NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.3% → 99.9%

Time: 12.9s

Alternatives: 14

Speedup: 2.0×

• 37.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-16}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 1e-16)
   (/ (+ (/ (+ x 1.0) (exp x)) (* (+ x 1.0) (exp (- x)))) 2.0)
   (/
    (+
     (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
     (* (exp (* x (- -1.0 eps))) (- 1.0 (/ 1.0 eps))))
    2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 1e-16) {
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 1d-16) then
        tmp = (((x + 1.0d0) / exp(x)) + ((x + 1.0d0) * exp(-x))) / 2.0d0
    else
        tmp = (((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 - (1.0d0 / eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (eps <= 1e-16) {
		tmp = (((x + 1.0) / Math.exp(x)) + ((x + 1.0) * Math.exp(-x))) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if eps <= 1e-16:
		tmp = (((x + 1.0) / math.exp(x)) + ((x + 1.0) * math.exp(-x))) / 2.0
	else:
		tmp = (((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) + (math.exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (eps <= 1e-16)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / exp(x)) + Float64(Float64(x + 1.0) * exp(Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 - Float64(1.0 / eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1e-16)
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	else
		tmp = (((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 1e-16], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 9.9999999999999998e-17

   1. Initial program 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in eps around 0 74.9%

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

   6. Simplified 75.5%

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

      1. exp-neg 75.5%

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

      2. un-div-inv 75.5%

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

   8. Applied egg-rr 75.5%

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

   if 9.9999999999999998e-17 < 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. Recombined 2 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-16}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 2: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-289}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+223}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1e-289)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (<= x 6.2e+33)
     (/ (+ 1.0 (exp (* eps x))) 2.0)
     (if (<= x 2.5e+88)
       (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0)
       (if (<= x 3.8e+223)
         (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)
         (/ (+ (/ (+ x 1.0) (exp x)) (* (+ x 1.0) (exp (- x)))) 2.0))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -1e-289) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 6.2e+33) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else if (x <= 2.5e+88) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (x <= 3.8e+223) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1d-289)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if (x <= 6.2d+33) then
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    else if (x <= 2.5d+88) then
        tmp = ((1.0d0 + (1.0d0 / eps)) + (1.0d0 - (1.0d0 / eps))) / 2.0d0
    else if (x <= 3.8d+223) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    else
        tmp = (((x + 1.0d0) / exp(x)) + ((x + 1.0d0) * exp(-x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -1e-289) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 6.2e+33) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else if (x <= 2.5e+88) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (x <= 3.8e+223) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = (((x + 1.0) / Math.exp(x)) + ((x + 1.0) * Math.exp(-x))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -1e-289:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif x <= 6.2e+33:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	elif x <= 2.5e+88:
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	elif x <= 3.8e+223:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	else:
		tmp = (((x + 1.0) / math.exp(x)) + ((x + 1.0) * math.exp(-x))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -1e-289)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 6.2e+33)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	elseif (x <= 2.5e+88)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	elseif (x <= 3.8e+223)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / exp(x)) + Float64(Float64(x + 1.0) * exp(Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1e-289)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif (x <= 6.2e+33)
		tmp = (1.0 + exp((eps * x))) / 2.0;
	elseif (x <= 2.5e+88)
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	elseif (x <= 3.8e+223)
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	else
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -1e-289], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.2e+33], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.5e+88], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.8e+223], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1e-289

   1. Initial program 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in x around 0 50.6%

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

   5. Taylor expanded in eps around inf 78.8%

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

      1. sub-neg 78.8%

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

      2. mul-1-neg 78.8%

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

      3. remove-double-neg 78.8%

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

      4. mul-1-neg 78.8%

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

      5. distribute-rgt-neg-in 78.8%

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

      6. mul-1-neg 78.8%

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

      7. distribute-lft-in 78.8%

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

      8. metadata-eval 78.8%

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

      9. neg-mul-1 78.8%

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

   7. Simplified 78.8%

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

   if -1e-289 < x < 6.2e33

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in x around 0 29.5%

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

   5. Taylor expanded in eps around inf 82.3%

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

      1. neg-mul-1 82.3%

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

      2. distribute-rgt-neg-in 82.3%

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

   7. Simplified 82.3%

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

   8. Taylor expanded in eps around inf 82.5%

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

   if 6.2e33 < x < 2.49999999999999999e88

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 17.4%

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

   5. Taylor expanded in x around 0 73.9%

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

   if 2.49999999999999999e88 < x < 3.8e223

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 39.3%

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

   5. Taylor expanded in eps around inf 39.5%

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

      1. neg-mul-1 39.5%

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

      2. distribute-rgt-neg-in 39.5%

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

   7. Simplified 39.5%

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

   if 3.8e223 < x

   1. Initial program 100.0%

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

   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. Taylor expanded in eps around 0 76.6%

