (eps_m * x))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) * exp(-x)
    if (eps_m <= 2d-40) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 / exp((x + (eps_m * x))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (x + 1.0) * Math.exp(-x);
	double tmp;
	if (eps_m <= 2e-40) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 / Math.exp((x + (eps_m * x))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x + 1.0) * math.exp(-x)
	tmp = 0
	if eps_m <= 2e-40:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 / math.exp((x + (eps_m * x))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x + 1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps_m <= 2e-40)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / exp(Float64(x + Float64(eps_m * x))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x + 1.0) * exp(-x);
	tmp = 0.0;
	if (eps_m <= 2e-40)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / exp((x + (eps_m * x))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 2e-40], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[N[(x + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.9999999999999999e-40

   1. Initial program 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in eps around 0 73.3%

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

      1. distribute-rgt1-in 73.3%

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

      2. mul-1-neg 73.3%

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

      3. distribute-lft-out 73.3%

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

      4. distribute-rgt1-in 73.9%

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

      5. mul-1-neg 73.9%

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

   7. Simplified 73.9%

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

   if 1.9999999999999999e-40 < eps

   1. Initial program 98.8%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in eps around inf 100.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{e^{x + \varepsilon \cdot x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps\_m \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\frac{t\_0 \cdot \left(x + 2\right) + x \cdot t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{e^{x + eps\_m \cdot x}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 2e-38)
     (/ (+ (* t_0 (+ x 2.0)) (* x t_0)) 2.0)
     (/ (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (exp (+ x (* eps_m x))))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 2e-38) {
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / exp((x + (eps_m * x))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 2d-38) then
        tmp = ((t_0 * (x + 2.0d0)) + (x * t_0)) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 / exp((x + (eps_m * x))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 2e-38) {
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 / Math.exp((x + (eps_m * x))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 2e-38:
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 / math.exp((x + (eps_m * x))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 2e-38)
		tmp = Float64(Float64(Float64(t_0 * Float64(x + 2.0)) + Float64(x * t_0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / exp(Float64(x + Float64(eps_m * x))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 2e-38)
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / exp((x + (eps_m * x))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 2e-38], N[(N[(N[(t$95$0 * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[N[(x + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.9999999999999999e-38

   1. Initial program 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in eps around 0 73.3%

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

      1. associate--r+ 73.3%

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

      2. associate-*r* 73.3%

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

      3. mul-1-neg 73.3%

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

      4. cancel-sign-sub 73.3%

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

      5. distribute-rgt1-in 73.3%

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

      6. distribute-rgt-out-- 73.9%

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

      7. mul-1-neg 73.9%

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

      8. mul-1-neg 73.9%

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

   7. Simplified 73.9%

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

   8. Taylor expanded in x around 0 73.9%

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

      1. +-commutative 73.9%

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

   10. Simplified 73.9%

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

   if 1.9999999999999999e-38 < eps

   1. Initial program 98.8%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in eps around inf 100.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(x + 2\right) + x \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{e^{x + \varepsilon \cdot x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps\_m \leq 10^{-37}:\\ \;\;\;\;\frac{t\_0 \cdot \left(x + 2\right) + x \cdot t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 1e-37)
     (/ (+ (* t_0 (+ x 2.0)) (* x t_0)) 2.0)
     (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* eps_m (- x)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 1e-37) {
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + exp((eps_m * -x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 1d-37) then
        tmp = ((t_0 * (x + 2.0d0)) + (x * t_0)) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + exp((eps_m * -x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 1e-37) {
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + Math.exp((eps_m * -x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 1e-37:
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + math.exp((eps_m * -x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 1e-37)
		tmp = Float64(Float64(Float64(t_0 * Float64(x + 2.0)) + Float64(x * t_0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 1e-37)
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + exp((eps_m * -x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 1e-37], N[(N[(N[(t$95$0 * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.00000000000000007e-37

   1. Initial program 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in eps around 0 73.3%

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

      1. associate--r+ 73.3%

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

      2. associate-*r* 73.3%

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

      3. mul-1-neg 73.3%

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

      4. cancel-sign-sub 73.3%

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

      5. distribute-rgt1-in 73.3%

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

      6. distribute-rgt-out-- 73.9%

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

      7. mul-1-neg 73.9%

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

      8. mul-1-neg 73.9%

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

   7. Simplified 73.9%

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

   8. Taylor expanded in x around 0 73.9%

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

      1. +-commutative 73.9%

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

   10. Simplified 73.9%

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

   if 1.00000000000000007e-37 < eps

   1. Initial program 98.8%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. rec-exp 100.0%

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

       2. *-commutative 100.0%

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

       3. distribute-rgt-neg-in 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-37}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(x + 2\right) + x \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 10^{-37}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1e-37)
   (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* eps_m (- x)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1e-37) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + exp((eps_m * -x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1d-37) then
        tmp = ((eps_m * (exp(-x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + exp((eps_m * -x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1e-37) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + Math.exp((eps_m * -x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1e-37:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + math.exp((eps_m * -x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1e-37)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1e-37)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + exp((eps_m * -x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1e-37], N[(N[(N[(eps$95$m * N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.00000000000000007e-37

