NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 99.7%

Time: 17.8s

Alternatives: 12

Speedup: 2.0×

• 38.4% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \mathbf{if}\;\varepsilon \leq 1.529 \cdot 10^{-38}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ x 1.0) (exp (- x)))))
   (if (<= eps 1.529e-38)
     (/ (+ t_0 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 = (x + 1.0) * exp(-x);
	double tmp;
	if (eps <= 1.529e-38) {
		tmp = (t_0 + 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 = (x + 1.0d0) * exp(-x)
    if (eps <= 1.529d-38) then
        tmp = (t_0 + 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 = (x + 1.0) * Math.exp(-x);
	double tmp;
	if (eps <= 1.529e-38) {
		tmp = (t_0 + 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 = (x + 1.0) * math.exp(-x)
	tmp = 0
	if eps <= 1.529e-38:
		tmp = (t_0 + 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 = Float64(Float64(x + 1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps <= 1.529e-38)
		tmp = Float64(Float64(t_0 + 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 = (x + 1.0) * exp(-x);
	tmp = 0.0;
	if (eps <= 1.529e-38)
		tmp = (t_0 + 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[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, 1.529e-38], N[(N[(t$95$0 + t$95$0), $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 < 1.529e-38

   1. Initial program 68.2%

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

      1. div-sub 68.2%

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

      2. +-rgt-identity 68.2%

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

      3. div-sub 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in eps around 0 66.6%

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

      1. *-commutative 66.6%

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

      2. distribute-lft1-in 66.6%

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

      3. neg-mul-1 66.6%

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

      4. distribute-lft-out 66.6%

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

      5. mul-1-neg 66.6%

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

      6. *-commutative 66.6%

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

      7. distribute-lft1-in 67.2%

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

      8. neg-mul-1 67.2%

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

   6. Simplified 67.2%

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

   if 1.529e-38 < eps

   1. Initial program 97.3%

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

      1. div-sub 97.3%

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

      2. +-rgt-identity 97.3%

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

      3. div-sub 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.529 \cdot 10^{-38}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (x + 1.0) * exp(-x);
	double tmp;
	if (eps <= 1.0) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp((eps * -x)) + 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 = (x + 1.0d0) * exp(-x)
    if (eps <= 1.0d0) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp((eps * -x)) + 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 = (x + 1.0) * Math.exp(-x);
	double tmp;
	if (eps <= 1.0) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp((eps * -x)) + Math.exp((eps * x))) / 2.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = (x + 1.0) * exp(-x);
	tmp = 0.0;
	if (eps <= 1.0)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp((eps * -x)) + 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[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, 1.0], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 68.2%

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

      1. div-sub 68.2%

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

      2. +-rgt-identity 68.2%

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

      3. div-sub 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in eps around 0 67.7%

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

      1. *-commutative 67.7%

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

      2. distribute-lft1-in 67.7%

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

      3. neg-mul-1 67.7%

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

      4. distribute-lft-out 67.7%

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

      5. mul-1-neg 67.7%

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

      6. *-commutative 67.7%

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

      7. distribute-lft1-in 68.2%

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

      8. neg-mul-1 68.2%

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

   6. Simplified 68.2%

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

   if 1 < eps

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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 eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 76.5%

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

Alternative 3: 83.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-293}:\\ \;\;\;\;\frac{t_0 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 10^{+14} \lor \neg \left(x \leq 10^{+46}\right) \land x \leq 1.5 \cdot 10^{+159}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{x \cdot \left(\varepsilon + 1\right)} + 1\right) + \varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot 2}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -2e-293)
     (/ (+ t_0 (exp (* x (- -1.0 eps)))) 2.0)
     (if (or (<= x 1e+14) (and (not (<= x 1e+46)) (<= x 1.5e+159)))
       (* 0.5 (+ (+ (exp (* x (+ eps 1.0))) 1.0) (* eps x)))
       (/ (* t_0 2.0) 2.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -2e-293) {
		tmp = (t_0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 1e+14) || (!(x <= 1e+46) && (x <= 1.5e+159))) {
		tmp = 0.5 * ((exp((x * (eps + 1.0))) + 1.0) + (eps * x));
	} else {
		tmp = (t_0 * 2.0) / 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 (x <= (-2d-293)) then
        tmp = (t_0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if ((x <= 1d+14) .or. (.not. (x <= 1d+46)) .and. (x <= 1.5d+159)) then
        tmp = 0.5d0 * ((exp((x * (eps + 1.0d0))) + 1.0d0) + (eps * x))
    else
        tmp = (t_0 * 2.0d0) / 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 (x <= -2e-293) {
		tmp = (t_0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 1e+14) || (!(x <= 1e+46) && (x <= 1.5e+159))) {
		tmp = 0.5 * ((Math.exp((x * (eps + 1.0))) + 1.0) + (eps * x));
	} else {
		tmp = (t_0 * 2.0) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -2e-293:
		tmp = (t_0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif (x <= 1e+14) or (not (x <= 1e+46) and (x <= 1.5e+159)):
		tmp = 0.5 * ((math.exp((x * (eps + 1.0))) + 1.0) + (eps * x))
	else:
		tmp = (t_0 * 2.0) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -2e-293)
		tmp = Float64(Float64(t_0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif ((x <= 1e+14) || (!(x <= 1e+46) && (x <= 1.5e+159)))
		tmp = Float64(0.5 * Float64(Float64(exp(Float64(x * Float64(eps + 1.0))) + 1.0) + Float64(eps * x)));
	else
		tmp = Float64(Float64(t_0 * 2.0) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = exp(-x);
	tmp = 0.0;
	if (x <= -2e-293)
		tmp = (t_0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif ((x <= 1e+14) || (~((x <= 1e+46)) && (x <= 1.5e+159)))
		tmp = 0.5 * ((exp((x * (eps + 1.0))) + 1.0) + (eps * x));
	else
		tmp = (t_0 * 2.0) / 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[x, -2e-293], N[(N[(t$95$0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1e+14], And[N[Not[LessEqual[x, 1e+46]], $MachinePrecision], LessEqual[x, 1.5e+159]]], N[(0.5 * N[(N[(N[Exp[N[(x * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] + N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0000000000000001e-293

