NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.2% → 97.4%

Time: 18.1s

Alternatives: 11

Speedup: 1.7×

• 38.5% 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: 73.2% 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: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 + eps\_m\right)}\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-188}:\\ \;\;\;\;\frac{t\_0 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ -1.0 eps_m)))))
   (if (<= x -3.8e-188)
     (/ (+ t_0 (exp (* x (- eps_m)))) 2.0)
     (/ (+ t_0 (exp (- x))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (-1.0 + eps_m)));
	double tmp;
	if (x <= -3.8e-188) {
		tmp = (t_0 + exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (t_0 + exp(-x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * ((-1.0d0) + eps_m)))
    if (x <= (-3.8d-188)) then
        tmp = (t_0 + exp((x * -eps_m))) / 2.0d0
    else
        tmp = (t_0 + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (-1.0 + eps_m)));
	double tmp;
	if (x <= -3.8e-188) {
		tmp = (t_0 + Math.exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (t_0 + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (-1.0 + eps_m)))
	tmp = 0
	if x <= -3.8e-188:
		tmp = (t_0 + math.exp((x * -eps_m))) / 2.0
	else:
		tmp = (t_0 + math.exp(-x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(-1.0 + eps_m)))
	tmp = 0.0
	if (x <= -3.8e-188)
		tmp = Float64(Float64(t_0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(t_0 + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (-1.0 + eps_m)));
	tmp = 0.0;
	if (x <= -3.8e-188)
		tmp = (t_0 + exp((x * -eps_m))) / 2.0;
	else
		tmp = (t_0 + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.8e-188], N[(N[(t$95$0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e-188

   1. Initial program 66.2%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in eps around inf 96.2%

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

   6. Taylor expanded in eps around inf 96.3%

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

      1. *-commutative 96.3%

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

   8. Simplified 96.3%

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

   if -3.8e-188 < x

   1. Initial program 68.1%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 82.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-188}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-188}:\\ \;\;\;\;\frac{e^{x \cdot \left(-eps\_m\right)} + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -3.8e-188)
     (/ (+ (exp (* x (- eps_m))) t_0) 2.0)
     (/ (+ (exp (* x (+ -1.0 eps_m))) t_0) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -3.8e-188) {
		tmp = (exp((x * -eps_m)) + t_0) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) + t_0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-3.8d-188)) then
        tmp = (exp((x * -eps_m)) + t_0) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps_m))) + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (x <= -3.8e-188) {
		tmp = (Math.exp((x * -eps_m)) + t_0) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + t_0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -3.8e-188:
		tmp = (math.exp((x * -eps_m)) + t_0) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) + t_0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -3.8e-188)
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps_m))) + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + t_0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (x <= -3.8e-188)
		tmp = (exp((x * -eps_m)) + t_0) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -3.8e-188], N[(N[(N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e-188

