NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.8%

Time: 20.8s

Alternatives: 11

Speedup: 2.0×

• 38.0% 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: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.0066:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cosh \left(x \cdot eps\_m\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.0066)
   (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
   (/ (* 2.0 (cosh (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.0066) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (2.0 * cosh((x * eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.0066d0) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else
        tmp = (2.0d0 * cosh((x * eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.0066) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (2.0 * Math.cosh((x * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.0066:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	else:
		tmp = (2.0 * math.cosh((x * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.0066)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	else
		tmp = Float64(Float64(2.0 * cosh(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.0066)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	else
		tmp = (2.0 * cosh((x * eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.0066], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * N[Cosh[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0066

   1. Initial program 59.1%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in eps around 0 31.0%

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

   7. Simplified 72.5%

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

   8. Taylor expanded in eps around 0 72.5%

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

   if 0.0066 < eps

   1. Initial program 100.0%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. *-un-lft-identity 100.0%

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

       2. *-commutative 100.0%

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

       3. rec-exp 100.0%

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

       4. cosh-undef 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0066:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cosh \left(x \cdot \varepsilon\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

3. Simplified 65.2%

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

5. Taylor expanded in eps around inf 98.5%

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

6. Final simplification 98.5%

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

Alternative 3: 85.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 760:\\ \;\;\;\;\frac{2 \cdot \cosh \left(x \cdot eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+101}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+246}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 760.0)
   (/ (* 2.0 (cosh (* x eps_m))) 2.0)
   (if (<= x 6.6e+101)
     0.0
     (if (<= x 2.65e+246) (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 760.0) {
		tmp = (2.0 * cosh((x * eps_m))) / 2.0;
	} else if (x <= 6.6e+101) {
		tmp = 0.0;
	} else if (x <= 2.65e+246) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 760.0d0) then
        tmp = (2.0d0 * cosh((x * eps_m))) / 2.0d0
    else if (x <= 6.6d+101) then
        tmp = 0.0d0
    else if (x <= 2.65d+246) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 760.0) {
		tmp = (2.0 * Math.cosh((x * eps_m))) / 2.0;
	} else if (x <= 6.6e+101) {
		tmp = 0.0;
	} else if (x <= 2.65e+246) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 760.0:
		tmp = (2.0 * math.cosh((x * eps_m))) / 2.0
	elif x <= 6.6e+101:
		tmp = 0.0
	elif x <= 2.65e+246:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 760.0)
		tmp = Float64(Float64(2.0 * cosh(Float64(x * eps_m))) / 2.0);
	elseif (x <= 6.6e+101)
		tmp = 0.0;
	elseif (x <= 2.65e+246)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 760.0)
		tmp = (2.0 * cosh((x * eps_m))) / 2.0;
	elseif (x <= 6.6e+101)
		tmp = 0.0;
	elseif (x <= 2.65e+246)
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 760.0], N[(N[(2.0 * N[Cosh[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.6e+101], 0.0, If[LessEqual[x, 2.65e+246], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 760

   1. Initial program 62.6%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in eps around inf 98.1%

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

   6. Taylor expanded in eps around inf 98.1%

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

      1. *-commutative 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in eps around inf 98.3%

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

       1. *-un-lft-identity 98.3%

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

       2. *-commutative 98.3%

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

       3. rec-exp 98.3%

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

       4. cosh-undef 98.3%

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

   11. Applied egg-rr 98.3%

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

       1. *-lft-identity 98.3%

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

   13. Simplified 98.3%

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

   if 760 < x < 6.60000000000000022e101 or 2.64999999999999988e246 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 77.8%

