NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.7% → 99.8%

Time: 18.8s

Alternatives: 18

Speedup: 1.9×

• 38.5% 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: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps\_m \leq 4 \cdot 10^{-42}:\\ \;\;\;\;\frac{t\_0 \cdot \left(x + 2\right) + x \cdot t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 4e-42)
     (/ (+ (* t_0 (+ x 2.0)) (* x t_0)) 2.0)
     (/ (+ (exp (* x (+ 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 t_0 = exp(-x);
	double tmp;
	if (eps_m <= 4e-42) {
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (exp((x * (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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 4d-42) then
        tmp = ((t_0 * (x + 2.0d0)) + (x * t_0)) / 2.0d0
    else
        tmp = (exp((x * (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 t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 4e-42) {
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (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):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 4e-42:
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0
	else:
		tmp = (math.exp((x * (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)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 4e-42)
		tmp = Float64(Float64(Float64(t_0 * Float64(x + 2.0)) + Float64(x * t_0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(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)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 4e-42)
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	else
		tmp = (exp((x * (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_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 4e-42], N[(N[(N[(t$95$0 * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.00000000000000015e-42

   1. Initial program 62.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in eps around 0 67.5%

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

      1. associate--r+ 67.4%

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

      2. associate-*r* 67.4%

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

      3. mul-1-neg 67.4%

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

      4. cancel-sign-sub 67.4%

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

      5. distribute-rgt1-in 67.4%

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

      6. distribute-rgt-out-- 69.1%

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

      7. mul-1-neg 69.1%

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

      8. mul-1-neg 69.1%

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

   7. Simplified 69.1%

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

   8. Taylor expanded in x around 0 69.1%

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

      1. +-commutative 69.1%

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

   10. Simplified 69.1%

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

   if 4.00000000000000015e-42 < eps

   1. Initial program 96.2%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in eps around inf 100.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4 \cdot 10^{-42}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(x + 2\right) + x \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps\_m \leq 10^{-38}:\\ \;\;\;\;\frac{t\_0 \cdot \left(x + 2\right) + x \cdot t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 1e-38)
     (/ (+ (* t_0 (+ x 2.0)) (* x t_0)) 2.0)
     (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* eps_m (- x)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 1e-38) {
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 1d-38) then
        tmp = ((t_0 * (x + 2.0d0)) + (x * t_0)) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-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 t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 1e-38) {
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + Math.exp((eps_m * -x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 1e-38:
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + math.exp((eps_m * -x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 1e-38)
		tmp = Float64(Float64(Float64(t_0 * Float64(x + 2.0)) + Float64(x * t_0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 1e-38)
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	else
		tmp = (exp((x * (eps_m + -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_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 1e-38], N[(N[(N[(t$95$0 * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 9.9999999999999996e-39

   1. Initial program 62.8%

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

   3. Simplified 62.8%

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

   5. Taylor expanded in eps around 0 67.6%

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

      1. associate--r+ 67.6%

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

      2. associate-*r* 67.6%

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

      3. mul-1-neg 67.6%

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

      4. cancel-sign-sub 67.6%

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

      5. distribute-rgt1-in 67.6%

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

      6. distribute-rgt-out-- 69.3%

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

      7. mul-1-neg 69.3%

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

      8. mul-1-neg 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in x around 0 69.3%

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

      1. +-commutative 69.3%

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

   10. Simplified 69.3%

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

   if 9.9999999999999996e-39 < eps

   1. Initial program 96.2%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 78.4%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.7e-38)
   (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
   (/ (+ (exp (* x (+ eps_m -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 <= 1.7e-38) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -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 <= 1.7d-38) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-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 <= 1.7e-38) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -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 <= 1.7e-38:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -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 <= 1.7e-38)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(eps_m * Float64(-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 <= 1.7e-38)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	else
		tmp = (exp((x * (eps_m + -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, 1.7e-38], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.7000000000000001e-38

   1. Initial program 62.8%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in eps around 0 32.0%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in eps around 0 69.3%

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

   if 1.7000000000000001e-38 < eps

   1. Initial program 96.2%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(\frac{-1}{eps\_m} - -1\right)}{2}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 1800000000:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+91}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)))
   (if (<= x -0.95)
     (/
      (+
       (+ 1.0 (/ 1.0 eps_m))
       (* (exp (* x (- -1.0 eps_m))) (- (/ -1.0 eps_m) -1.0)))
      2.0)
     (if (<= x -1.2e-18)
       t_0
       (if (<= x -1e-285)
         (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
         (if (<= x 1800000000.0)
           (/ (+ 1.0 (exp (* eps_m x))) 2.0)
           (if (<= x 9e+91)
             t_0
             (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	double tmp;
	if (x <= -0.95) {
		tmp = ((1.0 + (1.0 / eps_m)) + (exp((x * (-1.0 - eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0;
	} else if (x <= -1.2e-18) {
		tmp = t_0;
	} else if (x <= -1e-285) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if (x <= 1800000000.0) {
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	} else if (x <= 9e+91) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    if (x <= (-0.95d0)) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (exp((x * ((-1.0d0) - eps_m))) * (((-1.0d0) / eps_m) - (-1.0d0)))) / 2.0d0
    else if (x <= (-1.2d-18)) then
        tmp = t_0
    else if (x <= (-1d-285)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if (x <= 1800000000.0d0) then
        tmp = (1.0d0 + exp((eps_m * x))) / 2.0d0
    else if (x <= 9d+91) then
        tmp = t_0
    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 t_0 = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	double tmp;
	if (x <= -0.95) {
		tmp = ((1.0 + (1.0 / eps_m)) + (Math.exp((x * (-1.0 - eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0;
	} else if (x <= -1.2e-18) {
		tmp = t_0;
	} else if (x <= -1e-285) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if (x <= 1800000000.0) {
		tmp = (1.0 + Math.exp((eps_m * x))) / 2.0;
	} else if (x <= 9e+91) {
		tmp = t_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):
	t_0 = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	tmp = 0
	if x <= -0.95:
		tmp = ((1.0 + (1.0 / eps_m)) + (math.exp((x * (-1.0 - eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0
	elif x <= -1.2e-18:
		tmp = t_0
	elif x <= -1e-285:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif x <= 1800000000.0:
		tmp = (1.0 + math.exp((eps_m * x))) / 2.0
	elif x <= 9e+91:
		tmp = t_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)
	t_0 = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0)
	tmp = 0.0
	if (x <= -0.95)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(Float64(-1.0 / eps_m) - -1.0))) / 2.0);
	elseif (x <= -1.2e-18)
		tmp = t_0;
	elseif (x <= -1e-285)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif (x <= 1800000000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * x))) / 2.0);
	elseif (x <= 9e+91)
		tmp = t_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)
	t_0 = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	tmp = 0.0;
	if (x <= -0.95)
		tmp = ((1.0 + (1.0 / eps_m)) + (exp((x * (-1.0 - eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0;
	elseif (x <= -1.2e-18)
		tmp = t_0;
	elseif (x <= -1e-285)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif (x <= 1800000000.0)
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	elseif (x <= 9e+91)
		tmp = t_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_] := Block[{t$95$0 = N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.95], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(-1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.2e-18], t$95$0, If[LessEqual[x, -1e-285], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1800000000.0], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9e+91], t$95$0, 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 5 regimes

