NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.7% → 100.0%

Time: 22.5s

Alternatives: 13

Speedup: 2.1×

• 39.0% 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: 73.7% 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

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\varepsilon \leq 0.1:\\ \;\;\;\;\frac{t_0 \cdot \left(\left(x + 1\right) - -1\right) + x \cdot t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps 0.1)
     (/ (+ (* t_0 (- (+ x 1.0) -1.0)) (* x t_0)) 2.0)
     (/ (+ (exp (* x (+ eps -1.0))) (exp (* x (- eps)))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (eps <= 0.1) {
		tmp = ((t_0 * ((x + 1.0) - -1.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (exp((x * (eps + -1.0))) + exp((x * -eps))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps <= 0.1d0) then
        tmp = ((t_0 * ((x + 1.0d0) - (-1.0d0))) + (x * t_0)) / 2.0d0
    else
        tmp = (exp((x * (eps + (-1.0d0)))) + exp((x * -eps))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps <= 0.1) {
		tmp = ((t_0 * ((x + 1.0) - -1.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp((x * -eps))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = math.exp(-x)
	tmp = 0
	if eps <= 0.1:
		tmp = ((t_0 * ((x + 1.0) - -1.0)) + (x * t_0)) / 2.0
	else:
		tmp = (math.exp((x * (eps + -1.0))) + math.exp((x * -eps))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps <= 0.1)
		tmp = Float64(Float64(Float64(t_0 * Float64(Float64(x + 1.0) - -1.0)) + Float64(x * t_0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(x * Float64(-eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps <= 0.1)
		tmp = ((t_0 * ((x + 1.0) - -1.0)) + (x * t_0)) / 2.0;
	else
		tmp = (exp((x * (eps + -1.0))) + exp((x * -eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps, 0.1], N[(N[(N[(t$95$0 * N[(N[(x + 1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.10000000000000001

   1. Initial program 60.7%

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

      1. sub-neg 60.7%

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

      2. neg-sub0 60.7%

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

      3. associate-+r- 60.7%

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

   3. Simplified 60.7%

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

   4. Taylor expanded in eps around 0 73.3%

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

      1. associate--r+ 73.3%

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

      2. associate-*r* 73.3%

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

      3. neg-mul-1 73.3%

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

      4. cancel-sign-sub 73.3%

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

      5. distribute-rgt1-in 73.3%

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

      6. distribute-rgt-out-- 73.3%

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

      7. neg-mul-1 73.3%

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

      8. neg-mul-1 73.3%

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

   6. Simplified 73.3%

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

   if 0.10000000000000001 < eps

   1. Initial program 100.0%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. neg-mul-1 100.0%

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

      4. sub-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. associate-*r* 100.0%

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

      7. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.1:\\ \;\;\;\;\frac{e^{-x} \cdot \left(\left(x + 1\right) - -1\right) + x \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Simplified 53.4%

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

4. Taylor expanded in eps around inf 98.6%

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

5. Final simplification 98.6%

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

Alternative 3: 84.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 8.2e+108) (/ (+ (exp (* x (- eps))) (exp (* x eps))) 2.0) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 8.2e+108) {
		tmp = (exp((x * -eps)) + exp((x * eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= 8.2d+108) then
        tmp = (exp((x * -eps)) + exp((x * eps))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 8.2e+108) {
		tmp = (Math.exp((x * -eps)) + Math.exp((x * eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 8.2e+108:
		tmp = (math.exp((x * -eps)) + math.exp((x * eps))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 8.2e+108)
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps))) + exp(Float64(x * eps))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 8.2e+108)
		tmp = (exp((x * -eps)) + exp((x * eps))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 8.2e+108], N[(N[(N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.1999999999999998e108

   1. Initial program 68.5%

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

   3. Simplified 45.1%

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

   4. Taylor expanded in eps around inf 98.3%

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

   5. Taylor expanded in eps around inf 94.3%

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

      1. *-commutative 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in eps around -inf 94.3%