   1. Initial program 60.0%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in eps around 0 34.0%

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

   7. Simplified 73.9%

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

   if 1.00000000000000007e-37 < eps

   1. Initial program 98.8%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. rec-exp 100.0%

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

       2. *-commutative 100.0%

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

       3. distribute-rgt-neg-in 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-37}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{elif}\;eps\_m \leq 4 \cdot 10^{+145} \lor \neg \left(eps\_m \leq 6.8 \cdot 10^{+247}\right):\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{eps\_m} + x \cdot \left(\left(-1 + \frac{-1}{eps\_m}\right) \cdot \left(1 - eps\_m\right)\right)\right)\right) + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(\frac{-1}{eps\_m} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \left(1 - eps\_m \cdot x\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2.35e-5)
   (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   (if (or (<= eps_m 4e+145) (not (<= eps_m 6.8e+247)))
     (/
      (+
       (+
        1.0
        (+ (/ 1.0 eps_m) (* x (* (+ -1.0 (/ -1.0 eps_m)) (- 1.0 eps_m)))))
       (* (exp (* x (- -1.0 eps_m))) (- (/ -1.0 eps_m) -1.0)))
      2.0)
     (/ (+ (exp (* x (+ eps_m -1.0))) (- 1.0 (* eps_m x))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2.35e-5) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else if ((eps_m <= 4e+145) || !(eps_m <= 6.8e+247)) {
		tmp = ((1.0 + ((1.0 / eps_m) + (x * ((-1.0 + (-1.0 / eps_m)) * (1.0 - eps_m))))) + (exp((x * (-1.0 - eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 - (eps_m * x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 2.35d-5) then
        tmp = ((eps_m * (exp(-x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else if ((eps_m <= 4d+145) .or. (.not. (eps_m <= 6.8d+247))) then
        tmp = ((1.0d0 + ((1.0d0 / eps_m) + (x * (((-1.0d0) + ((-1.0d0) / eps_m)) * (1.0d0 - eps_m))))) + (exp((x * ((-1.0d0) - eps_m))) * (((-1.0d0) / eps_m) - (-1.0d0)))) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 - (eps_m * x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2.35e-5) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else if ((eps_m <= 4e+145) || !(eps_m <= 6.8e+247)) {
		tmp = ((1.0 + ((1.0 / eps_m) + (x * ((-1.0 + (-1.0 / eps_m)) * (1.0 - eps_m))))) + (Math.exp((x * (-1.0 - eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 - (eps_m * x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 2.35e-5:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	elif (eps_m <= 4e+145) or not (eps_m <= 6.8e+247):
		tmp = ((1.0 + ((1.0 / eps_m) + (x * ((-1.0 + (-1.0 / eps_m)) * (1.0 - eps_m))))) + (math.exp((x * (-1.0 - eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 - (eps_m * x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 2.35e-5)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	elseif ((eps_m <= 4e+145) || !(eps_m <= 6.8e+247))
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(1.0 / eps_m) + Float64(x * Float64(Float64(-1.0 + Float64(-1.0 / eps_m)) * Float64(1.0 - eps_m))))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(Float64(-1.0 / eps_m) - -1.0))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 - Float64(eps_m * x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 2.35e-5)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	elseif ((eps_m <= 4e+145) || ~((eps_m <= 6.8e+247)))
		tmp = ((1.0 + ((1.0 / eps_m) + (x * ((-1.0 + (-1.0 / eps_m)) * (1.0 - eps_m))))) + (exp((x * (-1.0 - eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 - (eps_m * x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 2.35e-5], N[(N[(N[(eps$95$m * N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[eps$95$m, 4e+145], N[Not[LessEqual[eps$95$m, 6.8e+247]], $MachinePrecision]], N[(N[(N[(1.0 + N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(x * N[(N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(-1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 2.34999999999999986e-5

   1. Initial program 60.8%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in eps around 0 34.9%

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

   7. Simplified 74.7%

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

   if 2.34999999999999986e-5 < eps < 4e145 or 6.79999999999999961e247 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 78.9%

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

      1. +-commutative 78.9%

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

      2. mul-1-neg 78.9%

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

      3. distribute-rgt-neg-in 78.9%

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

      4. *-commutative 78.9%

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

      5. distribute-rgt-neg-in 78.9%

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

      6. distribute-neg-in 78.9%

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

      7. metadata-eval 78.9%

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

      8. distribute-neg-frac 78.9%

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

      9. metadata-eval 78.9%

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

   7. Simplified 78.9%

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

   if 4e145 < eps < 6.79999999999999961e247

   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 80.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 80.0%

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

       1. mul-1-neg 80.0%

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

       2. *-commutative 80.0%

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

       3. unsub-neg 80.0%

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

       4. *-commutative 80.0%

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

   11. Simplified 80.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{+145} \lor \neg \left(\varepsilon \leq 6.8 \cdot 10^{+247}\right):\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{\varepsilon} + x \cdot \left(\left(-1 + \frac{-1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-275}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 3600000000000 \lor \neg \left(x \leq 6.2 \cdot 10^{+37} \lor \neg \left(x \leq 5.2 \cdot 10^{+91}\right) \land x \leq 9 \cdot 10^{+271}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-275)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (or (<= x 3600000000000.0)
           (not (or (<= x 6.2e+37) (and (not (<= x 5.2e+91)) (<= x 9e+271)))))
     (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-275) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if ((x <= 3600000000000.0) || !((x <= 6.2e+37) || (!(x <= 5.2e+91) && (x <= 9e+271)))) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} 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 <= (-2d-275)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if ((x <= 3600000000000.0d0) .or. (.not. (x <= 6.2d+37) .or. (.not. (x <= 5.2d+91)) .and. (x <= 9d+271))) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    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 <= -2e-275) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if ((x <= 3600000000000.0) || !((x <= 6.2e+37) || (!(x <= 5.2e+91) && (x <= 9e+271)))) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-275:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif (x <= 3600000000000.0) or not ((x <= 6.2e+37) or (not (x <= 5.2e+91) and (x <= 9e+271))):
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-275)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif ((x <= 3600000000000.0) || !((x <= 6.2e+37) || (!(x <= 5.2e+91) && (x <= 9e+271))))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	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 <= -2e-275)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif ((x <= 3600000000000.0) || ~(((x <= 6.2e+37) || (~((x <= 5.2e+91)) && (x <= 9e+271)))))
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-275], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3600000000000.0], N[Not[Or[LessEqual[x, 6.2e+37], And[N[Not[LessEqual[x, 5.2e+91]], $MachinePrecision], LessEqual[x, 9e+271]]]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999987e-275

   1. Initial program 64.8%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in eps around inf 97.0%

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

   6. Taylor expanded in eps around inf 97.0%

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

      1. *-commutative 97.0%

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

   8. Simplified 97.0%

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

   9. Taylor expanded in eps around 0 80.8%

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

       1. neg-mul-1 80.8%

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

   11. Simplified 80.8%

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

   if -1.99999999999999987e-275 < x < 3.6e12 or 6.2000000000000004e37 < x < 5.2000000000000001e91 or 8.9999999999999994e271 < x

   1. Initial program 66.8%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in eps around inf 97.7%

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

   6. Taylor expanded in x around 0 73.7%

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

   if 3.6e12 < x < 6.2000000000000004e37 or 5.2000000000000001e91 < x < 8.9999999999999994e271

