NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.6% → 99.7%

Time: 15.9s

Alternatives: 13

Speedup: 2.0×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 - \varepsilon\right)}\\ t_1 := \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)}\\ \mathbf{if}\;t_1 + t_0 \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\varepsilon} + -1\right)\right) \cdot t_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x (- -1.0 eps))))
        (t_1 (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))))
   (if (<= (+ t_1 (* t_0 (+ 1.0 (/ -1.0 eps)))) 2.0)
     (/ (+ (/ (+ 1.0 x) (exp x)) (* (+ 1.0 x) (exp (- x)))) 2.0)
     (/ (- t_1 (* (log1p (expm1 (+ (/ 1.0 eps) -1.0))) t_0)) 2.0))))

C

double code(double x, double eps) {
	double t_0 = exp((x * (-1.0 - eps)));
	double t_1 = (1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)));
	double tmp;
	if ((t_1 + (t_0 * (1.0 + (-1.0 / eps)))) <= 2.0) {
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	} else {
		tmp = (t_1 - (log1p(expm1(((1.0 / eps) + -1.0))) * t_0)) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp((x * (-1.0 - eps)));
	double t_1 = (1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)));
	double tmp;
	if ((t_1 + (t_0 * (1.0 + (-1.0 / eps)))) <= 2.0) {
		tmp = (((1.0 + x) / Math.exp(x)) + ((1.0 + x) * Math.exp(-x))) / 2.0;
	} else {
		tmp = (t_1 - (Math.log1p(Math.expm1(((1.0 / eps) + -1.0))) * t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp((x * (-1.0 - eps)))
	t_1 = (1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))
	tmp = 0
	if (t_1 + (t_0 * (1.0 + (-1.0 / eps)))) <= 2.0:
		tmp = (((1.0 + x) / math.exp(x)) + ((1.0 + x) * math.exp(-x))) / 2.0
	else:
		tmp = (t_1 - (math.log1p(math.expm1(((1.0 / eps) + -1.0))) * t_0)) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(x * Float64(-1.0 - eps)))
	t_1 = Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0))))
	tmp = 0.0
	if (Float64(t_1 + Float64(t_0 * Float64(1.0 + Float64(-1.0 / eps)))) <= 2.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + x) / exp(x)) + Float64(Float64(1.0 + x) * exp(Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(t_1 - Float64(log1p(expm1(Float64(Float64(1.0 / eps) + -1.0))) * t_0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(t$95$0 * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$1 - N[(N[Log[1 + N[(Exp[N[(N[(1.0 / eps), $MachinePrecision] + -1.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x))))) < 2

   1. Initial program 43.7%

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

      1. div-sub 43.7%

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

      2. +-rgt-identity 43.7%

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

      3. div-sub 43.7%

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

   3. Simplified 43.7%

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

   4. Taylor expanded in eps around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. neg-mul-1 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. mul-1-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-lft1-in 100.0%

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

      8. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

      1. exp-neg 100.0%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 2 < (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x)))))

   1. Initial program 98.1%

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

      1. div-sub 98.1%

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

      2. +-rgt-identity 98.1%

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

      3. div-sub 98.1%

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

   3. Simplified 98.1%

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

      1. log1p-expm1-u 98.1%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\varepsilon} + -1\right)\right)} \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   5. Applied egg-rr 98.1%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\varepsilon} + -1\right)\right)} \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\varepsilon} + -1\right)\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{\varepsilon}\\ t_1 := e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\\ \mathbf{if}\;t_0 \cdot e^{x \cdot \left(\varepsilon + -1\right)} + t_1 \leq 0:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot e^{\varepsilon \cdot x - x} + t_1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps)))
        (t_1 (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps)))))
   (if (<= (+ (* t_0 (exp (* x (+ eps -1.0)))) t_1) 0.0)
     (/ (+ (/ (+ 1.0 x) (exp x)) (* (+ 1.0 x) (exp (- x)))) 2.0)
     (/ (+ (* t_0 (exp (- (* eps x) x))) t_1) 2.0))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double t_1 = exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps));
	double tmp;
	if (((t_0 * exp((x * (eps + -1.0)))) + t_1) <= 0.0) {
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	} else {
		tmp = ((t_0 * exp(((eps * x) - x))) + t_1) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (1.0d0 / eps)
    t_1 = exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps))
    if (((t_0 * exp((x * (eps + (-1.0d0))))) + t_1) <= 0.0d0) then
        tmp = (((1.0d0 + x) / exp(x)) + ((1.0d0 + x) * exp(-x))) / 2.0d0
    else
        tmp = ((t_0 * exp(((eps * x) - x))) + t_1) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double t_1 = Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps));
	double tmp;
	if (((t_0 * Math.exp((x * (eps + -1.0)))) + t_1) <= 0.0) {
		tmp = (((1.0 + x) / Math.exp(x)) + ((1.0 + x) * Math.exp(-x))) / 2.0;
	} else {
		tmp = ((t_0 * Math.exp(((eps * x) - x))) + t_1) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 + (1.0 / eps)
	t_1 = math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps))
	tmp = 0
	if ((t_0 * math.exp((x * (eps + -1.0)))) + t_1) <= 0.0:
		tmp = (((1.0 + x) / math.exp(x)) + ((1.0 + x) * math.exp(-x))) / 2.0
	else:
		tmp = ((t_0 * math.exp(((eps * x) - x))) + t_1) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(1.0 / eps))
	t_1 = Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(x * Float64(eps + -1.0)))) + t_1) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + x) / exp(x)) + Float64(Float64(1.0 + x) * exp(Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(Float64(eps * x) - x))) + t_1) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 + (1.0 / eps);
	t_1 = exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps));
	tmp = 0.0;
	if (((t_0 * exp((x * (eps + -1.0)))) + t_1) <= 0.0)
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	else
		tmp = ((t_0 * exp(((eps * x) - x))) + t_1) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], 0.0], N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$0 * N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x))))) < 0.0