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

      1. mul-1-neg 77.8%

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

      2. mul-1-neg 77.8%

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

      3. rec-exp 77.8%

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

      4. sub-neg 77.8%

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

      5. div-sub 77.8%

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

      6. mul-1-neg 77.8%

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

      7. rec-exp 77.8%

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

      8. +-inverses 77.8%

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

   7. Simplified 77.8%

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

   if 6.60000000000000022e101 < x < 2.64999999999999988e246

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 37.7%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 760:\\ \;\;\;\;\frac{2 \cdot \cosh \left(x \cdot \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+101}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+246}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 950 \lor \neg \left(x \leq 2.9 \cdot 10^{+94}\right) \land x \leq 10^{+252}:\\ \;\;\;\;\frac{2 \cdot \cosh \left(x \cdot eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (or (<= x 950.0) (and (not (<= x 2.9e+94)) (<= x 1e+252)))
   (/ (* 2.0 (cosh (* x eps_m))) 2.0)
   0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((x <= 950.0) || (!(x <= 2.9e+94) && (x <= 1e+252))) {
		tmp = (2.0 * cosh((x * eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if ((x <= 950.0d0) .or. (.not. (x <= 2.9d+94)) .and. (x <= 1d+252)) then
        tmp = (2.0d0 * cosh((x * eps_m))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if ((x <= 950.0) || (!(x <= 2.9e+94) && (x <= 1e+252))) {
		tmp = (2.0 * Math.cosh((x * eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if (x <= 950.0) or (not (x <= 2.9e+94) and (x <= 1e+252)):
		tmp = (2.0 * math.cosh((x * eps_m))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if ((x <= 950.0) || (!(x <= 2.9e+94) && (x <= 1e+252)))
		tmp = Float64(Float64(2.0 * cosh(Float64(x * eps_m))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if ((x <= 950.0) || (~((x <= 2.9e+94)) && (x <= 1e+252)))
		tmp = (2.0 * cosh((x * eps_m))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[Or[LessEqual[x, 950.0], And[N[Not[LessEqual[x, 2.9e+94]], $MachinePrecision], LessEqual[x, 1e+252]]], N[(N[(2.0 * N[Cosh[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 950 or 2.8999999999999998e94 < x < 1.0000000000000001e252

   1. Initial program 67.3%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in eps around inf 98.3%

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

   6. Taylor expanded in eps around inf 94.8%

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

      1. *-commutative 94.8%

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

   8. Simplified 94.8%

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

   9. Taylor expanded in eps around inf 94.2%

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

       1. *-un-lft-identity 94.2%

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

       2. *-commutative 94.2%

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

       3. rec-exp 94.2%

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

       4. cosh-undef 94.2%

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

   11. Applied egg-rr 94.2%

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

       1. *-lft-identity 94.2%

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

   13. Simplified 94.2%

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

   if 950 < x < 2.8999999999999998e94 or 1.0000000000000001e252 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 77.8%

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

      1. mul-1-neg 77.8%

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

      2. mul-1-neg 77.8%

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

      3. rec-exp 77.8%

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

      4. sub-neg 77.8%

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

      5. div-sub 77.8%

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

      6. mul-1-neg 77.8%

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

      7. rec-exp 77.8%

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

      8. +-inverses 77.8%

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

   7. Simplified 77.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 950 \lor \neg \left(x \leq 2.9 \cdot 10^{+94}\right) \land x \leq 10^{+252}:\\ \;\;\;\;\frac{2 \cdot \cosh \left(x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+95}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+253}:\\ \;\;\;\;\frac{2 \cdot e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.0)
   (/ (+ 2.0 (* x (- -1.0 eps_m))) 2.0)
   (if (<= x 1.02e+95) 0.0 (if (<= x 1e+253) (/ (* 2.0 (exp x)) 2.0) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1.02e+95) {
		tmp = 0.0;
	} else if (x <= 1e+253) {
		tmp = (2.0 * exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps_m))) / 2.0d0
    else if (x <= 1.02d+95) then
        tmp = 0.0d0
    else if (x <= 1d+253) then
        tmp = (2.0d0 * exp(x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1.02e+95) {
		tmp = 0.0;
	} else if (x <= 1e+253) {
		tmp = (2.0 * Math.exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0
	elif x <= 1.02e+95:
		tmp = 0.0
	elif x <= 1e+253:
		tmp = (2.0 * math.exp(x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 1.02e+95)
		tmp = 0.0;
	elseif (x <= 1e+253)
		tmp = Float64(Float64(2.0 * exp(x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	elseif (x <= 1.02e+95)
		tmp = 0.0;
	elseif (x <= 1e+253)
		tmp = (2.0 * exp(x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.0], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.02e+95], 0.0, If[LessEqual[x, 1e+253], N[(N[(2.0 * N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 2

   1. Initial program 62.9%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in eps around inf 98.5%

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

   6. Taylor expanded in eps around inf 98.5%

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

      1. *-commutative 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in x around 0 76.4%

