NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 97.9%

Time: 19.1s

Alternatives: 15

Speedup: 1.8×

• 38.8% 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: 73.8% 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: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 + eps\_m\right)}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-245}:\\ \;\;\;\;\frac{t\_0 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ -1.0 eps_m)))))
   (if (<= x -1.25e-245)
     (/ (+ t_0 (exp (* x (- eps_m)))) 2.0)
     (/ (+ t_0 (exp (- x))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (-1.0 + eps_m)));
	double tmp;
	if (x <= -1.25e-245) {
		tmp = (t_0 + exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (t_0 + exp(-x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * ((-1.0d0) + eps_m)))
    if (x <= (-1.25d-245)) then
        tmp = (t_0 + exp((x * -eps_m))) / 2.0d0
    else
        tmp = (t_0 + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (-1.0 + eps_m)));
	double tmp;
	if (x <= -1.25e-245) {
		tmp = (t_0 + Math.exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (t_0 + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (-1.0 + eps_m)))
	tmp = 0
	if x <= -1.25e-245:
		tmp = (t_0 + math.exp((x * -eps_m))) / 2.0
	else:
		tmp = (t_0 + math.exp(-x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(-1.0 + eps_m)))
	tmp = 0.0
	if (x <= -1.25e-245)
		tmp = Float64(Float64(t_0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(t_0 + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (-1.0 + eps_m)));
	tmp = 0.0;
	if (x <= -1.25e-245)
		tmp = (t_0 + exp((x * -eps_m))) / 2.0;
	else
		tmp = (t_0 + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.25e-245], N[(N[(t$95$0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2499999999999999e-245

   1. Initial program 67.4%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in eps around inf 98.6%

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

   6. Taylor expanded in eps around inf 98.6%

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

      1. *-commutative 98.6%

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

   8. Simplified 98.6%

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

   if -1.2499999999999999e-245 < x

   1. Initial program 73.9%

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

   3. Simplified 60.2%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 82.6%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-245}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-245}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.25e-245)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (/ (+ (exp (* x (+ -1.0 eps_m))) (exp (- x))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-245) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) + exp(-x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.25d-245)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps_m))) + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-245) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.25e-245:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) + math.exp(-x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.25e-245)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.25e-245)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.25e-245], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2499999999999999e-245

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in x around 0 42.6%

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

   6. Taylor expanded in eps around inf 73.7%

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

   7. Taylor expanded in eps around inf 74.2%

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

      1. *-commutative 98.6%

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

   9. Simplified 74.2%

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

   if -1.2499999999999999e-245 < x

   1. Initial program 73.9%

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

   3. Simplified 60.2%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 82.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-245}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (exp((x * (-1.0 - eps_m))) + exp((x * (-1.0 + eps_m)))) / 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 * ((-1.0d0) - eps_m))) + exp((x * ((-1.0d0) + eps_m)))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

3. Simplified 59.1%

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

5. Taylor expanded in eps around inf 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 78.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-245}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+68} \lor \neg \left(x \leq 2.4 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.25e-245)
   (/ (* 2.0 (exp (- x))) 2.0)
   (if (or (<= x 6e+68) (not (<= x 2.4e+161)))
     (/ (+ 1.0 (exp (* x (+ -1.0 eps_m)))) 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 <= -1.25e-245) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else if ((x <= 6e+68) || !(x <= 2.4e+161)) {
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 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 <= (-1.25d-245)) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else if ((x <= 6d+68) .or. (.not. (x <= 2.4d+161))) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps_m)))) / 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 <= -1.25e-245) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else if ((x <= 6e+68) || !(x <= 2.4e+161)) {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps_m)))) / 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 <= -1.25e-245:
		tmp = (2.0 * math.exp(-x)) / 2.0
	elif (x <= 6e+68) or not (x <= 2.4e+161):
		tmp = (1.0 + math.exp((x * (-1.0 + eps_m)))) / 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 <= -1.25e-245)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	elseif ((x <= 6e+68) || !(x <= 2.4e+161))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps_m)))) / 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 <= -1.25e-245)
		tmp = (2.0 * exp(-x)) / 2.0;
	elseif ((x <= 6e+68) || ~((x <= 2.4e+161)))
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 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, -1.25e-245], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 6e+68], N[Not[LessEqual[x, 2.4e+161]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps$95$m), $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 < -1.2499999999999999e-245