2. if x < -0.94999999999999996

   1. Initial program 93.3%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in x around 0 50.3%

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

      1. metadata-eval 50.3%

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

      2. distribute-neg-frac 50.3%

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

      3. metadata-eval 50.3%

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

      4. associate-*l/ 50.3%

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

      5. *-commutative 50.3%

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

      6. distribute-lft-neg-in 50.3%

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

      7. cancel-sign-sub-inv 50.3%

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

      8. *-commutative 50.3%

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

      9. associate-*l/ 50.3%

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

      10. metadata-eval 50.3%

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

   7. Simplified 50.3%

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

   if -0.94999999999999996 < x < -1.19999999999999997e-18 or 1.8e9 < x < 9e91

   1. Initial program 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in eps around 0 60.1%

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

   7. Simplified 81.8%

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

   8. Taylor expanded in eps around 0 81.9%

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

   if -1.19999999999999997e-18 < x < -1.00000000000000007e-285

   1. Initial program 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in x around 0 35.1%

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

      1. metadata-eval 35.1%

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

      2. distribute-neg-frac 35.1%

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

      3. metadata-eval 35.1%

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

      4. associate-*l/ 35.1%

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

      5. *-commutative 35.1%

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

      6. distribute-lft-neg-in 35.1%

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

      7. cancel-sign-sub-inv 35.1%

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

      8. *-commutative 35.1%

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

      9. associate-*l/ 35.1%

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

      10. metadata-eval 35.1%

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

   7. Simplified 35.1%

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

   8. Taylor expanded in eps around inf 89.4%

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

      1. associate-*r* 89.4%

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

      2. *-lft-identity 89.4%

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

      3. metadata-eval 89.4%

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

      4. cancel-sign-sub-inv 89.4%

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

      5. associate-*r* 89.4%

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

      6. mul-1-neg 89.4%

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

      7. associate-*r* 89.4%

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

      8. neg-mul-1 89.4%

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

      9. cancel-sign-sub-inv 89.4%

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

      10. metadata-eval 89.4%

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

      11. *-lft-identity 89.4%

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

      12. +-commutative 89.4%

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

   10. Simplified 89.4%

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

   11. Taylor expanded in eps around inf 89.4%

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

       1. associate-*r* 89.4%

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

       2. neg-mul-1 89.4%

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

       3. *-commutative 89.4%

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

   13. Simplified 89.4%

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

   if -1.00000000000000007e-285 < x < 1.8e9

   1. Initial program 60.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in x around 0 42.7%

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

   6. Taylor expanded in eps around inf 80.9%

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

      1. neg-mul-1 80.9%

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

      2. distribute-rgt-neg-in 80.9%

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

   8. Simplified 80.9%

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

   9. Taylor expanded in eps around inf 81.6%

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

       1. *-commutative 81.6%

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

   11. Simplified 81.6%

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

   if 9e91 < 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%

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

   5. Taylor expanded in x around 0 33.4%

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

   6. Taylor expanded in eps around inf 33.7%

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

      1. neg-mul-1 33.7%

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

      2. distribute-rgt-neg-in 33.7%

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

   8. Simplified 33.7%

      \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-\left(1 - \varepsilon\right)\right)}}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 1800000000:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+91}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 60000000000000 \lor \neg \left(x \leq 2.9 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-296)
   (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
   (if (or (<= x 60000000000000.0) (not (<= x 2.9e+92)))
     (/ (+ 1.0 (exp (* eps_m x))) 2.0)
     (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-296) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if ((x <= 60000000000000.0) || !(x <= 2.9e+92)) {
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	} else {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 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 <= (-2d-296)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if ((x <= 60000000000000.0d0) .or. (.not. (x <= 2.9d+92))) then
        tmp = (1.0d0 + exp((eps_m * x))) / 2.0d0
    else
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 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 <= -2e-296) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if ((x <= 60000000000000.0) || !(x <= 2.9e+92)) {
		tmp = (1.0 + Math.exp((eps_m * x))) / 2.0;
	} else {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-296:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif (x <= 60000000000000.0) or not (x <= 2.9e+92):
		tmp = (1.0 + math.exp((eps_m * x))) / 2.0
	else:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-296)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif ((x <= 60000000000000.0) || !(x <= 2.9e+92))
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2e-296)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif ((x <= 60000000000000.0) || ~((x <= 2.9e+92)))
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	else
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-296], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 60000000000000.0], N[Not[LessEqual[x, 2.9e+92]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e-296

