NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.9% → 99.9%

Time: 20.1s

Alternatives: 18

Speedup: 6.7×

• 37.9% 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: 73.9% 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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{x + 1}{e^{x}}\\ \mathbf{if}\;eps\_m \leq 3.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{t\_0 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ x 1.0) (exp x))))
   (if (<= eps_m 3.4e-17)
     (/ (+ t_0 t_0) 2.0)
     (/
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x + 1.0) / exp(x);
	double tmp;
	if (eps_m <= 3.4e-17) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-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 = (x + 1.0d0) / exp(x)
    if (eps_m <= 3.4d-17) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (((1.0d0 + (1.0d0 / eps_m)) * exp((x * (eps_m + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps_m))) * (1.0d0 + ((-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 = (x + 1.0) / Math.exp(x);
	double tmp;
	if (eps_m <= 3.4e-17) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * Math.exp((x * (eps_m + -1.0)))) + (Math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x + 1.0) / math.exp(x)
	tmp = 0
	if eps_m <= 3.4e-17:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (((1.0 + (1.0 / eps_m)) * math.exp((x * (eps_m + -1.0)))) + (math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x + 1.0) / exp(x))
	tmp = 0.0
	if (eps_m <= 3.4e-17)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + 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 = (x + 1.0) / exp(x);
	tmp = 0.0;
	if (eps_m <= 3.4e-17)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-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[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 3.4e-17], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 3.3999999999999998e-17

   1. Initial program 60.5%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in eps around 0 73.0%

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

   7. Simplified 73.6%

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

      1. exp-neg 73.6%

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

      2. un-div-inv 73.6%

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

   9. Applied egg-rr 73.6%

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

       1. exp-neg 73.6%

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

       2. un-div-inv 73.6%

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

   11. Applied egg-rr 73.6%

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

   if 3.3999999999999998e-17 < eps

   1. Initial program 99.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \frac{x + 1}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ t_1 := x \cdot \left(-1 - eps\_m\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{-232}:\\ \;\;\;\;\frac{1 + e^{t\_1}}{2}\\ \mathbf{elif}\;x \leq 0.64:\\ \;\;\;\;\frac{t\_0 + \left(1 + t\_1\right)}{2}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + x \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + \frac{1}{x + 1}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))) (t_1 (* x (- -1.0 eps_m))))
   (if (<= x -1e-232)
     (/ (+ 1.0 (exp t_1)) 2.0)
     (if (<= x 0.64)
       (/ (+ t_0 (+ 1.0 t_1)) 2.0)
       (if (<= x 5.4e+69)
         (/ (+ (/ (+ x 1.0) (exp x)) (* x (exp (- x)))) 2.0)
         (/ (+ t_0 (/ 1.0 (+ x 1.0))) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double t_1 = x * (-1.0 - eps_m);
	double tmp;
	if (x <= -1e-232) {
		tmp = (1.0 + exp(t_1)) / 2.0;
	} else if (x <= 0.64) {
		tmp = (t_0 + (1.0 + t_1)) / 2.0;
	} else if (x <= 5.4e+69) {
		tmp = (((x + 1.0) / exp(x)) + (x * exp(-x))) / 2.0;
	} else {
		tmp = (t_0 + (1.0 / (x + 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) :: t_1
    real(8) :: tmp
    t_0 = exp((x * (eps_m + (-1.0d0))))
    t_1 = x * ((-1.0d0) - eps_m)
    if (x <= (-1d-232)) then
        tmp = (1.0d0 + exp(t_1)) / 2.0d0
    else if (x <= 0.64d0) then
        tmp = (t_0 + (1.0d0 + t_1)) / 2.0d0
    else if (x <= 5.4d+69) then
        tmp = (((x + 1.0d0) / exp(x)) + (x * exp(-x))) / 2.0d0
    else
        tmp = (t_0 + (1.0d0 / (x + 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 * (eps_m + -1.0)));
	double t_1 = x * (-1.0 - eps_m);
	double tmp;
	if (x <= -1e-232) {
		tmp = (1.0 + Math.exp(t_1)) / 2.0;
	} else if (x <= 0.64) {
		tmp = (t_0 + (1.0 + t_1)) / 2.0;
	} else if (x <= 5.4e+69) {
		tmp = (((x + 1.0) / Math.exp(x)) + (x * Math.exp(-x))) / 2.0;
	} else {
		tmp = (t_0 + (1.0 / (x + 1.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	t_1 = x * (-1.0 - eps_m)
	tmp = 0
	if x <= -1e-232:
		tmp = (1.0 + math.exp(t_1)) / 2.0
	elif x <= 0.64:
		tmp = (t_0 + (1.0 + t_1)) / 2.0
	elif x <= 5.4e+69:
		tmp = (((x + 1.0) / math.exp(x)) + (x * math.exp(-x))) / 2.0
	else:
		tmp = (t_0 + (1.0 / (x + 1.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	t_1 = Float64(x * Float64(-1.0 - eps_m))
	tmp = 0.0
	if (x <= -1e-232)
		tmp = Float64(Float64(1.0 + exp(t_1)) / 2.0);
	elseif (x <= 0.64)
		tmp = Float64(Float64(t_0 + Float64(1.0 + t_1)) / 2.0);
	elseif (x <= 5.4e+69)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / exp(x)) + Float64(x * exp(Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(t_0 + Float64(1.0 / Float64(x + 1.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	t_1 = x * (-1.0 - eps_m);
	tmp = 0.0;
	if (x <= -1e-232)
		tmp = (1.0 + exp(t_1)) / 2.0;
	elseif (x <= 0.64)
		tmp = (t_0 + (1.0 + t_1)) / 2.0;
	elseif (x <= 5.4e+69)
		tmp = (((x + 1.0) / exp(x)) + (x * exp(-x))) / 2.0;
	else
		tmp = (t_0 + (1.0 / (x + 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[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e-232], N[(N[(1.0 + N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.64], N[(N[(t$95$0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.4e+69], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(x * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.00000000000000002e-232