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

      1. associate-*r* 94.3%

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

      2. neg-mul-1 94.3%

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

      3. neg-mul-1 94.3%

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

      4. sub-neg 94.3%

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

      5. mul-1-neg 94.3%

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

      6. associate-*r* 94.3%

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

      7. neg-mul-1 94.3%

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

   10. Simplified 94.3%

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

   11. Taylor expanded in eps around inf 94.7%

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

   if 8.1999999999999998e108 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 59.6%

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

      1. neg-mul-1 59.6%

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

      2. rec-exp 59.6%

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

      3. neg-mul-1 59.6%

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

      4. div-sub 59.6%

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

      5. +-inverses 59.6%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 59.6%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 92.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Simplified 53.4%

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

4. Taylor expanded in eps around inf 98.6%

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

5. Taylor expanded in eps around inf 88.7%

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

   1. *-commutative 88.7%

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

7. Simplified 88.7%

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

8. Taylor expanded in eps around -inf 88.7%

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

   1. associate-*r* 88.7%

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

   2. neg-mul-1 88.7%

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

   3. neg-mul-1 88.7%

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

   4. sub-neg 88.7%

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

   5. mul-1-neg 88.7%

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

   6. associate-*r* 88.7%

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

   7. neg-mul-1 88.7%

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

10. Simplified 88.7%

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

11. Final simplification 88.7%

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

Alternative 5: 83.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-268}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+109}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-268)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (<= x 2.8e+109) (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0) 0.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -2e-268) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 2.8e+109) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= (-2d-268)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if (x <= 2.8d+109) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -2e-268) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 2.8e+109) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -2e-268:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif x <= 2.8e+109:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -2e-268)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 2.8e+109)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2e-268)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif (x <= 2.8e+109)
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -2e-268], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.8e+109], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999992e-268

   1. Initial program 72.0%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in eps around inf 97.6%

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

   5. Taylor expanded in x around 0 73.0%

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

   6. Taylor expanded in eps around -inf 73.0%

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

      1. sub-neg 73.0%

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

      2. mul-1-neg 73.0%

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

      3. remove-double-neg 73.0%

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

      4. associate-*r* 73.0%

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

      5. exp-prod 64.9%

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

      6. cancel-sign-sub-inv 64.9%

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

      7. metadata-eval 64.9%

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

      8. *-lft-identity 64.9%

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

      9. +-commutative 64.9%

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

      10. exp-prod 73.0%

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

      11. *-commutative 73.0%

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

      12. associate-*l* 73.0%

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

      13. neg-mul-1 73.0%

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

   8. Simplified 73.0%

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

   if -1.99999999999999992e-268 < x < 2.8000000000000002e109

   1. Initial program 65.3%

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

      1. sub-neg 65.3%

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

      2. neg-sub0 65.3%

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

      3. associate-+r- 65.3%

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

   3. Simplified 65.3%

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

   4. Taylor expanded in x around 0 43.9%

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

   5. Taylor expanded in eps around inf 78.3%

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

      1. *-commutative 78.3%

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

      2. sub-neg 78.3%

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

      3. mul-1-neg 78.3%

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

      4. *-commutative 78.3%

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

      5. associate-*r* 78.3%

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

      6. mul-1-neg 78.3%

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

      7. mul-1-neg 78.3%

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

      8. sub-neg 78.3%

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

   7. Simplified 78.3%

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

   if 2.8000000000000002e109 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 59.6%

