NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.4% → 99.1%

Time: 15.2s

Alternatives: 14

Speedup: 2.0×

• 37.1% 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: 72.4% 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: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 4e-81)
   (/ (/ (fma x 2.0 2.0) (exp x)) 2.0)
   (/ (+ (exp (* x (+ -1.0 eps))) (exp (* x (- eps)))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 4e-81) {
		tmp = (fma(x, 2.0, 2.0) / exp(x)) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps))) + exp((x * -eps))) / 2.0;
	}
	return tmp;
}

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (eps <= 4e-81)
		tmp = Float64(Float64(fma(x, 2.0, 2.0) / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps))) + exp(Float64(x * Float64(-eps)))) / 2.0);
	end
	return tmp
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 4e-81], N[(N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 3.9999999999999998e-81

   1. Initial program 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in eps around 0 68.1%

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

      1. associate--r+ 68.1%

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

      2. associate-*r* 68.1%

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

      3. mul-1-neg 68.1%

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

      4. cancel-sign-sub 68.1%

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

      5. distribute-rgt1-in 68.1%

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

      6. distribute-rgt-out-- 68.1%

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

      7. mul-1-neg 68.1%

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

      8. mul-1-neg 68.1%

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

   6. Simplified 68.1%

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

   7. Taylor expanded in x around inf 68.1%

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

      1. associate-*r* 68.1%

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

      2. distribute-rgt-out 68.1%

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

   9. Simplified 68.1%

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

       1. *-commutative 68.1%

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

       2. exp-neg 68.1%

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

       3. un-div-inv 68.1%

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

       4. +-commutative 68.1%

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

       5. *-commutative 68.1%

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

       6. fma-def 68.1%

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

   11. Applied egg-rr 68.1%

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

   if 3.9999999999999998e-81 < eps

   1. Initial program 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. mul-1-neg 100.0%

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

      7. mul-1-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. mul-1-neg 100.0%

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

      10. associate-*r* 100.0%

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

      11. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]

Alternative 2: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

3. Simplified 73.5%

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

4. Taylor expanded in eps around inf 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 92.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in eps around 0 70.3%