   1. Initial program 66.2%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in eps around inf 96.2%

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

   6. Taylor expanded in eps around inf 96.3%

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

      1. *-commutative 96.3%

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

   8. Simplified 96.3%

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

   9. Taylor expanded in eps around 0 82.8%

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

       1. neg-mul-1 82.8%

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

   11. Simplified 82.8%

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

   if -3.8e-188 < x

   1. Initial program 68.1%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 82.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-188}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq 2.05 \cdot 10^{-230}:\\ \;\;\;\;\frac{e^{x \cdot \left(-eps\_m\right)} + t\_0}{2}\\ \mathbf{elif}\;x \leq 1020000000000:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{eps\_m} + \mathsf{fma}\left(eps\_m, eps\_m, -1\right) \cdot \frac{1 + \frac{1}{eps\_m}}{1 + eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+127} \lor \neg \left(x \leq 10^{+274}\right):\\ \;\;\;\;\frac{2 \cdot t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x 2.05e-230)
     (/ (+ (exp (* x (- eps_m))) t_0) 2.0)
     (if (<= x 1020000000000.0)
       (/
        (+
         2.0
         (*
          x
          (+
           (/ 1.0 eps_m)
           (*
            (fma eps_m eps_m -1.0)
            (/ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 eps_m))))))
        2.0)
       (if (or (<= x 8e+127) (not (<= x 1e+274)))
         (/ (* 2.0 t_0) 2.0)
         (/ (exp x) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= 2.05e-230) {
		tmp = (exp((x * -eps_m)) + t_0) / 2.0;
	} else if (x <= 1020000000000.0) {
		tmp = (2.0 + (x * ((1.0 / eps_m) + (fma(eps_m, eps_m, -1.0) * ((1.0 + (1.0 / eps_m)) / (1.0 + eps_m)))))) / 2.0;
	} else if ((x <= 8e+127) || !(x <= 1e+274)) {
		tmp = (2.0 * t_0) / 2.0;
	} else {
		tmp = exp(x) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= 2.05e-230)
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps_m))) + t_0) / 2.0);
	elseif (x <= 1020000000000.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps_m) + Float64(fma(eps_m, eps_m, -1.0) * Float64(Float64(1.0 + Float64(1.0 / eps_m)) / Float64(1.0 + eps_m)))))) / 2.0);
	elseif ((x <= 8e+127) || !(x <= 1e+274))
		tmp = Float64(Float64(2.0 * t_0) / 2.0);
	else
		tmp = Float64(exp(x) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, 2.05e-230], N[(N[(N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1020000000000.0], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] * N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 8e+127], N[Not[LessEqual[x, 1e+274]], $MachinePrecision]], N[(N[(2.0 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[x], $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.0500000000000001e-230

   1. Initial program 62.2%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in eps around inf 97.3%

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

   6. Taylor expanded in eps around inf 97.3%

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

      1. *-commutative 97.3%

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

   8. Simplified 97.3%

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

   9. Taylor expanded in eps around 0 87.5%

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

       1. neg-mul-1 87.5%

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

   11. Simplified 87.5%

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

   if 2.0500000000000001e-230 < x < 1.02e12

   1. Initial program 44.4%

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

   3. Simplified 39.9%

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

   5. Taylor expanded in x around 0 68.2%

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

      1. flip-- 72.6%

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

      2. +-commutative 72.6%

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

      3. associate-*r/ 72.6%

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

      4. metadata-eval 72.6%

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

      5. fma-neg 72.6%

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

      6. metadata-eval 72.6%

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

   7. Applied egg-rr 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-/l* 72.5%

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

      3. +-commutative 72.5%

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

   9. Simplified 72.5%

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

   10. Taylor expanded in eps around 0 72.4%

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

   if 1.02e12 < x < 7.99999999999999964e127 or 9.99999999999999921e273 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 74.7%

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

   7. Taylor expanded in eps around 0 60.6%

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

      1. mul-1-neg 60.6%

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

   9. Simplified 60.6%

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

   if 7.99999999999999964e127 < x < 9.99999999999999921e273

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 42.3%

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

   7. Simplified 42.3%

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

   8. Taylor expanded in x around inf 42.3%

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

      1. exp-neg 42.3%

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

      2. associate-*r/ 42.3%

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

      3. *-rgt-identity 42.3%

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

   10. Simplified 42.3%

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

       1. cancel-sign-sub-inv 42.3%

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

       2. div-inv 42.3%

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

       3. exp-neg 42.3%

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

       4. metadata-eval 42.3%

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

       5. *-un-lft-identity 42.3%

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

       6. distribute-rgt-out 42.3%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 59.3%

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

       9. sqr-neg 59.3%

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

       10. sqrt-unprod 59.3%

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

       11. add-sqr-sqrt 59.3%

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

       12. +-commutative 59.3%

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

   12. Applied egg-rr 59.3%

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

   13. Taylor expanded in x around 0 59.3%

       \[\leadsto \frac{e^{x} \cdot \color{blue}{1}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-230}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1020000000000:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \mathsf{fma}\left(\varepsilon, \varepsilon, -1\right) \cdot \frac{1 + \frac{1}{\varepsilon}}{1 + \varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+127} \lor \neg \left(x \leq 10^{+274}\right):\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

