NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.5% → 100.0%

Time: 12.7s

Alternatives: 14

Speedup: 2.0×

• 38.1% 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: 73.5% 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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 1e-10)
   (/ (+ (/ (+ 1.0 x) (exp x)) (* (+ 1.0 x) (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-10) {
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * 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-10) then
        tmp = (((1.0d0 + x) / exp(x)) + ((1.0d0 + x) * 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-10) {
		tmp = (((1.0 + x) / Math.exp(x)) + ((1.0 + x) * 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-10:
		tmp = (((1.0 + x) / math.exp(x)) + ((1.0 + x) * 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-10)
		tmp = Float64(Float64(Float64(Float64(1.0 + x) / exp(x)) + Float64(Float64(1.0 + x) * 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-10)
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * 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-10], N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $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 < 1.00000000000000004e-10

   1. Initial program 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in eps around 0 70.0%

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

   6. Simplified 71.1%

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

      1. exp-neg 71.1%

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

      2. un-div-inv 71.1%

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

      3. +-commutative 71.1%

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

   8. Applied egg-rr 71.1%

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

   if 1.00000000000000004e-10 < 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 78.9%

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

Alternative 2: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+193}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+238}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (+ 1.0 x) (exp x)) (* (+ 1.0 x) (exp (- x)))) 2.0)))
   (if (<= x -1.1e-228)
     (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
     (if (<= x 1.2e+85)
       (/ (+ 1.0 (exp (* eps x))) 2.0)
       (if (<= x 3.4e+161)
         t_0
         (if (<= x 6e+193)
           (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)
           (if (<= x 1.28e+238)
             (/ (+ (- 1.0 (/ -1.0 eps)) (+ 1.0 (/ -1.0 eps))) 2.0)
             t_0)))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	double tmp;
	if (x <= -1.1e-228) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else if (x <= 1.2e+85) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else if (x <= 3.4e+161) {
		tmp = t_0;
	} else if (x <= 6e+193) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else if (x <= 1.28e+238) {
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	} else {
		tmp = t_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 + x) / exp(x)) + ((1.0d0 + x) * exp(-x))) / 2.0d0
    if (x <= (-1.1d-228)) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else if (x <= 1.2d+85) then
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    else if (x <= 3.4d+161) then
        tmp = t_0
    else if (x <= 6d+193) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    else if (x <= 1.28d+238) then
        tmp = ((1.0d0 - ((-1.0d0) / eps)) + (1.0d0 + ((-1.0d0) / eps))) / 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = (((1.0 + x) / Math.exp(x)) + ((1.0 + x) * Math.exp(-x))) / 2.0;
	double tmp;
	if (x <= -1.1e-228) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else if (x <= 1.2e+85) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else if (x <= 3.4e+161) {
		tmp = t_0;
	} else if (x <= 6e+193) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else if (x <= 1.28e+238) {
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = (((1.0 + x) / math.exp(x)) + ((1.0 + x) * math.exp(-x))) / 2.0
	tmp = 0
	if x <= -1.1e-228:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	elif x <= 1.2e+85:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	elif x <= 3.4e+161:
		tmp = t_0
	elif x <= 6e+193:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	elif x <= 1.28e+238:
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0
	else:
		tmp = t_0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(1.0 + x) / exp(x)) + Float64(Float64(1.0 + x) * exp(Float64(-x)))) / 2.0)
	tmp = 0.0
	if (x <= -1.1e-228)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	elseif (x <= 1.2e+85)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	elseif (x <= 3.4e+161)
		tmp = t_0;
	elseif (x <= 6e+193)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	elseif (x <= 1.28e+238)
		tmp = Float64(Float64(Float64(1.0 - Float64(-1.0 / eps)) + Float64(1.0 + Float64(-1.0 / eps))) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	tmp = 0.0;
	if (x <= -1.1e-228)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	elseif (x <= 1.2e+85)
		tmp = (1.0 + exp((eps * x))) / 2.0;
	elseif (x <= 3.4e+161)
		tmp = t_0;
	elseif (x <= 6e+193)
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	elseif (x <= 1.28e+238)
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1.1e-228], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.2e+85], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.4e+161], t$95$0, If[LessEqual[x, 6e+193], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.28e+238], N[(N[(N[(1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.1e-228