   1. Initial program 78.7%

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

      1. div-sub 78.7%

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

      2. +-rgt-identity 78.7%

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

      3. div-sub 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in eps around inf 96.6%

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

   5. Taylor expanded in eps around 0 82.0%

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

   if -2.0000000000000001e-293 < x < 1e14 or 9.9999999999999999e45 < x < 1.5000000000000001e159

   1. Initial program 64.7%

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

      1. div-sub 64.7%

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

      2. +-rgt-identity 64.7%

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

      3. div-sub 64.7%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in x around 0 35.9%

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

   5. Taylor expanded in eps around inf 70.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 + \left(-1 \cdot x + x\right)\right) - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]

   6. Taylor expanded in x around inf 70.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-out 70.2%

         \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + \varepsilon \cdot x\right)} \]

      2. mul-1-neg 70.2%

         \[\leadsto 0.5 \cdot \left(\left(1 - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}\right) + \varepsilon \cdot x\right) \]

      3. mul-1-neg 70.2%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{\color{blue}{-\left(\varepsilon + 1\right) \cdot x}}\right)\right) + \varepsilon \cdot x\right) \]

      4. *-commutative 70.2%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{x \cdot \left(\varepsilon + 1\right)}}\right)\right) + \varepsilon \cdot x\right) \]

   8. Simplified 70.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)\right) + \varepsilon \cdot x\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 48.7%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\sqrt{x \cdot \left(\varepsilon + 1\right)} \cdot \sqrt{x \cdot \left(\varepsilon + 1\right)}}}\right)\right) + \varepsilon \cdot x\right) \]

      2. sqrt-unprod 58.1%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\sqrt{\left(x \cdot \left(\varepsilon + 1\right)\right) \cdot \left(x \cdot \left(\varepsilon + 1\right)\right)}}}\right)\right) + \varepsilon \cdot x\right) \]

      3. sqr-neg 58.1%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\sqrt{\color{blue}{\left(-x \cdot \left(\varepsilon + 1\right)\right) \cdot \left(-x \cdot \left(\varepsilon + 1\right)\right)}}}\right)\right) + \varepsilon \cdot x\right) \]

      4. sqrt-unprod 9.4%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\sqrt{-x \cdot \left(\varepsilon + 1\right)} \cdot \sqrt{-x \cdot \left(\varepsilon + 1\right)}}}\right)\right) + \varepsilon \cdot x\right) \]

      5. add-sqr-sqrt 69.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\left(-x \cdot \left(\varepsilon + 1\right)\right)}}\right)\right) + \varepsilon \cdot x\right) \]

      6. neg-sub0 69.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\left(0 - x \cdot \left(\varepsilon + 1\right)\right)}}\right)\right) + \varepsilon \cdot x\right) \]

      7. distribute-rgt-in 69.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\left(0 - \color{blue}{\left(\varepsilon \cdot x + 1 \cdot x\right)}\right)}\right)\right) + \varepsilon \cdot x\right) \]

      8. *-un-lft-identity 69.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\left(0 - \left(\varepsilon \cdot x + \color{blue}{x}\right)\right)}\right)\right) + \varepsilon \cdot x\right) \]

      9. fma-def 69.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\left(0 - \color{blue}{\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}\right)\right) + \varepsilon \cdot x\right) \]

   10. Applied egg-rr 69.9%

       \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\left(0 - \mathsf{fma}\left(\varepsilon, x, x\right)\right)}}\right)\right) + \varepsilon \cdot x\right) \]
   11. Step-by-step derivation

