NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.3% → 99.8%

Time: 13.3s

Alternatives: 15

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: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + x\right) \cdot e^{-x}\\ t_1 := e^{\varepsilon \cdot \left(-x\right)}\\ \mathbf{if}\;\varepsilon \leq -14.5:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + t_1}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + t_1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ 1.0 x) (exp (- x)))) (t_1 (exp (* eps (- x)))))
   (if (<= eps -14.5)
     (/ (+ (exp (* eps x)) t_1) 2.0)
     (if (<= eps 5e-41)
       (/ (+ t_0 t_0) 2.0)
       (/ (+ (exp (* x (+ -1.0 eps))) t_1) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = (1.0 + x) * exp(-x);
	double t_1 = exp((eps * -x));
	double tmp;
	if (eps <= -14.5) {
		tmp = (exp((eps * x)) + t_1) / 2.0;
	} else if (eps <= 5e-41) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps))) + t_1) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 + x) * exp(-x)
    t_1 = exp((eps * -x))
    if (eps <= (-14.5d0)) then
        tmp = (exp((eps * x)) + t_1) / 2.0d0
    else if (eps <= 5d-41) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps))) + t_1) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 + x) * Math.exp(-x);
	double t_1 = Math.exp((eps * -x));
	double tmp;
	if (eps <= -14.5) {
		tmp = (Math.exp((eps * x)) + t_1) / 2.0;
	} else if (eps <= 5e-41) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps))) + t_1) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 + x) * math.exp(-x)
	t_1 = math.exp((eps * -x))
	tmp = 0
	if eps <= -14.5:
		tmp = (math.exp((eps * x)) + t_1) / 2.0
	elif eps <= 5e-41:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps))) + t_1) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 + x) * exp(Float64(-x)))
	t_1 = exp(Float64(eps * Float64(-x)))
	tmp = 0.0
	if (eps <= -14.5)
		tmp = Float64(Float64(exp(Float64(eps * x)) + t_1) / 2.0);
	elseif (eps <= 5e-41)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps))) + t_1) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 + x) * exp(-x);
	t_1 = exp((eps * -x));
	tmp = 0.0;
	if (eps <= -14.5)
		tmp = (exp((eps * x)) + t_1) / 2.0;
	elseif (eps <= 5e-41)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps))) + t_1) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eps, -14.5], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 5e-41], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -14.5

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -14.5 < eps < 4.9999999999999996e-41

   1. Initial program 43.7%

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

      1. div-sub 43.7%

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

      2. +-rgt-identity 43.7%

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

      3. div-sub 43.7%

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

   3. Simplified 43.7%

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

   4. Taylor expanded in eps around 0 98.1%

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

      1. *-commutative 98.1%

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

      2. distribute-lft1-in 98.1%

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

      3. mul-1-neg 98.1%

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

      4. distribute-lft-out 98.1%

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

      5. mul-1-neg 98.1%

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

      6. *-commutative 98.1%

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

      7. distribute-lft1-in 100.0%

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

      8. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if 4.9999999999999996e-41 < eps

   1. Initial program 97.2%

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

      1. div-sub 97.2%

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

      2. +-rgt-identity 97.2%

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

      3. div-sub 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in eps around inf 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 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -14.5:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot e^{-x} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + x\right) \cdot e^{-x}\\ \mathbf{if}\;\varepsilon \leq -14.5 \lor \neg \left(\varepsilon \leq 0.00033\right):\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ 1.0 x) (exp (- x)))))
   (if (or (<= eps -14.5) (not (<= eps 0.00033)))
     (/ (+ (exp (* eps x)) (exp (* eps (- x)))) 2.0)
     (/ (+ t_0 t_0) 2.0))))

C

double code(double x, double eps) {
	double t_0 = (1.0 + x) * exp(-x);
	double tmp;
	if ((eps <= -14.5) || !(eps <= 0.00033)) {
		tmp = (exp((eps * x)) + exp((eps * -x))) / 2.0;
	} else {
		tmp = (t_0 + t_0) / 2.0;
	}
	return tmp;
}

Fortran

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 = (1.0d0 + x) * exp(-x)
    if ((eps <= (-14.5d0)) .or. (.not. (eps <= 0.00033d0))) then
        tmp = (exp((eps * x)) + exp((eps * -x))) / 2.0d0
    else
        tmp = (t_0 + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 + x) * Math.exp(-x);
	double tmp;
	if ((eps <= -14.5) || !(eps <= 0.00033)) {
		tmp = (Math.exp((eps * x)) + Math.exp((eps * -x))) / 2.0;
	} else {
		tmp = (t_0 + t_0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 + x) * math.exp(-x)
	tmp = 0
	if (eps <= -14.5) or not (eps <= 0.00033):
		tmp = (math.exp((eps * x)) + math.exp((eps * -x))) / 2.0
	else:
		tmp = (t_0 + t_0) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 + x) * exp(Float64(-x)))
	tmp = 0.0
	if ((eps <= -14.5) || !(eps <= 0.00033))
		tmp = Float64(Float64(exp(Float64(eps * x)) + exp(Float64(eps * Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 + x) * exp(-x);
	tmp = 0.0;
	if ((eps <= -14.5) || ~((eps <= 0.00033)))
		tmp = (exp((eps * x)) + exp((eps * -x))) / 2.0;
	else
		tmp = (t_0 + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -14.5], N[Not[LessEqual[eps, 0.00033]], $MachinePrecision]], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -14.5 or 3.3e-4 < eps