   1. Initial program 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in x around 0 41.5%

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

   5. Taylor expanded in eps around inf 68.4%

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

      1. associate-*r* 68.4%

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

      2. exp-prod 61.7%

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

      3. remove-double-neg 61.7%

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

      4. neg-mul-1 61.7%

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

      5. sub-neg 61.7%

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

      6. exp-prod 68.4%

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

      7. associate-*r* 68.4%

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

      8. mul-1-neg 68.4%

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

      9. associate-*r* 68.4%

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

      10. exp-prod 61.7%

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

      11. cancel-sign-sub-inv 61.7%

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

      12. metadata-eval 61.7%

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

      13. *-lft-identity 61.7%

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

      14. exp-prod 68.4%

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

      15. neg-mul-1 68.4%

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

   7. Simplified 68.4%

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

   8. Taylor expanded in eps around inf 69.0%

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

   if -1.1e-228 < x < 1.19999999999999998e85

   1. Initial program 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in x around 0 46.7%

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

   5. Taylor expanded in eps around inf 87.4%

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

      1. associate-*r* 87.4%

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

      2. exp-prod 77.0%

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

      3. remove-double-neg 77.0%

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

      4. neg-mul-1 77.0%

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

      5. sub-neg 77.0%

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

      6. exp-prod 87.4%

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

      7. associate-*r* 87.4%

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

      8. mul-1-neg 87.4%

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

      9. associate-*r* 87.4%

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

      10. exp-prod 77.0%

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

      11. cancel-sign-sub-inv 77.0%

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

      12. metadata-eval 77.0%

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

      13. *-lft-identity 77.0%

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

      14. exp-prod 87.4%

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

      15. neg-mul-1 87.4%

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

   7. Simplified 87.4%

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

      1. add-sqr-sqrt 20.6%

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

      2. sqrt-unprod 77.9%

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

      3. sqr-neg 77.9%

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

      4. sqrt-unprod 61.1%

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

      5. add-sqr-sqrt 81.7%

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

      6. +-commutative 81.7%

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

      7. distribute-lft-in 81.7%

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

      8. *-rgt-identity 81.7%

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

   9. Applied egg-rr 81.7%

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

   10. Taylor expanded in eps around inf 81.9%

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

       1. *-commutative 81.9%

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

   12. Simplified 81.9%

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

   if 1.19999999999999998e85 < x < 3.39999999999999993e161 or 1.28000000000000007e238 < 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 69.9%

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

   6. Simplified 69.9%

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

      1. exp-neg 69.9%

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

      2. un-div-inv 69.9%

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

      3. +-commutative 69.9%

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

   8. Applied egg-rr 69.9%

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

   if 3.39999999999999993e161 < x < 6e193

   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 51.4%

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

   5. Taylor expanded in eps around inf 51.6%

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

      1. neg-mul-1 51.6%

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

      2. distribute-lft-neg-in 51.6%

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

   7. Simplified 51.6%

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

   if 6e193 < x < 1.28000000000000007e238

   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 29.0%

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

   5. Taylor expanded in x around 0 56.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+193}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+238}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \end{array} \]

Alternative 3: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \frac{\frac{e^{-x} + \frac{-1}{e^{x}}}{\varepsilon}}{2}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+80}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+194}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;x \leq 4.25 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (/ (+ (exp (- x)) (/ -1.0 (exp x))) eps) 2.0)))
   (if (<= x -1.1e-228)
     (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
     (if (<= x 4.6e+80)
       (/ (+ 1.0 (exp (* eps x))) 2.0)
       (if (<= x 4e+161)
         t_0
         (if (<= x 1.9e+194)
           (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)
           (if (<= x 4.25e+241)
             (/ (+ (- 1.0 (/ -1.0 eps)) (+ 1.0 (/ -1.0 eps))) 2.0)
             t_0)))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = ((exp(-x) + (-1.0 / exp(x))) / eps) / 2.0;
	double tmp;
	if (x <= -1.1e-228) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else if (x <= 4.6e+80) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else if (x <= 4e+161) {
		tmp = t_0;
	} else if (x <= 1.9e+194) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else if (x <= 4.25e+241) {
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	} else {
		tmp = t_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 = ((exp(-x) + ((-1.0d0) / exp(x))) / eps) / 2.0d0
    if (x <= (-1.1d-228)) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else if (x <= 4.6d+80) then
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    else if (x <= 4d+161) then
        tmp = t_0
    else if (x <= 1.9d+194) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    else if (x <= 4.25d+241) then
        tmp = ((1.0d0 - ((-1.0d0) / eps)) + (1.0d0 + ((-1.0d0) / eps))) / 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = ((Math.exp(-x) + (-1.0 / Math.exp(x))) / eps) / 2.0;
	double tmp;
	if (x <= -1.1e-228) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else if (x <= 4.6e+80) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else if (x <= 4e+161) {
		tmp = t_0;
	} else if (x <= 1.9e+194) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else if (x <= 4.25e+241) {
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = ((math.exp(-x) + (-1.0 / math.exp(x))) / eps) / 2.0
	tmp = 0
	if x <= -1.1e-228:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	elif x <= 4.6e+80:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	elif x <= 4e+161:
		tmp = t_0
	elif x <= 1.9e+194:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	elif x <= 4.25e+241:
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0
	else:
		tmp = t_0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(Float64(exp(Float64(-x)) + Float64(-1.0 / exp(x))) / eps) / 2.0)
	tmp = 0.0
	if (x <= -1.1e-228)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	elseif (x <= 4.6e+80)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	elseif (x <= 4e+161)
		tmp = t_0;
	elseif (x <= 1.9e+194)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	elseif (x <= 4.25e+241)
		tmp = Float64(Float64(Float64(1.0 - Float64(-1.0 / eps)) + Float64(1.0 + Float64(-1.0 / eps))) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = ((exp(-x) + (-1.0 / exp(x))) / eps) / 2.0;
	tmp = 0.0;
	if (x <= -1.1e-228)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	elseif (x <= 4.6e+80)
		tmp = (1.0 + exp((eps * x))) / 2.0;
	elseif (x <= 4e+161)
		tmp = t_0;
	elseif (x <= 1.9e+194)
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	elseif (x <= 4.25e+241)
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[Exp[(-x)], $MachinePrecision] + N[(-1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1.1e-228], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.6e+80], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4e+161], t$95$0, If[LessEqual[x, 1.9e+194], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.25e+241], N[(N[(N[(1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.1e-228

