NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.1% → 99.2%

Time: 19.3s

Alternatives: 12

Speedup: 2.1×

• 38.2% 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: 73.1% 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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \mathbf{if}\;\varepsilon \leq 3 \cdot 10^{-79}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

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 3e-79)
     (/ (+ 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 <= 3e-79) {
		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 <= 3d-79) 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 <= 3e-79) {
		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 <= 3e-79:
		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 <= 3e-79)
		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 <= 3e-79)
		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, 3e-79], 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 < 3e-79

   1. Initial program 60.9%

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

      1. div-sub 60.9%

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

      2. +-rgt-identity 60.9%

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

      3. div-sub 60.9%

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

   3. Simplified 60.9%

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

   4. Taylor expanded in eps around 0 71.1%

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

      1. *-commutative 71.1%

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

      2. distribute-lft1-in 71.1%

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

      3. neg-mul-1 71.1%

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

      4. distribute-lft-out 71.1%

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

      5. mul-1-neg 71.1%

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

      6. *-commutative 71.1%

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

      7. distribute-lft1-in 72.3%

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

      8. neg-mul-1 72.3%

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

   6. Simplified 72.3%

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

   if 3e-79 < eps

   1. Initial program 87.4%

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

      1. div-sub 87.4%

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

      2. +-rgt-identity 87.4%

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

      3. div-sub 87.4%

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

   3. Simplified 87.4%

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

   4. Taylor expanded in eps around inf 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 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3 \cdot 10^{-79}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]

Alternative 2: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-261}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-261)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (/ (+ (exp (- x)) (exp (* x (+ eps -1.0)))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -2e-261) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (exp(-x) + exp((x * (eps + -1.0)))) / 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 <= (-2d-261)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else
        tmp = (exp(-x) + exp((x * (eps + (-1.0d0))))) / 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 <= -2e-261) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (Math.exp(-x) + Math.exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -2e-261:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = (math.exp(-x) + math.exp((x * (eps + -1.0)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -2e-261)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2e-261)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	else
		tmp = (exp(-x) + exp((x * (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -2e-261], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999997e-261

   1. Initial program 66.3%

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

      1. div-sub 66.3%

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

      2. +-rgt-identity 66.3%

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

      3. div-sub 66.3%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in x around 0 41.8%

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

   5. Taylor expanded in eps around inf 70.7%

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

   if -1.99999999999999997e-261 < x

   1. Initial program 71.8%

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

      1. div-sub 71.8%

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

      2. +-rgt-identity 71.8%

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

      3. div-sub 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in eps around inf 99.8%

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

   5. Taylor expanded in eps around 0 85.1%

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

   6. Taylor expanded in eps around -inf 85.1%

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

      1. cancel-sign-sub-inv 85.1%

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

      2. associate-*r* 85.1%

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

      3. neg-mul-1 85.1%

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

      4. sub-neg 85.1%

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

      5. mul-1-neg 85.1%

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

      6. metadata-eval 85.1%

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

      7. neg-mul-1 85.1%

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

      8. *-lft-identity 85.1%

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

   8. Simplified 85.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-261}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]

Alternative 3: 91.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\varepsilon \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0 \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + e^{\varepsilon \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps 1.4e-6) (/ (* t_0 2.0) 2.0) (/ (+ t_0 (exp (* eps x))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (eps <= 1.4e-6) {
		tmp = (t_0 * 2.0) / 2.0;
	} else {
		tmp = (t_0 + exp((eps * x))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps <= 1.4d-6) then
        tmp = (t_0 * 2.0d0) / 2.0d0
    else
        tmp = (t_0 + exp((eps * x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps <= 1.4e-6) {
		tmp = (t_0 * 2.0) / 2.0;
	} else {
		tmp = (t_0 + Math.exp((eps * x))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = math.exp(-x)
	tmp = 0
	if eps <= 1.4e-6:
		tmp = (t_0 * 2.0) / 2.0
	else:
		tmp = (t_0 + math.exp((eps * x))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps <= 1.4e-6)
		tmp = Float64(Float64(t_0 * 2.0) / 2.0);
	else
		tmp = Float64(Float64(t_0 + exp(Float64(eps * x))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps <= 1.4e-6)
		tmp = (t_0 * 2.0) / 2.0;
	else
		tmp = (t_0 + exp((eps * x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps, 1.4e-6], N[(N[(t$95$0 * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.39999999999999994e-6