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 72.2%

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

      1. mul-1-neg 72.2%

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

      2. mul-1-neg 72.2%

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

      3. rec-exp 72.2%

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

      4. sub-neg 72.2%

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

      5. div-sub 72.2%

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

      6. mul-1-neg 72.2%

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

      7. rec-exp 72.2%

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

      8. +-inverses 72.2%

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

   7. Simplified 72.2%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-275}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 3600000000000 \lor \neg \left(x \leq 6.2 \cdot 10^{+37} \lor \neg \left(x \leq 5.2 \cdot 10^{+91}\right) \land x \leq 9 \cdot 10^{+271}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-275}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 2800000000000 \lor \neg \left(x \leq 6 \cdot 10^{+37}\right) \land \left(x \leq 1.9 \cdot 10^{+91} \lor \neg \left(x \leq 1.6 \cdot 10^{+271}\right)\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-275)
   (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
   (if (or (<= x 2800000000000.0)
           (and (not (<= x 6e+37)) (or (<= x 1.9e+91) (not (<= x 1.6e+271)))))
     (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-275) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if ((x <= 2800000000000.0) || (!(x <= 6e+37) && ((x <= 1.9e+91) || !(x <= 1.6e+271)))) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} 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 <= (-2d-275)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if ((x <= 2800000000000.0d0) .or. (.not. (x <= 6d+37)) .and. (x <= 1.9d+91) .or. (.not. (x <= 1.6d+271))) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    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 <= -2e-275) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if ((x <= 2800000000000.0) || (!(x <= 6e+37) && ((x <= 1.9e+91) || !(x <= 1.6e+271)))) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-275:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif (x <= 2800000000000.0) or (not (x <= 6e+37) and ((x <= 1.9e+91) or not (x <= 1.6e+271))):
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-275)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif ((x <= 2800000000000.0) || (!(x <= 6e+37) && ((x <= 1.9e+91) || !(x <= 1.6e+271))))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	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 <= -2e-275)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif ((x <= 2800000000000.0) || (~((x <= 6e+37)) && ((x <= 1.9e+91) || ~((x <= 1.6e+271)))))
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-275], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 2800000000000.0], And[N[Not[LessEqual[x, 6e+37]], $MachinePrecision], Or[LessEqual[x, 1.9e+91], N[Not[LessEqual[x, 1.6e+271]], $MachinePrecision]]]], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999987e-275

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in x around 0 43.0%

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

   6. Taylor expanded in eps around inf 75.1%

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

      1. sub-neg 75.1%

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

      2. mul-1-neg 75.1%

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

      3. remove-double-neg 75.1%

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

      4. associate-*r* 75.1%

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

      5. exp-prod 69.4%

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

      6. remove-double-neg 69.4%

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

      7. mul-1-neg 69.4%

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

      8. sub-neg 69.4%

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

      9. exp-prod 75.1%

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

      10. associate-*r* 75.1%

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

      11. remove-double-neg 75.1%

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

      12. mul-1-neg 75.1%

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

      13. sub-neg 75.1%

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

      14. mul-1-neg 75.1%

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

      15. mul-1-neg 75.1%

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

      16. distribute-rgt-neg-in 75.1%

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

   8. Simplified 75.1%

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

   9. Taylor expanded in eps around inf 75.6%

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

       1. mul-1-neg 75.6%

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

       2. *-commutative 75.6%

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

       3. distribute-rgt-neg-in 75.6%

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

   11. Simplified 75.6%

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

   if -1.99999999999999987e-275 < x < 2.8e12 or 6.00000000000000043e37 < x < 1.8999999999999999e91 or 1.6000000000000001e271 < x

   1. Initial program 66.8%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in eps around inf 97.7%

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

   6. Taylor expanded in x around 0 73.7%

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

   if 2.8e12 < x < 6.00000000000000043e37 or 1.8999999999999999e91 < x < 1.6000000000000001e271

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 72.2%

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

      1. mul-1-neg 72.2%

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

      2. mul-1-neg 72.2%

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

      3. rec-exp 72.2%

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

      4. sub-neg 72.2%

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

      5. div-sub 72.2%

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

      6. mul-1-neg 72.2%

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

      7. rec-exp 72.2%

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

      8. +-inverses 72.2%

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

   7. Simplified 72.2%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-275}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 2800000000000 \lor \neg \left(x \leq 6 \cdot 10^{+37}\right) \land \left(x \leq 1.9 \cdot 10^{+91} \lor \neg \left(x \leq 1.6 \cdot 10^{+271}\right)\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-275}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{t\_0 + \left(1 - eps\_m \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+37}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+91} \lor \neg \left(x \leq 7.6 \cdot 10^{+270}\right):\\ \;\;\;\;\frac{1 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x -2e-275)
     (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
     (if (<= x 4.3e+15)
       (/ (+ t_0 (- 1.0 (* eps_m x))) 2.0)
       (if (<= x 6.5e+37)
         0.0
         (if (or (<= x 3.5e+91) (not (<= x 7.6e+270)))
           (/ (+ 1.0 t_0) 2.0)
           0.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -2e-275) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if (x <= 4.3e+15) {
		tmp = (t_0 + (1.0 - (eps_m * x))) / 2.0;
	} else if (x <= 6.5e+37) {
		tmp = 0.0;
	} else if ((x <= 3.5e+91) || !(x <= 7.6e+270)) {
		tmp = (1.0 + t_0) / 2.0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = exp((x * (eps_m + (-1.0d0))))
    if (x <= (-2d-275)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if (x <= 4.3d+15) then
        tmp = (t_0 + (1.0d0 - (eps_m * x))) / 2.0d0
    else if (x <= 6.5d+37) then
        tmp = 0.0d0
    else if ((x <= 3.5d+91) .or. (.not. (x <= 7.6d+270))) then
        tmp = (1.0d0 + t_0) / 2.0d0
    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 t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -2e-275) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if (x <= 4.3e+15) {
		tmp = (t_0 + (1.0 - (eps_m * x))) / 2.0;
	} else if (x <= 6.5e+37) {
		tmp = 0.0;
	} else if ((x <= 3.5e+91) || !(x <= 7.6e+270)) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= -2e-275:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif x <= 4.3e+15:
		tmp = (t_0 + (1.0 - (eps_m * x))) / 2.0
	elif x <= 6.5e+37:
		tmp = 0.0
	elif (x <= 3.5e+91) or not (x <= 7.6e+270):
		tmp = (1.0 + t_0) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= -2e-275)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif (x <= 4.3e+15)
		tmp = Float64(Float64(t_0 + Float64(1.0 - Float64(eps_m * x))) / 2.0);
	elseif (x <= 6.5e+37)
		tmp = 0.0;
	elseif ((x <= 3.5e+91) || !(x <= 7.6e+270))
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= -2e-275)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif (x <= 4.3e+15)
		tmp = (t_0 + (1.0 - (eps_m * x))) / 2.0;
	elseif (x <= 6.5e+37)
		tmp = 0.0;
	elseif ((x <= 3.5e+91) || ~((x <= 7.6e+270)))
		tmp = (1.0 + t_0) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2e-275], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.3e+15], N[(N[(t$95$0 + N[(1.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.5e+37], 0.0, If[Or[LessEqual[x, 3.5e+91], N[Not[LessEqual[x, 7.6e+270]], $MachinePrecision]], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.99999999999999987e-275

