NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.7% → 99.8%

Time: 23.1s

Alternatives: 14

Speedup: 2.0×

• 38.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% 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

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2.2e-23)
   (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
   (/ (+ (exp (* x (+ -1.0 eps_m))) (exp (* eps_m (- x)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2.2e-23) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) + 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 <= 2.2d-23) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps_m))) + 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 <= 2.2e-23) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + 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 <= 2.2e-23:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) + 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 <= 2.2e-23)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + 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 <= 2.2e-23)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) + 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, 2.2e-23], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $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 < 2.1999999999999999e-23

   1. Initial program 56.0%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in eps around 0 26.8%

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

   7. Simplified 70.9%

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

   8. Taylor expanded in eps around 0 70.9%

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

   if 2.1999999999999999e-23 < eps

   1. Initial program 97.6%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. mul-1-neg 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

3. Simplified 58.5%

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

5. Taylor expanded in eps around inf 99.2%

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

6. Final simplification 99.2%

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

Alternative 3: 84.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-253}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-13} \lor \neg \left(x \leq 5.2 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.6e-253)
   (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
   (if (or (<= x 2.7e-13) (not (<= x 5.2e+36)))
     (/ (+ 1.0 (exp (* x (+ -1.0 eps_m)))) 2.0)
     (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.6e-253) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if ((x <= 2.7e-13) || !(x <= 5.2e+36)) {
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 2.0;
	} else {
		tmp = (exp(-x) * (2.0 + (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 <= (-1.6d-253)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if ((x <= 2.7d-13) .or. (.not. (x <= 5.2d+36))) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps_m)))) / 2.0d0
    else
        tmp = (exp(-x) * (2.0d0 + (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 <= -1.6e-253) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if ((x <= 2.7e-13) || !(x <= 5.2e+36)) {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps_m)))) / 2.0;
	} else {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.6e-253:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif (x <= 2.7e-13) or not (x <= 5.2e+36):
		tmp = (1.0 + math.exp((x * (-1.0 + eps_m)))) / 2.0
	else:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.6e-253)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif ((x <= 2.7e-13) || !(x <= 5.2e+36))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + 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 <= -1.6e-253)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif ((x <= 2.7e-13) || ~((x <= 5.2e+36)))
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 2.0;
	else
		tmp = (exp(-x) * (2.0 + (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, -1.6e-253], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 2.7e-13], N[Not[LessEqual[x, 5.2e+36]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5999999999999999e-253

   1. Initial program 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in x around 0 44.3%

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

      1. metadata-eval 44.3%

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

      2. distribute-neg-frac 44.3%

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

      3. metadata-eval 44.3%

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

      4. associate-*l/ 44.3%

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

      5. *-commutative 44.3%

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

      6. distribute-lft-neg-in 44.3%

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

      7. cancel-sign-sub-inv 44.3%

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

      8. *-commutative 44.3%

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

      9. associate-*l/ 44.3%

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

      10. metadata-eval 44.3%

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

   7. Simplified 44.3%

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

   8. Taylor expanded in eps around inf 74.3%

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

      1. mul-1-neg 74.3%

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

      2. associate-*r* 74.3%

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

      3. *-lft-identity 74.3%

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

      4. metadata-eval 74.3%

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

      5. cancel-sign-sub-inv 74.3%

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

      6. associate-*r* 74.3%

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

      7. mul-1-neg 74.3%

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

      8. distribute-rgt-neg-in 74.3%

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

      9. sub-neg 74.3%

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

      10. mul-1-neg 74.3%

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

      11. remove-double-neg 74.3%

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

      12. +-commutative 74.3%

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

   10. Simplified 74.3%

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

   11. Taylor expanded in eps around inf 74.3%

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

       1. associate-*r* 74.3%

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

       2. mul-1-neg 74.3%

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

   13. Simplified 74.3%

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

   if -1.5999999999999999e-253 < x < 2.70000000000000011e-13 or 5.2000000000000003e36 < x

