NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 99.1%

Time: 13.2s

Alternatives: 14

Speedup: 2.0×

• 38.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.8% 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: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\frac{e^{x \cdot \left(-eps\_m\right)} + e^{x \cdot eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.4)
   (/ (+ (exp (* x (- eps_m))) (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 <= 2.4) {
		tmp = (exp((x * -eps_m)) + 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 <= 2.4d0) then
        tmp = (exp((x * -eps_m)) + 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 <= 2.4) {
		tmp = (Math.exp((x * -eps_m)) + 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 <= 2.4:
		tmp = (math.exp((x * -eps_m)) + 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 <= 2.4)
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps_m))) + exp(Float64(x * 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 <= 2.4)
		tmp = (exp((x * -eps_m)) + 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, 2.4], N[(N[(N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision] + 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 < 2.39999999999999991

   1. Initial program 63.0%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in eps around inf 98.3%

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

   6. Taylor expanded in eps around inf 98.3%

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

      1. associate-*r* 98.3%

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

      2. neg-mul-1 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in eps around inf 99.0%

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

       1. *-commutative 99.0%

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

   11. Simplified 99.0%

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

   if 2.39999999999999991 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 73.0%

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

      1. neg-mul-1 73.0%

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

   8. Simplified 73.0%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{e^{x \cdot \left(-eps\_m\right)} + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -5e-308)
     (/ (+ (exp (* x (- eps_m))) t_0) 2.0)
     (/ (+ (exp (* x (+ -1.0 eps_m))) t_0) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -5e-308) {
		tmp = (exp((x * -eps_m)) + t_0) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) + t_0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-5d-308)) then
        tmp = (exp((x * -eps_m)) + t_0) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps_m))) + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (x <= -5e-308) {
		tmp = (Math.exp((x * -eps_m)) + t_0) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + t_0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -5e-308:
		tmp = (math.exp((x * -eps_m)) + t_0) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) + t_0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -5e-308)
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps_m))) + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + t_0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (x <= -5e-308)
		tmp = (exp((x * -eps_m)) + t_0) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -5e-308], N[(N[(N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999955e-308

   1. Initial program 68.0%

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

   3. Simplified 50.9%

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

   5. Taylor expanded in eps around inf 97.6%

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

   6. Taylor expanded in eps around inf 97.6%

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

      1. associate-*r* 97.6%

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

      2. neg-mul-1 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in eps around 0 85.5%

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

   if -4.99999999999999955e-308 < x

   1. Initial program 75.1%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in eps around inf 99.6%

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

   6. Taylor expanded in eps around 0 80.8%

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

      1. neg-mul-1 80.8%

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

   8. Simplified 80.8%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{1 + e^{x - x \cdot eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-308)
   (/ (+ 1.0 (exp (- x (* 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 <= -5e-308) {
		tmp = (1.0 + exp((x - (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 <= (-5d-308)) then
        tmp = (1.0d0 + exp((x - (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 <= -5e-308) {
		tmp = (1.0 + Math.exp((x - (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 <= -5e-308:
		tmp = (1.0 + math.exp((x - (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 <= -5e-308)
		tmp = Float64(Float64(1.0 + exp(Float64(x - Float64(x * 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 <= -5e-308)
		tmp = (1.0 + exp((x - (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, -5e-308], N[(N[(1.0 + N[Exp[N[(x - N[(x * eps$95$m), $MachinePrecision]), $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 < -4.99999999999999955e-308

   1. Initial program 68.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in x around 0 46.3%

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

      1. *-commutative 46.3%

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

      2. sub-neg 46.3%

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

      3. distribute-rgt-in 46.3%

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

      4. *-un-lft-identity 46.3%

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

      5. add-sqr-sqrt 46.3%

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

      6. sqrt-unprod 47.8%

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

      7. sqr-neg 47.8%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 46.4%

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

      10. add-sqr-sqrt 46.4%

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

      11. sqrt-unprod 45.5%

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

      12. sqr-neg 45.5%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 40.4%

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

   7. Applied egg-rr 40.4%

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

   8. Taylor expanded in eps around inf 69.9%

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

      1. neg-mul-1 69.9%

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

      2. sub-neg 69.9%

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

   10. Simplified 69.9%

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

   if -4.99999999999999955e-308 < x

   1. Initial program 75.1%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in eps around inf 99.6%

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

   6. Taylor expanded in eps around 0 80.8%

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

      1. neg-mul-1 80.8%

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

   8. Simplified 80.8%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{1 + e^{x - x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% 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 72.0%

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

3. Simplified 56.0%

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

5. Taylor expanded in eps around inf 98.7%

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

6. Final simplification 98.7%

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

Alternative 5: 69.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-7}:\\ \;\;\;\;eps\_m \cdot \left(1 + eps\_m \cdot \left(-1 + eps\_m\right)\right)\\ \mathbf{elif}\;x \leq 680:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+185}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.32e+66)
   (/
    (/
     (*
      x
      (+
       -1.0
       (* x (+ 0.5 (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
     eps_m)
    2.0)
   (if (<= x -9e-7)
     (* eps_m (+ 1.0 (* eps_m (+ -1.0 eps_m))))
     (if (<= x 680.0)
       1.0
       (if (<= x 8.8e+33)
         (/ (/ (expm1 x) eps_m) 2.0)
         (if (<= x 2.3e+185)
           (/ (+ (+ 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 <= -1.32e+66) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	} else if (x <= -9e-7) {
		tmp = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	} else if (x <= 680.0) {
		tmp = 1.0;
	} else if (x <= 8.8e+33) {
		tmp = (expm1(x) / eps_m) / 2.0;
	} else if (x <= 2.3e+185) {
		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;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.32e+66) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	} else if (x <= -9e-7) {
		tmp = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	} else if (x <= 680.0) {
		tmp = 1.0;
	} else if (x <= 8.8e+33) {
		tmp = (Math.expm1(x) / eps_m) / 2.0;
	} else if (x <= 2.3e+185) {
		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 <= -1.32e+66:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0
	elif x <= -9e-7:
		tmp = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)))
	elif x <= 680.0:
		tmp = 1.0
	elif x <= 8.8e+33:
		tmp = (math.expm1(x) / eps_m) / 2.0
	elif x <= 2.3e+185:
		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 <= -1.32e+66)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0);
	elseif (x <= -9e-7)
		tmp = Float64(eps_m * Float64(1.0 + Float64(eps_m * Float64(-1.0 + eps_m))));
	elseif (x <= 680.0)
		tmp = 1.0;
	elseif (x <= 8.8e+33)
		tmp = Float64(Float64(expm1(x) / eps_m) / 2.0);
	elseif (x <= 2.3e+185)
		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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.32e+66], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -9e-7], N[(eps$95$m * N[(1.0 + N[(eps$95$m * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 680.0], 1.0, If[LessEqual[x, 8.8e+33], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.3e+185], 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 6 regimes