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -14.5 < eps < 3.3e-4

   1. Initial program 45.0%

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

      1. div-sub 45.0%

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

      2. +-rgt-identity 45.0%

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

      3. div-sub 45.0%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in eps around 0 98.2%

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

      1. *-commutative 98.2%

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

      2. distribute-lft1-in 98.2%

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

      3. mul-1-neg 98.2%

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

      4. distribute-lft-out 98.2%

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

      5. mul-1-neg 98.2%

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

      6. *-commutative 98.2%

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

      7. distribute-lft1-in 100.0%

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

      8. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -14.5 \lor \neg \left(\varepsilon \leq 0.00033\right):\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot e^{-x} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \end{array} \]

Alternative 3: 82.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{+221} \lor \neg \left(\varepsilon \leq 9.5 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.8e+221) (not (<= eps 9.5e+68)))
   (/ (+ 1.0 (exp (* x (+ 1.0 eps)))) 2.0)
   (/ (+ (exp (* x (- -1.0 eps))) (exp (- x))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.8e+221) || !(eps <= 9.5e+68)) {
		tmp = (1.0 + exp((x * (1.0 + eps)))) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 - eps))) + exp(-x)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-4.8d+221)) .or. (.not. (eps <= 9.5d+68))) then
        tmp = (1.0d0 + exp((x * (1.0d0 + eps)))) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) - eps))) + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.8e+221) || !(eps <= 9.5e+68)) {
		tmp = (1.0 + Math.exp((x * (1.0 + eps)))) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 - eps))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.8e+221) or not (eps <= 9.5e+68):
		tmp = (1.0 + math.exp((x * (1.0 + eps)))) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 - eps))) + math.exp(-x)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.8e+221) || !(eps <= 9.5e+68))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(1.0 + eps)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 - eps))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.8e+221) || ~((eps <= 9.5e+68)))
		tmp = (1.0 + exp((x * (1.0 + eps)))) / 2.0;
	else
		tmp = (exp((x * (-1.0 - eps))) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.8e+221], N[Not[LessEqual[eps, 9.5e+68]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.80000000000000038e221 or 9.50000000000000069e68 < 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 x around 0 49.8%

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

   5. Taylor expanded in eps around inf 49.8%

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

      1. +-commutative 49.8%

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

      2. add-sqr-sqrt 15.4%

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

      3. sqrt-unprod 39.5%

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

      4. sqr-neg 39.5%

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

      5. sqrt-unprod 31.4%

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

      6. add-sqr-sqrt 76.6%

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

      7. distribute-rgt-neg-out 76.6%

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

      8. *-commutative 76.6%

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

   7. Applied egg-rr 76.6%

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

   if -4.80000000000000038e221 < eps < 9.50000000000000069e68

   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 98.4%

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

   5. Taylor expanded in eps around 0 96.1%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{+221} \lor \neg \left(\varepsilon \leq 9.5 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]

Alternative 4: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{+119} \lor \neg \left(\varepsilon \leq 0.00033\right):\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4e+119) (not (<= eps 0.00033)))
   (/ (+ (exp (* eps x)) (exp (* eps (- x)))) 2.0)
   (/ (+ (exp (* x (- -1.0 eps))) (exp (- x))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4e+119) || !(eps <= 0.00033)) {
		tmp = (exp((eps * x)) + exp((eps * -x))) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 - eps))) + exp(-x)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-4d+119)) .or. (.not. (eps <= 0.00033d0))) then
        tmp = (exp((eps * x)) + exp((eps * -x))) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) - eps))) + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4e+119) || !(eps <= 0.00033)) {
		tmp = (Math.exp((eps * x)) + Math.exp((eps * -x))) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 - eps))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4e+119) or not (eps <= 0.00033):
		tmp = (math.exp((eps * x)) + math.exp((eps * -x))) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 - eps))) + math.exp(-x)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4e+119) || !(eps <= 0.00033))
		tmp = Float64(Float64(exp(Float64(eps * x)) + exp(Float64(eps * Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 - eps))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4e+119) || ~((eps <= 0.00033)))
		tmp = (exp((eps * x)) + exp((eps * -x))) / 2.0;
	else
		tmp = (exp((x * (-1.0 - eps))) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4e+119], N[Not[LessEqual[eps, 0.00033]], $MachinePrecision]], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.99999999999999978e119 or 3.3e-4 < eps

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -3.99999999999999978e119 < eps < 3.3e-4

   1. Initial program 56.4%

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

      1. div-sub 56.4%

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

      2. +-rgt-identity 56.4%

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

      3. div-sub 56.4%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in eps around inf 97.8%

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

   5. Taylor expanded in eps around 0 97.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{+119} \lor \neg \left(\varepsilon \leq 0.00033\right):\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]

