NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 100.0%

Time: 19.4s

Alternatives: 13

Speedup: 1.9×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% 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: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{2}{e^{x}} \cdot \left(x + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} - \left(\frac{1}{eps\_m} + -1\right) \cdot e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 6e-6)
   (/ (* (/ 2.0 (exp x)) (+ x 1.0)) 2.0)
   (/
    (-
     (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
     (* (+ (/ 1.0 eps_m) -1.0) (exp (* x (- -1.0 eps_m)))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 6e-6) {
		tmp = ((2.0 / exp(x)) * (x + 1.0)) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) - (((1.0 / eps_m) + -1.0) * exp((x * (-1.0 - eps_m))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 6d-6) then
        tmp = ((2.0d0 / exp(x)) * (x + 1.0d0)) / 2.0d0
    else
        tmp = (((1.0d0 + (1.0d0 / eps_m)) * exp((x * (eps_m + (-1.0d0))))) - (((1.0d0 / eps_m) + (-1.0d0)) * exp((x * ((-1.0d0) - eps_m))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 6e-6) {
		tmp = ((2.0 / Math.exp(x)) * (x + 1.0)) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * Math.exp((x * (eps_m + -1.0)))) - (((1.0 / eps_m) + -1.0) * Math.exp((x * (-1.0 - eps_m))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 6e-6:
		tmp = ((2.0 / math.exp(x)) * (x + 1.0)) / 2.0
	else:
		tmp = (((1.0 + (1.0 / eps_m)) * math.exp((x * (eps_m + -1.0)))) - (((1.0 / eps_m) + -1.0) * math.exp((x * (-1.0 - eps_m))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 6e-6)
		tmp = Float64(Float64(Float64(2.0 / exp(x)) * Float64(x + 1.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) - Float64(Float64(Float64(1.0 / eps_m) + -1.0) * exp(Float64(x * Float64(-1.0 - eps_m))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 6e-6)
		tmp = ((2.0 / exp(x)) * (x + 1.0)) / 2.0;
	else
		tmp = (((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) - (((1.0 / eps_m) + -1.0) * exp((x * (-1.0 - eps_m))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 6e-6], N[(N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 6.0000000000000002e-6

   1. Initial program 61.9%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in eps around 0 31.1%

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

   7. Simplified 70.6%

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

   8. Taylor expanded in eps around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. exp-neg 70.6%

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

      3. associate-*r/ 70.6%

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

      4. metadata-eval 70.6%

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

   10. Simplified 70.6%

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

   if 6.0000000000000002e-6 < eps

   1. Initial program 100.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{2}{e^{x}} \cdot \left(x + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-187}:\\ \;\;\;\;\frac{1 + e^{x - eps\_m \cdot x}}{2}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+148}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+219} \lor \neg \left(x \leq 2.9 \cdot 10^{+296}\right):\\ \;\;\;\;\frac{\frac{2}{e^{x}} \cdot \left(x + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -3.6e-187)
   (/ (+ 1.0 (exp (- x (* eps_m x)))) 2.0)
   (if (<= x 3.9e+148)
     (/ (+ 1.0 (exp (* eps_m x))) 2.0)
     (if (or (<= x 9.8e+219) (not (<= x 2.9e+296)))
       (/ (* (/ 2.0 (exp x)) (+ x 1.0)) 2.0)
       (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -3.6e-187) {
		tmp = (1.0 + exp((x - (eps_m * x)))) / 2.0;
	} else if (x <= 3.9e+148) {
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	} else if ((x <= 9.8e+219) || !(x <= 2.9e+296)) {
		tmp = ((2.0 / exp(x)) * (x + 1.0)) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-3.6d-187)) then
        tmp = (1.0d0 + exp((x - (eps_m * x)))) / 2.0d0
    else if (x <= 3.9d+148) then
        tmp = (1.0d0 + exp((eps_m * x))) / 2.0d0
    else if ((x <= 9.8d+219) .or. (.not. (x <= 2.9d+296))) then
        tmp = ((2.0d0 / exp(x)) * (x + 1.0d0)) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -3.6e-187) {
		tmp = (1.0 + Math.exp((x - (eps_m * x)))) / 2.0;
	} else if (x <= 3.9e+148) {
		tmp = (1.0 + Math.exp((eps_m * x))) / 2.0;
	} else if ((x <= 9.8e+219) || !(x <= 2.9e+296)) {
		tmp = ((2.0 / Math.exp(x)) * (x + 1.0)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -3.6e-187:
		tmp = (1.0 + math.exp((x - (eps_m * x)))) / 2.0
	elif x <= 3.9e+148:
		tmp = (1.0 + math.exp((eps_m * x))) / 2.0
	elif (x <= 9.8e+219) or not (x <= 2.9e+296):
		tmp = ((2.0 / math.exp(x)) * (x + 1.0)) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -3.6e-187)
		tmp = Float64(Float64(1.0 + exp(Float64(x - Float64(eps_m * x)))) / 2.0);
	elseif (x <= 3.9e+148)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * x))) / 2.0);
	elseif ((x <= 9.8e+219) || !(x <= 2.9e+296))
		tmp = Float64(Float64(Float64(2.0 / exp(x)) * Float64(x + 1.0)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -3.6e-187)
		tmp = (1.0 + exp((x - (eps_m * x)))) / 2.0;
	elseif (x <= 3.9e+148)
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	elseif ((x <= 9.8e+219) || ~((x <= 2.9e+296)))
		tmp = ((2.0 / exp(x)) * (x + 1.0)) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -3.6e-187], N[(N[(1.0 + N[Exp[N[(x - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.9e+148], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 9.8e+219], N[Not[LessEqual[x, 2.9e+296]], $MachinePrecision]], N[(N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.59999999999999994e-187