       1. neg-sub0 69.9%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\left(-\mathsf{fma}\left(\varepsilon, x, x\right)\right)}}\right)\right) + \varepsilon \cdot x\right) \]

       2. fma-def 69.9%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\left(-\color{blue}{\left(\varepsilon \cdot x + x\right)}\right)}\right)\right) + \varepsilon \cdot x\right) \]

       3. distribute-lft1-in 69.9%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\left(-\color{blue}{\left(\varepsilon + 1\right) \cdot x}\right)}\right)\right) + \varepsilon \cdot x\right) \]

       4. distribute-lft-neg-in 69.9%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\left(-\left(\varepsilon + 1\right)\right) \cdot x}}\right)\right) + \varepsilon \cdot x\right) \]

       5. *-commutative 69.9%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{x \cdot \left(-\left(\varepsilon + 1\right)\right)}}\right)\right) + \varepsilon \cdot x\right) \]

       6. neg-sub0 69.9%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \color{blue}{\left(0 - \left(\varepsilon + 1\right)\right)}}\right)\right) + \varepsilon \cdot x\right) \]

       7. +-commutative 69.9%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \left(0 - \color{blue}{\left(1 + \varepsilon\right)}\right)}\right)\right) + \varepsilon \cdot x\right) \]

       8. associate--r+ 69.9%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \color{blue}{\left(\left(0 - 1\right) - \varepsilon\right)}}\right)\right) + \varepsilon \cdot x\right) \]

       9. metadata-eval 69.9%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \left(\color{blue}{-1} - \varepsilon\right)}\right)\right) + \varepsilon \cdot x\right) \]

   12. Simplified 69.9%

       \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{x \cdot \left(-1 - \varepsilon\right)}}\right)\right) + \varepsilon \cdot x\right) \]

   if 1e14 < x < 9.9999999999999999e45 or 1.5000000000000001e159 < 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. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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 eps around inf 100.0%

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

   5. Taylor expanded in eps around 0 84.3%

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

   6. Taylor expanded in eps around 0 68.7%

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

      1. cancel-sign-sub-inv 68.7%

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

      2. metadata-eval 68.7%

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

      3. distribute-rgt1-in 68.7%

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

      4. metadata-eval 68.7%

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

      5. neg-mul-1 68.7%

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

   8. Simplified 68.7%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-293}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 10^{+14} \lor \neg \left(x \leq 10^{+46}\right) \land x \leq 1.5 \cdot 10^{+159}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{x \cdot \left(\varepsilon + 1\right)} + 1\right) + \varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \end{array} \]

Alternative 4: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 1.0) {
		tmp = (exp(-x) * 2.0) / 2.0;
	} else {
		tmp = (exp((eps * -x)) + 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) :: tmp
    if (eps <= 1.0d0) then
        tmp = (exp(-x) * 2.0d0) / 2.0d0
    else
        tmp = (exp((eps * -x)) + 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 tmp;
	if (eps <= 1.0) {
		tmp = (Math.exp(-x) * 2.0) / 2.0;
	} else {
		tmp = (Math.exp((eps * -x)) + Math.exp((eps * x))) / 2.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 68.2%

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

      1. div-sub 68.2%

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

      2. +-rgt-identity 68.2%

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

      3. div-sub 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in eps around inf 97.5%

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

   5. Taylor expanded in eps around 0 89.5%

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

   6. Taylor expanded in eps around 0 77.4%

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

      1. cancel-sign-sub-inv 77.4%

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

      2. metadata-eval 77.4%

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

      3. distribute-rgt1-in 77.4%

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

      4. metadata-eval 77.4%

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

      5. neg-mul-1 77.4%

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

   8. Simplified 77.4%

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

   if 1 < eps

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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 eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 83.3%

