NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.1% → 99.8%

Time: 27.8s

Alternatives: 13

Speedup: 2.1×

• 38.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% 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, 1.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \mathbf{if}\;\varepsilon \leq 10^{-29}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ x 1.0) (exp (- x)))))
   (if (<= eps 1e-29)
     (/ (+ t_0 t_0) 2.0)
     (/ (+ (exp (* x (+ eps -1.0))) (exp (* eps (- x)))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (x + 1.0) * exp(-x);
	double tmp;
	if (eps <= 1e-29) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp((x * (eps + -1.0))) + exp((eps * -x))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) * exp(-x)
    if (eps <= 1d-29) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp((x * (eps + (-1.0d0)))) + exp((eps * -x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = (x + 1.0) * Math.exp(-x);
	double tmp;
	if (eps <= 1e-29) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp((eps * -x))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = (x + 1.0) * math.exp(-x)
	tmp = 0
	if eps <= 1e-29:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (math.exp((x * (eps + -1.0))) + math.exp((eps * -x))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(x + 1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps <= 1e-29)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(eps * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = (x + 1.0) * exp(-x);
	tmp = 0.0;
	if (eps <= 1e-29)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp((x * (eps + -1.0))) + exp((eps * -x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, 1e-29], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 9.99999999999999943e-30

   1. Initial program 67.1%

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

      1. sub-neg 67.1%

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

      2. neg-sub0 67.1%

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

      3. associate-+r- 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in eps around 0 64.5%

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

      1. distribute-rgt1-in 64.5%

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

      2. neg-mul-1 64.5%

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

      3. distribute-lft-out 64.5%

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

      4. distribute-rgt1-in 65.0%

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

      5. neg-mul-1 65.0%

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

   6. Simplified 65.0%

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

   if 9.99999999999999943e-30 < eps

   1. Initial program 98.8%

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

   3. Simplified 72.6%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-29}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]

Alternative 2: 95.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -120000000:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-196}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\varepsilon + -1\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \frac{1 - \varepsilon \cdot \varepsilon}{\frac{1 - \varepsilon}{1 + \frac{-1}{\varepsilon}}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -120000000.0)
   (/ (/ 2.0 (exp x)) 2.0)
   (if (<= x -1.1e-196)
     (/
      (+
       2.0
       (*
        x
        (-
         (* (+ eps -1.0) (+ 1.0 (/ 1.0 eps)))
         (/ (- 1.0 (* eps eps)) (/ (- 1.0 eps) (+ 1.0 (/ -1.0 eps)))))))
      2.0)
     (/ (+ (exp (- x)) (exp (* x (+ eps -1.0)))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -120000000.0) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else if (x <= -1.1e-196) {
		tmp = (2.0 + (x * (((eps + -1.0) * (1.0 + (1.0 / eps))) - ((1.0 - (eps * eps)) / ((1.0 - eps) / (1.0 + (-1.0 / eps))))))) / 2.0;
	} else {
		tmp = (exp(-x) + exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-120000000.0d0)) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else if (x <= (-1.1d-196)) then
        tmp = (2.0d0 + (x * (((eps + (-1.0d0)) * (1.0d0 + (1.0d0 / eps))) - ((1.0d0 - (eps * eps)) / ((1.0d0 - eps) / (1.0d0 + ((-1.0d0) / eps))))))) / 2.0d0
    else
        tmp = (exp(-x) + exp((x * (eps + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -120000000.0) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else if (x <= -1.1e-196) {
		tmp = (2.0 + (x * (((eps + -1.0) * (1.0 + (1.0 / eps))) - ((1.0 - (eps * eps)) / ((1.0 - eps) / (1.0 + (-1.0 / eps))))))) / 2.0;
	} else {
		tmp = (Math.exp(-x) + Math.exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -120000000.0:
		tmp = (2.0 / math.exp(x)) / 2.0
	elif x <= -1.1e-196:
		tmp = (2.0 + (x * (((eps + -1.0) * (1.0 + (1.0 / eps))) - ((1.0 - (eps * eps)) / ((1.0 - eps) / (1.0 + (-1.0 / eps))))))) / 2.0
	else:
		tmp = (math.exp(-x) + math.exp((x * (eps + -1.0)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -120000000.0)
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	elseif (x <= -1.1e-196)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(eps + -1.0) * Float64(1.0 + Float64(1.0 / eps))) - Float64(Float64(1.0 - Float64(eps * eps)) / Float64(Float64(1.0 - eps) / Float64(1.0 + Float64(-1.0 / eps))))))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -120000000.0)
		tmp = (2.0 / exp(x)) / 2.0;
	elseif (x <= -1.1e-196)
		tmp = (2.0 + (x * (((eps + -1.0) * (1.0 + (1.0 / eps))) - ((1.0 - (eps * eps)) / ((1.0 - eps) / (1.0 + (-1.0 / eps))))))) / 2.0;
	else
		tmp = (exp(-x) + exp((x * (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -120000000.0], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.1e-196], N[(N[(2.0 + N[(x * N[(N[(N[(eps + -1.0), $MachinePrecision] * N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - eps), $MachinePrecision] / N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2e8

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around 0 100.0%

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

   6. Taylor expanded in eps around 0 100.0%

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

      1. cancel-sign-sub-inv 100.0%

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

      2. neg-mul-1 100.0%

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

      3. metadata-eval 100.0%

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

      4. neg-mul-1 100.0%

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

      5. distribute-rgt1-in 100.0%

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

      6. metadata-eval 100.0%

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

      7. exp-neg 100.0%

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

      8. associate-*r/ 100.0%

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

      9. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   if -1.2e8 < x < -1.10000000000000007e-196