   1. Initial program 25.7%

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

      1. div-sub 25.7%

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

      2. +-rgt-identity 25.7%

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

      3. div-sub 25.7%

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

   3. Simplified 25.7%

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

   4. Taylor expanded in eps around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. neg-mul-1 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. mul-1-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-lft1-in 100.0%

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

      8. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

      1. exp-neg 100.0%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x)))))

   1. Initial program 98.6%

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

      1. div-sub 98.6%

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

      2. +-rgt-identity 98.6%

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

      3. div-sub 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in x around inf 98.6%

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

      1. *-commutative 98.6%

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

      2. +-commutative 98.6%

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

      3. mul-1-neg 98.6%

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

      4. *-commutative 98.6%

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

      5. distribute-rgt-neg-in 98.6%

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

      6. mul-1-neg 98.6%

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

      7. sub-neg 98.6%

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

      8. mul-1-neg 98.6%

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

      9. distribute-lft-in 98.6%

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

      10. metadata-eval 98.6%

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

      11. neg-mul-1 98.6%

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

      12. mul-1-neg 98.6%

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

      13. remove-double-neg 98.6%

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

      14. +-commutative 98.6%

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

      15. distribute-rgt-in 98.6%

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

      16. neg-mul-1 98.6%

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

      17. unsub-neg 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

   1. div-sub 66.7%

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

   2. +-rgt-identity 66.7%

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

   3. div-sub 66.7%

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

3. Simplified 66.7%

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

4. Taylor expanded in eps around inf 64.1%

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

   1. mul-1-neg 64.1%

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

   2. *-commutative 64.1%

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

   3. distribute-rgt-neg-in 64.1%

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

   4. mul-1-neg 64.1%

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

   5. sub-neg 64.1%

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

   6. mul-1-neg 64.1%

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

   7. distribute-lft-in 64.1%

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

   8. metadata-eval 64.1%

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

   9. neg-mul-1 64.1%

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

   10. mul-1-neg 64.1%

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

   11. remove-double-neg 64.1%

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

   12. +-commutative 64.1%

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

   13. distribute-rgt-in 64.1%

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

   14. neg-mul-1 64.1%

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

   15. unsub-neg 64.1%

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

6. Simplified 64.1%

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

7. Taylor expanded in eps around inf 97.0%

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

   1. mul-1-neg 97.0%

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

   2. +-commutative 97.0%

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

   3. exp-prod 97.0%

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

   4. *-lft-identity 97.0%

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

   5. metadata-eval 97.0%

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

   6. cancel-sign-sub-inv 97.0%

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

   7. exp-prod 97.0%

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

   8. mul-1-neg 97.0%

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

   9. *-commutative 97.0%

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

   10. sub-neg 97.0%

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

   11. mul-1-neg 97.0%

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

   12. remove-double-neg 97.0%

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

   13. +-commutative 97.0%

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

9. Simplified 97.0%

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

10. Taylor expanded in eps around inf 97.0%

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

11. Taylor expanded in eps around -inf 97.0%

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

    1. *-commutative 97.0%

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

    2. distribute-rgt-neg-in 97.0%

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

    3. cancel-sign-sub-inv 97.0%

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

    4. metadata-eval 97.0%

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

    5. *-lft-identity 97.0%

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

    6. distribute-neg-in 97.0%

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

    7. metadata-eval 97.0%

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

    8. unsub-neg 97.0%

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

13. Simplified 97.0%

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

14. Final simplification 97.0%

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

Alternative 4: 77.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-1 - \varepsilon\right)\\ t_1 := \varepsilon \cdot x - x\\ t_2 := \frac{e^{t_0} + \left(1 + t_1\right)}{2}\\ t_3 := e^{t_1}\\ \mathbf{if}\;\varepsilon \leq -5.1 \cdot 10^{+131}:\\ \;\;\;\;\frac{1 + t_3}{2}\\ \mathbf{elif}\;\varepsilon \leq -1.6:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.85 \cdot 10^{+176} \lor \neg \left(\varepsilon \leq 10^{+261}\right):\\ \;\;\;\;\frac{t_3 + \left(1 + t_0\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (- -1.0 eps)))
        (t_1 (- (* eps x) x))
        (t_2 (/ (+ (exp t_0) (+ 1.0 t_1)) 2.0))
        (t_3 (exp t_1)))
   (if (<= eps -5.1e+131)
     (/ (+ 1.0 t_3) 2.0)
     (if (<= eps -1.6)
       t_2
       (if (<= eps 1.4e-9)
         (/ (* 2.0 (exp (- x))) 2.0)
         (if (or (<= eps 1.85e+176) (not (<= eps 1e+261)))
           (/ (+ t_3 (+ 1.0 t_0)) 2.0)
           t_2))))))