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

Alternative 5: 84.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -18:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \left(e^{\left(-x\right) - \varepsilon \cdot x} + \left(\varepsilon \cdot x + 1\right)\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+14} \lor \neg \left(x \leq 3 \cdot 10^{+45}\right) \land x \leq 10^{+159}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{x \cdot \left(\varepsilon + 1\right)} + 1\right) + \varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -18.0)
   (+ 0.5 (/ 0.5 (exp x)))
   (if (<= x 5e-295)
     (* 0.5 (+ (exp (- (- x) (* eps x))) (+ (* eps x) 1.0)))
     (if (or (<= x 3e+14) (and (not (<= x 3e+45)) (<= x 1e+159)))
       (* 0.5 (+ (+ (exp (* x (+ eps 1.0))) 1.0) (* eps x)))
       (/ (* (exp (- x)) 2.0) 2.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -18.0) {
		tmp = 0.5 + (0.5 / exp(x));
	} else if (x <= 5e-295) {
		tmp = 0.5 * (exp((-x - (eps * x))) + ((eps * x) + 1.0));
	} else if ((x <= 3e+14) || (!(x <= 3e+45) && (x <= 1e+159))) {
		tmp = 0.5 * ((exp((x * (eps + 1.0))) + 1.0) + (eps * x));
	} else {
		tmp = (exp(-x) * 2.0) / 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 <= (-18.0d0)) then
        tmp = 0.5d0 + (0.5d0 / exp(x))
    else if (x <= 5d-295) then
        tmp = 0.5d0 * (exp((-x - (eps * x))) + ((eps * x) + 1.0d0))
    else if ((x <= 3d+14) .or. (.not. (x <= 3d+45)) .and. (x <= 1d+159)) then
        tmp = 0.5d0 * ((exp((x * (eps + 1.0d0))) + 1.0d0) + (eps * x))
    else
        tmp = (exp(-x) * 2.0d0) / 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 <= -18.0) {
		tmp = 0.5 + (0.5 / Math.exp(x));
	} else if (x <= 5e-295) {
		tmp = 0.5 * (Math.exp((-x - (eps * x))) + ((eps * x) + 1.0));
	} else if ((x <= 3e+14) || (!(x <= 3e+45) && (x <= 1e+159))) {
		tmp = 0.5 * ((Math.exp((x * (eps + 1.0))) + 1.0) + (eps * x));
	} else {
		tmp = (Math.exp(-x) * 2.0) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -18.0:
		tmp = 0.5 + (0.5 / math.exp(x))
	elif x <= 5e-295:
		tmp = 0.5 * (math.exp((-x - (eps * x))) + ((eps * x) + 1.0))
	elif (x <= 3e+14) or (not (x <= 3e+45) and (x <= 1e+159)):
		tmp = 0.5 * ((math.exp((x * (eps + 1.0))) + 1.0) + (eps * x))
	else:
		tmp = (math.exp(-x) * 2.0) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -18.0)
		tmp = Float64(0.5 + Float64(0.5 / exp(x)));
	elseif (x <= 5e-295)
		tmp = Float64(0.5 * Float64(exp(Float64(Float64(-x) - Float64(eps * x))) + Float64(Float64(eps * x) + 1.0)));
	elseif ((x <= 3e+14) || (!(x <= 3e+45) && (x <= 1e+159)))
		tmp = Float64(0.5 * Float64(Float64(exp(Float64(x * Float64(eps + 1.0))) + 1.0) + Float64(eps * x)));
	else
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -18.0)
		tmp = 0.5 + (0.5 / exp(x));
	elseif (x <= 5e-295)
		tmp = 0.5 * (exp((-x - (eps * x))) + ((eps * x) + 1.0));
	elseif ((x <= 3e+14) || (~((x <= 3e+45)) && (x <= 1e+159)))
		tmp = 0.5 * ((exp((x * (eps + 1.0))) + 1.0) + (eps * x));
	else
		tmp = (exp(-x) * 2.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -18.0], N[(0.5 + N[(0.5 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-295], N[(0.5 * N[(N[Exp[N[((-x) - N[(eps * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(eps * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 3e+14], And[N[Not[LessEqual[x, 3e+45]], $MachinePrecision], LessEqual[x, 1e+159]]], N[(0.5 * N[(N[(N[Exp[N[(x * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] + N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -18

   1. Initial program 93.0%

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

      1. div-sub 93.0%

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

      2. +-rgt-identity 93.0%

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

      3. div-sub 93.0%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in x around 0 47.3%

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

   5. Taylor expanded in eps around inf 44.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 + \left(-1 \cdot x + x\right)\right) - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]

   6. Taylor expanded in x around inf 44.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-out 44.7%

         \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + \varepsilon \cdot x\right)} \]

      2. mul-1-neg 44.7%

         \[\leadsto 0.5 \cdot \left(\left(1 - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}\right) + \varepsilon \cdot x\right) \]

      3. mul-1-neg 44.7%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{\color{blue}{-\left(\varepsilon + 1\right) \cdot x}}\right)\right) + \varepsilon \cdot x\right) \]

      4. *-commutative 44.7%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{x \cdot \left(\varepsilon + 1\right)}}\right)\right) + \varepsilon \cdot x\right) \]

   8. Simplified 44.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)\right) + \varepsilon \cdot x\right)} \]

   9. Taylor expanded in eps around 0 92.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 + e^{-x}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in 92.9%

          \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{-x}} \]

       2. metadata-eval 92.9%

          \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{-x} \]

       3. exp-neg 92.9%

          \[\leadsto 0.5 + 0.5 \cdot \color{blue}{\frac{1}{e^{x}}} \]

       4. associate-*r/ 92.9%

          \[\leadsto 0.5 + \color{blue}{\frac{0.5 \cdot 1}{e^{x}}} \]

       5. metadata-eval 92.9%

          \[\leadsto 0.5 + \frac{\color{blue}{0.5}}{e^{x}} \]