   1. Initial program 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in x around 0 47.5%

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

   6. Taylor expanded in eps around inf 65.4%

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

      1. neg-mul-1 65.4%

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

      2. distribute-rgt-neg-in 65.4%

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

   8. Simplified 65.4%

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

      1. add-sqr-sqrt 3.5%

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

      2. sqrt-unprod 30.8%

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

      3. sqr-neg 30.8%

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

      4. sqrt-unprod 27.9%

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

      5. add-sqr-sqrt 62.9%

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

      6. sub-neg 62.9%

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

      7. distribute-rgt-in 62.9%

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

      8. *-un-lft-identity 62.9%

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

   10. Applied egg-rr 62.9%

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

   if -3.59999999999999994e-187 < x < 3.90000000000000002e148

   1. Initial program 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in x around 0 40.7%

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

   6. Taylor expanded in eps around inf 76.5%

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

      1. neg-mul-1 76.5%

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

      2. distribute-rgt-neg-in 76.5%

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

   8. Simplified 76.5%

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

   9. Taylor expanded in eps around inf 77.5%

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

       1. *-commutative 77.5%

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

   11. Simplified 77.5%

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

   if 3.90000000000000002e148 < x < 9.80000000000000007e219 or 2.90000000000000004e296 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in eps around 0 82.1%

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

      1. associate-*r* 82.1%

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

      2. exp-neg 82.1%

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

      3. associate-*r/ 82.1%

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

      4. metadata-eval 82.1%

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

   10. Simplified 82.1%

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

   if 9.80000000000000007e219 < x < 2.90000000000000004e296

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 28.8%

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

   6. Taylor expanded in eps around inf 29.0%

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

      1. neg-mul-1 29.0%

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

      2. distribute-rgt-neg-in 29.0%

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

   8. Simplified 29.0%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-187}:\\ \;\;\;\;\frac{1 + e^{x - \varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+148}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+219} \lor \neg \left(x \leq 2.9 \cdot 10^{+296}\right):\\ \;\;\;\;\frac{\frac{2}{e^{x}} \cdot \left(x + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1:\\ \;\;\;\;\frac{\frac{2}{e^{x}} \cdot \left(x + 1\right)}{2}\\ \mathbf{elif}\;eps\_m \leq 4.1 \cdot 10^{+242} \lor \neg \left(eps\_m \leq 1.35 \cdot 10^{+253}\right):\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{eps\_m \cdot \left(2 \cdot x - eps\_m \cdot x\right) - x}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.0)
   (/ (* (/ 2.0 (exp x)) (+ x 1.0)) 2.0)
   (if (or (<= eps_m 4.1e+242) (not (<= eps_m 1.35e+253)))
     (/ (+ 1.0 (exp (* eps_m x))) 2.0)
     (/ (+ 2.0 (/ (- (* eps_m (- (* 2.0 x) (* eps_m x))) x) eps_m)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = ((2.0 / exp(x)) * (x + 1.0)) / 2.0;
	} else if ((eps_m <= 4.1e+242) || !(eps_m <= 1.35e+253)) {
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	} else {
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.0d0) then
        tmp = ((2.0d0 / exp(x)) * (x + 1.0d0)) / 2.0d0
    else if ((eps_m <= 4.1d+242) .or. (.not. (eps_m <= 1.35d+253))) then
        tmp = (1.0d0 + exp((eps_m * x))) / 2.0d0
    else
        tmp = (2.0d0 + (((eps_m * ((2.0d0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = ((2.0 / Math.exp(x)) * (x + 1.0)) / 2.0;
	} else if ((eps_m <= 4.1e+242) || !(eps_m <= 1.35e+253)) {
		tmp = (1.0 + Math.exp((eps_m * x))) / 2.0;
	} else {
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.0:
		tmp = ((2.0 / math.exp(x)) * (x + 1.0)) / 2.0
	elif (eps_m <= 4.1e+242) or not (eps_m <= 1.35e+253):
		tmp = (1.0 + math.exp((eps_m * x))) / 2.0
	else:
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.0)
		tmp = Float64(Float64(Float64(2.0 / exp(x)) * Float64(x + 1.0)) / 2.0);
	elseif ((eps_m <= 4.1e+242) || !(eps_m <= 1.35e+253))
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * x))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(eps_m * Float64(Float64(2.0 * x) - Float64(eps_m * x))) - x) / eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.0)
		tmp = ((2.0 / exp(x)) * (x + 1.0)) / 2.0;
	elseif ((eps_m <= 4.1e+242) || ~((eps_m <= 1.35e+253)))
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	else
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.0], N[(N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[eps$95$m, 4.1e+242], N[Not[LessEqual[eps$95$m, 1.35e+253]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(eps$95$m * N[(N[(2.0 * x), $MachinePrecision] - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1

   1. Initial program 62.5%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in eps around 0 32.2%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in eps around 0 71.1%

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

      1. associate-*r* 71.1%

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

      2. exp-neg 71.1%

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

      3. associate-*r/ 71.1%

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

      4. metadata-eval 71.1%

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

   10. Simplified 71.1%

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

   if 1 < eps < 4.09999999999999979e242 or 1.35000000000000001e253 < eps

   1. Initial program 100.0%

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

   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. Add Preprocessing

   5. Taylor expanded in x around 0 69.1%

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

   6. Taylor expanded in eps around inf 69.1%

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

      1. neg-mul-1 69.1%

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

      2. distribute-rgt-neg-in 69.1%

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

   8. Simplified 69.1%

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

   9. Taylor expanded in eps around inf 69.1%

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

       1. *-commutative 69.1%

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

   11. Simplified 69.1%

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

   if 4.09999999999999979e242 < eps < 1.35000000000000001e253

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around 0 0.3%

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

      1. *-commutative 0.3%

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

      2. associate-*r* 0.3%

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

      3. associate-*r* 0.3%

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

      4. associate-*l* 0.3%

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

      5. distribute-lft-in 0.3%

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

      6. metadata-eval 0.3%

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

      7. associate-*r/ 0.3%

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

      8. metadata-eval 0.3%

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

      9. *-commutative 0.3%

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

   8. Simplified 0.3%

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

      1. *-commutative 0.3%

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

      2. distribute-lft-in 0.3%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod 5.9%

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

      7. add-sqr-sqrt 5.9%

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

      8. neg-sub0 5.9%

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

      9. associate--r- 5.9%

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

      10. metadata-eval 5.9%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 100.0%