   1. Initial program 62.3%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in x around 0 39.6%

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

      1. metadata-eval 39.6%

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

      2. distribute-neg-frac 39.6%

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

      3. metadata-eval 39.6%

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

      4. associate-*l/ 39.6%

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

      5. *-commutative 39.6%

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

      6. distribute-lft-neg-in 39.6%

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

      7. cancel-sign-sub-inv 39.6%

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

      8. *-commutative 39.6%

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

      9. associate-*l/ 39.6%

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

      10. metadata-eval 39.6%

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

   7. Simplified 39.6%

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

   8. Taylor expanded in eps around inf 72.0%

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

      1. associate-*r* 72.0%

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

      2. *-lft-identity 72.0%

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

      3. metadata-eval 72.0%

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

      4. cancel-sign-sub-inv 72.0%

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

      5. associate-*r* 72.0%

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

      6. mul-1-neg 72.0%

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

      7. associate-*r* 72.0%

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

      8. neg-mul-1 72.0%

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

      9. cancel-sign-sub-inv 72.0%

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

      10. metadata-eval 72.0%

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

      11. *-lft-identity 72.0%

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

      12. +-commutative 72.0%

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

   10. Simplified 72.0%

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

   11. Taylor expanded in eps around inf 72.2%

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

       1. associate-*r* 72.2%

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

       2. neg-mul-1 72.2%

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

       3. *-commutative 72.2%

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

   13. Simplified 72.2%

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

   if -2e-296 < x < 6e13 or 2.9000000000000001e92 < x

   1. Initial program 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in x around 0 38.4%

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

   6. Taylor expanded in eps around inf 59.0%

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

      1. neg-mul-1 59.0%

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

      2. distribute-rgt-neg-in 59.0%

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

   8. Simplified 59.0%

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

   9. Taylor expanded in eps around inf 59.3%

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

       1. *-commutative 59.3%

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

   11. Simplified 59.3%

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

   if 6e13 < x < 2.9000000000000001e92

   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 76.8%

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

   7. Simplified 76.8%

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

   8. Taylor expanded in eps around 0 76.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 60000000000000 \lor \neg \left(x \leq 2.9 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 60000000000000:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+91}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-285)
   (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
   (if (<= x 60000000000000.0)
     (/ (+ 1.0 (exp (* eps_m x))) 2.0)
     (if (<= x 9e+91)
       (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 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 <= -1e-285) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if (x <= 60000000000000.0) {
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	} else if (x <= 9e+91) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 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 <= (-1d-285)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if (x <= 60000000000000.0d0) then
        tmp = (1.0d0 + exp((eps_m * x))) / 2.0d0
    else if (x <= 9d+91) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 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 <= -1e-285) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if (x <= 60000000000000.0) {
		tmp = (1.0 + Math.exp((eps_m * x))) / 2.0;
	} else if (x <= 9e+91) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 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 <= -1e-285:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif x <= 60000000000000.0:
		tmp = (1.0 + math.exp((eps_m * x))) / 2.0
	elif x <= 9e+91:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 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 <= -1e-285)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif (x <= 60000000000000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * x))) / 2.0);
	elseif (x <= 9e+91)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 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 <= -1e-285)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif (x <= 60000000000000.0)
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	elseif (x <= 9e+91)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 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, -1e-285], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 60000000000000.0], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9e+91], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $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 < -1.00000000000000007e-285

   1. Initial program 62.3%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in x around 0 39.6%

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

      1. metadata-eval 39.6%

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

      2. distribute-neg-frac 39.6%

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

      3. metadata-eval 39.6%

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

      4. associate-*l/ 39.6%

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

      5. *-commutative 39.6%

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

      6. distribute-lft-neg-in 39.6%

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

      7. cancel-sign-sub-inv 39.6%

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

      8. *-commutative 39.6%

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

      9. associate-*l/ 39.6%

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

      10. metadata-eval 39.6%

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

   7. Simplified 39.6%

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

   8. Taylor expanded in eps around inf 72.0%

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

      1. associate-*r* 72.0%

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

      2. *-lft-identity 72.0%

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

      3. metadata-eval 72.0%

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

      4. cancel-sign-sub-inv 72.0%

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

      5. associate-*r* 72.0%

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

      6. mul-1-neg 72.0%

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

      7. associate-*r* 72.0%

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

      8. neg-mul-1 72.0%

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

      9. cancel-sign-sub-inv 72.0%

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

      10. metadata-eval 72.0%

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

      11. *-lft-identity 72.0%

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

      12. +-commutative 72.0%

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

   10. Simplified 72.0%

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

   11. Taylor expanded in eps around inf 72.2%

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

       1. associate-*r* 72.2%

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

       2. neg-mul-1 72.2%

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

       3. *-commutative 72.2%

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

   13. Simplified 72.2%

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

   if -1.00000000000000007e-285 < x < 6e13

   1. Initial program 60.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in x around 0 42.7%

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

   6. Taylor expanded in eps around inf 80.9%

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

      1. neg-mul-1 80.9%

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

      2. distribute-rgt-neg-in 80.9%

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

   8. Simplified 80.9%

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

   9. Taylor expanded in eps around inf 81.6%

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

       1. *-commutative 81.6%

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

   11. Simplified 81.6%

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

   if 6e13 < x < 9e91

   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 76.8%

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

   7. Simplified 76.8%

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

   8. Taylor expanded in eps around 0 76.8%

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

   if 9e91 < 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%

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

   5. Taylor expanded in x around 0 33.4%

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

   6. Taylor expanded in eps around inf 33.7%

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

      1. neg-mul-1 33.7%

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

      2. distribute-rgt-neg-in 33.7%

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

   8. Simplified 33.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 60000000000000:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+91}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-299}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;x \leq 210000000000 \lor \neg \left(x \leq 9 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-299)
   (/ (+ (exp (- x)) 1.0) 2.0)
   (if (or (<= x 210000000000.0) (not (<= x 9e+91)))
     (/ (+ 1.0 (exp (* eps_m x))) 2.0)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-299) {
		tmp = (exp(-x) + 1.0) / 2.0;
	} else if ((x <= 210000000000.0) || !(x <= 9e+91)) {
		tmp = (1.0 + exp((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 <= (-5d-299)) then
        tmp = (exp(-x) + 1.0d0) / 2.0d0
    else if ((x <= 210000000000.0d0) .or. (.not. (x <= 9d+91))) then
        tmp = (1.0d0 + exp((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 <= -5e-299) {
		tmp = (Math.exp(-x) + 1.0) / 2.0;
	} else if ((x <= 210000000000.0) || !(x <= 9e+91)) {
		tmp = (1.0 + Math.exp((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 <= -5e-299:
		tmp = (math.exp(-x) + 1.0) / 2.0
	elif (x <= 210000000000.0) or not (x <= 9e+91):
		tmp = (1.0 + math.exp((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 <= -5e-299)
		tmp = Float64(Float64(exp(Float64(-x)) + 1.0) / 2.0);
	elseif ((x <= 210000000000.0) || !(x <= 9e+91))
		tmp = Float64(Float64(1.0 + exp(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 <= -5e-299)
		tmp = (exp(-x) + 1.0) / 2.0;
	elseif ((x <= 210000000000.0) || ~((x <= 9e+91)))
		tmp = (1.0 + exp((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, -5e-299], N[(N[(N[Exp[(-x)], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 210000000000.0], N[Not[LessEqual[x, 9e+91]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -4.99999999999999956e-299

