NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 99.3%

Time: 11.5s

Alternatives: 7

Speedup: 1.9×

• 38.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -5.114597943280682e+282
  2. eps = 1.2368870015556245e-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: 73.4% 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: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{2 \cdot \cosh \left(x \cdot eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.5)
   (/ (* 2.0 (cosh (* x eps_m))) 2.0)
   (/ (+ (exp (* x (+ -1.0 eps_m))) (exp (- x))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.5) {
		tmp = (2.0 * cosh((x * eps_m))) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) + exp(-x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.5d0) then
        tmp = (2.0d0 * cosh((x * eps_m))) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps_m))) + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.5) {
		tmp = (2.0 * Math.cosh((x * eps_m))) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.5:
		tmp = (2.0 * math.cosh((x * eps_m))) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) + math.exp(-x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.5)
		tmp = Float64(Float64(2.0 * cosh(Float64(x * eps_m))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = (2.0 * cosh((x * eps_m))) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.5], N[(N[(2.0 * N[Cosh[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 65.8%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in eps around inf 98.0%

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

   6. Taylor expanded in eps around inf 98.1%

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

      1. associate-*r* 98.1%

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

      2. neg-mul-1 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in eps around inf 98.8%

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

       1. *-commutative 98.8%

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

   11. Simplified 98.8%

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

       1. +-commutative 98.8%

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

       2. distribute-lft-neg-out 98.8%

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

       3. *-commutative 98.8%

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

       4. cosh-undef 98.8%

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

   13. Applied egg-rr 98.8%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 70.0%

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

      1. neg-mul-1 70.0%

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

   8. Simplified 70.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{2 \cdot \cosh \left(x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Simplified 61.7%

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

5. Taylor expanded in eps around inf 98.6%

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

6. Final simplification 98.6%

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

Alternative 3: 87.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 4.5 \cdot 10^{-100}:\\ \;\;\;\;1\\ \mathbf{elif}\;eps\_m \leq 0.027:\\ \;\;\;\;\frac{x \cdot \left(\frac{2 - x \cdot \left(\left(-1 + \frac{1}{eps\_m}\right) \cdot \left(-1 - eps\_m\right)\right)}{x} + \left(-1 + eps\_m\right) \cdot \left(1 + \frac{1}{eps\_m}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cosh \left(x \cdot eps\_m\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 4.5e-100)
   1.0
   (if (<= eps_m 0.027)
     (/
      (*
       x
       (+
        (/ (- 2.0 (* x (* (+ -1.0 (/ 1.0 eps_m)) (- -1.0 eps_m)))) x)
        (* (+ -1.0 eps_m) (+ 1.0 (/ 1.0 eps_m)))))
      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 <= 4.5e-100) {
		tmp = 1.0;
	} else if (eps_m <= 0.027) {
		tmp = (x * (((2.0 - (x * ((-1.0 + (1.0 / eps_m)) * (-1.0 - eps_m)))) / x) + ((-1.0 + eps_m) * (1.0 + (1.0 / eps_m))))) / 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 <= 4.5d-100) then
        tmp = 1.0d0
    else if (eps_m <= 0.027d0) then
        tmp = (x * (((2.0d0 - (x * (((-1.0d0) + (1.0d0 / eps_m)) * ((-1.0d0) - eps_m)))) / x) + (((-1.0d0) + eps_m) * (1.0d0 + (1.0d0 / eps_m))))) / 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 <= 4.5e-100) {
		tmp = 1.0;
	} else if (eps_m <= 0.027) {
		tmp = (x * (((2.0 - (x * ((-1.0 + (1.0 / eps_m)) * (-1.0 - eps_m)))) / x) + ((-1.0 + eps_m) * (1.0 + (1.0 / eps_m))))) / 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 <= 4.5e-100:
		tmp = 1.0
	elif eps_m <= 0.027:
		tmp = (x * (((2.0 - (x * ((-1.0 + (1.0 / eps_m)) * (-1.0 - eps_m)))) / x) + ((-1.0 + eps_m) * (1.0 + (1.0 / eps_m))))) / 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 <= 4.5e-100)
		tmp = 1.0;
	elseif (eps_m <= 0.027)
		tmp = Float64(Float64(x * Float64(Float64(Float64(2.0 - Float64(x * Float64(Float64(-1.0 + Float64(1.0 / eps_m)) * Float64(-1.0 - eps_m)))) / x) + Float64(Float64(-1.0 + eps_m) * Float64(1.0 + Float64(1.0 / eps_m))))) / 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 <= 4.5e-100)
		tmp = 1.0;
	elseif (eps_m <= 0.027)
		tmp = (x * (((2.0 - (x * ((-1.0 + (1.0 / eps_m)) * (-1.0 - eps_m)))) / x) + ((-1.0 + eps_m) * (1.0 + (1.0 / eps_m))))) / 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, 4.5e-100], 1.0, If[LessEqual[eps$95$m, 0.027], N[(N[(x * N[(N[(N[(2.0 - N[(x * N[(N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(-1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if eps < 4.5000000000000001e-100