Alternative 5: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

   1. div-sub 76.1%

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

   2. +-rgt-identity 76.1%

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

   3. div-sub 76.1%

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

3. Simplified 76.1%

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

4. Taylor expanded in eps around inf 98.8%

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

5. Final simplification 98.8%

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

Alternative 6: 80.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{-1}{\varepsilon} + \left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.5e-12)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 2.7e+19)
     (/ (+ 1.0 (exp (* x (+ 1.0 eps)))) 2.0)
     (/ (* x (+ (/ -1.0 eps) (* (+ 1.0 eps) (+ -1.0 (/ 1.0 eps))))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.5e-12) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 2.7e+19) {
		tmp = (1.0 + exp((x * (1.0 + eps)))) / 2.0;
	} else {
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-6.5d-12)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 2.7d+19) then
        tmp = (1.0d0 + exp((x * (1.0d0 + eps)))) / 2.0d0
    else
        tmp = (x * (((-1.0d0) / eps) + ((1.0d0 + eps) * ((-1.0d0) + (1.0d0 / eps))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.5e-12) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 2.7e+19) {
		tmp = (1.0 + Math.exp((x * (1.0 + eps)))) / 2.0;
	} else {
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.5e-12:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 2.7e+19:
		tmp = (1.0 + math.exp((x * (1.0 + eps)))) / 2.0
	else:
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.5e-12)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 2.7e+19)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(1.0 + eps)))) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(-1.0 / eps) + Float64(Float64(1.0 + eps) * Float64(-1.0 + Float64(1.0 / eps))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.5e-12)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 2.7e+19)
		tmp = (1.0 + exp((x * (1.0 + eps)))) / 2.0;
	else
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.5e-12], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.7e+19], N[(N[(1.0 + N[Exp[N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(-1.0 / eps), $MachinePrecision] + N[(N[(1.0 + eps), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5000000000000002e-12

   1. Initial program 92.6%

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

      1. div-sub 92.6%

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

      2. +-rgt-identity 92.6%

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

      3. div-sub 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in x around 0 51.5%

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

      1. +-commutative 51.5%

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

      2. associate-+l+ 51.5%

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

      3. *-commutative 51.5%

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

      4. associate-*l* 51.5%

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

      5. +-commutative 51.5%

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

      6. *-commutative 51.5%

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

      7. associate-*l* 51.5%

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

      8. *-commutative 51.5%

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

      9. neg-mul-1 51.5%

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

      10. sub-neg 51.5%

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

      11. +-commutative 51.5%

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

      12. distribute-neg-in 51.5%

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

      13. remove-double-neg 51.5%

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

      14. metadata-eval 51.5%

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

      15. distribute-rgt-out 51.5%

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

      16. mul-1-neg 51.5%

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

      17. unsub-neg 51.5%

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

      18. *-commutative 51.5%

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

   6. Simplified 51.5%

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

   7. Taylor expanded in eps around inf 53.4%

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

   8. Taylor expanded in eps around 0 94.5%

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

      1. sub-neg 94.5%

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

      2. distribute-lft1-in 94.5%

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

      3. metadata-eval 94.5%

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

      4. mul0-lft 94.5%

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

      5. metadata-eval 94.5%

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

      6. mul-1-neg 94.5%

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

      7. remove-double-neg 94.5%

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

      8. mul-1-neg 94.5%

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

   10. Simplified 94.5%

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

   if -6.5000000000000002e-12 < x < 2.7e19

   1. Initial program 57.3%

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

      1. div-sub 57.3%

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

      2. +-rgt-identity 57.3%

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

      3. div-sub 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in x around 0 42.2%

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

   5. Taylor expanded in eps around inf 84.4%

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

      1. +-commutative 84.4%

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

      2. add-sqr-sqrt 44.7%

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

      3. sqrt-unprod 85.8%

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

      4. sqr-neg 85.8%

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

      5. sqrt-unprod 44.6%

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

      6. add-sqr-sqrt 90.7%

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

      7. distribute-rgt-neg-out 90.7%

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

      8. *-commutative 90.7%

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

   7. Applied egg-rr 90.7%

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

   if 2.7e19 < 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 x around 0 3.1%

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

   5. Taylor expanded in eps around 0 17.4%

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

   6. Taylor expanded in x around inf 69.5%

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

      1. mul-1-neg 69.5%

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

      2. mul-1-neg 69.5%

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

      3. sub-neg 69.5%

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

      4. metadata-eval 69.5%

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

      5. *-commutative 69.5%

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

      6. +-commutative 69.5%

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

      7. metadata-eval 69.5%

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

      8. sub-neg 69.5%

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

      9. sub-neg 69.5%

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

      10. metadata-eval 69.5%

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

      11. sub-neg 69.5%

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

   8. Simplified 69.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{-1}{\varepsilon} + \left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]

Alternative 7: 80.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+31}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 720:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{-1}{\varepsilon} + \left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.4e+31)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 720.0)
     (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
     (/ (* x (+ (/ -1.0 eps) (* (+ 1.0 eps) (+ -1.0 (/ 1.0 eps))))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.4e+31) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 720.0) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else {
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-3.4d+31)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 720.0d0) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else
        tmp = (x * (((-1.0d0) / eps) + ((1.0d0 + eps) * ((-1.0d0) + (1.0d0 / eps))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.4e+31) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 720.0) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else {
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.4e+31:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 720.0:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	else:
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.4e+31)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 720.0)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(-1.0 / eps) + Float64(Float64(1.0 + eps) * Float64(-1.0 + Float64(1.0 / eps))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.4e+31)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 720.0)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	else
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.4e+31], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 720.0], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(-1.0 / eps), $MachinePrecision] + N[(N[(1.0 + eps), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3999999999999998e31