   1. Initial program 67.4%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in eps around inf 98.6%

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

   6. Taylor expanded in eps around 0 88.6%

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

   7. Taylor expanded in eps around 0 79.0%

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

      1. mul-1-neg 79.0%

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

   9. Simplified 79.0%

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

   if -1.2499999999999999e-245 < x < 6.0000000000000004e68 or 2.3999999999999999e161 < x

   1. Initial program 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in x around 0 40.3%

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

   6. Taylor expanded in eps around inf 70.9%

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

      1. sub-neg 70.9%

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

      2. mul-1-neg 70.9%

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

      3. associate-*r* 70.9%

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

      4. mul-1-neg 70.9%

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

      5. mul-1-neg 70.9%

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

      6. sub-neg 70.9%

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

   8. Simplified 70.9%

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

   if 6.0000000000000004e68 < x < 2.3999999999999999e161

   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 20.1%

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

   6. Taylor expanded in x around 0 69.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-245}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+68} \lor \neg \left(x \leq 2.4 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-245}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+70} \lor \neg \left(x \leq 2.5 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.25e-245)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (or (<= x 9.2e+70) (not (<= x 2.5e+161)))
     (/ (+ 1.0 (exp (* x (+ -1.0 eps_m)))) 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 <= -1.25e-245) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if ((x <= 9.2e+70) || !(x <= 2.5e+161)) {
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 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 <= (-1.25d-245)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if ((x <= 9.2d+70) .or. (.not. (x <= 2.5d+161))) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps_m)))) / 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 <= -1.25e-245) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if ((x <= 9.2e+70) || !(x <= 2.5e+161)) {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps_m)))) / 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 <= -1.25e-245:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif (x <= 9.2e+70) or not (x <= 2.5e+161):
		tmp = (1.0 + math.exp((x * (-1.0 + eps_m)))) / 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 <= -1.25e-245)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif ((x <= 9.2e+70) || !(x <= 2.5e+161))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps_m)))) / 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 <= -1.25e-245)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif ((x <= 9.2e+70) || ~((x <= 2.5e+161)))
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 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, -1.25e-245], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 9.2e+70], N[Not[LessEqual[x, 2.5e+161]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps$95$m), $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 < -1.2499999999999999e-245

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in x around 0 42.6%

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

   6. Taylor expanded in eps around inf 73.7%

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

   7. Taylor expanded in eps around inf 74.2%

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

      1. *-commutative 98.6%

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

   9. Simplified 74.2%

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

   if -1.2499999999999999e-245 < x < 9.19999999999999975e70 or 2.4999999999999998e161 < x

   1. Initial program 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in x around 0 40.3%

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

   6. Taylor expanded in eps around inf 70.9%

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

      1. sub-neg 70.9%

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

      2. mul-1-neg 70.9%

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

      3. associate-*r* 70.9%

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

      4. mul-1-neg 70.9%

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

      5. mul-1-neg 70.9%

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

      6. sub-neg 70.9%

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

   8. Simplified 70.9%

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

   if 9.19999999999999975e70 < x < 2.4999999999999998e161

   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 20.1%

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

   6. Taylor expanded in x around 0 69.2%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-245}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+70} \lor \neg \left(x \leq 2.5 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1.85 \cdot 10^{+181}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(2 + x \cdot eps\_m\right) - x}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.85e+181)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (/ (- (* eps_m (+ 2.0 (* x eps_m))) x) eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.85e+181) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.85d+181) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (((eps_m * (2.0d0 + (x * eps_m))) - x) / eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.85e+181) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.85e+181:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.85e+181)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(eps_m * Float64(2.0 + Float64(x * eps_m))) - x) / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.85e+181)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.85e+181], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(eps$95$m * N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.8500000000000002e181

   1. Initial program 67.0%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in eps around inf 99.3%

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

   6. Taylor expanded in eps around 0 86.9%

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

   7. Taylor expanded in eps around 0 78.3%

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

      1. mul-1-neg 78.3%

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

   9. Simplified 78.3%

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

   if 1.8500000000000002e181 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 61.0%