C

double code(double x, double eps) {
	double t_0 = x * (-1.0 - eps);
	double t_1 = (eps * x) - x;
	double t_2 = (exp(t_0) + (1.0 + t_1)) / 2.0;
	double t_3 = exp(t_1);
	double tmp;
	if (eps <= -5.1e+131) {
		tmp = (1.0 + t_3) / 2.0;
	} else if (eps <= -1.6) {
		tmp = t_2;
	} else if (eps <= 1.4e-9) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else if ((eps <= 1.85e+176) || !(eps <= 1e+261)) {
		tmp = (t_3 + (1.0 + t_0)) / 2.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = x * ((-1.0d0) - eps)
    t_1 = (eps * x) - x
    t_2 = (exp(t_0) + (1.0d0 + t_1)) / 2.0d0
    t_3 = exp(t_1)
    if (eps <= (-5.1d+131)) then
        tmp = (1.0d0 + t_3) / 2.0d0
    else if (eps <= (-1.6d0)) then
        tmp = t_2
    else if (eps <= 1.4d-9) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else if ((eps <= 1.85d+176) .or. (.not. (eps <= 1d+261))) then
        tmp = (t_3 + (1.0d0 + t_0)) / 2.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (-1.0 - eps);
	double t_1 = (eps * x) - x;
	double t_2 = (Math.exp(t_0) + (1.0 + t_1)) / 2.0;
	double t_3 = Math.exp(t_1);
	double tmp;
	if (eps <= -5.1e+131) {
		tmp = (1.0 + t_3) / 2.0;
	} else if (eps <= -1.6) {
		tmp = t_2;
	} else if (eps <= 1.4e-9) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else if ((eps <= 1.85e+176) || !(eps <= 1e+261)) {
		tmp = (t_3 + (1.0 + t_0)) / 2.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (-1.0 - eps)
	t_1 = (eps * x) - x
	t_2 = (math.exp(t_0) + (1.0 + t_1)) / 2.0
	t_3 = math.exp(t_1)
	tmp = 0
	if eps <= -5.1e+131:
		tmp = (1.0 + t_3) / 2.0
	elif eps <= -1.6:
		tmp = t_2
	elif eps <= 1.4e-9:
		tmp = (2.0 * math.exp(-x)) / 2.0
	elif (eps <= 1.85e+176) or not (eps <= 1e+261):
		tmp = (t_3 + (1.0 + t_0)) / 2.0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(-1.0 - eps))
	t_1 = Float64(Float64(eps * x) - x)
	t_2 = Float64(Float64(exp(t_0) + Float64(1.0 + t_1)) / 2.0)
	t_3 = exp(t_1)
	tmp = 0.0
	if (eps <= -5.1e+131)
		tmp = Float64(Float64(1.0 + t_3) / 2.0);
	elseif (eps <= -1.6)
		tmp = t_2;
	elseif (eps <= 1.4e-9)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	elseif ((eps <= 1.85e+176) || !(eps <= 1e+261))
		tmp = Float64(Float64(t_3 + Float64(1.0 + t_0)) / 2.0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (-1.0 - eps);
	t_1 = (eps * x) - x;
	t_2 = (exp(t_0) + (1.0 + t_1)) / 2.0;
	t_3 = exp(t_1);
	tmp = 0.0;
	if (eps <= -5.1e+131)
		tmp = (1.0 + t_3) / 2.0;
	elseif (eps <= -1.6)
		tmp = t_2;
	elseif (eps <= 1.4e-9)
		tmp = (2.0 * exp(-x)) / 2.0;
	elseif ((eps <= 1.85e+176) || ~((eps <= 1e+261)))
		tmp = (t_3 + (1.0 + t_0)) / 2.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[t$95$0], $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[Exp[t$95$1], $MachinePrecision]}, If[LessEqual[eps, -5.1e+131], N[(N[(1.0 + t$95$3), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, -1.6], t$95$2, If[LessEqual[eps, 1.4e-9], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[eps, 1.85e+176], N[Not[LessEqual[eps, 1e+261]], $MachinePrecision]], N[(N[(t$95$3 + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if eps < -5.1000000000000004e131

   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. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. mul-1-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. mul-1-neg 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. neg-mul-1 100.0%

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

      10. mul-1-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-rgt-in 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. exp-prod 100.0%

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

      4. *-lft-identity 100.0%

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

      5. metadata-eval 100.0%

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

      6. cancel-sign-sub-inv 100.0%

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

      7. exp-prod 100.0%

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

      8. mul-1-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. mul-1-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in eps around inf 100.0%

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

   11. Taylor expanded in x around 0 67.5%

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

   if -5.1000000000000004e131 < eps < -1.6000000000000001 or 1.8499999999999999e176 < eps < 9.9999999999999993e260

   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. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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 97.6%

