NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.2% → 99.3%

Time: 15.2s

Alternatives: 16

Speedup: 2.0×

• 38.4% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\varepsilon \leq 1.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{x \cdot t_0 - t_0 \cdot \left(-1 + \left(-1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps 1.6e-15)
     (/ (- (* x t_0) (* t_0 (+ -1.0 (- -1.0 x)))) 2.0)
     (/ (+ (exp (* eps x)) (exp (* x (- -1.0 eps)))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (eps <= 1.6e-15) {
		tmp = ((x * t_0) - (t_0 * (-1.0 + (-1.0 - x)))) / 2.0;
	} else {
		tmp = (exp((eps * x)) + exp((x * (-1.0 - eps)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps <= 1.6d-15) then
        tmp = ((x * t_0) - (t_0 * ((-1.0d0) + ((-1.0d0) - x)))) / 2.0d0
    else
        tmp = (exp((eps * x)) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps <= 1.6e-15) {
		tmp = ((x * t_0) - (t_0 * (-1.0 + (-1.0 - x)))) / 2.0;
	} else {
		tmp = (Math.exp((eps * x)) + Math.exp((x * (-1.0 - eps)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = math.exp(-x)
	tmp = 0
	if eps <= 1.6e-15:
		tmp = ((x * t_0) - (t_0 * (-1.0 + (-1.0 - x)))) / 2.0
	else:
		tmp = (math.exp((eps * x)) + math.exp((x * (-1.0 - eps)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps <= 1.6e-15)
		tmp = Float64(Float64(Float64(x * t_0) - Float64(t_0 * Float64(-1.0 + Float64(-1.0 - x)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(eps * x)) + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps <= 1.6e-15)
		tmp = ((x * t_0) - (t_0 * (-1.0 + (-1.0 - x)))) / 2.0;
	else
		tmp = (exp((eps * x)) + exp((x * (-1.0 - eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps, 1.6e-15], N[(N[(N[(x * t$95$0), $MachinePrecision] - N[(t$95$0 * N[(-1.0 + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.6e-15

   1. Initial program 65.6%

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

      1. sub-neg 65.6%

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

      2. neg-sub0 65.6%

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

      3. associate-+r- 65.6%

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

   3. Simplified 65.6%

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

   4. Taylor expanded in eps around 0 72.9%

      \[\leadsto \frac{\color{blue}{\left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right) - \left(-1 \cdot e^{-1 \cdot x} + -1 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}}{2} \]
   5. 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. neg-mul-1 72.9%

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

      4. cancel-sign-sub 72.9%

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

      5. distribute-rgt1-in 72.8%

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

      6. distribute-rgt-out-- 73.4%

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

      7. neg-mul-1 73.4%

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

      8. neg-mul-1 73.4%

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

   6. Simplified 73.4%

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

   if 1.6e-15 < eps

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. *-commutative 100.0%

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

      4. mul-1-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. remove-double-neg 100.0%

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

      8. mul-1-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. mul-1-neg 100.0%

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

      12. mul-1-neg 100.0%

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

      13. *-commutative 100.0%

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

      14. distribute-rgt-neg-in 100.0%

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

      15. sub-neg 100.0%

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

      16. mul-1-neg 100.0%

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

      17. remove-double-neg 100.0%

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

      18. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{x \cdot e^{-x} - e^{-x} \cdot \left(-1 + \left(-1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]

Alternative 2: 85.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.9 \cdot 10^{+90}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+184}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+251}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 4.9e+90)
   (/ (+ (exp (* eps x)) (exp (* x (- -1.0 eps)))) 2.0)
   (if (<= x 1.8e+184)
     0.0
     (if (<= x 3.7e+251) (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0) 0.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 4.9e+90) {
		tmp = (exp((eps * x)) + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 1.8e+184) {
		tmp = 0.0;
	} else if (x <= 3.7e+251) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 4.9d+90) then
        tmp = (exp((eps * x)) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if (x <= 1.8d+184) then
        tmp = 0.0d0
    else if (x <= 3.7d+251) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 4.9e+90) {
		tmp = (Math.exp((eps * x)) + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 1.8e+184) {
		tmp = 0.0;
	} else if (x <= 3.7e+251) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 4.9e+90:
		tmp = (math.exp((eps * x)) + math.exp((x * (-1.0 - eps)))) / 2.0
	elif x <= 1.8e+184:
		tmp = 0.0
	elif x <= 3.7e+251:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 4.9e+90)
		tmp = Float64(Float64(exp(Float64(eps * x)) + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 1.8e+184)
		tmp = 0.0;
	elseif (x <= 3.7e+251)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 4.9e+90)
		tmp = (exp((eps * x)) + exp((x * (-1.0 - eps)))) / 2.0;
	elseif (x <= 1.8e+184)
		tmp = 0.0;
	elseif (x <= 3.7e+251)
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 4.9e+90], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.8e+184], 0.0, If[LessEqual[x, 3.7e+251], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.9000000000000003e90

   1. Initial program 64.3%

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

      1. sub-neg 64.3%

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

      2. neg-sub0 64.3%

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

      3. associate-+r- 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in eps around inf 98.8%

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

      1. associate-*r* 98.8%

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

      2. neg-mul-1 98.8%

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

      3. *-commutative 98.8%

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

      4. mul-1-neg 98.8%

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

      5. mul-1-neg 98.8%

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

      6. distribute-rgt-neg-in 98.8%

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

      7. remove-double-neg 98.8%

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

      8. mul-1-neg 98.8%

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

      9. sub-neg 98.8%

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

      10. distribute-rgt-neg-in 98.8%

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

      11. mul-1-neg 98.8%

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

      12. mul-1-neg 98.8%

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

      13. *-commutative 98.8%

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

      14. distribute-rgt-neg-in 98.8%

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

      15. sub-neg 98.8%

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

      16. mul-1-neg 98.8%

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

      17. remove-double-neg 98.8%

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

      18. +-commutative 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in eps around inf 94.8%

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

      1. *-commutative 94.8%

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

   9. Simplified 94.8%

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

   if 4.9000000000000003e90 < x < 1.80000000000000007e184 or 3.6999999999999999e251 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 74.3%

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

      1. neg-mul-1 74.3%

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

      2. rec-exp 74.3%

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

      3. neg-mul-1 74.3%

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

      4. div-sub 74.3%

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

      5. +-inverses 74.3%

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

   6. Simplified 74.3%

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

   if 1.80000000000000007e184 < x < 3.6999999999999999e251

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 31.9%

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

   5. Taylor expanded in eps around inf 32.2%

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

      1. *-commutative 32.2%

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

      2. neg-mul-1 32.2%

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

      3. neg-sub0 32.2%

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

      4. *-commutative 32.2%

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

      5. sub-neg 32.2%

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

      6. mul-1-neg 32.2%

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

      7. distribute-rgt-in 32.2%

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

      8. *-lft-identity 32.2%

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

      9. mul-1-neg 32.2%

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

      10. cancel-sign-sub-inv 32.2%

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

      11. associate-+l- 32.2%

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

      12. neg-sub0 32.2%

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

      13. neg-mul-1 32.2%

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

      14. distribute-rgt-out 32.2%

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

   7. Simplified 32.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.9 \cdot 10^{+90}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+184}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+251}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.1× speedup

Math

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

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (/ (+ (exp (* x (+ eps -1.0))) (exp (* x (- -1.0 eps)))) 2.0))

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (exp((x * (eps + (-1.0d0)))) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. sub-neg 73.5%