   1. Initial program 62.3%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in eps around inf 95.5%

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

   6. Taylor expanded in eps around inf 95.6%

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

      1. *-commutative 95.6%

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

   8. Simplified 95.6%

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

   9. Taylor expanded in eps around 0 83.3%

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

       1. mul-1-neg 83.3%

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

   11. Simplified 83.3%

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

   if -4.99999999999999956e-299 < x < 2.1e11 or 9e91 < x

   1. Initial program 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in x around 0 38.4%

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

   6. Taylor expanded in eps around inf 59.0%

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

      1. neg-mul-1 59.0%

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

      2. distribute-rgt-neg-in 59.0%

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

   8. Simplified 59.0%

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

   9. Taylor expanded in eps around inf 59.3%

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

       1. *-commutative 59.3%

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

   11. Simplified 59.3%

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

   if 2.1e11 < x < 9e91

   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 76.8%

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

      1. mul-1-neg 76.8%

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

      2. mul-1-neg 76.8%

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

      3. rec-exp 76.8%

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

      4. sub-neg 76.8%

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

      5. div-sub 76.8%

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

      6. mul-1-neg 76.8%

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

      7. rec-exp 76.8%

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

      8. +-inverses 76.8%

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

   7. Simplified 76.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-299}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;x \leq 210000000000 \lor \neg \left(x \leq 9 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 46000000000000 \lor \neg \left(x \leq 9.5 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-285)
   (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
   (if (or (<= x 46000000000000.0) (not (<= x 9.5e+91)))
     (/ (+ 1.0 (exp (* eps_m x))) 2.0)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-285) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if ((x <= 46000000000000.0) || !(x <= 9.5e+91)) {
		tmp = (1.0 + exp((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 <= (-1d-285)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if ((x <= 46000000000000.0d0) .or. (.not. (x <= 9.5d+91))) then
        tmp = (1.0d0 + exp((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 <= -1e-285) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if ((x <= 46000000000000.0) || !(x <= 9.5e+91)) {
		tmp = (1.0 + Math.exp((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 <= -1e-285:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif (x <= 46000000000000.0) or not (x <= 9.5e+91):
		tmp = (1.0 + math.exp((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 <= -1e-285)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif ((x <= 46000000000000.0) || !(x <= 9.5e+91))
		tmp = Float64(Float64(1.0 + exp(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 <= -1e-285)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif ((x <= 46000000000000.0) || ~((x <= 9.5e+91)))
		tmp = (1.0 + exp((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, -1e-285], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 46000000000000.0], N[Not[LessEqual[x, 9.5e+91]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000007e-285

   1. Initial program 62.3%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in x around 0 39.6%

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

      1. metadata-eval 39.6%

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

      2. distribute-neg-frac 39.6%

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

      3. metadata-eval 39.6%

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

      4. associate-*l/ 39.6%

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

      5. *-commutative 39.6%

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

      6. distribute-lft-neg-in 39.6%

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

      7. cancel-sign-sub-inv 39.6%

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

      8. *-commutative 39.6%

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

      9. associate-*l/ 39.6%

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

      10. metadata-eval 39.6%

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

   7. Simplified 39.6%

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

   8. Taylor expanded in eps around inf 72.0%

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

      1. associate-*r* 72.0%

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

      2. *-lft-identity 72.0%

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

      3. metadata-eval 72.0%

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

      4. cancel-sign-sub-inv 72.0%

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

      5. associate-*r* 72.0%

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

      6. mul-1-neg 72.0%

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

      7. associate-*r* 72.0%

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

      8. neg-mul-1 72.0%

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

      9. cancel-sign-sub-inv 72.0%

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

      10. metadata-eval 72.0%

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

      11. *-lft-identity 72.0%

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

      12. +-commutative 72.0%

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

   10. Simplified 72.0%

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

   11. Taylor expanded in eps around inf 72.2%

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

       1. associate-*r* 72.2%

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

       2. neg-mul-1 72.2%

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

       3. *-commutative 72.2%

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

   13. Simplified 72.2%

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

   if -1.00000000000000007e-285 < x < 4.6e13 or 9.5000000000000001e91 < x

   1. Initial program 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in x around 0 38.4%

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

   6. Taylor expanded in eps around inf 59.0%

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

      1. neg-mul-1 59.0%

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

      2. distribute-rgt-neg-in 59.0%

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

   8. Simplified 59.0%

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

   9. Taylor expanded in eps around inf 59.3%

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

       1. *-commutative 59.3%

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

   11. Simplified 59.3%

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

   if 4.6e13 < x < 9.5000000000000001e91

   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 76.8%

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

      1. mul-1-neg 76.8%

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

      2. mul-1-neg 76.8%

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

      3. rec-exp 76.8%

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

      4. sub-neg 76.8%

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

      5. div-sub 76.8%

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

      6. mul-1-neg 76.8%

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

      7. rec-exp 76.8%

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

      8. +-inverses 76.8%

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

   7. Simplified 76.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 46000000000000 \lor \neg \left(x \leq 9.5 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1950:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(2 + \left(x + \left(eps\_m \cdot x - x\right)\right)\right) - x}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+92}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.6e-85)
   (/ (+ 2.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666)))))) 2.0)
   (if (<= x 1950.0)
     (/ (/ (- (* eps_m (+ 2.0 (+ x (- (* eps_m x) x)))) x) eps_m) 2.0)
     (if (<= x 1.05e+92) 0.0 (/ (+ 1.0 (exp x)) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.6e-85) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 1950.0) {
		tmp = (((eps_m * (2.0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0;
	} else if (x <= 1.05e+92) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + exp(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 (x <= 1.6d-85) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))) / 2.0d0
    else if (x <= 1950.0d0) then
        tmp = (((eps_m * (2.0d0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0d0
    else if (x <= 1.05d+92) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 + exp(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 (x <= 1.6e-85) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 1950.0) {
		tmp = (((eps_m * (2.0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0;
	} else if (x <= 1.05e+92) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.6e-85:
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0
	elif x <= 1950.0:
		tmp = (((eps_m * (2.0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0
	elif x <= 1.05e+92:
		tmp = 0.0
	else:
		tmp = (1.0 + math.exp(x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.6e-85)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666)))))) / 2.0);
	elseif (x <= 1950.0)
		tmp = Float64(Float64(Float64(Float64(eps_m * Float64(2.0 + Float64(x + Float64(Float64(eps_m * x) - x)))) - x) / eps_m) / 2.0);
	elseif (x <= 1.05e+92)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.6e-85)
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	elseif (x <= 1950.0)
		tmp = (((eps_m * (2.0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0;
	elseif (x <= 1.05e+92)
		tmp = 0.0;
	else
		tmp = (1.0 + exp(x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.6e-85], 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, 1950.0], N[(N[(N[(N[(eps$95$m * N[(2.0 + N[(x + N[(N[(eps$95$m * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.05e+92], 0.0, N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.60000000000000014e-85

   1. Initial program 58.6%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in eps around inf 96.7%

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

   6. Taylor expanded in eps around inf 96.7%

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

      1. *-commutative 96.7%

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

   8. Simplified 96.7%

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

   9. Taylor expanded in eps around 0 85.8%

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

       1. mul-1-neg 85.8%

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

   11. Simplified 85.8%

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

   12. Taylor expanded in x around 0 78.0%

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

   if 1.60000000000000014e-85 < x < 1950

   1. Initial program 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in x around 0 40.8%

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

   6. Taylor expanded in x around 0 17.4%

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

      1. mul-1-neg 17.4%

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

      2. associate-*r* 17.4%

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

      3. distribute-rgt-neg-in 17.4%

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

      4. distribute-lft-in 17.4%

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

      5. *-rgt-identity 17.4%

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

      6. associate-*r/ 17.4%

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

      7. *-rgt-identity 17.4%

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

      8. sub-neg 17.4%

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

      9. distribute-neg-in 17.4%

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

      10. metadata-eval 17.4%

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

      11. remove-double-neg 17.4%

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

      12. +-commutative 17.4%

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

   8. Simplified 17.4%

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

   9. Taylor expanded in eps around 0 29.7%

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

   if 1950 < x < 1.04999999999999993e92

   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 74.1%

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

      1. mul-1-neg 74.1%

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

      2. mul-1-neg 74.1%

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

      3. rec-exp 74.1%

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

      4. sub-neg 74.1%

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

      5. div-sub 74.1%

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

      6. mul-1-neg 74.1%

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

      7. rec-exp 74.1%

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

      8. +-inverses 74.1%

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

   7. Simplified 74.1%

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

   if 1.04999999999999993e92 < 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 inf 100.0%