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

   6. Taylor expanded in x around 0 31.2%

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

      1. mul-1-neg 31.2%

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

      2. unsub-neg 31.2%

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

      3. *-commutative 31.2%

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

      4. associate-*r* 31.2%

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

   8. Simplified 31.2%

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

   9. Taylor expanded in eps around 0 51.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.85 \cdot 10^{+181}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 + x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.8% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+161} \lor \neg \left(x \leq 2.15 \cdot 10^{+241}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e-6)
   (* (* x eps_m) -0.5)
   (if (<= x 350.0)
     1.0
     (if (or (<= x 3e+161) (not (<= x 2.15e+241)))
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
       (/ (* x eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 350.0) {
		tmp = 1.0;
	} else if ((x <= 3e+161) || !(x <= 2.15e+241)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.35d-6)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 350.0d0) then
        tmp = 1.0d0
    else if ((x <= 3d+161) .or. (.not. (x <= 2.15d+241))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 350.0) {
		tmp = 1.0;
	} else if ((x <= 3e+161) || !(x <= 2.15e+241)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.35e-6:
		tmp = (x * eps_m) * -0.5
	elif x <= 350.0:
		tmp = 1.0
	elif (x <= 3e+161) or not (x <= 2.15e+241):
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e-6)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 350.0)
		tmp = 1.0;
	elseif ((x <= 3e+161) || !(x <= 2.15e+241))
		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(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.35e-6)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 350.0)
		tmp = 1.0;
	elseif ((x <= 3e+161) || ~((x <= 2.15e+241)))
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e-6], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 350.0], 1.0, If[Or[LessEqual[x, 3e+161], N[Not[LessEqual[x, 2.15e+241]], $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[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.34999999999999999e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 50.2%

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

   6. Taylor expanded in x around 0 24.2%

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

      1. mul-1-neg 24.2%

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

      2. unsub-neg 24.2%

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

      3. *-commutative 24.2%

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

      4. associate-*r* 24.2%

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

   8. Simplified 24.2%

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

   9. Taylor expanded in eps around inf 24.1%

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

       1. frac-2neg 24.1%

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

       2. div-inv 24.1%

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

       3. distribute-rgt-neg-in 24.1%

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

       4. mul-1-neg 24.1%

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

       5. add-sqr-sqrt 24.1%

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

       6. sqrt-unprod 26.8%

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

       7. mul-1-neg 26.8%

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

       8. mul-1-neg 26.8%

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

       9. sqr-neg 26.8%

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

       10. sqrt-unprod 0.0%

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

       11. add-sqr-sqrt 24.9%

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

       12. *-commutative 24.9%

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

       13. metadata-eval 24.9%

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

       14. metadata-eval 24.9%

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

   11. Applied egg-rr 24.9%

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

   if -1.34999999999999999e-6 < x < 350

   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 76.2%

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

   if 350 < x < 3.00000000000000011e161 or 2.15000000000000002e241 < 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 21.1%

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

   6. Taylor expanded in x around 0 54.7%

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

   if 3.00000000000000011e161 < x < 2.15000000000000002e241

   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 53.9%

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

   6. Taylor expanded in x around 0 33.3%

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

      1. mul-1-neg 33.3%

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

      2. unsub-neg 33.3%

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

      3. *-commutative 33.3%

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

      4. associate-*r* 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in eps around inf 33.7%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+161} \lor \neg \left(x \leq 2.15 \cdot 10^{+241}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+161} \lor \neg \left(x \leq 5.3 \cdot 10^{+241}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(2 + x \cdot eps\_m\right) - x}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e-6)
   (* (* x eps_m) -0.5)
   (if (<= x 350.0)
     1.0
     (if (or (<= x 3e+161) (not (<= x 5.3e+241)))
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
       (/ (/ (- (* eps_m (+ 2.0 (* x eps_m))) x) eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 350.0) {
		tmp = 1.0;
	} else if ((x <= 3e+161) || !(x <= 5.3e+241)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.35d-6)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 350.0d0) then
        tmp = 1.0d0
    else if ((x <= 3d+161) .or. (.not. (x <= 5.3d+241))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (((eps_m * (2.0d0 + (x * eps_m))) - x) / eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 350.0) {
		tmp = 1.0;
	} else if ((x <= 3e+161) || !(x <= 5.3e+241)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.35e-6:
		tmp = (x * eps_m) * -0.5
	elif x <= 350.0:
		tmp = 1.0
	elif (x <= 3e+161) or not (x <= 5.3e+241):
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e-6)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 350.0)
		tmp = 1.0;
	elseif ((x <= 3e+161) || !(x <= 5.3e+241))
		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(Float64(Float64(eps_m * Float64(2.0 + Float64(x * eps_m))) - x) / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.35e-6)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 350.0)
		tmp = 1.0;
	elseif ((x <= 3e+161) || ~((x <= 5.3e+241)))
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e-6], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 350.0], 1.0, If[Or[LessEqual[x, 3e+161], N[Not[LessEqual[x, 5.3e+241]], $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[(N[(N[(eps$95$m * N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.34999999999999999e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 50.2%