   1. Initial program 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in x around 0 41.5%

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

   5. Taylor expanded in eps around inf 68.4%

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

      1. associate-*r* 68.4%

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

      2. exp-prod 61.7%

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

      3. remove-double-neg 61.7%

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

      4. neg-mul-1 61.7%

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

      5. sub-neg 61.7%

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

      6. exp-prod 68.4%

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

      7. associate-*r* 68.4%

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

      8. mul-1-neg 68.4%

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

      9. associate-*r* 68.4%

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

      10. exp-prod 61.7%

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

      11. cancel-sign-sub-inv 61.7%

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

      12. metadata-eval 61.7%

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

      13. *-lft-identity 61.7%

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

      14. exp-prod 68.4%

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

      15. neg-mul-1 68.4%

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

   7. Simplified 68.4%

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

   8. Taylor expanded in eps around inf 69.0%

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

   if -1.1e-228 < x < 4.60000000000000008e80

   1. Initial program 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in x around 0 46.7%

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

   5. Taylor expanded in eps around inf 87.4%

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

      1. associate-*r* 87.4%

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

      2. exp-prod 77.0%

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

      3. remove-double-neg 77.0%

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

      4. neg-mul-1 77.0%

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

      5. sub-neg 77.0%

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

      6. exp-prod 87.4%

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

      7. associate-*r* 87.4%

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

      8. mul-1-neg 87.4%

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

      9. associate-*r* 87.4%

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

      10. exp-prod 77.0%

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

      11. cancel-sign-sub-inv 77.0%

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

      12. metadata-eval 77.0%

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

      13. *-lft-identity 77.0%

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

      14. exp-prod 87.4%

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

      15. neg-mul-1 87.4%

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

   7. Simplified 87.4%

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

      1. add-sqr-sqrt 20.6%

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

      2. sqrt-unprod 77.9%

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

      3. sqr-neg 77.9%

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

      4. sqrt-unprod 61.1%

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

      5. add-sqr-sqrt 81.7%

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

      6. +-commutative 81.7%

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

      7. distribute-lft-in 81.7%

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

      8. *-rgt-identity 81.7%

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

   9. Applied egg-rr 81.7%

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

   10. Taylor expanded in eps around inf 81.9%

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

       1. *-commutative 81.9%

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

   12. Simplified 81.9%

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

   if 4.60000000000000008e80 < x < 4.0000000000000002e161 or 4.24999999999999977e241 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 69.9%

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

   if 4.0000000000000002e161 < x < 1.8999999999999999e194

   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 51.4%

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

   5. Taylor expanded in eps around inf 51.6%

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

      1. neg-mul-1 51.6%

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

      2. distribute-lft-neg-in 51.6%

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

   7. Simplified 51.6%

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

   if 1.8999999999999999e194 < x < 4.24999999999999977e241

   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 29.0%

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

   5. Taylor expanded in x around 0 56.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+80}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{e^{-x} + \frac{-1}{e^{x}}}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+194}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;x \leq 4.25 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{-x} + \frac{-1}{e^{x}}}{\varepsilon}}{2}\\ \end{array} \]

Alternative 4: 70.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -22000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -22000000.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (<= x 520.0)
     1.0
     (if (<= x 3e+80)
       (/ (/ (expm1 x) 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 <= -22000000.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else if (x <= 3e+80) {
		tmp = (expm1(x) / eps) / 2.0;
	} else {
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -22000000.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else if (x <= 3e+80) {
		tmp = (Math.expm1(x) / 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 <= -22000000.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 520.0:
		tmp = 1.0
	elif x <= 3e+80:
		tmp = (math.expm1(x) / 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 <= -22000000.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 520.0)
		tmp = 1.0;
	elseif (x <= 3e+80)
		tmp = Float64(Float64(expm1(x) / 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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -22000000.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 520.0], 1.0, If[LessEqual[x, 3e+80], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $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 4 regimes

2. if x < -2.2e7

   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 50.3%

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

   5. Taylor expanded in eps around 0 51.3%

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

      1. expm1-def 51.3%

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

      2. neg-mul-1 51.3%

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

   7. Simplified 51.3%

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

   if -2.2e7 < x < 520

   1. Initial program 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in x around 0 76.1%

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

   if 520 < x < 2.99999999999999987e80

   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 40.2%

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

   5. Taylor expanded in eps around 0 1.8%

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

      1. expm1-def 1.8%

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

      2. neg-mul-1 1.8%

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

   7. Simplified 1.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 1.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}\right)\right)}}{2} \]