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

3. Simplified 59.0%

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

5. Taylor expanded in eps around inf 98.6%

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

6. Final simplification 98.6%

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

Alternative 5: 78.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{2 \cdot e^{-x}}{2}\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-188}:\\ \;\;\;\;\frac{2 - x \cdot \left(\left(-1 + eps\_m\right) \cdot \left(-1 + \frac{-1}{eps\_m}\right) - \frac{\mathsf{fma}\left(eps\_m, eps\_m, -1\right) \cdot \left(-1 - \frac{-1}{eps\_m}\right)}{-1 + eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 840000000000:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{eps\_m} + \mathsf{fma}\left(eps\_m, eps\_m, -1\right) \cdot \frac{1 + \frac{1}{eps\_m}}{1 + eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+126} \lor \neg \left(x \leq 1.1 \cdot 10^{+274}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 (exp (- x))) 2.0)))
   (if (<= x -2.3e-6)
     t_0
     (if (<= x -4e-188)
       (/
        (-
         2.0
         (*
          x
          (-
           (* (+ -1.0 eps_m) (+ -1.0 (/ -1.0 eps_m)))
           (/
            (* (fma eps_m eps_m -1.0) (- -1.0 (/ -1.0 eps_m)))
            (+ -1.0 eps_m)))))
        2.0)
       (if (<= x 2.2e-231)
         (/ (+ 2.0 (* x eps_m)) 2.0)
         (if (<= x 840000000000.0)
           (/
            (+
             2.0
             (*
              x
              (+
               (/ 1.0 eps_m)
               (*
                (fma eps_m eps_m -1.0)
                (/ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 eps_m))))))
            2.0)
           (if (or (<= x 5e+126) (not (<= x 1.1e+274)))
             t_0
             (/ (exp x) 2.0))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (2.0 * exp(-x)) / 2.0;
	double tmp;
	if (x <= -2.3e-6) {
		tmp = t_0;
	} else if (x <= -4e-188) {
		tmp = (2.0 - (x * (((-1.0 + eps_m) * (-1.0 + (-1.0 / eps_m))) - ((fma(eps_m, eps_m, -1.0) * (-1.0 - (-1.0 / eps_m))) / (-1.0 + eps_m))))) / 2.0;
	} else if (x <= 2.2e-231) {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	} else if (x <= 840000000000.0) {
		tmp = (2.0 + (x * ((1.0 / eps_m) + (fma(eps_m, eps_m, -1.0) * ((1.0 + (1.0 / eps_m)) / (1.0 + eps_m)))))) / 2.0;
	} else if ((x <= 5e+126) || !(x <= 1.1e+274)) {
		tmp = t_0;
	} else {
		tmp = exp(x) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0)
	tmp = 0.0
	if (x <= -2.3e-6)
		tmp = t_0;
	elseif (x <= -4e-188)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(Float64(Float64(-1.0 + eps_m) * Float64(-1.0 + Float64(-1.0 / eps_m))) - Float64(Float64(fma(eps_m, eps_m, -1.0) * Float64(-1.0 - Float64(-1.0 / eps_m))) / Float64(-1.0 + eps_m))))) / 2.0);
	elseif (x <= 2.2e-231)
		tmp = Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0);
	elseif (x <= 840000000000.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps_m) + Float64(fma(eps_m, eps_m, -1.0) * Float64(Float64(1.0 + Float64(1.0 / eps_m)) / Float64(1.0 + eps_m)))))) / 2.0);
	elseif ((x <= 5e+126) || !(x <= 1.1e+274))
		tmp = t_0;
	else
		tmp = Float64(exp(x) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2.3e-6], t$95$0, If[LessEqual[x, -4e-188], N[(N[(2.0 - N[(x * N[(N[(N[(-1.0 + eps$95$m), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] * N[(-1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.2e-231], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 840000000000.0], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] * N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 5e+126], N[Not[LessEqual[x, 1.1e+274]], $MachinePrecision]], t$95$0, N[(N[Exp[x], $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.3e-6 or 8.4e11 < x < 4.99999999999999977e126 or 1.1e274 < x