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

      1. mul-1-neg 97.6%

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

      2. *-commutative 97.6%

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

      3. distribute-rgt-neg-in 97.6%

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

      4. mul-1-neg 97.6%

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

      5. sub-neg 97.6%

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

      6. mul-1-neg 97.6%

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

      7. distribute-lft-in 97.6%

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

      8. metadata-eval 97.6%

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

      9. neg-mul-1 97.6%

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

      10. mul-1-neg 97.6%

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

      11. remove-double-neg 97.6%

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

      12. +-commutative 97.6%

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

      13. distribute-rgt-in 97.6%

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

      14. neg-mul-1 97.6%

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

      15. unsub-neg 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. exp-prod 100.0%

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

      4. *-lft-identity 100.0%

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

      5. metadata-eval 100.0%

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

      6. cancel-sign-sub-inv 100.0%

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

      7. exp-prod 100.0%

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

      8. mul-1-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. mul-1-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in eps around inf 100.0%

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

   11. Taylor expanded in x around 0 81.6%

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

       1. *-commutative 81.6%

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

       2. sub-neg 81.6%

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

       3. metadata-eval 81.6%

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

       4. distribute-rgt-out 81.6%

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

       5. neg-mul-1 81.6%

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

       6. sub-neg 81.6%

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

       7. *-commutative 81.6%

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

   13. Simplified 81.6%

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

   if -1.6000000000000001 < eps < 1.39999999999999992e-9

   1. Initial program 25.9%

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

      1. div-sub 25.9%

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

      2. +-rgt-identity 25.9%

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

      3. div-sub 25.9%

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

   3. Simplified 25.9%

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

   4. Taylor expanded in eps around inf 93.7%

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

   5. Taylor expanded in eps around 0 93.7%

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

      1. cancel-sign-sub-inv 93.7%

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

      2. metadata-eval 93.7%

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

      3. distribute-rgt1-in 93.7%

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

      4. metadata-eval 93.7%

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

      5. neg-mul-1 93.7%

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

   7. Simplified 93.7%

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

   if 1.39999999999999992e-9 < eps < 1.8499999999999999e176 or 9.9999999999999993e260 < 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. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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 92.3%

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

      1. mul-1-neg 92.3%

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

      2. *-commutative 92.3%

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

      3. distribute-rgt-neg-in 92.3%

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

      4. mul-1-neg 92.3%

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

      5. sub-neg 92.3%

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

      6. mul-1-neg 92.3%

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

      7. distribute-lft-in 92.3%

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

      8. metadata-eval 92.3%

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

      9. neg-mul-1 92.3%

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

      10. mul-1-neg 92.3%

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

      11. remove-double-neg 92.3%

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

      12. +-commutative 92.3%

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

      13. distribute-rgt-in 92.3%

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

      14. neg-mul-1 92.3%

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

      15. unsub-neg 92.3%

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

   6. Simplified 92.3%

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

   7. Taylor expanded in eps around inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. exp-prod 99.3%

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

      4. *-lft-identity 99.3%

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

      5. metadata-eval 99.3%

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

      6. cancel-sign-sub-inv 99.3%

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

      7. exp-prod 99.3%

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

      8. mul-1-neg 99.3%

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

      9. *-commutative 99.3%

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

      10. sub-neg 99.3%

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

      11. mul-1-neg 99.3%

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

      12. remove-double-neg 99.3%

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

      13. +-commutative 99.3%

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

   9. Simplified 99.3%

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

   10. Taylor expanded in eps around inf 99.3%

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

   11. Taylor expanded in x around 0 80.7%

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

       1. associate-*r* 80.7%

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

       2. +-commutative 80.7%

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

       3. mul-1-neg 80.7%

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

       4. cancel-sign-sub-inv 80.7%

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

       5. *-commutative 80.7%

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

       6. cancel-sign-sub-inv 80.7%

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

       7. distribute-lft-neg-in 80.7%

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

       8. *-commutative 80.7%

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

       9. distribute-lft-neg-in 80.7%

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

       10. +-commutative 80.7%

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

       11. distribute-neg-in 80.7%

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

       12. metadata-eval 80.7%

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

       13. unsub-neg 80.7%

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

   13. Simplified 80.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.1 \cdot 10^{+131}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x - x}}{2}\\ \mathbf{elif}\;\varepsilon \leq -1.6:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 + \left(\varepsilon \cdot x - x\right)\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.85 \cdot 10^{+176} \lor \neg \left(\varepsilon \leq 10^{+261}\right):\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 + \left(\varepsilon \cdot x - x\right)\right)}{2}\\ \end{array} \]

Alternative 5: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\varepsilon \cdot x - x}\\ \mathbf{if}\;\varepsilon \leq -1.85 \cdot 10^{+58}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- (* eps x) x))))
   (if (<= eps -1.85e+58)
     (/ (+ 1.0 t_0) 2.0)
     (if (<= eps 1.4e-9)
       (/ (* 2.0 (exp (- x))) 2.0)
       (/ (+ t_0 (+ 1.0 (* x (- -1.0 eps)))) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = exp(((eps * x) - x));
	double tmp;
	if (eps <= -1.85e+58) {
		tmp = (1.0 + t_0) / 2.0;
	} else if (eps <= 1.4e-9) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (t_0 + (1.0 + (x * (-1.0 - eps)))) / 2.0;
	}
	return tmp;
}