   1. Initial program 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in x around 0 54.7%

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

      1. *-commutative 54.7%

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

      2. flip-+ 69.2%

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

      3. associate-*r/ 69.2%

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

      4. sub-neg 69.2%

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

      5. distribute-neg-frac 69.2%

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

      6. metadata-eval 69.2%

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

      7. metadata-eval 69.2%

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

   6. Applied egg-rr 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-/l* 69.2%

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

      3. +-commutative 69.2%

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

   8. Simplified 69.2%

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

   if -1.10000000000000007e-196 < x

   1. Initial program 75.6%

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

   3. Simplified 55.7%

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

   4. Taylor expanded in eps around inf 98.3%

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

   5. Taylor expanded in eps around 0 81.2%

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

   6. Taylor expanded in eps around -inf 81.2%

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

      1. cancel-sign-sub-inv 81.2%

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

      2. associate-*r* 81.2%

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

      3. neg-mul-1 81.2%

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

      4. mul-1-neg 81.2%

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

      5. metadata-eval 81.2%

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

      6. neg-mul-1 81.2%

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

      7. *-lft-identity 81.2%

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

   8. Simplified 81.2%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -120000000:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-196}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\varepsilon + -1\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \frac{1 - \varepsilon \cdot \varepsilon}{\frac{1 - \varepsilon}{1 + \frac{-1}{\varepsilon}}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-29}:\\ \;\;\;\;\frac{\frac{x + 2}{e^{x}} + \frac{x}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 1e-29)
   (/ (+ (/ (+ x 2.0) (exp x)) (/ x (exp x))) 2.0)
   (/ (+ (exp (* x (+ eps -1.0))) (exp (* eps (- x)))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 1e-29) {
		tmp = (((x + 2.0) / exp(x)) + (x / exp(x))) / 2.0;
	} else {
		tmp = (exp((x * (eps + -1.0))) + exp((eps * -x))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 1d-29) then
        tmp = (((x + 2.0d0) / exp(x)) + (x / exp(x))) / 2.0d0
    else
        tmp = (exp((x * (eps + (-1.0d0)))) + exp((eps * -x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (eps <= 1e-29) {
		tmp = (((x + 2.0) / Math.exp(x)) + (x / Math.exp(x))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp((eps * -x))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if eps <= 1e-29:
		tmp = (((x + 2.0) / math.exp(x)) + (x / math.exp(x))) / 2.0
	else:
		tmp = (math.exp((x * (eps + -1.0))) + math.exp((eps * -x))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (eps <= 1e-29)
		tmp = Float64(Float64(Float64(Float64(x + 2.0) / exp(x)) + Float64(x / exp(x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(eps * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1e-29)
		tmp = (((x + 2.0) / exp(x)) + (x / exp(x))) / 2.0;
	else
		tmp = (exp((x * (eps + -1.0))) + exp((eps * -x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 1e-29], N[(N[(N[(N[(x + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 9.99999999999999943e-30

   1. Initial program 67.1%

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

      1. sub-neg 67.1%

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

      2. neg-sub0 67.1%

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

      3. associate-+r- 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in eps around 0 64.5%

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

      1. associate--r+ 64.5%

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

      2. cancel-sign-sub-inv 64.5%

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

      3. distribute-rgt1-in 64.4%

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

      4. distribute-rgt-out-- 65.0%

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

      5. neg-mul-1 65.0%

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

      6. neg-mul-1 65.0%

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

      7. rec-exp 65.0%

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

      8. associate-*r/ 65.0%

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

      9. *-rgt-identity 65.0%

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

      10. metadata-eval 65.0%

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

      11. *-lft-identity 65.0%

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

   6. Simplified 65.0%

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

   7. Taylor expanded in x around inf 64.4%

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

      1. distribute-rgt-out 65.0%

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

      2. rec-exp 65.0%

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

      3. +-commutative 65.0%

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

      4. associate-*l/ 65.0%

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

      5. *-lft-identity 65.0%

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

   9. Simplified 65.0%

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

   if 9.99999999999999943e-30 < eps

   1. Initial program 98.8%

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

   3. Simplified 72.6%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-29}:\\ \;\;\;\;\frac{\frac{x + 2}{e^{x}} + \frac{x}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]

Alternative 4: 85.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.76:\\ \;\;\;\;\frac{\frac{x + 2}{e^{x}} + \frac{x}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right) - x\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.76)
   (/ (+ (/ (+ x 2.0) (exp x)) (/ x (exp x))) 2.0)
   (/ (+ 2.0 (- (* (* eps eps) (* x x)) x)) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 0.76) {
		tmp = (((x + 2.0) / exp(x)) + (x / exp(x))) / 2.0;
	} else {
		tmp = (2.0 + (((eps * eps) * (x * x)) - x)) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 0.76d0) then
        tmp = (((x + 2.0d0) / exp(x)) + (x / exp(x))) / 2.0d0
    else
        tmp = (2.0d0 + (((eps * eps) * (x * x)) - x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (eps <= 0.76) {
		tmp = (((x + 2.0) / Math.exp(x)) + (x / Math.exp(x))) / 2.0;
	} else {
		tmp = (2.0 + (((eps * eps) * (x * x)) - x)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if eps <= 0.76:
		tmp = (((x + 2.0) / math.exp(x)) + (x / math.exp(x))) / 2.0
	else:
		tmp = (2.0 + (((eps * eps) * (x * x)) - x)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (eps <= 0.76)
		tmp = Float64(Float64(Float64(Float64(x + 2.0) / exp(x)) + Float64(x / exp(x))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(eps * eps) * Float64(x * x)) - x)) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 0.76)
		tmp = (((x + 2.0) / exp(x)) + (x / exp(x))) / 2.0;
	else
		tmp = (2.0 + (((eps * eps) * (x * x)) - x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 0.76], N[(N[(N[(N[(x + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(eps * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.76000000000000001