   11. Simplified 92.9%

       \[\leadsto \color{blue}{0.5 + \frac{0.5}{e^{x}}} \]

   if -18 < x < 5.00000000000000008e-295

   1. Initial program 67.7%

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

      1. div-sub 67.7%

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

      2. +-rgt-identity 67.7%

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

      3. div-sub 67.7%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in x around 0 46.5%

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

   5. Taylor expanded in eps around inf 77.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 + \left(-1 \cdot x + x\right)\right) - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]

   6. Taylor expanded in x around inf 77.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-out 77.9%

         \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + \varepsilon \cdot x\right)} \]

      2. mul-1-neg 77.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}\right) + \varepsilon \cdot x\right) \]

      3. mul-1-neg 77.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{\color{blue}{-\left(\varepsilon + 1\right) \cdot x}}\right)\right) + \varepsilon \cdot x\right) \]

      4. *-commutative 77.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{x \cdot \left(\varepsilon + 1\right)}}\right)\right) + \varepsilon \cdot x\right) \]

   8. Simplified 77.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)\right) + \varepsilon \cdot x\right)} \]

   9. Taylor expanded in x around inf 77.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{-\left(\varepsilon \cdot x + x\right)} + \left(1 + \varepsilon \cdot x\right)\right)} \]

   if 5.00000000000000008e-295 < x < 3e14 or 3.00000000000000011e45 < x < 9.9999999999999993e158

   1. Initial program 65.5%

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

      1. div-sub 65.5%

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

      2. +-rgt-identity 65.5%

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

      3. div-sub 65.5%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in x around 0 35.3%

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

   5. Taylor expanded in eps around inf 68.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 + \left(-1 \cdot x + x\right)\right) - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]

   6. Taylor expanded in x around inf 68.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-out 68.7%

         \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + \varepsilon \cdot x\right)} \]

      2. mul-1-neg 68.7%

         \[\leadsto 0.5 \cdot \left(\left(1 - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}\right) + \varepsilon \cdot x\right) \]

      3. mul-1-neg 68.7%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{\color{blue}{-\left(\varepsilon + 1\right) \cdot x}}\right)\right) + \varepsilon \cdot x\right) \]

      4. *-commutative 68.7%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{x \cdot \left(\varepsilon + 1\right)}}\right)\right) + \varepsilon \cdot x\right) \]

   8. Simplified 68.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)\right) + \varepsilon \cdot x\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 48.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\sqrt{x \cdot \left(\varepsilon + 1\right)} \cdot \sqrt{x \cdot \left(\varepsilon + 1\right)}}}\right)\right) + \varepsilon \cdot x\right) \]

      2. sqrt-unprod 55.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\sqrt{\left(x \cdot \left(\varepsilon + 1\right)\right) \cdot \left(x \cdot \left(\varepsilon + 1\right)\right)}}}\right)\right) + \varepsilon \cdot x\right) \]

      3. sqr-neg 55.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\sqrt{\color{blue}{\left(-x \cdot \left(\varepsilon + 1\right)\right) \cdot \left(-x \cdot \left(\varepsilon + 1\right)\right)}}}\right)\right) + \varepsilon \cdot x\right) \]

      4. sqrt-unprod 7.0%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\sqrt{-x \cdot \left(\varepsilon + 1\right)} \cdot \sqrt{-x \cdot \left(\varepsilon + 1\right)}}}\right)\right) + \varepsilon \cdot x\right) \]

      5. add-sqr-sqrt 68.7%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\left(-x \cdot \left(\varepsilon + 1\right)\right)}}\right)\right) + \varepsilon \cdot x\right) \]

      6. neg-sub0 68.7%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\left(0 - x \cdot \left(\varepsilon + 1\right)\right)}}\right)\right) + \varepsilon \cdot x\right) \]

      7. distribute-rgt-in 68.7%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\left(0 - \color{blue}{\left(\varepsilon \cdot x + 1 \cdot x\right)}\right)}\right)\right) + \varepsilon \cdot x\right) \]

      8. *-un-lft-identity 68.7%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\left(0 - \left(\varepsilon \cdot x + \color{blue}{x}\right)\right)}\right)\right) + \varepsilon \cdot x\right) \]

      9. fma-def 68.7%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\left(0 - \color{blue}{\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}\right)\right) + \varepsilon \cdot x\right) \]

   10. Applied egg-rr 68.7%

       \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\left(0 - \mathsf{fma}\left(\varepsilon, x, x\right)\right)}}\right)\right) + \varepsilon \cdot x\right) \]
   11. Step-by-step derivation

       1. neg-sub0 68.7%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\left(-\mathsf{fma}\left(\varepsilon, x, x\right)\right)}}\right)\right) + \varepsilon \cdot x\right) \]

       2. fma-def 68.7%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\left(-\color{blue}{\left(\varepsilon \cdot x + x\right)}\right)}\right)\right) + \varepsilon \cdot x\right) \]

       3. distribute-lft1-in 68.7%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\left(-\color{blue}{\left(\varepsilon + 1\right) \cdot x}\right)}\right)\right) + \varepsilon \cdot x\right) \]

