NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.3% → 99.9%

Time: 17.1s

Alternatives: 10

Speedup: 2.1×

• 39.3% 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: 74.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\varepsilon \leq 10^{-5}:\\ \;\;\;\;\frac{t_0 \cdot \left(\left(x + 1\right) - -1\right) + x \cdot t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps 1e-5)
     (/ (+ (* t_0 (- (+ x 1.0) -1.0)) (* x t_0)) 2.0)
     (/ (+ (exp (* x (+ eps -1.0))) (exp (* eps (- x)))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (eps <= 1e-5) {
		tmp = ((t_0 * ((x + 1.0) - -1.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (exp((x * (eps + -1.0))) + exp((eps * -x))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps <= 1d-5) then
        tmp = ((t_0 * ((x + 1.0d0) - (-1.0d0))) + (x * t_0)) / 2.0d0
    else
        tmp = (exp((x * (eps + (-1.0d0)))) + exp((eps * -x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps <= 1e-5) {
		tmp = ((t_0 * ((x + 1.0) - -1.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp((eps * -x))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = math.exp(-x)
	tmp = 0
	if eps <= 1e-5:
		tmp = ((t_0 * ((x + 1.0) - -1.0)) + (x * t_0)) / 2.0
	else:
		tmp = (math.exp((x * (eps + -1.0))) + math.exp((eps * -x))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps <= 1e-5)
		tmp = Float64(Float64(Float64(t_0 * Float64(Float64(x + 1.0) - -1.0)) + Float64(x * t_0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(eps * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps <= 1e-5)
		tmp = ((t_0 * ((x + 1.0) - -1.0)) + (x * t_0)) / 2.0;
	else
		tmp = (exp((x * (eps + -1.0))) + exp((eps * -x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps, 1e-5], N[(N[(N[(t$95$0 * N[(N[(x + 1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.00000000000000008e-5

   1. Initial program 63.9%

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

      1. sub-neg 63.9%

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

      2. neg-sub0 63.9%

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

      3. associate-+r- 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in eps around 0 68.4%

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

      1. associate--r+ 68.4%

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

      2. associate-*r* 68.4%

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

      3. neg-mul-1 68.4%

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

      4. cancel-sign-sub 68.4%

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

      5. distribute-rgt1-in 68.4%

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

      6. distribute-rgt-out-- 68.4%

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

      7. neg-mul-1 68.4%

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

      8. neg-mul-1 68.4%

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

   6. Simplified 68.4%

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

   if 1.00000000000000008e-5 < eps

   1. Initial program 100.0%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 77.6%

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

Alternative 2: 90.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-273}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)}{\left(\varepsilon + -1\right) + \frac{\varepsilon + -1}{\varepsilon}}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -2.2e-273)
   (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
   (if (<= x 6.8e+79)
     (/ (+ 1.0 (exp (- (* eps x) x))) 2.0)
     (/
      (*
       x
       (/
        (+ -1.0 (* (- 1.0 eps) (- 1.0 eps)))
        (+ (+ eps -1.0) (/ (+ eps -1.0) eps))))
      2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -2.2e-273) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else if (x <= 6.8e+79) {
		tmp = (1.0 + exp(((eps * x) - x))) / 2.0;
	} else {
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.2d-273)) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else if (x <= 6.8d+79) then
        tmp = (1.0d0 + exp(((eps * x) - x))) / 2.0d0
    else
        tmp = (x * (((-1.0d0) + ((1.0d0 - eps) * (1.0d0 - eps))) / ((eps + (-1.0d0)) + ((eps + (-1.0d0)) / eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.2e-273) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else if (x <= 6.8e+79) {
		tmp = (1.0 + Math.exp(((eps * x) - x))) / 2.0;
	} else {
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -2.2e-273:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	elif x <= 6.8e+79:
		tmp = (1.0 + math.exp(((eps * x) - x))) / 2.0
	else:
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -2.2e-273)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	elseif (x <= 6.8e+79)
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(eps * x) - x))) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(-1.0 + Float64(Float64(1.0 - eps) * Float64(1.0 - eps))) / Float64(Float64(eps + -1.0) + Float64(Float64(eps + -1.0) / eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.2e-273)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	elseif (x <= 6.8e+79)
		tmp = (1.0 + exp(((eps * x) - x))) / 2.0;
	else
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -2.2e-273], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.8e+79], N[(N[(1.0 + N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(-1.0 + N[(N[(1.0 - eps), $MachinePrecision] * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(eps + -1.0), $MachinePrecision] + N[(N[(eps + -1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1999999999999998e-273