   1. Initial program 67.1%

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

      1. sub-neg 67.1%

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

      2. neg-sub0 67.1%

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

      3. associate-+r- 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in eps around 0 65.1%

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

      1. associate--r+ 65.1%

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

      2. cancel-sign-sub-inv 65.1%

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

      3. distribute-rgt1-in 65.0%

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

      4. distribute-rgt-out-- 65.6%

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

      5. neg-mul-1 65.6%

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

      6. neg-mul-1 65.6%

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

      7. rec-exp 65.6%

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

      8. associate-*r/ 65.6%

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

      9. *-rgt-identity 65.6%

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

      10. metadata-eval 65.6%

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

      11. *-lft-identity 65.6%

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

   6. Simplified 65.6%

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

   7. Taylor expanded in x around inf 65.0%

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

      1. distribute-rgt-out 65.6%

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

      2. rec-exp 65.6%

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

      3. +-commutative 65.6%

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

      4. associate-*l/ 65.6%

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

      5. *-lft-identity 65.6%

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

   9. Simplified 65.6%

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

   if 0.76000000000000001 < eps

   1. Initial program 100.0%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 75.2%

      \[\leadsto \frac{\color{blue}{2 + \left(-1 \cdot x + {x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right)\right)}}{2} \]
   9. Step-by-step derivation

      1. +-commutative 75.2%

         \[\leadsto \frac{2 + \color{blue}{\left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) + -1 \cdot x\right)}}{2} \]

      2. mul-1-neg 75.2%

         \[\leadsto \frac{2 + \left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) + \color{blue}{\left(-x\right)}\right)}{2} \]

      3. unsub-neg 75.2%

         \[\leadsto \frac{2 + \color{blue}{\left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) - x\right)}}{2} \]

      4. unpow2 75.2%

         \[\leadsto \frac{2 + \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) - x\right)}{2} \]

      5. cancel-sign-sub-inv 75.2%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \color{blue}{\left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} + \left(--0.5\right) \cdot {\varepsilon}^{2}\right)} - x\right)}{2} \]

      6. metadata-eval 75.2%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} + \color{blue}{0.5} \cdot {\varepsilon}^{2}\right) - x\right)}{2} \]

      7. distribute-lft-out 75.2%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \color{blue}{\left(0.5 \cdot \left({\left(\varepsilon - 1\right)}^{2} + {\varepsilon}^{2}\right)\right)} - x\right)}{2} \]

      8. sub-neg 75.2%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\color{blue}{\left(\varepsilon + \left(-1\right)\right)}}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

      9. metadata-eval 75.2%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(\varepsilon + \color{blue}{-1}\right)}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

      10. +-commutative 75.2%

          \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\color{blue}{\left(-1 + \varepsilon\right)}}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

      11. unpow2 75.2%

          \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(-1 + \varepsilon\right)}^{2} + \color{blue}{\varepsilon \cdot \varepsilon}\right)\right) - x\right)}{2} \]

   10. Simplified 75.2%

       \[\leadsto \frac{\color{blue}{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(-1 + \varepsilon\right)}^{2} + \varepsilon \cdot \varepsilon\right)\right) - x\right)}}{2} \]

   11. Taylor expanded in eps around inf 75.2%

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

       1. unpow2 75.2%

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

       2. *-commutative 75.2%

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

       3. unpow2 75.2%

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

   13. Simplified 75.2%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.76:\\ \;\;\;\;\frac{\frac{x + 2}{e^{x}} + \frac{x}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right) - x\right)}{2}\\ \end{array} \]

Alternative 5: 84.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4.7 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right) - x\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 4.7e+30)
   (/ (/ 2.0 (exp x)) 2.0)
   (/ (+ 2.0 (- (* (* eps eps) (* x x)) x)) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 4.7e+30) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else {
		tmp = (2.0 + (((eps * eps) * (x * x)) - x)) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 4.7d+30) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else
        tmp = (2.0d0 + (((eps * eps) * (x * x)) - x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (eps <= 4.7e+30) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else {
		tmp = (2.0 + (((eps * eps) * (x * x)) - x)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if eps <= 4.7e+30:
		tmp = (2.0 / math.exp(x)) / 2.0
	else:
		tmp = (2.0 + (((eps * eps) * (x * x)) - x)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (eps <= 4.7e+30)
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(eps * eps) * Float64(x * x)) - x)) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 4.7e+30)
		tmp = (2.0 / exp(x)) / 2.0;
	else
		tmp = (2.0 + (((eps * eps) * (x * x)) - x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 4.7e+30], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(eps * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.6999999999999999e30

   1. Initial program 68.2%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in eps around inf 97.7%

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

   5. Taylor expanded in eps around 0 83.3%

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

   6. Taylor expanded in eps around 0 78.1%

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

      1. cancel-sign-sub-inv 78.1%

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

      2. neg-mul-1 78.1%

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

      3. metadata-eval 78.1%

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

      4. neg-mul-1 78.1%

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

      5. distribute-rgt1-in 78.1%

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

      6. metadata-eval 78.1%

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

      7. exp-neg 78.1%

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

      8. associate-*r/ 78.1%

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

      9. metadata-eval 78.1%

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

   8. Simplified 78.1%

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

   if 4.6999999999999999e30 < eps

   1. Initial program 100.0%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 75.7%

      \[\leadsto \frac{\color{blue}{2 + \left(-1 \cdot x + {x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right)\right)}}{2} \]
   9. Step-by-step derivation