   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 x around 0 53.0%

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

      1. +-commutative 53.0%

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

      2. associate-+l+ 53.0%

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

      3. *-commutative 53.0%

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

      4. associate-*l* 53.0%

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

      5. +-commutative 53.0%

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

      6. *-commutative 53.0%

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

      7. associate-*l* 53.0%

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

      8. *-commutative 53.0%

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

      9. neg-mul-1 53.0%

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

      10. sub-neg 53.0%

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

      11. +-commutative 53.0%

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

      12. distribute-neg-in 53.0%

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

      13. remove-double-neg 53.0%

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

      14. metadata-eval 53.0%

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

      15. distribute-rgt-out 53.0%

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

      16. mul-1-neg 53.0%

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

      17. unsub-neg 53.0%

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

      18. *-commutative 53.0%

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

   6. Simplified 53.0%

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

   7. Taylor expanded in eps around inf 53.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul-1-neg 100.0%

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

      7. remove-double-neg 100.0%

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

      8. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   if -3.3999999999999998e31 < x < 720

   1. Initial program 56.0%

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

      1. div-sub 56.0%

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

      2. +-rgt-identity 56.0%

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

      3. div-sub 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in x around 0 42.7%

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

   5. Taylor expanded in eps around inf 84.5%

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

   6. Taylor expanded in eps around inf 84.6%

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

      1. *-commutative 97.8%

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

   8. Simplified 84.6%

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

   if 720 < 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 x around 0 3.1%

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

   5. Taylor expanded in eps around 0 17.0%

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

   6. Taylor expanded in x around inf 69.0%

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

      1. mul-1-neg 69.0%

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

      2. mul-1-neg 69.0%

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

      3. sub-neg 69.0%

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

      4. metadata-eval 69.0%

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

      5. *-commutative 69.0%

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

      6. +-commutative 69.0%

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

      7. metadata-eval 69.0%

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

      8. sub-neg 69.0%

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

      9. sub-neg 69.0%

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

      10. metadata-eval 69.0%

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

      11. sub-neg 69.0%

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

   8. Simplified 69.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+31}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 720:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{-1}{\varepsilon} + \left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]

Alternative 8: 74.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 460:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{-1}{\varepsilon} + \left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 460.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (/ (* x (+ (/ -1.0 eps) (* (+ 1.0 eps) (+ -1.0 (/ 1.0 eps))))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 460.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 460.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = (x * (((-1.0d0) / eps) + ((1.0d0 + eps) * ((-1.0d0) + (1.0d0 / eps))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 460.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 460.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 460.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(-1.0 / eps) + Float64(Float64(1.0 + eps) * Float64(-1.0 + Float64(1.0 / eps))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 460.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 460.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(-1.0 / eps), $MachinePrecision] + N[(N[(1.0 + eps), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 460

   1. Initial program 64.9%

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

      1. div-sub 64.9%

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

      2. +-rgt-identity 64.9%

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

      3. div-sub 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in x around 0 45.2%

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

      1. +-commutative 45.2%

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

      2. associate-+l+ 45.2%

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

      3. *-commutative 45.2%

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

      4. associate-*l* 45.2%

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

      5. +-commutative 45.2%

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

      6. *-commutative 45.2%

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

      7. associate-*l* 45.2%

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

      8. *-commutative 45.2%

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

      9. neg-mul-1 45.2%

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

      10. sub-neg 45.2%

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

      11. +-commutative 45.2%

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

      12. distribute-neg-in 45.2%

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

      13. remove-double-neg 45.2%

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

      14. metadata-eval 45.2%

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

      15. distribute-rgt-out 45.2%

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

      16. mul-1-neg 45.2%

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

      17. unsub-neg 45.2%

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

      18. *-commutative 45.2%

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

   6. Simplified 45.2%

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

   7. Taylor expanded in eps around inf 78.6%

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

   8. Taylor expanded in eps around 0 82.4%

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

      1. sub-neg 82.4%

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

      2. distribute-lft1-in 82.4%

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

      3. metadata-eval 82.4%

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

      4. mul0-lft 82.4%

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

      5. metadata-eval 82.4%

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

      6. mul-1-neg 82.4%

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

      7. remove-double-neg 82.4%

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

      8. mul-1-neg 82.4%

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

   10. Simplified 82.4%

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

   if 460 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 x around 0 3.1%

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

   5. Taylor expanded in eps around 0 17.0%

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

   6. Taylor expanded in x around inf 69.0%

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

      1. mul-1-neg 69.0%

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

      2. mul-1-neg 69.0%

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

      3. sub-neg 69.0%

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

      4. metadata-eval 69.0%

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

      5. *-commutative 69.0%

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

      6. +-commutative 69.0%

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

      7. metadata-eval 69.0%

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

      8. sub-neg 69.0%

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

      9. sub-neg 69.0%

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

      10. metadata-eval 69.0%

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

      11. sub-neg 69.0%

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

   8. Simplified 69.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 460:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{-1}{\varepsilon} + \left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]

Alternative 9: 64.2% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\frac{-1}{\varepsilon} + \left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)\\ \mathbf{if}\;x \leq -7.3 \cdot 10^{+207}:\\ \;\;\;\;\frac{\frac{x}{\varepsilon} + -0.5 \cdot \frac{x \cdot x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 400:\\ \;\;\;\;\frac{2 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (+ (/ -1.0 eps) (* (+ 1.0 eps) (+ -1.0 (/ 1.0 eps)))))))
   (if (<= x -7.3e+207)
     (/ (+ (/ x eps) (* -0.5 (/ (* x x) eps))) 2.0)
     (if (<= x 400.0) (/ (+ 2.0 t_0) 2.0) (/ t_0 2.0)))))