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

   6. Simplified 76.6%

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

      1. exp-neg 76.6%

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

      2. un-div-inv 76.6%

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

   8. Applied egg-rr 76.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-289}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+223}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \end{array} \]

Alternative 3: 77.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := 1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\\ \mathbf{if}\;x \leq -21500:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+33}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+225}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + t_0 \cdot t_0}{\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (- (/ 1.0 eps) (/ 1.0 eps)))))
   (if (<= x -21500.0)
     (/ (/ (expm1 (- x)) eps) 2.0)
     (if (<= x 2.1e+33)
       (/ (+ 1.0 (exp (* eps x))) 2.0)
       (if (<= x 2.7e+87)
         (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0)
         (if (<= x 4.2e+225)
           (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)
           (/
            (/
             (+ -1.0 (* t_0 t_0))
             (+ (/ 1.0 eps) (+ 1.0 (+ -1.0 (/ 1.0 eps)))))
            2.0)))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps));
	double tmp;
	if (x <= -21500.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 2.1e+33) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else if (x <= 2.7e+87) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (x <= 4.2e+225) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps));
	double tmp;
	if (x <= -21500.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 2.1e+33) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else if (x <= 2.7e+87) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (x <= 4.2e+225) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps))
	tmp = 0
	if x <= -21500.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 2.1e+33:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	elif x <= 2.7e+87:
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	elif x <= 4.2e+225:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	else:
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + (-1.0 + (1.0 / eps))))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(1.0 + Float64(Float64(1.0 / eps) - Float64(1.0 / eps)))
	tmp = 0.0
	if (x <= -21500.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 2.1e+33)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	elseif (x <= 2.7e+87)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	elseif (x <= 4.2e+225)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(-1.0 + Float64(t_0 * t_0)) / Float64(Float64(1.0 / eps) + Float64(1.0 + Float64(-1.0 + Float64(1.0 / eps))))) / 2.0);
	end
	return tmp
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(N[(1.0 / eps), $MachinePrecision] - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -21500.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.1e+33], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.7e+87], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.2e+225], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / eps), $MachinePrecision] + N[(1.0 + N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -21500

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 37.7%

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

   5. Taylor expanded in eps around 0 64.3%

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

      1. expm1-def 64.3%

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

      2. neg-mul-1 64.3%

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

   7. Simplified 64.3%

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

   if -21500 < x < 2.1000000000000001e33

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in x around 0 33.8%

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

   5. Taylor expanded in eps around inf 81.5%

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

      1. neg-mul-1 81.5%

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

      2. distribute-rgt-neg-in 81.5%

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

   7. Simplified 81.5%

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

   8. Taylor expanded in eps around inf 81.9%

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

   if 2.1000000000000001e33 < x < 2.70000000000000007e87

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 17.4%

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

   5. Taylor expanded in x around 0 73.9%

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

   if 2.70000000000000007e87 < x < 4.2e225

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 39.3%

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

   5. Taylor expanded in eps around inf 39.5%

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

      1. neg-mul-1 39.5%

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

      2. distribute-rgt-neg-in 39.5%

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

   7. Simplified 39.5%

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

   if 4.2e225 < x

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 16.3%

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

   5. Taylor expanded in x around 0 72.3%

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

      1. associate--r- 3.1%

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

      2. flip-+ 2.2%

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

      3. associate--l+ 2.2%

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

      4. associate--l+ 71.4%

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

      5. metadata-eval 71.4%

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

      6. *-un-lft-identity 71.4%

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

      7. cancel-sign-sub-inv 71.4%

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

      8. metadata-eval 71.4%

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

      9. add-sqr-sqrt 38.1%

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

      10. sqrt-unprod 71.4%

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

      11. sqr-neg 71.4%

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

      12. sqrt-unprod 38.1%

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

      13. add-sqr-sqrt 76.2%

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

      14. div-inv 76.2%

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

      15. metadata-eval 76.2%

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

      16. frac-2neg 76.2%

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

   7. Applied egg-rr 76.6%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -21500:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+33}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+225}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + \left(1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\right) \cdot \left(1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\right)}{\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)}}{2}\\ \end{array} \]