2. if x < -1.32000000000000009e66

   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 56.6%

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

   6. Taylor expanded in eps around 0 44.8%

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

      1. expm1-define 44.8%

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

      2. neg-mul-1 44.8%

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

   8. Simplified 44.8%

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

   9. Taylor expanded in x around 0 44.8%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}}{\varepsilon}}{2} \]

   if -1.32000000000000009e66 < x < -8.99999999999999959e-7

   1. Initial program 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in x around 0 22.5%

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

   6. Taylor expanded in x around 0 3.0%

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

      1. add-sqr-sqrt 2.3%

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

      2. sqrt-unprod 2.8%

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

      3. frac-times 2.8%

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

      4. metadata-eval 2.8%

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

      5. metadata-eval 2.8%

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

      6. frac-times 2.8%

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

      7. sqrt-unprod 0.8%

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

      8. add-sqr-sqrt 3.1%

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

      9. div-inv 3.1%

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

      10. mul-1-neg 3.1%

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

      11. sub-neg 3.1%

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

      12. flip-- 3.1%

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

   8. Applied egg-rr 3.0%

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

   9. Taylor expanded in eps around 0 41.9%

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

   if -8.99999999999999959e-7 < x < 680

   1. Initial program 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in x around 0 44.5%

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

   6. Taylor expanded in x around 0 32.6%

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

   7. Taylor expanded in eps around 0 77.6%

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

   if 680 < x < 8.79999999999999975e33

   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 44.5%

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

      1. *-commutative 44.5%

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

      2. sub-neg 44.5%

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

      3. distribute-rgt-in 44.5%

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

      4. *-un-lft-identity 44.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 44.4%

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

      7. sqr-neg 44.4%

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

      8. sqrt-unprod 44.4%

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

      9. add-sqr-sqrt 44.4%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 44.4%

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

      12. sqr-neg 44.4%

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

      13. sqrt-unprod 44.4%

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

      14. add-sqr-sqrt 44.4%

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

   7. Applied egg-rr 44.4%

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

   8. Taylor expanded in eps around 0 43.1%

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

      1. expm1-define 43.1%

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

   10. Simplified 43.1%

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

   if 8.79999999999999975e33 < x < 2.3000000000000001e185

   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 24.4%

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

   6. Taylor expanded in x around 0 65.6%

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

   if 2.3000000000000001e185 < 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 31.2%

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

      1. *-commutative 31.2%

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

      2. sub-neg 31.2%

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

      3. distribute-rgt-in 31.2%

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

      4. *-un-lft-identity 31.2%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 37.9%

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

      7. sqr-neg 37.9%

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

      8. sqrt-unprod 31.0%

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

      9. add-sqr-sqrt 31.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 47.2%

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

      12. sqr-neg 47.2%

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

      13. sqrt-unprod 47.1%

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

      14. add-sqr-sqrt 47.1%

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

   7. Applied egg-rr 47.1%

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

   8. Taylor expanded in eps around 0 29.6%

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

      1. expm1-define 29.6%

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

   10. Simplified 29.6%

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

   11. Taylor expanded in x around 0 29.6%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)}}{\varepsilon}}{2} \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \varepsilon \cdot \left(-1 + \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 680:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+185}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{1 + e^{x - x \cdot eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{+33} \lor \neg \left(x \leq 7 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-308)
   (/ (+ 1.0 (exp (- x (* x eps_m)))) 2.0)
   (if (or (<= x 1.08e+33) (not (<= x 7e+184)))
     (/ (+ 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 <= -2e-308) {
		tmp = (1.0 + exp((x - (x * eps_m)))) / 2.0;
	} else if ((x <= 1.08e+33) || !(x <= 7e+184)) {
		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 <= (-2d-308)) then
        tmp = (1.0d0 + exp((x - (x * eps_m)))) / 2.0d0
    else if ((x <= 1.08d+33) .or. (.not. (x <= 7d+184))) 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 <= -2e-308) {
		tmp = (1.0 + Math.exp((x - (x * eps_m)))) / 2.0;
	} else if ((x <= 1.08e+33) || !(x <= 7e+184)) {
		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 <= -2e-308:
		tmp = (1.0 + math.exp((x - (x * eps_m)))) / 2.0
	elif (x <= 1.08e+33) or not (x <= 7e+184):
		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 <= -2e-308)
		tmp = Float64(Float64(1.0 + exp(Float64(x - Float64(x * eps_m)))) / 2.0);
	elseif ((x <= 1.08e+33) || !(x <= 7e+184))
		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 <= -2e-308)
		tmp = (1.0 + exp((x - (x * eps_m)))) / 2.0;
	elseif ((x <= 1.08e+33) || ~((x <= 7e+184)))
		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, -2e-308], N[(N[(1.0 + N[Exp[N[(x - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.08e+33], N[Not[LessEqual[x, 7e+184]], $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 < -1.9999999999999998e-308