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

   6. Taylor expanded in eps around inf 79.5%

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

      1. *-commutative 79.5%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in eps around 0 3.1%

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

       1. mul-1-neg 3.1%

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

   11. Simplified 3.1%

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

       1. *-un-lft-identity 3.1%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 60.3%

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

       4. sqr-neg 60.3%

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

       5. sqrt-unprod 60.3%

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

       6. add-sqr-sqrt 60.3%

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

   13. Applied egg-rr 60.3%

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

       1. *-lft-identity 60.3%

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

   15. Simplified 60.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1950:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 + \left(x + \left(\varepsilon \cdot x - x\right)\right)\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+92}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-92}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;x \leq 1950:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(2 + \left(x + \left(eps\_m \cdot x - x\right)\right)\right) - x}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+92}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 9e-92)
   (/ (+ (exp (- x)) 1.0) 2.0)
   (if (<= x 1950.0)
     (/ (/ (- (* eps_m (+ 2.0 (+ x (- (* eps_m x) x)))) x) eps_m) 2.0)
     (if (<= x 2.1e+92) 0.0 (/ (+ 1.0 (exp x)) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 9e-92) {
		tmp = (exp(-x) + 1.0) / 2.0;
	} else if (x <= 1950.0) {
		tmp = (((eps_m * (2.0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0;
	} else if (x <= 2.1e+92) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + exp(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 (x <= 9d-92) then
        tmp = (exp(-x) + 1.0d0) / 2.0d0
    else if (x <= 1950.0d0) then
        tmp = (((eps_m * (2.0d0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0d0
    else if (x <= 2.1d+92) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 + exp(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 (x <= 9e-92) {
		tmp = (Math.exp(-x) + 1.0) / 2.0;
	} else if (x <= 1950.0) {
		tmp = (((eps_m * (2.0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0;
	} else if (x <= 2.1e+92) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 9e-92:
		tmp = (math.exp(-x) + 1.0) / 2.0
	elif x <= 1950.0:
		tmp = (((eps_m * (2.0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0
	elif x <= 2.1e+92:
		tmp = 0.0
	else:
		tmp = (1.0 + math.exp(x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 9e-92)
		tmp = Float64(Float64(exp(Float64(-x)) + 1.0) / 2.0);
	elseif (x <= 1950.0)
		tmp = Float64(Float64(Float64(Float64(eps_m * Float64(2.0 + Float64(x + Float64(Float64(eps_m * x) - x)))) - x) / eps_m) / 2.0);
	elseif (x <= 2.1e+92)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 9e-92)
		tmp = (exp(-x) + 1.0) / 2.0;
	elseif (x <= 1950.0)
		tmp = (((eps_m * (2.0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0;
	elseif (x <= 2.1e+92)
		tmp = 0.0;
	else
		tmp = (1.0 + exp(x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 9e-92], N[(N[(N[Exp[(-x)], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1950.0], N[(N[(N[(N[(eps$95$m * N[(2.0 + N[(x + N[(N[(eps$95$m * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.1e+92], 0.0, N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 9.0000000000000001e-92

   1. Initial program 58.6%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in eps around inf 96.7%

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

   6. Taylor expanded in eps around inf 96.7%

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

      1. *-commutative 96.7%

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

   8. Simplified 96.7%

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

   9. Taylor expanded in eps around 0 85.8%

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

       1. mul-1-neg 85.8%

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

   11. Simplified 85.8%

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

   if 9.0000000000000001e-92 < x < 1950

   1. Initial program 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in x around 0 40.8%

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

   6. Taylor expanded in x around 0 17.4%

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

      1. mul-1-neg 17.4%

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

      2. associate-*r* 17.4%

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

      3. distribute-rgt-neg-in 17.4%

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

      4. distribute-lft-in 17.4%

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

      5. *-rgt-identity 17.4%

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

      6. associate-*r/ 17.4%

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

      7. *-rgt-identity 17.4%

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

      8. sub-neg 17.4%

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

      9. distribute-neg-in 17.4%

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

      10. metadata-eval 17.4%

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

      11. remove-double-neg 17.4%

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

      12. +-commutative 17.4%

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

   8. Simplified 17.4%

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

   9. Taylor expanded in eps around 0 29.7%

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

   if 1950 < x < 2.09999999999999986e92

   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 74.1%

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

      1. mul-1-neg 74.1%

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

      2. mul-1-neg 74.1%

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

      3. rec-exp 74.1%

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

      4. sub-neg 74.1%

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

      5. div-sub 74.1%

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

      6. mul-1-neg 74.1%

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

      7. rec-exp 74.1%

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

      8. +-inverses 74.1%

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

   7. Simplified 74.1%

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

   if 2.09999999999999986e92 < 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 inf 100.0%

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

   6. Taylor expanded in eps around inf 79.5%

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

      1. *-commutative 79.5%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in eps around 0 3.1%

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

       1. mul-1-neg 3.1%

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

   11. Simplified 3.1%

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

       1. *-un-lft-identity 3.1%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 60.3%

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

       4. sqr-neg 60.3%

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

       5. sqrt-unprod 60.3%

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

       6. add-sqr-sqrt 60.3%

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

   13. Applied egg-rr 60.3%

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

       1. *-lft-identity 60.3%

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

   15. Simplified 60.3%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-92}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;x \leq 1950:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 + \left(x + \left(\varepsilon \cdot x - x\right)\right)\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+92}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.9% accurate, 8.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1750:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(2 + \left(x + \left(eps\_m \cdot x - x\right)\right)\right) - x}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.92 \cdot 10^{+154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.3e-95)
   (/ (+ 2.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666)))))) 2.0)
   (if (<= x 1750.0)
     (/ (/ (- (* eps_m (+ 2.0 (+ x (- (* eps_m x) x)))) x) eps_m) 2.0)
     (if (<= x 1.92e+154) 0.0 (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.3e-95) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 1750.0) {
		tmp = (((eps_m * (2.0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0;
	} else if (x <= 1.92e+154) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 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 <= 1.3d-95) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))) / 2.0d0
    else if (x <= 1750.0d0) then
        tmp = (((eps_m * (2.0d0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0d0
    else if (x <= 1.92d+154) then
        tmp = 0.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 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 <= 1.3e-95) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 1750.0) {
		tmp = (((eps_m * (2.0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0;
	} else if (x <= 1.92e+154) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.3e-95:
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0
	elif x <= 1750.0:
		tmp = (((eps_m * (2.0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0
	elif x <= 1.92e+154:
		tmp = 0.0
	else:
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.3e-95)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666)))))) / 2.0);
	elseif (x <= 1750.0)
		tmp = Float64(Float64(Float64(Float64(eps_m * Float64(2.0 + Float64(x + Float64(Float64(eps_m * x) - x)))) - x) / eps_m) / 2.0);
	elseif (x <= 1.92e+154)
		tmp = 0.0;
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.3e-95)
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	elseif (x <= 1750.0)
		tmp = (((eps_m * (2.0 + (x + ((eps_m * x) - x)))) - x) / eps_m) / 2.0;
	elseif (x <= 1.92e+154)
		tmp = 0.0;
	else
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.3e-95], 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, 1750.0], N[(N[(N[(N[(eps$95$m * N[(2.0 + N[(x + N[(N[(eps$95$m * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.92e+154], 0.0, N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.3e-95