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

   10. Taylor expanded in x around 0 62.6%

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

       1. associate-*r* 62.6%

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

       2. neg-mul-1 62.6%

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

   12. Simplified 62.6%

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

   if 2 < x < 1.0200000000000001e95 or 9.9999999999999994e252 < x

   1. Initial program 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in eps around 0 75.5%

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

      1. mul-1-neg 75.5%

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

      2. mul-1-neg 75.5%

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

      3. rec-exp 75.4%

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

      4. sub-neg 75.4%

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

      5. div-sub 75.4%

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

      6. mul-1-neg 75.4%

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

      7. rec-exp 75.5%

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

      8. +-inverses 75.5%

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

   7. Simplified 75.5%

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

   if 1.0200000000000001e95 < x < 9.9999999999999994e252

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 36.7%

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

      1. associate--r+ 36.7%

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

      2. associate-*r* 36.7%

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

      3. mul-1-neg 36.7%

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

      4. cancel-sign-sub 36.7%

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

      5. distribute-rgt1-in 36.7%

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

      6. distribute-rgt-out-- 36.7%

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

      7. mul-1-neg 36.7%

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

      8. mul-1-neg 36.7%

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

   7. Simplified 36.7%

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

      1. add-sqr-sqrt 36.7%

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

      2. sqrt-unprod 3.8%

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

      3. sqr-neg 3.8%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 36.7%

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

      6. cancel-sign-sub-inv 36.7%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 64.8%

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

      9. sqr-neg 64.8%

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

      10. sqrt-unprod 64.8%

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

      11. add-sqr-sqrt 64.8%

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

      12. associate--l+ 64.8%

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

      13. metadata-eval 64.8%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 0.0%

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

      16. sqr-neg 0.0%

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

      17. sqrt-unprod 0.0%

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

      18. add-sqr-sqrt 0.0%

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

   9. Applied egg-rr 0.0%

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

       1. *-commutative 0.0%

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

       2. distribute-rgt-out-- 0.0%

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

       3. *-commutative 0.0%

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

       4. sub-neg 0.0%

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

       5. +-commutative 0.0%

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

       6. associate-+l+ 64.8%

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

       7. neg-mul-1 64.8%

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

       8. distribute-rgt1-in 64.8%

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

       9. metadata-eval 64.8%

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

       10. mul0-lft 64.8%

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

       11. metadata-eval 64.8%

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

   11. Simplified 64.8%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+95}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+253}:\\ \;\;\;\;\frac{2 \cdot e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+98}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+252}:\\ \;\;\;\;\frac{2 \cdot e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 480.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 2e+98) 0.0 (if (<= x 1e+252) (/ (* 2.0 (exp x)) 2.0) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 480.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 2e+98) {
		tmp = 0.0;
	} else if (x <= 1e+252) {
		tmp = (2.0 * exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 480.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 2d+98) then
        tmp = 0.0d0
    else if (x <= 1d+252) then
        tmp = (2.0d0 * exp(x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 480.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 2e+98) {
		tmp = 0.0;
	} else if (x <= 1e+252) {
		tmp = (2.0 * Math.exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 480.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 2e+98:
		tmp = 0.0
	elif x <= 1e+252:
		tmp = (2.0 * math.exp(x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 480.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 2e+98)
		tmp = 0.0;
	elseif (x <= 1e+252)
		tmp = Float64(Float64(2.0 * exp(x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 480.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 2e+98)
		tmp = 0.0;
	elseif (x <= 1e+252)
		tmp = (2.0 * exp(x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 480.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+98], 0.0, If[LessEqual[x, 1e+252], N[(N[(2.0 * N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 480

   1. Initial program 62.6%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in eps around inf 98.1%

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

   6. Taylor expanded in eps around inf 98.1%

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

      1. *-commutative 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in x around 0 76.0%

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

   10. Taylor expanded in eps around 0 83.0%

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

       1. rec-exp 83.0%

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

   12. Simplified 83.0%

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

   if 480 < x < 2e98 or 1.0000000000000001e252 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 77.8%

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

      1. mul-1-neg 77.8%

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

      2. mul-1-neg 77.8%

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

      3. rec-exp 77.8%

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

      4. sub-neg 77.8%

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

      5. div-sub 77.8%

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

      6. mul-1-neg 77.8%

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

      7. rec-exp 77.8%

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

      8. +-inverses 77.8%

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

   7. Simplified 77.8%

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

   if 2e98 < x < 1.0000000000000001e252

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 36.7%

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

      1. associate--r+ 36.7%

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

      2. associate-*r* 36.7%

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

      3. mul-1-neg 36.7%

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

      4. cancel-sign-sub 36.7%

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

      5. distribute-rgt1-in 36.7%

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

      6. distribute-rgt-out-- 36.7%

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

      7. mul-1-neg 36.7%

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

      8. mul-1-neg 36.7%

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

   7. Simplified 36.7%

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

      1. add-sqr-sqrt 36.7%

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

      2. sqrt-unprod 3.8%

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

      3. sqr-neg 3.8%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 36.7%

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

      6. cancel-sign-sub-inv 36.7%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 64.8%