   1. Initial program 57.6%

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

      1. div-sub 57.6%

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

      2. +-rgt-identity 57.6%

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

      3. div-sub 57.6%

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

   3. Simplified 57.6%

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

   4. 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 85.1%

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

   6. Taylor expanded in eps around 0 81.7%

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

      1. cancel-sign-sub-inv 81.7%

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

      2. neg-mul-1 81.7%

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

      3. metadata-eval 81.7%

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

      4. neg-mul-1 81.7%

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

      5. *-lft-identity 81.7%

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

      6. count-2 81.7%

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

   8. Simplified 81.7%

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

   if 1.39999999999999994e-6 < 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 0 84.5%

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

   6. Taylor expanded in eps around -inf 84.5%

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

      1. cancel-sign-sub-inv 84.5%

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

      2. associate-*r* 84.5%

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

      3. neg-mul-1 84.5%

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

      4. sub-neg 84.5%

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

      5. mul-1-neg 84.5%

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

      6. metadata-eval 84.5%

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

      7. neg-mul-1 84.5%

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

      8. *-lft-identity 84.5%

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

   8. Simplified 84.5%

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

   9. Taylor expanded in eps around inf 84.5%

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

       1. *-commutative 84.5%

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

   11. Simplified 84.5%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{\varepsilon \cdot x}}{2}\\ \end{array} \]

Alternative 4: 80.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{2 - x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(1 + \frac{-1}{\varepsilon}\right) + \frac{-1 + \frac{-1}{\varepsilon}}{\frac{\varepsilon + 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 7.5e-228)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (<= x 8.4e+38)
     (/
      (-
       2.0
       (*
        x
        (+
         (* (+ eps 1.0) (+ 1.0 (/ -1.0 eps)))
         (/ (+ -1.0 (/ -1.0 eps)) (/ (+ eps 1.0) (fma eps eps -1.0))))))
      2.0)
     (/ (* (exp (- x)) 2.0) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 7.5e-228) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 8.4e+38) {
		tmp = (2.0 - (x * (((eps + 1.0) * (1.0 + (-1.0 / eps))) + ((-1.0 + (-1.0 / eps)) / ((eps + 1.0) / fma(eps, eps, -1.0)))))) / 2.0;
	} else {
		tmp = (exp(-x) * 2.0) / 2.0;
	}
	return tmp;
}

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 7.5e-228)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 8.4e+38)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(Float64(Float64(eps + 1.0) * Float64(1.0 + Float64(-1.0 / eps))) + Float64(Float64(-1.0 + Float64(-1.0 / eps)) / Float64(Float64(eps + 1.0) / fma(eps, eps, -1.0)))))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) / 2.0);
	end
	return tmp
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 7.5e-228], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.4e+38], N[(N[(2.0 - N[(x * N[(N[(N[(eps + 1.0), $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] / N[(N[(eps + 1.0), $MachinePrecision] / N[(eps * eps + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.4999999999999999e-228