   1. Initial program 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in x around 0 46.3%

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

   6. Taylor expanded in eps around inf 69.6%

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

      1. cancel-sign-sub-inv 69.6%

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

      2. metadata-eval 69.6%

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

      3. *-lft-identity 69.6%

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

      4. associate-*r* 69.6%

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

      5. exp-prod 60.5%

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

      6. remove-double-neg 60.5%

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

      7. mul-1-neg 60.5%

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

      8. sub-neg 60.5%

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

      9. exp-prod 69.6%

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

      10. associate-*r* 69.6%

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

      11. remove-double-neg 69.6%

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

      12. mul-1-neg 69.6%

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

      13. sub-neg 69.6%

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

      14. mul-1-neg 69.6%

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

      15. associate-*r* 69.6%

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

      16. exp-prod 60.5%

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

   8. Simplified 69.6%

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

   if -1.00000000000000002e-232 < x < 0.640000000000000013

   1. Initial program 52.6%

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

   3. Simplified 44.1%

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

   5. Taylor expanded in eps around inf 99.7%

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

   6. Taylor expanded in x around 0 93.4%

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

      1. associate-*r* 93.4%

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

      2. neg-mul-1 93.4%

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

   8. Simplified 93.4%

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

   if 0.640000000000000013 < x < 5.3999999999999996e69

   1. Initial program 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in eps around 0 82.5%

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

   7. Simplified 82.4%

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

      1. exp-neg 82.4%

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

      2. un-div-inv 82.4%

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

   9. Applied egg-rr 82.4%

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

   10. Taylor expanded in x around inf 68.9%

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

   if 5.3999999999999996e69 < 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%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 30.2%

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

   7. Taylor expanded in eps around 0 30.3%

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

      1. +-commutative 30.3%

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

   9. Simplified 30.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-232}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 0.64:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + x \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{x + 1}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{x + 1}{e^{x}}\\ \mathbf{if}\;eps\_m \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\frac{t\_0 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{e^{x + eps\_m \cdot x}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ x 1.0) (exp x))))
   (if (<= eps_m 2e-37)
     (/ (+ t_0 t_0) 2.0)
     (/ (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (exp (+ x (* eps_m x))))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x + 1.0) / exp(x);
	double tmp;
	if (eps_m <= 2e-37) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / exp((x + (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 = (x + 1.0d0) / exp(x)
    if (eps_m <= 2d-37) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 / exp((x + (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 = (x + 1.0) / Math.exp(x);
	double tmp;
	if (eps_m <= 2e-37) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 / Math.exp((x + (eps_m * x))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x + 1.0) / math.exp(x)
	tmp = 0
	if eps_m <= 2e-37:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 / math.exp((x + (eps_m * x))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x + 1.0) / exp(x))
	tmp = 0.0
	if (eps_m <= 2e-37)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / exp(Float64(x + Float64(eps_m * x))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x + 1.0) / exp(x);
	tmp = 0.0;
	if (eps_m <= 2e-37)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / exp((x + (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[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 2e-37], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[N[(x + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.00000000000000013e-37

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in eps around 0 72.3%

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

   7. Simplified 72.8%

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

      1. exp-neg 72.8%

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

      2. un-div-inv 72.8%

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

   9. Applied egg-rr 72.8%

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

       1. exp-neg 72.8%

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

       2. un-div-inv 72.8%

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

   11. Applied egg-rr 72.8%

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

   if 2.00000000000000013e-37 < eps

   1. Initial program 94.0%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in eps around inf 99.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \frac{x + 1}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{e^{x + \varepsilon \cdot x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

3. Simplified 64.7%

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

5. Taylor expanded in eps around inf 98.4%

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

6. Final simplification 98.4%

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

Alternative 5: 76.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3900000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1150:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot \left(1 + eps\_m \cdot \left(eps\_m + 2\right)\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+87} \lor \neg \left(x \leq 2.1 \cdot 10^{+242}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.5 \cdot {\left(eps\_m \cdot x\right)}^{2}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -3900000.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 1150.0)
     (/
      (+ 2.0 (* x (+ -1.0 (* 0.5 (* x (+ 1.0 (* eps_m (+ eps_m 2.0))))))))
      2.0)
     (if (or (<= x 1.8e+87) (not (<= x 2.1e+242)))
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
       (/ (+ 2.0 (* 0.5 (pow (* eps_m x) 2.0))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -3900000.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 1150.0) {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0;
	} else if ((x <= 1.8e+87) || !(x <= 2.1e+242)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (0.5 * pow((eps_m * 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 <= (-3900000.0d0)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 1150.0d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (0.5d0 * (x * (1.0d0 + (eps_m * (eps_m + 2.0d0)))))))) / 2.0d0
    else if ((x <= 1.8d+87) .or. (.not. (x <= 2.1d+242))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (2.0d0 + (0.5d0 * ((eps_m * 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 <= -3900000.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 1150.0) {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0;
	} else if ((x <= 1.8e+87) || !(x <= 2.1e+242)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (0.5 * Math.pow((eps_m * x), 2.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -3900000.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 1150.0:
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0
	elif (x <= 1.8e+87) or not (x <= 2.1e+242):
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (2.0 + (0.5 * math.pow((eps_m * x), 2.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -3900000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 1150.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(0.5 * Float64(x * Float64(1.0 + Float64(eps_m * Float64(eps_m + 2.0)))))))) / 2.0);
	elseif ((x <= 1.8e+87) || !(x <= 2.1e+242))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(0.5 * (Float64(eps_m * 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 <= -3900000.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 1150.0)
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0;
	elseif ((x <= 1.8e+87) || ~((x <= 2.1e+242)))
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (2.0 + (0.5 * ((eps_m * 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, -3900000.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1150.0], N[(N[(2.0 + N[(x * N[(-1.0 + N[(0.5 * N[(x * N[(1.0 + N[(eps$95$m * N[(eps$95$m + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.8e+87], N[Not[LessEqual[x, 2.1e+242]], $MachinePrecision]], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(0.5 * N[Power[N[(eps$95$m * x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.9e6