Fortran

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(((eps * x) - x))
    if (eps <= (-1.85d+58)) then
        tmp = (1.0d0 + t_0) / 2.0d0
    else if (eps <= 1.4d-9) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (t_0 + (1.0d0 + (x * ((-1.0d0) - eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp(((eps * x) - x));
	double tmp;
	if (eps <= -1.85e+58) {
		tmp = (1.0 + t_0) / 2.0;
	} else if (eps <= 1.4e-9) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (t_0 + (1.0 + (x * (-1.0 - eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp(((eps * x) - x))
	tmp = 0
	if eps <= -1.85e+58:
		tmp = (1.0 + t_0) / 2.0
	elif eps <= 1.4e-9:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (t_0 + (1.0 + (x * (-1.0 - eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(Float64(eps * x) - x))
	tmp = 0.0
	if (eps <= -1.85e+58)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	elseif (eps <= 1.4e-9)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(t_0 + Float64(1.0 + Float64(x * Float64(-1.0 - eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp(((eps * x) - x));
	tmp = 0.0;
	if (eps <= -1.85e+58)
		tmp = (1.0 + t_0) / 2.0;
	elseif (eps <= 1.4e-9)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (t_0 + (1.0 + (x * (-1.0 - eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eps, -1.85e+58], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 1.4e-9], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[(1.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.8500000000000001e58

   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. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. mul-1-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. mul-1-neg 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. neg-mul-1 100.0%

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

      10. mul-1-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-rgt-in 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. exp-prod 100.0%

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

      4. *-lft-identity 100.0%

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

      5. metadata-eval 100.0%

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

      6. cancel-sign-sub-inv 100.0%

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

      7. exp-prod 100.0%

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

      8. mul-1-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. mul-1-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in eps around inf 100.0%

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

   11. Taylor expanded in x around 0 63.2%

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

   if -1.8500000000000001e58 < eps < 1.39999999999999992e-9

   1. Initial program 34.9%

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

      1. div-sub 34.9%

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

      2. +-rgt-identity 34.9%

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

      3. div-sub 34.9%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in eps around inf 94.4%

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

   5. Taylor expanded in eps around 0 91.0%

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

      1. cancel-sign-sub-inv 91.0%

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

      2. metadata-eval 91.0%

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

      3. distribute-rgt1-in 91.0%

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

      4. metadata-eval 91.0%

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

      5. neg-mul-1 91.0%

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

   7. Simplified 91.0%

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

   if 1.39999999999999992e-9 < 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. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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 94.3%

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

      1. mul-1-neg 94.3%

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

      2. *-commutative 94.3%

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

      3. distribute-rgt-neg-in 94.3%

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

      4. mul-1-neg 94.3%

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

      5. sub-neg 94.3%

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

      6. mul-1-neg 94.3%

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

      7. distribute-lft-in 94.3%

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

      8. metadata-eval 94.3%

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

      9. neg-mul-1 94.3%

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

      10. mul-1-neg 94.3%

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

      11. remove-double-neg 94.3%

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

      12. +-commutative 94.3%

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

      13. distribute-rgt-in 94.3%

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

      14. neg-mul-1 94.3%

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

      15. unsub-neg 94.3%

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

   6. Simplified 94.3%

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

   7. Taylor expanded in eps around inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. exp-prod 99.5%

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

      4. *-lft-identity 99.5%

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

      5. metadata-eval 99.5%

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

      6. cancel-sign-sub-inv 99.5%

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

      7. exp-prod 99.5%

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

      8. mul-1-neg 99.5%

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

      9. *-commutative 99.5%

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

      10. sub-neg 99.5%

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

      11. mul-1-neg 99.5%

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

      12. remove-double-neg 99.5%

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

      13. +-commutative 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in eps around inf 99.5%

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

   11. Taylor expanded in x around 0 67.1%

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

       1. associate-*r* 67.1%

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

       2. +-commutative 67.1%

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

       3. mul-1-neg 67.1%

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

       4. cancel-sign-sub-inv 67.1%

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

       5. *-commutative 67.1%

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

       6. cancel-sign-sub-inv 67.1%

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

       7. distribute-lft-neg-in 67.1%

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

       8. *-commutative 67.1%

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

       9. distribute-lft-neg-in 67.1%

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

       10. +-commutative 67.1%

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

       11. distribute-neg-in 67.1%

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

       12. metadata-eval 67.1%

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

       13. unsub-neg 67.1%

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

   13. Simplified 67.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.85 \cdot 10^{+58}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x - x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \end{array} \]