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

      1. neg-mul-1 59.6%

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

      2. rec-exp 59.6%

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

      3. neg-mul-1 59.6%

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

      4. div-sub 59.6%

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

      5. +-inverses 59.6%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 59.6%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-268}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+109}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 79.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.95 \cdot 10^{-272}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+56}:\\ \;\;\;\;\frac{2 - x \cdot \left(\frac{1}{\varepsilon} + \frac{-1 - \frac{-1}{\varepsilon}}{\varepsilon + -1} \cdot \left(1 + \varepsilon \cdot \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 2.95e-272)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (<= x 1.6e+56)
     (/
      (-
       2.0
       (*
        x
        (+
         (/ 1.0 eps)
         (* (/ (- -1.0 (/ -1.0 eps)) (+ eps -1.0)) (+ 1.0 (* eps eps))))))
      2.0)
     0.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 2.95e-272) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 1.6e+56) {
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.95d-272) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if (x <= 1.6d+56) then
        tmp = (2.0d0 - (x * ((1.0d0 / eps) + ((((-1.0d0) - ((-1.0d0) / eps)) / (eps + (-1.0d0))) * (1.0d0 + (eps * eps)))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.95e-272) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 1.6e+56) {
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 2.95e-272:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif x <= 1.6e+56:
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 2.95e-272)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 1.6e+56)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(Float64(-1.0 - Float64(-1.0 / eps)) / Float64(eps + -1.0)) * Float64(1.0 + Float64(eps * eps)))))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.95e-272)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif (x <= 1.6e+56)
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 2.95e-272], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.6e+56], N[(N[(2.0 - N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(N[(-1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] / N[(eps + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 2.95e-272

   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 44.2%

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

   4. Taylor expanded in eps around inf 98.0%

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

   5. Taylor expanded in x around 0 77.7%

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

   6. Taylor expanded in eps around -inf 77.7%

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

      1. sub-neg 77.7%

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

      2. mul-1-neg 77.7%

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

      3. remove-double-neg 77.7%

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

      4. associate-*r* 77.7%

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

      5. exp-prod 71.0%

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

      6. cancel-sign-sub-inv 71.0%

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

      7. metadata-eval 71.0%

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

      8. *-lft-identity 71.0%

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

      9. +-commutative 71.0%

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

      10. exp-prod 77.7%

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

      11. *-commutative 77.7%

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

      12. associate-*l* 77.7%

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

      13. neg-mul-1 77.7%

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

   8. Simplified 77.7%

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

   if 2.95e-272 < x < 1.60000000000000002e56

   1. Initial program 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in x around 0 62.9%

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

   5. Taylor expanded in eps around 0 64.9%

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

      1. *-commutative 64.9%

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

      2. flip-+ 73.5%

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

      3. associate-*r/ 73.5%

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

      4. sub-neg 73.5%

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

      5. distribute-neg-frac 73.5%

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

      6. metadata-eval 73.5%

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

      7. metadata-eval 73.5%

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

   7. Applied egg-rr 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/l* 73.5%

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

   9. Simplified 73.5%

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

       1. div-inv 73.5%

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

       2. sub-neg 73.5%

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

       3. distribute-rgt-neg-in 73.5%

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

       4. add-sqr-sqrt 41.6%

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

       5. sqrt-unprod 90.5%

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

       6. sqr-neg 90.5%

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

       7. sqrt-prod 48.9%

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

       8. add-sqr-sqrt 77.5%

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

       9. clear-num 77.5%

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

       10. frac-2neg 77.5%

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

   11. Applied egg-rr 77.5%

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

       1. *-commutative 77.5%

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

   13. Simplified 77.5%

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

   if 1.60000000000000002e56 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 55.4%