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in x around 0 43.0%

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

   6. Taylor expanded in eps around inf 75.1%

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

      1. sub-neg 75.1%

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

      2. mul-1-neg 75.1%

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

      3. remove-double-neg 75.1%

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

      4. associate-*r* 75.1%

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

      5. exp-prod 69.4%

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

      6. remove-double-neg 69.4%

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

      7. mul-1-neg 69.4%

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

      8. sub-neg 69.4%

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

      9. exp-prod 75.1%

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

      10. associate-*r* 75.1%

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

      11. remove-double-neg 75.1%

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

      12. mul-1-neg 75.1%

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

      13. sub-neg 75.1%

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

      14. mul-1-neg 75.1%

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

      15. mul-1-neg 75.1%

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

      16. distribute-rgt-neg-in 75.1%

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

   8. Simplified 75.1%

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

   9. Taylor expanded in eps around inf 75.6%

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

       1. mul-1-neg 75.6%

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

       2. *-commutative 75.6%

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

       3. distribute-rgt-neg-in 75.6%

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

   11. Simplified 75.6%

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

   if -1.99999999999999987e-275 < x < 4.3e15

   1. Initial program 56.3%

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

   3. Simplified 48.2%

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

   5. Taylor expanded in eps around inf 96.9%

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

   6. Taylor expanded in eps around inf 95.8%

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

      1. *-commutative 95.8%

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

   8. Simplified 95.8%

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

   9. Taylor expanded in x around 0 85.8%

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

       1. mul-1-neg 85.8%

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

       2. *-commutative 85.8%

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

       3. unsub-neg 85.8%

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

       4. *-commutative 85.8%

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

   11. Simplified 85.8%

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

   if 4.3e15 < x < 6.4999999999999998e37 or 3.50000000000000001e91 < x < 7.60000000000000036e270

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 72.2%

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

      1. mul-1-neg 72.2%

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

      2. mul-1-neg 72.2%

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

      3. rec-exp 72.2%

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

      4. sub-neg 72.2%

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

      5. div-sub 72.2%

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

      6. mul-1-neg 72.2%

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

      7. rec-exp 72.2%

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

      8. +-inverses 72.2%

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

   7. Simplified 72.2%

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

   if 6.4999999999999998e37 < x < 3.50000000000000001e91 or 7.60000000000000036e270 < 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 inf 100.0%

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

   6. Taylor expanded in x around 0 36.7%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-275}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 - \varepsilon \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+37}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+91} \lor \neg \left(x \leq 7.6 \cdot 10^{+270}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+37}:\\ \;\;\;\;\frac{x \cdot \frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+91}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot e^{x}}{2}\\ \mathbf{elif}\;x \leq 10^{+271}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.05)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 6e+37)
     (/ (* x (/ 2.0 (exp x))) 2.0)
     (if (<= x 1.6e+91)
       (/ (* (* x 2.0) (exp x)) 2.0)
       (if (<= x 1e+271) 0.0 (/ (* x (+ 2.0 (* x (+ x 2.0)))) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.05) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 6e+37) {
		tmp = (x * (2.0 / exp(x))) / 2.0;
	} else if (x <= 1.6e+91) {
		tmp = ((x * 2.0) * exp(x)) / 2.0;
	} else if (x <= 1e+271) {
		tmp = 0.0;
	} else {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 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) :: tmp
    if (x <= 2.05d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 6d+37) then
        tmp = (x * (2.0d0 / exp(x))) / 2.0d0
    else if (x <= 1.6d+91) then
        tmp = ((x * 2.0d0) * exp(x)) / 2.0d0
    else if (x <= 1d+271) then
        tmp = 0.0d0
    else
        tmp = (x * (2.0d0 + (x * (x + 2.0d0)))) / 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 <= 2.05) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 6e+37) {
		tmp = (x * (2.0 / Math.exp(x))) / 2.0;
	} else if (x <= 1.6e+91) {
		tmp = ((x * 2.0) * Math.exp(x)) / 2.0;
	} else if (x <= 1e+271) {
		tmp = 0.0;
	} else {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.05:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 6e+37:
		tmp = (x * (2.0 / math.exp(x))) / 2.0
	elif x <= 1.6e+91:
		tmp = ((x * 2.0) * math.exp(x)) / 2.0
	elif x <= 1e+271:
		tmp = 0.0
	else:
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.05)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 6e+37)
		tmp = Float64(Float64(x * Float64(2.0 / exp(x))) / 2.0);
	elseif (x <= 1.6e+91)
		tmp = Float64(Float64(Float64(x * 2.0) * exp(x)) / 2.0);
	elseif (x <= 1e+271)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(x + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.05)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 6e+37)
		tmp = (x * (2.0 / exp(x))) / 2.0;
	elseif (x <= 1.6e+91)
		tmp = ((x * 2.0) * exp(x)) / 2.0;
	elseif (x <= 1e+271)
		tmp = 0.0;
	else
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.05], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6e+37], N[(N[(x * N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.6e+91], N[(N[(N[(x * 2.0), $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+271], 0.0, N[(N[(x * N[(2.0 + N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 2.0499999999999998

   1. Initial program 60.5%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in eps around inf 97.3%

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

   6. Taylor expanded in eps around inf 97.4%

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

      1. *-commutative 97.4%

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

   8. Simplified 97.4%

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

   9. Taylor expanded in eps around 0 80.3%

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

       1. neg-mul-1 80.3%

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

   11. Simplified 80.3%

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

   if 2.0499999999999998 < x < 6.00000000000000043e37

   1. Initial program 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in eps around 0 72.0%

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

      1. associate--r+ 72.0%

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

      2. associate-*r* 72.0%

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

      3. mul-1-neg 72.0%

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

      4. cancel-sign-sub 72.0%

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

      5. distribute-rgt1-in 72.0%

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

      6. distribute-rgt-out-- 72.0%

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

      7. mul-1-neg 72.0%

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

      8. mul-1-neg 72.0%

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

   7. Simplified 72.0%

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

   8. Taylor expanded in x around inf 66.7%

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

      1. rec-exp 66.7%

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

      2. associate-*r* 66.7%

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

      3. *-commutative 66.7%

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

      4. rec-exp 66.7%

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

      5. neg-mul-1 66.7%

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

      6. associate-*r* 66.7%

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

      7. neg-mul-1 66.7%

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

   10. Simplified 66.7%

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

   11. Taylor expanded in x around inf 66.7%

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

       1. associate-*r* 66.7%

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

       2. *-commutative 66.7%

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

       3. exp-neg 66.7%

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

       4. associate-/l* 66.7%

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

       5. *-rgt-identity 66.7%

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

       6. associate-/l* 66.7%

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

   13. Simplified 66.7%

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

   if 6.00000000000000043e37 < x < 1.59999999999999995e91

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 27.8%

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

      1. associate--r+ 27.8%

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

      2. associate-*r* 27.8%

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

      3. mul-1-neg 27.8%

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

      4. cancel-sign-sub 27.8%

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

      5. distribute-rgt1-in 27.8%

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

      6. distribute-rgt-out-- 27.8%

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

      7. mul-1-neg 27.8%

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

      8. mul-1-neg 27.8%

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

   7. Simplified 27.8%

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

   8. Taylor expanded in x around inf 27.8%

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

      1. rec-exp 27.8%

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

      2. associate-*r* 27.8%

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

      3. *-commutative 27.8%

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

      4. rec-exp 27.8%

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

      5. neg-mul-1 27.8%

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

      6. associate-*r* 27.8%

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

      7. neg-mul-1 27.8%

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

   10. Simplified 27.8%

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

       1. pow1 27.8%

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

       2. associate-*r* 27.8%

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

       3. *-commutative 27.8%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 73.8%