Alternative 6: 76.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-8} \lor \neg \left(x \leq 1900000000000\right):\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x - x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.12e-8) (not (<= x 1900000000000.0)))
   (/ (/ 2.0 (exp x)) 2.0)
   (/ (+ 1.0 (exp (- (* eps x) x))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.12e-8) || !(x <= 1900000000000.0)) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else {
		tmp = (1.0 + exp(((eps * x) - x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-1.12d-8)) .or. (.not. (x <= 1900000000000.0d0))) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else
        tmp = (1.0d0 + exp(((eps * x) - x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -1.12e-8) || !(x <= 1900000000000.0)) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp(((eps * x) - x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.12e-8) or not (x <= 1900000000000.0):
		tmp = (2.0 / math.exp(x)) / 2.0
	else:
		tmp = (1.0 + math.exp(((eps * x) - x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.12e-8) || !(x <= 1900000000000.0))
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(eps * x) - x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.12e-8) || ~((x <= 1900000000000.0)))
		tmp = (2.0 / exp(x)) / 2.0;
	else
		tmp = (1.0 + exp(((eps * x) - x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.12e-8], N[Not[LessEqual[x, 1900000000000.0]], $MachinePrecision]], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.11999999999999994e-8 or 1.9e12 < x

   1. Initial program 96.0%

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

      1. div-sub 96.0%

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

      2. +-rgt-identity 96.0%

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

      3. div-sub 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in eps around inf 95.8%

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

      1. mul-1-neg 95.8%

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

      2. *-commutative 95.8%

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

      3. distribute-rgt-neg-in 95.8%

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

      4. mul-1-neg 95.8%

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

      5. sub-neg 95.8%

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

      6. mul-1-neg 95.8%

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

      7. distribute-lft-in 95.8%

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

      8. metadata-eval 95.8%

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

      9. neg-mul-1 95.8%

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

      10. mul-1-neg 95.8%

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

      11. remove-double-neg 95.8%

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

      12. +-commutative 95.8%

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

      13. distribute-rgt-in 95.8%

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

      14. neg-mul-1 95.8%

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

      15. unsub-neg 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in eps around inf 95.6%

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

      1. mul-1-neg 95.6%

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

      2. +-commutative 95.6%

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

      3. exp-prod 95.6%

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

      4. *-lft-identity 95.6%

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

      5. metadata-eval 95.6%

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

      6. cancel-sign-sub-inv 95.6%

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

      7. exp-prod 95.6%

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

      8. mul-1-neg 95.6%

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

      9. *-commutative 95.6%

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

      10. sub-neg 95.6%

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

      11. mul-1-neg 95.6%

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

      12. remove-double-neg 95.6%

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

      13. +-commutative 95.6%

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

   9. Simplified 95.6%

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

   10. Taylor expanded in eps around 0 64.0%

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

       1. exp-neg 64.0%

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

       2. associate-*r/ 64.0%

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

       3. metadata-eval 64.0%

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

   12. Simplified 64.0%

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

   if -1.11999999999999994e-8 < x < 1.9e12

   1. Initial program 48.5%

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

      1. div-sub 48.5%

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

      2. +-rgt-identity 48.5%

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

      3. div-sub 48.5%

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

   3. Simplified 48.5%

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

   4. Taylor expanded in eps around inf 44.4%

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

      1. mul-1-neg 44.4%

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

      2. *-commutative 44.4%

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

      3. distribute-rgt-neg-in 44.4%

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

      4. mul-1-neg 44.4%

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

      5. sub-neg 44.4%

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

      6. mul-1-neg 44.4%

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

      7. distribute-lft-in 44.4%

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

      8. metadata-eval 44.4%

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

      9. neg-mul-1 44.4%

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

      10. mul-1-neg 44.4%

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

      11. remove-double-neg 44.4%

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

      12. +-commutative 44.4%

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

      13. distribute-rgt-in 44.4%

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

      14. neg-mul-1 44.4%

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

      15. unsub-neg 44.4%

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

   6. Simplified 44.4%

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

   7. Taylor expanded in eps around inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. +-commutative 97.9%

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

      3. exp-prod 97.9%

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

      4. *-lft-identity 97.9%

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

      5. metadata-eval 97.9%

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

      6. cancel-sign-sub-inv 97.9%

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

      7. exp-prod 97.9%

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

      8. mul-1-neg 97.9%

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

      9. *-commutative 97.9%

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

      10. sub-neg 97.9%

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

      11. mul-1-neg 97.9%

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

      12. remove-double-neg 97.9%

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

      13. +-commutative 97.9%

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

   9. Simplified 97.9%

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

   10. Taylor expanded in eps around inf 97.9%

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

   11. Taylor expanded in x around 0 86.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-8} \lor \neg \left(x \leq 1900000000000\right):\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x - x}}{2}\\ \end{array} \]

Alternative 7: 69.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot e^{-x}}{2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

   1. div-sub 66.7%

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

   2. +-rgt-identity 66.7%

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

   3. div-sub 66.7%

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

3. Simplified 66.7%

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

4. Taylor expanded in eps around inf 97.0%

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

5. Taylor expanded in eps around 0 70.8%

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

   1. cancel-sign-sub-inv 70.8%

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

   2. metadata-eval 70.8%

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

   3. distribute-rgt1-in 70.8%

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

   4. metadata-eval 70.8%

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

   5. neg-mul-1 70.8%

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

7. Simplified 70.8%

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

8. Final simplification 70.8%

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

Alternative 8: 69.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{e^{x}}}{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (/ 2.0 (exp x)) 2.0))