   1. Initial program 62.2%

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

      1. div-sub 62.2%

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

      2. +-rgt-identity 62.2%

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

      3. div-sub 62.2%

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

   3. Simplified 44.7%

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

   4. Taylor expanded in x around 0 43.1%

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

   5. Taylor expanded in eps around inf 77.3%

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

   if 7.4999999999999999e-228 < x < 8.4e38

   1. Initial program 57.4%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in x around 0 61.0%

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

      1. +-commutative 61.0%

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

      2. *-rgt-identity 61.0%

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

      3. flip-- 73.7%

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

      4. +-commutative 73.7%

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

      5. associate-*r/ 73.7%

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

      6. *-rgt-identity 73.7%

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

      7. inv-pow 73.7%

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

      8. metadata-eval 73.7%

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

      9. fma-neg 73.7%

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

      10. metadata-eval 73.7%

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

      11. +-commutative 73.7%

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

   6. Applied egg-rr 73.7%

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

      1. associate-/l* 73.7%

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

      2. unpow-1 73.7%

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

   8. Simplified 73.7%

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

   if 8.4e38 < 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 81.6%

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

   6. Taylor expanded in eps around 0 56.6%

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

      1. cancel-sign-sub-inv 56.6%

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

      2. neg-mul-1 56.6%

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

      3. metadata-eval 56.6%

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

      4. neg-mul-1 56.6%

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

      5. *-lft-identity 56.6%

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

      6. count-2 56.6%

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

   8. Simplified 56.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{2 - x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(1 + \frac{-1}{\varepsilon}\right) + \frac{-1 + \frac{-1}{\varepsilon}}{\frac{\varepsilon + 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \end{array} \]

Alternative 5: 80.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+39}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \frac{1 - \frac{-1}{\varepsilon}}{\frac{\varepsilon + 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 6.2e-228)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (<= x 1.12e+39)
     (/
      (+
       2.0
       (*
        x
        (+
         (/ 1.0 eps)
         (/ (- 1.0 (/ -1.0 eps)) (/ (+ eps 1.0) (fma eps eps -1.0))))))
      2.0)
     (/ (* (exp (- x)) 2.0) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 6.2e-228) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 1.12e+39) {
		tmp = (2.0 + (x * ((1.0 / eps) + ((1.0 - (-1.0 / eps)) / ((eps + 1.0) / fma(eps, eps, -1.0)))))) / 2.0;
	} else {
		tmp = (exp(-x) * 2.0) / 2.0;
	}
	return tmp;
}

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 6.2e-228)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 1.12e+39)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(1.0 - Float64(-1.0 / eps)) / Float64(Float64(eps + 1.0) / fma(eps, eps, -1.0)))))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) / 2.0);
	end
	return tmp
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 6.2e-228], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.12e+39], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] / N[(N[(eps + 1.0), $MachinePrecision] / N[(eps * eps + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.1999999999999996e-228

   1. Initial program 62.2%

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

      1. div-sub 62.2%

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

      2. +-rgt-identity 62.2%

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

      3. div-sub 62.2%

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

   3. Simplified 44.7%

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

   4. Taylor expanded in x around 0 43.1%

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

   5. Taylor expanded in eps around inf 77.3%

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

   if 6.1999999999999996e-228 < x < 1.12e39

   1. Initial program 57.4%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in x around 0 61.0%

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

      1. +-commutative 61.0%

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

      2. *-rgt-identity 61.0%

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

      3. flip-- 73.7%

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

      4. +-commutative 73.7%

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

      5. associate-*r/ 73.7%

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

      6. *-rgt-identity 73.7%

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

      7. inv-pow 73.7%

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

      8. metadata-eval 73.7%

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

      9. fma-neg 73.7%

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

      10. metadata-eval 73.7%

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

      11. +-commutative 73.7%

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

   6. Applied egg-rr 73.7%

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

      1. associate-/l* 73.7%

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

      2. unpow-1 73.7%

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

   8. Simplified 73.7%

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

   9. Taylor expanded in eps around 0 73.5%

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

   if 1.12e39 < 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 81.6%

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

   6. Taylor expanded in eps around 0 56.6%

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

      1. cancel-sign-sub-inv 56.6%

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

      2. neg-mul-1 56.6%

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

      3. metadata-eval 56.6%

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

      4. neg-mul-1 56.6%

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

      5. *-lft-identity 56.6%

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

      6. count-2 56.6%

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

   8. Simplified 56.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+39}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \frac{1 - \frac{-1}{\varepsilon}}{\frac{\varepsilon + 1}{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \end{array} \]

Alternative 6: 77.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{-241}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -2.45e-241)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (/ (* (exp (- x)) 2.0) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -2.45e-241) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} 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 <= (-2.45d-241)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    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 <= -2.45e-241) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (Math.exp(-x) * 2.0) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -2.45e-241:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = (math.exp(-x) * 2.0) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -2.45e-241)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	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 <= -2.45e-241)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	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, -2.45e-241], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4499999999999999e-241

   1. Initial program 68.0%

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

      1. div-sub 68.0%

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

      2. +-rgt-identity 68.0%

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

      3. div-sub 68.0%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in x around 0 42.7%