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

   2. neg-sub0 73.5%

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

   3. associate-+r- 73.5%

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

3. Simplified 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. Taylor expanded in eps around inf 99.1%

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

   1. associate-*r* 99.1%

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

   2. neg-mul-1 99.1%

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

   3. *-commutative 99.1%

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

   4. mul-1-neg 99.1%

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

   5. mul-1-neg 99.1%

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

   6. distribute-rgt-neg-in 99.1%

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

   7. remove-double-neg 99.1%

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

   8. mul-1-neg 99.1%

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

   9. sub-neg 99.1%

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

   10. distribute-rgt-neg-in 99.1%

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

   11. mul-1-neg 99.1%

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

   12. mul-1-neg 99.1%

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

   13. *-commutative 99.1%

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

   14. distribute-rgt-neg-in 99.1%

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

   15. sub-neg 99.1%

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

   16. mul-1-neg 99.1%

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

   17. remove-double-neg 99.1%

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

   18. +-commutative 99.1%

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

6. Simplified 99.1%

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

7. Final simplification 99.1%

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

Alternative 4: 85.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 - \varepsilon\right)}\\ \mathbf{if}\;x \leq -470:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right) \cdot t_0}{2}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-259}:\\ \;\;\;\;\frac{\left(1 + \left(x + \left(\varepsilon \cdot x - x\right)\right)\right) + t_0}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+186}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+252}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x (- -1.0 eps)))))
   (if (<= x -470.0)
     (/ (- (+ 1.0 (/ 1.0 eps)) (* (+ -1.0 (/ 1.0 eps)) t_0)) 2.0)
     (if (<= x -5e-259)
       (/ (+ (+ 1.0 (+ x (- (* eps x) x))) t_0) 2.0)
       (if (<= x 3.8e+90)
         (/ (+ 1.0 (exp (* eps x))) 2.0)
         (if (<= x 1.05e+186)
           0.0
           (if (<= x 1.55e+252)
             (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)
             0.0)))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = exp((x * (-1.0 - eps)));
	double tmp;
	if (x <= -470.0) {
		tmp = ((1.0 + (1.0 / eps)) - ((-1.0 + (1.0 / eps)) * t_0)) / 2.0;
	} else if (x <= -5e-259) {
		tmp = ((1.0 + (x + ((eps * x) - x))) + t_0) / 2.0;
	} else if (x <= 3.8e+90) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else if (x <= 1.05e+186) {
		tmp = 0.0;
	} else if (x <= 1.55e+252) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * ((-1.0d0) - eps)))
    if (x <= (-470.0d0)) then
        tmp = ((1.0d0 + (1.0d0 / eps)) - (((-1.0d0) + (1.0d0 / eps)) * t_0)) / 2.0d0
    else if (x <= (-5d-259)) then
        tmp = ((1.0d0 + (x + ((eps * x) - x))) + t_0) / 2.0d0
    else if (x <= 3.8d+90) then
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    else if (x <= 1.05d+186) then
        tmp = 0.0d0
    else if (x <= 1.55d+252) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = Math.exp((x * (-1.0 - eps)));
	double tmp;
	if (x <= -470.0) {
		tmp = ((1.0 + (1.0 / eps)) - ((-1.0 + (1.0 / eps)) * t_0)) / 2.0;
	} else if (x <= -5e-259) {
		tmp = ((1.0 + (x + ((eps * x) - x))) + t_0) / 2.0;
	} else if (x <= 3.8e+90) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else if (x <= 1.05e+186) {
		tmp = 0.0;
	} else if (x <= 1.55e+252) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = math.exp((x * (-1.0 - eps)))
	tmp = 0
	if x <= -470.0:
		tmp = ((1.0 + (1.0 / eps)) - ((-1.0 + (1.0 / eps)) * t_0)) / 2.0
	elif x <= -5e-259:
		tmp = ((1.0 + (x + ((eps * x) - x))) + t_0) / 2.0
	elif x <= 3.8e+90:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	elif x <= 1.05e+186:
		tmp = 0.0
	elif x <= 1.55e+252:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = exp(Float64(x * Float64(-1.0 - eps)))
	tmp = 0.0
	if (x <= -470.0)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) - Float64(Float64(-1.0 + Float64(1.0 / eps)) * t_0)) / 2.0);
	elseif (x <= -5e-259)
		tmp = Float64(Float64(Float64(1.0 + Float64(x + Float64(Float64(eps * x) - x))) + t_0) / 2.0);
	elseif (x <= 3.8e+90)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	elseif (x <= 1.05e+186)
		tmp = 0.0;
	elseif (x <= 1.55e+252)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = exp((x * (-1.0 - eps)));
	tmp = 0.0;
	if (x <= -470.0)
		tmp = ((1.0 + (1.0 / eps)) - ((-1.0 + (1.0 / eps)) * t_0)) / 2.0;
	elseif (x <= -5e-259)
		tmp = ((1.0 + (x + ((eps * x) - x))) + t_0) / 2.0;
	elseif (x <= 3.8e+90)
		tmp = (1.0 + exp((eps * x))) / 2.0;
	elseif (x <= 1.05e+186)
		tmp = 0.0;
	elseif (x <= 1.55e+252)
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -470.0], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -5e-259], N[(N[(N[(1.0 + N[(x + N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.8e+90], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.05e+186], 0.0, If[LessEqual[x, 1.55e+252], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -470

   1. Initial program 96.7%

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

      1. sub-neg 96.7%

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

      2. neg-sub0 96.7%

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

      3. associate-+r- 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in x around 0 45.0%

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

   if -470 < x < -4.99999999999999977e-259

   1. Initial program 44.9%

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

      1. sub-neg 44.9%

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

      2. neg-sub0 44.9%

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

      3. associate-+r- 44.9%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in x around 0 34.6%

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

      1. +-commutative 34.6%

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

      2. associate-+r+ 34.6%

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

      3. associate-*r* 34.6%

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

      4. neg-mul-1 34.6%

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

      5. *-commutative 34.6%

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

      6. associate-*r* 34.6%

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

      7. distribute-lft-neg-in 34.6%

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

      8. distribute-rgt-neg-in 34.6%

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

      9. sub-neg 34.6%

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

      10. mul-1-neg 34.6%

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

      11. distribute-neg-in 34.6%

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

      12. metadata-eval 34.6%

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

      13. mul-1-neg 34.6%

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

      14. remove-double-neg 34.6%

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

      15. +-commutative 34.6%

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

      16. distribute-rgt-in 34.6%

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

      17. neg-mul-1 34.6%

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

      18. sub-neg 34.6%

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

      19. *-commutative 34.6%

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

   6. Simplified 34.6%

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

   7. Taylor expanded in eps around inf 89.0%

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

   if -4.99999999999999977e-259 < x < 3.8000000000000001e90

   1. Initial program 65.8%

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

      1. sub-neg 65.8%

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

      2. neg-sub0 65.8%

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

      3. associate-+r- 65.8%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in x around 0 41.6%

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

   5. Taylor expanded in eps around inf 75.2%

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

      1. *-commutative 75.2%

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

      2. neg-mul-1 75.2%

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

      3. neg-sub0 75.2%

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

      4. *-commutative 75.2%

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

      5. sub-neg 75.2%

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

      6. mul-1-neg 75.2%

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

      7. distribute-rgt-in 75.2%

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

      8. *-lft-identity 75.2%

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

      9. mul-1-neg 75.2%

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

      10. cancel-sign-sub-inv 75.2%

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

      11. associate-+l- 75.2%

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

      12. neg-sub0 75.2%

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

      13. neg-mul-1 75.2%

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

      14. distribute-rgt-out 75.2%

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

   7. Simplified 75.2%

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

   8. Taylor expanded in eps around inf 75.3%

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

      1. *-commutative 75.3%

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

   10. Simplified 75.3%

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

   if 3.8000000000000001e90 < x < 1.05e186 or 1.54999999999999991e252 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 74.3%