       4. distribute-lft-neg-in 68.7%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{\left(-\left(\varepsilon + 1\right)\right) \cdot x}}\right)\right) + \varepsilon \cdot x\right) \]

       5. *-commutative 68.7%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{x \cdot \left(-\left(\varepsilon + 1\right)\right)}}\right)\right) + \varepsilon \cdot x\right) \]

       6. neg-sub0 68.7%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \color{blue}{\left(0 - \left(\varepsilon + 1\right)\right)}}\right)\right) + \varepsilon \cdot x\right) \]

       7. +-commutative 68.7%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \left(0 - \color{blue}{\left(1 + \varepsilon\right)}\right)}\right)\right) + \varepsilon \cdot x\right) \]

       8. associate--r+ 68.7%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \color{blue}{\left(\left(0 - 1\right) - \varepsilon\right)}}\right)\right) + \varepsilon \cdot x\right) \]

       9. metadata-eval 68.7%

          \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \left(\color{blue}{-1} - \varepsilon\right)}\right)\right) + \varepsilon \cdot x\right) \]

   12. Simplified 68.7%

       \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{x \cdot \left(-1 - \varepsilon\right)}}\right)\right) + \varepsilon \cdot x\right) \]

   if 3e14 < x < 3.00000000000000011e45 or 9.9999999999999993e158 < 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. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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 eps around inf 100.0%

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

   5. Taylor expanded in eps around 0 84.3%

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

   6. Taylor expanded in eps around 0 68.7%

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

      1. cancel-sign-sub-inv 68.7%

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

      2. metadata-eval 68.7%

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

      3. distribute-rgt1-in 68.7%

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

      4. metadata-eval 68.7%

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

      5. neg-mul-1 68.7%

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

   8. Simplified 68.7%

      \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -18:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \left(e^{\left(-x\right) - \varepsilon \cdot x} + \left(\varepsilon \cdot x + 1\right)\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+14} \lor \neg \left(x \leq 3 \cdot 10^{+45}\right) \land x \leq 10^{+159}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{x \cdot \left(\varepsilon + 1\right)} + 1\right) + \varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \end{array} \]

Alternative 6: 78.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 1.0) {
		tmp = (exp(-x) * 2.0) / 2.0;
	} else {
		tmp = 0.5 * (exp((-x - (eps * x))) + ((eps * x) + 1.0));
	}
	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 (eps <= 1.0d0) then
        tmp = (exp(-x) * 2.0d0) / 2.0d0
    else
        tmp = 0.5d0 * (exp((-x - (eps * x))) + ((eps * x) + 1.0d0))
    end if
    code = tmp
end function

Java

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

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if eps <= 1.0:
		tmp = (math.exp(-x) * 2.0) / 2.0
	else:
		tmp = 0.5 * (math.exp((-x - (eps * x))) + ((eps * x) + 1.0))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 1.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.5 * N[(N[Exp[N[((-x) - N[(eps * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(eps * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 68.2%

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

      1. div-sub 68.2%

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

      2. +-rgt-identity 68.2%

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

      3. div-sub 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in eps around inf 97.5%

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

   5. Taylor expanded in eps around 0 89.5%

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

   6. Taylor expanded in eps around 0 77.4%

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

      1. cancel-sign-sub-inv 77.4%

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

      2. metadata-eval 77.4%

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

      3. distribute-rgt1-in 77.4%

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

      4. metadata-eval 77.4%

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

      5. neg-mul-1 77.4%

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

   8. Simplified 77.4%

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

   if 1 < eps

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in x around 0 65.5%

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

   5. Taylor expanded in eps around inf 65.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 + \left(-1 \cdot x + x\right)\right) - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]

   6. Taylor expanded in x around inf 65.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-out 65.5%

         \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + \varepsilon \cdot x\right)} \]

      2. mul-1-neg 65.5%

         \[\leadsto 0.5 \cdot \left(\left(1 - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}\right) + \varepsilon \cdot x\right) \]

      3. mul-1-neg 65.5%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{\color{blue}{-\left(\varepsilon + 1\right) \cdot x}}\right)\right) + \varepsilon \cdot x\right) \]

      4. *-commutative 65.5%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{x \cdot \left(\varepsilon + 1\right)}}\right)\right) + \varepsilon \cdot x\right) \]

   8. Simplified 65.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)\right) + \varepsilon \cdot x\right)} \]

   9. Taylor expanded in x around inf 65.5%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{\left(-x\right) - \varepsilon \cdot x} + \left(\varepsilon \cdot x + 1\right)\right)\\ \end{array} \]