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

   5. Taylor expanded in eps around inf 69.7%

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

   if -2.4499999999999999e-241 < x

   1. Initial program 70.7%

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

      1. div-sub 70.7%

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

      2. +-rgt-identity 70.7%

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

      3. div-sub 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in eps around inf 99.8%

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

   5. Taylor expanded in eps around 0 85.3%

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

   6. Taylor expanded in eps around 0 68.1%

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

      1. cancel-sign-sub-inv 68.1%

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

      2. neg-mul-1 68.1%

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

      3. metadata-eval 68.1%

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

      4. neg-mul-1 68.1%

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

      5. *-lft-identity 68.1%

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

      6. count-2 68.1%

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

   8. Simplified 68.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{-241}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \end{array} \]

Alternative 7: 74.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+20} \lor \neg \left(x \leq -2 \cdot 10^{-240}\right):\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\varepsilon + -1\right) \cdot \left(1 - \frac{-1}{\varepsilon}\right) + \frac{-1 + \varepsilon \cdot \varepsilon}{-\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.15e+20) (not (<= x -2e-240)))
   (/ (* (exp (- x)) 2.0) 2.0)
   (/
    (+
     2.0
     (*
      x
      (+
       (* (+ eps -1.0) (- 1.0 (/ -1.0 eps)))
       (/ (+ -1.0 (* eps eps)) (- eps)))))
    2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if ((x <= -1.15e+20) || !(x <= -2e-240)) {
		tmp = (exp(-x) * 2.0) / 2.0;
	} else {
		tmp = (2.0 + (x * (((eps + -1.0) * (1.0 - (-1.0 / eps))) + ((-1.0 + (eps * eps)) / -eps)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-1.15d+20)) .or. (.not. (x <= (-2d-240)))) then
        tmp = (exp(-x) * 2.0d0) / 2.0d0
    else
        tmp = (2.0d0 + (x * (((eps + (-1.0d0)) * (1.0d0 - ((-1.0d0) / eps))) + (((-1.0d0) + (eps * eps)) / -eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if ((x <= -1.15e+20) || !(x <= -2e-240)) {
		tmp = (Math.exp(-x) * 2.0) / 2.0;
	} else {
		tmp = (2.0 + (x * (((eps + -1.0) * (1.0 - (-1.0 / eps))) + ((-1.0 + (eps * eps)) / -eps)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if (x <= -1.15e+20) or not (x <= -2e-240):
		tmp = (math.exp(-x) * 2.0) / 2.0
	else:
		tmp = (2.0 + (x * (((eps + -1.0) * (1.0 - (-1.0 / eps))) + ((-1.0 + (eps * eps)) / -eps)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if ((x <= -1.15e+20) || !(x <= -2e-240))
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(eps + -1.0) * Float64(1.0 - Float64(-1.0 / eps))) + Float64(Float64(-1.0 + Float64(eps * eps)) / Float64(-eps))))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.15e+20) || ~((x <= -2e-240)))
		tmp = (exp(-x) * 2.0) / 2.0;
	else
		tmp = (2.0 + (x * (((eps + -1.0) * (1.0 - (-1.0 / eps))) + ((-1.0 + (eps * eps)) / -eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[Or[LessEqual[x, -1.15e+20], N[Not[LessEqual[x, -2e-240]], $MachinePrecision]], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(N[(eps + -1.0), $MachinePrecision] * N[(1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision] / (-eps)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e20 or -1.9999999999999999e-240 < x

   1. Initial program 75.5%

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

      1. div-sub 75.5%

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

      2. +-rgt-identity 75.5%

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

      3. div-sub 75.5%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in eps around inf 99.8%

