NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.2% → 99.8%

Time: 10.5s

Alternatives: 12

Speedup: 1.9×

• 39.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -3.763566011279035e+147
  2. eps = 4.162489598552965e-293

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: 74.2% 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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \mathbf{if}\;eps\_m \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (+ x 1.0) (exp (- x)))))
   (if (<= eps_m 6.2e-5)
     (/ (+ t_0 t_0) 2.0)
     (/ (+ (exp (* x eps_m)) (exp (* x (- eps_m)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x + 1.0) * exp(-x);
	double tmp;
	if (eps_m <= 6.2e-5) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) * exp(-x)
    if (eps_m <= 6.2d-5) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (x + 1.0) * Math.exp(-x);
	double tmp;
	if (eps_m <= 6.2e-5) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp((x * eps_m)) + Math.exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x + 1.0) * math.exp(-x)
	tmp = 0
	if eps_m <= 6.2e-5:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (math.exp((x * eps_m)) + math.exp((x * -eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x + 1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps_m <= 6.2e-5)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x + 1.0) * exp(-x);
	tmp = 0.0;
	if (eps_m <= 6.2e-5)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 6.2e-5], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 6.20000000000000027e-5

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in eps around 0 66.9%

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

      1. distribute-rgt1-in 66.9%

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

      2. mul-1-neg 66.9%

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

      3. distribute-lft-out 66.9%

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

      4. distribute-rgt1-in 67.5%

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

      5. mul-1-neg 67.5%

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

   7. Simplified 67.5%

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

   if 6.20000000000000027e-5 < eps

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

   7. Taylor expanded in eps around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. associate-*r* 100.0%

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

      3. sub-neg 100.0%

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

      4. mul-1-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. mul-1-neg 100.0%

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

      11. unsub-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in eps around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-lft-neg-out 100.0%

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

       3. *-commutative 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps\_m \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0 \cdot \left(\left(x + 1\right) + 1\right) + x \cdot t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 6.2e-5)
     (/ (+ (* t_0 (+ (+ x 1.0) 1.0)) (* x t_0)) 2.0)
     (/ (+ (exp (* x eps_m)) (exp (* x (- eps_m)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 6.2e-5) {
		tmp = ((t_0 * ((x + 1.0) + 1.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 6.2d-5) then
        tmp = ((t_0 * ((x + 1.0d0) + 1.0d0)) + (x * t_0)) / 2.0d0
    else
        tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 6.2e-5) {
		tmp = ((t_0 * ((x + 1.0) + 1.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (Math.exp((x * eps_m)) + Math.exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 6.2e-5:
		tmp = ((t_0 * ((x + 1.0) + 1.0)) + (x * t_0)) / 2.0
	else:
		tmp = (math.exp((x * eps_m)) + math.exp((x * -eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 6.2e-5)
		tmp = Float64(Float64(Float64(t_0 * Float64(Float64(x + 1.0) + 1.0)) + Float64(x * t_0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + exp(Float64(x * Float64(-eps_m)))) / 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 (eps_m <= 6.2e-5)
		tmp = ((t_0 * ((x + 1.0) + 1.0)) + (x * t_0)) / 2.0;
	else
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 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[eps$95$m, 6.2e-5], N[(N[(N[(t$95$0 * N[(N[(x + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 6.20000000000000027e-5

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in eps around 0 66.9%

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

      1. associate--r+ 66.9%

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

      2. associate-*r* 66.9%

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

      3. cancel-sign-sub-inv 66.9%

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

      4. distribute-rgt1-in 66.9%

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

      5. distribute-rgt-out-- 67.5%

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

      6. mul-1-neg 67.5%

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

      7. mul-1-neg 67.5%

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

      8. remove-double-neg 67.5%

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

      9. mul-1-neg 67.5%

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

   7. Simplified 67.5%

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

   if 6.20000000000000027e-5 < eps

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

   7. Taylor expanded in eps around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. associate-*r* 100.0%

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

      3. sub-neg 100.0%

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

      4. mul-1-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. mul-1-neg 100.0%

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

      11. unsub-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in eps around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-lft-neg-out 100.0%