   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 41.7%

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

   6. Taylor expanded in eps around inf 70.3%

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

   7. Taylor expanded in eps around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-rgt1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul-1-neg 100.0%

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

      7. remove-double-neg 100.0%

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

      8. neg-mul-1 100.0%

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

   9. Simplified 100.0%

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

   if -3.9e6 < x < 1150

   1. Initial program 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in x around 0 41.6%

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

   6. Taylor expanded in eps around inf 84.1%

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

   7. Taylor expanded in x around 0 88.6%

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

   8. Taylor expanded in eps around 0 88.6%

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

   if 1150 < x < 1.79999999999999997e87 or 2.0999999999999999e242 < 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 15.2%

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

   6. Taylor expanded in x around 0 66.9%

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

   if 1.79999999999999997e87 < x < 2.0999999999999999e242

   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 26.0%

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

   6. Taylor expanded in eps around inf 46.8%

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

   7. Taylor expanded in x around 0 62.7%

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

   8. Taylor expanded in eps around inf 62.8%

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

      1. *-commutative 62.8%

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

      2. unpow2 62.8%

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

      3. unpow2 62.8%

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

      4. swap-sqr 63.1%

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

      5. unpow2 63.1%

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

   10. Simplified 63.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3900000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1150:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot \left(1 + \varepsilon \cdot \left(\varepsilon + 2\right)\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+87} \lor \neg \left(x \leq 2.1 \cdot 10^{+242}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.5 \cdot {\left(\varepsilon \cdot x\right)}^{2}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-256}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 2000:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot \left(1 + eps\_m \cdot \left(eps\_m + 2\right)\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+87} \lor \neg \left(x \leq 10^{+242}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.5 \cdot {\left(eps\_m \cdot x\right)}^{2}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 3e-256)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (if (<= x 2000.0)
     (/
      (+ 2.0 (* x (+ -1.0 (* 0.5 (* x (+ 1.0 (* eps_m (+ eps_m 2.0))))))))
      2.0)
     (if (or (<= x 1.3e+87) (not (<= x 1e+242)))
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
       (/ (+ 2.0 (* 0.5 (pow (* eps_m x) 2.0))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 3e-256) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 2000.0) {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0;
	} else if ((x <= 1.3e+87) || !(x <= 1e+242)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (0.5 * pow((eps_m * 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 <= 3d-256) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 2000.0d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (0.5d0 * (x * (1.0d0 + (eps_m * (eps_m + 2.0d0)))))))) / 2.0d0
    else if ((x <= 1.3d+87) .or. (.not. (x <= 1d+242))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (2.0d0 + (0.5d0 * ((eps_m * 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 <= 3e-256) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 2000.0) {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0;
	} else if ((x <= 1.3e+87) || !(x <= 1e+242)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (0.5 * Math.pow((eps_m * x), 2.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 3e-256:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif x <= 2000.0:
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0
	elif (x <= 1.3e+87) or not (x <= 1e+242):
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (2.0 + (0.5 * math.pow((eps_m * x), 2.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 3e-256)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 2000.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(0.5 * Float64(x * Float64(1.0 + Float64(eps_m * Float64(eps_m + 2.0)))))))) / 2.0);
	elseif ((x <= 1.3e+87) || !(x <= 1e+242))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(0.5 * (Float64(eps_m * 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 <= 3e-256)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 2000.0)
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0;
	elseif ((x <= 1.3e+87) || ~((x <= 1e+242)))
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (2.0 + (0.5 * ((eps_m * 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, 3e-256], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2000.0], N[(N[(2.0 + N[(x * N[(-1.0 + N[(0.5 * N[(x * N[(1.0 + N[(eps$95$m * N[(eps$95$m + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.3e+87], N[Not[LessEqual[x, 1e+242]], $MachinePrecision]], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(0.5 * N[Power[N[(eps$95$m * x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.9999999999999998e-256

   1. Initial program 67.1%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in x around 0 46.4%

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

   6. Taylor expanded in eps around inf 77.4%

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

      1. cancel-sign-sub-inv 77.4%

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

      2. metadata-eval 77.4%

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

      3. *-lft-identity 77.4%

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

      4. associate-*r* 77.4%

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

      5. exp-prod 70.8%

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

      6. remove-double-neg 70.8%

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

      7. mul-1-neg 70.8%

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

      8. sub-neg 70.8%

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

      9. exp-prod 77.4%

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

      10. associate-*r* 77.4%

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

      11. remove-double-neg 77.4%

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

      12. mul-1-neg 77.4%

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

      13. sub-neg 77.4%

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

      14. mul-1-neg 77.4%

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

      15. associate-*r* 77.4%

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

      16. exp-prod 70.8%

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

   8. Simplified 77.4%

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

   if 2.9999999999999998e-256 < x < 2e3

   1. Initial program 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in x around 0 39.2%

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

   6. Taylor expanded in eps around inf 81.5%

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

   7. Taylor expanded in x around 0 89.4%

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

   8. Taylor expanded in eps around 0 89.4%

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

   if 2e3 < x < 1.29999999999999999e87 or 1.00000000000000005e242 < 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 15.2%