Alternative 7: 69.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.1 \cdot 10^{+243}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 3.1e+243) (/ (* (exp (- x)) 2.0) 2.0) (* 0.5 (* eps x))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 3.1e+243) {
		tmp = (exp(-x) * 2.0) / 2.0;
	} else {
		tmp = 0.5 * (eps * x);
	}
	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 (eps <= 3.1d+243) then
        tmp = (exp(-x) * 2.0d0) / 2.0d0
    else
        tmp = 0.5d0 * (eps * x)
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (eps <= 3.1e+243) {
		tmp = (Math.exp(-x) * 2.0) / 2.0;
	} else {
		tmp = 0.5 * (eps * x);
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if eps <= 3.1e+243:
		tmp = (math.exp(-x) * 2.0) / 2.0
	else:
		tmp = 0.5 * (eps * x)
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (eps <= 3.1e+243)
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) / 2.0);
	else
		tmp = Float64(0.5 * Float64(eps * x));
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 3.1e+243)
		tmp = (exp(-x) * 2.0) / 2.0;
	else
		tmp = 0.5 * (eps * x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 3.1e+243], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.5 * N[(eps * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 3.1e243

   1. Initial program 74.4%

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

      1. div-sub 74.4%

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

      2. +-rgt-identity 74.4%

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

      3. div-sub 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in eps around inf 98.0%

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

   5. Taylor expanded in eps around 0 85.0%

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

   6. Taylor expanded in eps around 0 73.7%

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

      1. cancel-sign-sub-inv 73.7%

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

      2. metadata-eval 73.7%

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

      3. distribute-rgt1-in 73.7%

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

      4. metadata-eval 73.7%

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

      5. neg-mul-1 73.7%

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

   8. Simplified 73.7%

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

   if 3.1e243 < eps

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in x around 0 70.7%

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

   5. Taylor expanded in eps around inf 39.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 39.7%

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

      2. *-commutative 39.7%

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

   7. Simplified 39.7%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.1 \cdot 10^{+243}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \]

Alternative 8: 70.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 980000000:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 980000000.0) (+ 0.5 (/ 0.5 (exp x))) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 980000000.0) {
		tmp = 0.5 + (0.5 / exp(x));
	} 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 <= 980000000.0d0) then
        tmp = 0.5d0 + (0.5d0 / exp(x))
    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 <= 980000000.0) {
		tmp = 0.5 + (0.5 / Math.exp(x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 980000000.0:
		tmp = 0.5 + (0.5 / math.exp(x))
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 980000000.0)
		tmp = Float64(0.5 + Float64(0.5 / exp(x)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 980000000.0)
		tmp = 0.5 + (0.5 / exp(x));
	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, 980000000.0], N[(0.5 + N[(0.5 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 9.8e8

   1. Initial program 66.6%

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

      1. div-sub 66.6%

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

      2. +-rgt-identity 66.6%

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

      3. div-sub 66.6%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in x around 0 42.8%

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

   5. Taylor expanded in eps around inf 72.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 + \left(-1 \cdot x + x\right)\right) - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]

   6. Taylor expanded in x around inf 72.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-out 72.9%

         \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + \varepsilon \cdot x\right)} \]

      2. mul-1-neg 72.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}\right) + \varepsilon \cdot x\right) \]

      3. mul-1-neg 72.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{\color{blue}{-\left(\varepsilon + 1\right) \cdot x}}\right)\right) + \varepsilon \cdot x\right) \]

      4. *-commutative 72.9%

         \[\leadsto 0.5 \cdot \left(\left(1 - \left(-e^{-\color{blue}{x \cdot \left(\varepsilon + 1\right)}}\right)\right) + \varepsilon \cdot x\right) \]

   8. Simplified 72.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 - \left(-e^{-x \cdot \left(\varepsilon + 1\right)}\right)\right) + \varepsilon \cdot x\right)} \]

   9. Taylor expanded in eps around 0 74.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 + e^{-x}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in 74.3%

          \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{-x}} \]

       2. metadata-eval 74.3%

          \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{-x} \]

       3. exp-neg 74.3%

          \[\leadsto 0.5 + 0.5 \cdot \color{blue}{\frac{1}{e^{x}}} \]

       4. associate-*r/ 74.3%

          \[\leadsto 0.5 + \color{blue}{\frac{0.5 \cdot 1}{e^{x}}} \]

       5. metadata-eval 74.3%

          \[\leadsto 0.5 + \frac{\color{blue}{0.5}}{e^{x}} \]

   11. Simplified 74.3%

       \[\leadsto \color{blue}{0.5 + \frac{0.5}{e^{x}}} \]

   if 9.8e8 < 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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 57.3%