C

double code(double x, double eps) {
	double t_0 = x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))));
	double tmp;
	if (x <= -7.3e+207) {
		tmp = ((x / eps) + (-0.5 * ((x * x) / eps))) / 2.0;
	} else if (x <= 400.0) {
		tmp = (2.0 + t_0) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Fortran

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) / eps) + ((1.0d0 + eps) * ((-1.0d0) + (1.0d0 / eps))))
    if (x <= (-7.3d+207)) then
        tmp = ((x / eps) + ((-0.5d0) * ((x * x) / eps))) / 2.0d0
    else if (x <= 400.0d0) then
        tmp = (2.0d0 + t_0) / 2.0d0
    else
        tmp = t_0 / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))));
	double tmp;
	if (x <= -7.3e+207) {
		tmp = ((x / eps) + (-0.5 * ((x * x) / eps))) / 2.0;
	} else if (x <= 400.0) {
		tmp = (2.0 + t_0) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))
	tmp = 0
	if x <= -7.3e+207:
		tmp = ((x / eps) + (-0.5 * ((x * x) / eps))) / 2.0
	elif x <= 400.0:
		tmp = (2.0 + t_0) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(Float64(-1.0 / eps) + Float64(Float64(1.0 + eps) * Float64(-1.0 + Float64(1.0 / eps)))))
	tmp = 0.0
	if (x <= -7.3e+207)
		tmp = Float64(Float64(Float64(x / eps) + Float64(-0.5 * Float64(Float64(x * x) / eps))) / 2.0);
	elseif (x <= 400.0)
		tmp = Float64(Float64(2.0 + t_0) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))));
	tmp = 0.0;
	if (x <= -7.3e+207)
		tmp = ((x / eps) + (-0.5 * ((x * x) / eps))) / 2.0;
	elseif (x <= 400.0)
		tmp = (2.0 + t_0) / 2.0;
	else
		tmp = t_0 / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(N[(-1.0 / eps), $MachinePrecision] + N[(N[(1.0 + eps), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.3e+207], N[(N[(N[(x / eps), $MachinePrecision] + N[(-0.5 * N[(N[(x * x), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 400.0], N[(N[(2.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.30000000000000014e207

   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 x around 0 33.7%

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

   5. Taylor expanded in eps around 0 68.4%

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

      1. mul-1-neg 68.4%

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

   7. Simplified 68.4%

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

   8. Taylor expanded in x around 0 68.4%

      \[\leadsto \frac{\color{blue}{\frac{x}{\varepsilon} + -0.5 \cdot \frac{{x}^{2}}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. unpow2 68.4%

         \[\leadsto \frac{\frac{x}{\varepsilon} + -0.5 \cdot \frac{\color{blue}{x \cdot x}}{\varepsilon}}{2} \]

   10. Simplified 68.4%

       \[\leadsto \frac{\color{blue}{\frac{x}{\varepsilon} + -0.5 \cdot \frac{x \cdot x}{\varepsilon}}}{2} \]

   if -7.30000000000000014e207 < x < 400

   1. Initial program 60.6%

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

      1. div-sub 60.6%

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

      2. +-rgt-identity 60.6%

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

      3. div-sub 60.6%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in x around 0 69.1%

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

   5. Taylor expanded in eps around 0 70.4%

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

   if 400 < 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 x around 0 3.1%

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

   5. Taylor expanded in eps around 0 17.0%

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

   6. Taylor expanded in x around inf 69.0%

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

      1. mul-1-neg 69.0%

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

      2. mul-1-neg 69.0%

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

      3. sub-neg 69.0%

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

      4. metadata-eval 69.0%

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

      5. *-commutative 69.0%

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

      6. +-commutative 69.0%

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

      7. metadata-eval 69.0%

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

      8. sub-neg 69.0%

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

      9. sub-neg 69.0%

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

      10. metadata-eval 69.0%

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

      11. sub-neg 69.0%

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

   8. Simplified 69.0%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.3 \cdot 10^{+207}:\\ \;\;\;\;\frac{\frac{x}{\varepsilon} + -0.5 \cdot \frac{x \cdot x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 400:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{\varepsilon} + \left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{-1}{\varepsilon} + \left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]

Alternative 10: 63.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{x}{\varepsilon} + -0.5 \cdot \frac{x \cdot x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 430:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{-1}{\varepsilon} + \left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.2e+183)
   (/ (+ (/ x eps) (* -0.5 (/ (* x x) eps))) 2.0)
   (if (<= x 430.0)
     1.0
     (/ (* x (+ (/ -1.0 eps) (* (+ 1.0 eps) (+ -1.0 (/ 1.0 eps))))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e+183) {
		tmp = ((x / eps) + (-0.5 * ((x * x) / eps))) / 2.0;
	} else if (x <= 430.0) {
		tmp = 1.0;
	} else {
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5.2d+183)) then
        tmp = ((x / eps) + ((-0.5d0) * ((x * x) / eps))) / 2.0d0
    else if (x <= 430.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * (((-1.0d0) / eps) + ((1.0d0 + eps) * ((-1.0d0) + (1.0d0 / eps))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e+183) {
		tmp = ((x / eps) + (-0.5 * ((x * x) / eps))) / 2.0;
	} else if (x <= 430.0) {
		tmp = 1.0;
	} else {
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.2e+183:
		tmp = ((x / eps) + (-0.5 * ((x * x) / eps))) / 2.0
	elif x <= 430.0:
		tmp = 1.0
	else:
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.2e+183)
		tmp = Float64(Float64(Float64(x / eps) + Float64(-0.5 * Float64(Float64(x * x) / eps))) / 2.0);
	elseif (x <= 430.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * Float64(Float64(-1.0 / eps) + Float64(Float64(1.0 + eps) * Float64(-1.0 + Float64(1.0 / eps))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.2e+183)
		tmp = ((x / eps) + (-0.5 * ((x * x) / eps))) / 2.0;
	elseif (x <= 430.0)
		tmp = 1.0;
	else
		tmp = (x * ((-1.0 / eps) + ((1.0 + eps) * (-1.0 + (1.0 / eps))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.2e+183], N[(N[(N[(x / eps), $MachinePrecision] + N[(-0.5 * N[(N[(x * x), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 430.0], 1.0, N[(N[(x * N[(N[(-1.0 / eps), $MachinePrecision] + N[(N[(1.0 + eps), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1999999999999999e183