   1. Initial program 68.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in x around 0 46.3%

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

      1. *-commutative 46.3%

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

      2. sub-neg 46.3%

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

      3. distribute-rgt-in 46.3%

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

      4. *-un-lft-identity 46.3%

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

      5. add-sqr-sqrt 46.3%

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

      6. sqrt-unprod 47.8%

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

      7. sqr-neg 47.8%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 46.4%

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

      10. add-sqr-sqrt 46.4%

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

      11. sqrt-unprod 45.5%

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

      12. sqr-neg 45.5%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 40.4%

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

   7. Applied egg-rr 40.4%

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

   8. Taylor expanded in eps around inf 69.9%

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

      1. neg-mul-1 69.9%

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

      2. sub-neg 69.9%

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

   10. Simplified 69.9%

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

   if -1.9999999999999998e-308 < x < 1.08000000000000005e33 or 6.99999999999999956e184 < x

   1. Initial program 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in x around 0 41.0%

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

   6. Taylor expanded in eps around inf 72.3%

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

      1. mul-1-neg 72.3%

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

      2. distribute-lft-neg-in 72.3%

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

   8. Simplified 72.3%

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

   if 1.08000000000000005e33 < x < 6.99999999999999956e184

   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 24.4%

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

   6. Taylor expanded in x around 0 65.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{1 + e^{x - x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{+33} \lor \neg \left(x \leq 7 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-268}:\\ \;\;\;\;\frac{1 + e^{x - x \cdot eps\_m}}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+33}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + \left(1 - x \cdot \left(1 + eps\_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+185}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.6e-268)
   (/ (+ 1.0 (exp (- x (* x eps_m)))) 2.0)
   (if (<= x 6e+33)
     (/ (+ (exp (* x eps_m)) (- 1.0 (* x (+ 1.0 eps_m)))) 2.0)
     (if (<= x 1.1e+185)
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)
       (/ (+ 1.0 (exp (* x (+ -1.0 eps_m)))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.6e-268) {
		tmp = (1.0 + exp((x - (x * eps_m)))) / 2.0;
	} else if (x <= 6e+33) {
		tmp = (exp((x * eps_m)) + (1.0 - (x * (1.0 + eps_m)))) / 2.0;
	} else if (x <= 1.1e+185) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.6d-268)) then
        tmp = (1.0d0 + exp((x - (x * eps_m)))) / 2.0d0
    else if (x <= 6d+33) then
        tmp = (exp((x * eps_m)) + (1.0d0 - (x * (1.0d0 + eps_m)))) / 2.0d0
    else if (x <= 1.1d+185) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.6e-268) {
		tmp = (1.0 + Math.exp((x - (x * eps_m)))) / 2.0;
	} else if (x <= 6e+33) {
		tmp = (Math.exp((x * eps_m)) + (1.0 - (x * (1.0 + eps_m)))) / 2.0;
	} else if (x <= 1.1e+185) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.6e-268:
		tmp = (1.0 + math.exp((x - (x * eps_m)))) / 2.0
	elif x <= 6e+33:
		tmp = (math.exp((x * eps_m)) + (1.0 - (x * (1.0 + eps_m)))) / 2.0
	elif x <= 1.1e+185:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.6e-268)
		tmp = Float64(Float64(1.0 + exp(Float64(x - Float64(x * eps_m)))) / 2.0);
	elseif (x <= 6e+33)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + Float64(1.0 - Float64(x * Float64(1.0 + eps_m)))) / 2.0);
	elseif (x <= 1.1e+185)
		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(1.0 + exp(Float64(x * Float64(-1.0 + eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.6e-268)
		tmp = (1.0 + exp((x - (x * eps_m)))) / 2.0;
	elseif (x <= 6e+33)
		tmp = (exp((x * eps_m)) + (1.0 - (x * (1.0 + eps_m)))) / 2.0;
	elseif (x <= 1.1e+185)
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.6e-268], N[(N[(1.0 + N[Exp[N[(x - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6e+33], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.1e+185], 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[(1.0 + N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.5999999999999999e-268

   1. Initial program 68.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in x around 0 44.2%

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

      1. *-commutative 44.2%

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

      2. sub-neg 44.2%

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

      3. distribute-rgt-in 44.2%

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

      4. *-un-lft-identity 44.2%

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

      5. add-sqr-sqrt 44.2%

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

      6. sqrt-unprod 45.9%

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

      7. sqr-neg 45.9%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 44.4%

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

      10. add-sqr-sqrt 44.4%

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

      11. sqrt-unprod 42.8%

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

      12. sqr-neg 42.8%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 37.6%

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

   7. Applied egg-rr 37.6%

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

   8. Taylor expanded in eps around inf 66.8%

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

      1. neg-mul-1 66.8%

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

      2. sub-neg 66.8%

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

   10. Simplified 66.8%

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

   if -1.5999999999999999e-268 < x < 5.99999999999999967e33

   1. Initial program 60.7%

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

   3. Simplified 32.3%

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

   5. Taylor expanded in eps around inf 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. Taylor expanded in x around 0 86.3%