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

   6. Taylor expanded in x around 0 66.9%

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

   if 1.29999999999999999e87 < x < 1.00000000000000005e242

   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 26.0%

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

   6. Taylor expanded in eps around inf 46.8%

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

   7. Taylor expanded in x around 0 62.7%

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

   8. Taylor expanded in eps around inf 62.8%

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

      1. *-commutative 62.8%

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

      2. unpow2 62.8%

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

      3. unpow2 62.8%

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

      4. swap-sqr 63.1%

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

      5. unpow2 63.1%

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

   10. Simplified 63.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-256}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2000:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot \left(1 + \varepsilon \cdot \left(\varepsilon + 2\right)\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+87} \lor \neg \left(x \leq 10^{+242}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.5 \cdot {\left(\varepsilon \cdot x\right)}^{2}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-234}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 4.2:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \left(1 - x\right)}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+87} \lor \neg \left(x \leq 2.4 \cdot 10^{+242}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.5 \cdot {\left(eps\_m \cdot x\right)}^{2}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-234)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (if (<= x 4.2)
     (/ (+ (exp (* x (+ eps_m -1.0))) (- 1.0 x)) 2.0)
     (if (or (<= x 6.8e+87) (not (<= x 2.4e+242)))
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
       (/ (+ 2.0 (* 0.5 (pow (* eps_m x) 2.0))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-234) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 4.2) {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 - x)) / 2.0;
	} else if ((x <= 6.8e+87) || !(x <= 2.4e+242)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (0.5 * pow((eps_m * 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 <= (-1d-234)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 4.2d0) then
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 - x)) / 2.0d0
    else if ((x <= 6.8d+87) .or. (.not. (x <= 2.4d+242))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (2.0d0 + (0.5d0 * ((eps_m * 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 <= -1e-234) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 4.2) {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 - x)) / 2.0;
	} else if ((x <= 6.8e+87) || !(x <= 2.4e+242)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (0.5 * Math.pow((eps_m * x), 2.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e-234:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif x <= 4.2:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 - x)) / 2.0
	elif (x <= 6.8e+87) or not (x <= 2.4e+242):
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (2.0 + (0.5 * math.pow((eps_m * x), 2.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-234)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 4.2)
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 - x)) / 2.0);
	elseif ((x <= 6.8e+87) || !(x <= 2.4e+242))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(0.5 * (Float64(eps_m * 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 <= -1e-234)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 4.2)
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 - x)) / 2.0;
	elseif ((x <= 6.8e+87) || ~((x <= 2.4e+242)))
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (2.0 + (0.5 * ((eps_m * 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, -1e-234], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.2], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 6.8e+87], N[Not[LessEqual[x, 2.4e+242]], $MachinePrecision]], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(0.5 * N[Power[N[(eps$95$m * x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.9999999999999996e-235

   1. Initial program 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in x around 0 46.3%

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

   6. Taylor expanded in eps around inf 69.6%

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

      1. cancel-sign-sub-inv 69.6%

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

      2. metadata-eval 69.6%

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

      3. *-lft-identity 69.6%

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

      4. associate-*r* 69.6%

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

      5. exp-prod 60.5%

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

      6. remove-double-neg 60.5%

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

      7. mul-1-neg 60.5%

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

      8. sub-neg 60.5%

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

      9. exp-prod 69.6%

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

      10. associate-*r* 69.6%

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

      11. remove-double-neg 69.6%

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

      12. mul-1-neg 69.6%

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

      13. sub-neg 69.6%

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

      14. mul-1-neg 69.6%

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

      15. associate-*r* 69.6%

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

      16. exp-prod 60.5%

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

   8. Simplified 69.6%

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

   if -9.9999999999999996e-235 < x < 4.20000000000000018

   1. Initial program 52.1%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in eps around inf 98.8%

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

   6. Taylor expanded in x around 0 92.6%

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

      1. associate-*r* 92.6%

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

      2. neg-mul-1 92.6%

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

   8. Simplified 92.6%

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

   9. Taylor expanded in eps around 0 92.4%

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

       1. neg-mul-1 92.4%

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

   11. Simplified 92.4%

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

   if 4.20000000000000018 < x < 6.8000000000000004e87 or 2.40000000000000012e242 < x

   1. Initial program 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in x around 0 14.7%

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

   6. Taylor expanded in x around 0 63.7%

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

   if 6.8000000000000004e87 < x < 2.40000000000000012e242

   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 26.0%

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

   6. Taylor expanded in eps around inf 46.8%

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

   7. Taylor expanded in x around 0 62.7%

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

   8. Taylor expanded in eps around inf 62.8%

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

      1. *-commutative 62.8%

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

      2. unpow2 62.8%

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

      3. unpow2 62.8%

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

      4. swap-sqr 63.1%

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

      5. unpow2 63.1%

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

   10. Simplified 63.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-234}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 4.2:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 - x\right)}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+87} \lor \neg \left(x \leq 2.4 \cdot 10^{+242}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 0.5 \cdot {\left(\varepsilon \cdot x\right)}^{2}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 8100000000000 \lor \neg \left(x \leq 2.05 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{x + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -3.5e-236)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (if (or (<= x 8100000000000.0) (not (<= x 2.05e+71)))
     (/ (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (+ x 1.0))) 2.0)
     (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -3.5e-236) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if ((x <= 8100000000000.0) || !(x <= 2.05e+71)) {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / (x + 1.0))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-3.5d-236)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if ((x <= 8100000000000.0d0) .or. (.not. (x <= 2.05d+71))) then
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 / (x + 1.0d0))) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -3.5e-236) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if ((x <= 8100000000000.0) || !(x <= 2.05e+71)) {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 / (x + 1.0))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -3.5e-236:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif (x <= 8100000000000.0) or not (x <= 2.05e+71):
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 / (x + 1.0))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -3.5e-236)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif ((x <= 8100000000000.0) || !(x <= 2.05e+71))
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / Float64(x + 1.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -3.5e-236)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif ((x <= 8100000000000.0) || ~((x <= 2.05e+71)))
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / (x + 1.0))) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -3.5e-236], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 8100000000000.0], N[Not[LessEqual[x, 2.05e+71]], $MachinePrecision]], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.49999999999999994e-236