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

       6. sqr-neg 73.8%

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

       7. sqrt-unprod 73.8%

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

       8. add-sqr-sqrt 73.8%

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

   12. Applied egg-rr 73.8%

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

       1. unpow1 73.8%

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

   14. Simplified 73.8%

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

   if 1.59999999999999995e91 < x < 9.99999999999999953e270

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 68.1%

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

      1. mul-1-neg 68.1%

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

      2. mul-1-neg 68.1%

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

      3. rec-exp 68.1%

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

      4. sub-neg 68.1%

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

      5. div-sub 68.1%

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

      6. mul-1-neg 68.1%

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

      7. rec-exp 68.1%

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

      8. +-inverses 68.1%

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

   7. Simplified 68.1%

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

   if 9.99999999999999953e270 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 28.4%

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

      1. associate--r+ 28.4%

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

      2. associate-*r* 28.4%

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

      3. mul-1-neg 28.4%

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

      4. cancel-sign-sub 28.4%

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

      5. distribute-rgt1-in 28.4%

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

      6. distribute-rgt-out-- 28.4%

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

      7. mul-1-neg 28.4%

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

      8. mul-1-neg 28.4%

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

   7. Simplified 28.4%

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

   8. Taylor expanded in x around inf 28.4%

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

      1. rec-exp 28.4%

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

      2. associate-*r* 28.4%

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

      3. *-commutative 28.4%

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

      4. rec-exp 28.4%

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

      5. neg-mul-1 28.4%

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

      6. associate-*r* 28.4%

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

      7. neg-mul-1 28.4%

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

   10. Simplified 28.4%

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

       1. pow1 28.4%

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

       2. associate-*r* 28.4%

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

       3. *-commutative 28.4%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 73.2%

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

       6. sqr-neg 73.2%

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

       7. sqrt-unprod 73.2%

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

       8. add-sqr-sqrt 73.2%

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

   12. Applied egg-rr 73.2%

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

       1. unpow1 73.2%

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

   14. Simplified 73.2%

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

   15. Taylor expanded in x around 0 73.2%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+37}:\\ \;\;\;\;\frac{x \cdot \frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+91}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot e^{x}}{2}\\ \mathbf{elif}\;x \leq 10^{+271}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+270}:\\ \;\;\;\;\frac{x \cdot \frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.05)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 6.5e+270)
     (/ (* x (/ 2.0 (exp x))) 2.0)
     (/ (* x (+ 2.0 (* x (+ x 2.0)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.05) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 6.5e+270) {
		tmp = (x * (2.0 / exp(x))) / 2.0;
	} else {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 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) :: tmp
    if (x <= 2.05d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 6.5d+270) then
        tmp = (x * (2.0d0 / exp(x))) / 2.0d0
    else
        tmp = (x * (2.0d0 + (x * (x + 2.0d0)))) / 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 <= 2.05) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 6.5e+270) {
		tmp = (x * (2.0 / Math.exp(x))) / 2.0;
	} else {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.05:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 6.5e+270:
		tmp = (x * (2.0 / math.exp(x))) / 2.0
	else:
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.05)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 6.5e+270)
		tmp = Float64(Float64(x * Float64(2.0 / exp(x))) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(x + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.05)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 6.5e+270)
		tmp = (x * (2.0 / exp(x))) / 2.0;
	else
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.05], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.5e+270], N[(N[(x * N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(2.0 + N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.0499999999999998

   1. Initial program 60.5%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in eps around inf 97.3%

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

   6. Taylor expanded in eps around inf 97.4%

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

      1. *-commutative 97.4%

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

   8. Simplified 97.4%

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

   9. Taylor expanded in eps around 0 80.3%

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

       1. neg-mul-1 80.3%

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

   11. Simplified 80.3%

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

   if 2.0499999999999998 < x < 6.4999999999999996e270

   1. Initial program 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in eps around 0 59.8%

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

      1. associate--r+ 59.8%

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

      2. associate-*r* 59.8%

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

      3. mul-1-neg 59.8%

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

      4. cancel-sign-sub 59.8%

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

      5. distribute-rgt1-in 59.8%

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

      6. distribute-rgt-out-- 59.8%

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

      7. mul-1-neg 59.8%

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

      8. mul-1-neg 59.8%

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

   7. Simplified 59.8%

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

   8. Taylor expanded in x around inf 58.6%

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

      1. rec-exp 58.6%

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

      2. associate-*r* 58.6%

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

      3. *-commutative 58.6%

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

      4. rec-exp 58.6%

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

      5. neg-mul-1 58.6%

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

      6. associate-*r* 58.6%

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

      7. neg-mul-1 58.6%

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

   10. Simplified 58.6%

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

   11. Taylor expanded in x around inf 58.6%

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

       1. associate-*r* 58.6%

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

       2. *-commutative 58.6%

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

       3. exp-neg 58.6%

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

       4. associate-/l* 58.6%

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

       5. *-rgt-identity 58.6%

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

       6. associate-/l* 58.6%

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

   13. Simplified 58.6%

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

   if 6.4999999999999996e270 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 28.4%

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

      1. associate--r+ 28.4%

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

      2. associate-*r* 28.4%

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

      3. mul-1-neg 28.4%

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

      4. cancel-sign-sub 28.4%

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

      5. distribute-rgt1-in 28.4%

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

      6. distribute-rgt-out-- 28.4%

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

      7. mul-1-neg 28.4%

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

      8. mul-1-neg 28.4%

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

   7. Simplified 28.4%

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

   8. Taylor expanded in x around inf 28.4%

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

      1. rec-exp 28.4%

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

      2. associate-*r* 28.4%

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

      3. *-commutative 28.4%

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

      4. rec-exp 28.4%

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

      5. neg-mul-1 28.4%

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

      6. associate-*r* 28.4%

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

      7. neg-mul-1 28.4%

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

   10. Simplified 28.4%

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

       1. pow1 28.4%

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

       2. associate-*r* 28.4%

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

       3. *-commutative 28.4%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 73.2%