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

      1. neg-mul-1 86.3%

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

      2. distribute-rgt-neg-in 86.3%

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

      3. distribute-neg-in 86.3%

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

      4. metadata-eval 86.3%

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

      5. unsub-neg 86.3%

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

   8. Simplified 86.3%

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

   9. Taylor expanded in eps around inf 86.3%

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

       1. *-commutative 99.0%

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

   11. Simplified 86.3%

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

   if 5.99999999999999967e33 < x < 1.1e185

   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 24.4%

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

   6. Taylor expanded in x around 0 65.6%

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

   if 1.1e185 < 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 31.2%

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

   6. Taylor expanded in eps around inf 31.4%

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

      1. mul-1-neg 31.4%

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

      2. distribute-lft-neg-in 31.4%

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

   8. Simplified 31.4%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-268}:\\ \;\;\;\;\frac{1 + e^{x - x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+33}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \left(1 - x \cdot \left(1 + \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+185}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5200000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+184}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5200000.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (<= x 600.0)
     1.0
     (if (<= x 5e+34)
       (/ (/ (expm1 x) eps_m) 2.0)
       (if (<= x 7e+184)
         (/ (+ (+ 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 <= -5200000.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= 600.0) {
		tmp = 1.0;
	} else if (x <= 5e+34) {
		tmp = (expm1(x) / eps_m) / 2.0;
	} else if (x <= 7e+184) {
		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;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -5200000.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= 600.0) {
		tmp = 1.0;
	} else if (x <= 5e+34) {
		tmp = (Math.expm1(x) / eps_m) / 2.0;
	} else if (x <= 7e+184) {
		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 <= -5200000.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= 600.0:
		tmp = 1.0
	elif x <= 5e+34:
		tmp = (math.expm1(x) / eps_m) / 2.0
	elif x <= 7e+184:
		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 <= -5200000.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 600.0)
		tmp = 1.0;
	elseif (x <= 5e+34)
		tmp = Float64(Float64(expm1(x) / eps_m) / 2.0);
	elseif (x <= 7e+184)
		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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5200000.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 600.0], 1.0, If[LessEqual[x, 5e+34], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7e+184], 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 5 regimes

2. if x < -5.2e6

   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 50.3%

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

   6. Taylor expanded in eps around 0 51.4%

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

      1. expm1-define 51.4%

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

      2. neg-mul-1 51.4%

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

   8. Simplified 51.4%

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

   if -5.2e6 < x < 600

   1. Initial program 54.3%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in x around 0 44.0%

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

   6. Taylor expanded in x around 0 32.3%

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

   7. Taylor expanded in eps around 0 76.6%

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

   if 600 < x < 4.9999999999999998e34

   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 44.5%

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

      1. *-commutative 44.5%

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

      2. sub-neg 44.5%

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

      3. distribute-rgt-in 44.5%

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

      4. *-un-lft-identity 44.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 44.4%

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

      7. sqr-neg 44.4%

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

      8. sqrt-unprod 44.4%

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

      9. add-sqr-sqrt 44.4%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 44.4%

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

      12. sqr-neg 44.4%

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

      13. sqrt-unprod 44.4%

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

      14. add-sqr-sqrt 44.4%

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

   7. Applied egg-rr 44.4%

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

   8. Taylor expanded in eps around 0 43.1%

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

      1. expm1-define 43.1%

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

   10. Simplified 43.1%

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

   if 4.9999999999999998e34 < x < 6.99999999999999956e184

   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 24.4%

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

   6. Taylor expanded in x around 0 65.6%

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

   if 6.99999999999999956e184 < 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 31.2%

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

      1. *-commutative 31.2%

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

      2. sub-neg 31.2%

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

      3. distribute-rgt-in 31.2%

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

      4. *-un-lft-identity 31.2%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 37.9%

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

      7. sqr-neg 37.9%

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

      8. sqrt-unprod 31.0%

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

      9. add-sqr-sqrt 31.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 47.2%

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

      12. sqr-neg 47.2%

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

      13. sqrt-unprod 47.1%

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

      14. add-sqr-sqrt 47.1%

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

   7. Applied egg-rr 47.1%

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

   8. Taylor expanded in eps around 0 29.6%

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

      1. expm1-define 29.6%

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

   10. Simplified 29.6%

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

   11. Taylor expanded in x around 0 29.6%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)}}{\varepsilon}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5200000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+184}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 440:\\ \;\;\;\;\frac{1 + e^{x - x \cdot eps\_m}}{2}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+184}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 440.0)
   (/ (+ 1.0 (exp (- x (* x eps_m)))) 2.0)
   (if (<= x 2.1e+34)
     (/ (/ (expm1 x) eps_m) 2.0)
     (if (<= x 9.5e+184)
       (/ (+ (+ 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 <= 440.0) {
		tmp = (1.0 + exp((x - (x * eps_m)))) / 2.0;
	} else if (x <= 2.1e+34) {
		tmp = (expm1(x) / eps_m) / 2.0;
	} else if (x <= 9.5e+184) {
		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;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 440.0) {
		tmp = (1.0 + Math.exp((x - (x * eps_m)))) / 2.0;
	} else if (x <= 2.1e+34) {
		tmp = (Math.expm1(x) / eps_m) / 2.0;
	} else if (x <= 9.5e+184) {
		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 <= 440.0:
		tmp = (1.0 + math.exp((x - (x * eps_m)))) / 2.0
	elif x <= 2.1e+34:
		tmp = (math.expm1(x) / eps_m) / 2.0
	elif x <= 9.5e+184:
		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 <= 440.0)
		tmp = Float64(Float64(1.0 + exp(Float64(x - Float64(x * eps_m)))) / 2.0);
	elseif (x <= 2.1e+34)
		tmp = Float64(Float64(expm1(x) / eps_m) / 2.0);
	elseif (x <= 9.5e+184)
		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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 440.0], N[(N[(1.0 + N[Exp[N[(x - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.1e+34], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.5e+184], 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 < 440

   1. Initial program 63.0%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in x around 0 45.2%

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

      1. *-commutative 45.2%

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

      2. sub-neg 45.2%

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

      3. distribute-rgt-in 45.2%

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

      4. *-un-lft-identity 45.2%

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

      5. add-sqr-sqrt 27.2%

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

      6. sqrt-unprod 46.0%

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

      7. sqr-neg 46.0%

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

      8. sqrt-unprod 17.9%

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

      9. add-sqr-sqrt 45.2%

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

      10. add-sqr-sqrt 27.3%

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

      11. sqrt-unprod 44.1%

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

      12. sqr-neg 44.1%

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

      13. sqrt-unprod 18.4%

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

      14. add-sqr-sqrt 42.2%

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

   7. Applied egg-rr 42.2%

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

   8. Taylor expanded in eps around inf 77.5%

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

      1. neg-mul-1 77.5%

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

      2. sub-neg 77.5%

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

   10. Simplified 77.5%

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

   if 440 < x < 2.10000000000000017e34

   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 44.5%

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

      1. *-commutative 44.5%

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

      2. sub-neg 44.5%

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

      3. distribute-rgt-in 44.5%

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

      4. *-un-lft-identity 44.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 44.4%