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

      13. sqr-neg 100.0%

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

      14. sqrt-unprod 5.9%

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

      15. add-sqr-sqrt 5.9%

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

      16. neg-sub0 5.9%

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

      17. associate--r- 5.9%

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

      18. metadata-eval 5.9%

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

      19. add-sqr-sqrt 0.0%

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

   10. Applied egg-rr 5.9%

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

       1. distribute-lft-out 5.9%

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

       2. *-commutative 5.9%

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

       3. associate-*l* 5.9%

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

       4. +-commutative 5.9%

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

       5. +-commutative 5.9%

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

       6. distribute-rgt-in 5.9%

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

       7. associate-*l/ 5.9%

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

       8. *-lft-identity 5.9%

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

       9. neg-mul-1 5.9%

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

       10. unsub-neg 5.9%

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

   12. Simplified 5.9%

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

   13. Taylor expanded in eps around 0 100.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{\frac{2}{e^{x}} \cdot \left(x + 1\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.1 \cdot 10^{+242} \lor \neg \left(\varepsilon \leq 1.35 \cdot 10^{+253}\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{\varepsilon \cdot \left(2 \cdot x - \varepsilon \cdot x\right) - x}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1:\\ \;\;\;\;\frac{\frac{2}{e^{x}} \cdot \left(x + 1\right)}{2}\\ \mathbf{elif}\;eps\_m \leq 2.6 \cdot 10^{+242}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{elif}\;eps\_m \leq 2.05 \cdot 10^{+253}:\\ \;\;\;\;\frac{2 + \frac{eps\_m \cdot \left(2 \cdot x - eps\_m \cdot x\right) - x}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.0)
   (/ (* (/ 2.0 (exp x)) (+ x 1.0)) 2.0)
   (if (<= eps_m 2.6e+242)
     (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)
     (if (<= eps_m 2.05e+253)
       (/ (+ 2.0 (/ (- (* eps_m (- (* 2.0 x) (* eps_m x))) x) eps_m)) 2.0)
       (/ (+ 1.0 (exp (* eps_m x))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = ((2.0 / exp(x)) * (x + 1.0)) / 2.0;
	} else if (eps_m <= 2.6e+242) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} else if (eps_m <= 2.05e+253) {
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0;
	} else {
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.0d0) then
        tmp = ((2.0d0 / exp(x)) * (x + 1.0d0)) / 2.0d0
    else if (eps_m <= 2.6d+242) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    else if (eps_m <= 2.05d+253) then
        tmp = (2.0d0 + (((eps_m * ((2.0d0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0d0
    else
        tmp = (1.0d0 + exp((eps_m * x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = ((2.0 / Math.exp(x)) * (x + 1.0)) / 2.0;
	} else if (eps_m <= 2.6e+242) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else if (eps_m <= 2.05e+253) {
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((eps_m * x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.0:
		tmp = ((2.0 / math.exp(x)) * (x + 1.0)) / 2.0
	elif eps_m <= 2.6e+242:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	elif eps_m <= 2.05e+253:
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0
	else:
		tmp = (1.0 + math.exp((eps_m * x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.0)
		tmp = Float64(Float64(Float64(2.0 / exp(x)) * Float64(x + 1.0)) / 2.0);
	elseif (eps_m <= 2.6e+242)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	elseif (eps_m <= 2.05e+253)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(eps_m * Float64(Float64(2.0 * x) - Float64(eps_m * x))) - x) / eps_m)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.0)
		tmp = ((2.0 / exp(x)) * (x + 1.0)) / 2.0;
	elseif (eps_m <= 2.6e+242)
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	elseif (eps_m <= 2.05e+253)
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0;
	else
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.0], N[(N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 2.6e+242], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 2.05e+253], N[(N[(2.0 + N[(N[(N[(eps$95$m * N[(N[(2.0 * x), $MachinePrecision] - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 1

   1. Initial program 62.5%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in eps around 0 32.2%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in eps around 0 71.1%

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

      1. associate-*r* 71.1%

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

      2. exp-neg 71.1%

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

      3. associate-*r/ 71.1%

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

      4. metadata-eval 71.1%

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

   10. Simplified 71.1%

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

   if 1 < eps < 2.5999999999999998e242

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 68.3%

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

   6. Taylor expanded in eps around inf 68.3%

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

      1. neg-mul-1 68.3%

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

      2. distribute-rgt-neg-in 68.3%

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

   8. Simplified 68.3%

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

   if 2.5999999999999998e242 < eps < 2.05e253

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around 0 0.3%

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

      1. *-commutative 0.3%

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

      2. associate-*r* 0.3%

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

      3. associate-*r* 0.3%

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

      4. associate-*l* 0.3%

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

      5. distribute-lft-in 0.3%

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

      6. metadata-eval 0.3%

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

      7. associate-*r/ 0.3%

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

      8. metadata-eval 0.3%

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

      9. *-commutative 0.3%

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

   8. Simplified 0.3%

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

      1. *-commutative 0.3%

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

      2. distribute-lft-in 0.3%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod 5.9%

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

      7. add-sqr-sqrt 5.9%

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

      8. neg-sub0 5.9%

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

      9. associate--r- 5.9%

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

      10. metadata-eval 5.9%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 100.0%