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

      1. neg-mul-1 55.4%

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

      2. rec-exp 55.4%

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

      3. neg-mul-1 55.4%

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

      4. div-sub 55.4%

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

      5. +-inverses 55.4%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 55.4%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.95 \cdot 10^{-272}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+56}:\\ \;\;\;\;\frac{2 - x \cdot \left(\frac{1}{\varepsilon} + \frac{-1 - \frac{-1}{\varepsilon}}{\varepsilon + -1} \cdot \left(1 + \varepsilon \cdot \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 77.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -350000000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-206}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\varepsilon + -1\right) \cdot \left(1 - \frac{-1}{\varepsilon}\right) + \frac{-1 + \varepsilon \cdot \varepsilon}{\frac{1 - \varepsilon}{1 + \frac{-1}{\varepsilon}}}\right)}{2}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-272}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+56}:\\ \;\;\;\;\frac{2 - x \cdot \left(\frac{1}{\varepsilon} + \frac{-1 - \frac{-1}{\varepsilon}}{\varepsilon + -1} \cdot \left(1 + \varepsilon \cdot \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -350000000000.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (<= x -1.4e-206)
     (/
      (+
       2.0
       (*
        x
        (+
         (* (+ eps -1.0) (- 1.0 (/ -1.0 eps)))
         (/ (+ -1.0 (* eps eps)) (/ (- 1.0 eps) (+ 1.0 (/ -1.0 eps)))))))
      2.0)
     (if (<= x 2.9e-272)
       (/ (- 2.0 (* x eps)) 2.0)
       (if (<= x 1.1e+56)
         (/
          (-
           2.0
           (*
            x
            (+
             (/ 1.0 eps)
             (* (/ (- -1.0 (/ -1.0 eps)) (+ eps -1.0)) (+ 1.0 (* eps eps))))))
          2.0)
         0.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -350000000000.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= -1.4e-206) {
		tmp = (2.0 + (x * (((eps + -1.0) * (1.0 - (-1.0 / eps))) + ((-1.0 + (eps * eps)) / ((1.0 - eps) / (1.0 + (-1.0 / eps))))))) / 2.0;
	} else if (x <= 2.9e-272) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else if (x <= 1.1e+56) {
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -350000000000.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= -1.4e-206) {
		tmp = (2.0 + (x * (((eps + -1.0) * (1.0 - (-1.0 / eps))) + ((-1.0 + (eps * eps)) / ((1.0 - eps) / (1.0 + (-1.0 / eps))))))) / 2.0;
	} else if (x <= 2.9e-272) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else if (x <= 1.1e+56) {
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -350000000000.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= -1.4e-206:
		tmp = (2.0 + (x * (((eps + -1.0) * (1.0 - (-1.0 / eps))) + ((-1.0 + (eps * eps)) / ((1.0 - eps) / (1.0 + (-1.0 / eps))))))) / 2.0
	elif x <= 2.9e-272:
		tmp = (2.0 - (x * eps)) / 2.0
	elif x <= 1.1e+56:
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -350000000000.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= -1.4e-206)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(eps + -1.0) * Float64(1.0 - Float64(-1.0 / eps))) + Float64(Float64(-1.0 + Float64(eps * eps)) / Float64(Float64(1.0 - eps) / Float64(1.0 + Float64(-1.0 / eps))))))) / 2.0);
	elseif (x <= 2.9e-272)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	elseif (x <= 1.1e+56)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(Float64(-1.0 - Float64(-1.0 / eps)) / Float64(eps + -1.0)) * Float64(1.0 + Float64(eps * eps)))))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -350000000000.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.4e-206], N[(N[(2.0 + N[(x * N[(N[(N[(eps + -1.0), $MachinePrecision] * N[(1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - eps), $MachinePrecision] / N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.9e-272], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.1e+56], N[(N[(2.0 - N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(N[(-1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] / N[(eps + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 5 regimes

2. if x < -3.5e11

   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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 40.4%