   1. Initial program 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in x around 0 46.3%

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

   6. Taylor expanded in eps around inf 69.6%

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

      1. cancel-sign-sub-inv 69.6%

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

      2. metadata-eval 69.6%

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

      3. *-lft-identity 69.6%

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

      4. associate-*r* 69.6%

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

      5. exp-prod 60.5%

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

      6. remove-double-neg 60.5%

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

      7. mul-1-neg 60.5%

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

      8. sub-neg 60.5%

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

      9. exp-prod 69.6%

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

      10. associate-*r* 69.6%

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

      11. remove-double-neg 69.6%

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

      12. mul-1-neg 69.6%

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

      13. sub-neg 69.6%

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

      14. mul-1-neg 69.6%

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

      15. associate-*r* 69.6%

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

      16. exp-prod 60.5%

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

   8. Simplified 69.6%

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

   if -3.49999999999999994e-236 < x < 8.1e12 or 2.0500000000000001e71 < x

   1. Initial program 68.6%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in eps around inf 98.1%

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

   6. Taylor expanded in x around 0 69.1%

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

   7. Taylor expanded in eps around 0 69.2%

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

      1. +-commutative 69.2%

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

   9. Simplified 69.2%

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

   if 8.1e12 < x < 2.0500000000000001e71

   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 2.4%

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

   6. Taylor expanded in x around 0 79.2%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 8100000000000 \lor \neg \left(x \leq 2.05 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{x + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ t_1 := x \cdot \left(-1 - eps\_m\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{-235}:\\ \;\;\;\;\frac{1 + e^{t\_1}}{2}\\ \mathbf{elif}\;x \leq 800000000000:\\ \;\;\;\;\frac{t\_0 + \frac{1}{1 - t\_1}}{2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + \frac{1}{x + 1}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))) (t_1 (* x (- -1.0 eps_m))))
   (if (<= x -2e-235)
     (/ (+ 1.0 (exp t_1)) 2.0)
     (if (<= x 800000000000.0)
       (/ (+ t_0 (/ 1.0 (- 1.0 t_1))) 2.0)
       (if (<= x 5.5e+72)
         (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
         (/ (+ t_0 (/ 1.0 (+ x 1.0))) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double t_1 = x * (-1.0 - eps_m);
	double tmp;
	if (x <= -2e-235) {
		tmp = (1.0 + exp(t_1)) / 2.0;
	} else if (x <= 800000000000.0) {
		tmp = (t_0 + (1.0 / (1.0 - t_1))) / 2.0;
	} else if (x <= 5.5e+72) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (t_0 + (1.0 / (x + 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) :: t_1
    real(8) :: tmp
    t_0 = exp((x * (eps_m + (-1.0d0))))
    t_1 = x * ((-1.0d0) - eps_m)
    if (x <= (-2d-235)) then
        tmp = (1.0d0 + exp(t_1)) / 2.0d0
    else if (x <= 800000000000.0d0) then
        tmp = (t_0 + (1.0d0 / (1.0d0 - t_1))) / 2.0d0
    else if (x <= 5.5d+72) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (t_0 + (1.0d0 / (x + 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 * (eps_m + -1.0)));
	double t_1 = x * (-1.0 - eps_m);
	double tmp;
	if (x <= -2e-235) {
		tmp = (1.0 + Math.exp(t_1)) / 2.0;
	} else if (x <= 800000000000.0) {
		tmp = (t_0 + (1.0 / (1.0 - t_1))) / 2.0;
	} else if (x <= 5.5e+72) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (t_0 + (1.0 / (x + 1.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	t_1 = x * (-1.0 - eps_m)
	tmp = 0
	if x <= -2e-235:
		tmp = (1.0 + math.exp(t_1)) / 2.0
	elif x <= 800000000000.0:
		tmp = (t_0 + (1.0 / (1.0 - t_1))) / 2.0
	elif x <= 5.5e+72:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (t_0 + (1.0 / (x + 1.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	t_1 = Float64(x * Float64(-1.0 - eps_m))
	tmp = 0.0
	if (x <= -2e-235)
		tmp = Float64(Float64(1.0 + exp(t_1)) / 2.0);
	elseif (x <= 800000000000.0)
		tmp = Float64(Float64(t_0 + Float64(1.0 / Float64(1.0 - t_1))) / 2.0);
	elseif (x <= 5.5e+72)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(t_0 + Float64(1.0 / Float64(x + 1.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	t_1 = x * (-1.0 - eps_m);
	tmp = 0.0;
	if (x <= -2e-235)
		tmp = (1.0 + exp(t_1)) / 2.0;
	elseif (x <= 800000000000.0)
		tmp = (t_0 + (1.0 / (1.0 - t_1))) / 2.0;
	elseif (x <= 5.5e+72)
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (t_0 + (1.0 / (x + 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[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e-235], N[(N[(1.0 + N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 800000000000.0], N[(N[(t$95$0 + N[(1.0 / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.5e+72], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.9999999999999999e-235