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

       3. *-commutative 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(\left(x + 1\right) + 1\right) + x \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 4.8e+127)
   (/ (+ (exp (* x eps_m)) (exp (* x (- eps_m)))) 2.0)
   (* x (exp (- x)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 4.8e+127) {
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	} else {
		tmp = x * exp(-x);
	}
	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 <= 4.8d+127) then
        tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0d0
    else
        tmp = x * exp(-x)
    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 <= 4.8e+127) {
		tmp = (Math.exp((x * eps_m)) + Math.exp((x * -eps_m))) / 2.0;
	} else {
		tmp = x * Math.exp(-x);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 4.8e+127:
		tmp = (math.exp((x * eps_m)) + math.exp((x * -eps_m))) / 2.0
	else:
		tmp = x * math.exp(-x)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 4.8e+127)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	else
		tmp = Float64(x * exp(Float64(-x)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 4.8e+127)
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	else
		tmp = x * exp(-x);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 4.8e+127], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(x * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.8000000000000004e127

   1. Initial program 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in eps around inf 99.2%

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

   6. Taylor expanded in eps around inf 94.4%

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

   7. Taylor expanded in eps around -inf 94.4%

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

      1. mul-1-neg 94.4%

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

      2. associate-*r* 94.4%

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

      3. sub-neg 94.4%

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

      4. mul-1-neg 94.4%

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

      5. remove-double-neg 94.4%

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

      6. *-commutative 94.4%

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

      7. associate-*l* 94.4%

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

      8. distribute-lft-in 94.4%

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

      9. metadata-eval 94.4%

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

      10. mul-1-neg 94.4%

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

      11. unsub-neg 94.4%

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

   9. Simplified 94.4%

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

   10. Taylor expanded in eps around inf 94.5%

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

       1. mul-1-neg 94.5%

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

       2. distribute-lft-neg-out 94.5%

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

       3. *-commutative 94.5%

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

   12. Simplified 94.5%

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

   if 4.8000000000000004e127 < 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%

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

   5. Taylor expanded in eps around 0 60.6%

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

      1. associate--r+ 60.6%

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

      2. associate-*r* 60.6%

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

      3. cancel-sign-sub-inv 60.6%

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

      4. distribute-rgt1-in 60.6%

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

      5. distribute-rgt-out-- 60.6%

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

      6. mul-1-neg 60.6%

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

      7. mul-1-neg 60.6%

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

      8. remove-double-neg 60.6%

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

      9. mul-1-neg 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in x around inf 60.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% 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 73.5%

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

3. Simplified 73.5%

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

5. Taylor expanded in eps around inf 99.3%

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

6. Final simplification 99.3%

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

Alternative 5: 84.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -10000000:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-279}:\\ \;\;\;\;\frac{\left(x \cdot eps\_m + 1\right) + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+127}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -10000000.0)
     (/ (+ t_0 1.0) 2.0)
     (if (<= x -1e-279)
       (/ (+ (+ (* x eps_m) 1.0) (exp (* x (- -1.0 eps_m)))) 2.0)
       (if (<= x 2.6e+127) (/ (+ (exp (* x eps_m)) 1.0) 2.0) (* x t_0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -10000000.0) {
		tmp = (t_0 + 1.0) / 2.0;
	} else if (x <= -1e-279) {
		tmp = (((x * eps_m) + 1.0) + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 2.6e+127) {
		tmp = (exp((x * eps_m)) + 1.0) / 2.0;
	} else {
		tmp = x * t_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 <= (-10000000.0d0)) then
        tmp = (t_0 + 1.0d0) / 2.0d0
    else if (x <= (-1d-279)) then
        tmp = (((x * eps_m) + 1.0d0) + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 2.6d+127) then
        tmp = (exp((x * eps_m)) + 1.0d0) / 2.0d0
    else
        tmp = x * t_0
    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 <= -10000000.0) {
		tmp = (t_0 + 1.0) / 2.0;
	} else if (x <= -1e-279) {
		tmp = (((x * eps_m) + 1.0) + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 2.6e+127) {
		tmp = (Math.exp((x * eps_m)) + 1.0) / 2.0;
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -10000000.0:
		tmp = (t_0 + 1.0) / 2.0
	elif x <= -1e-279:
		tmp = (((x * eps_m) + 1.0) + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif x <= 2.6e+127:
		tmp = (math.exp((x * eps_m)) + 1.0) / 2.0
	else:
		tmp = x * t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -10000000.0)
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	elseif (x <= -1e-279)
		tmp = Float64(Float64(Float64(Float64(x * eps_m) + 1.0) + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 2.6e+127)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + 1.0) / 2.0);
	else
		tmp = Float64(x * t_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 <= -10000000.0)
		tmp = (t_0 + 1.0) / 2.0;
	elseif (x <= -1e-279)
		tmp = (((x * eps_m) + 1.0) + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 2.6e+127)
		tmp = (exp((x * eps_m)) + 1.0) / 2.0;
	else
		tmp = x * t_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, -10000000.0], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1e-279], N[(N[(N[(N[(x * eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.6e+127], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1e7