      2. expm1-udef 1.7%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}\right)} - 1}}{2} \]

      3. expm1-udef 1.7%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{e^{-x} - 1}}{\varepsilon}\right)} - 1}{2} \]

      4. expm1-udef 1.7%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{expm1}\left(-x\right)}}{\varepsilon}\right)} - 1}{2} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      6. sqrt-unprod 38.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)}{\varepsilon}\right)} - 1}{2} \]

      7. sqr-neg 38.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\sqrt{\color{blue}{x \cdot x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      8. sqrt-unprod 38.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      9. add-sqr-sqrt 38.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{x}\right)}{\varepsilon}\right)} - 1}{2} \]

   9. Applied egg-rr 38.5%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)} - 1}}{2} \]
   10. Step-by-step derivation

       1. expm1-def 38.5%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)\right)}}{2} \]

       2. expm1-log1p 38.7%

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

   11. Simplified 38.7%

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

   if 2.99999999999999987e80 < 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 26.3%

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

   5. Taylor expanded in x around 0 56.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -22000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 5: 77.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -22000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+83}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -22000000.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (<= x 1.85e+83)
     (/ (+ 1.0 (exp (* eps x))) 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 <= -22000000.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 1.85e+83) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else {
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -22000000.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 1.85e+83) {
		tmp = (1.0 + Math.exp((eps * x))) / 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 <= -22000000.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 1.85e+83:
		tmp = (1.0 + math.exp((eps * x))) / 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 <= -22000000.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 1.85e+83)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -22000000.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.85e+83], N[(N[(1.0 + N[Exp[N[(eps * x), $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 3 regimes

2. if x < -2.2e7

   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 50.3%

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

   5. Taylor expanded in eps around 0 51.3%

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

      1. expm1-def 51.3%

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

      2. neg-mul-1 51.3%

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

   7. Simplified 51.3%

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

   if -2.2e7 < x < 1.8500000000000001e83

   1. Initial program 55.7%

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

   3. Simplified 55.7%

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

   4. Taylor expanded in x around 0 42.0%

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

   5. Taylor expanded in eps around inf 83.9%

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

      1. associate-*r* 83.9%

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

      2. exp-prod 73.3%

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

      3. remove-double-neg 73.3%

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

      4. neg-mul-1 73.3%

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

      5. sub-neg 73.3%

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

      6. exp-prod 83.9%

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

      7. associate-*r* 83.9%

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

      8. mul-1-neg 83.9%

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

      9. associate-*r* 83.9%

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

      10. exp-prod 73.3%

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

      11. cancel-sign-sub-inv 73.3%

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

      12. metadata-eval 73.3%

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

      13. *-lft-identity 73.3%

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

      14. exp-prod 83.9%

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

      15. neg-mul-1 83.9%

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

   7. Simplified 83.9%

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

      1. add-sqr-sqrt 42.4%

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

      2. sqrt-unprod 76.8%

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

      3. sqr-neg 76.8%

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

      4. sqrt-unprod 38.0%

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

      5. add-sqr-sqrt 82.7%

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

      6. +-commutative 82.7%

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

      7. distribute-lft-in 82.7%

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

      8. *-rgt-identity 82.7%

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

   9. Applied egg-rr 82.7%

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

   10. Taylor expanded in eps around inf 83.2%

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

       1. *-commutative 83.2%

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

   12. Simplified 83.2%

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

   if 1.8500000000000001e83 < 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 26.3%

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

   5. Taylor expanded in x around 0 56.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -22000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+83}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 6: 84.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1.1e-228)
   (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
   (if (<= x 3.2e+83)
     (/ (+ 1.0 (exp (* eps x))) 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 <= -1.1e-228) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else if (x <= 3.2e+83) {
		tmp = (1.0 + exp((eps * x))) / 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 <= (-1.1d-228)) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else if (x <= 3.2d+83) then
        tmp = (1.0d0 + exp((eps * x))) / 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 <= -1.1e-228) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else if (x <= 3.2e+83) {
		tmp = (1.0 + Math.exp((eps * x))) / 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 <= -1.1e-228:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	elif x <= 3.2e+83:
		tmp = (1.0 + math.exp((eps * x))) / 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 <= -1.1e-228)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	elseif (x <= 3.2e+83)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 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 <= -1.1e-228)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	elseif (x <= 3.2e+83)
		tmp = (1.0 + exp((eps * x))) / 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, -1.1e-228], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.2e+83], N[(N[(1.0 + N[Exp[N[(eps * x), $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 3 regimes