   1. Initial program 96.1%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in eps around inf 95.4%

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

   6. Taylor expanded in eps around 0 83.9%

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

   7. Taylor expanded in eps around 0 75.3%

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

      1. mul-1-neg 75.3%

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

   9. Simplified 75.3%

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

   if -2.3e-6 < x < -3.9999999999999998e-188

   1. Initial program 44.1%

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

   3. Simplified 35.1%

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

   5. Taylor expanded in x around 0 68.3%

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

      1. flip-- 34.4%

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

      2. clear-num 34.4%

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

      3. metadata-eval 34.4%

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

      4. inv-pow 34.4%

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

      5. inv-pow 34.4%

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

      6. pow-prod-up 32.6%

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

      7. metadata-eval 32.6%

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

   7. Applied egg-rr 32.6%

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

      1. +-commutative 32.6%

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

      2. flip-+ 37.7%

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

      3. metadata-eval 37.7%

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

      4. fma-neg 37.7%

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

      5. metadata-eval 37.7%

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

      6. associate-*l/ 37.7%

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

      7. clear-num 37.7%

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

      8. metadata-eval 37.7%

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

      9. metadata-eval 37.7%

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

      10. pow-prod-up 39.5%

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

      11. inv-pow 39.5%

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

      12. inv-pow 39.5%

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

      13. flip-- 73.4%

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

      14. sub-neg 73.4%

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

      15. metadata-eval 73.4%

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

   9. Applied egg-rr 73.4%

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

   if -3.9999999999999998e-188 < x < 2.20000000000000009e-231

   1. Initial program 51.6%

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

   3. Simplified 12.4%

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

   5. Taylor expanded in x around 0 97.3%

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

   6. Taylor expanded in eps around 0 97.2%

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

   7. Taylor expanded in eps around 0 97.2%

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

   if 2.20000000000000009e-231 < x < 8.4e11

   1. Initial program 44.4%

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

   3. Simplified 39.9%

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

   5. Taylor expanded in x around 0 68.2%

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

      1. flip-- 72.6%

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

      2. +-commutative 72.6%

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

      3. associate-*r/ 72.6%

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

      4. metadata-eval 72.6%

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

      5. fma-neg 72.6%

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

      6. metadata-eval 72.6%

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

   7. Applied egg-rr 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-/l* 72.5%

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

      3. +-commutative 72.5%

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

   9. Simplified 72.5%

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

   10. Taylor expanded in eps around 0 72.4%

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

   if 4.99999999999999977e126 < x < 1.1e274

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 42.3%

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

   7. Simplified 42.3%

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

   8. Taylor expanded in x around inf 42.3%

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

      1. exp-neg 42.3%

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

      2. associate-*r/ 42.3%

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

      3. *-rgt-identity 42.3%

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

   10. Simplified 42.3%

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

       1. cancel-sign-sub-inv 42.3%

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

       2. div-inv 42.3%

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

       3. exp-neg 42.3%

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

       4. metadata-eval 42.3%

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

       5. *-un-lft-identity 42.3%

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

       6. distribute-rgt-out 42.3%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 59.3%