   1. Initial program 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in eps around inf 97.4%

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

   6. Taylor expanded in eps around inf 97.4%

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

   7. Taylor expanded in eps around 0 97.4%

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

      1. sub-neg 97.4%

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

      2. mul-1-neg 97.4%

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

      3. remove-double-neg 97.4%

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

      4. neg-mul-1 97.4%

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

   9. Simplified 97.4%

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

   if -1e7 < x < -1.00000000000000006e-279

   1. Initial program 52.4%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in eps around inf 98.9%

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

   7. Taylor expanded in eps around -inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. associate-*r* 98.9%

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

      3. sub-neg 98.9%

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

      4. mul-1-neg 98.9%

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

      5. remove-double-neg 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*l* 98.9%

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

      8. distribute-lft-in 98.9%

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

      9. metadata-eval 98.9%

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

      10. mul-1-neg 98.9%

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

      11. unsub-neg 98.9%

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

   9. Simplified 98.9%

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

   10. Taylor expanded in eps around 0 85.2%

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

   if -1.00000000000000006e-279 < x < 2.6000000000000002e127

   1. Initial program 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 91.4%

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

   7. Taylor expanded in x around 0 71.2%

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

   if 2.6000000000000002e127 < 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%

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

   5. Taylor expanded in eps around 0 60.6%

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

      1. associate--r+ 60.6%

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

      2. associate-*r* 60.6%

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

      3. cancel-sign-sub-inv 60.6%

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

      4. distribute-rgt1-in 60.6%

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

      5. distribute-rgt-out-- 60.6%

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

      6. mul-1-neg 60.6%

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

      7. mul-1-neg 60.6%

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

      8. remove-double-neg 60.6%

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

      9. mul-1-neg 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in x around inf 60.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10000000:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-279}:\\ \;\;\;\;\frac{\left(x \cdot \varepsilon + 1\right) + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+127}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-280}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+127}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -2e-280)
     (/ (+ t_0 1.0) 2.0)
     (if (<= x 6e+127) (/ (+ (exp (* x eps_m)) 1.0) 2.0) (* x t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -2e-280) {
		tmp = (t_0 + 1.0) / 2.0;
	} else if (x <= 6e+127) {
		tmp = (exp((x * eps_m)) + 1.0) / 2.0;
	} else {
		tmp = x * t_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 <= (-2d-280)) then
        tmp = (t_0 + 1.0d0) / 2.0d0
    else if (x <= 6d+127) then
        tmp = (exp((x * eps_m)) + 1.0d0) / 2.0d0
    else
        tmp = x * t_0
    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 <= -2e-280) {
		tmp = (t_0 + 1.0) / 2.0;
	} else if (x <= 6e+127) {
		tmp = (Math.exp((x * eps_m)) + 1.0) / 2.0;
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -2e-280:
		tmp = (t_0 + 1.0) / 2.0
	elif x <= 6e+127:
		tmp = (math.exp((x * eps_m)) + 1.0) / 2.0
	else:
		tmp = x * t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -2e-280)
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	elseif (x <= 6e+127)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + 1.0) / 2.0);
	else
		tmp = Float64(x * t_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 <= -2e-280)
		tmp = (t_0 + 1.0) / 2.0;
	elseif (x <= 6e+127)
		tmp = (exp((x * eps_m)) + 1.0) / 2.0;
	else
		tmp = x * t_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, -2e-280], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6e+127], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9999999999999999e-280