   1. Initial program 78.9%

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

      1. sub-neg 78.9%

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

      2. neg-sub0 78.9%

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

      3. associate-+r- 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in x around 0 43.6%

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

      1. metadata-eval 43.6%

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

      2. distribute-neg-frac 43.6%

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

      3. sub-neg 43.6%

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

   6. Simplified 43.6%

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

   7. Taylor expanded in eps around inf 62.2%

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

      1. cancel-sign-sub-inv 62.2%

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

      2. associate-*r* 62.2%

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

      3. exp-prod 50.9%

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

      4. remove-double-neg 50.9%

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

      5. neg-mul-1 50.9%

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

      6. sub-neg 50.9%

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

      7. exp-prod 62.2%

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

      8. associate-*r* 62.2%

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

      9. remove-double-neg 62.2%

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

      10. mul-1-neg 62.2%

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

      11. metadata-eval 62.2%

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

      12. *-lft-identity 62.2%

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

      13. mul-1-neg 62.2%

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

      14. remove-double-neg 62.2%

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

      15. associate-*r* 62.2%

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

      16. neg-mul-1 62.2%

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

   9. Simplified 62.2%

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

   10. Taylor expanded in eps around inf 62.5%

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

       1. associate-*r* 62.5%

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

       2. mul-1-neg 62.5%

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

   12. Simplified 62.5%

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

   if -2.1999999999999998e-273 < x < 6.80000000000000063e79

   1. Initial program 56.3%

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

      1. sub-neg 56.3%

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

      2. neg-sub0 56.3%

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

      3. associate-+r- 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in x around 0 34.9%

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

      1. metadata-eval 34.9%

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

      2. distribute-neg-frac 34.9%

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

      3. sub-neg 34.9%

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

   6. Simplified 34.9%

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

   7. Taylor expanded in eps around inf 77.3%

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

      1. cancel-sign-sub-inv 77.3%

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

      2. associate-*r* 77.3%

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

      3. exp-prod 73.7%

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

      4. remove-double-neg 73.7%

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

      5. neg-mul-1 73.7%

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

      6. sub-neg 73.7%

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

      7. exp-prod 77.3%

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

      8. associate-*r* 77.3%

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

      9. remove-double-neg 77.3%

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

      10. mul-1-neg 77.3%

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

      11. metadata-eval 77.3%

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

      12. *-lft-identity 77.3%

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

      13. mul-1-neg 77.3%

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

      14. remove-double-neg 77.3%

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

      15. associate-*r* 77.3%

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

      16. neg-mul-1 77.3%

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

   9. Simplified 77.3%

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

       1. distribute-rgt-in 77.3%

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

       2. *-un-lft-identity 77.3%

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

       3. unsub-neg 77.3%

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

       4. add-sqr-sqrt 10.5%

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

       5. sqrt-unprod 78.3%

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

       6. sqr-neg 78.3%

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

       7. sqrt-unprod 68.7%

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

       8. add-sqr-sqrt 79.1%

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

   11. Applied egg-rr 79.1%

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

       1. *-commutative 79.1%

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

   13. Simplified 79.1%

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

   if 6.80000000000000063e79 < x

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 28.5%

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

   5. Taylor expanded in x around inf 18.3%

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

      1. mul-1-neg 18.3%

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

      2. distribute-rgt-neg-in 18.3%

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

      3. *-commutative 18.3%

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

      4. distribute-rgt-neg-in 18.3%

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

      5. mul-1-neg 18.3%

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

      6. distribute-lft-in 18.3%

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

      7. metadata-eval 18.3%

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

      8. associate-*r/ 18.3%

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

      9. metadata-eval 18.3%

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

   7. Simplified 18.3%

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

      1. distribute-lft-in 18.3%

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

      2. flip-+ 18.3%

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

   9. Applied egg-rr 18.3%

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

       1. *-commutative 18.3%

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

       2. neg-mul-1 18.3%

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

       3. *-commutative 18.3%

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

       4. neg-mul-1 18.3%

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

       5. sqr-neg 18.3%

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

       6. *-commutative 18.3%

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

       7. neg-mul-1 18.3%

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

   11. Simplified 18.3%

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

   12. Taylor expanded in eps around inf 71.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-273}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)}{\left(\varepsilon + -1\right) + \frac{\varepsilon + -1}{\varepsilon}}}{2}\\ \end{array} \]