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

      1. neg-mul-1 74.3%

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

      2. rec-exp 74.3%

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

      3. neg-mul-1 74.3%

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

      4. div-sub 74.3%

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

      5. +-inverses 74.3%

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

   6. Simplified 74.3%

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

   if 1.05e186 < x < 1.54999999999999991e252

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 31.9%

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

   5. Taylor expanded in eps around inf 32.2%

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

      1. *-commutative 32.2%

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

      2. neg-mul-1 32.2%

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

      3. neg-sub0 32.2%

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

      4. *-commutative 32.2%

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

      5. sub-neg 32.2%

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

      6. mul-1-neg 32.2%

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

      7. distribute-rgt-in 32.2%

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

      8. *-lft-identity 32.2%

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

      9. mul-1-neg 32.2%

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

      10. cancel-sign-sub-inv 32.2%

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

      11. associate-+l- 32.2%

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

      12. neg-sub0 32.2%

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

      13. neg-mul-1 32.2%

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

      14. distribute-rgt-out 32.2%

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

   7. Simplified 32.2%

      \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -470:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-259}:\\ \;\;\;\;\frac{\left(1 + \left(x + \left(\varepsilon \cdot x - x\right)\right)\right) + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+186}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+252}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 78.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-27}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+187}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+251}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1.7e-27)
   (/
    (- (+ 1.0 (/ 1.0 eps)) (* (+ -1.0 (/ 1.0 eps)) (exp (* x (- -1.0 eps)))))
    2.0)
   (if (<= x 4.6e+90)
     (/ (+ 1.0 (exp (* eps x))) 2.0)
     (if (<= x 1.55e+187)
       0.0
       (if (<= x 9e+251) (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0) 0.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -1.7e-27) {
		tmp = ((1.0 + (1.0 / eps)) - ((-1.0 + (1.0 / eps)) * exp((x * (-1.0 - eps))))) / 2.0;
	} else if (x <= 4.6e+90) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else if (x <= 1.55e+187) {
		tmp = 0.0;
	} else if (x <= 9e+251) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.7d-27)) then
        tmp = ((1.0d0 + (1.0d0 / eps)) - (((-1.0d0) + (1.0d0 / eps)) * exp((x * ((-1.0d0) - eps))))) / 2.0d0
    else if (x <= 4.6d+90) then
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    else if (x <= 1.55d+187) then
        tmp = 0.0d0
    else if (x <= 9d+251) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.7e-27) {
		tmp = ((1.0 + (1.0 / eps)) - ((-1.0 + (1.0 / eps)) * Math.exp((x * (-1.0 - eps))))) / 2.0;
	} else if (x <= 4.6e+90) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else if (x <= 1.55e+187) {
		tmp = 0.0;
	} else if (x <= 9e+251) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -1.7e-27:
		tmp = ((1.0 + (1.0 / eps)) - ((-1.0 + (1.0 / eps)) * math.exp((x * (-1.0 - eps))))) / 2.0
	elif x <= 4.6e+90:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	elif x <= 1.55e+187:
		tmp = 0.0
	elif x <= 9e+251:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -1.7e-27)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) - Float64(Float64(-1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(-1.0 - eps))))) / 2.0);
	elseif (x <= 4.6e+90)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	elseif (x <= 1.55e+187)
		tmp = 0.0;
	elseif (x <= 9e+251)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.7e-27)
		tmp = ((1.0 + (1.0 / eps)) - ((-1.0 + (1.0 / eps)) * exp((x * (-1.0 - eps))))) / 2.0;
	elseif (x <= 4.6e+90)
		tmp = (1.0 + exp((eps * x))) / 2.0;
	elseif (x <= 1.55e+187)
		tmp = 0.0;
	elseif (x <= 9e+251)
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -1.7e-27], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.6e+90], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.55e+187], 0.0, If[LessEqual[x, 9e+251], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.69999999999999985e-27

   1. Initial program 94.2%

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

      1. sub-neg 94.2%

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

      2. neg-sub0 94.2%

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

      3. associate-+r- 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in x around 0 42.9%

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

   if -1.69999999999999985e-27 < x < 4.6e90

   1. Initial program 57.8%

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

      1. sub-neg 57.8%

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

      2. neg-sub0 57.8%

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

      3. associate-+r- 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in x around 0 37.7%

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

   5. Taylor expanded in eps around inf 79.5%

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

      1. *-commutative 79.5%

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

      2. neg-mul-1 79.5%

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

      3. neg-sub0 79.5%

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

      4. *-commutative 79.5%

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

      5. sub-neg 79.5%

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

      6. mul-1-neg 79.5%

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

      7. distribute-rgt-in 79.5%

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

      8. *-lft-identity 79.5%

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

      9. mul-1-neg 79.5%

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

      10. cancel-sign-sub-inv 79.5%

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

      11. associate-+l- 79.5%

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

      12. neg-sub0 79.5%

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

      13. neg-mul-1 79.5%

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

      14. distribute-rgt-out 79.5%

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

   7. Simplified 79.5%

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

   8. Taylor expanded in eps around inf 79.6%

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

      1. *-commutative 79.6%

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

   10. Simplified 79.6%

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

   if 4.6e90 < x < 1.55000000000000006e187 or 8.9999999999999997e251 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 74.3%

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

      1. neg-mul-1 74.3%

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

      2. rec-exp 74.3%

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

      3. neg-mul-1 74.3%

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

      4. div-sub 74.3%

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

      5. +-inverses 74.3%

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

   6. Simplified 74.3%

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

   if 1.55000000000000006e187 < x < 8.9999999999999997e251

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 31.9%

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

   5. Taylor expanded in eps around inf 32.2%

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

      1. *-commutative 32.2%

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

      2. neg-mul-1 32.2%

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

      3. neg-sub0 32.2%

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

      4. *-commutative 32.2%

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

      5. sub-neg 32.2%

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

      6. mul-1-neg 32.2%

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

      7. distribute-rgt-in 32.2%

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

      8. *-lft-identity 32.2%

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

      9. mul-1-neg 32.2%

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

      10. cancel-sign-sub-inv 32.2%

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

      11. associate-+l- 32.2%

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

      12. neg-sub0 32.2%

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

      13. neg-mul-1 32.2%

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

      14. distribute-rgt-out 32.2%

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

   7. Simplified 32.2%

      \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-27}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+187}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+251}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 77.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-262}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+89}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+191}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+252}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -2.6e-262)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 1.02e+89)
     (/ (+ 1.0 (exp (* eps x))) 2.0)
     (if (<= x 9.8e+191)
       0.0
       (if (<= x 1.35e+252) (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0) 0.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -2.6e-262) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 1.02e+89) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else if (x <= 9.8e+191) {
		tmp = 0.0;
	} else if (x <= 1.35e+252) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.6d-262)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 1.02d+89) then
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    else if (x <= 9.8d+191) then
        tmp = 0.0d0
    else if (x <= 1.35d+252) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.6e-262) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 1.02e+89) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else if (x <= 9.8e+191) {
		tmp = 0.0;
	} else if (x <= 1.35e+252) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -2.6e-262:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 1.02e+89:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	elif x <= 9.8e+191:
		tmp = 0.0
	elif x <= 1.35e+252:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -2.6e-262)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 1.02e+89)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	elseif (x <= 9.8e+191)
		tmp = 0.0;
	elseif (x <= 1.35e+252)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.6e-262)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 1.02e+89)
		tmp = (1.0 + exp((eps * x))) / 2.0;
	elseif (x <= 9.8e+191)
		tmp = 0.0;
	elseif (x <= 1.35e+252)
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -2.6e-262], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.02e+89], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.8e+191], 0.0, If[LessEqual[x, 1.35e+252], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.5999999999999999e-262

   1. Initial program 62.5%

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

      1. sub-neg 62.5%

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

      2. neg-sub0 62.5%

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

      3. associate-+r- 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in x around 0 40.6%

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

   5. Taylor expanded in eps around inf 75.4%

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

      1. *-commutative 75.4%

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

      2. neg-mul-1 75.4%

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

      3. neg-sub0 75.4%

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

      4. *-commutative 75.4%

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

      5. sub-neg 75.4%

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

      6. mul-1-neg 75.4%

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

      7. distribute-rgt-in 75.4%

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

      8. *-lft-identity 75.4%

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

      9. mul-1-neg 75.4%

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

      10. cancel-sign-sub-inv 75.4%

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

      11. associate-+l- 75.4%

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

      12. neg-sub0 75.4%

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

      13. neg-mul-1 75.4%

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

      14. distribute-rgt-out 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in eps around 0 83.9%

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

      1. neg-mul-1 83.9%

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

   10. Simplified 83.9%

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

   if -2.5999999999999999e-262 < x < 1.0199999999999999e89

   1. Initial program 65.8%

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

      1. sub-neg 65.8%

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

      2. neg-sub0 65.8%

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

      3. associate-+r- 65.8%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in x around 0 41.6%

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

   5. Taylor expanded in eps around inf 75.2%

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

      1. *-commutative 75.2%

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

      2. neg-mul-1 75.2%

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

      3. neg-sub0 75.2%

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

      4. *-commutative 75.2%

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

      5. sub-neg 75.2%

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

      6. mul-1-neg 75.2%

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

      7. distribute-rgt-in 75.2%

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

      8. *-lft-identity 75.2%

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

      9. mul-1-neg 75.2%

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

      10. cancel-sign-sub-inv 75.2%

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

      11. associate-+l- 75.2%

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

      12. neg-sub0 75.2%

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

      13. neg-mul-1 75.2%

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

      14. distribute-rgt-out 75.2%

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

   7. Simplified 75.2%

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

   8. Taylor expanded in eps around inf 75.3%

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

      1. *-commutative 75.3%

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

   10. Simplified 75.3%

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

   if 1.0199999999999999e89 < x < 9.7999999999999999e191 or 1.35000000000000005e252 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 74.3%