2. if x < -1.1e-228

   1. Initial program 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in x around 0 41.5%

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

   5. Taylor expanded in eps around inf 68.4%

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

      1. associate-*r* 68.4%

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

      2. exp-prod 61.7%

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

      3. remove-double-neg 61.7%

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

      4. neg-mul-1 61.7%

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

      5. sub-neg 61.7%

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

      6. exp-prod 68.4%

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

      7. associate-*r* 68.4%

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

      8. mul-1-neg 68.4%

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

      9. associate-*r* 68.4%

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

      10. exp-prod 61.7%

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

      11. cancel-sign-sub-inv 61.7%

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

      12. metadata-eval 61.7%

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

      13. *-lft-identity 61.7%

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

      14. exp-prod 68.4%

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

      15. neg-mul-1 68.4%

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

   7. Simplified 68.4%

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

   8. Taylor expanded in eps around inf 69.0%

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

   if -1.1e-228 < x < 3.1999999999999999e83

   1. Initial program 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in x around 0 46.7%

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

   5. Taylor expanded in eps around inf 87.4%

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

      1. associate-*r* 87.4%

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

      2. exp-prod 77.0%

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

      3. remove-double-neg 77.0%

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

      4. neg-mul-1 77.0%

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

      5. sub-neg 77.0%

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

      6. exp-prod 87.4%

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

      7. associate-*r* 87.4%

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

      8. mul-1-neg 87.4%

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

      9. associate-*r* 87.4%

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

      10. exp-prod 77.0%

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

      11. cancel-sign-sub-inv 77.0%

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

      12. metadata-eval 77.0%

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

      13. *-lft-identity 77.0%

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

      14. exp-prod 87.4%

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

      15. neg-mul-1 87.4%

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

   7. Simplified 87.4%

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

      1. add-sqr-sqrt 20.6%

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

      2. sqrt-unprod 77.9%

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

      3. sqr-neg 77.9%

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

      4. sqrt-unprod 61.1%

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

      5. add-sqr-sqrt 81.7%

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

      6. +-commutative 81.7%

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

      7. distribute-lft-in 81.7%

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

      8. *-rgt-identity 81.7%

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

   9. Applied egg-rr 81.7%

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

   10. Taylor expanded in eps around inf 81.9%

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

       1. *-commutative 81.9%

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

   12. Simplified 81.9%

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

   if 3.1999999999999999e83 < 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 26.3%

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

   5. Taylor expanded in x around 0 56.1%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 7: 64.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 180:\\ \;\;\;\;\frac{2 + \left(x - \varepsilon \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 180.0)
   (/ (+ 2.0 (- x (* eps x))) 2.0)
   (if (<= x 5.9e+84)
     (/ (/ (expm1 x) 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 <= 180.0) {
		tmp = (2.0 + (x - (eps * x))) / 2.0;
	} else if (x <= 5.9e+84) {
		tmp = (expm1(x) / eps) / 2.0;
	} else {
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 180.0) {
		tmp = (2.0 + (x - (eps * x))) / 2.0;
	} else if (x <= 5.9e+84) {
		tmp = (Math.expm1(x) / 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 <= 180.0:
		tmp = (2.0 + (x - (eps * x))) / 2.0
	elif x <= 5.9e+84:
		tmp = (math.expm1(x) / 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 <= 180.0)
		tmp = Float64(Float64(2.0 + Float64(x - Float64(eps * x))) / 2.0);
	elseif (x <= 5.9e+84)
		tmp = Float64(Float64(expm1(x) / 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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 180.0], N[(N[(2.0 + N[(x - N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.9e+84], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $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 3 regimes

2. if x < 180

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

   4. Taylor expanded in x around 0 42.8%

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

   5. Taylor expanded in x around 0 49.4%

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

      1. mul-1-neg 49.4%

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

      2. unsub-neg 49.4%

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

      3. associate-*r* 49.4%

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

      4. *-commutative 49.4%

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

   7. Simplified 49.4%

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

   8. Taylor expanded in eps around inf 65.4%

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

      1. *-commutative 65.4%

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

      2. sub-neg 65.4%

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

      3. distribute-rgt-in 65.4%

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

      4. *-un-lft-identity 65.4%

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

      5. distribute-lft-neg-in 65.4%

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

      6. distribute-rgt-neg-in 65.4%

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

      7. add-sqr-sqrt 38.6%

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

      8. sqrt-unprod 67.1%

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

      9. sqr-neg 67.1%

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

      10. sqrt-unprod 27.0%

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

      11. add-sqr-sqrt 67.0%

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

      12. *-commutative 67.0%

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

      13. +-commutative 67.0%

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

      14. add-sqr-sqrt 27.0%

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

      15. sqrt-unprod 63.2%

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

      16. sqr-neg 63.2%

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

      17. sqrt-unprod 40.0%

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

      18. add-sqr-sqrt 67.0%

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

      19. sub-neg 67.0%

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

      20. *-commutative 67.0%

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

   10. Applied egg-rr 67.0%

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

   if 180 < x < 5.89999999999999984e84

   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 40.2%

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

   5. Taylor expanded in eps around 0 1.8%

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

      1. expm1-def 1.8%

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

      2. neg-mul-1 1.8%

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

   7. Simplified 1.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 1.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}\right)\right)}}{2} \]

      2. expm1-udef 1.7%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}\right)} - 1}}{2} \]

      3. expm1-udef 1.7%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{e^{-x} - 1}}{\varepsilon}\right)} - 1}{2} \]