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

       9. sqr-neg 59.3%

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

       10. sqrt-unprod 59.3%

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

       11. add-sqr-sqrt 59.3%

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

       12. +-commutative 59.3%

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

   12. Applied egg-rr 59.3%

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

   13. Taylor expanded in x around 0 59.3%

       \[\leadsto \frac{e^{x} \cdot \color{blue}{1}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-188}:\\ \;\;\;\;\frac{2 - x \cdot \left(\left(-1 + \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right) - \frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right) \cdot \left(-1 - \frac{-1}{\varepsilon}\right)}{-1 + \varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 840000000000:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \mathsf{fma}\left(\varepsilon, \varepsilon, -1\right) \cdot \frac{1 + \frac{1}{\varepsilon}}{1 + \varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+126} \lor \neg \left(x \leq 1.1 \cdot 10^{+274}\right):\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{2 \cdot e^{-x}}{2}\\ \mathbf{if}\;x \leq 5 \cdot 10^{-232}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 540000000000:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{eps\_m} + \mathsf{fma}\left(eps\_m, eps\_m, -1\right) \cdot \frac{1 + \frac{1}{eps\_m}}{1 + eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+126} \lor \neg \left(x \leq 10^{+274}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 (exp (- x))) 2.0)))
   (if (<= x 5e-232)
     t_0
     (if (<= x 540000000000.0)
       (/
        (+
         2.0
         (*
          x
          (+
           (/ 1.0 eps_m)
           (*
            (fma eps_m eps_m -1.0)
            (/ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 eps_m))))))
        2.0)
       (if (or (<= x 5e+126) (not (<= x 1e+274))) t_0 (/ (exp x) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (2.0 * exp(-x)) / 2.0;
	double tmp;
	if (x <= 5e-232) {
		tmp = t_0;
	} else if (x <= 540000000000.0) {
		tmp = (2.0 + (x * ((1.0 / eps_m) + (fma(eps_m, eps_m, -1.0) * ((1.0 + (1.0 / eps_m)) / (1.0 + eps_m)))))) / 2.0;
	} else if ((x <= 5e+126) || !(x <= 1e+274)) {
		tmp = t_0;
	} else {
		tmp = exp(x) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0)
	tmp = 0.0
	if (x <= 5e-232)
		tmp = t_0;
	elseif (x <= 540000000000.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps_m) + Float64(fma(eps_m, eps_m, -1.0) * Float64(Float64(1.0 + Float64(1.0 / eps_m)) / Float64(1.0 + eps_m)))))) / 2.0);
	elseif ((x <= 5e+126) || !(x <= 1e+274))
		tmp = t_0;
	else
		tmp = Float64(exp(x) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, 5e-232], t$95$0, If[LessEqual[x, 540000000000.0], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] * N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 5e+126], N[Not[LessEqual[x, 1e+274]], $MachinePrecision]], t$95$0, N[(N[Exp[x], $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.9999999999999999e-232 or 5.4e11 < x < 4.99999999999999977e126 or 9.99999999999999921e273 < x

   1. Initial program 70.2%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in eps around inf 97.8%

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

   6. Taylor expanded in eps around 0 88.3%

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

   7. Taylor expanded in eps around 0 77.9%

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

      1. mul-1-neg 77.9%

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

   9. Simplified 77.9%

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

   if 4.9999999999999999e-232 < x < 5.4e11

   1. Initial program 44.4%

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

   3. Simplified 39.9%

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

   5. Taylor expanded in x around 0 68.2%

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

      1. flip-- 72.6%

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

      2. +-commutative 72.6%

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

      3. associate-*r/ 72.6%

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

      4. metadata-eval 72.6%

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

      5. fma-neg 72.6%

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

      6. metadata-eval 72.6%

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

   7. Applied egg-rr 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-/l* 72.5%

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

      3. +-commutative 72.5%

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

   9. Simplified 72.5%

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

   10. Taylor expanded in eps around 0 72.4%

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

   if 4.99999999999999977e126 < x < 9.99999999999999921e273

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 42.3%

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

   7. Simplified 42.3%

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

   8. Taylor expanded in x around inf 42.3%

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

      1. exp-neg 42.3%

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

      2. associate-*r/ 42.3%

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

      3. *-rgt-identity 42.3%

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

   10. Simplified 42.3%

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

       1. cancel-sign-sub-inv 42.3%

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

       2. div-inv 42.3%

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

       3. exp-neg 42.3%

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

       4. metadata-eval 42.3%

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

       5. *-un-lft-identity 42.3%

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

       6. distribute-rgt-out 42.3%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 59.3%