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

      13. sqr-neg 100.0%

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

      14. sqrt-unprod 5.9%

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

      15. add-sqr-sqrt 5.9%

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

      16. neg-sub0 5.9%

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

      17. associate--r- 5.9%

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

      18. metadata-eval 5.9%

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

      19. add-sqr-sqrt 0.0%

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

   10. Applied egg-rr 5.9%

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

       1. distribute-lft-out 5.9%

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

       2. *-commutative 5.9%

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

       3. associate-*l* 5.9%

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

       4. +-commutative 5.9%

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

       5. +-commutative 5.9%

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

       6. distribute-rgt-in 5.9%

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

       7. associate-*l/ 5.9%

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

       8. *-lft-identity 5.9%

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

       9. neg-mul-1 5.9%

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

       10. unsub-neg 5.9%

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

   12. Simplified 5.9%

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

   13. Taylor expanded in eps around 0 100.0%

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

   if 2.05e253 < eps

   1. Initial program 100.0%

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

   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. Add Preprocessing

   5. Taylor expanded in x around 0 72.3%

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

   6. Taylor expanded in eps around inf 72.3%

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

      1. neg-mul-1 72.3%

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

      2. distribute-rgt-neg-in 72.3%

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

   8. Simplified 72.3%

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

   9. Taylor expanded in eps around inf 72.3%

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

       1. *-commutative 72.3%

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

   11. Simplified 72.3%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{\frac{2}{e^{x}} \cdot \left(x + 1\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{+242}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.05 \cdot 10^{+253}:\\ \;\;\;\;\frac{2 + \frac{\varepsilon \cdot \left(2 \cdot x - \varepsilon \cdot x\right) - x}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \frac{eps\_m \cdot \left(2 \cdot x - eps\_m \cdot x\right) - x}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 580:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+146}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+227}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + eps\_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4.2e-5)
   (/ (+ 2.0 (/ (- (* eps_m (- (* 2.0 x) (* eps_m x))) x) eps_m)) 2.0)
   (if (<= x 580.0)
     1.0
     (if (<= x 1e+146)
       (/ (/ (expm1 x) eps_m) 2.0)
       (if (<= x 2.5e+227)
         0.0
         (if (<= x 2.25e+296) (/ (+ 2.0 (* eps_m x)) 2.0) 0.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -4.2e-5) {
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0;
	} else if (x <= 580.0) {
		tmp = 1.0;
	} else if (x <= 1e+146) {
		tmp = (expm1(x) / eps_m) / 2.0;
	} else if (x <= 2.5e+227) {
		tmp = 0.0;
	} else if (x <= 2.25e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -4.2e-5) {
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0;
	} else if (x <= 580.0) {
		tmp = 1.0;
	} else if (x <= 1e+146) {
		tmp = (Math.expm1(x) / eps_m) / 2.0;
	} else if (x <= 2.5e+227) {
		tmp = 0.0;
	} else if (x <= 2.25e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -4.2e-5:
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0
	elif x <= 580.0:
		tmp = 1.0
	elif x <= 1e+146:
		tmp = (math.expm1(x) / eps_m) / 2.0
	elif x <= 2.5e+227:
		tmp = 0.0
	elif x <= 2.25e+296:
		tmp = (2.0 + (eps_m * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -4.2e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(eps_m * Float64(Float64(2.0 * x) - Float64(eps_m * x))) - x) / eps_m)) / 2.0);
	elseif (x <= 580.0)
		tmp = 1.0;
	elseif (x <= 1e+146)
		tmp = Float64(Float64(expm1(x) / eps_m) / 2.0);
	elseif (x <= 2.5e+227)
		tmp = 0.0;
	elseif (x <= 2.25e+296)
		tmp = Float64(Float64(2.0 + Float64(eps_m * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -4.2e-5], N[(N[(2.0 + N[(N[(N[(eps$95$m * N[(N[(2.0 * x), $MachinePrecision] - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 580.0], 1.0, If[LessEqual[x, 1e+146], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.5e+227], 0.0, If[LessEqual[x, 2.25e+296], N[(N[(2.0 + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -4.19999999999999977e-5

   1. Initial program 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 66.0%

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

   6. Taylor expanded in x around 0 24.5%

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

      1. *-commutative 24.5%

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

      2. associate-*r* 24.5%

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

      3. associate-*r* 24.5%

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

      4. associate-*l* 24.5%

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

      5. distribute-lft-in 24.5%

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

      6. metadata-eval 24.5%

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

      7. associate-*r/ 24.5%

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

      8. metadata-eval 24.5%

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

      9. *-commutative 24.5%

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

   8. Simplified 24.5%

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

      1. *-commutative 24.5%

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

      2. distribute-lft-in 2.9%

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

      3. add-sqr-sqrt 2.9%

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

      4. sqrt-unprod 31.7%

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

      5. sqr-neg 31.7%

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

      6. sqrt-unprod 1.5%

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

      7. add-sqr-sqrt 1.6%

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

      8. neg-sub0 1.6%

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

      9. associate--r- 1.6%

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

      10. metadata-eval 1.6%

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

      11. add-sqr-sqrt 0.2%

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

      12. sqrt-unprod 22.6%

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

      13. sqr-neg 22.6%

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

      14. sqrt-unprod 15.0%

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

      15. add-sqr-sqrt 15.1%

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

      16. neg-sub0 15.1%

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

      17. associate--r- 15.1%

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

      18. metadata-eval 15.1%

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

      19. add-sqr-sqrt 0.1%

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

   10. Applied egg-rr 1.6%

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

       1. distribute-lft-out 15.2%

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

       2. *-commutative 15.2%

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

       3. associate-*l* 15.2%

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

       4. +-commutative 15.2%

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

       5. +-commutative 15.2%

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

       6. distribute-rgt-in 15.2%

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

       7. associate-*l/ 15.2%

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

       8. *-lft-identity 15.2%

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

       9. neg-mul-1 15.2%

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

       10. unsub-neg 15.2%

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

   12. Simplified 15.2%

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

   13. Taylor expanded in eps around 0 27.5%

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

   if -4.19999999999999977e-5 < x < 580

   1. Initial program 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in x around 0 74.0%

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

   if 580 < x < 9.99999999999999934e145

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 38.5%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)}}{\varepsilon}}{2} \]

      2. neg-mul-1 1.9%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right)}{\varepsilon}}{2} \]

   8. Simplified 1.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. expm1-undefine 1.9%

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

      2. div-sub 1.9%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 37.5%

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

      5. sqr-neg 37.5%

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

      6. sqrt-unprod 37.5%

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

      7. add-sqr-sqrt 37.5%

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

   10. Applied egg-rr 37.5%

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

       1. div-sub 37.5%

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

       2. expm1-undefine 37.5%

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{\varepsilon}}{2} \]

   12. Simplified 37.5%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]

   if 9.99999999999999934e145 < x < 2.4999999999999998e227 or 2.2499999999999998e296 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 79.5%