      4. expm1-udef 1.7%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{expm1}\left(-x\right)}}{\varepsilon}\right)} - 1}{2} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      6. sqrt-unprod 38.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)}{\varepsilon}\right)} - 1}{2} \]

      7. sqr-neg 38.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\sqrt{\color{blue}{x \cdot x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      8. sqrt-unprod 38.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      9. add-sqr-sqrt 38.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{x}\right)}{\varepsilon}\right)} - 1}{2} \]

   9. Applied egg-rr 38.5%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)} - 1}}{2} \]
   10. Step-by-step derivation

       1. expm1-def 38.5%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)\right)}}{2} \]

       2. expm1-log1p 38.7%

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

   11. Simplified 38.7%

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

   if 5.89999999999999984e84 < 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 26.3%

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

   5. Taylor expanded in x around 0 56.1%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 180:\\ \;\;\;\;\frac{2 + \left(x - \varepsilon \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 8: 70.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 660:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 660.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 4.6e+82)
     (/ (/ (expm1 x) 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 <= 660.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 4.6e+82) {
		tmp = (expm1(x) / eps) / 2.0;
	} else {
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 660.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 4.6e+82) {
		tmp = (Math.expm1(x) / 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 <= 660.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 4.6e+82:
		tmp = (math.expm1(x) / 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 <= 660.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 4.6e+82)
		tmp = Float64(Float64(expm1(x) / 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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 660.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.6e+82], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $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 3 regimes

2. if x < 660

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

   4. Taylor expanded in x around 0 43.9%

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

   5. Taylor expanded in eps around inf 80.0%

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

      1. associate-*r* 80.0%

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

      2. exp-prod 70.9%

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

      3. remove-double-neg 70.9%

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

      4. neg-mul-1 70.9%

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

      5. sub-neg 70.9%

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

      6. exp-prod 80.0%

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

      7. associate-*r* 80.0%

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

      8. mul-1-neg 80.0%

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

      9. associate-*r* 80.0%

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

      10. exp-prod 70.9%

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

      11. cancel-sign-sub-inv 70.9%

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

      12. metadata-eval 70.9%

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

      13. *-lft-identity 70.9%

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

      14. exp-prod 80.0%

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

      15. neg-mul-1 80.0%

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

   7. Simplified 80.0%

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

   8. Taylor expanded in eps around 0 80.6%

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

   if 660 < x < 4.59999999999999976e82

   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 40.2%

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

   5. Taylor expanded in eps around 0 1.8%

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

      1. expm1-def 1.8%

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

      2. neg-mul-1 1.8%

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

   7. Simplified 1.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 1.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}\right)\right)}}{2} \]

      2. expm1-udef 1.7%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}\right)} - 1}}{2} \]

      3. expm1-udef 1.7%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{e^{-x} - 1}}{\varepsilon}\right)} - 1}{2} \]

      4. expm1-udef 1.7%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{expm1}\left(-x\right)}}{\varepsilon}\right)} - 1}{2} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      6. sqrt-unprod 38.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)}{\varepsilon}\right)} - 1}{2} \]

      7. sqr-neg 38.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\sqrt{\color{blue}{x \cdot x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      8. sqrt-unprod 38.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      9. add-sqr-sqrt 38.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{x}\right)}{\varepsilon}\right)} - 1}{2} \]

   9. Applied egg-rr 38.5%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)} - 1}}{2} \]
   10. Step-by-step derivation

       1. expm1-def 38.5%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)\right)}}{2} \]

       2. expm1-log1p 38.7%

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

   11. Simplified 38.7%

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

   if 4.59999999999999976e82 < 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 26.3%

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

   5. Taylor expanded in x around 0 56.1%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 660:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 9: 63.6% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 135:\\ \;\;\;\;\frac{2 + \left(x - \varepsilon \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 135.0)
   (/ (+ 2.0 (- x (* eps x))) 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 <= 135.0) {
		tmp = (2.0 + (x - (eps * x))) / 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 <= 135.0d0) then
        tmp = (2.0d0 + (x - (eps * x))) / 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 <= 135.0) {
		tmp = (2.0 + (x - (eps * x))) / 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 <= 135.0:
		tmp = (2.0 + (x - (eps * x))) / 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 <= 135.0)
		tmp = Float64(Float64(2.0 + Float64(x - Float64(eps * x))) / 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 <= 135.0)
		tmp = (2.0 + (x - (eps * x))) / 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, 135.0], N[(N[(2.0 + N[(x - N[(eps * x), $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 < 135