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

      1. div-sub 57.3%

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

      2. rec-exp 57.3%

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

      3. neg-mul-1 57.3%

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

      4. +-inverses 57.3%

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

   6. Simplified 57.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 980000000:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 63.6% accurate, 10.8× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 150:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) \cdot \left(\frac{-1}{\varepsilon} + 1\right) + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 150.0)
   (/ (+ 2.0 (* x (+ (* (- -1.0 eps) (+ (/ -1.0 eps) 1.0)) (/ -1.0 eps)))) 2.0)
   0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 150.0) {
		tmp = (2.0 + (x * (((-1.0 - eps) * ((-1.0 / eps) + 1.0)) + (-1.0 / eps)))) / 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 <= 150.0d0) then
        tmp = (2.0d0 + (x * ((((-1.0d0) - eps) * (((-1.0d0) / eps) + 1.0d0)) + ((-1.0d0) / eps)))) / 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 <= 150.0) {
		tmp = (2.0 + (x * (((-1.0 - eps) * ((-1.0 / eps) + 1.0)) + (-1.0 / eps)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 150.0:
		tmp = (2.0 + (x * (((-1.0 - eps) * ((-1.0 / eps) + 1.0)) + (-1.0 / eps)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 150.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(-1.0 - eps) * Float64(Float64(-1.0 / eps) + 1.0)) + Float64(-1.0 / eps)))) / 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 <= 150.0)
		tmp = (2.0 + (x * (((-1.0 - eps) * ((-1.0 / eps) + 1.0)) + (-1.0 / eps)))) / 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, 150.0], N[(N[(2.0 + N[(x * N[(N[(N[(-1.0 - eps), $MachinePrecision] * N[(N[(-1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 150

   1. Initial program 66.4%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in x around 0 54.1%

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

   5. Taylor expanded in eps around 0 57.9%

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

   if 150 < 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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 56.5%

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

      1. div-sub 56.5%

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

      2. rec-exp 56.5%

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

      3. neg-mul-1 56.5%

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

      4. +-inverses 56.5%

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

   6. Simplified 56.5%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 150:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) \cdot \left(\frac{-1}{\varepsilon} + 1\right) + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 63.1% accurate, 20.5× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 4.4e-9) (/ (- 2.0 (* x (+ eps 2.0))) 2.0) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 4.4e-9) {
		tmp = (2.0 - (x * (eps + 2.0))) / 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 <= 4.4d-9) then
        tmp = (2.0d0 - (x * (eps + 2.0d0))) / 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 <= 4.4e-9) {
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 4.4e-9:
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 4.4e-9)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(eps + 2.0))) / 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 <= 4.4e-9)
		tmp = (2.0 - (x * (eps + 2.0))) / 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, 4.4e-9], N[(N[(2.0 - N[(x * N[(eps + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.3999999999999997e-9

   1. Initial program 66.0%

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

      1. div-sub 66.0%

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

      2. +-rgt-identity 66.0%

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

      3. div-sub 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in eps around inf 97.7%

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

   5. Taylor expanded in eps around 0 84.9%

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

   6. Taylor expanded in x around 0 58.9%

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

      1. mul-1-neg 58.9%

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

      2. unsub-neg 58.9%

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

      3. *-commutative 58.9%

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

      4. +-commutative 58.9%

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

   8. Simplified 58.9%

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

   if 4.3999999999999997e-9 < x

   1. Initial program 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in eps around 0 53.2%

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

      1. div-sub 53.2%

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

      2. rec-exp 53.2%

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

      3. neg-mul-1 53.2%

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

      4. +-inverses 53.2%

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

   6. Simplified 53.2%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 56.9% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 980000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 980000000.0) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 980000000.0) {
		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 <= 980000000.0d0) 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 <= 980000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 980000000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 980000000.0)
		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 <= 980000000.0)
		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, 980000000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 9.8e8

   1. Initial program 66.6%

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

      1. div-sub 66.6%

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

      2. +-rgt-identity 66.6%

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

      3. div-sub 66.6%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in x around 0 42.8%

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

   5. Taylor expanded in eps around inf 72.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 + \left(-1 \cdot x + x\right)\right) - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]

   6. Taylor expanded in x around 0 53.8%

      \[\leadsto \color{blue}{1} \]

   if 9.8e8 < 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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 57.3%

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

      1. div-sub 57.3%

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

      2. rec-exp 57.3%

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

      3. neg-mul-1 57.3%

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

      4. +-inverses 57.3%

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

   6. Simplified 57.3%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 980000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 43.8% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ 1 \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 1.0)

C

eps = abs(eps);
double code(double x, double eps) {
	return 1.0;
}

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 = 1.0d0
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	return 1.0;
}

Python

eps = abs(eps)
def code(x, eps):
	return 1.0

Julia

eps = abs(eps)
function code(x, eps)
	return 1.0
end

MATLAB

eps = abs(eps)
function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := 1.0

Derivation

1. Initial program 76.5%

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

   1. div-sub 76.5%

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

   2. +-rgt-identity 76.5%

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

   3. div-sub 76.5%

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

3. Simplified 64.2%

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

4. Taylor expanded in x around 0 38.3%

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

5. Taylor expanded in eps around inf 59.7%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 + \left(-1 \cdot x + x\right)\right) - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right) + 0.5 \cdot \left(\varepsilon \cdot x\right)} \]

6. Taylor expanded in x around 0 38.8%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 38.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023200 
(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))