      1. +-commutative 75.7%

         \[\leadsto \frac{2 + \color{blue}{\left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) + -1 \cdot x\right)}}{2} \]

      2. mul-1-neg 75.7%

         \[\leadsto \frac{2 + \left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) + \color{blue}{\left(-x\right)}\right)}{2} \]

      3. unsub-neg 75.7%

         \[\leadsto \frac{2 + \color{blue}{\left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) - x\right)}}{2} \]

      4. unpow2 75.7%

         \[\leadsto \frac{2 + \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) - x\right)}{2} \]

      5. cancel-sign-sub-inv 75.7%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \color{blue}{\left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} + \left(--0.5\right) \cdot {\varepsilon}^{2}\right)} - x\right)}{2} \]

      6. metadata-eval 75.7%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} + \color{blue}{0.5} \cdot {\varepsilon}^{2}\right) - x\right)}{2} \]

      7. distribute-lft-out 75.7%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \color{blue}{\left(0.5 \cdot \left({\left(\varepsilon - 1\right)}^{2} + {\varepsilon}^{2}\right)\right)} - x\right)}{2} \]

      8. sub-neg 75.7%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\color{blue}{\left(\varepsilon + \left(-1\right)\right)}}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

      9. metadata-eval 75.7%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(\varepsilon + \color{blue}{-1}\right)}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

      10. +-commutative 75.7%

          \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\color{blue}{\left(-1 + \varepsilon\right)}}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

      11. unpow2 75.7%

          \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(-1 + \varepsilon\right)}^{2} + \color{blue}{\varepsilon \cdot \varepsilon}\right)\right) - x\right)}{2} \]

   10. Simplified 75.7%

       \[\leadsto \frac{\color{blue}{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(-1 + \varepsilon\right)}^{2} + \varepsilon \cdot \varepsilon\right)\right) - x\right)}}{2} \]

   11. Taylor expanded in eps around inf 75.7%

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

       1. unpow2 75.7%

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

       2. *-commutative 75.7%

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

       3. unpow2 75.7%

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

   13. Simplified 75.7%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4.7 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right) - x\right)}{2}\\ \end{array} \]

Alternative 6: 56.9% accurate, 13.2× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 580:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{+102}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+234}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + -2\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 580.0)
   1.0
   (if (<= x 6.3e+102)
     0.0
     (if (<= x 1.7e+234) (/ (* x (+ 2.0 (* x (+ x -2.0)))) 2.0) 0.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 580.0) {
		tmp = 1.0;
	} else if (x <= 6.3e+102) {
		tmp = 0.0;
	} else if (x <= 1.7e+234) {
		tmp = (x * (2.0 + (x * (x + -2.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 580.0d0) then
        tmp = 1.0d0
    else if (x <= 6.3d+102) then
        tmp = 0.0d0
    else if (x <= 1.7d+234) then
        tmp = (x * (2.0d0 + (x * (x + (-2.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 580.0) {
		tmp = 1.0;
	} else if (x <= 6.3e+102) {
		tmp = 0.0;
	} else if (x <= 1.7e+234) {
		tmp = (x * (2.0 + (x * (x + -2.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 580.0:
		tmp = 1.0
	elif x <= 6.3e+102:
		tmp = 0.0
	elif x <= 1.7e+234:
		tmp = (x * (2.0 + (x * (x + -2.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 580.0)
		tmp = 1.0;
	elseif (x <= 6.3e+102)
		tmp = 0.0;
	elseif (x <= 1.7e+234)
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(x + -2.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 580.0)
		tmp = 1.0;
	elseif (x <= 6.3e+102)
		tmp = 0.0;
	elseif (x <= 1.7e+234)
		tmp = (x * (2.0 + (x * (x + -2.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 580.0], 1.0, If[LessEqual[x, 6.3e+102], 0.0, If[LessEqual[x, 1.7e+234], N[(N[(x * N[(2.0 + N[(x * N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 580

   1. Initial program 68.6%

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

      1. sub-neg 68.6%

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

      2. neg-sub0 68.6%

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

      3. associate-+r- 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in x around 0 50.6%

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

   if 580 < x < 6.30000000000000029e102 or 1.7e234 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 61.6%

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

      1. neg-mul-1 61.6%

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

      2. rec-exp 61.6%

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

      3. neg-mul-1 61.6%

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

      4. div-sub 61.6%

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

      5. +-inverses 61.6%

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

   6. Simplified 61.6%

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

   if 6.30000000000000029e102 < x < 1.7e234

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 37.7%

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

      1. associate--r+ 37.7%

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

      2. cancel-sign-sub-inv 37.7%

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

      3. distribute-rgt1-in 37.7%

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

      4. distribute-rgt-out-- 37.7%

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

      5. neg-mul-1 37.7%

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

      6. neg-mul-1 37.7%

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

      7. rec-exp 37.7%

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

      8. associate-*r/ 37.7%

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

      9. *-rgt-identity 37.7%

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

      10. metadata-eval 37.7%

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

      11. *-lft-identity 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in x around inf 37.7%