   1. Initial program 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in eps around inf 98.3%

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

   6. Taylor expanded in eps around inf 98.3%

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

   7. Taylor expanded in eps around 0 83.9%

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

      1. sub-neg 83.9%

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

      2. mul-1-neg 83.9%

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

      3. remove-double-neg 83.9%

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

      4. neg-mul-1 83.9%

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

   9. Simplified 83.9%

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

   if -1.9999999999999999e-280 < x < 6.0000000000000005e127

   1. Initial program 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 91.4%

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

   7. Taylor expanded in x around 0 71.2%

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

   if 6.0000000000000005e127 < 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%

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

   5. Taylor expanded in eps around 0 60.6%

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

      1. associate--r+ 60.6%

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

      2. associate-*r* 60.6%

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

      3. cancel-sign-sub-inv 60.6%

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

      4. distribute-rgt1-in 60.6%

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

      5. distribute-rgt-out-- 60.6%

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

      6. mul-1-neg 60.6%

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

      7. mul-1-neg 60.6%

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

      8. remove-double-neg 60.6%

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

      9. mul-1-neg 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in x around inf 60.6%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-280}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+127}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot \frac{1 - eps\_m \cdot eps\_m}{-1 + eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+32}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right) + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -3.8e-32)
   (/ (* x (/ (- 1.0 (* eps_m eps_m)) (+ -1.0 eps_m))) 2.0)
   (if (<= x 1.5e+32)
     (+ (* (* x x) (- (* x 0.3333333333333333) 0.5)) 1.0)
     (* x (exp (- x))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -3.8e-32) {
		tmp = (x * ((1.0 - (eps_m * eps_m)) / (-1.0 + eps_m))) / 2.0;
	} else if (x <= 1.5e+32) {
		tmp = ((x * x) * ((x * 0.3333333333333333) - 0.5)) + 1.0;
	} else {
		tmp = x * exp(-x);
	}
	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 <= (-3.8d-32)) then
        tmp = (x * ((1.0d0 - (eps_m * eps_m)) / ((-1.0d0) + eps_m))) / 2.0d0
    else if (x <= 1.5d+32) then
        tmp = ((x * x) * ((x * 0.3333333333333333d0) - 0.5d0)) + 1.0d0
    else
        tmp = x * exp(-x)
    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 <= -3.8e-32) {
		tmp = (x * ((1.0 - (eps_m * eps_m)) / (-1.0 + eps_m))) / 2.0;
	} else if (x <= 1.5e+32) {
		tmp = ((x * x) * ((x * 0.3333333333333333) - 0.5)) + 1.0;
	} else {
		tmp = x * Math.exp(-x);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -3.8e-32:
		tmp = (x * ((1.0 - (eps_m * eps_m)) / (-1.0 + eps_m))) / 2.0
	elif x <= 1.5e+32:
		tmp = ((x * x) * ((x * 0.3333333333333333) - 0.5)) + 1.0
	else:
		tmp = x * math.exp(-x)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -3.8e-32)
		tmp = Float64(Float64(x * Float64(Float64(1.0 - Float64(eps_m * eps_m)) / Float64(-1.0 + eps_m))) / 2.0);
	elseif (x <= 1.5e+32)
		tmp = Float64(Float64(Float64(x * x) * Float64(Float64(x * 0.3333333333333333) - 0.5)) + 1.0);
	else
		tmp = Float64(x * exp(Float64(-x)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -3.8e-32)
		tmp = (x * ((1.0 - (eps_m * eps_m)) / (-1.0 + eps_m))) / 2.0;
	elseif (x <= 1.5e+32)
		tmp = ((x * x) * ((x * 0.3333333333333333) - 0.5)) + 1.0;
	else
		tmp = x * exp(-x);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -3.8e-32], N[(N[(x * N[(N[(1.0 - N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.5e+32], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.80000000000000008e-32