   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 x around 0 38.4%

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

   5. Taylor expanded in eps around 0 63.6%

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

      1. mul-1-neg 63.6%

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

   7. Simplified 63.6%

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

   8. Taylor expanded in x around 0 63.6%

      \[\leadsto \frac{\color{blue}{\frac{x}{\varepsilon} + -0.5 \cdot \frac{{x}^{2}}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. unpow2 63.6%

         \[\leadsto \frac{\frac{x}{\varepsilon} + -0.5 \cdot \frac{\color{blue}{x \cdot x}}{\varepsilon}}{2} \]

   10. Simplified 63.6%

       \[\leadsto \frac{\color{blue}{\frac{x}{\varepsilon} + -0.5 \cdot \frac{x \cdot x}{\varepsilon}}}{2} \]

   if -5.1999999999999999e183 < x < 430

   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 x around 0 70.4%

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

   if 430 < 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 x around 0 3.1%

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

   5. Taylor expanded in eps around 0 17.0%

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

   6. Taylor expanded in x around inf 69.0%

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

      1. mul-1-neg 69.0%

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

      2. mul-1-neg 69.0%

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

      3. sub-neg 69.0%

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

      4. metadata-eval 69.0%

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

      5. *-commutative 69.0%

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

      6. +-commutative 69.0%

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

      7. metadata-eval 69.0%

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

      8. sub-neg 69.0%

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

      9. sub-neg 69.0%

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

      10. metadata-eval 69.0%

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

      11. sub-neg 69.0%

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

   8. Simplified 69.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{x}{\varepsilon} + -0.5 \cdot \frac{x \cdot x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 430:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{-1}{\varepsilon} + \left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]

Alternative 11: 61.0% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\frac{\left(\left(1 - \varepsilon\right) \cdot x\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.2)
   (/ (* (* (- 1.0 eps) x) (+ -1.0 (/ -1.0 eps))) 2.0)
   (if (<= x 500.0) 1.0 (/ (/ 0.0 eps) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.2) {
		tmp = (((1.0 - eps) * x) * (-1.0 + (-1.0 / eps))) / 2.0;
	} else if (x <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = (0.0 / eps) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.2d0)) then
        tmp = (((1.0d0 - eps) * x) * ((-1.0d0) + ((-1.0d0) / eps))) / 2.0d0
    else if (x <= 500.0d0) then
        tmp = 1.0d0
    else
        tmp = (0.0d0 / eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.2) {
		tmp = (((1.0 - eps) * x) * (-1.0 + (-1.0 / eps))) / 2.0;
	} else if (x <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = (0.0 / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.2:
		tmp = (((1.0 - eps) * x) * (-1.0 + (-1.0 / eps))) / 2.0
	elif x <= 500.0:
		tmp = 1.0
	else:
		tmp = (0.0 / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.2)
		tmp = Float64(Float64(Float64(Float64(1.0 - eps) * x) * Float64(-1.0 + Float64(-1.0 / eps))) / 2.0);
	elseif (x <= 500.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(0.0 / eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.2)
		tmp = (((1.0 - eps) * x) * (-1.0 + (-1.0 / eps))) / 2.0;
	elseif (x <= 500.0)
		tmp = 1.0;
	else
		tmp = (0.0 / eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.2], N[(N[(N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 500.0], 1.0, N[(N[(0.0 / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2000000000000002

   1. Initial program 94.9%

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

      1. div-sub 94.9%

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

      2. +-rgt-identity 94.9%

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

      3. div-sub 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in x around 0 52.7%

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

      1. +-commutative 52.7%

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

      2. associate-+l+ 52.7%

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

      3. *-commutative 52.7%

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

      4. associate-*l* 52.7%

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

      5. +-commutative 52.7%

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

      6. *-commutative 52.7%

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

      7. associate-*l* 52.7%

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

      8. *-commutative 52.7%

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

      9. neg-mul-1 52.7%

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

      10. sub-neg 52.7%

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

      11. +-commutative 52.7%

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

      12. distribute-neg-in 52.7%

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

      13. remove-double-neg 52.7%

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

      14. metadata-eval 52.7%

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

      15. distribute-rgt-out 52.7%

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

      16. mul-1-neg 52.7%

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

      17. unsub-neg 52.7%

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

      18. *-commutative 52.7%

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

   6. Simplified 52.7%

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

   7. Taylor expanded in x around inf 32.7%

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

   if -2.2000000000000002 < x < 500

   1. Initial program 56.2%

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

      1. div-sub 56.2%

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

      2. +-rgt-identity 56.2%

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

      3. div-sub 56.2%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in x around 0 78.9%