Alternative 4: 84.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := 1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\\ \mathbf{if}\;x \leq -1.56 \cdot 10^{-291}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+227}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + t_0 \cdot t_0}{\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (- (/ 1.0 eps) (/ 1.0 eps)))))
   (if (<= x -1.56e-291)
     (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
     (if (<= x 8.8e+33)
       (/ (+ 1.0 (exp (* eps x))) 2.0)
       (if (<= x 2.9e+88)
         (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0)
         (if (<= x 1.75e+227)
           (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)
           (/
            (/
             (+ -1.0 (* t_0 t_0))
             (+ (/ 1.0 eps) (+ 1.0 (+ -1.0 (/ 1.0 eps)))))
            2.0)))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps));
	double tmp;
	if (x <= -1.56e-291) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 8.8e+33) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else if (x <= 2.9e+88) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (x <= 1.75e+227) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((1.0d0 / eps) - (1.0d0 / eps))
    if (x <= (-1.56d-291)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if (x <= 8.8d+33) then
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    else if (x <= 2.9d+88) then
        tmp = ((1.0d0 + (1.0d0 / eps)) + (1.0d0 - (1.0d0 / eps))) / 2.0d0
    else if (x <= 1.75d+227) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    else
        tmp = (((-1.0d0) + (t_0 * t_0)) / ((1.0d0 / eps) + (1.0d0 + ((-1.0d0) + (1.0d0 / eps))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps));
	double tmp;
	if (x <= -1.56e-291) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 8.8e+33) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else if (x <= 2.9e+88) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (x <= 1.75e+227) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps))
	tmp = 0
	if x <= -1.56e-291:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif x <= 8.8e+33:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	elif x <= 2.9e+88:
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	elif x <= 1.75e+227:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	else:
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + (-1.0 + (1.0 / eps))))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(1.0 + Float64(Float64(1.0 / eps) - Float64(1.0 / eps)))
	tmp = 0.0
	if (x <= -1.56e-291)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 8.8e+33)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	elseif (x <= 2.9e+88)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	elseif (x <= 1.75e+227)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(-1.0 + Float64(t_0 * t_0)) / Float64(Float64(1.0 / eps) + Float64(1.0 + Float64(-1.0 + Float64(1.0 / eps))))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps));
	tmp = 0.0;
	if (x <= -1.56e-291)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif (x <= 8.8e+33)
		tmp = (1.0 + exp((eps * x))) / 2.0;
	elseif (x <= 2.9e+88)
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	elseif (x <= 1.75e+227)
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	else
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + (-1.0 + (1.0 / eps))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(N[(1.0 / eps), $MachinePrecision] - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.56e-291], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.8e+33], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.9e+88], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.75e+227], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / eps), $MachinePrecision] + N[(1.0 + N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.56e-291