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

   4. Taylor expanded in x around 0 42.8%

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

   5. Taylor expanded in x around 0 49.4%

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

      1. mul-1-neg 49.4%

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

      2. unsub-neg 49.4%

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

      3. associate-*r* 49.4%

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

      4. *-commutative 49.4%

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

   7. Simplified 49.4%

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

   8. Taylor expanded in eps around inf 65.4%

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

      1. *-commutative 65.4%

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

      2. sub-neg 65.4%

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

      3. distribute-rgt-in 65.4%

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

      4. *-un-lft-identity 65.4%

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

      5. distribute-lft-neg-in 65.4%

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

      6. distribute-rgt-neg-in 65.4%

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

      7. add-sqr-sqrt 38.6%

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

      8. sqrt-unprod 67.1%

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

      9. sqr-neg 67.1%

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

      10. sqrt-unprod 27.0%

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

      11. add-sqr-sqrt 67.0%

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

      12. *-commutative 67.0%

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

      13. +-commutative 67.0%

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

      14. add-sqr-sqrt 27.0%

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

      15. sqrt-unprod 63.2%

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

      16. sqr-neg 63.2%

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

      17. sqrt-unprod 40.0%

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

      18. add-sqr-sqrt 67.0%

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

      19. sub-neg 67.0%

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

      20. *-commutative 67.0%

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

   10. Applied egg-rr 67.0%

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

   if 135 < 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 29.1%

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

   5. Taylor expanded in x around 0 48.6%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 135:\\ \;\;\;\;\frac{2 + \left(x - \varepsilon \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 10: 58.4% accurate, 17.4× speedup

Math

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

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1.0)
   (/ (+ 2.0 (* x (- (/ 1.0 eps) eps))) 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 = (2.0 + (x * ((1.0 / eps) - 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 <= (-1.0d0)) then
        tmp = (2.0d0 + (x * ((1.0d0 / eps) - 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 <= -1.0) {
		tmp = (2.0 + (x * ((1.0 / eps) - 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 <= -1.0:
		tmp = (2.0 + (x * ((1.0 / eps) - 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 <= -1.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps) - 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 <= -1.0)
		tmp = (2.0 + (x * ((1.0 / eps) - 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, -1.0], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps), $MachinePrecision] - 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 < -1

   1. Initial program 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in x around 0 50.3%

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

   5. Taylor expanded in x around 0 26.0%

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

      1. mul-1-neg 26.0%

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

      2. unsub-neg 26.0%

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

      3. associate-*r* 26.0%

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

      4. *-commutative 26.0%

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

   7. Simplified 26.0%

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

   8. Taylor expanded in eps around 0 26.0%

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

      1. div-inv 26.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 33.0%

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

      4. sqr-neg 33.0%

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

      5. sqrt-unprod 26.1%

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

      6. add-sqr-sqrt 26.1%

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

      7. cancel-sign-sub-inv 26.1%

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

      8. div-inv 26.1%

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

      9. mul-1-neg 26.1%

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

      10. distribute-rgt-neg-in 26.1%

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

      11. add-sqr-sqrt 26.1%

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

      12. sqrt-unprod 33.0%

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

      13. sqr-neg 33.0%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 33.1%

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

   10. Applied egg-rr 33.1%

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

       1. *-lft-identity 33.1%

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

       2. associate-*l/ 33.1%

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

       3. cancel-sign-sub-inv 33.1%

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

       4. distribute-rgt-in 33.1%

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

       5. distribute-neg-frac 33.1%

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

       6. metadata-eval 33.1%

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

   12. Simplified 33.1%

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

   if -1 < x

   1. Initial program 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in x around 0 37.2%

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

   5. Taylor expanded in x around 0 43.8%

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

      1. mul-1-neg 43.8%

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

      2. unsub-neg 43.8%

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

      3. associate-*r* 43.8%

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

      4. *-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in eps around inf 58.1%

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

   9. Taylor expanded in eps around inf 58.6%

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

       1. associate-*r* 58.6%

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

       2. neg-mul-1 58.6%

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

   11. Simplified 58.6%

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

4. Final simplification 54.5%

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

Alternative 11: 58.3% accurate, 20.5× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -22000000:\\ \;\;\;\;\frac{2 - \left(x + \varepsilon \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -22000000.0)
   (/ (- 2.0 (+ x (* eps x))) 2.0)
   (/ (+ 2.0 (* eps x)) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -22000000.0) {
		tmp = (2.0 - (x + (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 <= (-22000000.0d0)) then
        tmp = (2.0d0 - (x + (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 <= -22000000.0) {
		tmp = (2.0 - (x + (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 <= -22000000.0:
		tmp = (2.0 - (x + (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 <= -22000000.0)
		tmp = Float64(Float64(2.0 - Float64(x + Float64(eps * 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 <= -22000000.0)
		tmp = (2.0 - (x + (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, -22000000.0], N[(N[(2.0 - N[(x + N[(eps * x), $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 < -2.2e7