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

      9. sqr-neg 64.8%

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

      10. sqrt-unprod 64.8%

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

      11. add-sqr-sqrt 64.8%

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

      12. associate--l+ 64.8%

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

      13. metadata-eval 64.8%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 0.0%

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

      16. sqr-neg 0.0%

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

      17. sqrt-unprod 0.0%

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

      18. add-sqr-sqrt 0.0%

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

   9. Applied egg-rr 0.0%

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

       1. *-commutative 0.0%

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

       2. distribute-rgt-out-- 0.0%

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

       3. *-commutative 0.0%

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

       4. sub-neg 0.0%

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

       5. +-commutative 0.0%

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

       6. associate-+l+ 64.8%

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

       7. neg-mul-1 64.8%

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

       8. distribute-rgt1-in 64.8%

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

       9. metadata-eval 64.8%

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

       10. mul0-lft 64.8%

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

       11. metadata-eval 64.8%

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

   11. Simplified 64.8%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+98}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+252}:\\ \;\;\;\;\frac{2 \cdot e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+97}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+247}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(x \cdot eps\_m - -2\right) - x}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.0)
   (/ (+ 2.0 (* x (- -1.0 eps_m))) 2.0)
   (if (<= x 7.2e+97)
     0.0
     (if (<= x 9e+247)
       (/ (/ (- (* eps_m (- (* x eps_m) -2.0)) x) eps_m) 2.0)
       0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 7.2e+97) {
		tmp = 0.0;
	} else if (x <= 9e+247) {
		tmp = (((eps_m * ((x * eps_m) - -2.0)) - x) / eps_m) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps_m))) / 2.0d0
    else if (x <= 7.2d+97) then
        tmp = 0.0d0
    else if (x <= 9d+247) then
        tmp = (((eps_m * ((x * eps_m) - (-2.0d0))) - x) / eps_m) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 7.2e+97) {
		tmp = 0.0;
	} else if (x <= 9e+247) {
		tmp = (((eps_m * ((x * eps_m) - -2.0)) - x) / eps_m) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0
	elif x <= 7.2e+97:
		tmp = 0.0
	elif x <= 9e+247:
		tmp = (((eps_m * ((x * eps_m) - -2.0)) - x) / eps_m) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 7.2e+97)
		tmp = 0.0;
	elseif (x <= 9e+247)
		tmp = Float64(Float64(Float64(Float64(eps_m * Float64(Float64(x * eps_m) - -2.0)) - x) / eps_m) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	elseif (x <= 7.2e+97)
		tmp = 0.0;
	elseif (x <= 9e+247)
		tmp = (((eps_m * ((x * eps_m) - -2.0)) - x) / eps_m) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.0], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7.2e+97], 0.0, If[LessEqual[x, 9e+247], N[(N[(N[(N[(eps$95$m * N[(N[(x * eps$95$m), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 2

   1. Initial program 62.9%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in eps around inf 98.5%

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

   6. Taylor expanded in eps around inf 98.5%

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

      1. *-commutative 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in x around 0 76.4%