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

   6. Taylor expanded in x around 0 24.2%

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

      1. mul-1-neg 24.2%

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

      2. unsub-neg 24.2%

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

      3. *-commutative 24.2%

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

      4. associate-*r* 24.2%

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

   8. Simplified 24.2%

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

   9. Taylor expanded in eps around inf 24.1%

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

       1. frac-2neg 24.1%

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

       2. div-inv 24.1%

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

       3. distribute-rgt-neg-in 24.1%

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

       4. mul-1-neg 24.1%

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

       5. add-sqr-sqrt 24.1%

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

       6. sqrt-unprod 26.8%

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

       7. mul-1-neg 26.8%

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

       8. mul-1-neg 26.8%

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

       9. sqr-neg 26.8%

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

       10. sqrt-unprod 0.0%

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

       11. add-sqr-sqrt 24.9%

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

       12. *-commutative 24.9%

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

       13. metadata-eval 24.9%

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

       14. metadata-eval 24.9%

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

   11. Applied egg-rr 24.9%

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

   if -1.34999999999999999e-6 < x < 350

   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 76.2%

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

   if 350 < x < 3.00000000000000011e161 or 5.3000000000000001e241 < 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 21.1%

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

   6. Taylor expanded in x around 0 54.7%

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

   if 3.00000000000000011e161 < x < 5.3000000000000001e241

   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 53.9%

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

   6. Taylor expanded in x around 0 33.3%

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

      1. mul-1-neg 33.3%

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

      2. unsub-neg 33.3%

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

      3. *-commutative 33.3%

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

      4. associate-*r* 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in eps around 0 38.2%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+161} \lor \neg \left(x \leq 5.3 \cdot 10^{+241}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 + x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.4% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + eps\_m\right) \cdot \left(-1 + \frac{1}{eps\_m}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+161} \lor \neg \left(x \leq 5.2 \cdot 10^{+233}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(2 + x \cdot eps\_m\right) - x}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e-6)
   (/ (+ 2.0 (* x (* (+ 1.0 eps_m) (+ -1.0 (/ 1.0 eps_m))))) 2.0)
   (if (<= x 350.0)
     1.0
     (if (or (<= x 2.4e+161) (not (<= x 5.2e+233)))
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
       (/ (/ (- (* eps_m (+ 2.0 (* x eps_m))) x) eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0;
	} else if (x <= 350.0) {
		tmp = 1.0;
	} else if ((x <= 2.4e+161) || !(x <= 5.2e+233)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.35d-6)) then
        tmp = (2.0d0 + (x * ((1.0d0 + eps_m) * ((-1.0d0) + (1.0d0 / eps_m))))) / 2.0d0
    else if (x <= 350.0d0) then
        tmp = 1.0d0
    else if ((x <= 2.4d+161) .or. (.not. (x <= 5.2d+233))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (((eps_m * (2.0d0 + (x * eps_m))) - x) / eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0;
	} else if (x <= 350.0) {
		tmp = 1.0;
	} else if ((x <= 2.4e+161) || !(x <= 5.2e+233)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.35e-6:
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0
	elif x <= 350.0:
		tmp = 1.0
	elif (x <= 2.4e+161) or not (x <= 5.2e+233):
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e-6)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 + eps_m) * Float64(-1.0 + Float64(1.0 / eps_m))))) / 2.0);
	elseif (x <= 350.0)
		tmp = 1.0;
	elseif ((x <= 2.4e+161) || !(x <= 5.2e+233))
		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(Float64(Float64(eps_m * Float64(2.0 + Float64(x * eps_m))) - x) / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.35e-6)
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0;
	elseif (x <= 350.0)
		tmp = 1.0;
	elseif ((x <= 2.4e+161) || ~((x <= 5.2e+233)))
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e-6], N[(N[(2.0 + N[(x * N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 350.0], 1.0, If[Or[LessEqual[x, 2.4e+161], N[Not[LessEqual[x, 5.2e+233]], $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[(N[(N[(eps$95$m * N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.34999999999999999e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 52.9%