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

      1. neg-mul-1 74.3%

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

      2. rec-exp 74.3%

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

      3. neg-mul-1 74.3%

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

      4. div-sub 74.3%

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

      5. +-inverses 74.3%

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

   6. Simplified 74.3%

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

   if 9.7999999999999999e191 < x < 1.35000000000000005e252

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 31.9%

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

   5. Taylor expanded in eps around inf 32.2%

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

      1. *-commutative 32.2%

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

      2. neg-mul-1 32.2%

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

      3. neg-sub0 32.2%

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

      4. *-commutative 32.2%

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

      5. sub-neg 32.2%

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

      6. mul-1-neg 32.2%

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

      7. distribute-rgt-in 32.2%

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

      8. *-lft-identity 32.2%

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

      9. mul-1-neg 32.2%

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

      10. cancel-sign-sub-inv 32.2%

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

      11. associate-+l- 32.2%

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

      12. neg-sub0 32.2%

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

      13. neg-mul-1 32.2%

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

      14. distribute-rgt-out 32.2%

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

   7. Simplified 32.2%

      \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-262}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+89}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+191}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+252}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 77.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-262}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+90} \lor \neg \left(x \leq 8.2 \cdot 10^{+184}\right) \land x \leq 10^{+252}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -2.6e-262)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (or (<= x 1.85e+90) (and (not (<= x 8.2e+184)) (<= x 1e+252)))
     (/ (+ 1.0 (exp (* eps x))) 2.0)
     0.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -2.6e-262) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if ((x <= 1.85e+90) || (!(x <= 8.2e+184) && (x <= 1e+252))) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.6d-262)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if ((x <= 1.85d+90) .or. (.not. (x <= 8.2d+184)) .and. (x <= 1d+252)) then
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.6e-262) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if ((x <= 1.85e+90) || (!(x <= 8.2e+184) && (x <= 1e+252))) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -2.6e-262:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif (x <= 1.85e+90) or (not (x <= 8.2e+184) and (x <= 1e+252)):
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -2.6e-262)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif ((x <= 1.85e+90) || (!(x <= 8.2e+184) && (x <= 1e+252)))
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.6e-262)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif ((x <= 1.85e+90) || (~((x <= 8.2e+184)) && (x <= 1e+252)))
		tmp = (1.0 + exp((eps * x))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -2.6e-262], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.85e+90], And[N[Not[LessEqual[x, 8.2e+184]], $MachinePrecision], LessEqual[x, 1e+252]]], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5999999999999999e-262

   1. Initial program 62.5%

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

      1. sub-neg 62.5%

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

      2. neg-sub0 62.5%

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

      3. associate-+r- 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in x around 0 40.6%

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

   5. Taylor expanded in eps around inf 75.4%

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

      1. *-commutative 75.4%

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

      2. neg-mul-1 75.4%

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

      3. neg-sub0 75.4%

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

      4. *-commutative 75.4%

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

      5. sub-neg 75.4%

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

      6. mul-1-neg 75.4%

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

      7. distribute-rgt-in 75.4%

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

      8. *-lft-identity 75.4%

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

      9. mul-1-neg 75.4%

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

      10. cancel-sign-sub-inv 75.4%

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

      11. associate-+l- 75.4%

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

      12. neg-sub0 75.4%

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

      13. neg-mul-1 75.4%

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

      14. distribute-rgt-out 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in eps around 0 83.9%

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

      1. neg-mul-1 83.9%

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

   10. Simplified 83.9%

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

   if -2.5999999999999999e-262 < x < 1.85e90 or 8.1999999999999993e184 < x < 1.0000000000000001e252

   1. Initial program 71.4%

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

      1. sub-neg 71.4%

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

      2. neg-sub0 71.4%

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

      3. associate-+r- 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in x around 0 40.0%

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

   5. Taylor expanded in eps around inf 68.1%

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

      1. *-commutative 68.1%

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

      2. neg-mul-1 68.1%

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

      3. neg-sub0 68.1%

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

      4. *-commutative 68.1%

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

      5. sub-neg 68.1%

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

      6. mul-1-neg 68.1%

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

      7. distribute-rgt-in 68.1%

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

      8. *-lft-identity 68.1%

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

      9. mul-1-neg 68.1%

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

      10. cancel-sign-sub-inv 68.1%

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

      11. associate-+l- 68.1%

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

      12. neg-sub0 68.1%

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

      13. neg-mul-1 68.1%

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

      14. distribute-rgt-out 68.1%

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

   7. Simplified 68.1%

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

   8. Taylor expanded in eps around inf 68.2%

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

      1. *-commutative 68.2%

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

   10. Simplified 68.2%

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

   if 1.85e90 < x < 8.1999999999999993e184 or 1.0000000000000001e252 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 74.3%

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

      1. neg-mul-1 74.3%

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

      2. rec-exp 74.3%

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

      3. neg-mul-1 74.3%

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

      4. div-sub 74.3%

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

      5. +-inverses 74.3%

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

   6. Simplified 74.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-262}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+90} \lor \neg \left(x \leq 8.2 \cdot 10^{+184}\right) \land x \leq 10^{+252}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 70.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1400000000000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+185}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+247}:\\ \;\;\;\;\frac{2 + \left(1 + \frac{1}{\varepsilon}\right) \cdot \left(0.5 \cdot \left(x \cdot x\right) - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 1400000000000.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 3.4e+185)
     0.0
     (if (<= x 9.8e+247)
       (/ (+ 2.0 (* (+ 1.0 (/ 1.0 eps)) (- (* 0.5 (* x x)) x))) 2.0)
       0.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 1400000000000.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 3.4e+185) {
		tmp = 0.0;
	} else if (x <= 9.8e+247) {
		tmp = (2.0 + ((1.0 + (1.0 / eps)) * ((0.5 * (x * x)) - x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1400000000000.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 3.4d+185) then
        tmp = 0.0d0
    else if (x <= 9.8d+247) then
        tmp = (2.0d0 + ((1.0d0 + (1.0d0 / eps)) * ((0.5d0 * (x * x)) - x))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 1400000000000.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 3.4e+185) {
		tmp = 0.0;
	} else if (x <= 9.8e+247) {
		tmp = (2.0 + ((1.0 + (1.0 / eps)) * ((0.5 * (x * x)) - x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 1400000000000.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 3.4e+185:
		tmp = 0.0
	elif x <= 9.8e+247:
		tmp = (2.0 + ((1.0 + (1.0 / eps)) * ((0.5 * (x * x)) - x))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 1400000000000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 3.4e+185)
		tmp = 0.0;
	elseif (x <= 9.8e+247)
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 + Float64(1.0 / eps)) * Float64(Float64(0.5 * Float64(x * x)) - x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1400000000000.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 3.4e+185)
		tmp = 0.0;
	elseif (x <= 9.8e+247)
		tmp = (2.0 + ((1.0 + (1.0 / eps)) * ((0.5 * (x * x)) - x))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 1400000000000.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.4e+185], 0.0, If[LessEqual[x, 9.8e+247], N[(N[(2.0 + N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.4e12

   1. Initial program 59.6%

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

      1. sub-neg 59.6%

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

      2. neg-sub0 59.6%

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

      3. associate-+r- 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in x around 0 42.7%

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

   5. Taylor expanded in eps around inf 81.2%

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

      1. *-commutative 81.2%

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

      2. neg-mul-1 81.2%

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

      3. neg-sub0 81.2%

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

      4. *-commutative 81.2%

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

      5. sub-neg 81.2%

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

      6. mul-1-neg 81.2%

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

      7. distribute-rgt-in 81.2%

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

      8. *-lft-identity 81.2%

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

      9. mul-1-neg 81.2%

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

      10. cancel-sign-sub-inv 81.2%

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

      11. associate-+l- 81.2%

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

      12. neg-sub0 81.2%

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

      13. neg-mul-1 81.2%

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

      14. distribute-rgt-out 81.2%

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

   7. Simplified 81.2%

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

   8. Taylor expanded in eps around 0 81.9%

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

      1. neg-mul-1 81.9%

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

   10. Simplified 81.9%

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

   if 1.4e12 < x < 3.40000000000000017e185 or 9.7999999999999996e247 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 63.4%