   1. Initial program 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in x around 0 50.6%

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

   5. Taylor expanded in eps around inf 78.8%

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

      1. sub-neg 78.8%

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

      2. mul-1-neg 78.8%

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

      3. remove-double-neg 78.8%

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

      4. mul-1-neg 78.8%

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

      5. distribute-rgt-neg-in 78.8%

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

      6. mul-1-neg 78.8%

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

      7. distribute-lft-in 78.8%

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

      8. metadata-eval 78.8%

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

      9. neg-mul-1 78.8%

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

   7. Simplified 78.8%

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

   if -1.56e-291 < x < 8.79999999999999975e33

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in x around 0 29.5%

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

   5. Taylor expanded in eps around inf 82.3%

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

      1. neg-mul-1 82.3%

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

      2. distribute-rgt-neg-in 82.3%

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

   7. Simplified 82.3%

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

   8. Taylor expanded in eps around inf 82.5%

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

   if 8.79999999999999975e33 < x < 2.9e88

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 17.4%

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

   5. Taylor expanded in x around 0 73.9%

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

   if 2.9e88 < x < 1.75e227

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 39.3%

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

   5. Taylor expanded in eps around inf 39.5%

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

      1. neg-mul-1 39.5%

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

      2. distribute-rgt-neg-in 39.5%

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

   7. Simplified 39.5%

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

   if 1.75e227 < x

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 16.3%

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

   5. Taylor expanded in x around 0 72.3%

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

      1. associate--r- 3.1%

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

      2. flip-+ 2.2%

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

      3. associate--l+ 2.2%

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

      4. associate--l+ 71.4%

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

      5. metadata-eval 71.4%

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

      6. *-un-lft-identity 71.4%

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

      7. cancel-sign-sub-inv 71.4%

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

      8. metadata-eval 71.4%

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

      9. add-sqr-sqrt 38.1%

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

      10. sqrt-unprod 71.4%

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

      11. sqr-neg 71.4%

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

      12. sqrt-unprod 38.1%

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

      13. add-sqr-sqrt 76.2%

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

      14. div-inv 76.2%

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

      15. metadata-eval 76.2%

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

      16. frac-2neg 76.2%

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

   7. Applied egg-rr 76.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.56 \cdot 10^{-291}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+227}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + \left(1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\right) \cdot \left(1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\right)}{\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)}}{2}\\ \end{array} \]

Alternative 5: 77.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := 1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\\ t_1 := \frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{if}\;x \leq -21500:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+224}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + t_0 \cdot t_0}{\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (- (/ 1.0 eps) (/ 1.0 eps))))
        (t_1 (/ (+ 1.0 (exp (* eps x))) 2.0)))
   (if (<= x -21500.0)
     (/ (/ (expm1 (- x)) eps) 2.0)
     (if (<= x 1.3e+33)
       t_1
       (if (<= x 7.5e+88)
         (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0)
         (if (<= x 1.3e+224)
           t_1
           (/
            (/
             (+ -1.0 (* t_0 t_0))
             (+ (/ 1.0 eps) (+ 1.0 (+ -1.0 (/ 1.0 eps)))))
            2.0)))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps));
	double t_1 = (1.0 + exp((eps * x))) / 2.0;
	double tmp;
	if (x <= -21500.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 1.3e+33) {
		tmp = t_1;
	} else if (x <= 7.5e+88) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (x <= 1.3e+224) {
		tmp = t_1;
	} else {
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps));
	double t_1 = (1.0 + Math.exp((eps * x))) / 2.0;
	double tmp;
	if (x <= -21500.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 1.3e+33) {
		tmp = t_1;
	} else if (x <= 7.5e+88) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (x <= 1.3e+224) {
		tmp = t_1;
	} else {
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps))
	t_1 = (1.0 + math.exp((eps * x))) / 2.0
	tmp = 0
	if x <= -21500.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 1.3e+33:
		tmp = t_1
	elif x <= 7.5e+88:
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	elif x <= 1.3e+224:
		tmp = t_1
	else:
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + (-1.0 + (1.0 / eps))))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(1.0 + Float64(Float64(1.0 / eps) - Float64(1.0 / eps)))
	t_1 = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0)
	tmp = 0.0
	if (x <= -21500.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 1.3e+33)
		tmp = t_1;
	elseif (x <= 7.5e+88)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	elseif (x <= 1.3e+224)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-1.0 + Float64(t_0 * t_0)) / Float64(Float64(1.0 / eps) + Float64(1.0 + Float64(-1.0 + Float64(1.0 / eps))))) / 2.0);
	end
	return tmp
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(N[(1.0 / eps), $MachinePrecision] - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -21500.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.3e+33], t$95$1, If[LessEqual[x, 7.5e+88], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.3e+224], t$95$1, N[(N[(N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / eps), $MachinePrecision] + N[(1.0 + N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -21500

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 37.7%

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

   5. Taylor expanded in eps around 0 64.3%

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

      1. expm1-def 64.3%

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

      2. neg-mul-1 64.3%

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

   7. Simplified 64.3%

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

   if -21500 < x < 1.2999999999999999e33 or 7.50000000000000031e88 < x < 1.3e224

   1. Initial program 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in x around 0 34.6%

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

   5. Taylor expanded in eps around inf 75.9%

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

      1. neg-mul-1 75.9%

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

      2. distribute-rgt-neg-in 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in eps around inf 76.3%