   1. Initial program 68.2%

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

   3. Simplified 55.2%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in x around 0 65.7%

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

   if 2.70000000000000011e-13 < x < 5.2000000000000003e36

   1. Initial program 80.6%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in eps around 0 60.9%

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

   7. Simplified 80.2%

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

   8. Taylor expanded in eps around 0 80.2%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-253}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-13} \lor \neg \left(x \leq 5.2 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 + eps\_m\right)}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-253}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{t\_0 + \left(1 - x \cdot eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+37}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ -1.0 eps_m)))))
   (if (<= x -2e-253)
     (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
     (if (<= x 2.7e-13)
       (/ (+ t_0 (- 1.0 (* x eps_m))) 2.0)
       (if (<= x 8e+37)
         (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
         (/ (+ 1.0 t_0) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (-1.0 + eps_m)));
	double tmp;
	if (x <= -2e-253) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if (x <= 2.7e-13) {
		tmp = (t_0 + (1.0 - (x * eps_m))) / 2.0;
	} else if (x <= 8e+37) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * ((-1.0d0) + eps_m)))
    if (x <= (-2d-253)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if (x <= 2.7d-13) then
        tmp = (t_0 + (1.0d0 - (x * eps_m))) / 2.0d0
    else if (x <= 8d+37) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else
        tmp = (1.0d0 + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (-1.0 + eps_m)));
	double tmp;
	if (x <= -2e-253) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if (x <= 2.7e-13) {
		tmp = (t_0 + (1.0 - (x * eps_m))) / 2.0;
	} else if (x <= 8e+37) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (-1.0 + eps_m)))
	tmp = 0
	if x <= -2e-253:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif x <= 2.7e-13:
		tmp = (t_0 + (1.0 - (x * eps_m))) / 2.0
	elif x <= 8e+37:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	else:
		tmp = (1.0 + t_0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(-1.0 + eps_m)))
	tmp = 0.0
	if (x <= -2e-253)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif (x <= 2.7e-13)
		tmp = Float64(Float64(t_0 + Float64(1.0 - Float64(x * eps_m))) / 2.0);
	elseif (x <= 8e+37)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (-1.0 + eps_m)));
	tmp = 0.0;
	if (x <= -2e-253)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif (x <= 2.7e-13)
		tmp = (t_0 + (1.0 - (x * eps_m))) / 2.0;
	elseif (x <= 8e+37)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	else
		tmp = (1.0 + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2e-253], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.7e-13], N[(N[(t$95$0 + N[(1.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8e+37], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.0000000000000001e-253

   1. Initial program 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in x around 0 44.3%

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

      1. metadata-eval 44.3%

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

      2. distribute-neg-frac 44.3%

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

      3. metadata-eval 44.3%

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

      4. associate-*l/ 44.3%

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

      5. *-commutative 44.3%

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

      6. distribute-lft-neg-in 44.3%

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

      7. cancel-sign-sub-inv 44.3%

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

      8. *-commutative 44.3%

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

      9. associate-*l/ 44.3%

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

      10. metadata-eval 44.3%

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

   7. Simplified 44.3%

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

   8. Taylor expanded in eps around inf 74.3%

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

      1. mul-1-neg 74.3%

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

      2. associate-*r* 74.3%

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

      3. *-lft-identity 74.3%

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

      4. metadata-eval 74.3%

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

      5. cancel-sign-sub-inv 74.3%

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

      6. associate-*r* 74.3%

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

      7. mul-1-neg 74.3%

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

      8. distribute-rgt-neg-in 74.3%

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

      9. sub-neg 74.3%

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

      10. mul-1-neg 74.3%

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

      11. remove-double-neg 74.3%

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

      12. +-commutative 74.3%

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

   10. Simplified 74.3%

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

   11. Taylor expanded in eps around inf 74.3%

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

       1. associate-*r* 74.3%

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

       2. mul-1-neg 74.3%

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

   13. Simplified 74.3%

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

   if -2.0000000000000001e-253 < x < 2.70000000000000011e-13

   1. Initial program 45.6%

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

   3. Simplified 23.6%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 88.1%

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

       1. mul-1-neg 88.1%

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

       2. *-commutative 88.1%

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

       3. unsub-neg 88.1%

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

       4. *-commutative 88.1%

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

   11. Simplified 88.1%

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

   if 2.70000000000000011e-13 < x < 7.99999999999999963e37

   1. Initial program 80.6%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in eps around 0 60.9%

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

   7. Simplified 80.2%

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

   8. Taylor expanded in eps around 0 80.2%

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

   if 7.99999999999999963e37 < 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 inf 100.0%