   1. Initial program 58.6%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in eps around inf 96.7%

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

   6. Taylor expanded in eps around inf 96.7%

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

      1. *-commutative 96.7%

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

   8. Simplified 96.7%

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

   9. Taylor expanded in eps around 0 85.8%

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

       1. mul-1-neg 85.8%

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

   11. Simplified 85.8%

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

   12. Taylor expanded in x around 0 78.0%

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

   if 1.3e-95 < x < 1750

   1. Initial program 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in x around 0 40.8%

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

   6. Taylor expanded in x around 0 17.4%

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

      1. mul-1-neg 17.4%

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

      2. associate-*r* 17.4%

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

      3. distribute-rgt-neg-in 17.4%

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

      4. distribute-lft-in 17.4%

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

      5. *-rgt-identity 17.4%

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

      6. associate-*r/ 17.4%

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

      7. *-rgt-identity 17.4%

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

      8. sub-neg 17.4%

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

      9. distribute-neg-in 17.4%

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

      10. metadata-eval 17.4%

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

      11. remove-double-neg 17.4%

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

      12. +-commutative 17.4%

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

   8. Simplified 17.4%

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

   9. Taylor expanded in eps around 0 29.7%

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

   if 1750 < x < 1.91999999999999994e154

   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 62.8%

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

      1. mul-1-neg 62.8%

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

      2. mul-1-neg 62.8%

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

      3. rec-exp 62.8%

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

      4. sub-neg 62.8%

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

      5. div-sub 62.8%

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

      6. mul-1-neg 62.8%

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

      7. rec-exp 62.8%

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

      8. +-inverses 62.8%

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

   7. Simplified 62.8%

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

   if 1.91999999999999994e154 < 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 inf 100.0%

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

   6. Taylor expanded in eps around inf 87.4%

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

      1. *-commutative 87.4%

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

   8. Simplified 87.4%

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

   9. Taylor expanded in eps around 0 3.1%

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

       1. mul-1-neg 3.1%

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

   11. Simplified 3.1%

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

   12. Taylor expanded in x around 0 64.7%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(0.5 \cdot x - 1\right)}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1750:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 + \left(x + \left(\varepsilon \cdot x - x\right)\right)\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.92 \cdot 10^{+154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0)))
   (if (<= x -1.9e+154)
     t_0
     (if (<= x 2.0)
       (/ (+ 2.0 (* x (- -1.0 eps_m))) 2.0)
       (if (<= x 1.9e+154) 0.0 t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	double tmp;
	if (x <= -1.9e+154) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1.9e+154) {
		tmp = 0.0;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 2.0d0
    if (x <= (-1.9d+154)) then
        tmp = t_0
    else if (x <= 2.0d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps_m))) / 2.0d0
    else if (x <= 1.9d+154) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	double tmp;
	if (x <= -1.9e+154) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1.9e+154) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	tmp = 0
	if x <= -1.9e+154:
		tmp = t_0
	elif x <= 2.0:
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0
	elif x <= 1.9e+154:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0)
	tmp = 0.0
	if (x <= -1.9e+154)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 1.9e+154)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	tmp = 0.0;
	if (x <= -1.9e+154)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	elseif (x <= 1.9e+154)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1.9e+154], t$95$0, If[LessEqual[x, 2.0], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.9e+154], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8999999999999999e154 or 1.8999999999999999e154 < 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 inf 100.0%

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

   6. Taylor expanded in eps around inf 91.7%

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

      1. *-commutative 91.7%

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

   8. Simplified 91.7%

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

   9. Taylor expanded in eps around 0 36.0%

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

       1. mul-1-neg 36.0%

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

   11. Simplified 36.0%

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

   12. Taylor expanded in x around 0 76.6%

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

   if -1.8999999999999999e154 < x < 2

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in x around 0 40.1%

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

      1. metadata-eval 40.1%

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

      2. distribute-neg-frac 40.1%

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

      3. metadata-eval 40.1%

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

      4. associate-*l/ 40.1%

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

      5. *-commutative 40.1%

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

      6. distribute-lft-neg-in 40.1%

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

      7. cancel-sign-sub-inv 40.1%

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

      8. *-commutative 40.1%

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

      9. associate-*l/ 40.1%

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

      10. metadata-eval 40.1%

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

   7. Simplified 40.1%

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

   8. Taylor expanded in eps around inf 79.6%

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

      1. associate-*r* 79.6%

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

      2. *-lft-identity 79.6%

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

      3. metadata-eval 79.6%

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

      4. cancel-sign-sub-inv 79.6%

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

      5. associate-*r* 79.6%

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

      6. mul-1-neg 79.6%

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

      7. associate-*r* 79.6%

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

      8. neg-mul-1 79.6%

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

      9. cancel-sign-sub-inv 79.6%

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

      10. metadata-eval 79.6%

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

      11. *-lft-identity 79.6%

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

      12. +-commutative 79.6%

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

   10. Simplified 79.6%

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

   11. Taylor expanded in x around 0 65.9%

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

   if 2 < x < 1.8999999999999999e154

   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 61.1%

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

      1. mul-1-neg 61.1%

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

      2. mul-1-neg 61.1%

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

      3. rec-exp 61.1%

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

      4. sub-neg 61.1%

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

      5. div-sub 61.1%

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

      6. mul-1-neg 61.1%

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

      7. rec-exp 61.1%

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

      8. +-inverses 61.1%

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

   7. Simplified 61.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.8% accurate, 10.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.55:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.55)
   (/ (+ 2.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666)))))) 2.0)
   (if (<= x 1.9e+154) 0.0 (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.55) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 1.9e+154) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 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 <= 2.55d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))) / 2.0d0
    else if (x <= 1.9d+154) then
        tmp = 0.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 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 <= 2.55) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 1.9e+154) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.55:
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0
	elif x <= 1.9e+154:
		tmp = 0.0
	else:
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.55)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666)))))) / 2.0);
	elseif (x <= 1.9e+154)
		tmp = 0.0;
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.55)
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	elseif (x <= 1.9e+154)
		tmp = 0.0;
	else
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.55], 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, 1.9e+154], 0.0, N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.5499999999999998