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

      1. neg-mul-1 63.4%

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

      2. rec-exp 63.4%

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

      3. neg-mul-1 63.4%

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

      4. div-sub 63.4%

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

      5. +-inverses 63.4%

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

   6. Simplified 63.4%

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

   if 3.40000000000000017e185 < x < 9.7999999999999996e247

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 29.8%

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

   5. Taylor expanded in eps around 0 2.9%

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

      1. neg-mul-1 2.9%

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

   7. Simplified 2.9%

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

   8. Taylor expanded in x around 0 72.4%

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

      1. +-commutative 72.4%

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

      2. associate-*r* 72.4%

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

      3. neg-mul-1 72.4%

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

      4. distribute-lft-neg-in 72.4%

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

      5. distribute-rgt-out 72.7%

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

      6. unpow2 72.7%

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

   10. Simplified 72.7%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1400000000000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+185}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+247}:\\ \;\;\;\;\frac{2 + \left(1 + \frac{1}{\varepsilon}\right) \cdot \left(0.5 \cdot \left(x \cdot x\right) - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 63.6% accurate, 9.8× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.45:\\ \;\;\;\;\frac{\left(2 - x\right) - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+189}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+249}:\\ \;\;\;\;\frac{2 + \left(1 + \frac{1}{\varepsilon}\right) \cdot \left(0.5 \cdot \left(x \cdot x\right) - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 0.45)
   (/ (- (- 2.0 x) (* eps x)) 2.0)
   (if (<= x 1.5e+189)
     0.0
     (if (<= x 2e+249)
       (/ (+ 2.0 (* (+ 1.0 (/ 1.0 eps)) (- (* 0.5 (* x x)) x))) 2.0)
       0.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 0.45) {
		tmp = ((2.0 - x) - (eps * x)) / 2.0;
	} else if (x <= 1.5e+189) {
		tmp = 0.0;
	} else if (x <= 2e+249) {
		tmp = (2.0 + ((1.0 + (1.0 / eps)) * ((0.5 * (x * x)) - x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 0.45d0) then
        tmp = ((2.0d0 - x) - (eps * x)) / 2.0d0
    else if (x <= 1.5d+189) then
        tmp = 0.0d0
    else if (x <= 2d+249) then
        tmp = (2.0d0 + ((1.0d0 + (1.0d0 / eps)) * ((0.5d0 * (x * x)) - x))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 0.45) {
		tmp = ((2.0 - x) - (eps * x)) / 2.0;
	} else if (x <= 1.5e+189) {
		tmp = 0.0;
	} else if (x <= 2e+249) {
		tmp = (2.0 + ((1.0 + (1.0 / eps)) * ((0.5 * (x * x)) - x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 0.45:
		tmp = ((2.0 - x) - (eps * x)) / 2.0
	elif x <= 1.5e+189:
		tmp = 0.0
	elif x <= 2e+249:
		tmp = (2.0 + ((1.0 + (1.0 / eps)) * ((0.5 * (x * x)) - x))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 0.45)
		tmp = Float64(Float64(Float64(2.0 - x) - Float64(eps * x)) / 2.0);
	elseif (x <= 1.5e+189)
		tmp = 0.0;
	elseif (x <= 2e+249)
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 + Float64(1.0 / eps)) * Float64(Float64(0.5 * Float64(x * x)) - x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 0.45)
		tmp = ((2.0 - x) - (eps * x)) / 2.0;
	elseif (x <= 1.5e+189)
		tmp = 0.0;
	elseif (x <= 2e+249)
		tmp = (2.0 + ((1.0 + (1.0 / eps)) * ((0.5 * (x * x)) - x))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 0.45], N[(N[(N[(2.0 - x), $MachinePrecision] - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.5e+189], 0.0, If[LessEqual[x, 2e+249], N[(N[(2.0 + N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.450000000000000011

   1. Initial program 58.9%

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

      1. sub-neg 58.9%

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

      2. neg-sub0 58.9%

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

      3. associate-+r- 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in x around 0 46.5%

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

      1. +-commutative 46.5%

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

      2. associate-+r+ 46.5%

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

      3. associate-*r* 46.5%

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

      4. neg-mul-1 46.5%

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

      5. *-commutative 46.5%

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

      6. associate-*r* 46.5%

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

      7. distribute-lft-neg-in 46.5%

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

      8. distribute-rgt-neg-in 46.5%

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

      9. sub-neg 46.5%

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

      10. mul-1-neg 46.5%

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

      11. distribute-neg-in 46.5%

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

      12. metadata-eval 46.5%

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

      13. mul-1-neg 46.5%

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

      14. remove-double-neg 46.5%

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

      15. +-commutative 46.5%

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

      16. distribute-rgt-in 46.5%

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

      17. neg-mul-1 46.5%

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

      18. sub-neg 46.5%

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

      19. *-commutative 46.5%

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

   6. Simplified 46.5%

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

   7. Taylor expanded in x around 0 25.8%

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

   8. Taylor expanded in eps around 0 28.4%

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

      1. neg-mul-1 28.4%

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

   10. Simplified 28.4%

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

   11. Taylor expanded in eps around 0 68.2%

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

       1. neg-mul-1 68.2%

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

       2. associate-+r+ 68.2%

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

       3. mul-1-neg 68.2%

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

       4. *-commutative 68.2%

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

       5. unsub-neg 68.2%

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

       6. unsub-neg 68.2%

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

   13. Simplified 68.2%

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

   if 0.450000000000000011 < x < 1.4999999999999999e189 or 1.9999999999999998e249 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 60.9%

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

      1. neg-mul-1 60.9%

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

      2. rec-exp 60.9%

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

      3. neg-mul-1 60.9%

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

      4. div-sub 60.9%

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

      5. +-inverses 60.9%

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

   6. Simplified 60.9%

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

   if 1.4999999999999999e189 < x < 1.9999999999999998e249

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 29.8%

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

   5. Taylor expanded in eps around 0 2.9%

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

      1. neg-mul-1 2.9%

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

   7. Simplified 2.9%

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

   8. Taylor expanded in x around 0 72.4%

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

      1. +-commutative 72.4%

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

      2. associate-*r* 72.4%

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

      3. neg-mul-1 72.4%

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

      4. distribute-lft-neg-in 72.4%

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

      5. distribute-rgt-out 72.7%

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

      6. unpow2 72.7%

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

   10. Simplified 72.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.45:\\ \;\;\;\;\frac{\left(2 - x\right) - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+189}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+249}:\\ \;\;\;\;\frac{2 + \left(1 + \frac{1}{\varepsilon}\right) \cdot \left(0.5 \cdot \left(x \cdot x\right) - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 63.0% accurate, 17.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{elif}\;x \leq 1400000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+193}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+251}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -0.00065)
   (/ (* eps (- x)) 2.0)
   (if (<= x 1400000000000.0)
     1.0
     (if (<= x 2.6e+193) 0.0 (if (<= x 9e+251) (/ (* eps x) 2.0) 0.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -0.00065) {
		tmp = (eps * -x) / 2.0;
	} else if (x <= 1400000000000.0) {
		tmp = 1.0;
	} else if (x <= 2.6e+193) {
		tmp = 0.0;
	} else if (x <= 9e+251) {
		tmp = (eps * x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.00065d0)) then
        tmp = (eps * -x) / 2.0d0
    else if (x <= 1400000000000.0d0) then
        tmp = 1.0d0
    else if (x <= 2.6d+193) then
        tmp = 0.0d0
    else if (x <= 9d+251) then
        tmp = (eps * x) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.00065) {
		tmp = (eps * -x) / 2.0;
	} else if (x <= 1400000000000.0) {
		tmp = 1.0;
	} else if (x <= 2.6e+193) {
		tmp = 0.0;
	} else if (x <= 9e+251) {
		tmp = (eps * x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -0.00065:
		tmp = (eps * -x) / 2.0
	elif x <= 1400000000000.0:
		tmp = 1.0
	elif x <= 2.6e+193:
		tmp = 0.0
	elif x <= 9e+251:
		tmp = (eps * x) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -0.00065)
		tmp = Float64(Float64(eps * Float64(-x)) / 2.0);
	elseif (x <= 1400000000000.0)
		tmp = 1.0;
	elseif (x <= 2.6e+193)
		tmp = 0.0;
	elseif (x <= 9e+251)
		tmp = Float64(Float64(eps * x) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.00065)
		tmp = (eps * -x) / 2.0;
	elseif (x <= 1400000000000.0)
		tmp = 1.0;
	elseif (x <= 2.6e+193)
		tmp = 0.0;
	elseif (x <= 9e+251)
		tmp = (eps * x) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -0.00065], N[(N[(eps * (-x)), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1400000000000.0], 1.0, If[LessEqual[x, 2.6e+193], 0.0, If[LessEqual[x, 9e+251], N[(N[(eps * x), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.4999999999999997e-4

   1. Initial program 96.9%

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

      1. sub-neg 96.9%

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

      2. neg-sub0 96.9%

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

      3. associate-+r- 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in x around 0 66.6%

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

      1. +-commutative 66.6%

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

      2. associate-+r+ 66.6%

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

      3. associate-*r* 66.6%

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

      4. neg-mul-1 66.6%

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

      5. *-commutative 66.6%

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

      6. associate-*r* 66.6%

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

      7. distribute-lft-neg-in 66.6%

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

      8. distribute-rgt-neg-in 66.6%

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

      9. sub-neg 66.6%

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

      10. mul-1-neg 66.6%

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

      11. distribute-neg-in 66.6%

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

      12. metadata-eval 66.6%

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

      13. mul-1-neg 66.6%

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

      14. remove-double-neg 66.6%

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

      15. +-commutative 66.6%

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

      16. distribute-rgt-in 66.6%

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

      17. neg-mul-1 66.6%

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

      18. sub-neg 66.6%

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

      19. *-commutative 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in x around 0 0.8%

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

   8. Taylor expanded in eps around 0 17.2%

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

      1. neg-mul-1 17.2%

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

   10. Simplified 17.2%

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

   11. Taylor expanded in eps around inf 17.2%

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

       1. associate-*r* 17.2%

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

       2. mul-1-neg 17.2%

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

   13. Simplified 17.2%

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

   if -6.4999999999999997e-4 < x < 1.4e12

   1. Initial program 50.9%

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

      1. sub-neg 50.9%

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

      2. neg-sub0 50.9%

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

      3. associate-+r- 50.9%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in x around 0 80.1%

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

   if 1.4e12 < x < 2.60000000000000013e193 or 8.9999999999999997e251 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 63.8%