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

   if 500 < 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 x around 0 26.3%

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

   5. Taylor expanded in eps around 0 1.9%

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

      1. mul-1-neg 1.9%

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

   7. Simplified 1.9%

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

      1. flip-- 1.9%

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

      2. metadata-eval 1.9%

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

      3. neg-mul-1 1.9%

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

      4. neg-mul-1 1.9%

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

      5. prod-exp 1.9%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 41.0%

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

      8. neg-mul-1 41.0%

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

      9. neg-mul-1 41.0%

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

      10. sqr-neg 41.0%

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

      11. sqrt-unprod 44.8%

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

      12. add-sqr-sqrt 56.8%

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

      13. distribute-lft1-in 56.8%

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

      14. metadata-eval 56.8%

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

      15. metadata-eval 56.8%

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

      16. exp-prod 56.8%

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

      17. add-exp-log 56.8%

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

      18. neg-mul-1 56.8%

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

   9. Applied egg-rr 56.8%

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

       1. pow-base-1 56.8%

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

       2. div-sub 56.8%

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

       3. +-inverses 56.8%

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

   11. Simplified 56.8%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\frac{\left(\left(1 - \varepsilon\right) \cdot x\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 12: 60.3% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{x}{\varepsilon} + -0.5 \cdot \frac{x \cdot x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.2e+183)
   (/ (+ (/ x eps) (* -0.5 (/ (* x x) eps))) 2.0)
   (if (<= x 500.0) 1.0 (/ (/ 0.0 eps) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e+183) {
		tmp = ((x / eps) + (-0.5 * ((x * x) / eps))) / 2.0;
	} else if (x <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = (0.0 / eps) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5.2d+183)) then
        tmp = ((x / eps) + ((-0.5d0) * ((x * x) / eps))) / 2.0d0
    else if (x <= 500.0d0) then
        tmp = 1.0d0
    else
        tmp = (0.0d0 / eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e+183) {
		tmp = ((x / eps) + (-0.5 * ((x * x) / eps))) / 2.0;
	} else if (x <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = (0.0 / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.2e+183:
		tmp = ((x / eps) + (-0.5 * ((x * x) / eps))) / 2.0
	elif x <= 500.0:
		tmp = 1.0
	else:
		tmp = (0.0 / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.2e+183)
		tmp = Float64(Float64(Float64(x / eps) + Float64(-0.5 * Float64(Float64(x * x) / eps))) / 2.0);
	elseif (x <= 500.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(0.0 / eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.2e+183)
		tmp = ((x / eps) + (-0.5 * ((x * x) / eps))) / 2.0;
	elseif (x <= 500.0)
		tmp = 1.0;
	else
		tmp = (0.0 / eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.2e+183], N[(N[(N[(x / eps), $MachinePrecision] + N[(-0.5 * N[(N[(x * x), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 500.0], 1.0, N[(N[(0.0 / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1999999999999999e183

   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 x around 0 38.4%

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

   5. Taylor expanded in eps around 0 63.6%

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

      1. mul-1-neg 63.6%

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

   7. Simplified 63.6%

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

   8. Taylor expanded in x around 0 63.6%

      \[\leadsto \frac{\color{blue}{\frac{x}{\varepsilon} + -0.5 \cdot \frac{{x}^{2}}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. unpow2 63.6%

         \[\leadsto \frac{\frac{x}{\varepsilon} + -0.5 \cdot \frac{\color{blue}{x \cdot x}}{\varepsilon}}{2} \]

   10. Simplified 63.6%

       \[\leadsto \frac{\color{blue}{\frac{x}{\varepsilon} + -0.5 \cdot \frac{x \cdot x}{\varepsilon}}}{2} \]

   if -5.1999999999999999e183 < x < 500

   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 x around 0 70.4%

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

   if 500 < 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 x around 0 26.3%

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

   5. Taylor expanded in eps around 0 1.9%

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

      1. mul-1-neg 1.9%

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

   7. Simplified 1.9%

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

      1. flip-- 1.9%

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

      2. metadata-eval 1.9%

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

      3. neg-mul-1 1.9%

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

      4. neg-mul-1 1.9%

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

      5. prod-exp 1.9%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 41.0%

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

      8. neg-mul-1 41.0%

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

      9. neg-mul-1 41.0%

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

      10. sqr-neg 41.0%

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

      11. sqrt-unprod 44.8%

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

      12. add-sqr-sqrt 56.8%

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

      13. distribute-lft1-in 56.8%

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

      14. metadata-eval 56.8%

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

      15. metadata-eval 56.8%

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

      16. exp-prod 56.8%

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

      17. add-exp-log 56.8%

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

      18. neg-mul-1 56.8%

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

   9. Applied egg-rr 56.8%

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

       1. pow-base-1 56.8%

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

       2. div-sub 56.8%

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

       3. +-inverses 56.8%

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

   11. Simplified 56.8%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{x}{\varepsilon} + -0.5 \cdot \frac{x \cdot x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 13: 49.7% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+183} \lor \neg \left(x \leq 320\right):\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5.2e+183) (not (<= x 320.0))) (/ (* eps x) 2.0) 1.0))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5.2e+183) || !(x <= 320.0)) {
		tmp = (eps * x) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-5.2d+183)) .or. (.not. (x <= 320.0d0))) then
        tmp = (eps * x) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -5.2e+183) || !(x <= 320.0)) {
		tmp = (eps * x) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -5.2e+183) or not (x <= 320.0):
		tmp = (eps * x) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5.2e+183) || !(x <= 320.0))
		tmp = Float64(Float64(eps * x) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -5.2e+183) || ~((x <= 320.0)))
		tmp = (eps * x) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5.2e+183], N[Not[LessEqual[x, 320.0]], $MachinePrecision]], N[(N[(eps * x), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -5.1999999999999999e183 or 320 < 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 x around 0 29.2%