   1. Initial program 61.0%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in eps around inf 96.5%

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

   6. Taylor expanded in eps around inf 96.6%

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

      1. *-commutative 96.6%

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

   8. Simplified 96.6%

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

   9. Taylor expanded in eps around 0 79.4%

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

       1. mul-1-neg 79.4%

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

   11. Simplified 79.4%

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

   12. Taylor expanded in x around 0 72.6%

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

   if 2.5499999999999998 < x < 1.8999999999999999e154

   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 61.1%

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

      1. mul-1-neg 61.1%

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

      2. mul-1-neg 61.1%

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

      3. rec-exp 61.1%

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

      4. sub-neg 61.1%

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

      5. div-sub 61.1%

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

      6. mul-1-neg 61.1%

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

      7. rec-exp 61.1%

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

      8. +-inverses 61.1%

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

   7. Simplified 61.1%

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

   if 1.8999999999999999e154 < 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 inf 100.0%

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

   6. Taylor expanded in eps around inf 87.4%

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

      1. *-commutative 87.4%

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

   8. Simplified 87.4%

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

   9. Taylor expanded in eps around 0 3.1%

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

       1. mul-1-neg 3.1%

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

   11. Simplified 3.1%

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

   12. Taylor expanded in x around 0 64.7%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.55:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.0% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1750:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+233}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;eps\_m \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1750.0) 1.0 (if (<= x 1.35e+233) 0.0 (* eps_m (* x 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1750.0) {
		tmp = 1.0;
	} else if (x <= 1.35e+233) {
		tmp = 0.0;
	} else {
		tmp = eps_m * (x * 0.5);
	}
	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 <= 1750.0d0) then
        tmp = 1.0d0
    else if (x <= 1.35d+233) then
        tmp = 0.0d0
    else
        tmp = eps_m * (x * 0.5d0)
    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 <= 1750.0) {
		tmp = 1.0;
	} else if (x <= 1.35e+233) {
		tmp = 0.0;
	} else {
		tmp = eps_m * (x * 0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1750.0:
		tmp = 1.0
	elif x <= 1.35e+233:
		tmp = 0.0
	else:
		tmp = eps_m * (x * 0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1750.0)
		tmp = 1.0;
	elseif (x <= 1.35e+233)
		tmp = 0.0;
	else
		tmp = Float64(eps_m * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1750.0)
		tmp = 1.0;
	elseif (x <= 1.35e+233)
		tmp = 0.0;
	else
		tmp = eps_m * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1750.0], 1.0, If[LessEqual[x, 1.35e+233], 0.0, N[(eps$95$m * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1750

   1. Initial program 61.2%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in x around 0 41.7%

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

   6. Taylor expanded in x around 0 43.7%

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

      1. mul-1-neg 43.7%

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

      2. associate-*r* 43.7%

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

      3. distribute-rgt-neg-in 43.7%

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

      4. distribute-lft-in 43.7%

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

      5. *-rgt-identity 43.7%

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

      6. associate-*r/ 43.7%

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

      7. *-rgt-identity 43.7%

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

      8. sub-neg 43.7%

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

      9. distribute-neg-in 43.7%

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

      10. metadata-eval 43.7%

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

      11. remove-double-neg 43.7%

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

      12. +-commutative 43.7%

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

   8. Simplified 43.7%

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

   9. Taylor expanded in x around 0 57.9%

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

   if 1750 < x < 1.35000000000000004e233

   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 57.3%

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

      1. mul-1-neg 57.3%

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

      2. mul-1-neg 57.3%

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

      3. rec-exp 57.3%

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

      4. sub-neg 57.3%

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

      5. div-sub 57.3%

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

      6. mul-1-neg 57.3%

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

      7. rec-exp 57.3%

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

      8. +-inverses 57.3%

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

   7. Simplified 57.3%

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

   if 1.35000000000000004e233 < 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%

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

   5. Taylor expanded in x around 0 40.9%

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

   6. Taylor expanded in x around 0 31.9%

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

      1. mul-1-neg 31.9%

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

      2. associate-*r* 31.9%

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

      3. distribute-rgt-neg-in 31.9%

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

      4. distribute-lft-in 31.9%

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

      5. *-rgt-identity 31.9%

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

      6. associate-*r/ 31.9%

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

      7. *-rgt-identity 31.9%

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

      8. sub-neg 31.9%

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

      9. distribute-neg-in 31.9%

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

      10. metadata-eval 31.9%

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

      11. remove-double-neg 31.9%

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

      12. +-commutative 31.9%

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

   8. Simplified 31.9%

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

   9. Taylor expanded in eps around inf 32.1%

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

       1. *-commutative 32.1%

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

       2. associate-*r* 32.1%

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

   11. Simplified 32.1%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1750:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+233}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.0% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+233}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;eps\_m \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.0)
   (/ (- 2.0 x) 2.0)
   (if (<= x 1.15e+233) 0.0 (* eps_m (* x 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else if (x <= 1.15e+233) {
		tmp = 0.0;
	} else {
		tmp = eps_m * (x * 0.5);
	}
	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 <= 2.0d0) then
        tmp = (2.0d0 - x) / 2.0d0
    else if (x <= 1.15d+233) then
        tmp = 0.0d0
    else
        tmp = eps_m * (x * 0.5d0)
    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 <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else if (x <= 1.15e+233) {
		tmp = 0.0;
	} else {
		tmp = eps_m * (x * 0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 - x) / 2.0
	elif x <= 1.15e+233:
		tmp = 0.0
	else:
		tmp = eps_m * (x * 0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 - x) / 2.0);
	elseif (x <= 1.15e+233)
		tmp = 0.0;
	else
		tmp = Float64(eps_m * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 - x) / 2.0;
	elseif (x <= 1.15e+233)
		tmp = 0.0;
	else
		tmp = eps_m * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.0], N[(N[(2.0 - x), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.15e+233], 0.0, N[(eps$95$m * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2