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

       6. sqr-neg 73.2%

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

       7. sqrt-unprod 73.2%

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

       8. add-sqr-sqrt 73.2%

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

   12. Applied egg-rr 73.2%

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

       1. unpow1 73.2%

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

   14. Simplified 73.2%

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

   15. Taylor expanded in x around 0 73.2%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+270}:\\ \;\;\;\;\frac{x \cdot \frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 800000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 10^{+271}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 800000.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 1e+271) 0.0 (/ (* x (+ 2.0 (* x (+ x 2.0)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 800000.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 1e+271) {
		tmp = 0.0;
	} else {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 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) :: tmp
    if (x <= 800000.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 1d+271) then
        tmp = 0.0d0
    else
        tmp = (x * (2.0d0 + (x * (x + 2.0d0)))) / 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 <= 800000.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 1e+271) {
		tmp = 0.0;
	} else {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 800000.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 1e+271:
		tmp = 0.0
	else:
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 800000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 1e+271)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(x + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 800000.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 1e+271)
		tmp = 0.0;
	else
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 800000.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+271], 0.0, N[(N[(x * N[(2.0 + N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 8e5

   1. Initial program 60.6%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in eps around inf 96.9%

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

   6. Taylor expanded in eps around inf 96.9%

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

      1. *-commutative 96.9%

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

   8. Simplified 96.9%

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

   9. Taylor expanded in eps around 0 79.0%

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

       1. neg-mul-1 79.0%

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

   11. Simplified 79.0%

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

   if 8e5 < x < 9.99999999999999953e270

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 60.9%

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

      1. mul-1-neg 60.9%

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

      2. mul-1-neg 60.9%

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

      3. rec-exp 60.9%

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

      4. sub-neg 60.9%

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

      5. div-sub 60.9%

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

      6. mul-1-neg 60.9%

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

      7. rec-exp 60.9%

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

      8. +-inverses 60.9%

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

   7. Simplified 60.9%

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

   if 9.99999999999999953e270 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 28.4%

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

      1. associate--r+ 28.4%

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

      2. associate-*r* 28.4%

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

      3. mul-1-neg 28.4%

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

      4. cancel-sign-sub 28.4%

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

      5. distribute-rgt1-in 28.4%

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

      6. distribute-rgt-out-- 28.4%

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

      7. mul-1-neg 28.4%

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

      8. mul-1-neg 28.4%

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

   7. Simplified 28.4%

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

   8. Taylor expanded in x around inf 28.4%

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

      1. rec-exp 28.4%

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

      2. associate-*r* 28.4%

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

      3. *-commutative 28.4%

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

      4. rec-exp 28.4%

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

      5. neg-mul-1 28.4%

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

      6. associate-*r* 28.4%

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

      7. neg-mul-1 28.4%

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

   10. Simplified 28.4%

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

       1. pow1 28.4%

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

       2. associate-*r* 28.4%

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

       3. *-commutative 28.4%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 73.2%

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

       6. sqr-neg 73.2%

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

       7. sqrt-unprod 73.2%

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

       8. add-sqr-sqrt 73.2%

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

   12. Applied egg-rr 73.2%

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

       1. unpow1 73.2%

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

   14. Simplified 73.2%

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

   15. Taylor expanded in x around 0 73.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 800000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 10^{+271}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;x \leq 85:\\ \;\;\;\;\frac{2 - eps\_m \cdot x}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+271}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2.6e+143)
   (/ (* x (+ 2.0 (* x 2.0))) 2.0)
   (if (<= x 85.0)
     (/ (- 2.0 (* eps_m x)) 2.0)
     (if (<= x 4e+271) 0.0 (/ (* x (+ 2.0 (* x (+ x 2.0)))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2.6e+143) {
		tmp = (x * (2.0 + (x * 2.0))) / 2.0;
	} else if (x <= 85.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 4e+271) {
		tmp = 0.0;
	} else {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 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) :: tmp
    if (x <= (-2.6d+143)) then
        tmp = (x * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else if (x <= 85.0d0) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else if (x <= 4d+271) then
        tmp = 0.0d0
    else
        tmp = (x * (2.0d0 + (x * (x + 2.0d0)))) / 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 <= -2.6e+143) {
		tmp = (x * (2.0 + (x * 2.0))) / 2.0;
	} else if (x <= 85.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 4e+271) {
		tmp = 0.0;
	} else {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2.6e+143:
		tmp = (x * (2.0 + (x * 2.0))) / 2.0
	elif x <= 85.0:
		tmp = (2.0 - (eps_m * x)) / 2.0
	elif x <= 4e+271:
		tmp = 0.0
	else:
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2.6e+143)
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	elseif (x <= 85.0)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	elseif (x <= 4e+271)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(x + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2.6e+143)
		tmp = (x * (2.0 + (x * 2.0))) / 2.0;
	elseif (x <= 85.0)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	elseif (x <= 4e+271)
		tmp = 0.0;
	else
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2.6e+143], N[(N[(x * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 85.0], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4e+271], 0.0, N[(N[(x * N[(2.0 + N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.5999999999999999e143

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 0.0%

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

      1. associate--r+ 0.0%

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

      2. associate-*r* 0.0%

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

      3. mul-1-neg 0.0%

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

      4. cancel-sign-sub 0.0%

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

      5. distribute-rgt1-in 0.0%

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

      6. distribute-rgt-out-- 0.0%

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

      7. mul-1-neg 0.0%

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

      8. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. rec-exp 0.0%

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

      2. associate-*r* 0.0%

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

      3. *-commutative 0.0%

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

      4. rec-exp 0.0%

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

      5. neg-mul-1 0.0%

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

      6. associate-*r* 0.0%

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

      7. neg-mul-1 0.0%

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

   10. Simplified 0.0%

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

       1. pow1 0.0%

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

       2. associate-*r* 0.0%

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

       3. *-commutative 0.0%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 0.0%

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

       6. sqr-neg 0.0%

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

       7. sqrt-unprod 0.0%

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

       8. add-sqr-sqrt 1.6%

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

   12. Applied egg-rr 1.6%

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

       1. unpow1 1.6%

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

   14. Simplified 1.6%

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

   15. Taylor expanded in x around 0 95.7%

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

   if -2.5999999999999999e143 < x < 85

   1. Initial program 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around 0 39.8%

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

   6. Taylor expanded in x around 0 45.6%

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

   7. Taylor expanded in eps around inf 70.1%

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

      1. associate-*r* 70.1%

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

      2. mul-1-neg 70.1%

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

   9. Simplified 70.1%

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

   if 85 < x < 3.99999999999999981e271

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 59.1%

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

      1. mul-1-neg 59.1%

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

      2. mul-1-neg 59.1%

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

      3. rec-exp 59.1%

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

      4. sub-neg 59.1%

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

      5. div-sub 59.1%

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

      6. mul-1-neg 59.1%

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

      7. rec-exp 59.1%

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

      8. +-inverses 59.1%

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

   7. Simplified 59.1%

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

   if 3.99999999999999981e271 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 28.4%