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

   5. Taylor expanded in eps around 0 61.5%

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

      1. expm1-def 61.5%

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

      2. mul-1-neg 61.5%

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

   7. Simplified 61.5%

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

   if -3.5e11 < x < -1.4000000000000001e-206

   1. Initial program 55.2%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around 0 58.3%

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

      1. *-commutative 57.2%

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

      2. flip-+ 68.6%

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

      3. associate-*r/ 68.6%

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

      4. sub-neg 68.6%

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

      5. distribute-neg-frac 68.6%

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

      6. metadata-eval 68.6%

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

      7. metadata-eval 68.6%

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

   6. Applied egg-rr 69.3%

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

      1. *-commutative 68.6%

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

      2. associate-/l* 68.6%

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

   8. Simplified 69.3%

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

   if -1.4000000000000001e-206 < x < 2.89999999999999995e-272

   1. Initial program 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around 0 97.4%

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

   5. Taylor expanded in eps around 0 97.4%

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

   6. Taylor expanded in eps around 0 97.4%

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

      1. neg-mul-1 97.4%

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

   8. Simplified 97.4%

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

   if 2.89999999999999995e-272 < x < 1.10000000000000008e56

   1. Initial program 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in x around 0 62.9%

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

   5. Taylor expanded in eps around 0 64.9%

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

      1. *-commutative 64.9%

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

      2. flip-+ 73.5%

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

      3. associate-*r/ 73.5%

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

      4. sub-neg 73.5%

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

      5. distribute-neg-frac 73.5%

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

      6. metadata-eval 73.5%

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

      7. metadata-eval 73.5%

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

   7. Applied egg-rr 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/l* 73.5%

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

   9. Simplified 73.5%

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

       1. div-inv 73.5%

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

       2. sub-neg 73.5%

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

       3. distribute-rgt-neg-in 73.5%

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

       4. add-sqr-sqrt 41.6%

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

       5. sqrt-unprod 90.5%

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

       6. sqr-neg 90.5%

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

       7. sqrt-prod 48.9%

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

       8. add-sqr-sqrt 77.5%

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

       9. clear-num 77.5%

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

       10. frac-2neg 77.5%

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

   11. Applied egg-rr 77.5%

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

       1. *-commutative 77.5%

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

   13. Simplified 77.5%

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

   if 1.10000000000000008e56 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 55.4%

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

      1. neg-mul-1 55.4%

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

      2. rec-exp 55.4%

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

      3. neg-mul-1 55.4%

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

      4. div-sub 55.4%

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

      5. +-inverses 55.4%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 55.4%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -350000000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-206}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\varepsilon + -1\right) \cdot \left(1 - \frac{-1}{\varepsilon}\right) + \frac{-1 + \varepsilon \cdot \varepsilon}{\frac{1 - \varepsilon}{1 + \frac{-1}{\varepsilon}}}\right)}{2}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-272}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+56}:\\ \;\;\;\;\frac{2 - x \cdot \left(\frac{1}{\varepsilon} + \frac{-1 - \frac{-1}{\varepsilon}}{\varepsilon + -1} \cdot \left(1 + \varepsilon \cdot \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 72.4% accurate, 7.3× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-206}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{\varepsilon} + \frac{-1 + \varepsilon \cdot \varepsilon}{\frac{1 - \varepsilon}{1 + \frac{-1}{\varepsilon}}}\right)}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-272}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{2 - x \cdot \left(\frac{1}{\varepsilon} + \frac{-1 - \frac{-1}{\varepsilon}}{\varepsilon + -1} \cdot \left(1 + \varepsilon \cdot \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-206)
   (/
    (+
     2.0
     (*
      x
      (+
       (/ -1.0 eps)
       (/ (+ -1.0 (* eps eps)) (/ (- 1.0 eps) (+ 1.0 (/ -1.0 eps)))))))
    2.0)
   (if (<= x 2.8e-272)
     (/ (- 2.0 (* x eps)) 2.0)
     (if (<= x 3.5e+56)
       (/
        (-
         2.0
         (*
          x
          (+
           (/ 1.0 eps)
           (* (/ (- -1.0 (/ -1.0 eps)) (+ eps -1.0)) (+ 1.0 (* eps eps))))))
        2.0)
       0.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -2e-206) {
		tmp = (2.0 + (x * ((-1.0 / eps) + ((-1.0 + (eps * eps)) / ((1.0 - eps) / (1.0 + (-1.0 / eps))))))) / 2.0;
	} else if (x <= 2.8e-272) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else if (x <= 3.5e+56) {
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= (-2d-206)) then
        tmp = (2.0d0 + (x * (((-1.0d0) / eps) + (((-1.0d0) + (eps * eps)) / ((1.0d0 - eps) / (1.0d0 + ((-1.0d0) / eps))))))) / 2.0d0
    else if (x <= 2.8d-272) then
        tmp = (2.0d0 - (x * eps)) / 2.0d0
    else if (x <= 3.5d+56) then
        tmp = (2.0d0 - (x * ((1.0d0 / eps) + ((((-1.0d0) - ((-1.0d0) / eps)) / (eps + (-1.0d0))) * (1.0d0 + (eps * eps)))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -2e-206) {
		tmp = (2.0 + (x * ((-1.0 / eps) + ((-1.0 + (eps * eps)) / ((1.0 - eps) / (1.0 + (-1.0 / eps))))))) / 2.0;
	} else if (x <= 2.8e-272) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else if (x <= 3.5e+56) {
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -2e-206:
		tmp = (2.0 + (x * ((-1.0 / eps) + ((-1.0 + (eps * eps)) / ((1.0 - eps) / (1.0 + (-1.0 / eps))))))) / 2.0
	elif x <= 2.8e-272:
		tmp = (2.0 - (x * eps)) / 2.0
	elif x <= 3.5e+56:
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -2e-206)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 / eps) + Float64(Float64(-1.0 + Float64(eps * eps)) / Float64(Float64(1.0 - eps) / Float64(1.0 + Float64(-1.0 / eps))))))) / 2.0);
	elseif (x <= 2.8e-272)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	elseif (x <= 3.5e+56)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(Float64(-1.0 - Float64(-1.0 / eps)) / Float64(eps + -1.0)) * Float64(1.0 + Float64(eps * eps)))))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2e-206)
		tmp = (2.0 + (x * ((-1.0 / eps) + ((-1.0 + (eps * eps)) / ((1.0 - eps) / (1.0 + (-1.0 / eps))))))) / 2.0;
	elseif (x <= 2.8e-272)
		tmp = (2.0 - (x * eps)) / 2.0;
	elseif (x <= 3.5e+56)
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -2e-206], N[(N[(2.0 + N[(x * N[(N[(-1.0 / eps), $MachinePrecision] + N[(N[(-1.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - eps), $MachinePrecision] / N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.8e-272], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.5e+56], N[(N[(2.0 - N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(N[(-1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] / N[(eps + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.00000000000000006e-206