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

   if 1.2999999999999999e33 < x < 7.50000000000000031e88

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 17.4%

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

   5. Taylor expanded in x around 0 73.9%

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

   if 1.3e224 < x

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 16.3%

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

   5. Taylor expanded in x around 0 72.3%

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

      1. associate--r- 3.1%

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

      2. flip-+ 2.2%

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

      3. associate--l+ 2.2%

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

      4. associate--l+ 71.4%

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

      5. metadata-eval 71.4%

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

      6. *-un-lft-identity 71.4%

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

      7. cancel-sign-sub-inv 71.4%

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

      8. metadata-eval 71.4%

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

      9. add-sqr-sqrt 38.1%

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

      10. sqrt-unprod 71.4%

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

      11. sqr-neg 71.4%

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

      12. sqrt-unprod 38.1%

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

      13. add-sqr-sqrt 76.2%

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

      14. div-inv 76.2%

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

      15. metadata-eval 76.2%

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

      16. frac-2neg 76.2%

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

   7. Applied egg-rr 76.6%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -21500:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+33}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+224}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + \left(1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\right) \cdot \left(1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\right)}{\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)}}{2}\\ \end{array} \]

Alternative 6: 70.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := 1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\\ \mathbf{if}\;x \leq -21500:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 750:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + t_0 \cdot t_0}{\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (- (/ 1.0 eps) (/ 1.0 eps)))))
   (if (<= x -21500.0)
     (/ (/ (expm1 (- x)) eps) 2.0)
     (if (<= x 750.0)
       1.0
       (/
        (/ (+ -1.0 (* t_0 t_0)) (+ (/ 1.0 eps) (+ 1.0 (+ -1.0 (/ 1.0 eps)))))
        2.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps));
	double tmp;
	if (x <= -21500.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 750.0) {
		tmp = 1.0;
	} else {
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps));
	double tmp;
	if (x <= -21500.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 750.0) {
		tmp = 1.0;
	} else {
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + (-1.0 + (1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps))
	tmp = 0
	if x <= -21500.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 750.0:
		tmp = 1.0
	else:
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + (-1.0 + (1.0 / eps))))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(1.0 + Float64(Float64(1.0 / eps) - Float64(1.0 / eps)))
	tmp = 0.0
	if (x <= -21500.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 750.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(-1.0 + Float64(t_0 * t_0)) / Float64(Float64(1.0 / eps) + Float64(1.0 + Float64(-1.0 + Float64(1.0 / eps))))) / 2.0);
	end
	return tmp
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(N[(1.0 / eps), $MachinePrecision] - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -21500.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 750.0], 1.0, N[(N[(N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / eps), $MachinePrecision] + N[(1.0 + N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -21500

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 37.7%

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

   5. Taylor expanded in eps around 0 64.3%

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

      1. expm1-def 64.3%

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

      2. neg-mul-1 64.3%

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

   7. Simplified 64.3%

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

   if -21500 < x < 750

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in x around 0 74.1%

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

   if 750 < x

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 23.6%

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

   5. Taylor expanded in x around 0 55.5%

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

      1. associate--r- 3.1%

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

      2. flip-+ 1.7%

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

      3. associate--l+ 1.7%

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

      4. associate--l+ 54.1%

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

      5. metadata-eval 54.1%

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

      6. *-un-lft-identity 54.1%

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

      7. cancel-sign-sub-inv 54.1%

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

      8. metadata-eval 54.1%

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

      9. add-sqr-sqrt 27.0%

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

      10. sqrt-unprod 55.5%

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

      11. sqr-neg 55.5%

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

      12. sqrt-unprod 29.8%

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

      13. add-sqr-sqrt 56.8%

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

      14. div-inv 56.8%

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

      15. metadata-eval 56.8%

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

      16. frac-2neg 56.8%

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

   7. Applied egg-rr 57.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -21500:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 750:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + \left(1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\right) \cdot \left(1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\right)}{\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)}}{2}\\ \end{array} \]