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

   6. Taylor expanded in x around 0 34.9%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-253}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+37}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.02e+103)
   (/ (+ 2.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666)))))) 2.0)
   (if (<= x -5e-253)
     (/ (+ 2.0 (* x (- -1.0 eps_m))) 2.0)
     (/ (+ 1.0 (exp x)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.02e+103) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= -5e-253) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else {
		tmp = (1.0 + exp(x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.02d+103)) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))) / 2.0d0
    else if (x <= (-5d-253)) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps_m))) / 2.0d0
    else
        tmp = (1.0d0 + exp(x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.02e+103) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= -5e-253) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.02e+103:
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0
	elif x <= -5e-253:
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0
	else:
		tmp = (1.0 + math.exp(x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.02e+103)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666)))))) / 2.0);
	elseif (x <= -5e-253)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps_m))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.02e+103)
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	elseif (x <= -5e-253)
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	else
		tmp = (1.0 + exp(x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.02e+103], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -5e-253], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.01999999999999991e103

   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 inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around 0 100.0%

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

       1. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0 100.0%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)}}{2} \]

   if -1.01999999999999991e103 < x < -4.99999999999999971e-253

   1. Initial program 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around 0 41.2%

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

      1. metadata-eval 41.2%

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

      2. distribute-neg-frac 41.2%

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

      3. metadata-eval 41.2%

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

      4. associate-*l/ 41.2%

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

      5. *-commutative 41.2%

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

      6. distribute-lft-neg-in 41.2%

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

      7. cancel-sign-sub-inv 41.2%

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

      8. *-commutative 41.2%

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

      9. associate-*l/ 41.2%

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

      10. metadata-eval 41.2%

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

   7. Simplified 41.2%

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

   8. Taylor expanded in eps around inf 81.5%

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

      1. mul-1-neg 81.5%

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

      2. associate-*r* 81.5%

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

      3. *-lft-identity 81.5%

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

      4. metadata-eval 81.5%

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

      5. cancel-sign-sub-inv 81.5%

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

      6. associate-*r* 81.5%

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

      7. mul-1-neg 81.5%

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

      8. distribute-rgt-neg-in 81.5%

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

      9. sub-neg 81.5%

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

      10. mul-1-neg 81.5%

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

      11. remove-double-neg 81.5%

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

      12. +-commutative 81.5%

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

   10. Simplified 81.5%

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

   11. Taylor expanded in x around 0 60.5%

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

       1. associate-*r* 60.5%

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

       2. neg-mul-1 60.5%

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

   13. Simplified 60.5%

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

   if -4.99999999999999971e-253 < x

   1. Initial program 69.0%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in eps around inf 99.3%

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

   6. Taylor expanded in eps around inf 87.1%

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

      1. *-commutative 87.1%

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

   8. Simplified 87.1%

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

   9. Taylor expanded in eps around 0 47.6%

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

       1. neg-mul-1 47.6%

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

   11. Simplified 47.6%

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

       1. *-un-lft-identity 47.6%

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

       2. add-sqr-sqrt 6.0%

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

       3. sqrt-unprod 69.9%

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

       4. sqr-neg 69.9%

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

       5. sqrt-unprod 63.9%

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

       6. add-sqr-sqrt 69.9%

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

   13. Applied egg-rr 69.9%

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

       1. *-lft-identity 69.9%

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

   15. Simplified 69.9%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-253}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.55e-253)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (/ (+ 1.0 (exp (* x (+ -1.0 eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.55e-253) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.55d-253)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.55e-253) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.55e-253:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.55e-253)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.55e-253)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.55e-253], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.54999999999999998e-253

   1. Initial program 68.9%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in eps around inf 99.1%

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

      1. *-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in eps around 0 78.0%

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

       1. neg-mul-1 78.0%

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

   11. Simplified 78.0%

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

   if -1.54999999999999998e-253 < x

   1. Initial program 69.0%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in eps around inf 99.3%

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

   6. Taylor expanded in x around 0 62.9%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-253}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-253}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-253)
   (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
   (/ (+ 1.0 (exp (* x (+ -1.0 eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-253) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2d-253)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-253) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-253:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-253)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2e-253)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-253], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e-253