   1. Initial program 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in x around 0 46.3%

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

   6. Taylor expanded in eps around inf 69.6%

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

      1. cancel-sign-sub-inv 69.6%

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

      2. metadata-eval 69.6%

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

      3. *-lft-identity 69.6%

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

      4. associate-*r* 69.6%

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

      5. exp-prod 60.5%

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

      6. remove-double-neg 60.5%

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

      7. mul-1-neg 60.5%

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

      8. sub-neg 60.5%

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

      9. exp-prod 69.6%

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

      10. associate-*r* 69.6%

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

      11. remove-double-neg 69.6%

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

      12. mul-1-neg 69.6%

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

      13. sub-neg 69.6%

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

      14. mul-1-neg 69.6%

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

      15. associate-*r* 69.6%

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

      16. exp-prod 60.5%

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

   8. Simplified 69.6%

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

   if -1.9999999999999999e-235 < x < 8e11

   1. Initial program 51.1%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in eps around inf 97.1%

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

   6. Taylor expanded in x around 0 90.9%

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

   if 8e11 < x < 5.5e72

   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 2.4%

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

   6. Taylor expanded in x around 0 79.2%

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

   if 5.5e72 < 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%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 30.2%

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

   7. Taylor expanded in eps around 0 30.3%

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

      1. +-commutative 30.3%

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

   9. Simplified 30.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-235}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 800000000000:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{1 - x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{x + 1}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3900000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 88000000000 \lor \neg \left(x \leq 1.7 \cdot 10^{+101}\right) \land x \leq 2.6 \cdot 10^{+242}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot \left(1 + eps\_m \cdot \left(eps\_m + 2\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -3900000.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (or (<= x 88000000000.0) (and (not (<= x 1.7e+101)) (<= x 2.6e+242)))
     (/
      (+ 2.0 (* x (+ -1.0 (* 0.5 (* x (+ 1.0 (* eps_m (+ eps_m 2.0))))))))
      2.0)
     (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -3900000.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if ((x <= 88000000000.0) || (!(x <= 1.7e+101) && (x <= 2.6e+242))) {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-3900000.0d0)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if ((x <= 88000000000.0d0) .or. (.not. (x <= 1.7d+101)) .and. (x <= 2.6d+242)) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (0.5d0 * (x * (1.0d0 + (eps_m * (eps_m + 2.0d0)))))))) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -3900000.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if ((x <= 88000000000.0) || (!(x <= 1.7e+101) && (x <= 2.6e+242))) {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -3900000.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif (x <= 88000000000.0) or (not (x <= 1.7e+101) and (x <= 2.6e+242)):
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -3900000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif ((x <= 88000000000.0) || (!(x <= 1.7e+101) && (x <= 2.6e+242)))
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(0.5 * Float64(x * Float64(1.0 + Float64(eps_m * Float64(eps_m + 2.0)))))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -3900000.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif ((x <= 88000000000.0) || (~((x <= 1.7e+101)) && (x <= 2.6e+242)))
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -3900000.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 88000000000.0], And[N[Not[LessEqual[x, 1.7e+101]], $MachinePrecision], LessEqual[x, 2.6e+242]]], N[(N[(2.0 + N[(x * N[(-1.0 + N[(0.5 * N[(x * N[(1.0 + N[(eps$95$m * N[(eps$95$m + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9e6

   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 41.7%

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

   6. Taylor expanded in eps around inf 70.3%

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

   7. Taylor expanded in eps around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-rgt1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul-1-neg 100.0%

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

      7. remove-double-neg 100.0%

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

      8. neg-mul-1 100.0%

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

   9. Simplified 100.0%

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

   if -3.9e6 < x < 8.8e10 or 1.70000000000000009e101 < x < 2.5999999999999998e242

   1. Initial program 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in x around 0 39.8%

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

   6. Taylor expanded in eps around inf 79.6%

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

   7. Taylor expanded in x around 0 85.7%

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

   8. Taylor expanded in eps around 0 85.7%

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

   if 8.8e10 < x < 1.70000000000000009e101 or 2.5999999999999998e242 < 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 17.9%

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

   6. Taylor expanded in x around 0 64.8%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3900000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 88000000000 \lor \neg \left(x \leq 1.7 \cdot 10^{+101}\right) \land x \leq 2.6 \cdot 10^{+242}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot \left(1 + \varepsilon \cdot \left(\varepsilon + 2\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 820 \lor \neg \left(x \leq 9.2 \cdot 10^{+108}\right) \land x \leq 7.4 \cdot 10^{+241}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot \left(1 + eps\_m \cdot \left(eps\_m + 2\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (or (<= x 820.0) (and (not (<= x 9.2e+108)) (<= x 7.4e+241)))
   (/ (+ 2.0 (* x (+ -1.0 (* 0.5 (* x (+ 1.0 (* eps_m (+ eps_m 2.0)))))))) 2.0)
   (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((x <= 820.0) || (!(x <= 9.2e+108) && (x <= 7.4e+241))) {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if ((x <= 820.0d0) .or. (.not. (x <= 9.2d+108)) .and. (x <= 7.4d+241)) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (0.5d0 * (x * (1.0d0 + (eps_m * (eps_m + 2.0d0)))))))) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if ((x <= 820.0) || (!(x <= 9.2e+108) && (x <= 7.4e+241))) {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if (x <= 820.0) or (not (x <= 9.2e+108) and (x <= 7.4e+241)):
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if ((x <= 820.0) || (!(x <= 9.2e+108) && (x <= 7.4e+241)))
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(0.5 * Float64(x * Float64(1.0 + Float64(eps_m * Float64(eps_m + 2.0)))))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if ((x <= 820.0) || (~((x <= 9.2e+108)) && (x <= 7.4e+241)))
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * (1.0 + (eps_m * (eps_m + 2.0)))))))) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[Or[LessEqual[x, 820.0], And[N[Not[LessEqual[x, 9.2e+108]], $MachinePrecision], LessEqual[x, 7.4e+241]]], N[(N[(2.0 + N[(x * N[(-1.0 + N[(0.5 * N[(x * N[(1.0 + N[(eps$95$m * N[(eps$95$m + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 820 or 9.1999999999999996e108 < x < 7.3999999999999995e241