   1. Initial program 67.9%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in x around 0 34.4%

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

   6. Taylor expanded in x around 0 20.0%

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

   7. Taylor expanded in eps around inf 51.1%

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

      1. associate-*r/ 51.1%

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

      2. metadata-eval 51.1%

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

   9. Simplified 51.1%

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

   10. Taylor expanded in eps around 0 46.3%

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

   if 4.5000000000000001e-100 < eps < 0.0269999999999999997

   1. Initial program 54.8%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in x around 0 2.5%

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

   6. Taylor expanded in x around -inf 3.2%

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

   7. Taylor expanded in x around 0 84.0%

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

      1. +-commutative 84.0%

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

      2. sub-neg 84.0%

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

      3. metadata-eval 84.0%

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

      4. +-commutative 84.0%

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

   9. Simplified 84.0%

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

   if 0.0269999999999999997 < 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 74.8%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

       1. +-commutative 100.0%

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

       2. distribute-lft-neg-out 100.0%

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

       3. *-commutative 100.0%

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

       4. cosh-undef 100.0%

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

   13. Applied egg-rr 100.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4.5 \cdot 10^{-100}:\\ \;\;\;\;1\\ \mathbf{elif}\;\varepsilon \leq 0.027:\\ \;\;\;\;\frac{x \cdot \left(\frac{2 - x \cdot \left(\left(-1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 - \varepsilon\right)\right)}{x} + \left(-1 + \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cosh \left(x \cdot \varepsilon\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.1% accurate, 12.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000116:\\ \;\;\;\;\frac{\frac{x + eps\_m \cdot \left(2 - x \cdot eps\_m\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.000116)
   (/ (/ (+ x (* eps_m (- 2.0 (* x eps_m)))) eps_m) 2.0)
   (if (<= x 550.0) 1.0 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.000116) {
		tmp = ((x + (eps_m * (2.0 - (x * eps_m)))) / eps_m) / 2.0;
	} else if (x <= 550.0) {
		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 <= (-0.000116d0)) then
        tmp = ((x + (eps_m * (2.0d0 - (x * eps_m)))) / eps_m) / 2.0d0
    else if (x <= 550.0d0) 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 <= -0.000116) {
		tmp = ((x + (eps_m * (2.0 - (x * eps_m)))) / eps_m) / 2.0;
	} else if (x <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.000116:
		tmp = ((x + (eps_m * (2.0 - (x * eps_m)))) / eps_m) / 2.0
	elif x <= 550.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.000116)
		tmp = Float64(Float64(Float64(x + Float64(eps_m * Float64(2.0 - Float64(x * eps_m)))) / eps_m) / 2.0);
	elseif (x <= 550.0)
		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 <= -0.000116)
		tmp = ((x + (eps_m * (2.0 - (x * eps_m)))) / eps_m) / 2.0;
	elseif (x <= 550.0)
		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, -0.000116], N[(N[(N[(x + N[(eps$95$m * N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 550.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.16e-4