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

      1. neg-mul-1 63.8%

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

      2. rec-exp 63.8%

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

      3. neg-mul-1 63.8%

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

      4. div-sub 63.8%

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

      5. +-inverses 63.8%

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

   6. Simplified 63.8%

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

   if 2.60000000000000013e193 < x < 8.9999999999999997e251

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 30.9%

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

      1. +-commutative 30.9%

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

      2. associate-+r+ 30.9%

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

      3. associate-*r* 30.9%

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

      4. neg-mul-1 30.9%

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

      5. *-commutative 30.9%

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

      6. associate-*r* 30.9%

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

      7. distribute-lft-neg-in 30.9%

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

      8. distribute-rgt-neg-in 30.9%

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

      9. sub-neg 30.9%

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

      10. mul-1-neg 30.9%

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

      11. distribute-neg-in 30.9%

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

      12. metadata-eval 30.9%

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

      13. mul-1-neg 30.9%

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

      14. remove-double-neg 30.9%

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

      15. +-commutative 30.9%

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

      16. distribute-rgt-in 30.9%

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

      17. neg-mul-1 30.9%

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

      18. sub-neg 30.9%

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

      19. *-commutative 30.9%

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

   6. Simplified 30.9%

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

   7. Taylor expanded in eps around inf 26.2%

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

      1. *-commutative 26.2%

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

   9. Simplified 26.2%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{elif}\;x \leq 1400000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+193}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+251}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 63.6% accurate, 17.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+191}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+251}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -0.00065)
   (/ (* eps (- x)) 2.0)
   (if (<= x 1.42)
     (/ (- 2.0 (* x x)) 2.0)
     (if (<= x 1.2e+191) 0.0 (if (<= x 7.8e+251) (/ (* eps x) 2.0) 0.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -0.00065) {
		tmp = (eps * -x) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 1.2e+191) {
		tmp = 0.0;
	} else if (x <= 7.8e+251) {
		tmp = (eps * x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.00065d0)) then
        tmp = (eps * -x) / 2.0d0
    else if (x <= 1.42d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else if (x <= 1.2d+191) then
        tmp = 0.0d0
    else if (x <= 7.8d+251) then
        tmp = (eps * x) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.00065) {
		tmp = (eps * -x) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 1.2e+191) {
		tmp = 0.0;
	} else if (x <= 7.8e+251) {
		tmp = (eps * x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -0.00065:
		tmp = (eps * -x) / 2.0
	elif x <= 1.42:
		tmp = (2.0 - (x * x)) / 2.0
	elif x <= 1.2e+191:
		tmp = 0.0
	elif x <= 7.8e+251:
		tmp = (eps * x) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -0.00065)
		tmp = Float64(Float64(eps * Float64(-x)) / 2.0);
	elseif (x <= 1.42)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	elseif (x <= 1.2e+191)
		tmp = 0.0;
	elseif (x <= 7.8e+251)
		tmp = Float64(Float64(eps * x) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.00065)
		tmp = (eps * -x) / 2.0;
	elseif (x <= 1.42)
		tmp = (2.0 - (x * x)) / 2.0;
	elseif (x <= 1.2e+191)
		tmp = 0.0;
	elseif (x <= 7.8e+251)
		tmp = (eps * x) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -0.00065], N[(N[(eps * (-x)), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.42], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.2e+191], 0.0, If[LessEqual[x, 7.8e+251], N[(N[(eps * x), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.4999999999999997e-4

   1. Initial program 96.9%

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

      1. sub-neg 96.9%

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

      2. neg-sub0 96.9%

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

      3. associate-+r- 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in x around 0 66.6%

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

      1. +-commutative 66.6%

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

      2. associate-+r+ 66.6%

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

      3. associate-*r* 66.6%

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

      4. neg-mul-1 66.6%

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

      5. *-commutative 66.6%

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

      6. associate-*r* 66.6%

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

      7. distribute-lft-neg-in 66.6%

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

      8. distribute-rgt-neg-in 66.6%

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

      9. sub-neg 66.6%

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

      10. mul-1-neg 66.6%

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

      11. distribute-neg-in 66.6%

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

      12. metadata-eval 66.6%

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

      13. mul-1-neg 66.6%

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

      14. remove-double-neg 66.6%

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

      15. +-commutative 66.6%

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

      16. distribute-rgt-in 66.6%

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

      17. neg-mul-1 66.6%

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

      18. sub-neg 66.6%

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

      19. *-commutative 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in x around 0 0.8%

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

   8. Taylor expanded in eps around 0 17.2%

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

      1. neg-mul-1 17.2%

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

   10. Simplified 17.2%

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

   11. Taylor expanded in eps around inf 17.2%

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

       1. associate-*r* 17.2%

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

       2. mul-1-neg 17.2%

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

   13. Simplified 17.2%

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

   if -6.4999999999999997e-4 < x < 1.4199999999999999

   1. Initial program 50.1%

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

      1. sub-neg 50.1%

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

      2. neg-sub0 50.1%

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

      3. associate-+r- 50.1%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in eps around 0 81.8%

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

      1. distribute-rgt1-in 81.7%

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

      2. neg-mul-1 81.7%

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

      3. distribute-lft-out 81.7%

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

      4. distribute-rgt1-in 81.7%

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

      5. neg-mul-1 81.7%

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

   6. Simplified 81.7%

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

   7. Taylor expanded in x around 0 81.4%

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

      1. *-commutative 81.4%

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

      2. unpow2 81.4%

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

   9. Simplified 81.4%

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

   10. Taylor expanded in x around 0 81.4%

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

       1. mul-1-neg 81.4%

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

       2. unsub-neg 81.4%

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

       3. unpow2 81.4%

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

   12. Simplified 81.4%

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

   if 1.4199999999999999 < x < 1.19999999999999993e191 or 7.79999999999999951e251 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 62.0%

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

      1. neg-mul-1 62.0%

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

      2. rec-exp 62.0%

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

      3. neg-mul-1 62.0%

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

      4. div-sub 62.0%

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

      5. +-inverses 62.0%

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

   6. Simplified 62.0%

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

   if 1.19999999999999993e191 < x < 7.79999999999999951e251

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 30.9%

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

      1. +-commutative 30.9%

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

      2. associate-+r+ 30.9%

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

      3. associate-*r* 30.9%

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

      4. neg-mul-1 30.9%

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

      5. *-commutative 30.9%

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

      6. associate-*r* 30.9%

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

      7. distribute-lft-neg-in 30.9%

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

      8. distribute-rgt-neg-in 30.9%

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

      9. sub-neg 30.9%

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

      10. mul-1-neg 30.9%

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

      11. distribute-neg-in 30.9%

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

      12. metadata-eval 30.9%

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

      13. mul-1-neg 30.9%

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

      14. remove-double-neg 30.9%

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

      15. +-commutative 30.9%

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

      16. distribute-rgt-in 30.9%

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

      17. neg-mul-1 30.9%

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

      18. sub-neg 30.9%

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

      19. *-commutative 30.9%

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

   6. Simplified 30.9%

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

   7. Taylor expanded in eps around inf 26.2%

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

      1. *-commutative 26.2%

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

   9. Simplified 26.2%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+191}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+251}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 55.7% accurate, 20.3× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1400000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+192}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+251}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 1400000000000.0)
   1.0
   (if (<= x 2.8e+192) 0.0 (if (<= x 8.4e+251) (/ (* eps x) 2.0) 0.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 1400000000000.0) {
		tmp = 1.0;
	} else if (x <= 2.8e+192) {
		tmp = 0.0;
	} else if (x <= 8.4e+251) {
		tmp = (eps * x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1400000000000.0d0) then
        tmp = 1.0d0
    else if (x <= 2.8d+192) then
        tmp = 0.0d0
    else if (x <= 8.4d+251) then
        tmp = (eps * x) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 1400000000000.0) {
		tmp = 1.0;
	} else if (x <= 2.8e+192) {
		tmp = 0.0;
	} else if (x <= 8.4e+251) {
		tmp = (eps * x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 1400000000000.0:
		tmp = 1.0
	elif x <= 2.8e+192:
		tmp = 0.0
	elif x <= 8.4e+251:
		tmp = (eps * x) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 1400000000000.0)
		tmp = 1.0;
	elseif (x <= 2.8e+192)
		tmp = 0.0;
	elseif (x <= 8.4e+251)
		tmp = Float64(Float64(eps * x) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1400000000000.0)
		tmp = 1.0;
	elseif (x <= 2.8e+192)
		tmp = 0.0;
	elseif (x <= 8.4e+251)
		tmp = (eps * x) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 1400000000000.0], 1.0, If[LessEqual[x, 2.8e+192], 0.0, If[LessEqual[x, 8.4e+251], N[(N[(eps * x), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.4e12

   1. Initial program 59.6%

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

      1. sub-neg 59.6%

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

      2. neg-sub0 59.6%

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

      3. associate-+r- 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in x around 0 65.4%