C

double code(double x, double eps) {
	return (2.0 / exp(x)) / 2.0;
}

Fortran

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

Java

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

Python

def code(x, eps):
	return (2.0 / math.exp(x)) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(2.0 / exp(x)) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (2.0 / exp(x)) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 66.7%

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

   1. div-sub 66.7%

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

   2. +-rgt-identity 66.7%

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

   3. div-sub 66.7%

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

3. Simplified 66.7%

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

4. Taylor expanded in eps around inf 64.1%

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

   1. mul-1-neg 64.1%

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

   2. *-commutative 64.1%

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

   3. distribute-rgt-neg-in 64.1%

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

   4. mul-1-neg 64.1%

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

   5. sub-neg 64.1%

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

   6. mul-1-neg 64.1%

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

   7. distribute-lft-in 64.1%

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

   8. metadata-eval 64.1%

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

   9. neg-mul-1 64.1%

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

   10. mul-1-neg 64.1%

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

   11. remove-double-neg 64.1%

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

   12. +-commutative 64.1%

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

   13. distribute-rgt-in 64.1%

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

   14. neg-mul-1 64.1%

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

   15. unsub-neg 64.1%

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

6. Simplified 64.1%

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

7. Taylor expanded in eps around inf 97.0%

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

   1. mul-1-neg 97.0%

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

   2. +-commutative 97.0%

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

   3. exp-prod 97.0%

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

   4. *-lft-identity 97.0%

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

   5. metadata-eval 97.0%

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

   6. cancel-sign-sub-inv 97.0%

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

   7. exp-prod 97.0%

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

   8. mul-1-neg 97.0%

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

   9. *-commutative 97.0%

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

   10. sub-neg 97.0%

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

   11. mul-1-neg 97.0%

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

   12. remove-double-neg 97.0%

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

   13. +-commutative 97.0%

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

9. Simplified 97.0%

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

10. Taylor expanded in eps around 0 70.8%

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

    1. exp-neg 70.8%

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

    2. associate-*r/ 70.8%

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

    3. metadata-eval 70.8%

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

12. Simplified 70.8%

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

13. Final simplification 70.8%

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

Alternative 9: 59.8% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot 0.5\right)}{\varepsilon} - \frac{x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 450:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.8e+216)
   (/ (- (/ (* x (* x 0.5)) eps) (/ x eps)) 2.0)
   (if (<= x 450.0) (/ (+ 2.0 (* eps x)) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.8e+216) {
		tmp = (((x * (x * 0.5)) / eps) - (x / eps)) / 2.0;
	} else if (x <= 450.0) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.8d+216)) then
        tmp = (((x * (x * 0.5d0)) / eps) - (x / eps)) / 2.0d0
    else if (x <= 450.0d0) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.8e+216) {
		tmp = (((x * (x * 0.5)) / eps) - (x / eps)) / 2.0;
	} else if (x <= 450.0) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.8e+216:
		tmp = (((x * (x * 0.5)) / eps) - (x / eps)) / 2.0
	elif x <= 450.0:
		tmp = (2.0 + (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.8e+216)
		tmp = Float64(Float64(Float64(Float64(x * Float64(x * 0.5)) / eps) - Float64(x / eps)) / 2.0);
	elseif (x <= 450.0)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.8e+216)
		tmp = (((x * (x * 0.5)) / eps) - (x / eps)) / 2.0;
	elseif (x <= 450.0)
		tmp = (2.0 + (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.8e+216], N[(N[(N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] - N[(x / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 450.0], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.79999999999999982e216

   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. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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 23.9%

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

   5. Taylor expanded in eps around 0 78.6%

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

      1. expm1-def 78.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)}}{\varepsilon}}{2} \]

      2. neg-mul-1 78.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right)}{\varepsilon}}{2} \]

   7. Simplified 78.6%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in x around 0 78.6%

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

      1. mul-1-neg 78.6%

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

      2. unsub-neg 78.6%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{{x}^{2}}{\varepsilon} - \frac{x}{\varepsilon}}}{2} \]

      3. associate-*r/ 78.6%

         \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot {x}^{2}}{\varepsilon}} - \frac{x}{\varepsilon}}{2} \]

      4. unpow2 78.6%

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

      5. associate-*r* 78.6%

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

   10. Simplified 78.6%

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

   if -2.79999999999999982e216 < x < 450

   1. Initial program 54.1%

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

      1. div-sub 54.1%

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

      2. +-rgt-identity 54.1%

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

      3. div-sub 54.1%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in x around 0 39.8%

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

   5. Taylor expanded in x around 0 43.5%

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

   6. Taylor expanded in eps around inf 67.7%

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

      1. associate-*r* 67.7%

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

      2. mul-1-neg 67.7%

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

   8. Simplified 67.7%

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

   if 450 < x

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in eps around 0 45.8%

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

      1. div-sub 45.8%

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

      2. rec-exp 45.8%

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

      3. neg-mul-1 45.8%

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

      4. +-inverses 45.8%

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

   6. Simplified 45.8%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot 0.5\right)}{\varepsilon} - \frac{x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 450:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 59.5% accurate, 20.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+216}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 490:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9.2e+216)
   (* (* eps x) -0.5)
   (if (<= x 490.0) (/ (+ 2.0 (* eps x)) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9.2e+216) {
		tmp = (eps * x) * -0.5;
	} else if (x <= 490.0) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-9.2d+216)) then
        tmp = (eps * x) * (-0.5d0)
    else if (x <= 490.0d0) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -9.2e+216) {
		tmp = (eps * x) * -0.5;
	} else if (x <= 490.0) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9.2e+216:
		tmp = (eps * x) * -0.5
	elif x <= 490.0:
		tmp = (2.0 + (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9.2e+216)
		tmp = Float64(Float64(eps * x) * -0.5);
	elseif (x <= 490.0)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9.2e+216)
		tmp = (eps * x) * -0.5;
	elseif (x <= 490.0)
		tmp = (2.0 + (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9.2e+216], N[(N[(eps * x), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 490.0], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -9.19999999999999983e216