   1. Initial program 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in x around 0 44.3%

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

      1. metadata-eval 44.3%

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

      2. distribute-neg-frac 44.3%

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

      3. metadata-eval 44.3%

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

      4. associate-*l/ 44.3%

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

      5. *-commutative 44.3%

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

      6. distribute-lft-neg-in 44.3%

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

      7. cancel-sign-sub-inv 44.3%

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

      8. *-commutative 44.3%

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

      9. associate-*l/ 44.3%

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

      10. metadata-eval 44.3%

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

   7. Simplified 44.3%

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

   8. Taylor expanded in eps around inf 74.3%

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

      1. mul-1-neg 74.3%

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

      2. associate-*r* 74.3%

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

      3. *-lft-identity 74.3%

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

      4. metadata-eval 74.3%

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

      5. cancel-sign-sub-inv 74.3%

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

      6. associate-*r* 74.3%

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

      7. mul-1-neg 74.3%

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

      8. distribute-rgt-neg-in 74.3%

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

      9. sub-neg 74.3%

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

      10. mul-1-neg 74.3%

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

      11. remove-double-neg 74.3%

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

      12. +-commutative 74.3%

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

   10. Simplified 74.3%

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

   11. Taylor expanded in eps around inf 74.3%

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

       1. associate-*r* 74.3%

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

       2. mul-1-neg 74.3%

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

   13. Simplified 74.3%

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

   if -2.0000000000000001e-253 < x

   1. Initial program 69.0%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in eps around inf 99.3%

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

   6. Taylor expanded in x around 0 62.9%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-253}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -600:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -600.0) (/ (+ 1.0 (exp (- x))) 2.0) (/ (+ 1.0 (exp x)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -600.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + exp(x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-600.0d0)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = (1.0d0 + exp(x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -600.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -600.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = (1.0 + math.exp(x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -600.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -600.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = (1.0 + exp(x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -600.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -600

   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 inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around 0 100.0%

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

       1. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   if -600 < x

   1. Initial program 63.5%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in eps around inf 90.6%

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

      1. *-commutative 90.6%

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

   8. Simplified 90.6%

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

   9. Taylor expanded in eps around 0 53.3%

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

       1. neg-mul-1 53.3%

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

   11. Simplified 53.3%

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

       1. *-un-lft-identity 53.3%

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

       2. add-sqr-sqrt 24.6%

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

       3. sqrt-unprod 68.6%

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

       4. sqr-neg 68.6%

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

       5. sqrt-unprod 44.0%

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

       6. add-sqr-sqrt 68.6%

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

   13. Applied egg-rr 68.6%

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

       1. *-lft-identity 68.6%

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

   15. Simplified 68.6%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -600:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+194}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.02e+103)
   (/ (+ 2.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666)))))) 2.0)
   (if (<= x 2.0)
     (/ (+ 2.0 (* x (- -1.0 eps_m))) 2.0)
     (if (<= x 9e+194) 0.0 (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.02e+103) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 9e+194) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.02d+103)) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))) / 2.0d0
    else if (x <= 2.0d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps_m))) / 2.0d0
    else if (x <= 9d+194) then
        tmp = 0.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.02e+103) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 9e+194) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.02e+103:
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0
	elif x <= 2.0:
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0
	elif x <= 9e+194:
		tmp = 0.0
	else:
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.02e+103)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666)))))) / 2.0);
	elseif (x <= 2.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 9e+194)
		tmp = 0.0;
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.02e+103)
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	elseif (x <= 2.0)
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	elseif (x <= 9e+194)
		tmp = 0.0;
	else
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.02e+103], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.0], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9e+194], 0.0, N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.01999999999999991e103

   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 inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around 0 100.0%

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

       1. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0 100.0%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)}}{2} \]