   1. Initial program 74.9%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in x around 0 34.2%

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

   5. Taylor expanded in eps around 0 48.6%

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

      1. *-commutative 48.6%

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

      2. flip-+ 56.0%

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

      3. associate-*r/ 56.0%

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

      4. sub-neg 56.0%

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

      5. distribute-neg-frac 56.0%

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

      6. metadata-eval 56.0%

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

      7. metadata-eval 56.0%

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

   7. Applied egg-rr 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-/l* 56.0%

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

   9. Simplified 56.0%

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

   if -2.00000000000000006e-206 < x < 2.79999999999999994e-272

   1. Initial program 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around 0 97.4%

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

   5. Taylor expanded in eps around 0 97.4%

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

   6. Taylor expanded in eps around 0 97.4%

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

      1. neg-mul-1 97.4%

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

   8. Simplified 97.4%

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

   if 2.79999999999999994e-272 < x < 3.49999999999999999e56

   1. Initial program 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in x around 0 62.9%

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

   5. Taylor expanded in eps around 0 64.9%

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

      1. *-commutative 64.9%

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

      2. flip-+ 73.5%

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

      3. associate-*r/ 73.5%

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

      4. sub-neg 73.5%

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

      5. distribute-neg-frac 73.5%

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

      6. metadata-eval 73.5%

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

      7. metadata-eval 73.5%

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

   7. Applied egg-rr 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/l* 73.5%

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

   9. Simplified 73.5%

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

       1. div-inv 73.5%

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

       2. sub-neg 73.5%

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

       3. distribute-rgt-neg-in 73.5%

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

       4. add-sqr-sqrt 41.6%

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

       5. sqrt-unprod 90.5%

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

       6. sqr-neg 90.5%

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

       7. sqrt-prod 48.9%

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

       8. add-sqr-sqrt 77.5%

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

       9. clear-num 77.5%

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

       10. frac-2neg 77.5%

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

   11. Applied egg-rr 77.5%

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

       1. *-commutative 77.5%

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

   13. Simplified 77.5%

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

   if 3.49999999999999999e56 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 55.4%