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

      1. mul-1-neg 79.5%

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

      2. mul-1-neg 79.5%

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

      3. rec-exp 79.5%

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

      4. sub-neg 79.5%

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

      5. div-sub 79.5%

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

      6. rec-exp 79.5%

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

      7. mul-1-neg 79.5%

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

      8. +-inverses 79.5%

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

   7. Simplified 79.5%

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

   if 2.4999999999999998e227 < x < 2.2499999999999998e296

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 25.4%

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

   6. Taylor expanded in eps around inf 25.5%

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

      1. neg-mul-1 25.5%

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

      2. distribute-rgt-neg-in 25.5%

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

   8. Simplified 25.5%

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

   9. Taylor expanded in eps around inf 25.2%

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

       1. *-commutative 25.2%

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

   11. Simplified 25.2%

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

   12. Taylor expanded in x around 0 23.5%

       \[\leadsto \frac{\color{blue}{2 + \varepsilon \cdot x}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \frac{\varepsilon \cdot \left(2 \cdot x - \varepsilon \cdot x\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 580:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+146}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+227}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -6200:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+148}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+228}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + eps\_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -6200.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 3.7e+148)
     (/ (+ 1.0 (exp (* eps_m x))) 2.0)
     (if (<= x 1.15e+228)
       0.0
       (if (<= x 2e+296) (/ (+ 2.0 (* eps_m x)) 2.0) 0.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -6200.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 3.7e+148) {
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	} else if (x <= 1.15e+228) {
		tmp = 0.0;
	} else if (x <= 2e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-6200.0d0)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 3.7d+148) then
        tmp = (1.0d0 + exp((eps_m * x))) / 2.0d0
    else if (x <= 1.15d+228) then
        tmp = 0.0d0
    else if (x <= 2d+296) then
        tmp = (2.0d0 + (eps_m * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -6200.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 3.7e+148) {
		tmp = (1.0 + Math.exp((eps_m * x))) / 2.0;
	} else if (x <= 1.15e+228) {
		tmp = 0.0;
	} else if (x <= 2e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -6200.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 3.7e+148:
		tmp = (1.0 + math.exp((eps_m * x))) / 2.0
	elif x <= 1.15e+228:
		tmp = 0.0
	elif x <= 2e+296:
		tmp = (2.0 + (eps_m * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -6200.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 3.7e+148)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * x))) / 2.0);
	elseif (x <= 1.15e+228)
		tmp = 0.0;
	elseif (x <= 2e+296)
		tmp = Float64(Float64(2.0 + Float64(eps_m * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -6200.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 3.7e+148)
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	elseif (x <= 1.15e+228)
		tmp = 0.0;
	elseif (x <= 2e+296)
		tmp = (2.0 + (eps_m * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -6200.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.7e+148], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.15e+228], 0.0, If[LessEqual[x, 2e+296], N[(N[(2.0 + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6200

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 63.0%

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

   6. Taylor expanded in eps around inf 63.0%

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

      1. neg-mul-1 63.0%

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

      2. distribute-rgt-neg-in 63.0%

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

   8. Simplified 63.0%

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

   9. Taylor expanded in eps around 0 100.0%

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

       1. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   if -6200 < x < 3.7000000000000002e148

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in x around 0 39.6%

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

   6. Taylor expanded in eps around inf 74.2%

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

      1. neg-mul-1 74.2%

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

      2. distribute-rgt-neg-in 74.2%

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

   8. Simplified 74.2%

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

   9. Taylor expanded in eps around inf 75.3%

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

       1. *-commutative 75.3%

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

   11. Simplified 75.3%

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

   if 3.7000000000000002e148 < x < 1.15000000000000006e228 or 1.99999999999999996e296 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 79.5%

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

      1. mul-1-neg 79.5%

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

      2. mul-1-neg 79.5%

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

      3. rec-exp 79.5%

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

      4. sub-neg 79.5%

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

      5. div-sub 79.5%

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

      6. rec-exp 79.5%

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

      7. mul-1-neg 79.5%

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

      8. +-inverses 79.5%

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

   7. Simplified 79.5%

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

   if 1.15000000000000006e228 < x < 1.99999999999999996e296

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 25.4%

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

   6. Taylor expanded in eps around inf 25.5%

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

      1. neg-mul-1 25.5%

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

      2. distribute-rgt-neg-in 25.5%

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

   8. Simplified 25.5%

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

   9. Taylor expanded in eps around inf 25.2%

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

       1. *-commutative 25.2%

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

   11. Simplified 25.2%

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

   12. Taylor expanded in x around 0 23.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6200:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+148}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+228}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 650:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 10^{+148}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+227}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + eps\_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 650.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 1e+148)
     (/ (/ (expm1 x) eps_m) 2.0)
     (if (<= x 8.2e+227)
       0.0
       (if (<= x 3e+296) (/ (+ 2.0 (* eps_m x)) 2.0) 0.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 650.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 1e+148) {
		tmp = (expm1(x) / eps_m) / 2.0;
	} else if (x <= 8.2e+227) {
		tmp = 0.0;
	} else if (x <= 3e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 650.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 1e+148) {
		tmp = (Math.expm1(x) / eps_m) / 2.0;
	} else if (x <= 8.2e+227) {
		tmp = 0.0;
	} else if (x <= 3e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 650.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 1e+148:
		tmp = (math.expm1(x) / eps_m) / 2.0
	elif x <= 8.2e+227:
		tmp = 0.0
	elif x <= 3e+296:
		tmp = (2.0 + (eps_m * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 650.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 1e+148)
		tmp = Float64(Float64(expm1(x) / eps_m) / 2.0);
	elseif (x <= 8.2e+227)
		tmp = 0.0;
	elseif (x <= 3e+296)
		tmp = Float64(Float64(2.0 + Float64(eps_m * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 650.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+148], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.2e+227], 0.0, If[LessEqual[x, 3e+296], N[(N[(2.0 + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < 650

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in x around 0 44.2%

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

   6. Taylor expanded in eps around inf 79.5%

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

      1. neg-mul-1 79.5%

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

      2. distribute-rgt-neg-in 79.5%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in eps around 0 76.5%

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

       1. neg-mul-1 76.5%

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

   11. Simplified 76.5%

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

   if 650 < x < 1e148

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 38.5%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)}}{\varepsilon}}{2} \]