   if -1.01999999999999991e103 < x < 2

   1. Initial program 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in x around 0 40.2%

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

      1. metadata-eval 40.2%

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

      2. distribute-neg-frac 40.2%

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

      3. metadata-eval 40.2%

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

      4. associate-*l/ 40.2%

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

      5. *-commutative 40.2%

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

      6. distribute-lft-neg-in 40.2%

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

      7. cancel-sign-sub-inv 40.2%

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

      8. *-commutative 40.2%

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

      9. associate-*l/ 40.2%

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

      10. metadata-eval 40.2%

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

   7. Simplified 40.2%

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

   8. Taylor expanded in eps around inf 87.8%

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

      1. mul-1-neg 87.8%

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

      2. associate-*r* 87.8%

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

      3. *-lft-identity 87.8%

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

      4. metadata-eval 87.8%

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

      5. cancel-sign-sub-inv 87.8%

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

      6. associate-*r* 87.8%

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

      7. mul-1-neg 87.8%

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

      8. distribute-rgt-neg-in 87.8%

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

      9. sub-neg 87.8%

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

      10. mul-1-neg 87.8%

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

      11. remove-double-neg 87.8%

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

      12. +-commutative 87.8%

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

   10. Simplified 87.8%

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

   11. Taylor expanded in x around 0 71.7%

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

       1. associate-*r* 71.7%

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

       2. neg-mul-1 71.7%

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

   13. Simplified 71.7%

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

   if 2 < x < 8.9999999999999997e194

   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 53.1%

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

      1. mul-1-neg 53.1%

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

      2. mul-1-neg 53.1%

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

      3. rec-exp 53.1%

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

      4. sub-neg 53.1%

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

      5. div-sub 53.1%

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

      6. mul-1-neg 53.1%

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

      7. rec-exp 53.1%

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

      8. +-inverses 53.1%

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

   7. Simplified 53.1%

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

   if 8.9999999999999997e194 < 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 inf 100.0%

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

   6. Taylor expanded in eps around inf 83.7%

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

      1. *-commutative 83.7%

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

   8. Simplified 83.7%

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

   9. Taylor expanded in eps around 0 3.1%

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

       1. neg-mul-1 3.1%

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

   11. Simplified 3.1%

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

   12. Taylor expanded in x around 0 63.1%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(0.5 \cdot x - 1\right)}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+194}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.6% accurate, 10.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+192}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.0)
   (/ (+ 2.0 (* x (- -1.0 eps_m))) 2.0)
   (if (<= x 1e+192) 0.0 (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1e+192) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 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 <= 2.0d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps_m))) / 2.0d0
    else if (x <= 1d+192) then
        tmp = 0.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 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 <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1e+192) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0
	elif x <= 1e+192:
		tmp = 0.0
	else:
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 1e+192)
		tmp = 0.0;
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	elseif (x <= 1e+192)
		tmp = 0.0;
	else
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.0], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+192], 0.0, N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2

   1. Initial program 58.2%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in x around 0 42.1%

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

      1. metadata-eval 42.1%

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

      2. distribute-neg-frac 42.1%

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

      3. metadata-eval 42.1%

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

      4. associate-*l/ 42.1%

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

      5. *-commutative 42.1%

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

      6. distribute-lft-neg-in 42.1%

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

      7. cancel-sign-sub-inv 42.1%

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

      8. *-commutative 42.1%

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

      9. associate-*l/ 42.1%

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

      10. metadata-eval 42.1%

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

   7. Simplified 42.1%

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

   8. Taylor expanded in eps around inf 82.9%

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

      1. mul-1-neg 82.9%

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

      2. associate-*r* 82.9%

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

      3. *-lft-identity 82.9%

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

      4. metadata-eval 82.9%

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

      5. cancel-sign-sub-inv 82.9%

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

      6. associate-*r* 82.9%

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

      7. mul-1-neg 82.9%

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

      8. distribute-rgt-neg-in 82.9%

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

      9. sub-neg 82.9%

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

      10. mul-1-neg 82.9%

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

      11. remove-double-neg 82.9%

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

      12. +-commutative 82.9%

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

   10. Simplified 82.9%

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

   11. Taylor expanded in x around 0 67.4%

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

       1. associate-*r* 67.4%

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

       2. neg-mul-1 67.4%

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

   13. Simplified 67.4%

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

   if 2 < x < 1.00000000000000004e192

   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 53.1%

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

      1. mul-1-neg 53.1%

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

      2. mul-1-neg 53.1%

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

      3. rec-exp 53.1%

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

      4. sub-neg 53.1%

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

      5. div-sub 53.1%

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

      6. mul-1-neg 53.1%

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

      7. rec-exp 53.1%

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

      8. +-inverses 53.1%

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

   7. Simplified 53.1%

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

   if 1.00000000000000004e192 < 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 inf 100.0%

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

   6. Taylor expanded in eps around inf 83.7%

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

      1. *-commutative 83.7%

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

   8. Simplified 83.7%

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

   9. Taylor expanded in eps around 0 3.1%

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

       1. neg-mul-1 3.1%

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

   11. Simplified 3.1%

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

   12. Taylor expanded in x around 0 63.1%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(0.5 \cdot x - 1\right)}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+192}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.4% accurate, 16.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 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 <= 2.0d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps_m))) / 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 <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps_m))) / 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 <= 2.0)
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 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, 2.0], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 58.2%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in x around 0 42.1%