Alternative 3: 85.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)}{\left(\varepsilon + -1\right) + \frac{\varepsilon + -1}{\varepsilon}}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 1.25e-5)
   (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
   (/
    (*
     x
     (/
      (+ -1.0 (* (- 1.0 eps) (- 1.0 eps)))
      (+ (+ eps -1.0) (/ (+ eps -1.0) eps))))
    2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 1.25e-5) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else {
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.25d-5) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else
        tmp = (x * (((-1.0d0) + ((1.0d0 - eps) * (1.0d0 - eps))) / ((eps + (-1.0d0)) + ((eps + (-1.0d0)) / eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.25e-5) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else {
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 1.25e-5:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	else:
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 1.25e-5)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(-1.0 + Float64(Float64(1.0 - eps) * Float64(1.0 - eps))) / Float64(Float64(eps + -1.0) + Float64(Float64(eps + -1.0) / eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.25e-5)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	else
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 1.25e-5], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(-1.0 + N[(N[(1.0 - eps), $MachinePrecision] * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(eps + -1.0), $MachinePrecision] + N[(N[(eps + -1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25000000000000006e-5

   1. Initial program 63.3%

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

      1. sub-neg 63.3%

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

      2. neg-sub0 63.3%

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

      3. associate-+r- 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in x around 0 38.5%

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

      1. metadata-eval 38.5%

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

      2. distribute-neg-frac 38.5%

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

      3. sub-neg 38.5%

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

   6. Simplified 38.5%

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

   7. Taylor expanded in eps around inf 73.0%

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

      1. cancel-sign-sub-inv 73.0%

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

      2. associate-*r* 73.0%

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

      3. exp-prod 65.1%

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

      4. remove-double-neg 65.1%

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

      5. neg-mul-1 65.1%

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

      6. sub-neg 65.1%

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

      7. exp-prod 73.0%

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

      8. associate-*r* 73.0%

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

      9. remove-double-neg 73.0%

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

      10. mul-1-neg 73.0%

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

      11. metadata-eval 73.0%

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

      12. *-lft-identity 73.0%

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

      13. mul-1-neg 73.0%

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

      14. remove-double-neg 73.0%

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

      15. associate-*r* 73.0%

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

      16. neg-mul-1 73.0%

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

   9. Simplified 73.0%

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

   10. Taylor expanded in eps around inf 74.2%

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

       1. associate-*r* 74.2%

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

       2. mul-1-neg 74.2%

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

   12. Simplified 74.2%

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

   if 1.25000000000000006e-5 < x

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 31.0%

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

   5. Taylor expanded in x around inf 14.4%

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

      1. mul-1-neg 14.4%

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

      2. distribute-rgt-neg-in 14.4%

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

      3. *-commutative 14.4%

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

      4. distribute-rgt-neg-in 14.4%

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

      5. mul-1-neg 14.4%

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

      6. distribute-lft-in 14.4%

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

      7. metadata-eval 14.4%

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

      8. associate-*r/ 14.4%

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

      9. metadata-eval 14.4%

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

   7. Simplified 14.4%

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

      1. distribute-lft-in 14.4%

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

      2. flip-+ 16.8%

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

   9. Applied egg-rr 16.8%

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

       1. *-commutative 16.8%

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

       2. neg-mul-1 16.8%

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

       3. *-commutative 16.8%

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

       4. neg-mul-1 16.8%

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

       5. sqr-neg 16.8%

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

       6. *-commutative 16.8%

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

       7. neg-mul-1 16.8%

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

   11. Simplified 16.8%

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

   12. Taylor expanded in eps around inf 65.9%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)}{\left(\varepsilon + -1\right) + \frac{\varepsilon + -1}{\varepsilon}}}{2}\\ \end{array} \]