   1. Initial program 61.0%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in eps around inf 96.5%

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

   6. Taylor expanded in eps around inf 96.6%

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

      1. *-commutative 96.6%

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

   8. Simplified 96.6%

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

   9. Taylor expanded in x around 0 58.3%

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

       1. *-commutative 58.3%

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

       2. distribute-rgt1-in 58.3%

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

       3. metadata-eval 58.3%

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

       4. mul0-lft 58.3%

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

       5. metadata-eval 58.3%

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

       6. mul-1-neg 58.3%

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

       7. unsub-neg 58.3%

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

   11. Simplified 58.3%

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

   if 2 < x < 1.15e233

   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 56.3%

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

      1. mul-1-neg 56.3%

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

      2. mul-1-neg 56.3%

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

      3. rec-exp 56.3%

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

      4. sub-neg 56.3%

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

      5. div-sub 56.3%

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

      6. mul-1-neg 56.3%

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

      7. rec-exp 56.3%

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

      8. +-inverses 56.3%

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

   7. Simplified 56.3%

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

   if 1.15e233 < 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%

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

   5. Taylor expanded in x around 0 40.9%

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

   6. Taylor expanded in x around 0 31.9%

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

      1. mul-1-neg 31.9%

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

      2. associate-*r* 31.9%

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

      3. distribute-rgt-neg-in 31.9%

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

      4. distribute-lft-in 31.9%

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

      5. *-rgt-identity 31.9%

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

      6. associate-*r/ 31.9%

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

      7. *-rgt-identity 31.9%

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

      8. sub-neg 31.9%

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

      9. distribute-neg-in 31.9%

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

      10. metadata-eval 31.9%

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

      11. remove-double-neg 31.9%

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

      12. +-commutative 31.9%

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

   8. Simplified 31.9%

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

   9. Taylor expanded in eps around inf 32.1%

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

       1. *-commutative 32.1%

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

       2. associate-*r* 32.1%

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

   11. Simplified 32.1%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+233}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.2% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+233}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;eps\_m \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.0)
   (/ (+ 2.0 (* x (- -1.0 eps_m))) 2.0)
   (if (<= x 1.2e+233) 0.0 (* eps_m (* x 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1.2e+233) {
		tmp = 0.0;
	} else {
		tmp = eps_m * (x * 0.5);
	}
	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 <= 2.0d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps_m))) / 2.0d0
    else if (x <= 1.2d+233) then
        tmp = 0.0d0
    else
        tmp = eps_m * (x * 0.5d0)
    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 <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1.2e+233) {
		tmp = 0.0;
	} else {
		tmp = eps_m * (x * 0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0
	elif x <= 1.2e+233:
		tmp = 0.0
	else:
		tmp = eps_m * (x * 0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 1.2e+233)
		tmp = 0.0;
	else
		tmp = Float64(eps_m * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	elseif (x <= 1.2e+233)
		tmp = 0.0;
	else
		tmp = eps_m * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.0], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.2e+233], 0.0, N[(eps$95$m * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2

   1. Initial program 61.0%

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

   3. Simplified 61.0%

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

   5. Taylor expanded in x around 0 41.9%

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

      1. metadata-eval 41.9%

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

      2. distribute-neg-frac 41.9%

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

      3. metadata-eval 41.9%

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

      4. associate-*l/ 41.9%

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

      5. *-commutative 41.9%

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

      6. distribute-lft-neg-in 41.9%

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

      7. cancel-sign-sub-inv 41.9%

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

      8. *-commutative 41.9%

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

      9. associate-*l/ 41.9%

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

      10. metadata-eval 41.9%

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

   7. Simplified 41.9%

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

   8. Taylor expanded in eps around inf 77.0%

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

      1. associate-*r* 77.0%

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

      2. *-lft-identity 77.0%

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

      3. metadata-eval 77.0%

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

      4. cancel-sign-sub-inv 77.0%

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

      5. associate-*r* 77.0%

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

      6. mul-1-neg 77.0%

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

      7. associate-*r* 77.0%

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

      8. neg-mul-1 77.0%

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

      9. cancel-sign-sub-inv 77.0%

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

      10. metadata-eval 77.0%

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

      11. *-lft-identity 77.0%

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

      12. +-commutative 77.0%

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

   10. Simplified 77.0%

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

   11. Taylor expanded in x around 0 63.6%

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

   if 2 < x < 1.20000000000000001e233

   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 56.3%

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

      1. mul-1-neg 56.3%

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

      2. mul-1-neg 56.3%

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

      3. rec-exp 56.3%

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

      4. sub-neg 56.3%

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

      5. div-sub 56.3%

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

      6. mul-1-neg 56.3%

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

      7. rec-exp 56.3%

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

      8. +-inverses 56.3%

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

   7. Simplified 56.3%

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

   if 1.20000000000000001e233 < 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%

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

   5. Taylor expanded in x around 0 40.9%

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

   6. Taylor expanded in x around 0 31.9%

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

      1. mul-1-neg 31.9%

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

      2. associate-*r* 31.9%

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

      3. distribute-rgt-neg-in 31.9%

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

      4. distribute-lft-in 31.9%

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

      5. *-rgt-identity 31.9%

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

      6. associate-*r/ 31.9%

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

      7. *-rgt-identity 31.9%

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

      8. sub-neg 31.9%

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

      9. distribute-neg-in 31.9%

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

      10. metadata-eval 31.9%

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

      11. remove-double-neg 31.9%

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

      12. +-commutative 31.9%

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

   8. Simplified 31.9%

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

   9. Taylor expanded in eps around inf 32.1%

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

       1. *-commutative 32.1%

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

       2. associate-*r* 32.1%

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

   11. Simplified 32.1%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+233}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.3% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1750:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 1750.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1750.0) {
		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 <= 1750.0d0) 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 <= 1750.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1750.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1750.0)
		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 <= 1750.0)
		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, 1750.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1750

   1. Initial program 61.2%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in x around 0 41.7%

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

   6. Taylor expanded in x around 0 43.7%

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

      1. mul-1-neg 43.7%

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

      2. associate-*r* 43.7%

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

      3. distribute-rgt-neg-in 43.7%

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

      4. distribute-lft-in 43.7%

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

      5. *-rgt-identity 43.7%

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

      6. associate-*r/ 43.7%

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

      7. *-rgt-identity 43.7%

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

      8. sub-neg 43.7%

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

      9. distribute-neg-in 43.7%

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

      10. metadata-eval 43.7%

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

      11. remove-double-neg 43.7%

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

      12. +-commutative 43.7%

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

   8. Simplified 43.7%

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

   9. Taylor expanded in x around 0 57.9%

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

   if 1750 < 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 49.5%

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

      1. mul-1-neg 49.5%

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

      2. mul-1-neg 49.5%

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

      3. rec-exp 49.5%

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

      4. sub-neg 49.5%

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

      5. div-sub 49.5%

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

      6. mul-1-neg 49.5%

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

      7. rec-exp 49.5%

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

      8. +-inverses 49.5%

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

   7. Simplified 49.5%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1750:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.6% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in x around 0 37.7%

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

6. Taylor expanded in x around 0 34.9%

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

   1. mul-1-neg 34.9%

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

   2. associate-*r* 34.9%

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

   3. distribute-rgt-neg-in 34.9%

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

   4. distribute-lft-in 34.9%

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

   5. *-rgt-identity 34.9%

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

   6. associate-*r/ 34.9%

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

   7. *-rgt-identity 34.9%

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

   8. sub-neg 34.9%

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

   9. distribute-neg-in 34.9%

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

   10. metadata-eval 34.9%

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

   11. remove-double-neg 34.9%

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

   12. +-commutative 34.9%

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

8. Simplified 34.9%

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

9. Taylor expanded in x around 0 41.6%

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

10. Final simplification 41.6%

    \[\leadsto 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(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))