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

      1. rec-exp 37.7%

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

      2. count-2 37.7%

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

      3. rec-exp 37.7%

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

      4. associate-*r/ 37.7%

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

      5. metadata-eval 37.7%

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

   9. Simplified 37.7%

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

   10. Taylor expanded in x around 0 63.9%

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

       1. +-commutative 63.9%

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

       2. unpow2 63.9%

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

       3. distribute-rgt-out 63.9%

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

   12. Simplified 63.9%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 580:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{+102}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+234}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + -2\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 63.9% accurate, 13.2× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 540:\\ \;\;\;\;\frac{2 + x \cdot \left(4 + x \cdot 3\right)}{2}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+102}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+232}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + -2\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 540.0)
   (/ (+ 2.0 (* x (+ 4.0 (* x 3.0)))) 2.0)
   (if (<= x 1.7e+102)
     0.0
     (if (<= x 4.4e+232) (/ (* x (+ 2.0 (* x (+ x -2.0)))) 2.0) 0.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 540.0) {
		tmp = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0;
	} else if (x <= 1.7e+102) {
		tmp = 0.0;
	} else if (x <= 4.4e+232) {
		tmp = (x * (2.0 + (x * (x + -2.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 540.0d0) then
        tmp = (2.0d0 + (x * (4.0d0 + (x * 3.0d0)))) / 2.0d0
    else if (x <= 1.7d+102) then
        tmp = 0.0d0
    else if (x <= 4.4d+232) then
        tmp = (x * (2.0d0 + (x * (x + (-2.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 540.0) {
		tmp = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0;
	} else if (x <= 1.7e+102) {
		tmp = 0.0;
	} else if (x <= 4.4e+232) {
		tmp = (x * (2.0 + (x * (x + -2.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 540.0:
		tmp = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0
	elif x <= 1.7e+102:
		tmp = 0.0
	elif x <= 4.4e+232:
		tmp = (x * (2.0 + (x * (x + -2.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 540.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(4.0 + Float64(x * 3.0)))) / 2.0);
	elseif (x <= 1.7e+102)
		tmp = 0.0;
	elseif (x <= 4.4e+232)
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(x + -2.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 540.0)
		tmp = (2.0 + (x * (4.0 + (x * 3.0)))) / 2.0;
	elseif (x <= 1.7e+102)
		tmp = 0.0;
	elseif (x <= 4.4e+232)
		tmp = (x * (2.0 + (x * (x + -2.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 540.0], N[(N[(2.0 + N[(x * N[(4.0 + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.7e+102], 0.0, If[LessEqual[x, 4.4e+232], N[(N[(x * N[(2.0 + N[(x * N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 540

   1. Initial program 68.6%

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

      1. sub-neg 68.6%

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

      2. neg-sub0 68.6%

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

      3. associate-+r- 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in eps around 0 50.8%

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

      1. associate--r+ 50.8%

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

      2. cancel-sign-sub-inv 50.8%

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

      3. distribute-rgt1-in 50.8%

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

      4. distribute-rgt-out-- 51.3%

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

      5. neg-mul-1 51.3%

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

      6. neg-mul-1 51.3%

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

      7. rec-exp 51.3%

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

      8. associate-*r/ 51.3%

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

      9. *-rgt-identity 51.3%

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

      10. metadata-eval 51.3%

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

      11. *-lft-identity 51.3%

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

   6. Simplified 51.3%

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

      1. *-commutative 51.3%

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

      2. div-inv 51.3%

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

      3. exp-neg 51.3%

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

      4. distribute-rgt-out 51.3%

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

      5. add-sqr-sqrt 24.1%

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

      6. sqrt-unprod 49.9%

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

      7. sqr-neg 49.9%

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

      8. sqrt-unprod 25.7%

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

      9. add-sqr-sqrt 49.4%

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

      10. associate--l+ 49.4%

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

      11. metadata-eval 49.4%

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

   8. Applied egg-rr 49.4%

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

   9. Taylor expanded in x around 0 64.2%

      \[\leadsto \frac{\color{blue}{2 + \left(3 \cdot {x}^{2} + 4 \cdot x\right)}}{2} \]
   10. Step-by-step derivation

       1. +-commutative 64.2%

          \[\leadsto \frac{2 + \color{blue}{\left(4 \cdot x + 3 \cdot {x}^{2}\right)}}{2} \]

       2. *-commutative 64.2%

          \[\leadsto \frac{2 + \left(\color{blue}{x \cdot 4} + 3 \cdot {x}^{2}\right)}{2} \]

       3. *-commutative 64.2%

          \[\leadsto \frac{2 + \left(x \cdot 4 + \color{blue}{{x}^{2} \cdot 3}\right)}{2} \]

       4. unpow2 64.2%

          \[\leadsto \frac{2 + \left(x \cdot 4 + \color{blue}{\left(x \cdot x\right)} \cdot 3\right)}{2} \]

       5. associate-*l* 64.2%

          \[\leadsto \frac{2 + \left(x \cdot 4 + \color{blue}{x \cdot \left(x \cdot 3\right)}\right)}{2} \]

       6. distribute-lft-out 64.2%

          \[\leadsto \frac{2 + \color{blue}{x \cdot \left(4 + x \cdot 3\right)}}{2} \]

   11. Simplified 64.2%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(4 + x \cdot 3\right)}}{2} \]

   if 540 < x < 1.7e102 or 4.4e232 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 61.6%

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

      1. neg-mul-1 61.6%

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

      2. rec-exp 61.6%

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

      3. neg-mul-1 61.6%

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

      4. div-sub 61.6%

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

      5. +-inverses 61.6%

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

   6. Simplified 61.6%

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

   if 1.7e102 < x < 4.4e232

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 37.7%

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

      1. associate--r+ 37.7%

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

      2. cancel-sign-sub-inv 37.7%

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

      3. distribute-rgt1-in 37.7%

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

      4. distribute-rgt-out-- 37.7%

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

      5. neg-mul-1 37.7%

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

      6. neg-mul-1 37.7%

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

      7. rec-exp 37.7%

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

      8. associate-*r/ 37.7%

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

      9. *-rgt-identity 37.7%

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

      10. metadata-eval 37.7%

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

      11. *-lft-identity 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in x around inf 37.7%