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

   if 1.4e12 < x < 2.79999999999999976e192 or 8.4000000000000001e251 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 63.8%

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

      1. neg-mul-1 63.8%

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

      2. rec-exp 63.8%

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

      3. neg-mul-1 63.8%

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

      4. div-sub 63.8%

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

      5. +-inverses 63.8%

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

   6. Simplified 63.8%

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

   if 2.79999999999999976e192 < x < 8.4000000000000001e251

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 30.9%

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

      1. +-commutative 30.9%

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

      2. associate-+r+ 30.9%

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

      3. associate-*r* 30.9%

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

      4. neg-mul-1 30.9%

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

      5. *-commutative 30.9%

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

      6. associate-*r* 30.9%

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

      7. distribute-lft-neg-in 30.9%

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

      8. distribute-rgt-neg-in 30.9%

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

      9. sub-neg 30.9%

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

      10. mul-1-neg 30.9%

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

      11. distribute-neg-in 30.9%

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

      12. metadata-eval 30.9%

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

      13. mul-1-neg 30.9%

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

      14. remove-double-neg 30.9%

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

      15. +-commutative 30.9%

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

      16. distribute-rgt-in 30.9%

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

      17. neg-mul-1 30.9%

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

      18. sub-neg 30.9%

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

      19. *-commutative 30.9%

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

   6. Simplified 30.9%

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

   7. Taylor expanded in eps around inf 26.2%

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

      1. *-commutative 26.2%

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

   9. Simplified 26.2%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1400000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+192}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+251}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 56.2% accurate, 20.3× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+188}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+251}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 2.0)
   (/ (- 2.0 x) 2.0)
   (if (<= x 2.5e+188) 0.0 (if (<= x 2.8e+251) (/ (* eps x) 2.0) 0.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else if (x <= 2.5e+188) {
		tmp = 0.0;
	} else if (x <= 2.8e+251) {
		tmp = (eps * x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = (2.0d0 - x) / 2.0d0
    else if (x <= 2.5d+188) then
        tmp = 0.0d0
    else if (x <= 2.8d+251) then
        tmp = (eps * x) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else if (x <= 2.5e+188) {
		tmp = 0.0;
	} else if (x <= 2.8e+251) {
		tmp = (eps * x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 - x) / 2.0
	elif x <= 2.5e+188:
		tmp = 0.0
	elif x <= 2.8e+251:
		tmp = (eps * x) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 - x) / 2.0);
	elseif (x <= 2.5e+188)
		tmp = 0.0;
	elseif (x <= 2.8e+251)
		tmp = Float64(Float64(eps * x) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 - x) / 2.0;
	elseif (x <= 2.5e+188)
		tmp = 0.0;
	elseif (x <= 2.8e+251)
		tmp = (eps * x) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 2.0], N[(N[(2.0 - x), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.5e+188], 0.0, If[LessEqual[x, 2.8e+251], N[(N[(eps * x), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 2

   1. Initial program 59.1%

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

      1. sub-neg 59.1%

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

      2. neg-sub0 59.1%

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

      3. associate-+r- 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in x around 0 46.8%

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

      1. +-commutative 46.8%

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

      2. associate-+r+ 46.8%

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

      3. associate-*r* 46.8%

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

      4. neg-mul-1 46.8%

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

      5. *-commutative 46.8%

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

      6. associate-*r* 46.8%

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

      7. distribute-lft-neg-in 46.8%

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

      8. distribute-rgt-neg-in 46.8%

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

      9. sub-neg 46.8%

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

      10. mul-1-neg 46.8%

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

      11. distribute-neg-in 46.8%

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

      12. metadata-eval 46.8%

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

      13. mul-1-neg 46.8%

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

      14. remove-double-neg 46.8%

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

      15. +-commutative 46.8%

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

      16. distribute-rgt-in 46.8%

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

      17. neg-mul-1 46.8%

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

      18. sub-neg 46.8%

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

      19. *-commutative 46.8%

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

   6. Simplified 46.8%

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

   7. Taylor expanded in x around 0 25.7%

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

   8. Taylor expanded in eps around 0 28.3%

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

      1. neg-mul-1 28.3%

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

   10. Simplified 28.3%

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

   11. Taylor expanded in eps around 0 66.3%

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

       1. neg-mul-1 66.3%

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

       2. unsub-neg 66.3%

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

   13. Simplified 66.3%

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

   if 2 < x < 2.5000000000000001e188 or 2.8e251 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 62.0%

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

      1. neg-mul-1 62.0%

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

      2. rec-exp 62.0%

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

      3. neg-mul-1 62.0%

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

      4. div-sub 62.0%

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

      5. +-inverses 62.0%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 62.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 2.5000000000000001e188 < x < 2.8e251

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. neg-sub0 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      3. associate-+r- 100.0%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in x around 0 30.9%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(-1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right) + \frac{1}{\varepsilon}\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
   5. Step-by-step derivation

      1. +-commutative 30.9%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\frac{1}{\varepsilon} + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      2. associate-+r+ 30.9%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      3. associate-*r* 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      4. neg-mul-1 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(-x\right)} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      5. *-commutative 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(-x\right) \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      6. associate-*r* 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(\left(-x\right) \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      7. distribute-lft-neg-in 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(-x \cdot \left(1 - \varepsilon\right)\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      8. distribute-rgt-neg-in 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(x \cdot \left(-\left(1 - \varepsilon\right)\right)\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      9. sub-neg 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(-\color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      10. mul-1-neg 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(-\left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      11. distribute-neg-in 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \color{blue}{\left(\left(-1\right) + \left(--1 \cdot \varepsilon\right)\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      12. metadata-eval 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(\color{blue}{-1} + \left(--1 \cdot \varepsilon\right)\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      13. mul-1-neg 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(-1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      14. remove-double-neg 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(-1 + \color{blue}{\varepsilon}\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      15. +-commutative 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \color{blue}{\left(\varepsilon + -1\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      16. distribute-rgt-in 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(\varepsilon \cdot x + -1 \cdot x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      17. neg-mul-1 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(\varepsilon \cdot x + \color{blue}{\left(-x\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      18. sub-neg 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(\varepsilon \cdot x - x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      19. *-commutative 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(\color{blue}{x \cdot \varepsilon} - x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   6. Simplified 30.9%

      \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \varepsilon - x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Taylor expanded in eps around inf 26.2%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   8. Step-by-step derivation

      1. *-commutative 26.2%

         \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   9. Simplified 26.2%

      \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+188}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+251}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14: 62.8% accurate, 20.3× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.45:\\ \;\;\;\;\frac{\left(2 - x\right) - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+193}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 0.45)
   (/ (- (- 2.0 x) (* eps x)) 2.0)
   (if (<= x 1.35e+193) 0.0 (if (<= x 2.2e+251) (/ (* eps x) 2.0) 0.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 0.45) {
		tmp = ((2.0 - x) - (eps * x)) / 2.0;
	} else if (x <= 1.35e+193) {
		tmp = 0.0;
	} else if (x <= 2.2e+251) {
		tmp = (eps * x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 0.45d0) then
        tmp = ((2.0d0 - x) - (eps * x)) / 2.0d0
    else if (x <= 1.35d+193) then
        tmp = 0.0d0
    else if (x <= 2.2d+251) then
        tmp = (eps * x) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 0.45) {
		tmp = ((2.0 - x) - (eps * x)) / 2.0;
	} else if (x <= 1.35e+193) {
		tmp = 0.0;
	} else if (x <= 2.2e+251) {
		tmp = (eps * x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 0.45:
		tmp = ((2.0 - x) - (eps * x)) / 2.0
	elif x <= 1.35e+193:
		tmp = 0.0
	elif x <= 2.2e+251:
		tmp = (eps * x) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 0.45)
		tmp = Float64(Float64(Float64(2.0 - x) - Float64(eps * x)) / 2.0);
	elseif (x <= 1.35e+193)
		tmp = 0.0;
	elseif (x <= 2.2e+251)
		tmp = Float64(Float64(eps * x) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 0.45)
		tmp = ((2.0 - x) - (eps * x)) / 2.0;
	elseif (x <= 1.35e+193)
		tmp = 0.0;
	elseif (x <= 2.2e+251)
		tmp = (eps * x) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 0.45], N[(N[(N[(2.0 - x), $MachinePrecision] - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.35e+193], 0.0, If[LessEqual[x, 2.2e+251], N[(N[(eps * x), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.450000000000000011

   1. Initial program 58.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 58.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. neg-sub0 58.9%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      3. associate-+r- 58.9%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   3. Simplified 58.9%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in x around 0 46.5%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(-1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right) + \frac{1}{\varepsilon}\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
   5. Step-by-step derivation