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

       9. sqr-neg 59.3%

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

       10. sqrt-unprod 59.3%

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

       11. add-sqr-sqrt 59.3%

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

       12. +-commutative 59.3%

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

   12. Applied egg-rr 59.3%

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

   13. Taylor expanded in x around 0 59.3%

       \[\leadsto \frac{e^{x} \cdot \color{blue}{1}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-232}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 540000000000:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \mathsf{fma}\left(\varepsilon, \varepsilon, -1\right) \cdot \frac{1 + \frac{1}{\varepsilon}}{1 + \varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+126} \lor \neg \left(x \leq 10^{+274}\right):\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{eps\_m} - \left(-1 - eps\_m\right) \cdot \left(-1 - \frac{-1}{eps\_m}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+63} \lor \neg \left(x \leq 9 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{\frac{0}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 3.2e-18)
   (/
    (+ 2.0 (* x (- (/ -1.0 eps_m) (* (- -1.0 eps_m) (- -1.0 (/ -1.0 eps_m))))))
    2.0)
   (if (or (<= x 5e+63) (not (<= x 9e+273)))
     (/ (/ 0.0 eps_m) 2.0)
     (/ (exp x) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 3.2e-18) {
		tmp = (2.0 + (x * ((-1.0 / eps_m) - ((-1.0 - eps_m) * (-1.0 - (-1.0 / eps_m)))))) / 2.0;
	} else if ((x <= 5e+63) || !(x <= 9e+273)) {
		tmp = (0.0 / eps_m) / 2.0;
	} else {
		tmp = exp(x) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 3.2d-18) then
        tmp = (2.0d0 + (x * (((-1.0d0) / eps_m) - (((-1.0d0) - eps_m) * ((-1.0d0) - ((-1.0d0) / eps_m)))))) / 2.0d0
    else if ((x <= 5d+63) .or. (.not. (x <= 9d+273))) then
        tmp = (0.0d0 / eps_m) / 2.0d0
    else
        tmp = exp(x) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 3.2e-18) {
		tmp = (2.0 + (x * ((-1.0 / eps_m) - ((-1.0 - eps_m) * (-1.0 - (-1.0 / eps_m)))))) / 2.0;
	} else if ((x <= 5e+63) || !(x <= 9e+273)) {
		tmp = (0.0 / eps_m) / 2.0;
	} else {
		tmp = Math.exp(x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 3.2e-18:
		tmp = (2.0 + (x * ((-1.0 / eps_m) - ((-1.0 - eps_m) * (-1.0 - (-1.0 / eps_m)))))) / 2.0
	elif (x <= 5e+63) or not (x <= 9e+273):
		tmp = (0.0 / eps_m) / 2.0
	else:
		tmp = math.exp(x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 3.2e-18)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 / eps_m) - Float64(Float64(-1.0 - eps_m) * Float64(-1.0 - Float64(-1.0 / eps_m)))))) / 2.0);
	elseif ((x <= 5e+63) || !(x <= 9e+273))
		tmp = Float64(Float64(0.0 / eps_m) / 2.0);
	else
		tmp = Float64(exp(x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 3.2e-18)
		tmp = (2.0 + (x * ((-1.0 / eps_m) - ((-1.0 - eps_m) * (-1.0 - (-1.0 / eps_m)))))) / 2.0;
	elseif ((x <= 5e+63) || ~((x <= 9e+273)))
		tmp = (0.0 / eps_m) / 2.0;
	else
		tmp = exp(x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 3.2e-18], N[(N[(2.0 + N[(x * N[(N[(-1.0 / eps$95$m), $MachinePrecision] - N[(N[(-1.0 - eps$95$m), $MachinePrecision] * N[(-1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 5e+63], N[Not[LessEqual[x, 9e+273]], $MachinePrecision]], N[(N[(0.0 / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[x], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.1999999999999999e-18