Alternative 4: 78.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)}{\left(\varepsilon + -1\right) + \frac{\varepsilon + -1}{\varepsilon}}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 2.4e-7)
   (/ (* (exp (- x)) 2.0) 2.0)
   (/
    (*
     x
     (/
      (+ -1.0 (* (- 1.0 eps) (- 1.0 eps)))
      (+ (+ eps -1.0) (/ (+ eps -1.0) eps))))
    2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 2.4e-7) {
		tmp = (exp(-x) * 2.0) / 2.0;
	} else {
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.4d-7) then
        tmp = (exp(-x) * 2.0d0) / 2.0d0
    else
        tmp = (x * (((-1.0d0) + ((1.0d0 - eps) * (1.0d0 - eps))) / ((eps + (-1.0d0)) + ((eps + (-1.0d0)) / eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.4e-7) {
		tmp = (Math.exp(-x) * 2.0) / 2.0;
	} else {
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 2.4e-7:
		tmp = (math.exp(-x) * 2.0) / 2.0
	else:
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 2.4e-7)
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(-1.0 + Float64(Float64(1.0 - eps) * Float64(1.0 - eps))) / Float64(Float64(eps + -1.0) + Float64(Float64(eps + -1.0) / eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.4e-7)
		tmp = (exp(-x) * 2.0) / 2.0;
	else
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 2.4e-7], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(-1.0 + N[(N[(1.0 - eps), $MachinePrecision] * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(eps + -1.0), $MachinePrecision] + N[(N[(eps + -1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999979e-7

   1. Initial program 63.3%

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

   3. Simplified 40.1%

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

   4. Taylor expanded in eps around inf 97.7%

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

   5. Taylor expanded in eps around 0 75.4%

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

      1. cancel-sign-sub-inv 75.4%

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

      2. metadata-eval 75.4%

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

      3. distribute-rgt1-in 75.4%

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

      4. metadata-eval 75.4%

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

      5. neg-mul-1 75.4%

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

   7. Simplified 75.4%

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

   if 2.39999999999999979e-7 < x

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 31.0%

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

   5. Taylor expanded in x around inf 14.4%

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

      1. mul-1-neg 14.4%

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

      2. distribute-rgt-neg-in 14.4%

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

      3. *-commutative 14.4%

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

      4. distribute-rgt-neg-in 14.4%

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

      5. mul-1-neg 14.4%

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

      6. distribute-lft-in 14.4%

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

      7. metadata-eval 14.4%

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

      8. associate-*r/ 14.4%

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

      9. metadata-eval 14.4%

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

   7. Simplified 14.4%

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

      1. distribute-lft-in 14.4%

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

      2. flip-+ 16.8%

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

   9. Applied egg-rr 16.8%

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

       1. *-commutative 16.8%

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

       2. neg-mul-1 16.8%

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

       3. *-commutative 16.8%

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

       4. neg-mul-1 16.8%

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

       5. sqr-neg 16.8%

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

       6. *-commutative 16.8%

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

       7. neg-mul-1 16.8%

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

   11. Simplified 16.8%

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

   12. Taylor expanded in eps around inf 65.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)}{\left(\varepsilon + -1\right) + \frac{\varepsilon + -1}{\varepsilon}}}{2}\\ \end{array} \]

Alternative 5: 78.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)}{\left(\varepsilon + -1\right) + \frac{\varepsilon + -1}{\varepsilon}}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 3.8e-7)
   (/ (+ (exp (- x)) 1.0) 2.0)
   (/
    (*
     x
     (/
      (+ -1.0 (* (- 1.0 eps) (- 1.0 eps)))
      (+ (+ eps -1.0) (/ (+ eps -1.0) eps))))
    2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 3.8e-7) {
		tmp = (exp(-x) + 1.0) / 2.0;
	} else {
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 3.8d-7) then
        tmp = (exp(-x) + 1.0d0) / 2.0d0
    else
        tmp = (x * (((-1.0d0) + ((1.0d0 - eps) * (1.0d0 - eps))) / ((eps + (-1.0d0)) + ((eps + (-1.0d0)) / eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 3.8e-7) {
		tmp = (Math.exp(-x) + 1.0) / 2.0;
	} else {
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 3.8e-7:
		tmp = (math.exp(-x) + 1.0) / 2.0
	else:
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 3.8e-7)
		tmp = Float64(Float64(exp(Float64(-x)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(-1.0 + Float64(Float64(1.0 - eps) * Float64(1.0 - eps))) / Float64(Float64(eps + -1.0) + Float64(Float64(eps + -1.0) / eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 3.8e-7)
		tmp = (exp(-x) + 1.0) / 2.0;
	else
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 3.8e-7], N[(N[(N[Exp[(-x)], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(-1.0 + N[(N[(1.0 - eps), $MachinePrecision] * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(eps + -1.0), $MachinePrecision] + N[(N[(eps + -1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.80000000000000015e-7