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

      1. rec-exp 37.7%

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

      2. count-2 37.7%

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

      3. rec-exp 37.7%

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

      4. associate-*r/ 37.7%

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

      5. metadata-eval 37.7%

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

   9. Simplified 37.7%

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

   10. Taylor expanded in x around 0 63.9%

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

       1. +-commutative 63.9%

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

       2. unpow2 63.9%

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

       3. distribute-rgt-out 63.9%

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

   12. Simplified 63.9%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 540:\\ \;\;\;\;\frac{2 + x \cdot \left(4 + x \cdot 3\right)}{2}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+102}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+232}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + -2\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 63.5% accurate, 13.2× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 540:\\ \;\;\;\;\frac{\left(2 + x \cdot \left(x \cdot 0.5\right)\right) - x}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+92}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+234}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + -2\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 540.0)
   (/ (- (+ 2.0 (* x (* x 0.5))) x) 2.0)
   (if (<= x 5e+92)
     0.0
     (if (<= x 1.05e+234) (/ (* x (+ 2.0 (* x (+ x -2.0)))) 2.0) 0.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 540.0) {
		tmp = ((2.0 + (x * (x * 0.5))) - x) / 2.0;
	} else if (x <= 5e+92) {
		tmp = 0.0;
	} else if (x <= 1.05e+234) {
		tmp = (x * (2.0 + (x * (x + -2.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 540.0d0) then
        tmp = ((2.0d0 + (x * (x * 0.5d0))) - x) / 2.0d0
    else if (x <= 5d+92) then
        tmp = 0.0d0
    else if (x <= 1.05d+234) then
        tmp = (x * (2.0d0 + (x * (x + (-2.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 540.0) {
		tmp = ((2.0 + (x * (x * 0.5))) - x) / 2.0;
	} else if (x <= 5e+92) {
		tmp = 0.0;
	} else if (x <= 1.05e+234) {
		tmp = (x * (2.0 + (x * (x + -2.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 540.0:
		tmp = ((2.0 + (x * (x * 0.5))) - x) / 2.0
	elif x <= 5e+92:
		tmp = 0.0
	elif x <= 1.05e+234:
		tmp = (x * (2.0 + (x * (x + -2.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 540.0)
		tmp = Float64(Float64(Float64(2.0 + Float64(x * Float64(x * 0.5))) - x) / 2.0);
	elseif (x <= 5e+92)
		tmp = 0.0;
	elseif (x <= 1.05e+234)
		tmp = Float64(Float64(x * Float64(2.0 + Float64(x * Float64(x + -2.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 540.0)
		tmp = ((2.0 + (x * (x * 0.5))) - x) / 2.0;
	elseif (x <= 5e+92)
		tmp = 0.0;
	elseif (x <= 1.05e+234)
		tmp = (x * (2.0 + (x * (x + -2.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 540.0], N[(N[(N[(2.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+92], 0.0, If[LessEqual[x, 1.05e+234], N[(N[(x * N[(2.0 + N[(x * N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 540

   1. Initial program 68.6%

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

   3. Simplified 44.4%

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

   4. Taylor expanded in eps around inf 97.7%

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

   5. Taylor expanded in eps around inf 97.7%

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

      1. *-commutative 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in x around 0 79.5%

      \[\leadsto \frac{\color{blue}{2 + \left(-1 \cdot x + {x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right)\right)}}{2} \]
   9. Step-by-step derivation

      1. +-commutative 79.5%

         \[\leadsto \frac{2 + \color{blue}{\left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) + -1 \cdot x\right)}}{2} \]

      2. mul-1-neg 79.5%

         \[\leadsto \frac{2 + \left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) + \color{blue}{\left(-x\right)}\right)}{2} \]

      3. unsub-neg 79.5%

         \[\leadsto \frac{2 + \color{blue}{\left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) - x\right)}}{2} \]

      4. unpow2 79.5%

         \[\leadsto \frac{2 + \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) - x\right)}{2} \]

      5. cancel-sign-sub-inv 79.5%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \color{blue}{\left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} + \left(--0.5\right) \cdot {\varepsilon}^{2}\right)} - x\right)}{2} \]

      6. metadata-eval 79.5%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} + \color{blue}{0.5} \cdot {\varepsilon}^{2}\right) - x\right)}{2} \]

      7. distribute-lft-out 79.5%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \color{blue}{\left(0.5 \cdot \left({\left(\varepsilon - 1\right)}^{2} + {\varepsilon}^{2}\right)\right)} - x\right)}{2} \]

      8. sub-neg 79.5%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\color{blue}{\left(\varepsilon + \left(-1\right)\right)}}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

      9. metadata-eval 79.5%

         \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(\varepsilon + \color{blue}{-1}\right)}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

      10. +-commutative 79.5%

          \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\color{blue}{\left(-1 + \varepsilon\right)}}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

      11. unpow2 79.5%

          \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(-1 + \varepsilon\right)}^{2} + \color{blue}{\varepsilon \cdot \varepsilon}\right)\right) - x\right)}{2} \]

   10. Simplified 79.5%

       \[\leadsto \frac{\color{blue}{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(-1 + \varepsilon\right)}^{2} + \varepsilon \cdot \varepsilon\right)\right) - x\right)}}{2} \]

   11. Taylor expanded in eps around 0 64.3%

       \[\leadsto \frac{\color{blue}{\left(2 + 0.5 \cdot {x}^{2}\right) - x}}{2} \]
   12. Step-by-step derivation

       1. *-commutative 64.3%

          \[\leadsto \frac{\left(2 + \color{blue}{{x}^{2} \cdot 0.5}\right) - x}{2} \]