   1. Initial program 55.2%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in x around 0 61.1%

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

      1. flip-- 65.4%

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

      2. +-commutative 65.4%

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

      3. associate-*r/ 65.4%

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

      4. metadata-eval 65.4%

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

      5. fma-neg 65.4%

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

      6. metadata-eval 65.4%

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

   7. Applied egg-rr 65.4%

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

      1. *-commutative 65.4%

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

      2. associate-/l* 65.8%

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

      3. +-commutative 65.8%

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

   9. Simplified 65.8%

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

   10. Taylor expanded in eps around 0 64.4%

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

   if 3.1999999999999999e-18 < x < 5.00000000000000011e63 or 8.99999999999999987e273 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 56.7%

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

   6. Taylor expanded in x around 0 56.7%

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

   if 5.00000000000000011e63 < x < 8.99999999999999987e273

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 45.3%

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

   7. Simplified 45.3%

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

   8. Taylor expanded in x around inf 45.3%

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

      1. exp-neg 45.3%

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

      2. associate-*r/ 45.3%

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

      3. *-rgt-identity 45.3%

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

   10. Simplified 45.3%

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

       1. cancel-sign-sub-inv 45.3%

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

       2. div-inv 45.3%

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

       3. exp-neg 45.3%

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

       4. metadata-eval 45.3%

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

       5. *-un-lft-identity 45.3%

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

       6. distribute-rgt-out 45.3%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 56.3%

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

       9. sqr-neg 56.3%

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

       10. sqrt-unprod 56.3%

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

       11. add-sqr-sqrt 56.3%

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

       12. +-commutative 56.3%

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

   12. Applied egg-rr 56.3%

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

   13. Taylor expanded in x around 0 56.3%

       \[\leadsto \frac{e^{x} \cdot \color{blue}{1}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{\varepsilon} - \left(-1 - \varepsilon\right) \cdot \left(-1 - \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+63} \lor \neg \left(x \leq 9 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+127} \lor \neg \left(x \leq 1.2 \cdot 10^{+274}\right):\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (or (<= x 1e+127) (not (<= x 1.2e+274)))
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (exp x) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((x <= 1e+127) || !(x <= 1.2e+274)) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = exp(x) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if ((x <= 1d+127) .or. (.not. (x <= 1.2d+274))) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = exp(x) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if ((x <= 1e+127) || !(x <= 1.2e+274)) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = Math.exp(x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if (x <= 1e+127) or not (x <= 1.2e+274):
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = math.exp(x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if ((x <= 1e+127) || !(x <= 1.2e+274))
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(exp(x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if ((x <= 1e+127) || ~((x <= 1.2e+274)))
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = exp(x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[Or[LessEqual[x, 1e+127], N[Not[LessEqual[x, 1.2e+274]], $MachinePrecision]], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[x], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999955e126 or 1.2e274 < x

   1. Initial program 63.3%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in eps around inf 98.4%

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

   6. Taylor expanded in eps around 0 86.3%

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

   7. Taylor expanded in eps around 0 75.7%

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

      1. mul-1-neg 75.7%

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

   9. Simplified 75.7%

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

   if 9.99999999999999955e126 < x < 1.2e274

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 42.3%

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

   7. Simplified 42.3%

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

   8. Taylor expanded in x around inf 42.3%

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

      1. exp-neg 42.3%

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

      2. associate-*r/ 42.3%

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

      3. *-rgt-identity 42.3%

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

   10. Simplified 42.3%

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

       1. cancel-sign-sub-inv 42.3%

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

       2. div-inv 42.3%

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

       3. exp-neg 42.3%

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

       4. metadata-eval 42.3%

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

       5. *-un-lft-identity 42.3%

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

       6. distribute-rgt-out 42.3%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 59.3%