      2. neg-mul-1 1.9%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right)}{\varepsilon}}{2} \]

   8. Simplified 1.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. expm1-undefine 1.9%

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

      2. div-sub 1.9%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 37.5%

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

      5. sqr-neg 37.5%

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

      6. sqrt-unprod 37.5%

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

      7. add-sqr-sqrt 37.5%

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

   10. Applied egg-rr 37.5%

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

       1. div-sub 37.5%

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

       2. expm1-undefine 37.5%

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{\varepsilon}}{2} \]

   12. Simplified 37.5%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]

   if 1e148 < x < 8.19999999999999992e227 or 3.00000000000000013e296 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 79.5%

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

      1. mul-1-neg 79.5%

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

      2. mul-1-neg 79.5%

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

      3. rec-exp 79.5%

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

      4. sub-neg 79.5%

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

      5. div-sub 79.5%

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

      6. rec-exp 79.5%

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

      7. mul-1-neg 79.5%

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

      8. +-inverses 79.5%

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

   7. Simplified 79.5%

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

   if 8.19999999999999992e227 < x < 3.00000000000000013e296

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 25.4%

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

   6. Taylor expanded in eps around inf 25.5%

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

      1. neg-mul-1 25.5%

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

      2. distribute-rgt-neg-in 25.5%

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

   8. Simplified 25.5%

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

   9. Taylor expanded in eps around inf 25.2%

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

       1. *-commutative 25.2%

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

   11. Simplified 25.2%

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

   12. Taylor expanded in x around 0 23.5%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 650:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 10^{+148}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+227}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.0% accurate, 8.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \frac{eps\_m \cdot \left(2 \cdot x - eps\_m \cdot x\right) - x}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+226}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + eps\_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -7e-5)
   (/ (+ 2.0 (/ (- (* eps_m (- (* 2.0 x) (* eps_m x))) x) eps_m)) 2.0)
   (if (<= x 2.7e+14)
     1.0
     (if (<= x 4.4e+226)
       0.0
       (if (<= x 2e+296) (/ (+ 2.0 (* eps_m x)) 2.0) 0.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -7e-5) {
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0;
	} else if (x <= 2.7e+14) {
		tmp = 1.0;
	} else if (x <= 4.4e+226) {
		tmp = 0.0;
	} else if (x <= 2e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-7d-5)) then
        tmp = (2.0d0 + (((eps_m * ((2.0d0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0d0
    else if (x <= 2.7d+14) then
        tmp = 1.0d0
    else if (x <= 4.4d+226) then
        tmp = 0.0d0
    else if (x <= 2d+296) then
        tmp = (2.0d0 + (eps_m * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -7e-5) {
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0;
	} else if (x <= 2.7e+14) {
		tmp = 1.0;
	} else if (x <= 4.4e+226) {
		tmp = 0.0;
	} else if (x <= 2e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -7e-5:
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0
	elif x <= 2.7e+14:
		tmp = 1.0
	elif x <= 4.4e+226:
		tmp = 0.0
	elif x <= 2e+296:
		tmp = (2.0 + (eps_m * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -7e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(eps_m * Float64(Float64(2.0 * x) - Float64(eps_m * x))) - x) / eps_m)) / 2.0);
	elseif (x <= 2.7e+14)
		tmp = 1.0;
	elseif (x <= 4.4e+226)
		tmp = 0.0;
	elseif (x <= 2e+296)
		tmp = Float64(Float64(2.0 + Float64(eps_m * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -7e-5)
		tmp = (2.0 + (((eps_m * ((2.0 * x) - (eps_m * x))) - x) / eps_m)) / 2.0;
	elseif (x <= 2.7e+14)
		tmp = 1.0;
	elseif (x <= 4.4e+226)
		tmp = 0.0;
	elseif (x <= 2e+296)
		tmp = (2.0 + (eps_m * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -7e-5], N[(N[(2.0 + N[(N[(N[(eps$95$m * N[(N[(2.0 * x), $MachinePrecision] - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.7e+14], 1.0, If[LessEqual[x, 4.4e+226], 0.0, If[LessEqual[x, 2e+296], N[(N[(2.0 + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.9999999999999994e-5

   1. Initial program 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 66.0%

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

   6. Taylor expanded in x around 0 24.5%

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

      1. *-commutative 24.5%

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

      2. associate-*r* 24.5%

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

      3. associate-*r* 24.5%

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

      4. associate-*l* 24.5%

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

      5. distribute-lft-in 24.5%

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

      6. metadata-eval 24.5%

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

      7. associate-*r/ 24.5%

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

      8. metadata-eval 24.5%

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

      9. *-commutative 24.5%

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

   8. Simplified 24.5%

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

      1. *-commutative 24.5%

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

      2. distribute-lft-in 2.9%

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

      3. add-sqr-sqrt 2.9%

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

      4. sqrt-unprod 31.7%

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

      5. sqr-neg 31.7%

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

      6. sqrt-unprod 1.5%

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

      7. add-sqr-sqrt 1.6%

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

      8. neg-sub0 1.6%

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

      9. associate--r- 1.6%

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

      10. metadata-eval 1.6%

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

      11. add-sqr-sqrt 0.2%

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

      12. sqrt-unprod 22.6%

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

      13. sqr-neg 22.6%

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

      14. sqrt-unprod 15.0%

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

      15. add-sqr-sqrt 15.1%

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

      16. neg-sub0 15.1%

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

      17. associate--r- 15.1%

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

      18. metadata-eval 15.1%

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

      19. add-sqr-sqrt 0.1%

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

   10. Applied egg-rr 1.6%

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

       1. distribute-lft-out 15.2%

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

       2. *-commutative 15.2%

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

       3. associate-*l* 15.2%

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

       4. +-commutative 15.2%

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

       5. +-commutative 15.2%

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

       6. distribute-rgt-in 15.2%

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

       7. associate-*l/ 15.2%

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

       8. *-lft-identity 15.2%

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

       9. neg-mul-1 15.2%

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

       10. unsub-neg 15.2%

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

   12. Simplified 15.2%

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

   13. Taylor expanded in eps around 0 27.5%

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

   if -6.9999999999999994e-5 < x < 2.7e14

   1. Initial program 53.9%

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

   3. Simplified 53.9%

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

   5. Taylor expanded in x around 0 73.5%

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

   if 2.7e14 < x < 4.39999999999999988e226 or 1.99999999999999996e296 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 56.4%