       2. unpow2 64.3%

          \[\leadsto \frac{\left(2 + \color{blue}{\left(x \cdot x\right)} \cdot 0.5\right) - x}{2} \]

       3. associate-*l* 64.3%

          \[\leadsto \frac{\left(2 + \color{blue}{x \cdot \left(x \cdot 0.5\right)}\right) - x}{2} \]

   13. Simplified 64.3%

       \[\leadsto \frac{\color{blue}{\left(2 + x \cdot \left(x \cdot 0.5\right)\right) - x}}{2} \]

   if 540 < x < 5.00000000000000022e92 or 1.05e234 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 61.6%

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

      1. neg-mul-1 61.6%

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

      2. rec-exp 61.6%

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

      3. neg-mul-1 61.6%

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

      4. div-sub 61.6%

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

      5. +-inverses 61.6%

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

   6. Simplified 61.6%

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

   if 5.00000000000000022e92 < x < 1.05e234

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 37.7%

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

      1. associate--r+ 37.7%

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

      2. cancel-sign-sub-inv 37.7%

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

      3. distribute-rgt1-in 37.7%

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

      4. distribute-rgt-out-- 37.7%

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

      5. neg-mul-1 37.7%

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

      6. neg-mul-1 37.7%

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

      7. rec-exp 37.7%

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

      8. associate-*r/ 37.7%

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

      9. *-rgt-identity 37.7%

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

      10. metadata-eval 37.7%

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

      11. *-lft-identity 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in x around inf 37.7%

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

      1. rec-exp 37.7%

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

      2. count-2 37.7%

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

      3. rec-exp 37.7%

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

      4. associate-*r/ 37.7%

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

      5. metadata-eval 37.7%

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

   9. Simplified 37.7%

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

   10. Taylor expanded in x around 0 63.9%

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

       1. +-commutative 63.9%

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

       2. unpow2 63.9%

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

       3. distribute-rgt-out 63.9%

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

   12. Simplified 63.9%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 540:\\ \;\;\;\;\frac{\left(2 + x \cdot \left(x \cdot 0.5\right)\right) - x}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+92}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+234}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x + -2\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 56.8% accurate, 17.2× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 580:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+195}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+232}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 580.0)
   1.0
   (if (<= x 1.25e+195)
     0.0
     (if (<= x 6.2e+232) (/ (+ 2.0 (* eps x)) 2.0) 0.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 580.0) {
		tmp = 1.0;
	} else if (x <= 1.25e+195) {
		tmp = 0.0;
	} else if (x <= 6.2e+232) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 580.0d0) then
        tmp = 1.0d0
    else if (x <= 1.25d+195) then
        tmp = 0.0d0
    else if (x <= 6.2d+232) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 580.0) {
		tmp = 1.0;
	} else if (x <= 1.25e+195) {
		tmp = 0.0;
	} else if (x <= 6.2e+232) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 580.0:
		tmp = 1.0
	elif x <= 1.25e+195:
		tmp = 0.0
	elif x <= 6.2e+232:
		tmp = (2.0 + (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 580.0)
		tmp = 1.0;
	elseif (x <= 1.25e+195)
		tmp = 0.0;
	elseif (x <= 6.2e+232)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 580.0)
		tmp = 1.0;
	elseif (x <= 1.25e+195)
		tmp = 0.0;
	elseif (x <= 6.2e+232)
		tmp = (2.0 + (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 580.0], 1.0, If[LessEqual[x, 1.25e+195], 0.0, If[LessEqual[x, 6.2e+232], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 580

   1. Initial program 68.6%

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

      1. sub-neg 68.6%

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

      2. neg-sub0 68.6%

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

      3. associate-+r- 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in x around 0 50.6%

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

   if 580 < x < 1.2499999999999999e195 or 6.19999999999999966e232 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 57.1%

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

      1. neg-mul-1 57.1%

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

      2. rec-exp 57.1%

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

      3. neg-mul-1 57.1%

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

      4. div-sub 57.1%

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

      5. +-inverses 57.1%

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

   6. Simplified 57.1%

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

   if 1.2499999999999999e195 < x < 6.19999999999999966e232

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 3.1%

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

      1. *-commutative 3.1%

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

      2. flip-+ 45.5%

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

      3. associate-*r/ 45.5%

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

      4. sub-neg 45.5%

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

      5. distribute-neg-frac 45.5%

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

      6. metadata-eval 45.5%

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

      7. metadata-eval 45.5%

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

   6. Applied egg-rr 45.5%

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

      1. *-commutative 45.5%

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

      2. associate-/l* 45.5%

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

      3. +-commutative 45.5%

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

   8. Simplified 45.5%

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

   9. Taylor expanded in eps around 0 24.2%

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

   10. Taylor expanded in eps around 0 24.1%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 580:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+195}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+232}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 70.9% accurate, 17.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

3. Simplified 59.8%

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

4. Taylor expanded in eps around inf 98.3%

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

5. Taylor expanded in eps around inf 88.8%

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

   1. *-commutative 88.8%

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

7. Simplified 88.8%

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

8. Taylor expanded in x around 0 69.9%

   \[\leadsto \frac{\color{blue}{2 + \left(-1 \cdot x + {x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right)\right)}}{2} \]
9. Step-by-step derivation

   1. +-commutative 69.9%

      \[\leadsto \frac{2 + \color{blue}{\left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) + -1 \cdot x\right)}}{2} \]

   2. mul-1-neg 69.9%

      \[\leadsto \frac{2 + \left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) + \color{blue}{\left(-x\right)}\right)}{2} \]