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

      1. metadata-eval 42.1%

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

      2. distribute-neg-frac 42.1%

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

      3. metadata-eval 42.1%

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

      4. associate-*l/ 42.1%

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

      5. *-commutative 42.1%

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

      6. distribute-lft-neg-in 42.1%

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

      7. cancel-sign-sub-inv 42.1%

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

      8. *-commutative 42.1%

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

      9. associate-*l/ 42.1%

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

      10. metadata-eval 42.1%

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

   7. Simplified 42.1%

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

   8. Taylor expanded in eps around inf 82.9%

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

      1. mul-1-neg 82.9%

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

      2. associate-*r* 82.9%

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

      3. *-lft-identity 82.9%

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

      4. metadata-eval 82.9%

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

      5. cancel-sign-sub-inv 82.9%

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

      6. associate-*r* 82.9%

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

      7. mul-1-neg 82.9%

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

      8. distribute-rgt-neg-in 82.9%

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

      9. sub-neg 82.9%

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

      10. mul-1-neg 82.9%

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

      11. remove-double-neg 82.9%

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

      12. +-commutative 82.9%

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

   10. Simplified 82.9%

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

   11. Taylor expanded in x around 0 67.4%

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

       1. associate-*r* 67.4%

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

       2. neg-mul-1 67.4%

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

   13. Simplified 67.4%

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

   if 2 < 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 47.8%

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

      1. mul-1-neg 47.8%

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

      2. mul-1-neg 47.8%

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

      3. rec-exp 47.8%

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

      4. sub-neg 47.8%

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

      5. div-sub 47.8%

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

      6. mul-1-neg 47.8%

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

      7. rec-exp 47.8%

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

      8. +-inverses 47.8%

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

   7. Simplified 47.8%

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

4. Final simplification 62.4%

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

Alternative 12: 57.3% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 2.0) (/ (- 2.0 x) 2.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - 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 <= 2.0d0) then
        tmp = (2.0d0 - 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 <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 - x) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 - 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 <= 2.0)
		tmp = (2.0 - 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, 2.0], N[(N[(2.0 - x), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 58.2%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in eps around inf 99.0%

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

   6. Taylor expanded in eps around inf 99.0%

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

      1. *-commutative 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in x around inf 99.0%

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

       1. associate-*r* 99.0%

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

       2. mul-1-neg 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in x around 0 61.2%

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

       1. distribute-rgt1-in 61.2%

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

       2. metadata-eval 61.2%

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

       3. mul0-lft 61.2%

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

       4. metadata-eval 61.2%

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

       5. *-commutative 61.2%

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

       6. mul-1-neg 61.2%

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

       7. unsub-neg 61.2%

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

   14. Simplified 61.2%

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

   if 2 < 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 47.8%

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

      1. mul-1-neg 47.8%

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

      2. mul-1-neg 47.8%

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

      3. rec-exp 47.8%

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

      4. sub-neg 47.8%

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

      5. div-sub 47.8%

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

      6. mul-1-neg 47.8%

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

      7. rec-exp 47.8%

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

      8. +-inverses 47.8%

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

   7. Simplified 47.8%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.3% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 460:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 460.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 460.0) {
		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 <= 460.0d0) 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 <= 460.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 460.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 460.0)
		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 <= 460.0)
		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, 460.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 460

   1. Initial program 58.2%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in x around 0 60.8%

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

   if 460 < 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 47.8%

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

      1. mul-1-neg 47.8%

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

      2. mul-1-neg 47.8%

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

      3. rec-exp 47.8%

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

      4. sub-neg 47.8%

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

      5. div-sub 47.8%

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

      6. mul-1-neg 47.8%

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

      7. rec-exp 47.8%

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

      8. +-inverses 47.8%

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

   7. Simplified 47.8%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 460:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 15.8% 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 69.0%

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

3. Simplified 58.5%

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

5. Taylor expanded in eps around 0 14.0%

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

   1. mul-1-neg 14.0%

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

   2. mul-1-neg 14.0%

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

   3. rec-exp 14.0%

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

   4. sub-neg 14.0%

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

   5. div-sub 14.0%

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

   6. mul-1-neg 14.0%

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

   7. rec-exp 14.0%

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

   8. +-inverses 14.2%

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

7. Simplified 14.2%

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

8. Final simplification 14.2%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(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))