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

   5. Taylor expanded in eps around 0 87.7%

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

   6. Taylor expanded in eps around 0 73.3%

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

      1. cancel-sign-sub-inv 73.3%

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

      2. neg-mul-1 73.3%

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

      3. metadata-eval 73.3%

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

      4. neg-mul-1 73.3%

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

      5. *-lft-identity 73.3%

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

      6. count-2 73.3%

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

   8. Simplified 73.3%

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

   if -1.15e20 < x < -1.9999999999999999e-240

   1. Initial program 51.4%

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

   3. Simplified 42.9%

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

   4. Taylor expanded in x around 0 64.2%

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

      1. +-commutative 64.2%

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

      2. *-commutative 64.2%

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

      3. flip-+ 77.9%

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

      4. associate-*r/ 77.9%

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

      5. sub-neg 77.9%

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

      6. distribute-neg-frac 77.9%

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

      7. metadata-eval 77.9%

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

      8. metadata-eval 77.9%

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

   6. Applied egg-rr 77.9%

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

      1. *-commutative 77.9%

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

      2. associate-/l* 77.9%

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

      3. metadata-eval 77.9%

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

      4. distribute-neg-frac 77.9%

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

      5. unpow-1 77.9%

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

      6. sub-neg 77.9%

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

      7. unpow-1 77.9%

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

   8. Simplified 77.9%

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

   9. Taylor expanded in eps around 0 77.9%

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

       1. neg-mul-1 77.9%

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

   11. Simplified 77.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+20} \lor \neg \left(x \leq -2 \cdot 10^{-240}\right):\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\varepsilon + -1\right) \cdot \left(1 - \frac{-1}{\varepsilon}\right) + \frac{-1 + \varepsilon \cdot \varepsilon}{-\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 8: 69.0% accurate, 13.3× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-241}:\\ \;\;\;\;\frac{2 - x \cdot \frac{\varepsilon \cdot \left(2 - \varepsilon\right)}{2 - \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -4.7e-241)
   (/ (- 2.0 (* x (/ (* eps (- 2.0 eps)) (- 2.0 eps)))) 2.0)
   (if (<= x 7.4e+24) 1.0 0.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -4.7e-241) {
		tmp = (2.0 - (x * ((eps * (2.0 - eps)) / (2.0 - eps)))) / 2.0;
	} else if (x <= 7.4e+24) {
		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 <= (-4.7d-241)) then
        tmp = (2.0d0 - (x * ((eps * (2.0d0 - eps)) / (2.0d0 - eps)))) / 2.0d0
    else if (x <= 7.4d+24) 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 <= -4.7e-241) {
		tmp = (2.0 - (x * ((eps * (2.0 - eps)) / (2.0 - eps)))) / 2.0;
	} else if (x <= 7.4e+24) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -4.7e-241:
		tmp = (2.0 - (x * ((eps * (2.0 - eps)) / (2.0 - eps)))) / 2.0
	elif x <= 7.4e+24:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -4.7e-241)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(Float64(eps * Float64(2.0 - eps)) / Float64(2.0 - eps)))) / 2.0);
	elseif (x <= 7.4e+24)
		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 <= -4.7e-241)
		tmp = (2.0 - (x * ((eps * (2.0 - eps)) / (2.0 - eps)))) / 2.0;
	elseif (x <= 7.4e+24)
		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, -4.7e-241], N[(N[(2.0 - N[(x * N[(N[(eps * N[(2.0 - eps), $MachinePrecision]), $MachinePrecision] / N[(2.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7.4e+24], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6999999999999999e-241

   1. Initial program 68.0%

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

      1. div-sub 68.0%

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

      2. +-rgt-identity 68.0%

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

      3. div-sub 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in eps around inf 96.0%