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

      7. sqr-neg 44.4%

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

      8. sqrt-unprod 44.4%

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

      9. add-sqr-sqrt 44.4%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 44.4%

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

      12. sqr-neg 44.4%

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

      13. sqrt-unprod 44.4%

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

      14. add-sqr-sqrt 44.4%

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

   7. Applied egg-rr 44.4%

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

   8. Taylor expanded in eps around 0 43.1%

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

      1. expm1-define 43.1%

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

   10. Simplified 43.1%

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

   if 2.10000000000000017e34 < x < 9.4999999999999995e184

   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 24.4%

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

   6. Taylor expanded in x around 0 65.6%

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

   if 9.4999999999999995e184 < 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 31.2%

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

      1. *-commutative 31.2%

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

      2. sub-neg 31.2%

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

      3. distribute-rgt-in 31.2%

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

      4. *-un-lft-identity 31.2%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 37.9%

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

      7. sqr-neg 37.9%

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

      8. sqrt-unprod 31.0%

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

      9. add-sqr-sqrt 31.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 47.2%

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

      12. sqr-neg 47.2%

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

      13. sqrt-unprod 47.1%

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

      14. add-sqr-sqrt 47.1%

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

   7. Applied egg-rr 47.1%

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

   8. Taylor expanded in eps around 0 29.6%

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

      1. expm1-define 29.6%

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

   10. Simplified 29.6%

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

   11. Taylor expanded in x around 0 29.6%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 440:\\ \;\;\;\;\frac{1 + e^{x - x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+184}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 - \frac{1}{eps\_m}\\ t_1 := eps\_m \cdot \left(1 + eps\_m \cdot \left(-1 + eps\_m\right)\right)\\ t_2 := 1 + \frac{1}{eps\_m}\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{2 - x \cdot \left(\left(1 + eps\_m\right) \cdot t\_0 - \left(-1 + eps\_m\right) \cdot t\_2\right)}{2}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+184}:\\ \;\;\;\;\frac{t\_2 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ 1.0 eps_m)))
        (t_1 (* eps_m (+ 1.0 (* eps_m (+ -1.0 eps_m)))))
        (t_2 (+ 1.0 (/ 1.0 eps_m))))
   (if (<= x -1.32e+66)
     (/
      (/
       (*
        x
        (+
         -1.0
         (*
          x
          (+ 0.5 (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
       eps_m)
      2.0)
     (if (<= x -6.6e-6)
       t_1
       (if (<= x 7.5e-105)
         (/ (- 2.0 (* x (- (* (+ 1.0 eps_m) t_0) (* (+ -1.0 eps_m) t_2)))) 2.0)
         (if (<= x 4.8e+34)
           t_1
           (if (<= x 8.8e+184)
             (/ (+ t_2 t_0) 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 t_0 = 1.0 - (1.0 / eps_m);
	double t_1 = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	double t_2 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -1.32e+66) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	} else if (x <= -6.6e-6) {
		tmp = t_1;
	} else if (x <= 7.5e-105) {
		tmp = (2.0 - (x * (((1.0 + eps_m) * t_0) - ((-1.0 + eps_m) * t_2)))) / 2.0;
	} else if (x <= 4.8e+34) {
		tmp = t_1;
	} else if (x <= 8.8e+184) {
		tmp = (t_2 + t_0) / 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (1.0d0 / eps_m)
    t_1 = eps_m * (1.0d0 + (eps_m * ((-1.0d0) + eps_m)))
    t_2 = 1.0d0 + (1.0d0 / eps_m)
    if (x <= (-1.32d+66)) then
        tmp = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / eps_m) / 2.0d0
    else if (x <= (-6.6d-6)) then
        tmp = t_1
    else if (x <= 7.5d-105) then
        tmp = (2.0d0 - (x * (((1.0d0 + eps_m) * t_0) - (((-1.0d0) + eps_m) * t_2)))) / 2.0d0
    else if (x <= 4.8d+34) then
        tmp = t_1
    else if (x <= 8.8d+184) then
        tmp = (t_2 + t_0) / 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 t_0 = 1.0 - (1.0 / eps_m);
	double t_1 = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	double t_2 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -1.32e+66) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	} else if (x <= -6.6e-6) {
		tmp = t_1;
	} else if (x <= 7.5e-105) {
		tmp = (2.0 - (x * (((1.0 + eps_m) * t_0) - ((-1.0 + eps_m) * t_2)))) / 2.0;
	} else if (x <= 4.8e+34) {
		tmp = t_1;
	} else if (x <= 8.8e+184) {
		tmp = (t_2 + t_0) / 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):
	t_0 = 1.0 - (1.0 / eps_m)
	t_1 = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)))
	t_2 = 1.0 + (1.0 / eps_m)
	tmp = 0
	if x <= -1.32e+66:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0
	elif x <= -6.6e-6:
		tmp = t_1
	elif x <= 7.5e-105:
		tmp = (2.0 - (x * (((1.0 + eps_m) * t_0) - ((-1.0 + eps_m) * t_2)))) / 2.0
	elif x <= 4.8e+34:
		tmp = t_1
	elif x <= 8.8e+184:
		tmp = (t_2 + t_0) / 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)
	t_0 = Float64(1.0 - Float64(1.0 / eps_m))
	t_1 = Float64(eps_m * Float64(1.0 + Float64(eps_m * Float64(-1.0 + eps_m))))
	t_2 = Float64(1.0 + Float64(1.0 / eps_m))
	tmp = 0.0
	if (x <= -1.32e+66)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0);
	elseif (x <= -6.6e-6)
		tmp = t_1;
	elseif (x <= 7.5e-105)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(Float64(Float64(1.0 + eps_m) * t_0) - Float64(Float64(-1.0 + eps_m) * t_2)))) / 2.0);
	elseif (x <= 4.8e+34)
		tmp = t_1;
	elseif (x <= 8.8e+184)
		tmp = Float64(Float64(t_2 + t_0) / 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)
	t_0 = 1.0 - (1.0 / eps_m);
	t_1 = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	t_2 = 1.0 + (1.0 / eps_m);
	tmp = 0.0;
	if (x <= -1.32e+66)
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	elseif (x <= -6.6e-6)
		tmp = t_1;
	elseif (x <= 7.5e-105)
		tmp = (2.0 - (x * (((1.0 + eps_m) * t_0) - ((-1.0 + eps_m) * t_2)))) / 2.0;
	elseif (x <= 4.8e+34)
		tmp = t_1;
	elseif (x <= 8.8e+184)
		tmp = (t_2 + t_0) / 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_] := Block[{t$95$0 = N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps$95$m * N[(1.0 + N[(eps$95$m * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.32e+66], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -6.6e-6], t$95$1, If[LessEqual[x, 7.5e-105], N[(N[(2.0 - N[(x * N[(N[(N[(1.0 + eps$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[(-1.0 + eps$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.8e+34], t$95$1, If[LessEqual[x, 8.8e+184], N[(N[(t$95$2 + t$95$0), $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 5 regimes