   1. Initial program 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around 0 38.8%

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

   6. Taylor expanded in x around 0 19.8%

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

      1. +-commutative 19.8%

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

      2. sub-neg 19.8%

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

      3. metadata-eval 19.8%

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

      4. +-commutative 19.8%

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

   8. Simplified 19.8%

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

   9. Taylor expanded in eps around 0 29.4%

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

       1. mul-1-neg 29.4%

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

       2. *-commutative 29.4%

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

       3. distribute-rgt-neg-in 29.4%

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

   11. Simplified 29.4%

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

   if -1.16e-4 < x < 550

   1. Initial program 57.7%

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

   3. Simplified 57.7%

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

   5. Taylor expanded in x around 0 44.0%

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

   6. Taylor expanded in x around 0 29.1%

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

   7. Taylor expanded in eps around inf 70.5%

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

      1. associate-*r/ 70.5%

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

      2. metadata-eval 70.5%

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

   9. Simplified 70.5%

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

   10. Taylor expanded in eps around 0 71.7%

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

   if 550 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 66.1%

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

      1. associate-*r* 66.1%

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

      2. neg-mul-1 66.1%

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

   8. Simplified 66.1%

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

   9. Taylor expanded in eps around inf 49.0%

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

       1. *-commutative 49.0%

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

   11. Simplified 49.0%

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

       1. flip-+ 0.2%

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

       2. div-sub 0.2%

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

   13. Applied egg-rr 0.0%

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

       1. +-inverses 52.9%

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

   15. Simplified 52.9%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000116:\\ \;\;\;\;\frac{\frac{x + \varepsilon \cdot \left(2 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.2% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 250:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 250.0) (/ (- 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 <= 250.0) {
		tmp = (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 <= 250.0d0) then
        tmp = (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 <= 250.0) {
		tmp = (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 <= 250.0:
		tmp = (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 <= 250.0)
		tmp = Float64(Float64(2.0 - 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 <= 250.0)
		tmp = (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, 250.0], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 250

   1. Initial program 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in x around 0 42.1%

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

   6. Taylor expanded in x around 0 43.7%

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

      1. +-commutative 43.7%

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

      2. sub-neg 43.7%

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

      3. metadata-eval 43.7%

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

      4. +-commutative 43.7%

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

   8. Simplified 43.7%

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

   9. Taylor expanded in eps around inf 60.3%

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

       1. mul-1-neg 60.3%

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

       2. *-commutative 60.3%

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

       3. distribute-rgt-neg-in 60.3%

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

   11. Simplified 60.3%

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

   if 250 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 66.1%

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

      1. associate-*r* 66.1%

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

      2. neg-mul-1 66.1%

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

   8. Simplified 66.1%

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

   9. Taylor expanded in eps around inf 49.0%

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

       1. *-commutative 49.0%

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

   11. Simplified 49.0%

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

       1. flip-+ 0.2%

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

       2. div-sub 0.2%

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

   13. Applied egg-rr 0.0%

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

       1. +-inverses 52.9%

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

   15. Simplified 52.9%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 250:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.4% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 550.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 550.0) {
		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 <= 550.0d0) 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 <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 550.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 550.0)
		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 <= 550.0)
		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, 550.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 550

   1. Initial program 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in x around 0 44.6%

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

   6. Taylor expanded in x around 0 29.0%

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

   7. Taylor expanded in eps around inf 62.0%

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

      1. associate-*r/ 62.0%

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

      2. metadata-eval 62.0%

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

   9. Simplified 62.0%

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

   10. Taylor expanded in eps around 0 57.8%

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

   if 550 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 66.1%

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

      1. associate-*r* 66.1%

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

      2. neg-mul-1 66.1%

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

   8. Simplified 66.1%

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

   9. Taylor expanded in eps around inf 49.0%

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

       1. *-commutative 49.0%

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

   11. Simplified 49.0%

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

       1. flip-+ 0.2%

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

       2. div-sub 0.2%

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

   13. Applied egg-rr 0.0%

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

       1. +-inverses 52.9%

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

   15. Simplified 52.9%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.5% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Simplified 75.0%

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

5. Taylor expanded in x around 0 40.8%

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

6. Taylor expanded in x around 0 24.3%

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

7. Taylor expanded in eps around inf 48.6%

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

   1. associate-*r/ 48.6%

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

   2. metadata-eval 48.6%

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

9. Simplified 48.6%

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

10. Taylor expanded in eps around 0 43.0%

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

Reproduce

herbie shell --seed 2024191 
(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))