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

   10. Taylor expanded in x around 0 62.6%

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

       1. associate-*r* 62.6%

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

       2. neg-mul-1 62.6%

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

   12. Simplified 62.6%

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

   if 2 < x < 7.19999999999999932e97 or 9.00000000000000004e247 < x

   1. Initial program 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in eps around 0 75.5%

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

      1. mul-1-neg 75.5%

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

      2. mul-1-neg 75.5%

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

      3. rec-exp 75.4%

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

      4. sub-neg 75.4%

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

      5. div-sub 75.4%

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

      6. mul-1-neg 75.4%

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

      7. rec-exp 75.5%

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

      8. +-inverses 75.5%

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

   7. Simplified 75.5%

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

   if 7.19999999999999932e97 < x < 9.00000000000000004e247

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 37.4%

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

   6. Taylor expanded in x around 0 19.9%

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

      1. mul-1-neg 19.9%

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

      2. unsub-neg 19.9%

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

      3. associate-*r* 19.9%

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

      4. distribute-rgt-in 19.9%

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

      5. *-lft-identity 19.9%

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

      6. associate-*l/ 19.9%

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

      7. *-lft-identity 19.9%

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

   8. Simplified 19.9%

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

   9. Taylor expanded in eps around 0 29.6%

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

       1. distribute-rgt1-in 29.6%

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

       2. metadata-eval 29.6%

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

       3. *-commutative 29.6%

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

       4. mul0-lft 29.6%

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

       5. associate-+r- 29.6%

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

       6. mul0-lft 29.6%

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

       7. *-commutative 29.6%

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

   11. Simplified 29.6%

       \[\leadsto \frac{\color{blue}{\frac{\varepsilon \cdot \left(2 + \left(\varepsilon \cdot x - 0 \cdot x\right)\right) - x}{\varepsilon}}}{2} \]
   12. Step-by-step derivation

       1. +-commutative 29.6%

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

       2. associate-+l- 29.6%

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

       3. mul0-lft 29.6%

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

       4. metadata-eval 29.6%

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

   13. Applied egg-rr 29.6%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+97}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+247}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon - -2\right) - x}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 16.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.0) (/ (+ 2.0 (* x (- -1.0 eps_m))) 2.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps_m))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

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

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.0], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 62.9%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in eps around inf 98.5%

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

   6. Taylor expanded in eps around inf 98.5%

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

      1. *-commutative 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in x around 0 76.4%

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

   10. Taylor expanded in x around 0 62.6%

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

       1. associate-*r* 62.6%

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

       2. neg-mul-1 62.6%

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

   12. Simplified 62.6%

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

   if 2 < x

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in eps around 0 57.4%

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

      1. mul-1-neg 57.4%

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

      2. mul-1-neg 57.4%

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

      3. rec-exp 57.3%

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

      4. sub-neg 57.3%

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

      5. div-sub 57.3%

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

      6. mul-1-neg 57.3%

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

      7. rec-exp 57.4%

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

      8. +-inverses 57.4%

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

   7. Simplified 57.4%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.9% accurate, 20.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot eps\_m}{-2}\\ \mathbf{elif}\;x \leq 560:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0) (/ (* x eps_m) (- 2.0)) (if (<= x 560.0) 1.0 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) / -2.0;
	} else if (x <= 560.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (x * eps_m) / -2.0d0
    else if (x <= 560.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) / -2.0;
	} else if (x <= 560.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (x * eps_m) / -2.0
	elif x <= 560.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * eps_m) / Float64(-2.0));
	elseif (x <= 560.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * eps_m) / -2.0;
	elseif (x <= 560.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(x * eps$95$m), $MachinePrecision] / (-2.0)), $MachinePrecision], If[LessEqual[x, 560.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in x around 0 47.5%

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

   6. Taylor expanded in eps around inf 14.2%

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

      1. associate-*r* 14.2%

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

      2. mul-1-neg 14.2%

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

   8. Simplified 14.2%

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

   if -1 < x < 560

   1. Initial program 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in x around 0 78.5%

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

   if 560 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 58.3%

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

      1. mul-1-neg 58.3%

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

      2. mul-1-neg 58.3%

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

      3. rec-exp 58.3%

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

      4. sub-neg 58.3%

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

      5. div-sub 58.3%

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

      6. mul-1-neg 58.3%

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

      7. rec-exp 58.3%

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

      8. +-inverses 58.3%

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

   7. Simplified 58.3%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \varepsilon}{-2}\\ \mathbf{elif}\;x \leq 560:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.5% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 520.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 520.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 520.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 520.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 520.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 520.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 520

   1. Initial program 62.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in x around 0 60.1%

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

   if 520 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 58.3%

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

      1. mul-1-neg 58.3%

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

      2. mul-1-neg 58.3%

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

      3. rec-exp 58.3%

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

      4. sub-neg 58.3%

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

      5. div-sub 58.3%

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

      6. mul-1-neg 58.3%

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

      7. rec-exp 58.3%

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

      8. +-inverses 58.3%

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

   7. Simplified 58.3%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 16.1% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

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

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.0d0
end function

Java

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

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 71.2%

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

3. Simplified 56.8%

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

5. Taylor expanded in eps around 0 15.1%

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

   1. mul-1-neg 15.1%

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

   2. mul-1-neg 15.1%

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

   3. rec-exp 15.1%

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

   4. sub-neg 15.1%

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

   5. div-sub 15.1%

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

   6. mul-1-neg 15.1%

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

   7. rec-exp 15.1%

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

   8. +-inverses 15.4%

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

7. Simplified 15.4%

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

8. Final simplification 15.4%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(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))