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

      1. neg-mul-1 55.4%

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

      2. rec-exp 55.4%

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

      3. neg-mul-1 55.4%

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

      4. div-sub 55.4%

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

      5. +-inverses 55.4%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 55.4%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-206}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{\varepsilon} + \frac{-1 + \varepsilon \cdot \varepsilon}{\frac{1 - \varepsilon}{1 + \frac{-1}{\varepsilon}}}\right)}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-272}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{2 - x \cdot \left(\frac{1}{\varepsilon} + \frac{-1 - \frac{-1}{\varepsilon}}{\varepsilon + -1} \cdot \left(1 + \varepsilon \cdot \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 66.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-276}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+56}:\\ \;\;\;\;\frac{2 - x \cdot \left(\frac{1}{\varepsilon} + \frac{-1 - \frac{-1}{\varepsilon}}{\varepsilon + -1} \cdot \left(1 + \varepsilon \cdot \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 1.7e-276)
   (/ (- 2.0 (* x eps)) 2.0)
   (if (<= x 1.75e+56)
     (/
      (-
       2.0
       (*
        x
        (+
         (/ 1.0 eps)
         (* (/ (- -1.0 (/ -1.0 eps)) (+ eps -1.0)) (+ 1.0 (* eps eps))))))
      2.0)
     0.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 1.7e-276) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else if (x <= 1.75e+56) {
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= 1.7d-276) then
        tmp = (2.0d0 - (x * eps)) / 2.0d0
    else if (x <= 1.75d+56) then
        tmp = (2.0d0 - (x * ((1.0d0 / eps) + ((((-1.0d0) - ((-1.0d0) / eps)) / (eps + (-1.0d0))) * (1.0d0 + (eps * eps)))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.7e-276) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else if (x <= 1.75e+56) {
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 1.7e-276:
		tmp = (2.0 - (x * eps)) / 2.0
	elif x <= 1.75e+56:
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 1.7e-276)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	elseif (x <= 1.75e+56)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(Float64(-1.0 - Float64(-1.0 / eps)) / Float64(eps + -1.0)) * Float64(1.0 + Float64(eps * eps)))))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.7e-276)
		tmp = (2.0 - (x * eps)) / 2.0;
	elseif (x <= 1.75e+56)
		tmp = (2.0 - (x * ((1.0 / eps) + (((-1.0 - (-1.0 / eps)) / (eps + -1.0)) * (1.0 + (eps * eps)))))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 1.7e-276], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.75e+56], N[(N[(2.0 - N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(N[(-1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] / N[(eps + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 1.69999999999999996e-276