   3. unsub-neg 69.9%

      \[\leadsto \frac{2 + \color{blue}{\left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) - x\right)}}{2} \]

   4. unpow2 69.9%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) - x\right)}{2} \]

   5. cancel-sign-sub-inv 69.9%

      \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \color{blue}{\left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} + \left(--0.5\right) \cdot {\varepsilon}^{2}\right)} - x\right)}{2} \]

   6. metadata-eval 69.9%

      \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} + \color{blue}{0.5} \cdot {\varepsilon}^{2}\right) - x\right)}{2} \]

   7. distribute-lft-out 69.9%

      \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \color{blue}{\left(0.5 \cdot \left({\left(\varepsilon - 1\right)}^{2} + {\varepsilon}^{2}\right)\right)} - x\right)}{2} \]

   8. sub-neg 69.9%

      \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\color{blue}{\left(\varepsilon + \left(-1\right)\right)}}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

   9. metadata-eval 69.9%

      \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(\varepsilon + \color{blue}{-1}\right)}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

   10. +-commutative 69.9%

       \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\color{blue}{\left(-1 + \varepsilon\right)}}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

   11. unpow2 69.9%

       \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(-1 + \varepsilon\right)}^{2} + \color{blue}{\varepsilon \cdot \varepsilon}\right)\right) - x\right)}{2} \]

10. Simplified 69.9%

    \[\leadsto \frac{\color{blue}{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(-1 + \varepsilon\right)}^{2} + \varepsilon \cdot \varepsilon\right)\right) - x\right)}}{2} \]

11. Taylor expanded in eps around inf 69.8%

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

    1. unpow2 69.8%

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

    2. *-commutative 69.8%

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

    3. unpow2 69.8%

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

    4. unswap-sqr 68.5%

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

13. Simplified 68.5%

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

14. Final simplification 68.5%

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

Alternative 11: 70.8% accurate, 17.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

3. Simplified 59.8%

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

4. Taylor expanded in eps around inf 98.3%

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

5. Taylor expanded in eps around inf 88.8%

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

   1. *-commutative 88.8%

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

7. Simplified 88.8%

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

8. Taylor expanded in x around 0 69.9%

   \[\leadsto \frac{\color{blue}{2 + \left(-1 \cdot x + {x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right)\right)}}{2} \]
9. Step-by-step derivation

   1. +-commutative 69.9%

      \[\leadsto \frac{2 + \color{blue}{\left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) + -1 \cdot x\right)}}{2} \]

   2. mul-1-neg 69.9%

      \[\leadsto \frac{2 + \left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) + \color{blue}{\left(-x\right)}\right)}{2} \]

   3. unsub-neg 69.9%

      \[\leadsto \frac{2 + \color{blue}{\left({x}^{2} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) - x\right)}}{2} \]

   4. unpow2 69.9%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} - -0.5 \cdot {\varepsilon}^{2}\right) - x\right)}{2} \]

   5. cancel-sign-sub-inv 69.9%

      \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \color{blue}{\left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} + \left(--0.5\right) \cdot {\varepsilon}^{2}\right)} - x\right)}{2} \]

   6. metadata-eval 69.9%

      \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot {\left(\varepsilon - 1\right)}^{2} + \color{blue}{0.5} \cdot {\varepsilon}^{2}\right) - x\right)}{2} \]

   7. distribute-lft-out 69.9%

      \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \color{blue}{\left(0.5 \cdot \left({\left(\varepsilon - 1\right)}^{2} + {\varepsilon}^{2}\right)\right)} - x\right)}{2} \]

   8. sub-neg 69.9%

      \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\color{blue}{\left(\varepsilon + \left(-1\right)\right)}}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

   9. metadata-eval 69.9%

      \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(\varepsilon + \color{blue}{-1}\right)}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

   10. +-commutative 69.9%

       \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\color{blue}{\left(-1 + \varepsilon\right)}}^{2} + {\varepsilon}^{2}\right)\right) - x\right)}{2} \]

   11. unpow2 69.9%

       \[\leadsto \frac{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(-1 + \varepsilon\right)}^{2} + \color{blue}{\varepsilon \cdot \varepsilon}\right)\right) - x\right)}{2} \]

10. Simplified 69.9%

    \[\leadsto \frac{\color{blue}{2 + \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \left({\left(-1 + \varepsilon\right)}^{2} + \varepsilon \cdot \varepsilon\right)\right) - x\right)}}{2} \]

11. Taylor expanded in eps around inf 69.8%

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

    1. unpow2 69.8%

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

    2. *-commutative 69.8%

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

    3. unpow2 69.8%

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

13. Simplified 69.8%

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

14. Final simplification 69.8%

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

Alternative 12: 57.1% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 450:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 450.0) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 450.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 450.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 450.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 450.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 450.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 450.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 450.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 450

   1. Initial program 68.6%

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

      1. sub-neg 68.6%

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

      2. neg-sub0 68.6%

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

      3. associate-+r- 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in x around 0 50.6%

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

   if 450 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 51.5%

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

      1. neg-mul-1 51.5%

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

      2. rec-exp 51.5%

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

      3. neg-mul-1 51.5%

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

      4. div-sub 51.5%

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

      5. +-inverses 51.5%

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

   6. Simplified 51.5%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 450:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 16.2% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

3. Simplified 77.3%

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

4. Taylor expanded in eps around 0 15.7%

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

   1. neg-mul-1 15.7%

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

   2. rec-exp 15.7%

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

   3. neg-mul-1 15.7%

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

   4. div-sub 15.7%

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

   5. +-inverses 16.0%

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

6. Simplified 16.0%

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

7. Final simplification 16.0%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023275 
(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))