Alternative 7: 63.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := 1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\\ t_1 := -1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;x \leq 120:\\ \;\;\;\;\frac{2 - x \cdot \left(\frac{1}{\varepsilon} + t_1 \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + t_0 \cdot t_0}{\frac{1}{\varepsilon} + \left(1 + t_1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (- (/ 1.0 eps) (/ 1.0 eps)))) (t_1 (+ -1.0 (/ 1.0 eps))))
   (if (<= x 120.0)
     (/ (- 2.0 (* x (+ (/ 1.0 eps) (* t_1 (- -1.0 eps))))) 2.0)
     (/ (/ (+ -1.0 (* t_0 t_0)) (+ (/ 1.0 eps) (+ 1.0 t_1))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps));
	double t_1 = -1.0 + (1.0 / eps);
	double tmp;
	if (x <= 120.0) {
		tmp = (2.0 - (x * ((1.0 / eps) + (t_1 * (-1.0 - eps))))) / 2.0;
	} else {
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + t_1))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((1.0d0 / eps) - (1.0d0 / eps))
    t_1 = (-1.0d0) + (1.0d0 / eps)
    if (x <= 120.0d0) then
        tmp = (2.0d0 - (x * ((1.0d0 / eps) + (t_1 * ((-1.0d0) - eps))))) / 2.0d0
    else
        tmp = (((-1.0d0) + (t_0 * t_0)) / ((1.0d0 / eps) + (1.0d0 + t_1))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps));
	double t_1 = -1.0 + (1.0 / eps);
	double tmp;
	if (x <= 120.0) {
		tmp = (2.0 - (x * ((1.0 / eps) + (t_1 * (-1.0 - eps))))) / 2.0;
	} else {
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + t_1))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps))
	t_1 = -1.0 + (1.0 / eps)
	tmp = 0
	if x <= 120.0:
		tmp = (2.0 - (x * ((1.0 / eps) + (t_1 * (-1.0 - eps))))) / 2.0
	else:
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + t_1))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(1.0 + Float64(Float64(1.0 / eps) - Float64(1.0 / eps)))
	t_1 = Float64(-1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (x <= 120.0)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(Float64(1.0 / eps) + Float64(t_1 * Float64(-1.0 - eps))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(-1.0 + Float64(t_0 * t_0)) / Float64(Float64(1.0 / eps) + Float64(1.0 + t_1))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = 1.0 + ((1.0 / eps) - (1.0 / eps));
	t_1 = -1.0 + (1.0 / eps);
	tmp = 0.0;
	if (x <= 120.0)
		tmp = (2.0 - (x * ((1.0 / eps) + (t_1 * (-1.0 - eps))))) / 2.0;
	else
		tmp = ((-1.0 + (t_0 * t_0)) / ((1.0 / eps) + (1.0 + t_1))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(N[(1.0 / eps), $MachinePrecision] - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 120.0], N[(N[(2.0 - N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(t$95$1 * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / eps), $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 120

   1. Initial program 56.4%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in x around 0 63.9%

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

   5. Taylor expanded in eps around 0 67.6%

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

   if 120 < x

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 24.3%

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

   5. Taylor expanded in x around 0 54.1%

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

      1. associate--r- 3.1%

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

      2. flip-+ 1.7%

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

      3. associate--l+ 1.7%

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

      4. associate--l+ 52.7%

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

      5. metadata-eval 52.7%

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

      6. *-un-lft-identity 52.7%

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

      7. cancel-sign-sub-inv 52.7%

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

      8. metadata-eval 52.7%

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

      9. add-sqr-sqrt 26.3%

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

      10. sqrt-unprod 54.0%

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

      11. sqr-neg 54.0%

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

      12. sqrt-unprod 29.0%

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

      13. add-sqr-sqrt 55.3%

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

      14. div-inv 55.3%

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

      15. metadata-eval 55.3%

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

      16. frac-2neg 55.3%

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

   7. Applied egg-rr 56.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 120:\\ \;\;\;\;\frac{2 - x \cdot \left(\frac{1}{\varepsilon} + \left(-1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + \left(1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\right) \cdot \left(1 + \left(\frac{1}{\varepsilon} - \frac{1}{\varepsilon}\right)\right)}{\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)}}{2}\\ \end{array} \]