   1. Initial program 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in eps around inf 96.0%

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

   6. Taylor expanded in x around 0 48.9%

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

   7. Taylor expanded in x around inf 28.9%

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

      1. mul-1-neg 28.9%

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

      2. distribute-rgt-neg-in 28.9%

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

      3. distribute-neg-in 28.9%

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

      4. metadata-eval 28.9%

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

      5. unsub-neg 28.9%

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

   9. Simplified 28.9%

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

       1. sub-neg 28.9%

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

       2. flip-+ 37.6%

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

       3. metadata-eval 37.6%

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

   11. Applied egg-rr 37.6%

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

   if -3.80000000000000008e-32 < x < 1.5e32

   1. Initial program 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in eps around 0 72.9%

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

      1. associate--r+ 72.9%

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

      2. associate-*r* 72.9%

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

      3. cancel-sign-sub-inv 72.9%

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

      4. distribute-rgt1-in 72.9%

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

      5. distribute-rgt-out-- 72.9%

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

      6. mul-1-neg 72.9%

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

      7. mul-1-neg 72.9%

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

      8. remove-double-neg 72.9%

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

      9. mul-1-neg 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in x around 0 73.0%

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   9. Step-by-step derivation

      1. unpow2 73.0%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   10. Applied egg-rr 73.0%

       \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   if 1.5e32 < 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%

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

   5. Taylor expanded in eps around 0 50.8%

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

      1. associate--r+ 50.8%

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

      2. associate-*r* 50.8%

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

      3. cancel-sign-sub-inv 50.8%

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

      4. distribute-rgt1-in 50.8%

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

      5. distribute-rgt-out-- 50.8%

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

      6. mul-1-neg 50.8%

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

      7. mul-1-neg 50.8%

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

      8. remove-double-neg 50.8%

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

      9. mul-1-neg 50.8%

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

   7. Simplified 50.8%

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

   8. Taylor expanded in x around inf 50.8%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot \frac{1 - \varepsilon \cdot \varepsilon}{-1 + \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+32}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right) + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq 1.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (if (<= x 1.5e+32) (/ (+ t_0 1.0) 2.0) (* x t_0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= 1.5e+32) {
		tmp = (t_0 + 1.0) / 2.0;
	} else {
		tmp = x * t_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 <= 1.5d+32) then
        tmp = (t_0 + 1.0d0) / 2.0d0
    else
        tmp = x * t_0
    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 <= 1.5e+32) {
		tmp = (t_0 + 1.0) / 2.0;
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= 1.5e+32:
		tmp = (t_0 + 1.0) / 2.0
	else:
		tmp = x * t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= 1.5e+32)
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	else
		tmp = Float64(x * t_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 <= 1.5e+32)
		tmp = (t_0 + 1.0) / 2.0;
	else
		tmp = x * t_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, 1.5e+32], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e32