   1. Initial program 69.2%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in x around 0 52.7%

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

   5. Taylor expanded in eps around 0 62.9%

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

   6. Taylor expanded in eps around 0 62.9%

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

      1. neg-mul-1 62.9%

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

   8. Simplified 62.9%

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

   if 1.69999999999999996e-276 < x < 1.75e56

   1. Initial program 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in x around 0 62.9%

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

   5. Taylor expanded in eps around 0 64.9%

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

      1. *-commutative 64.9%

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

      2. flip-+ 73.5%

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

      3. associate-*r/ 73.5%

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

      4. sub-neg 73.5%

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

      5. distribute-neg-frac 73.5%

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

      6. metadata-eval 73.5%

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

      7. metadata-eval 73.5%

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

   7. Applied egg-rr 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/l* 73.5%

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

   9. Simplified 73.5%

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

       1. div-inv 73.5%

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

       2. sub-neg 73.5%

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

       3. distribute-rgt-neg-in 73.5%

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

       4. add-sqr-sqrt 41.6%

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

       5. sqrt-unprod 90.5%

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

       6. sqr-neg 90.5%

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

       7. sqrt-prod 48.9%

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

       8. add-sqr-sqrt 77.5%

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

       9. clear-num 77.5%

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

       10. frac-2neg 77.5%

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

   11. Applied egg-rr 77.5%

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

       1. *-commutative 77.5%

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

   13. Simplified 77.5%

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

   if 1.75e56 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 55.4%

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

      1. neg-mul-1 55.4%

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

      2. rec-exp 55.4%

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

      3. neg-mul-1 55.4%

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

      4. div-sub 55.4%

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

      5. +-inverses 55.4%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 55.4%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-276}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+56}:\\ \;\;\;\;\frac{2 - x \cdot \left(\frac{1}{\varepsilon} + \frac{-1 - \frac{-1}{\varepsilon}}{\varepsilon + -1} \cdot \left(1 + \varepsilon \cdot \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 63.5% accurate, 25.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 185:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 185.0) (/ (- 2.0 (* x eps)) 2.0) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 185.0) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= 185.0d0) then
        tmp = (2.0d0 - (x * eps)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 185.0) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 185.0:
		tmp = (2.0 - (x * eps)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 185.0)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 185.0)
		tmp = (2.0 - (x * eps)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 185.0], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 185

   1. Initial program 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in x around 0 59.3%

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

   5. Taylor expanded in eps around 0 65.8%

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

   6. Taylor expanded in eps around 0 65.8%

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

      1. neg-mul-1 65.8%

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

   8. Simplified 65.8%

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

   if 185 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 51.6%

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

      1. neg-mul-1 51.6%

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

      2. rec-exp 51.6%

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

      3. neg-mul-1 51.6%

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

      4. div-sub 51.6%

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

      5. +-inverses 51.6%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 51.6%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 185:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 56.7% accurate, 32.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 2.0) (/ (- 2.0 x) 2.0) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = (2.0d0 - x) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 - x) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 2.0], N[(N[(2.0 - x), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 64.6%

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

   3. Simplified 38.2%

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

   4. Taylor expanded in eps around inf 98.1%

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

   5. Taylor expanded in eps around inf 98.1%

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

      1. *-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in x around 0 59.4%

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

      1. neg-mul-1 59.4%

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

      2. unsub-neg 59.4%

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

   10. Simplified 59.4%

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

   if 2 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 51.6%

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

      1. neg-mul-1 51.6%

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

      2. rec-exp 51.6%

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

      3. neg-mul-1 51.6%

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

      4. div-sub 51.6%

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

      5. +-inverses 51.6%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 51.6%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 56.7% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 500.0) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	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 <= 500.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 500.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 500.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 500.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 500.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 64.6%

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

      1. sub-neg 64.6%

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

      2. neg-sub0 64.6%

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

      3. associate-+r- 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in x around 0 59.3%

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

   if 500 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 51.6%

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

      1. neg-mul-1 51.6%

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

      2. rec-exp 51.6%

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

      3. neg-mul-1 51.6%

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

      4. div-sub 51.6%

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

      5. +-inverses 51.6%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 51.6%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 15.8% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ 0 \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 0.0)

C

eps = abs(eps);
double code(double x, double eps) {
	return 0.0;
}

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 = 0.0d0
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	return 0.0;
}

Python

eps = abs(eps)
def code(x, eps):
	return 0.0

Julia

eps = abs(eps)
function code(x, eps)
	return 0.0
end

MATLAB

eps = abs(eps)
function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := 0.0

Derivation

1. Initial program 73.3%

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

3. Simplified 73.3%

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

4. Taylor expanded in eps around 0 14.4%

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

   1. neg-mul-1 14.4%

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

   2. rec-exp 14.3%

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

   3. neg-mul-1 14.3%

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

   4. div-sub 14.3%

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

   5. +-inverses 14.6%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

6. Simplified 14.6%

   \[\leadsto \frac{\color{blue}{0}}{2} \]

7. Final simplification 14.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023278 
(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))