   1. Initial program 63.3%

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

   3. Simplified 40.1%

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

   4. Taylor expanded in eps around inf 97.7%

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

   5. Taylor expanded in eps around inf 97.7%

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

      1. *-commutative 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in eps around 0 75.4%

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

      1. mul-1-neg 75.4%

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

   10. Simplified 75.4%

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

   if 3.80000000000000015e-7 < x

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 31.0%

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

   5. Taylor expanded in x around inf 14.4%

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

      1. mul-1-neg 14.4%

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

      2. distribute-rgt-neg-in 14.4%

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

      3. *-commutative 14.4%

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

      4. distribute-rgt-neg-in 14.4%

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

      5. mul-1-neg 14.4%

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

      6. distribute-lft-in 14.4%

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

      7. metadata-eval 14.4%

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

      8. associate-*r/ 14.4%

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

      9. metadata-eval 14.4%

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

   7. Simplified 14.4%

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

      1. distribute-lft-in 14.4%

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

      2. flip-+ 16.8%

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

   9. Applied egg-rr 16.8%

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

       1. *-commutative 16.8%

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

       2. neg-mul-1 16.8%

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

       3. *-commutative 16.8%

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

       4. neg-mul-1 16.8%

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

       5. sqr-neg 16.8%

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

       6. *-commutative 16.8%

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

       7. neg-mul-1 16.8%

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

   11. Simplified 16.8%

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

   12. Taylor expanded in eps around inf 65.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)}{\left(\varepsilon + -1\right) + \frac{\varepsilon + -1}{\varepsilon}}}{2}\\ \end{array} \]

Alternative 6: 71.3% accurate, 9.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)}{\left(\varepsilon + -1\right) + \frac{\varepsilon + -1}{\varepsilon}}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 9.2e-7)
   (/ (- 2.0 (* eps x)) 2.0)
   (/
    (*
     x
     (/
      (+ -1.0 (* (- 1.0 eps) (- 1.0 eps)))
      (+ (+ eps -1.0) (/ (+ eps -1.0) eps))))
    2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 9.2e-7) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 9.2d-7) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else
        tmp = (x * (((-1.0d0) + ((1.0d0 - eps) * (1.0d0 - eps))) / ((eps + (-1.0d0)) + ((eps + (-1.0d0)) / eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 9.2e-7) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 9.2e-7:
		tmp = (2.0 - (eps * x)) / 2.0
	else:
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 9.2e-7)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(-1.0 + Float64(Float64(1.0 - eps) * Float64(1.0 - eps))) / Float64(Float64(eps + -1.0) + Float64(Float64(eps + -1.0) / eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 9.2e-7)
		tmp = (2.0 - (eps * x)) / 2.0;
	else
		tmp = (x * ((-1.0 + ((1.0 - eps) * (1.0 - eps))) / ((eps + -1.0) + ((eps + -1.0) / eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 9.2e-7], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(-1.0 + N[(N[(1.0 - eps), $MachinePrecision] * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(eps + -1.0), $MachinePrecision] + N[(N[(eps + -1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.1999999999999998e-7

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in x around 0 57.0%

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

   5. Taylor expanded in eps around 0 60.1%

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

   6. Taylor expanded in eps around 0 60.1%

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

      1. mul-1-neg 60.1%

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

   8. Simplified 60.1%

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

   if 9.1999999999999998e-7 < x

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 31.0%

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

   5. Taylor expanded in x around inf 14.4%

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

      1. mul-1-neg 14.4%

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

      2. distribute-rgt-neg-in 14.4%

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

      3. *-commutative 14.4%

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

      4. distribute-rgt-neg-in 14.4%

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

      5. mul-1-neg 14.4%

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

      6. distribute-lft-in 14.4%

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

      7. metadata-eval 14.4%

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

      8. associate-*r/ 14.4%

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

      9. metadata-eval 14.4%

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

   7. Simplified 14.4%

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

      1. distribute-lft-in 14.4%

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

      2. flip-+ 16.8%

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

   9. Applied egg-rr 16.8%

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

       1. *-commutative 16.8%

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

       2. neg-mul-1 16.8%

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

       3. *-commutative 16.8%

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

       4. neg-mul-1 16.8%

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

       5. sqr-neg 16.8%

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

       6. *-commutative 16.8%

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

       7. neg-mul-1 16.8%

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

   11. Simplified 16.8%

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

   12. Taylor expanded in eps around inf 65.9%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)}{\left(\varepsilon + -1\right) + \frac{\varepsilon + -1}{\varepsilon}}}{2}\\ \end{array} \]