   1. Initial program 64.6%

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

   3. Simplified 64.6%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in eps around inf 99.1%

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

   7. Taylor expanded in eps around 0 76.8%

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

      1. sub-neg 76.8%

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

      2. mul-1-neg 76.8%

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

      3. remove-double-neg 76.8%

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

      4. neg-mul-1 76.8%

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

   9. Simplified 76.8%

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

   if 1.5e32 < 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%

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

   5. Taylor expanded in eps around 0 50.8%

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

      1. associate--r+ 50.8%

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

      2. associate-*r* 50.8%

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

      3. cancel-sign-sub-inv 50.8%

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

      4. distribute-rgt1-in 50.8%

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

      5. distribute-rgt-out-- 50.8%

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

      6. mul-1-neg 50.8%

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

      7. mul-1-neg 50.8%

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

      8. remove-double-neg 50.8%

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

      9. mul-1-neg 50.8%

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

   7. Simplified 50.8%

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

   8. Taylor expanded in x around inf 50.8%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.0% accurate, 12.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1 - eps\_m \cdot eps\_m}{-1 + eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right) + 1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.02e-29)
   (/ (* x (/ (- 1.0 (* eps_m eps_m)) (+ -1.0 eps_m))) 2.0)
   (+ (* (* x x) (- (* x 0.3333333333333333) 0.5)) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.02e-29) {
		tmp = (x * ((1.0 - (eps_m * eps_m)) / (-1.0 + eps_m))) / 2.0;
	} else {
		tmp = ((x * x) * ((x * 0.3333333333333333) - 0.5)) + 1.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.02d-29)) then
        tmp = (x * ((1.0d0 - (eps_m * eps_m)) / ((-1.0d0) + eps_m))) / 2.0d0
    else
        tmp = ((x * x) * ((x * 0.3333333333333333d0) - 0.5d0)) + 1.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.02e-29) {
		tmp = (x * ((1.0 - (eps_m * eps_m)) / (-1.0 + eps_m))) / 2.0;
	} else {
		tmp = ((x * x) * ((x * 0.3333333333333333) - 0.5)) + 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.02e-29:
		tmp = (x * ((1.0 - (eps_m * eps_m)) / (-1.0 + eps_m))) / 2.0
	else:
		tmp = ((x * x) * ((x * 0.3333333333333333) - 0.5)) + 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.02e-29)
		tmp = Float64(Float64(x * Float64(Float64(1.0 - Float64(eps_m * eps_m)) / Float64(-1.0 + eps_m))) / 2.0);
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(Float64(x * 0.3333333333333333) - 0.5)) + 1.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.02e-29)
		tmp = (x * ((1.0 - (eps_m * eps_m)) / (-1.0 + eps_m))) / 2.0;
	else
		tmp = ((x * x) * ((x * 0.3333333333333333) - 0.5)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.02e-29], N[(N[(x * N[(N[(1.0 - N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.01999999999999994e-29

   1. Initial program 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in eps around inf 96.0%

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

   6. Taylor expanded in x around 0 48.9%

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

   7. Taylor expanded in x around inf 28.9%

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

      1. mul-1-neg 28.9%

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

      2. distribute-rgt-neg-in 28.9%

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

      3. distribute-neg-in 28.9%

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

      4. metadata-eval 28.9%

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

      5. unsub-neg 28.9%

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

   9. Simplified 28.9%

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

       1. sub-neg 28.9%

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

       2. flip-+ 37.6%

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

       3. metadata-eval 37.6%

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

   11. Applied egg-rr 37.6%

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

   if -1.01999999999999994e-29 < x

   1. Initial program 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in eps around 0 66.3%

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

      1. associate--r+ 66.3%

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

      2. associate-*r* 66.3%

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

      3. cancel-sign-sub-inv 66.3%

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

      4. distribute-rgt1-in 66.3%

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

      5. distribute-rgt-out-- 66.3%

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

      6. mul-1-neg 66.3%

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

      7. mul-1-neg 66.3%

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

      8. remove-double-neg 66.3%

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

      9. mul-1-neg 66.3%

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

   7. Simplified 66.3%

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

   8. Taylor expanded in x around 0 60.7%

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   9. Step-by-step derivation

      1. unpow2 60.7%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   10. Applied egg-rr 60.7%

       \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1 - \varepsilon \cdot \varepsilon}{-1 + \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.6% accurate, 14.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -14500000:\\ \;\;\;\;\frac{x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right) + 1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -14500000.0)
   (/ (* x (- -1.0 eps_m)) 2.0)
   (+ (* (* x x) (- (* x 0.3333333333333333) 0.5)) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -14500000.0) {
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	} else {
		tmp = ((x * x) * ((x * 0.3333333333333333) - 0.5)) + 1.0;
	}
	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 <= (-14500000.0d0)) then
        tmp = (x * ((-1.0d0) - eps_m)) / 2.0d0
    else
        tmp = ((x * x) * ((x * 0.3333333333333333d0) - 0.5d0)) + 1.0d0
    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 <= -14500000.0) {
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	} else {
		tmp = ((x * x) * ((x * 0.3333333333333333) - 0.5)) + 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -14500000.0:
		tmp = (x * (-1.0 - eps_m)) / 2.0
	else:
		tmp = ((x * x) * ((x * 0.3333333333333333) - 0.5)) + 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -14500000.0)
		tmp = Float64(Float64(x * Float64(-1.0 - eps_m)) / 2.0);
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(Float64(x * 0.3333333333333333) - 0.5)) + 1.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -14500000.0)
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	else
		tmp = ((x * x) * ((x * 0.3333333333333333) - 0.5)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -14500000.0], N[(N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45e7