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

      1. associate--r+ 28.4%

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

      2. associate-*r* 28.4%

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

      3. mul-1-neg 28.4%

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

      4. cancel-sign-sub 28.4%

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

      5. distribute-rgt1-in 28.4%

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

      6. distribute-rgt-out-- 28.4%

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

      7. mul-1-neg 28.4%

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

      8. mul-1-neg 28.4%

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

   7. Simplified 28.4%

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

   8. Taylor expanded in x around inf 28.4%

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

      1. rec-exp 28.4%

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

      2. associate-*r* 28.4%

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

      3. *-commutative 28.4%

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

      4. rec-exp 28.4%

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

      5. neg-mul-1 28.4%

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

      6. associate-*r* 28.4%

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

      7. neg-mul-1 28.4%

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

   10. Simplified 28.4%

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

       1. pow1 28.4%

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

       2. associate-*r* 28.4%

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

       3. *-commutative 28.4%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 73.2%

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

       6. sqr-neg 73.2%

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

       7. sqrt-unprod 73.2%

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

       8. add-sqr-sqrt 73.2%

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

   12. Applied egg-rr 73.2%

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

       1. unpow1 73.2%

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

   14. Simplified 73.2%

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

   15. Taylor expanded in x around 0 73.2%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;x \leq 85:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+271}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -820:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + x \cdot 0.3333333333333333\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 800000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+271}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -820.0)
   (/ (* x (+ 2.0 (* x (+ 2.0 (* x (+ 1.0 (* x 0.3333333333333333))))))) 2.0)
   (if (<= x 800000.0)
     1.0
     (if (<= x 1e+271) 0.0 (/ (* x (+ 2.0 (* x (+ x 2.0)))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -820.0) {
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0;
	} else if (x <= 800000.0) {
		tmp = 1.0;
	} else if (x <= 1e+271) {
		tmp = 0.0;
	} else {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 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) :: tmp
    if (x <= (-820.0d0)) then
        tmp = (x * (2.0d0 + (x * (2.0d0 + (x * (1.0d0 + (x * 0.3333333333333333d0))))))) / 2.0d0
    else if (x <= 800000.0d0) then
        tmp = 1.0d0
    else if (x <= 1d+271) then
        tmp = 0.0d0
    else
        tmp = (x * (2.0d0 + (x * (x + 2.0d0)))) / 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 <= -820.0) {
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0;
	} else if (x <= 800000.0) {
		tmp = 1.0;
	} else if (x <= 1e+271) {
		tmp = 0.0;
	} else {
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -820.0:
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0
	elif x <= 800000.0:
		tmp = 1.0
	elif x <= 1e+271:
		tmp = 0.0
	else:
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -820.0)
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(2.0 + Float64(x * Float64(1.0 + Float64(x * 0.3333333333333333))))))) / 2.0);
	elseif (x <= 800000.0)
		tmp = 1.0;
	elseif (x <= 1e+271)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(x + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -820.0)
		tmp = (x * (2.0 + (x * (2.0 + (x * (1.0 + (x * 0.3333333333333333))))))) / 2.0;
	elseif (x <= 800000.0)
		tmp = 1.0;
	elseif (x <= 1e+271)
		tmp = 0.0;
	else
		tmp = (x * (2.0 + (x * (x + 2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -820.0], N[(N[(x * N[(2.0 + N[(x * N[(2.0 + N[(x * N[(1.0 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 800000.0], 1.0, If[LessEqual[x, 1e+271], 0.0, N[(N[(x * N[(2.0 + N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -820

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 0.0%

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

      1. associate--r+ 0.0%

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

      2. associate-*r* 0.0%

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

      3. mul-1-neg 0.0%

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

      4. cancel-sign-sub 0.0%

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

      5. distribute-rgt1-in 0.0%

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

      6. distribute-rgt-out-- 0.0%

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

      7. mul-1-neg 0.0%

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

      8. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. rec-exp 0.0%

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

      2. associate-*r* 0.0%

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

      3. *-commutative 0.0%

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

      4. rec-exp 0.0%

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

      5. neg-mul-1 0.0%

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

      6. associate-*r* 0.0%

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

      7. neg-mul-1 0.0%

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

   10. Simplified 0.0%

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

       1. pow1 0.0%

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

       2. associate-*r* 0.0%

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

       3. *-commutative 0.0%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 0.0%

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

       6. sqr-neg 0.0%

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

       7. sqrt-unprod 0.0%

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

       8. add-sqr-sqrt 1.6%

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

   12. Applied egg-rr 1.6%

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

       1. unpow1 1.6%

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

   14. Simplified 1.6%

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

   15. Taylor expanded in x around 0 76.7%

       \[\leadsto \frac{\color{blue}{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + 0.3333333333333333 \cdot x\right)\right)\right)}}{2} \]
   16. Step-by-step derivation

       1. *-commutative 76.7%

          \[\leadsto \frac{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + \color{blue}{x \cdot 0.3333333333333333}\right)\right)\right)}{2} \]

   17. Simplified 76.7%

       \[\leadsto \frac{\color{blue}{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + x \cdot 0.3333333333333333\right)\right)\right)}}{2} \]

   if -820 < x < 8e5

   1. Initial program 50.5%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in x around 0 74.7%

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

   if 8e5 < x < 9.99999999999999953e270

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 60.9%

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

      1. mul-1-neg 60.9%

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

      2. mul-1-neg 60.9%

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

      3. rec-exp 60.9%

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

      4. sub-neg 60.9%

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

      5. div-sub 60.9%

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

      6. mul-1-neg 60.9%

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

      7. rec-exp 60.9%

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

      8. +-inverses 60.9%

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

   7. Simplified 60.9%

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

   if 9.99999999999999953e270 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 28.4%

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

      1. associate--r+ 28.4%

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

      2. associate-*r* 28.4%

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

      3. mul-1-neg 28.4%

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

      4. cancel-sign-sub 28.4%

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

      5. distribute-rgt1-in 28.4%

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

      6. distribute-rgt-out-- 28.4%

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

      7. mul-1-neg 28.4%

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

      8. mul-1-neg 28.4%

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

   7. Simplified 28.4%

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

   8. Taylor expanded in x around inf 28.4%

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

      1. rec-exp 28.4%

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

      2. associate-*r* 28.4%

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

      3. *-commutative 28.4%

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

      4. rec-exp 28.4%

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

      5. neg-mul-1 28.4%

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

      6. associate-*r* 28.4%

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

      7. neg-mul-1 28.4%

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

   10. Simplified 28.4%

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

       1. pow1 28.4%

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

       2. associate-*r* 28.4%

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

       3. *-commutative 28.4%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 73.2%