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

       9. sqr-neg 59.3%

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

       10. sqrt-unprod 59.3%

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

       11. add-sqr-sqrt 59.3%

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

       12. +-commutative 59.3%

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

   12. Applied egg-rr 59.3%

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

   13. Taylor expanded in x around 0 59.3%

       \[\leadsto \frac{e^{x} \cdot \color{blue}{1}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+127} \lor \neg \left(x \leq 1.2 \cdot 10^{+274}\right):\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{eps\_m} - \left(-1 - eps\_m\right) \cdot \left(-1 - \frac{-1}{eps\_m}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 3.2e-18)
   (/
    (+ 2.0 (* x (- (/ -1.0 eps_m) (* (- -1.0 eps_m) (- -1.0 (/ -1.0 eps_m))))))
    2.0)
   (/ (/ 0.0 eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 3.2e-18) {
		tmp = (2.0 + (x * ((-1.0 / eps_m) - ((-1.0 - eps_m) * (-1.0 - (-1.0 / eps_m)))))) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 3.2d-18) then
        tmp = (2.0d0 + (x * (((-1.0d0) / eps_m) - (((-1.0d0) - eps_m) * ((-1.0d0) - ((-1.0d0) / eps_m)))))) / 2.0d0
    else
        tmp = (0.0d0 / eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 3.2e-18) {
		tmp = (2.0 + (x * ((-1.0 / eps_m) - ((-1.0 - eps_m) * (-1.0 - (-1.0 / eps_m)))))) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 3.2e-18:
		tmp = (2.0 + (x * ((-1.0 / eps_m) - ((-1.0 - eps_m) * (-1.0 - (-1.0 / eps_m)))))) / 2.0
	else:
		tmp = (0.0 / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 3.2e-18)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 / eps_m) - Float64(Float64(-1.0 - eps_m) * Float64(-1.0 - Float64(-1.0 / eps_m)))))) / 2.0);
	else
		tmp = Float64(Float64(0.0 / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 3.2e-18)
		tmp = (2.0 + (x * ((-1.0 / eps_m) - ((-1.0 - eps_m) * (-1.0 - (-1.0 / eps_m)))))) / 2.0;
	else
		tmp = (0.0 / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 3.2e-18], N[(N[(2.0 + N[(x * N[(N[(-1.0 / eps$95$m), $MachinePrecision] - N[(N[(-1.0 - eps$95$m), $MachinePrecision] * N[(-1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.0 / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1999999999999999e-18

   1. Initial program 55.2%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in x around 0 61.1%

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

      1. flip-- 65.4%

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

      2. +-commutative 65.4%

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

      3. associate-*r/ 65.4%

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

      4. metadata-eval 65.4%

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

      5. fma-neg 65.4%

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

      6. metadata-eval 65.4%

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

   7. Applied egg-rr 65.4%

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

      1. *-commutative 65.4%

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

      2. associate-/l* 65.8%

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

      3. +-commutative 65.8%

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

   9. Simplified 65.8%

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

   10. Taylor expanded in eps around 0 64.4%

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

   if 3.1999999999999999e-18 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 49.4%

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

   6. Taylor expanded in x around 0 49.4%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{\varepsilon} - \left(-1 - \varepsilon\right) \cdot \left(-1 - \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.9% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 500.0) 1.0 (/ (/ 0.0 eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 500.0d0) then
        tmp = 1.0d0
    else
        tmp = (0.0d0 / eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 500.0:
		tmp = 1.0
	else:
		tmp = (0.0 / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 500.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(0.0 / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 500.0)
		tmp = 1.0;
	else
		tmp = (0.0 / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 500.0], 1.0, N[(N[(0.0 / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in x around 0 59.9%

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

   if 500 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 52.3%

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

   6. Taylor expanded in x around 0 52.3%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.4% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

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

3. Simplified 67.4%

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

5. Taylor expanded in x around 0 45.2%

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

6. Final simplification 45.2%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(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))