   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 inf 100.0%

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

   6. Taylor expanded in x around 0 50.4%

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

   7. Taylor expanded in x around inf 31.6%

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

      1. mul-1-neg 31.6%

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

      2. distribute-rgt-neg-in 31.6%

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

      3. distribute-neg-in 31.6%

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

      4. metadata-eval 31.6%

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

      5. unsub-neg 31.6%

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

   9. Simplified 31.6%

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

   if -1.45e7 < x

   1. Initial program 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in eps around 0 65.6%

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

      1. associate--r+ 65.6%

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

      2. associate-*r* 65.6%

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

      3. cancel-sign-sub-inv 65.6%

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

      4. distribute-rgt1-in 65.6%

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

      5. distribute-rgt-out-- 66.0%

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

      6. mul-1-neg 66.0%

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

      7. mul-1-neg 66.0%

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

      8. remove-double-neg 66.0%

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

      9. mul-1-neg 66.0%

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

   7. Simplified 66.0%

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

   8. Taylor expanded in x around 0 60.0%

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   9. Step-by-step derivation

      1. unpow2 60.0%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   10. Applied egg-rr 60.0%

       \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14500000:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.2% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot \left(eps\_m \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0) (* x (* eps_m -0.5)) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = x * (eps_m * -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x * (eps_m * (-0.5d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = x * (eps_m * -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = x * (eps_m * -0.5)
	else:
		tmp = 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x * Float64(eps_m * -0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x * (eps_m * -0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(x * N[(eps$95$m * -0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in eps around inf 97.4%

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

   6. Taylor expanded in x around 0 49.1%

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

   7. Taylor expanded in x around inf 30.7%

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

      1. mul-1-neg 30.7%

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

      2. distribute-rgt-neg-in 30.7%

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

      3. distribute-neg-in 30.7%

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

      4. metadata-eval 30.7%

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

      5. unsub-neg 30.7%

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

   9. Simplified 30.7%

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

   10. Taylor expanded in eps around inf 30.8%

       \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   11. Step-by-step derivation

       1. *-commutative 30.8%

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

       2. associate-*r* 30.8%

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

       3. *-commutative 30.8%

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

       4. associate-*l* 30.8%

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

   12. Simplified 30.8%

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

   if -1 < x

   1. Initial program 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in eps around 0 65.9%

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

      1. associate--r+ 65.9%

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

      2. associate-*r* 65.9%

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

      3. cancel-sign-sub-inv 65.9%

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

      4. distribute-rgt1-in 65.9%

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

      5. distribute-rgt-out-- 65.9%

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

      6. mul-1-neg 65.9%

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

      7. mul-1-neg 65.9%

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

      8. remove-double-neg 65.9%

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

      9. mul-1-neg 65.9%

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

   7. Simplified 65.9%

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

   8. Taylor expanded in x around 0 51.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.9% 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 73.5%

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

3. Simplified 73.5%

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

5. Taylor expanded in eps around 0 56.1%

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

   1. associate--r+ 56.1%

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

   2. associate-*r* 56.1%

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

   3. cancel-sign-sub-inv 56.1%

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

   4. distribute-rgt1-in 56.1%

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

   5. distribute-rgt-out-- 56.5%

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

   6. mul-1-neg 56.5%

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

   7. mul-1-neg 56.5%

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

   8. remove-double-neg 56.5%

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

   9. mul-1-neg 56.5%

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

7. Simplified 56.5%

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

8. Taylor expanded in x around 0 44.5%

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

Reproduce

herbie shell --seed 2024181 
(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))