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

      1. +-commutative 29.2%

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

      2. associate-+l+ 29.2%

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

      3. *-commutative 29.2%

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

      4. associate-*l* 29.2%

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

      5. +-commutative 29.2%

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

      6. *-commutative 29.2%

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

      7. associate-*l* 29.2%

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

      8. *-commutative 29.2%

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

      9. neg-mul-1 29.2%

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

      10. sub-neg 29.2%

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

      11. +-commutative 29.2%

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

      12. distribute-neg-in 29.2%

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

      13. remove-double-neg 29.2%

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

      14. metadata-eval 29.2%

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

      15. distribute-rgt-out 29.2%

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

      16. mul-1-neg 29.2%

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

      17. unsub-neg 29.2%

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

      18. *-commutative 29.2%

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

   6. Simplified 29.2%

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

   7. Taylor expanded in x around inf 21.6%

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

   8. Taylor expanded in eps around inf 22.2%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 22.2%

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

   10. Simplified 22.2%

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

   if -5.1999999999999999e183 < x < 320

   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 x around 0 70.4%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+183} \lor \neg \left(x \leq 320\right):\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 59.7% accurate, 24.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+183}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.2e+183)
   (/ (* eps x) 2.0)
   (if (<= x 550.0) 1.0 (/ (/ 0.0 eps) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e+183) {
		tmp = (eps * x) / 2.0;
	} else if (x <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = (0.0 / eps) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5.2d+183)) then
        tmp = (eps * x) / 2.0d0
    else if (x <= 550.0d0) then
        tmp = 1.0d0
    else
        tmp = (0.0d0 / eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e+183) {
		tmp = (eps * x) / 2.0;
	} else if (x <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = (0.0 / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.2e+183:
		tmp = (eps * x) / 2.0
	elif x <= 550.0:
		tmp = 1.0
	else:
		tmp = (0.0 / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.2e+183)
		tmp = Float64(Float64(eps * x) / 2.0);
	elseif (x <= 550.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(0.0 / eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.2e+183)
		tmp = (eps * x) / 2.0;
	elseif (x <= 550.0)
		tmp = 1.0;
	else
		tmp = (0.0 / eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.2e+183], N[(N[(eps * x), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 550.0], 1.0, N[(N[(0.0 / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1999999999999999e183

   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 x around 0 55.3%

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

      1. +-commutative 55.3%

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

      2. associate-+l+ 55.3%

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

      3. *-commutative 55.3%

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

      4. associate-*l* 55.3%

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

      5. +-commutative 55.3%

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

      6. *-commutative 55.3%

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

      7. associate-*l* 55.3%

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

      8. *-commutative 55.3%

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

      9. neg-mul-1 55.3%

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

      10. sub-neg 55.3%

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

      11. +-commutative 55.3%

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

      12. distribute-neg-in 55.3%

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

      13. remove-double-neg 55.3%

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

      14. metadata-eval 55.3%

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

      15. distribute-rgt-out 55.3%

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

      16. mul-1-neg 55.3%

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

      17. unsub-neg 55.3%

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

      18. *-commutative 55.3%

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

   6. Simplified 55.3%

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

   7. Taylor expanded in x around inf 55.3%

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

   8. Taylor expanded in eps around inf 55.3%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 55.3%

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

   10. Simplified 55.3%

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

   if -5.1999999999999999e183 < x < 550

   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 x around 0 70.4%

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

   if 550 < 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 x around 0 26.3%

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

   5. Taylor expanded in eps around 0 1.9%

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

      1. mul-1-neg 1.9%

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

   7. Simplified 1.9%

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

      1. flip-- 1.9%

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

      2. metadata-eval 1.9%

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

      3. neg-mul-1 1.9%

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

      4. neg-mul-1 1.9%

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

      5. prod-exp 1.9%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 41.0%

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

      8. neg-mul-1 41.0%

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

      9. neg-mul-1 41.0%

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

      10. sqr-neg 41.0%

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

      11. sqrt-unprod 44.8%

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

      12. add-sqr-sqrt 56.8%

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

      13. distribute-lft1-in 56.8%

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

      14. metadata-eval 56.8%

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

      15. metadata-eval 56.8%

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

      16. exp-prod 56.8%

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

      17. add-exp-log 56.8%

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

      18. neg-mul-1 56.8%

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

   9. Applied egg-rr 56.8%

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

       1. pow-base-1 56.8%

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

       2. div-sub 56.8%

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

       3. +-inverses 56.8%

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

   11. Simplified 56.8%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+183}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 15: 44.2% accurate, 227.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 76.1%

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

   1. div-sub 76.1%

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

   2. +-rgt-identity 76.1%

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

   3. div-sub 76.1%

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

3. Simplified 76.1%

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

4. Taylor expanded in x around 0 43.1%

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

5. Final simplification 43.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023178 
(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))