2. if x < -1.32000000000000009e66

   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 56.6%

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

   6. Taylor expanded in eps around 0 44.8%

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

      1. expm1-define 44.8%

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

      2. neg-mul-1 44.8%

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

   8. Simplified 44.8%

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

   9. Taylor expanded in x around 0 44.8%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}}{\varepsilon}}{2} \]

   if -1.32000000000000009e66 < x < -6.60000000000000034e-6 or 7.5000000000000006e-105 < x < 4.79999999999999974e34

   1. Initial program 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in x around 0 36.3%

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

   6. Taylor expanded in x around 0 11.4%

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

      1. add-sqr-sqrt 7.2%

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

      2. sqrt-unprod 8.7%

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

      3. frac-times 8.7%

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

      4. metadata-eval 8.7%

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

      5. metadata-eval 8.7%

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

      6. frac-times 8.7%

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

      7. sqrt-unprod 1.7%

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

      8. add-sqr-sqrt 7.9%

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

      9. div-inv 7.9%

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

      10. mul-1-neg 7.9%

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

      11. sub-neg 7.9%

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

      12. flip-- 7.8%

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

   8. Applied egg-rr 7.5%

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

   9. Taylor expanded in eps around 0 30.5%

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

   if -6.60000000000000034e-6 < x < 7.5000000000000006e-105

   1. Initial program 53.3%

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

   3. Simplified 24.6%

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

   5. Taylor expanded in x around 0 81.7%

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

   if 4.79999999999999974e34 < x < 8.8e184

   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 24.4%

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

   6. Taylor expanded in x around 0 65.6%

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

   if 8.8e184 < 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 31.2%

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

      1. *-commutative 31.2%

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

      2. sub-neg 31.2%

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

      3. distribute-rgt-in 31.2%

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

      4. *-un-lft-identity 31.2%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 37.9%

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

      7. sqr-neg 37.9%

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

      8. sqrt-unprod 31.0%

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

      9. add-sqr-sqrt 31.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 47.2%

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

      12. sqr-neg 47.2%

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

      13. sqrt-unprod 47.1%

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

      14. add-sqr-sqrt 47.1%

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

   7. Applied egg-rr 47.1%

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

   8. Taylor expanded in eps around 0 29.6%

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

      1. expm1-define 29.6%

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

   10. Simplified 29.6%

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

   11. Taylor expanded in x around 0 29.6%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)}}{\varepsilon}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \varepsilon \cdot \left(-1 + \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{2 - x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 - \frac{1}{\varepsilon}\right) - \left(-1 + \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+34}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \varepsilon \cdot \left(-1 + \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+184}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := eps\_m \cdot \left(1 + eps\_m \cdot \left(-1 + eps\_m\right)\right)\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+185}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* eps_m (+ 1.0 (* eps_m (+ -1.0 eps_m))))))
   (if (<= x -1.32e+66)
     (/
      (/
       (*
        x
        (+
         -1.0
         (*
          x
          (+ 0.5 (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
       eps_m)
      2.0)
     (if (<= x -5.8e-8)
       t_0
       (if (<= x 2.1e-103)
         1.0
         (if (<= x 1.12e+32)
           t_0
           (if (<= x 1.1e+185)
             (/ (+ (+ 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 t_0 = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	double tmp;
	if (x <= -1.32e+66) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	} else if (x <= -5.8e-8) {
		tmp = t_0;
	} else if (x <= 2.1e-103) {
		tmp = 1.0;
	} else if (x <= 1.12e+32) {
		tmp = t_0;
	} else if (x <= 1.1e+185) {
		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) :: t_0
    real(8) :: tmp
    t_0 = eps_m * (1.0d0 + (eps_m * ((-1.0d0) + eps_m)))
    if (x <= (-1.32d+66)) then
        tmp = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / eps_m) / 2.0d0
    else if (x <= (-5.8d-8)) then
        tmp = t_0
    else if (x <= 2.1d-103) then
        tmp = 1.0d0
    else if (x <= 1.12d+32) then
        tmp = t_0
    else if (x <= 1.1d+185) 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 t_0 = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	double tmp;
	if (x <= -1.32e+66) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	} else if (x <= -5.8e-8) {
		tmp = t_0;
	} else if (x <= 2.1e-103) {
		tmp = 1.0;
	} else if (x <= 1.12e+32) {
		tmp = t_0;
	} else if (x <= 1.1e+185) {
		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):
	t_0 = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)))
	tmp = 0
	if x <= -1.32e+66:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0
	elif x <= -5.8e-8:
		tmp = t_0
	elif x <= 2.1e-103:
		tmp = 1.0
	elif x <= 1.12e+32:
		tmp = t_0
	elif x <= 1.1e+185:
		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)
	t_0 = Float64(eps_m * Float64(1.0 + Float64(eps_m * Float64(-1.0 + eps_m))))
	tmp = 0.0
	if (x <= -1.32e+66)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0);
	elseif (x <= -5.8e-8)
		tmp = t_0;
	elseif (x <= 2.1e-103)
		tmp = 1.0;
	elseif (x <= 1.12e+32)
		tmp = t_0;
	elseif (x <= 1.1e+185)
		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)
	t_0 = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	tmp = 0.0;
	if (x <= -1.32e+66)
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	elseif (x <= -5.8e-8)
		tmp = t_0;
	elseif (x <= 2.1e-103)
		tmp = 1.0;
	elseif (x <= 1.12e+32)
		tmp = t_0;
	elseif (x <= 1.1e+185)
		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_] := Block[{t$95$0 = N[(eps$95$m * N[(1.0 + N[(eps$95$m * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.32e+66], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -5.8e-8], t$95$0, If[LessEqual[x, 2.1e-103], 1.0, If[LessEqual[x, 1.12e+32], t$95$0, If[LessEqual[x, 1.1e+185], 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 5 regimes