      1. +-commutative 46.5%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\frac{1}{\varepsilon} + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      2. associate-+r+ 46.5%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      3. associate-*r* 46.5%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      4. neg-mul-1 46.5%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(-x\right)} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      5. *-commutative 46.5%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(-x\right) \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      6. associate-*r* 46.5%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(\left(-x\right) \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      7. distribute-lft-neg-in 46.5%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(-x \cdot \left(1 - \varepsilon\right)\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      8. distribute-rgt-neg-in 46.5%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(x \cdot \left(-\left(1 - \varepsilon\right)\right)\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      9. sub-neg 46.5%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(-\color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      10. mul-1-neg 46.5%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(-\left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      11. distribute-neg-in 46.5%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \color{blue}{\left(\left(-1\right) + \left(--1 \cdot \varepsilon\right)\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      12. metadata-eval 46.5%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(\color{blue}{-1} + \left(--1 \cdot \varepsilon\right)\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      13. mul-1-neg 46.5%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(-1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      14. remove-double-neg 46.5%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(-1 + \color{blue}{\varepsilon}\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      15. +-commutative 46.5%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \color{blue}{\left(\varepsilon + -1\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      16. distribute-rgt-in 46.5%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(\varepsilon \cdot x + -1 \cdot x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      17. neg-mul-1 46.5%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(\varepsilon \cdot x + \color{blue}{\left(-x\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      18. sub-neg 46.5%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(\varepsilon \cdot x - x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      19. *-commutative 46.5%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(\color{blue}{x \cdot \varepsilon} - x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   6. Simplified 46.5%

      \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \varepsilon - x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Taylor expanded in x around 0 25.8%

      \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \varepsilon - x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \color{blue}{\left(\left(-1 \cdot \left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{\varepsilon}\right) - 1\right)}}{2} \]

   8. Taylor expanded in eps around 0 28.4%

      \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(-1 \cdot x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\left(-1 \cdot \left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{\varepsilon}\right) - 1\right)}{2} \]
   9. Step-by-step derivation

      1. neg-mul-1 28.4%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\left(-1 \cdot \left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{\varepsilon}\right) - 1\right)}{2} \]

   10. Simplified 28.4%

       \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\left(-1 \cdot \left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{\varepsilon}\right) - 1\right)}{2} \]

   11. Taylor expanded in eps around 0 68.2%

       \[\leadsto \frac{\color{blue}{2 + \left(-1 \cdot x + -1 \cdot \left(\varepsilon \cdot x\right)\right)}}{2} \]
   12. Step-by-step derivation

       1. neg-mul-1 68.2%

          \[\leadsto \frac{2 + \left(\color{blue}{\left(-x\right)} + -1 \cdot \left(\varepsilon \cdot x\right)\right)}{2} \]

       2. associate-+r+ 68.2%

          \[\leadsto \frac{\color{blue}{\left(2 + \left(-x\right)\right) + -1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]

       3. mul-1-neg 68.2%

          \[\leadsto \frac{\left(2 + \left(-x\right)\right) + \color{blue}{\left(-\varepsilon \cdot x\right)}}{2} \]

       4. *-commutative 68.2%

          \[\leadsto \frac{\left(2 + \left(-x\right)\right) + \left(-\color{blue}{x \cdot \varepsilon}\right)}{2} \]

       5. unsub-neg 68.2%

          \[\leadsto \frac{\color{blue}{\left(2 + \left(-x\right)\right) - x \cdot \varepsilon}}{2} \]

       6. unsub-neg 68.2%

          \[\leadsto \frac{\color{blue}{\left(2 - x\right)} - x \cdot \varepsilon}{2} \]

   13. Simplified 68.2%

       \[\leadsto \frac{\color{blue}{\left(2 - x\right) - x \cdot \varepsilon}}{2} \]

   if 0.450000000000000011 < x < 1.35e193 or 2.2e251 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\varepsilon \cdot x - x}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]

   4. Taylor expanded in eps around 0 61.2%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]
   5. Step-by-step derivation

      1. neg-mul-1 61.2%

         \[\leadsto \frac{\frac{e^{\color{blue}{-1 \cdot x}} - \frac{1}{e^{x}}}{\varepsilon}}{2} \]

      2. rec-exp 61.2%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} - \color{blue}{e^{-x}}}{\varepsilon}}{2} \]

      3. neg-mul-1 61.2%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} - e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]

      4. div-sub 61.2%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{-1 \cdot x}}{\varepsilon}}}{2} \]

      5. +-inverses 61.2%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 61.2%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 1.35e193 < x < 2.2e251

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. neg-sub0 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      3. associate-+r- 100.0%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in x around 0 30.9%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(-1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right) + \frac{1}{\varepsilon}\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
   5. Step-by-step derivation

      1. +-commutative 30.9%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(\frac{1}{\varepsilon} + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      2. associate-+r+ 30.9%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      3. associate-*r* 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(-1 \cdot x\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      4. neg-mul-1 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(-x\right)} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      5. *-commutative 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(-x\right) \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      6. associate-*r* 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(\left(-x\right) \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      7. distribute-lft-neg-in 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(-x \cdot \left(1 - \varepsilon\right)\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      8. distribute-rgt-neg-in 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(x \cdot \left(-\left(1 - \varepsilon\right)\right)\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      9. sub-neg 30.9%

         \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(-\color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      10. mul-1-neg 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(-\left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      11. distribute-neg-in 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \color{blue}{\left(\left(-1\right) + \left(--1 \cdot \varepsilon\right)\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      12. metadata-eval 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(\color{blue}{-1} + \left(--1 \cdot \varepsilon\right)\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      13. mul-1-neg 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(-1 + \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      14. remove-double-neg 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \left(-1 + \color{blue}{\varepsilon}\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      15. +-commutative 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \color{blue}{\left(\varepsilon + -1\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      16. distribute-rgt-in 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(\varepsilon \cdot x + -1 \cdot x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      17. neg-mul-1 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(\varepsilon \cdot x + \color{blue}{\left(-x\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      18. sub-neg 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \color{blue}{\left(\varepsilon \cdot x - x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      19. *-commutative 30.9%

          \[\leadsto \frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(\color{blue}{x \cdot \varepsilon} - x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   6. Simplified 30.9%

      \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) + \left(x \cdot \varepsilon - x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Taylor expanded in eps around inf 26.2%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   8. Step-by-step derivation

      1. *-commutative 26.2%

         \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   9. Simplified 26.2%

      \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.45:\\ \;\;\;\;\frac{\left(2 - x\right) - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+193}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15: 56.7% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1400000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 1400000000000.0) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 1400000000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1400000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 1400000000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 1400000000000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 1400000000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1400000000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 1400000000000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e12

   1. Initial program 59.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 59.6%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. neg-sub0 59.6%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      3. associate-+r- 59.6%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in x around 0 65.4%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 1.4e12 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\varepsilon \cdot x - x}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]

   4. Taylor expanded in eps around 0 56.4%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]
   5. Step-by-step derivation

      1. neg-mul-1 56.4%

         \[\leadsto \frac{\frac{e^{\color{blue}{-1 \cdot x}} - \frac{1}{e^{x}}}{\varepsilon}}{2} \]

      2. rec-exp 56.4%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} - \color{blue}{e^{-x}}}{\varepsilon}}{2} \]

      3. neg-mul-1 56.4%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} - e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]

      4. div-sub 56.4%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{-1 \cdot x}}{\varepsilon}}}{2} \]

      5. +-inverses 56.4%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 56.4%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1400000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16: 16.4% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ 0 \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 0.0)

C

eps = abs(eps);
double code(double x, double eps) {
	return 0.0;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	return 0.0;
}

Python

eps = abs(eps)
def code(x, eps):
	return 0.0

Julia

eps = abs(eps)
function code(x, eps)
	return 0.0
end

MATLAB

eps = abs(eps)
function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := 0.0

Derivation

1. Initial program 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%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\varepsilon \cdot x - x}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]

4. Taylor expanded in eps around 0 20.9%

   \[\leadsto \frac{\color{blue}{\frac{e^{-x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]
5. Step-by-step derivation

   1. neg-mul-1 20.9%

      \[\leadsto \frac{\frac{e^{\color{blue}{-1 \cdot x}} - \frac{1}{e^{x}}}{\varepsilon}}{2} \]

   2. rec-exp 20.9%

      \[\leadsto \frac{\frac{e^{-1 \cdot x} - \color{blue}{e^{-x}}}{\varepsilon}}{2} \]

   3. neg-mul-1 20.9%

      \[\leadsto \frac{\frac{e^{-1 \cdot x} - e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]

   4. div-sub 20.9%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{-1 \cdot x}}{\varepsilon}}}{2} \]

   5. +-inverses 21.1%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

6. Simplified 21.1%

   \[\leadsto \frac{\color{blue}{0}}{2} \]

7. Final simplification 21.1%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023293 
(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))