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

   5. Taylor expanded in eps around 0 84.2%

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

   6. Taylor expanded in x around 0 54.9%

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

      1. flip-- 31.9%

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

      2. neg-mul-1 31.9%

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

      3. neg-mul-1 31.9%

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

      4. sqr-neg 31.9%

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

      5. pow2 31.9%

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

      6. metadata-eval 31.9%

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

      7. add-sqr-sqrt 4.3%

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

      8. sqrt-unprod 41.3%

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

      9. neg-mul-1 41.3%

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

      10. neg-mul-1 41.3%

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

      11. sqr-neg 41.3%

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

      12. sqrt-unprod 37.1%

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

      13. add-sqr-sqrt 59.9%

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

   8. Applied egg-rr 59.9%

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

      1. unpow2 59.9%

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

      2. difference-of-sqr-1 59.9%

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

      3. +-commutative 59.9%

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

      4. associate-+r- 59.9%

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

      5. metadata-eval 59.9%

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

      6. rem-square-sqrt 37.1%

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

      7. fma-neg 37.1%

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

      8. metadata-eval 37.1%

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

      9. fma-udef 37.1%

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

      10. rem-square-sqrt 59.9%

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

      11. +-commutative 59.9%

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

      12. associate-+r- 59.9%

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

      13. metadata-eval 59.9%

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

      14. neg-sub0 59.9%

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

      15. +-commutative 59.9%

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

      16. associate-+r- 59.9%

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

      17. metadata-eval 59.9%

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

   10. Simplified 59.9%

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

   if -4.6999999999999999e-241 < x < 7.39999999999999998e24

   1. Initial program 52.5%

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

      1. div-sub 52.5%

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

      2. +-rgt-identity 52.5%

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

      3. div-sub 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in x around 0 76.2%

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

   if 7.39999999999999998e24 < 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 55.5%

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

      1. div-sub 55.5%

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

      2. rec-exp 55.5%

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

      3. neg-mul-1 55.5%

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

      4. +-inverses 55.5%

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

   6. Simplified 55.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-241}:\\ \;\;\;\;\frac{2 - x \cdot \frac{\varepsilon \cdot \left(2 - \varepsilon\right)}{2 - \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 57.5% accurate, 25.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 59.8%

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

      1. div-sub 59.8%

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

      2. +-rgt-identity 59.8%

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

      3. div-sub 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in eps around inf 97.9%

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

   5. Taylor expanded in eps around 0 86.1%

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

   6. Taylor expanded in x around 0 65.7%

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

   7. Taylor expanded in eps around 0 61.0%

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

   if 1 < 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 54.7%

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

      1. div-sub 54.7%

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

      2. rec-exp 54.7%

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

      3. neg-mul-1 54.7%

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

      4. +-inverses 54.7%

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

   6. Simplified 54.7%

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

4. Final simplification 59.5%

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

Alternative 10: 63.8% accurate, 25.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 8e-6) (/ (- 2.0 (* eps x)) 2.0) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 8e-6) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 8d-6) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 8e-6) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 8e-6:
		tmp = (2.0 - (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 8e-6)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 8e-6)
		tmp = (2.0 - (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 8e-6], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 7.99999999999999964e-6

   1. Initial program 59.6%

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

      1. div-sub 59.6%

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

      2. +-rgt-identity 59.6%

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

      3. div-sub 59.6%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in x around 0 40.7%

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

   5. Taylor expanded in x around 0 41.5%

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

   6. Taylor expanded in eps around inf 63.7%

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

      1. mul-1-neg 63.7%

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

      2. *-commutative 63.7%

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

      3. distribute-lft-neg-in 63.7%

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

   8. Simplified 63.7%

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

   if 7.99999999999999964e-6 < 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 53.9%

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

      1. div-sub 53.9%

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

      2. rec-exp 53.9%

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

      3. neg-mul-1 53.9%

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

      4. +-inverses 53.9%

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

   6. Simplified 53.9%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 56.6% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 7.4 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 7.4e+24) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 7.4e+24) {
		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 <= 7.4d+24) 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 <= 7.4e+24) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 7.4e+24:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 7.4e+24)
		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 <= 7.4e+24)
		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, 7.4e+24], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 7.39999999999999998e24

   1. Initial program 60.0%

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

      1. div-sub 60.0%

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

      2. +-rgt-identity 60.0%

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

      3. div-sub 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in x around 0 60.3%

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

   if 7.39999999999999998e24 < 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 55.5%

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

      1. div-sub 55.5%

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

      2. rec-exp 55.5%

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

      3. neg-mul-1 55.5%

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

      4. +-inverses 55.5%

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

   6. Simplified 55.5%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.4 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 15.4% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

3. Simplified 63.5%

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

4. Taylor expanded in eps around 0 15.1%

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

   1. div-sub 15.1%

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

   2. rec-exp 15.1%

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

   3. neg-mul-1 15.1%

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

   4. +-inverses 15.4%

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

6. Simplified 15.4%

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

7. Final simplification 15.4%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023208 
(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))