   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. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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 72.2%

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

      1. sub-neg 72.2%

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

      2. sub-neg 72.2%

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

      3. metadata-eval 72.2%

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

      4. associate-*r* 72.2%

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

      5. distribute-lft-in 72.2%

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

      6. neg-mul-1 72.2%

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

      7. metadata-eval 72.2%

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

      8. *-commutative 72.2%

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

      9. +-commutative 72.2%

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

      10. metadata-eval 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in eps around inf 72.2%

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

      1. *-commutative 72.2%

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

   9. Simplified 72.2%

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

   if -9.19999999999999983e216 < x < 490

   1. Initial program 54.1%

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

      1. div-sub 54.1%

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

      2. +-rgt-identity 54.1%

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

      3. div-sub 54.1%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in x around 0 39.8%

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

   5. Taylor expanded in x around 0 43.5%

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

   6. Taylor expanded in eps around inf 67.7%

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

      1. associate-*r* 67.7%

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

      2. mul-1-neg 67.7%

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

   8. Simplified 67.7%

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

   if 490 < x

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in eps around 0 45.8%

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

      1. div-sub 45.8%

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

      2. rec-exp 45.8%

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

      3. neg-mul-1 45.8%

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

      4. +-inverses 45.8%

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

   6. Simplified 45.8%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+216}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 490:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 59.6% accurate, 32.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-8}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 450:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.12e-8) (* (* eps x) -0.5) (if (<= x 450.0) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.12e-8) {
		tmp = (eps * x) * -0.5;
	} else if (x <= 450.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.12d-8)) then
        tmp = (eps * x) * (-0.5d0)
    else if (x <= 450.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.12e-8) {
		tmp = (eps * x) * -0.5;
	} else if (x <= 450.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.12e-8:
		tmp = (eps * x) * -0.5
	elif x <= 450.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.12e-8)
		tmp = Float64(Float64(eps * x) * -0.5);
	elseif (x <= 450.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.12e-8)
		tmp = (eps * x) * -0.5;
	elseif (x <= 450.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.12e-8], N[(N[(eps * x), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 450.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.11999999999999994e-8

   1. Initial program 90.6%

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

      1. div-sub 90.6%

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

      2. +-rgt-identity 90.6%

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

      3. div-sub 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in x around 0 58.7%

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

      1. sub-neg 58.7%

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

      2. sub-neg 58.7%

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

      3. metadata-eval 58.7%

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

      4. associate-*r* 58.7%

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

      5. distribute-lft-in 58.7%

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

      6. neg-mul-1 58.7%

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

      7. metadata-eval 58.7%

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

      8. *-commutative 58.7%

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

      9. +-commutative 58.7%

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

      10. metadata-eval 58.7%

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

   6. Simplified 58.7%

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

   7. Taylor expanded in eps around inf 35.0%

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

      1. *-commutative 35.0%

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

   9. Simplified 35.0%

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

   if -1.11999999999999994e-8 < x < 450

   1. Initial program 48.5%

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

      1. div-sub 48.5%

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

      2. +-rgt-identity 48.5%

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

      3. div-sub 48.5%

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

   3. Simplified 48.5%

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

   4. Taylor expanded in x around 0 76.5%

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

   if 450 < x

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in eps around 0 45.8%

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

      1. div-sub 45.8%

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

      2. rec-exp 45.8%

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

      3. neg-mul-1 45.8%

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

      4. +-inverses 45.8%

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

   6. Simplified 45.8%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-8}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 450:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 56.5% accurate, 74.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 490:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 490.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 490.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 490.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 490.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 490.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 490.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 490.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 490.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 490

   1. Initial program 57.4%

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

      1. div-sub 57.4%

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

      2. +-rgt-identity 57.4%

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

      3. div-sub 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in x around 0 61.0%

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

   if 490 < x

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in eps around 0 45.8%

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

      1. div-sub 45.8%

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

      2. rec-exp 45.8%

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

      3. neg-mul-1 45.8%

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

      4. +-inverses 45.8%

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

   6. Simplified 45.8%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 490:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 15.6% accurate, 227.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 66.7%

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

3. Simplified 59.6%

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

4. Taylor expanded in eps around 0 12.2%

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

   1. div-sub 12.1%

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

   2. rec-exp 12.1%

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

   3. neg-mul-1 12.1%

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

   4. +-inverses 12.3%

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

6. Simplified 12.3%

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

7. Final simplification 12.3%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023192 
(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))