   1. Initial program 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in x around 0 40.1%

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

   6. Taylor expanded in eps around inf 78.1%

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

   7. Taylor expanded in x around 0 84.5%

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

   8. Taylor expanded in eps around 0 84.5%

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

   if 820 < x < 9.1999999999999996e108 or 7.3999999999999995e241 < 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 17.9%

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

   6. Taylor expanded in x around 0 64.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 820 \lor \neg \left(x \leq 9.2 \cdot 10^{+108}\right) \land x \leq 7.4 \cdot 10^{+241}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot \left(1 + \varepsilon \cdot \left(\varepsilon + 2\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 510 \lor \neg \left(x \leq 1.6 \cdot 10^{+106}\right) \land x \leq 1.6 \cdot 10^{+242}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(eps\_m + 0.5\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (or (<= x 510.0) (and (not (<= x 1.6e+106)) (<= x 1.6e+242)))
   (/ (+ 2.0 (* x (+ -1.0 (* x (+ eps_m 0.5))))) 2.0)
   (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((x <= 510.0) || (!(x <= 1.6e+106) && (x <= 1.6e+242))) {
		tmp = (2.0 + (x * (-1.0 + (x * (eps_m + 0.5))))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if ((x <= 510.0d0) .or. (.not. (x <= 1.6d+106)) .and. (x <= 1.6d+242)) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * (eps_m + 0.5d0))))) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if ((x <= 510.0) || (!(x <= 1.6e+106) && (x <= 1.6e+242))) {
		tmp = (2.0 + (x * (-1.0 + (x * (eps_m + 0.5))))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if (x <= 510.0) or (not (x <= 1.6e+106) and (x <= 1.6e+242)):
		tmp = (2.0 + (x * (-1.0 + (x * (eps_m + 0.5))))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if ((x <= 510.0) || (!(x <= 1.6e+106) && (x <= 1.6e+242)))
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(eps_m + 0.5))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if ((x <= 510.0) || (~((x <= 1.6e+106)) && (x <= 1.6e+242)))
		tmp = (2.0 + (x * (-1.0 + (x * (eps_m + 0.5))))) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[Or[LessEqual[x, 510.0], And[N[Not[LessEqual[x, 1.6e+106]], $MachinePrecision], LessEqual[x, 1.6e+242]]], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * N[(eps$95$m + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 510 or 1.5999999999999999e106 < x < 1.6000000000000001e242

   1. Initial program 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in x around 0 40.1%

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

   6. Taylor expanded in eps around inf 78.1%

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

   7. Taylor expanded in x around 0 84.5%

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

   8. Taylor expanded in eps around 0 63.5%

      \[\leadsto \frac{2 + x \cdot \color{blue}{\left(\left(0.5 \cdot x + \varepsilon \cdot x\right) - 1\right)}}{2} \]
   9. Step-by-step derivation

      1. sub-neg 63.5%

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

      2. distribute-rgt-out 63.5%

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

      3. metadata-eval 63.5%

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

   10. Simplified 63.5%

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

   if 510 < x < 1.5999999999999999e106 or 1.6000000000000001e242 < 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 17.9%

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

   6. Taylor expanded in x around 0 64.8%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 510 \lor \neg \left(x \leq 1.6 \cdot 10^{+106}\right) \land x \leq 1.6 \cdot 10^{+242}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(\varepsilon + 0.5\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.1% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0055:\\ \;\;\;\;\frac{eps\_m \cdot x}{-2}\\ \mathbf{elif}\;x \leq 31:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.0055)
   (/ (* eps_m x) (- 2.0))
   (if (<= x 31.0) 1.0 (/ (* eps_m x) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.0055) {
		tmp = (eps_m * x) / -2.0;
	} else if (x <= 31.0) {
		tmp = 1.0;
	} else {
		tmp = (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 (x <= (-0.0055d0)) then
        tmp = (eps_m * x) / -2.0d0
    else if (x <= 31.0d0) then
        tmp = 1.0d0
    else
        tmp = (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 (x <= -0.0055) {
		tmp = (eps_m * x) / -2.0;
	} else if (x <= 31.0) {
		tmp = 1.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.0055:
		tmp = (eps_m * x) / -2.0
	elif x <= 31.0:
		tmp = 1.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.0055)
		tmp = Float64(Float64(eps_m * x) / Float64(-2.0));
	elseif (x <= 31.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.0055)
		tmp = (eps_m * x) / -2.0;
	elseif (x <= 31.0)
		tmp = 1.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.0055], N[(N[(eps$95$m * x), $MachinePrecision] / (-2.0)), $MachinePrecision], If[LessEqual[x, 31.0], 1.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0054999999999999997