Alternative 7: 63.1% accurate, 25.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0085:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 0.0085) (/ (- 2.0 (* eps x)) 2.0) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 0.0085) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 0.0085d0) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 0.0085) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 0.0085:
		tmp = (2.0 - (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 0.0085)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 0.0085)
		tmp = (2.0 - (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 0.0085], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.0085000000000000006

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in x around 0 57.0%

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

   5. Taylor expanded in eps around 0 60.1%

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

   6. Taylor expanded in eps around 0 60.1%

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

      1. mul-1-neg 60.1%

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

   8. Simplified 60.1%

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

   if 0.0085000000000000006 < 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%

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

   4. Taylor expanded in eps around 0 50.8%

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

      1. neg-mul-1 50.8%

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

      2. rec-exp 50.8%

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

      3. neg-mul-1 50.8%

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

      4. div-sub 50.8%

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

      5. +-inverses 50.8%

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

   6. Simplified 50.8%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0085:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 63.6% accurate, 28.2× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.6:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{elif}\;x \leq 530:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -0.6) (/ (* eps (- x)) 2.0) (if (<= x 530.0) 1.0 0.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -0.6) {
		tmp = (eps * -x) / 2.0;
	} else if (x <= 530.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.6d0)) then
        tmp = (eps * -x) / 2.0d0
    else if (x <= 530.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.6) {
		tmp = (eps * -x) / 2.0;
	} else if (x <= 530.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -0.6:
		tmp = (eps * -x) / 2.0
	elif x <= 530.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -0.6)
		tmp = Float64(Float64(eps * Float64(-x)) / 2.0);
	elseif (x <= 530.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.6)
		tmp = (eps * -x) / 2.0;
	elseif (x <= 530.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -0.6], N[(N[(eps * (-x)), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 530.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.599999999999999978

   1. Initial program 97.4%

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

      1. sub-neg 97.4%

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

      2. neg-sub0 97.4%

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

      3. associate-+r- 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in x around 0 56.7%

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

   5. Taylor expanded in eps around inf 19.9%

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

      1. mul-1-neg 19.9%

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

      2. *-commutative 19.9%

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

      3. distribute-rgt-neg-in 19.9%

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

   7. Simplified 19.9%

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

   if -0.599999999999999978 < x < 530

   1. Initial program 54.4%

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

      1. sub-neg 54.4%

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

      2. neg-sub0 54.4%

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

      3. associate-+r- 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in x around 0 71.1%

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

   if 530 < 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%

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

   4. Taylor expanded in eps around 0 51.4%

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

      1. neg-mul-1 51.4%

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

      2. rec-exp 51.4%

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

      3. neg-mul-1 51.4%

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

      4. div-sub 51.4%

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

      5. +-inverses 51.4%

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

   6. Simplified 51.4%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.6:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{elif}\;x \leq 530:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 56.4% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 520.0) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 520.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 520.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 520.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 520.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 520.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 520

   1. Initial program 63.5%

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

      1. sub-neg 63.5%

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

      2. neg-sub0 63.5%

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

      3. associate-+r- 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in x around 0 56.7%

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

   if 520 < 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%

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

   4. Taylor expanded in eps around 0 51.4%

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

      1. neg-mul-1 51.4%

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

      2. rec-exp 51.4%

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

      3. neg-mul-1 51.4%

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

      4. div-sub 51.4%

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

      5. +-inverses 51.4%

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

   6. Simplified 51.4%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 15.8% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ 0 \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 0.0)

C

eps = abs(eps);
double code(double x, double eps) {
	return 0.0;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	return 0.0;
}

Python

eps = abs(eps)
def code(x, eps):
	return 0.0

Julia

eps = abs(eps)
function code(x, eps)
	return 0.0
end

MATLAB

eps = abs(eps)
function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := 0.0

Derivation

1. Initial program 74.5%

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

3. Simplified 74.5%

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

4. Taylor expanded in eps around 0 17.0%

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

   1. neg-mul-1 17.0%

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

   2. rec-exp 17.0%

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

   3. neg-mul-1 17.0%

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

   4. div-sub 17.0%

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

   5. +-inverses 17.2%

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

6. Simplified 17.2%

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

7. Final simplification 17.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023271 
(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))