   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 50.3%

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

   5. Taylor expanded in x around 0 27.2%

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

      1. mul-1-neg 27.2%

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

      2. unsub-neg 27.2%

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

      3. associate-*r* 27.2%

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

      4. *-commutative 27.2%

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

   7. Simplified 27.2%

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

   8. Taylor expanded in eps around inf 27.2%

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

      1. *-commutative 27.2%

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

      2. sub-neg 27.2%

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

      3. distribute-rgt-in 27.2%

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

      4. *-un-lft-identity 27.2%

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

      5. distribute-lft-neg-in 27.2%

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

      6. distribute-rgt-neg-in 27.2%

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

      7. add-sqr-sqrt 27.2%

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

      8. sqrt-unprod 34.5%

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

      9. sqr-neg 34.5%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 34.6%

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

      12. *-commutative 34.6%

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

      13. +-commutative 34.6%

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

      14. *-commutative 34.6%

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

   10. Applied egg-rr 34.6%

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

   if -2.2e7 < x

   1. Initial program 66.5%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in x around 0 37.3%

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

   5. Taylor expanded in x around 0 43.4%

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

      1. mul-1-neg 43.4%

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

      2. unsub-neg 43.4%

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

      3. associate-*r* 43.4%

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

      4. *-commutative 43.4%

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

   7. Simplified 43.4%

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

   8. Taylor expanded in eps around inf 57.6%

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

   9. Taylor expanded in eps around inf 58.0%

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

       1. associate-*r* 58.0%

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

       2. neg-mul-1 58.0%

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

   11. Simplified 58.0%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -22000000:\\ \;\;\;\;\frac{2 - \left(x + \varepsilon \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \]

Alternative 12: 58.4% accurate, 20.5× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.013:\\ \;\;\;\;\frac{2 + \left(x - \varepsilon \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -0.013) (/ (+ 2.0 (- x (* eps x))) 2.0) (/ (+ 2.0 (* eps x)) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -0.013) {
		tmp = (2.0 + (x - (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 <= (-0.013d0)) then
        tmp = (2.0d0 + (x - (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 <= -0.013) {
		tmp = (2.0 + (x - (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 <= -0.013:
		tmp = (2.0 + (x - (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 <= -0.013)
		tmp = Float64(Float64(2.0 + Float64(x - Float64(eps * 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 <= -0.013)
		tmp = (2.0 + (x - (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, -0.013], N[(N[(2.0 + N[(x - N[(eps * x), $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 < -0.0129999999999999994

   1. Initial program 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in x around 0 50.3%

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

   5. Taylor expanded in x around 0 26.0%

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

      1. mul-1-neg 26.0%

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

      2. unsub-neg 26.0%

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

      3. associate-*r* 26.0%

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

      4. *-commutative 26.0%

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

   7. Simplified 26.0%

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

   8. Taylor expanded in eps around inf 25.9%

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

      1. *-commutative 25.9%

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

      2. sub-neg 25.9%

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

      3. distribute-rgt-in 25.9%

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

      4. *-un-lft-identity 25.9%

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

      5. distribute-lft-neg-in 25.9%

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

      6. distribute-rgt-neg-in 25.9%

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

      7. add-sqr-sqrt 25.9%

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

      8. sqrt-unprod 32.8%

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

      9. sqr-neg 32.8%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 33.0%

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

      12. *-commutative 33.0%

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

      13. +-commutative 33.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 15.7%

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

      16. sqr-neg 15.7%

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

      17. sqrt-unprod 33.1%

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

      18. add-sqr-sqrt 33.1%

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

      19. sub-neg 33.1%

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

      20. *-commutative 33.1%

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

   10. Applied egg-rr 33.1%

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

   if -0.0129999999999999994 < x

   1. Initial program 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in x around 0 37.2%

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

   5. Taylor expanded in x around 0 43.8%

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

      1. mul-1-neg 43.8%

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

      2. unsub-neg 43.8%

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

      3. associate-*r* 43.8%

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

      4. *-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in eps around inf 58.1%

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

   9. Taylor expanded in eps around inf 58.6%

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

       1. associate-*r* 58.6%

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

       2. neg-mul-1 58.6%

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

   11. Simplified 58.6%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.013:\\ \;\;\;\;\frac{2 + \left(x - \varepsilon \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \]

Alternative 13: 50.5% accurate, 32.4× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \frac{2 + \varepsilon \cdot x}{2} \end{array} \]

FPCore

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

C

eps = abs(eps);
double code(double x, double eps) {
	return (2.0 + (eps * x)) / 2.0;
}

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 = (2.0d0 + (eps * x)) / 2.0d0
end function

Java

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

Python

eps = abs(eps)
def code(x, eps):
	return (2.0 + (eps * x)) / 2.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Simplified 71.6%

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

4. Taylor expanded in x around 0 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 x around 0 41.0%

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

   1. mul-1-neg 41.0%

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

   2. unsub-neg 41.0%

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

   3. associate-*r* 41.0%

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

   4. *-commutative 41.0%

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

7. Simplified 41.0%

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

8. Taylor expanded in eps around inf 52.9%

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

9. Taylor expanded in eps around inf 53.4%

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

    1. associate-*r* 53.4%

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

    2. neg-mul-1 53.4%

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

11. Simplified 53.4%

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

12. Final simplification 53.4%

    \[\leadsto \frac{2 + \varepsilon \cdot x}{2} \]

Alternative 14: 44.4% 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 71.6%

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

3. Simplified 71.6%

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

4. Taylor expanded in x around 0 46.2%

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

5. Final simplification 46.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023314 
(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))