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

       6. sqr-neg 73.2%

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

       7. sqrt-unprod 73.2%

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

       8. add-sqr-sqrt 73.2%

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

   12. Applied egg-rr 73.2%

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

       1. unpow1 73.2%

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

   14. Simplified 73.2%

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

   15. Taylor expanded in x around 0 73.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -820:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(2 + x \cdot \left(1 + x \cdot 0.3333333333333333\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 800000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+271}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + 2\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{x \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+143}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 85:\\ \;\;\;\;\frac{2 - eps\_m \cdot x}{2}\\ \mathbf{elif}\;x \leq 10^{+271}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 2.0 (* x 2.0))) 2.0)))
   (if (<= x -2.6e+143)
     t_0
     (if (<= x 85.0) (/ (- 2.0 (* eps_m x)) 2.0) (if (<= x 1e+271) 0.0 t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x * (2.0 + (x * 2.0))) / 2.0;
	double tmp;
	if (x <= -2.6e+143) {
		tmp = t_0;
	} else if (x <= 85.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 1e+271) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (2.0d0 + (x * 2.0d0))) / 2.0d0
    if (x <= (-2.6d+143)) then
        tmp = t_0
    else if (x <= 85.0d0) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else if (x <= 1d+271) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (x * (2.0 + (x * 2.0))) / 2.0;
	double tmp;
	if (x <= -2.6e+143) {
		tmp = t_0;
	} else if (x <= 85.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 1e+271) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x * (2.0 + (x * 2.0))) / 2.0
	tmp = 0
	if x <= -2.6e+143:
		tmp = t_0
	elif x <= 85.0:
		tmp = (2.0 - (eps_m * x)) / 2.0
	elif x <= 1e+271:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x * Float64(2.0 + Float64(x * 2.0))) / 2.0)
	tmp = 0.0
	if (x <= -2.6e+143)
		tmp = t_0;
	elseif (x <= 85.0)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	elseif (x <= 1e+271)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x * (2.0 + (x * 2.0))) / 2.0;
	tmp = 0.0;
	if (x <= -2.6e+143)
		tmp = t_0;
	elseif (x <= 85.0)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	elseif (x <= 1e+271)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2.6e+143], t$95$0, If[LessEqual[x, 85.0], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+271], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5999999999999999e143 or 9.99999999999999953e270 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 9.8%

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

      1. associate--r+ 9.8%

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

      2. associate-*r* 9.8%

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

      3. mul-1-neg 9.8%

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

      4. cancel-sign-sub 9.8%

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

      5. distribute-rgt1-in 9.8%

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

      6. distribute-rgt-out-- 9.8%

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

      7. mul-1-neg 9.8%

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

      8. mul-1-neg 9.8%

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

   7. Simplified 9.8%

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

   8. Taylor expanded in x around inf 9.8%

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

      1. rec-exp 9.8%

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

      2. associate-*r* 9.8%

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

      3. *-commutative 9.8%

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

      4. rec-exp 9.8%

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

      5. neg-mul-1 9.8%

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

      6. associate-*r* 9.8%

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

      7. neg-mul-1 9.8%

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

   10. Simplified 9.8%

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

       1. pow1 9.8%

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

       2. associate-*r* 9.8%

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

       3. *-commutative 9.8%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 25.1%

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

       6. sqr-neg 25.1%

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

       7. sqrt-unprod 25.1%

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

       8. add-sqr-sqrt 26.2%

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

   12. Applied egg-rr 26.2%

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

       1. unpow1 26.2%

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

   14. Simplified 26.2%

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

   15. Taylor expanded in x around 0 87.9%

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

   if -2.5999999999999999e143 < x < 85

   1. Initial program 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around 0 39.8%

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

   6. Taylor expanded in x around 0 45.6%

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

   7. Taylor expanded in eps around inf 70.1%

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

      1. associate-*r* 70.1%

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

      2. mul-1-neg 70.1%

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

   9. Simplified 70.1%

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

   if 85 < x < 9.99999999999999953e270

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 59.1%

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

      1. mul-1-neg 59.1%

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

      2. mul-1-neg 59.1%

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

      3. rec-exp 59.1%

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

      4. sub-neg 59.1%

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

      5. div-sub 59.1%

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

      6. mul-1-neg 59.1%

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

      7. rec-exp 59.1%

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

      8. +-inverses 59.1%

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

   7. Simplified 59.1%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;x \leq 85:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 10^{+271}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.6% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 85:\\ \;\;\;\;\frac{2 - eps\_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 85.0) (/ (- 2.0 (* eps_m x)) 2.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 85.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} 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 <= 85.0d0) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    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 <= 85.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 85.0:
		tmp = (2.0 - (eps_m * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 85.0)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	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 <= 85.0)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 85.0], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 85

   1. Initial program 60.2%

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

   3. Simplified 60.2%

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

   5. Taylor expanded in x around 0 43.6%

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

   6. Taylor expanded in x around 0 45.6%

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

   7. Taylor expanded in eps around inf 67.2%

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

      1. associate-*r* 67.2%

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

      2. mul-1-neg 67.2%

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

   9. Simplified 67.2%

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

   if 85 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 54.7%

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

      1. mul-1-neg 54.7%

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

      2. mul-1-neg 54.7%

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

      3. rec-exp 54.7%

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

      4. sub-neg 54.7%

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

      5. div-sub 54.7%

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

      6. mul-1-neg 54.7%

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

      7. rec-exp 54.7%

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

      8. +-inverses 54.7%

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

   7. Simplified 54.7%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 85:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.8% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 800000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 800000.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 800000.0) {
		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 <= 800000.0d0) 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 <= 800000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 800000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 800000.0)
		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 <= 800000.0)
		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, 800000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8e5

   1. Initial program 60.6%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in x around 0 60.2%

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

   if 8e5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 56.1%

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

      1. mul-1-neg 56.1%

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

      2. mul-1-neg 56.1%

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

      3. rec-exp 56.1%

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

      4. sub-neg 56.1%

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

      5. div-sub 56.1%

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

      6. mul-1-neg 56.1%

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

      7. rec-exp 56.1%

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

      8. +-inverses 56.1%

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

   7. Simplified 56.1%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 800000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 16.0% 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 72.0%

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

3. Simplified 60.5%

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

5. Taylor expanded in eps around 0 17.8%

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

   1. mul-1-neg 17.8%

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

   2. mul-1-neg 17.8%

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

   3. rec-exp 17.8%

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

   4. sub-neg 17.8%

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

   5. div-sub 17.8%

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

   6. mul-1-neg 17.8%

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

   7. rec-exp 17.8%

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

   8. +-inverses 18.0%

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

7. Simplified 18.0%

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

8. Final simplification 18.0%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024081 
(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))