2. if x < -1.32000000000000009e66

   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 56.6%

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

   6. Taylor expanded in eps around 0 44.8%

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

      1. expm1-define 44.8%

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

      2. neg-mul-1 44.8%

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

   8. Simplified 44.8%

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

   9. Taylor expanded in x around 0 44.8%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}}{\varepsilon}}{2} \]

   if -1.32000000000000009e66 < x < -5.8000000000000003e-8 or 2.10000000000000005e-103 < x < 1.12000000000000007e32

   1. Initial program 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in x around 0 36.3%

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

   6. Taylor expanded in x around 0 11.4%

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

      1. add-sqr-sqrt 7.2%

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

      2. sqrt-unprod 8.7%

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

      3. frac-times 8.7%

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

      4. metadata-eval 8.7%

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

      5. metadata-eval 8.7%

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

      6. frac-times 8.7%

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

      7. sqrt-unprod 1.7%

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

      8. add-sqr-sqrt 7.9%

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

      9. div-inv 7.9%

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

      10. mul-1-neg 7.9%

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

      11. sub-neg 7.9%

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

      12. flip-- 7.8%

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

   8. Applied egg-rr 7.5%

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

   9. Taylor expanded in eps around 0 30.5%

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

   if -5.8000000000000003e-8 < x < 2.10000000000000005e-103

   1. Initial program 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in x around 0 45.0%

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

   6. Taylor expanded in x around 0 35.1%

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

   7. Taylor expanded in eps around 0 81.7%

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

   if 1.12000000000000007e32 < x < 1.1e185

   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 24.4%

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

   6. Taylor expanded in x around 0 65.6%

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

   if 1.1e185 < 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 31.2%

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

      1. *-commutative 31.2%

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

      2. sub-neg 31.2%

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

      3. distribute-rgt-in 31.2%

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

      4. *-un-lft-identity 31.2%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 37.9%

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

      7. sqr-neg 37.9%

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

      8. sqrt-unprod 31.0%

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

      9. add-sqr-sqrt 31.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 47.2%

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

      12. sqr-neg 47.2%

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

      13. sqrt-unprod 47.1%

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

      14. add-sqr-sqrt 47.1%

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

   7. Applied egg-rr 47.1%

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

   8. Taylor expanded in eps around 0 29.6%

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

      1. expm1-define 29.6%

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

   10. Simplified 29.6%

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

   11. Taylor expanded in x around 0 29.6%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)}}{\varepsilon}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \varepsilon \cdot \left(-1 + \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+32}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \varepsilon \cdot \left(-1 + \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+185}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := eps\_m \cdot \left(1 + eps\_m \cdot \left(-1 + eps\_m\right)\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+185}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* eps_m (+ 1.0 (* eps_m (+ -1.0 eps_m))))))
   (if (<= x -1e+103)
     (/ (/ (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666))))) eps_m) 2.0)
     (if (<= x -8e-8)
       t_0
       (if (<= x 9e-103)
         1.0
         (if (<= x 4.8e+34)
           t_0
           (if (<= x 1.1e+185)
             (/ (+ (+ 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 t_0 = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	double tmp;
	if (x <= -1e+103) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))) / eps_m) / 2.0;
	} else if (x <= -8e-8) {
		tmp = t_0;
	} else if (x <= 9e-103) {
		tmp = 1.0;
	} else if (x <= 4.8e+34) {
		tmp = t_0;
	} else if (x <= 1.1e+185) {
		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) :: t_0
    real(8) :: tmp
    t_0 = eps_m * (1.0d0 + (eps_m * ((-1.0d0) + eps_m)))
    if (x <= (-1d+103)) then
        tmp = ((x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0)))))) / eps_m) / 2.0d0
    else if (x <= (-8d-8)) then
        tmp = t_0
    else if (x <= 9d-103) then
        tmp = 1.0d0
    else if (x <= 4.8d+34) then
        tmp = t_0
    else if (x <= 1.1d+185) 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 t_0 = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	double tmp;
	if (x <= -1e+103) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))) / eps_m) / 2.0;
	} else if (x <= -8e-8) {
		tmp = t_0;
	} else if (x <= 9e-103) {
		tmp = 1.0;
	} else if (x <= 4.8e+34) {
		tmp = t_0;
	} else if (x <= 1.1e+185) {
		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):
	t_0 = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)))
	tmp = 0
	if x <= -1e+103:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))) / eps_m) / 2.0
	elif x <= -8e-8:
		tmp = t_0
	elif x <= 9e-103:
		tmp = 1.0
	elif x <= 4.8e+34:
		tmp = t_0
	elif x <= 1.1e+185:
		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)
	t_0 = Float64(eps_m * Float64(1.0 + Float64(eps_m * Float64(-1.0 + eps_m))))
	tmp = 0.0
	if (x <= -1e+103)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))))) / eps_m) / 2.0);
	elseif (x <= -8e-8)
		tmp = t_0;
	elseif (x <= 9e-103)
		tmp = 1.0;
	elseif (x <= 4.8e+34)
		tmp = t_0;
	elseif (x <= 1.1e+185)
		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)
	t_0 = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	tmp = 0.0;
	if (x <= -1e+103)
		tmp = ((x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))) / eps_m) / 2.0;
	elseif (x <= -8e-8)
		tmp = t_0;
	elseif (x <= 9e-103)
		tmp = 1.0;
	elseif (x <= 4.8e+34)
		tmp = t_0;
	elseif (x <= 1.1e+185)
		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_] := Block[{t$95$0 = N[(eps$95$m * N[(1.0 + N[(eps$95$m * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e+103], 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], If[LessEqual[x, -8e-8], t$95$0, If[LessEqual[x, 9e-103], 1.0, If[LessEqual[x, 4.8e+34], t$95$0, If[LessEqual[x, 1.1e+185], 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 5 regimes