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

      1. mul-1-neg 56.4%

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

      2. mul-1-neg 56.4%

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

      3. rec-exp 56.4%

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

      4. sub-neg 56.4%

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

      5. div-sub 56.4%

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

      6. rec-exp 56.4%

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

      7. mul-1-neg 56.4%

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

      8. +-inverses 56.4%

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

   7. Simplified 56.4%

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

   if 4.39999999999999988e226 < x < 1.99999999999999996e296

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 25.4%

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

   6. Taylor expanded in eps around inf 25.5%

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

      1. neg-mul-1 25.5%

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

      2. distribute-rgt-neg-in 25.5%

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

   8. Simplified 25.5%

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

   9. Taylor expanded in eps around inf 25.2%

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

       1. *-commutative 25.2%

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

   11. Simplified 25.2%

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

   12. Taylor expanded in x around 0 23.5%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \frac{\varepsilon \cdot \left(2 \cdot x - \varepsilon \cdot x\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+226}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+225}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + eps\_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.7e+14)
   1.0
   (if (<= x 4.5e+225)
     0.0
     (if (<= x 1.9e+296) (/ (+ 2.0 (* eps_m x)) 2.0) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.7e+14) {
		tmp = 1.0;
	} else if (x <= 4.5e+225) {
		tmp = 0.0;
	} else if (x <= 1.9e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.7d+14) then
        tmp = 1.0d0
    else if (x <= 4.5d+225) then
        tmp = 0.0d0
    else if (x <= 1.9d+296) then
        tmp = (2.0d0 + (eps_m * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.7e+14) {
		tmp = 1.0;
	} else if (x <= 4.5e+225) {
		tmp = 0.0;
	} else if (x <= 1.9e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.7e+14:
		tmp = 1.0
	elif x <= 4.5e+225:
		tmp = 0.0
	elif x <= 1.9e+296:
		tmp = (2.0 + (eps_m * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.7e+14)
		tmp = 1.0;
	elseif (x <= 4.5e+225)
		tmp = 0.0;
	elseif (x <= 1.9e+296)
		tmp = Float64(Float64(2.0 + Float64(eps_m * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.7e+14)
		tmp = 1.0;
	elseif (x <= 4.5e+225)
		tmp = 0.0;
	elseif (x <= 1.9e+296)
		tmp = (2.0 + (eps_m * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.7e+14], 1.0, If[LessEqual[x, 4.5e+225], 0.0, If[LessEqual[x, 1.9e+296], N[(N[(2.0 + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.7e14

   1. Initial program 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in x around 0 59.2%

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

   if 2.7e14 < x < 4.49999999999999976e225 or 1.89999999999999987e296 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 56.4%

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

      1. mul-1-neg 56.4%

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

      2. mul-1-neg 56.4%

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

      3. rec-exp 56.4%

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

      4. sub-neg 56.4%

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

      5. div-sub 56.4%

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

      6. rec-exp 56.4%

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

      7. mul-1-neg 56.4%

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

      8. +-inverses 56.4%

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

   7. Simplified 56.4%

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

   if 4.49999999999999976e225 < x < 1.89999999999999987e296

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 25.4%

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

   6. Taylor expanded in eps around inf 25.5%

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

      1. neg-mul-1 25.5%

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

      2. distribute-rgt-neg-in 25.5%

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

   8. Simplified 25.5%

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

   9. Taylor expanded in eps around inf 25.2%

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

       1. *-commutative 25.2%

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

   11. Simplified 25.2%

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

   12. Taylor expanded in x around 0 23.5%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+225}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 165:\\ \;\;\;\;\frac{2 - eps\_m \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{+228}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + eps\_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 165.0)
   (/ (- 2.0 (* eps_m x)) 2.0)
   (if (<= x 1.18e+228)
     0.0
     (if (<= x 2.95e+296) (/ (+ 2.0 (* eps_m x)) 2.0) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 165.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 1.18e+228) {
		tmp = 0.0;
	} else if (x <= 2.95e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 165.0d0) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else if (x <= 1.18d+228) then
        tmp = 0.0d0
    else if (x <= 2.95d+296) then
        tmp = (2.0d0 + (eps_m * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 165.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 1.18e+228) {
		tmp = 0.0;
	} else if (x <= 2.95e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 165.0:
		tmp = (2.0 - (eps_m * x)) / 2.0
	elif x <= 1.18e+228:
		tmp = 0.0
	elif x <= 2.95e+296:
		tmp = (2.0 + (eps_m * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 165.0)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	elseif (x <= 1.18e+228)
		tmp = 0.0;
	elseif (x <= 2.95e+296)
		tmp = Float64(Float64(2.0 + Float64(eps_m * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 165.0)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	elseif (x <= 1.18e+228)
		tmp = 0.0;
	elseif (x <= 2.95e+296)
		tmp = (2.0 + (eps_m * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 165.0], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.18e+228], 0.0, If[LessEqual[x, 2.95e+296], N[(N[(2.0 + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 165