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

   6. Taylor expanded in x around 0 24.8%

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

   if -1.34999999999999999e-6 < x < 350

   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 76.2%

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

   if 350 < x < 2.3999999999999999e161 or 5.20000000000000013e233 < 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 21.1%

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

   6. Taylor expanded in x around 0 54.7%

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

   if 2.3999999999999999e161 < x < 5.20000000000000013e233

   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 53.9%

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

   6. Taylor expanded in x around 0 33.3%

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

      1. mul-1-neg 33.3%

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

      2. unsub-neg 33.3%

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

      3. *-commutative 33.3%

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

      4. associate-*r* 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in eps around 0 38.2%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+161} \lor \neg \left(x \leq 5.2 \cdot 10^{+233}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 + x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+161} \lor \neg \left(x \leq 1.92 \cdot 10^{+233}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \frac{-1}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e-6)
   (* (* x eps_m) -0.5)
   (if (<= x 350.0)
     1.0
     (if (or (<= x 3.3e+161) (not (<= x 1.92e+233)))
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (/ -1.0 eps_m)) 2.0)
       (/ (* x eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 350.0) {
		tmp = 1.0;
	} else if ((x <= 3.3e+161) || !(x <= 1.92e+233)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (-1.0 / eps_m)) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.35d-6)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 350.0d0) then
        tmp = 1.0d0
    else if ((x <= 3.3d+161) .or. (.not. (x <= 1.92d+233))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + ((-1.0d0) / eps_m)) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 350.0) {
		tmp = 1.0;
	} else if ((x <= 3.3e+161) || !(x <= 1.92e+233)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (-1.0 / eps_m)) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.35e-6:
		tmp = (x * eps_m) * -0.5
	elif x <= 350.0:
		tmp = 1.0
	elif (x <= 3.3e+161) or not (x <= 1.92e+233):
		tmp = ((1.0 + (1.0 / eps_m)) + (-1.0 / eps_m)) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e-6)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 350.0)
		tmp = 1.0;
	elseif ((x <= 3.3e+161) || !(x <= 1.92e+233))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(-1.0 / eps_m)) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.35e-6)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 350.0)
		tmp = 1.0;
	elseif ((x <= 3.3e+161) || ~((x <= 1.92e+233)))
		tmp = ((1.0 + (1.0 / eps_m)) + (-1.0 / eps_m)) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e-6], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 350.0], 1.0, If[Or[LessEqual[x, 3.3e+161], N[Not[LessEqual[x, 1.92e+233]], $MachinePrecision]], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.34999999999999999e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 50.2%

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

   6. Taylor expanded in x around 0 24.2%

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

      1. mul-1-neg 24.2%

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

      2. unsub-neg 24.2%

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

      3. *-commutative 24.2%

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

      4. associate-*r* 24.2%

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

   8. Simplified 24.2%

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

   9. Taylor expanded in eps around inf 24.1%

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

       1. frac-2neg 24.1%

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

       2. div-inv 24.1%

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

       3. distribute-rgt-neg-in 24.1%

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

       4. mul-1-neg 24.1%

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

       5. add-sqr-sqrt 24.1%

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

       6. sqrt-unprod 26.8%

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

       7. mul-1-neg 26.8%

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

       8. mul-1-neg 26.8%

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

       9. sqr-neg 26.8%

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

       10. sqrt-unprod 0.0%

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

       11. add-sqr-sqrt 24.9%

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

       12. *-commutative 24.9%

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

       13. metadata-eval 24.9%

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

       14. metadata-eval 24.9%

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

   11. Applied egg-rr 24.9%

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

   if -1.34999999999999999e-6 < x < 350

   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 76.2%

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

   if 350 < x < 3.29999999999999997e161 or 1.9200000000000001e233 < 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 21.1%