Alternative 8: 63.0% accurate, 10.8× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 106:\\ \;\;\;\;\frac{2 - x \cdot \left(\frac{1}{\varepsilon} + \left(-1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 106.0)
   (/ (- 2.0 (* x (+ (/ 1.0 eps) (* (+ -1.0 (/ 1.0 eps)) (- -1.0 eps))))) 2.0)
   (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 106.0) {
		tmp = (2.0 - (x * ((1.0 / eps) + ((-1.0 + (1.0 / eps)) * (-1.0 - eps))))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 106.0d0) then
        tmp = (2.0d0 - (x * ((1.0d0 / eps) + (((-1.0d0) + (1.0d0 / eps)) * ((-1.0d0) - eps))))) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps)) + (1.0d0 - (1.0d0 / eps))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 106.0) {
		tmp = (2.0 - (x * ((1.0 / eps) + ((-1.0 + (1.0 / eps)) * (-1.0 - eps))))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 106.0:
		tmp = (2.0 - (x * ((1.0 / eps) + ((-1.0 + (1.0 / eps)) * (-1.0 - eps))))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 106.0)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(-1.0 + Float64(1.0 / eps)) * Float64(-1.0 - eps))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 106.0)
		tmp = (2.0 - (x * ((1.0 / eps) + ((-1.0 + (1.0 / eps)) * (-1.0 - eps))))) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 106.0], N[(N[(2.0 - N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 106

   1. Initial program 56.4%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in x around 0 63.9%

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

   5. Taylor expanded in eps around 0 67.6%

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

   if 106 < x

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 24.3%

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

   5. Taylor expanded in x around 0 54.1%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 106:\\ \;\;\;\;\frac{2 - x \cdot \left(\frac{1}{\varepsilon} + \left(-1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 9: 62.7% accurate, 15.1× speedup

Math

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

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 2.0)
   (/ (+ 2.0 (* x (- -1.0 eps))) 2.0)
   (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps))) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps)) + (1.0d0 - (1.0d0 / eps))) / 2.0d0
    end if
    code = tmp
end function

Java

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

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 2.0], N[(N[(2.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 56.4%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in x around 0 42.4%

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

   5. Taylor expanded in eps around inf 84.4%

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

      1. sub-neg 84.4%

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

      2. mul-1-neg 84.4%

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

      3. remove-double-neg 84.4%

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

      4. mul-1-neg 84.4%

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

      5. distribute-rgt-neg-in 84.4%

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

      6. mul-1-neg 84.4%

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

      7. distribute-lft-in 84.4%

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

      8. metadata-eval 84.4%

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

      9. neg-mul-1 84.4%

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

   7. Simplified 84.4%

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

   8. Taylor expanded in x around 0 67.2%

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

   if 2 < x

   1. Initial program 100.0%

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

   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. Taylor expanded in x around 0 24.3%

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

   5. Taylor expanded in x around 0 54.1%

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

4. Final simplification 63.3%

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

Alternative 10: 57.0% accurate, 20.5× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -5e-288)
   (/ (- 2.0 (* x (+ eps 2.0))) 2.0)
   (/ (+ 2.0 (* eps x)) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -5e-288) {
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5d-288)) then
        tmp = (2.0d0 - (x * (eps + 2.0d0))) / 2.0d0
    else
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -5e-288) {
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -5e-288:
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0
	else:
		tmp = (2.0 + (eps * x)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -5e-288)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(eps + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5e-288)
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	else
		tmp = (2.0 + (eps * x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -5e-288], N[(N[(2.0 - N[(x * N[(eps + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000011e-288

   1. Initial program 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in x around 0 39.7%

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

   5. Taylor expanded in x around 0 33.7%

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

      1. mul-1-neg 33.7%

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

      2. unsub-neg 33.7%

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

      3. *-commutative 33.7%

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

      4. associate-*r* 33.7%

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

   7. Simplified 33.7%

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

      1. +-commutative 33.7%

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

      2. distribute-rgt-in 29.2%

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

   9. Applied egg-rr 38.8%

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

   10. Taylor expanded in eps around inf 57.1%

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

       1. distribute-rgt-out 57.1%

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

   12. Simplified 57.1%

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

   if -5.00000000000000011e-288 < x

   1. Initial program 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in x around 0 27.4%

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

   5. Taylor expanded in eps around inf 57.9%

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

      1. neg-mul-1 57.9%

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

      2. distribute-rgt-neg-in 57.9%

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

   7. Simplified 57.9%

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

   8. Taylor expanded in eps around inf 57.9%

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

   9. Taylor expanded in eps around 0 48.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \]

Alternative 11: 57.0% accurate, 20.5× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -5e-288)
   (/ (+ 2.0 (* x (- -1.0 eps))) 2.0)
   (/ (+ 2.0 (* eps x)) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -5e-288) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5d-288)) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps))) / 2.0d0
    else
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -5e-288) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -5e-288:
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0
	else:
		tmp = (2.0 + (eps * x)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -5e-288)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5e-288)
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	else
		tmp = (2.0 + (eps * x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -5e-288], N[(N[(2.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000011e-288