2. if x < -1e103

   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 55.3%

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

   6. Taylor expanded in eps around 0 46.2%

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

      1. expm1-define 46.2%

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

      2. neg-mul-1 46.2%

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

   8. Simplified 46.2%

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

   9. Taylor expanded in x around 0 46.2%

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

   if -1e103 < x < -8.0000000000000002e-8 or 9e-103 < x < 4.79999999999999974e34

   1. Initial program 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in x around 0 38.8%

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

   6. Taylor expanded in x around 0 10.7%

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

      1. add-sqr-sqrt 6.8%

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

      2. sqrt-unprod 8.3%

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

      3. frac-times 8.3%

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

      4. metadata-eval 8.3%

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

      5. metadata-eval 8.3%

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

      6. frac-times 8.3%

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

      7. sqrt-unprod 1.7%

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

      8. add-sqr-sqrt 7.5%

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

      9. div-inv 7.5%

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

      10. mul-1-neg 7.5%

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

      11. sub-neg 7.5%

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

      12. flip-- 7.5%

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

   8. Applied egg-rr 7.1%

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

   9. Taylor expanded in eps around 0 30.8%

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

   if -8.0000000000000002e-8 < x < 9e-103

   1. Initial program 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in x around 0 45.0%

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

   6. Taylor expanded in x around 0 35.1%

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

   7. Taylor expanded in eps around 0 81.7%

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

   if 4.79999999999999974e34 < x < 1.1e185

   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 24.4%

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

   6. Taylor expanded in x around 0 65.6%

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

   if 1.1e185 < 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 31.2%

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

      1. *-commutative 31.2%

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

      2. sub-neg 31.2%

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

      3. distribute-rgt-in 31.2%

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

      4. *-un-lft-identity 31.2%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 37.9%

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

      7. sqr-neg 37.9%

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

      8. sqrt-unprod 31.0%

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

      9. add-sqr-sqrt 31.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 47.2%

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

      12. sqr-neg 47.2%

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

      13. sqrt-unprod 47.1%

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

      14. add-sqr-sqrt 47.1%

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

   7. Applied egg-rr 47.1%

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

   8. Taylor expanded in eps around 0 29.6%

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

      1. expm1-define 29.6%

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

   10. Simplified 29.6%

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

   11. Taylor expanded in x around 0 29.6%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)}}{\varepsilon}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \varepsilon \cdot \left(-1 + \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+34}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \varepsilon \cdot \left(-1 + \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+185}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.6% accurate, 16.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2 \cdot 10^{+103}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;eps\_m \cdot \left(1 + eps\_m \cdot \left(-1 + eps\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2e+103) 1.0 (* eps_m (+ 1.0 (* eps_m (+ -1.0 eps_m))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2e+103) {
		tmp = 1.0;
	} else {
		tmp = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	}
	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 <= 2d+103) then
        tmp = 1.0d0
    else
        tmp = eps_m * (1.0d0 + (eps_m * ((-1.0d0) + eps_m)))
    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 <= 2e+103) {
		tmp = 1.0;
	} else {
		tmp = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 2e+103:
		tmp = 1.0
	else:
		tmp = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 2e+103)
		tmp = 1.0;
	else
		tmp = Float64(eps_m * Float64(1.0 + Float64(eps_m * Float64(-1.0 + eps_m))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 2e+103)
		tmp = 1.0;
	else
		tmp = eps_m * (1.0 + (eps_m * (-1.0 + eps_m)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 2e+103], 1.0, N[(eps$95$m * N[(1.0 + N[(eps$95$m * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2e103

   1. Initial program 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in x around 0 35.2%

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

   6. Taylor expanded in x around 0 30.9%

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

   7. Taylor expanded in eps around 0 52.8%

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

   if 2e103 < 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 64.8%

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

   6. Taylor expanded in x around 0 30.5%

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

      1. add-sqr-sqrt 30.5%

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

      2. sqrt-unprod 30.5%

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

      3. frac-times 30.5%

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

      4. metadata-eval 30.5%

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

      5. metadata-eval 30.5%

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

      6. frac-times 30.5%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 30.5%

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

      9. div-inv 30.5%

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

      10. mul-1-neg 30.5%

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

      11. sub-neg 30.5%

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

      12. flip-- 30.5%

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

   8. Applied egg-rr 30.5%

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

   9. Taylor expanded in eps around 0 72.6%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{+103}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \varepsilon \cdot \left(-1 + \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.5% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

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

Fortran

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in x around 0 41.3%

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

6. Taylor expanded in x around 0 30.8%

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

7. Taylor expanded in eps around 0 48.2%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(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))