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

   6. Taylor expanded in x around 0 54.7%

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

   7. Taylor expanded in eps around 0 54.7%

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

   if 3.29999999999999997e161 < x < 1.9200000000000001e233

   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 53.9%

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

   6. Taylor expanded in x around 0 33.3%

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

      1. mul-1-neg 33.3%

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

      2. unsub-neg 33.3%

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

      3. *-commutative 33.3%

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

      4. associate-*r* 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in eps around inf 33.7%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+161} \lor \neg \left(x \leq 1.92 \cdot 10^{+233}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \frac{-1}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + eps\_m\right) \cdot \left(-1 + \frac{1}{eps\_m}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - x \cdot \left(0.5 - x \cdot 0.16666666666666666\right)\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e-6)
   (/ (+ 2.0 (* x (* (+ 1.0 eps_m) (+ -1.0 (/ 1.0 eps_m))))) 2.0)
   (if (<= x 350.0)
     1.0
     (if (<= x 2.4e+161)
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
       (/
        (/ (* x (- 1.0 (* x (- 0.5 (* x 0.16666666666666666))))) eps_m)
        2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0;
	} else if (x <= 350.0) {
		tmp = 1.0;
	} else if (x <= 2.4e+161) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = ((x * (1.0 - (x * (0.5 - (x * 0.16666666666666666))))) / 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 <= (-1.35d-6)) then
        tmp = (2.0d0 + (x * ((1.0d0 + eps_m) * ((-1.0d0) + (1.0d0 / eps_m))))) / 2.0d0
    else if (x <= 350.0d0) then
        tmp = 1.0d0
    else if (x <= 2.4d+161) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = ((x * (1.0d0 - (x * (0.5d0 - (x * 0.16666666666666666d0))))) / 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 <= -1.35e-6) {
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0;
	} else if (x <= 350.0) {
		tmp = 1.0;
	} else if (x <= 2.4e+161) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = ((x * (1.0 - (x * (0.5 - (x * 0.16666666666666666))))) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.35e-6:
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0
	elif x <= 350.0:
		tmp = 1.0
	elif x <= 2.4e+161:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = ((x * (1.0 - (x * (0.5 - (x * 0.16666666666666666))))) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e-6)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 + eps_m) * Float64(-1.0 + Float64(1.0 / eps_m))))) / 2.0);
	elseif (x <= 350.0)
		tmp = 1.0;
	elseif (x <= 2.4e+161)
		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(Float64(x * Float64(1.0 - Float64(x * Float64(0.5 - Float64(x * 0.16666666666666666))))) / 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 <= -1.35e-6)
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0;
	elseif (x <= 350.0)
		tmp = 1.0;
	elseif (x <= 2.4e+161)
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = ((x * (1.0 - (x * (0.5 - (x * 0.16666666666666666))))) / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e-6], N[(N[(2.0 + N[(x * N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 350.0], 1.0, If[LessEqual[x, 2.4e+161], 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[(N[(x * N[(1.0 - N[(x * N[(0.5 - N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.34999999999999999e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 52.9%

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

   6. Taylor expanded in x around 0 24.8%

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

   if -1.34999999999999999e-6 < x < 350

   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 76.2%

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

   if 350 < x < 2.3999999999999999e161

   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 22.6%

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

   6. Taylor expanded in x around 0 54.4%

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

   if 2.3999999999999999e161 < 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 27.0%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. mul-1-neg 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in x around 0 38.1%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(1 + x \cdot \left(0.16666666666666666 \cdot x - 0.5\right)\right)}}{\varepsilon}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 - x \cdot \left(0.5 - x \cdot 0.16666666666666666\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.4% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 145:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e-6)
   (* (* x eps_m) -0.5)
   (if (<= x 145.0) 1.0 (/ (* x eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 145.0) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.35d-6)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 145.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 145.0) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.35e-6:
		tmp = (x * eps_m) * -0.5
	elif x <= 145.0:
		tmp = 1.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e-6)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 145.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.35e-6)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 145.0)
		tmp = 1.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e-6], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 145.0], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.34999999999999999e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 50.2%