   1. Initial program 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 58.1%

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

   6. Taylor expanded in x around 0 29.0%

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

   7. Taylor expanded in eps around inf 28.9%

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

      1. mul-1-neg 28.9%

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

      2. distribute-rgt-neg-out 28.9%

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

   9. Simplified 28.9%

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

   if -0.0054999999999999997 < x < 31

   1. Initial program 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in x around 0 74.2%

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

   if 31 < 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 18.3%

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

   6. Taylor expanded in x around inf 13.8%

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

      1. mul-1-neg 13.8%

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

      2. distribute-rgt-neg-in 13.8%

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

      3. *-commutative 13.8%

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

      4. distribute-rgt-neg-in 13.8%

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

      5. neg-mul-1 13.8%

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

      6. distribute-rgt-in 13.8%

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

      7. metadata-eval 13.8%

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

      8. associate-*l/ 13.8%

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

      9. metadata-eval 13.8%

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

   8. Simplified 13.8%

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

   9. Taylor expanded in eps around inf 14.6%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0055:\\ \;\;\;\;\frac{\varepsilon \cdot x}{-2}\\ \mathbf{elif}\;x \leq 31:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.4% accurate, 17.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{2 + x \cdot \left(-1 + x \cdot \left(eps\_m + 0.5\right)\right)}{2} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (/ (+ 2.0 (* x (+ -1.0 (* x (+ eps_m 0.5))))) 2.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (2.0 + (x * (-1.0 + (x * (eps_m + 0.5))))) / 2.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = (2.0d0 + (x * ((-1.0d0) + (x * (eps_m + 0.5d0))))) / 2.0d0
end function

Java

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

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (2.0 + (x * (-1.0 + (x * (eps_m + 0.5))))) / 2.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(eps_m + 0.5))))) / 2.0)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (2.0 + (x * (-1.0 + (x * (eps_m + 0.5))))) / 2.0;
end

Wolfram

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

Derivation

1. Initial program 72.3%

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

3. Simplified 72.3%

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

5. Taylor expanded in x around 0 35.5%

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

6. Taylor expanded in eps around inf 69.3%

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

7. Taylor expanded in x around 0 76.1%

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

8. Taylor expanded in eps around 0 55.2%

   \[\leadsto \frac{2 + x \cdot \color{blue}{\left(\left(0.5 \cdot x + \varepsilon \cdot x\right) - 1\right)}}{2} \]
9. Step-by-step derivation

   1. sub-neg 55.2%

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

   2. distribute-rgt-out 55.2%

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

   3. metadata-eval 55.2%

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

10. Simplified 55.2%

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

11. Final simplification 55.2%

    \[\leadsto \frac{2 + x \cdot \left(-1 + x \cdot \left(\varepsilon + 0.5\right)\right)}{2} \]
12. Add Preprocessing

Alternative 15: 57.7% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 11.2:\\ \;\;\;\;\frac{2 - eps\_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 11.2) (/ (- 2.0 (* eps_m x)) 2.0) (/ (* eps_m x) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 11.2) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else {
		tmp = (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 (x <= 11.2d0) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else
        tmp = (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 (x <= 11.2) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 11.2:
		tmp = (2.0 - (eps_m * x)) / 2.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 11.2)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 11.2)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 11.2], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 11.199999999999999

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in x around 0 44.0%

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

   6. Taylor expanded in x around 0 29.5%

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

   7. Taylor expanded in eps around -inf 64.9%

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

      1. associate-*r* 64.9%

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

      2. distribute-lft-out-- 64.9%

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

      3. associate-*r* 64.9%

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

      4. mul-1-neg 64.9%

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

      5. distribute-lft-neg-in 64.9%

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

      6. *-commutative 64.9%

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

      7. mul-1-neg 64.9%

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

      8. remove-double-neg 64.9%

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

   9. Simplified 64.9%

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

   10. Taylor expanded in eps around 0 65.0%

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

       1. mul-1-neg 65.0%

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

       2. unsub-neg 65.0%

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

       3. *-commutative 65.0%

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

   12. Simplified 65.0%

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

   if 11.199999999999999 < 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 18.3%

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

   6. Taylor expanded in x around inf 13.8%

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

      1. mul-1-neg 13.8%

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

      2. distribute-rgt-neg-in 13.8%

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

      3. *-commutative 13.8%

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

      4. distribute-rgt-neg-in 13.8%

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

      5. neg-mul-1 13.8%

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

      6. distribute-rgt-in 13.8%

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

      7. metadata-eval 13.8%

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

      8. associate-*l/ 13.8%

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

      9. metadata-eval 13.8%

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

   8. Simplified 13.8%

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

   9. Taylor expanded in eps around inf 14.6%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 11.2:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.6% accurate, 20.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 2.0d0
end function

Java

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

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 72.3%

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

3. Simplified 72.3%

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

5. Taylor expanded in x around 0 35.5%

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

6. Taylor expanded in eps around inf 69.3%

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

7. Taylor expanded in x around 0 76.1%

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

8. Taylor expanded in eps around 0 58.1%

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

9. Final simplification 58.1%

   \[\leadsto \frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2} \]
10. Add Preprocessing

Alternative 17: 51.3% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 24.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 24.5) 1.0 (/ (* eps_m x) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 24.5) {
		tmp = 1.0;
	} else {
		tmp = (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 (x <= 24.5d0) then
        tmp = 1.0d0
    else
        tmp = (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 (x <= 24.5) {
		tmp = 1.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 24.5:
		tmp = 1.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 24.5)
		tmp = 1.0;
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 24.5)
		tmp = 1.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 24.5], 1.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 24.5

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in x around 0 60.2%

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

   if 24.5 < 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 18.3%

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

   6. Taylor expanded in x around inf 13.8%

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

      1. mul-1-neg 13.8%

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

      2. distribute-rgt-neg-in 13.8%

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

      3. *-commutative 13.8%

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

      4. distribute-rgt-neg-in 13.8%

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

      5. neg-mul-1 13.8%

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

      6. distribute-rgt-in 13.8%

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

      7. metadata-eval 13.8%

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

      8. associate-*l/ 13.8%

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

      9. metadata-eval 13.8%

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

   8. Simplified 13.8%

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

   9. Taylor expanded in eps around inf 14.6%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 24.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.0% 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 72.3%

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

3. Simplified 72.3%

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

5. Taylor expanded in x around 0 45.3%

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

6. Final simplification 45.3%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(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))