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in x around 0 44.1%

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

   6. Taylor expanded in x around 0 44.9%

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

      1. *-commutative 44.9%

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

      2. associate-*r* 44.9%

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

      3. associate-*r* 44.9%

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

      4. associate-*l* 44.9%

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

      5. distribute-lft-in 44.9%

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

      6. metadata-eval 44.9%

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

      7. associate-*r/ 44.9%

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

      8. metadata-eval 44.9%

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

      9. *-commutative 44.9%

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

   8. Simplified 44.9%

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

      1. *-commutative 44.9%

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

      2. distribute-lft-in 40.4%

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

      3. add-sqr-sqrt 30.6%

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

      4. sqrt-unprod 49.1%

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

      5. sqr-neg 49.1%

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

      6. sqrt-unprod 10.1%

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

      7. add-sqr-sqrt 40.2%

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

      8. neg-sub0 40.2%

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

      9. associate--r- 40.2%

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

      10. metadata-eval 40.2%

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

      11. add-sqr-sqrt 30.1%

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

      12. sqrt-unprod 48.4%

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

      13. sqr-neg 48.4%

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

      14. sqrt-unprod 12.9%

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

      15. add-sqr-sqrt 43.1%

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

      16. neg-sub0 43.1%

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

      17. associate--r- 43.1%

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

      18. metadata-eval 43.1%

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

      19. add-sqr-sqrt 17.6%

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

   10. Applied egg-rr 40.2%

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

       1. distribute-lft-out 43.0%

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

       2. *-commutative 43.0%

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

       3. associate-*l* 43.0%

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

       4. +-commutative 43.0%

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

       5. +-commutative 43.0%

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

       6. distribute-rgt-in 43.0%

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

       7. associate-*l/ 43.0%

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

       8. *-lft-identity 43.0%

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

       9. neg-mul-1 43.0%

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

       10. unsub-neg 43.0%

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

   12. Simplified 43.0%

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

   13. Taylor expanded in eps around inf 62.2%

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

       1. mul-1-neg 62.2%

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

       2. distribute-rgt-neg-in 62.2%

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

   15. Simplified 62.2%

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

   if 165 < x < 1.18e228 or 2.95000000000000008e296 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 53.9%

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

      1. mul-1-neg 53.9%

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

      2. mul-1-neg 53.9%

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

      3. rec-exp 53.9%

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

      4. sub-neg 53.9%

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

      5. div-sub 53.9%

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

      6. rec-exp 53.9%

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

      7. mul-1-neg 53.9%

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

      8. +-inverses 53.9%

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

   7. Simplified 53.9%

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

   if 1.18e228 < x < 2.95000000000000008e296

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 25.4%

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

   6. Taylor expanded in eps around inf 25.5%

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

      1. neg-mul-1 25.5%

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

      2. distribute-rgt-neg-in 25.5%

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

   8. Simplified 25.5%

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

   9. Taylor expanded in eps around inf 25.2%

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

       1. *-commutative 25.2%

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

   11. Simplified 25.2%

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

   12. Taylor expanded in x around 0 23.5%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 165:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{+228}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+225}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + eps\_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.5)
   (/ (+ 2.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666)))))) 2.0)
   (if (<= x 4.4e+225)
     0.0
     (if (<= x 3e+296) (/ (+ 2.0 (* eps_m x)) 2.0) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.5) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 4.4e+225) {
		tmp = 0.0;
	} else if (x <= 3e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.5d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))) / 2.0d0
    else if (x <= 4.4d+225) then
        tmp = 0.0d0
    else if (x <= 3d+296) then
        tmp = (2.0d0 + (eps_m * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.5) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 4.4e+225) {
		tmp = 0.0;
	} else if (x <= 3e+296) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.5:
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0
	elif x <= 4.4e+225:
		tmp = 0.0
	elif x <= 3e+296:
		tmp = (2.0 + (eps_m * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.5)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666)))))) / 2.0);
	elseif (x <= 4.4e+225)
		tmp = 0.0;
	elseif (x <= 3e+296)
		tmp = Float64(Float64(2.0 + Float64(eps_m * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	elseif (x <= 4.4e+225)
		tmp = 0.0;
	elseif (x <= 3e+296)
		tmp = (2.0 + (eps_m * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.5], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.4e+225], 0.0, If[LessEqual[x, 3e+296], N[(N[(2.0 + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.5

   1. Initial program 62.4%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in x around 0 44.4%

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

   6. Taylor expanded in eps around 0 41.9%

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

      1. neg-mul-1 41.9%

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

   8. Simplified 41.9%

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

   9. Taylor expanded in x around 0 51.1%

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

   10. Taylor expanded in eps around inf 69.2%

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

   if 2.5 < x < 4.40000000000000028e225 or 3.00000000000000013e296 < x

   1. Initial program 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in eps around 0 53.1%

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

      1. mul-1-neg 53.1%

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

      2. mul-1-neg 53.1%

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

      3. rec-exp 53.1%

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

      4. sub-neg 53.1%

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

      5. div-sub 53.1%

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

      6. rec-exp 53.1%

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

      7. mul-1-neg 53.1%

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

      8. +-inverses 53.1%

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

   7. Simplified 53.1%

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

   if 4.40000000000000028e225 < x < 3.00000000000000013e296

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 25.4%

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

   6. Taylor expanded in eps around inf 25.5%

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

      1. neg-mul-1 25.5%

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

      2. distribute-rgt-neg-in 25.5%

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

   8. Simplified 25.5%

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

   9. Taylor expanded in eps around inf 25.2%

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

       1. *-commutative 25.2%

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

   11. Simplified 25.2%

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

   12. Taylor expanded in x around 0 23.5%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+225}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+296}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.7% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 2.7e+14) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.7e+14) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.7d+14) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.7e+14) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.7e+14:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.7e+14)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.7e+14)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.7e+14], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.7e14

   1. Initial program 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in x around 0 59.2%

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

   if 2.7e14 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 50.8%

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

      1. mul-1-neg 50.8%

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

      2. mul-1-neg 50.8%

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

      3. rec-exp 50.8%

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

      4. sub-neg 50.8%

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

      5. div-sub 50.8%

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

      6. rec-exp 50.8%

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

      7. mul-1-neg 50.8%

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

      8. +-inverses 50.8%

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

   7. Simplified 50.8%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 15.9% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

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

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.0d0
end function

Java

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

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 73.5%

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

3. Simplified 62.1%

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

5. Taylor expanded in eps around 0 16.2%

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

   1. mul-1-neg 16.2%

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

   2. mul-1-neg 16.2%

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

   3. rec-exp 16.3%

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

   4. sub-neg 16.3%

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

   5. div-sub 16.3%

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

   6. rec-exp 16.2%

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

   7. mul-1-neg 16.2%

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

   8. +-inverses 16.4%

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

7. Simplified 16.4%

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

8. Final simplification 16.4%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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))