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

   6. Taylor expanded in x around 0 24.2%

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

      1. mul-1-neg 24.2%

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

      2. unsub-neg 24.2%

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

      3. *-commutative 24.2%

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

      4. associate-*r* 24.2%

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

   8. Simplified 24.2%

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

   9. Taylor expanded in eps around inf 24.1%

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

       1. frac-2neg 24.1%

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

       2. div-inv 24.1%

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

       3. distribute-rgt-neg-in 24.1%

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

       4. mul-1-neg 24.1%

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

       5. add-sqr-sqrt 24.1%

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

       6. sqrt-unprod 26.8%

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

       7. mul-1-neg 26.8%

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

       8. mul-1-neg 26.8%

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

       9. sqr-neg 26.8%

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

       10. sqrt-unprod 0.0%

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

       11. add-sqr-sqrt 24.9%

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

       12. *-commutative 24.9%

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

       13. metadata-eval 24.9%

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

       14. metadata-eval 24.9%

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

   11. Applied egg-rr 24.9%

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

   if -1.34999999999999999e-6 < x < 145

   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 76.2%

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

   if 145 < 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 30.6%

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

   6. Taylor expanded in x around 0 16.9%

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

      1. mul-1-neg 16.9%

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

      2. unsub-neg 16.9%

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

      3. *-commutative 16.9%

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

      4. associate-*r* 16.9%

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

   8. Simplified 16.9%

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

   9. Taylor expanded in eps around inf 17.7%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 145:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.0% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e-6) (* (* x eps_m) -0.5) (/ (+ 2.0 (* x eps_m)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (x * eps_m) * -0.5;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.35d-6)) then
        tmp = (x * eps_m) * (-0.5d0)
    else
        tmp = (2.0d0 + (x * eps_m)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (x * eps_m) * -0.5;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.35e-6:
		tmp = (x * eps_m) * -0.5
	else:
		tmp = (2.0 + (x * eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e-6)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.35e-6)
		tmp = (x * eps_m) * -0.5;
	else
		tmp = (2.0 + (x * eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e-6], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.34999999999999999e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 50.2%

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

   6. Taylor expanded in x around 0 24.2%

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

      1. mul-1-neg 24.2%

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

      2. unsub-neg 24.2%

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

      3. *-commutative 24.2%

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

      4. associate-*r* 24.2%

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

   8. Simplified 24.2%

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

   9. Taylor expanded in eps around inf 24.1%

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

       1. frac-2neg 24.1%

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

       2. div-inv 24.1%

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

       3. distribute-rgt-neg-in 24.1%

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

       4. mul-1-neg 24.1%

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

       5. add-sqr-sqrt 24.1%

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

       6. sqrt-unprod 26.8%

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

       7. mul-1-neg 26.8%

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

       8. mul-1-neg 26.8%

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

       9. sqr-neg 26.8%

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

       10. sqrt-unprod 0.0%

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

       11. add-sqr-sqrt 24.9%

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

       12. *-commutative 24.9%

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

       13. metadata-eval 24.9%

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

       14. metadata-eval 24.9%

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

   11. Applied egg-rr 24.9%

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

   if -1.34999999999999999e-6 < x

   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 37.3%

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

   6. Taylor expanded in x around 0 38.9%

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

      1. mul-1-neg 38.9%

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

      2. unsub-neg 38.9%

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

      3. *-commutative 38.9%

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

      4. associate-*r* 38.9%

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

   8. Simplified 38.9%

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

   9. Taylor expanded in eps around inf 58.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.9% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e-6) (* (* x eps_m) -0.5) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (x * eps_m) * -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.35d-6)) then
        tmp = (x * eps_m) * (-0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-6) {
		tmp = (x * eps_m) * -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.35e-6:
		tmp = (x * eps_m) * -0.5
	else:
		tmp = 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e-6)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.35e-6)
		tmp = (x * eps_m) * -0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e-6], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.34999999999999999e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 50.2%

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

   6. Taylor expanded in x around 0 24.2%

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

      1. mul-1-neg 24.2%

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

      2. unsub-neg 24.2%

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

      3. *-commutative 24.2%

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

      4. associate-*r* 24.2%

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

   8. Simplified 24.2%

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

   9. Taylor expanded in eps around inf 24.1%

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

       1. frac-2neg 24.1%

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

       2. div-inv 24.1%

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

       3. distribute-rgt-neg-in 24.1%

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

       4. mul-1-neg 24.1%

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

       5. add-sqr-sqrt 24.1%

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

       6. sqrt-unprod 26.8%

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

       7. mul-1-neg 26.8%

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

       8. mul-1-neg 26.8%

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

       9. sqr-neg 26.8%

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

       10. sqrt-unprod 0.0%

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

       11. add-sqr-sqrt 24.9%

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

       12. *-commutative 24.9%

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

       13. metadata-eval 24.9%

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

       14. metadata-eval 24.9%

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

   11. Applied egg-rr 24.9%

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

   if -1.34999999999999999e-6 < x

   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 54.4%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.4% 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 71.2%

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

3. Simplified 71.2%

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

5. Taylor expanded in x around 0 47.4%

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

6. Final simplification 47.4%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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))