   1. Initial program 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in x around 0 50.6%

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

   5. Taylor expanded in eps around inf 78.8%

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

      1. sub-neg 78.8%

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

      2. mul-1-neg 78.8%

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

      3. remove-double-neg 78.8%

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

      4. mul-1-neg 78.8%

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

      5. distribute-rgt-neg-in 78.8%

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

      6. mul-1-neg 78.8%

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

      7. distribute-lft-in 78.8%

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

      8. metadata-eval 78.8%

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

      9. neg-mul-1 78.8%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in x around 0 57.2%

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

   if -5.00000000000000011e-288 < x

   1. Initial program 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in x around 0 27.4%

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

   5. Taylor expanded in eps around inf 57.9%

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

      1. neg-mul-1 57.9%

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

      2. distribute-rgt-neg-in 57.9%

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

   7. Simplified 57.9%

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

   8. Taylor expanded in eps around inf 57.9%

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

   9. Taylor expanded in eps around 0 48.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \]

Alternative 12: 56.8% accurate, 25.0× speedup

Math

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

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1.0) (/ (* eps (- x)) 2.0) (/ (+ 2.0 (* eps x)) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = (eps * -x) / 2.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (eps * -x) / 2.0d0
    else
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    end if
    code = tmp
end function

Java

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

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -1.0:
		tmp = (eps * -x) / 2.0
	else:
		tmp = (2.0 + (eps * x)) / 2.0
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -1.0], N[(N[(eps * (-x)), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in x around 0 63.1%

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

   5. Taylor expanded in eps around inf 63.1%

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

      1. sub-neg 63.1%

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

      2. mul-1-neg 63.1%

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

      3. remove-double-neg 63.1%

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

      4. mul-1-neg 63.1%

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

      5. distribute-rgt-neg-in 63.1%

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

      6. mul-1-neg 63.1%

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

      7. distribute-lft-in 63.1%

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

      8. metadata-eval 63.1%

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

      9. neg-mul-1 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in x around 0 30.5%

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

   9. Taylor expanded in eps around inf 30.6%

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

       1. neg-mul-1 30.6%

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

       2. distribute-rgt-neg-in 30.6%

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

   11. Simplified 30.6%

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

   if -1 < x

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in x around 0 30.6%

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

   5. Taylor expanded in eps around inf 64.2%

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

      1. neg-mul-1 64.2%

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

      2. distribute-rgt-neg-in 64.2%

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

   7. Simplified 64.2%

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

   8. Taylor expanded in eps around inf 64.4%

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

   9. Taylor expanded in eps around 0 54.3%

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

4. Final simplification 51.6%

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

Alternative 13: 50.3% accurate, 28.2× speedup

Math

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

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x -1.0) (/ (* eps (- x)) 2.0) 1.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = (eps * -x) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (eps * -x) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -1.0:
		tmp = (eps * -x) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -1.0], N[(N[(eps * (-x)), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in x around 0 63.1%

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

   5. Taylor expanded in eps around inf 63.1%

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

      1. sub-neg 63.1%

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

      2. mul-1-neg 63.1%

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

      3. remove-double-neg 63.1%

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

      4. mul-1-neg 63.1%

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

      5. distribute-rgt-neg-in 63.1%

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

      6. mul-1-neg 63.1%

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

      7. distribute-lft-in 63.1%

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

      8. metadata-eval 63.1%

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

      9. neg-mul-1 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in x around 0 30.5%

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

   9. Taylor expanded in eps around inf 30.6%

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

       1. neg-mul-1 30.6%

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

       2. distribute-rgt-neg-in 30.6%

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

   11. Simplified 30.6%

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

   if -1 < x

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in x around 0 51.3%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 43.5% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ 1 \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

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

Python

eps = abs(eps)
def code(x, eps):
	return 1.0

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := 1.0

Derivation

1. Initial program 69.3%

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

3. Simplified 69.3%

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

4. Taylor expanded in x around 0 45.8%

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

5. Final simplification 45.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023309 
(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))