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

      1. associate--r+ 70.2%

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

      2. associate-*r* 70.2%

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

      3. mul-1-neg 70.2%

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

      4. cancel-sign-sub 70.2%

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

      5. distribute-rgt1-in 70.2%

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

      6. distribute-rgt-out-- 70.2%

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

      7. mul-1-neg 70.2%

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

      8. mul-1-neg 70.2%

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

   6. Simplified 70.2%

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

   7. Taylor expanded in x around inf 70.3%

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

      1. associate-*r* 70.3%

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

      2. distribute-rgt-out 70.2%

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

   9. Simplified 70.2%

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

       1. *-commutative 70.2%

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

       2. exp-neg 70.2%

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

       3. un-div-inv 70.3%

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

       4. +-commutative 70.3%

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

       5. *-commutative 70.3%

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

       6. fma-def 70.3%

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

   11. Applied egg-rr 70.3%

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

   if 1 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around 0 93.2%

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

   6. Taylor expanded in eps around inf 93.2%

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

      1. mul-1-neg 93.2%

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

      2. distribute-lft-neg-out 93.2%

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

      3. *-commutative 93.2%

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

   8. Simplified 93.2%

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

4. Final simplification 75.9%

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

Alternative 4: 73.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := -1 + \frac{-1}{\varepsilon}\\ t_1 := \left(-1 - \varepsilon\right) + t_0\\ \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{+139}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.15 \cdot 10^{+279}:\\ \;\;\;\;\frac{2 + \frac{x \cdot {\left(\varepsilon + \left(1 - \varepsilon\right) \cdot t_0\right)}^{2}}{\varepsilon \cdot 2}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.22 \cdot 10^{+294}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{\left(\frac{1}{\varepsilon} - \varepsilon\right) \cdot t_1}{t_1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 2 \cdot \left(x \cdot \varepsilon\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ -1.0 eps))) (t_1 (+ (- -1.0 eps) t_0)))
   (if (<= eps 1.0)
     (/ (/ (fma x 2.0 2.0) (exp x)) 2.0)
     (if (<= eps 3.2e+139)
       (/ (+ (exp (- x)) 1.0) 2.0)
       (if (<= eps 2.15e+279)
         (/
          (+ 2.0 (/ (* x (pow (+ eps (* (- 1.0 eps) t_0)) 2.0)) (* eps 2.0)))
          2.0)
         (if (<= eps 1.22e+294)
           (/ (+ 2.0 (* x (+ eps (/ (* (- (/ 1.0 eps) eps) t_1) t_1)))) 2.0)
           (/ (+ 2.0 (* 2.0 (* x eps))) 2.0)))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = -1.0 + (-1.0 / eps);
	double t_1 = (-1.0 - eps) + t_0;
	double tmp;
	if (eps <= 1.0) {
		tmp = (fma(x, 2.0, 2.0) / exp(x)) / 2.0;
	} else if (eps <= 3.2e+139) {
		tmp = (exp(-x) + 1.0) / 2.0;
	} else if (eps <= 2.15e+279) {
		tmp = (2.0 + ((x * pow((eps + ((1.0 - eps) * t_0)), 2.0)) / (eps * 2.0))) / 2.0;
	} else if (eps <= 1.22e+294) {
		tmp = (2.0 + (x * (eps + ((((1.0 / eps) - eps) * t_1) / t_1)))) / 2.0;
	} else {
		tmp = (2.0 + (2.0 * (x * eps))) / 2.0;
	}
	return tmp;
}

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(-1.0 + Float64(-1.0 / eps))
	t_1 = Float64(Float64(-1.0 - eps) + t_0)
	tmp = 0.0
	if (eps <= 1.0)
		tmp = Float64(Float64(fma(x, 2.0, 2.0) / exp(x)) / 2.0);
	elseif (eps <= 3.2e+139)
		tmp = Float64(Float64(exp(Float64(-x)) + 1.0) / 2.0);
	elseif (eps <= 2.15e+279)
		tmp = Float64(Float64(2.0 + Float64(Float64(x * (Float64(eps + Float64(Float64(1.0 - eps) * t_0)) ^ 2.0)) / Float64(eps * 2.0))) / 2.0);
	elseif (eps <= 1.22e+294)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(eps + Float64(Float64(Float64(Float64(1.0 / eps) - eps) * t_1) / t_1)))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(2.0 * Float64(x * eps))) / 2.0);
	end
	return tmp
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 - eps), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[eps, 1.0], N[(N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 3.2e+139], N[(N[(N[Exp[(-x)], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 2.15e+279], N[(N[(2.0 + N[(N[(x * N[Power[N[(eps + N[(N[(1.0 - eps), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(eps * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 1.22e+294], N[(N[(2.0 + N[(x * N[(eps + N[(N[(N[(N[(1.0 / eps), $MachinePrecision] - eps), $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(2.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if eps < 1

   1. Initial program 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in eps around 0 70.3%

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

      1. associate--r+ 70.2%

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

      2. associate-*r* 70.2%

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

      3. mul-1-neg 70.2%

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

      4. cancel-sign-sub 70.2%

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

      5. distribute-rgt1-in 70.2%

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

      6. distribute-rgt-out-- 70.2%

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

      7. mul-1-neg 70.2%

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

      8. mul-1-neg 70.2%

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

   6. Simplified 70.2%

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

   7. Taylor expanded in x around inf 70.3%

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

      1. associate-*r* 70.3%

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

      2. distribute-rgt-out 70.2%

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

   9. Simplified 70.2%

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

       1. *-commutative 70.2%

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

       2. exp-neg 70.2%

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

       3. un-div-inv 70.3%

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

       4. +-commutative 70.3%

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

       5. *-commutative 70.3%

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

       6. fma-def 70.3%

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

   11. Applied egg-rr 70.3%

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

   if 1 < eps < 3.2000000000000001e139

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 72.2%

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

      1. mul-1-neg 72.2%

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

   10. Simplified 72.2%

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

   if 3.2000000000000001e139 < eps < 2.1499999999999999e279

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 20.7%

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

      1. associate-*r* 20.7%

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

      2. neg-mul-1 20.7%

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

      3. distribute-neg-in 20.7%

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

      4. metadata-eval 20.7%

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

      5. sub-neg 20.7%

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

      6. mul-1-neg 20.7%

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

      7. +-commutative 20.7%

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

      8. sub-neg 20.7%

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

      9. metadata-eval 20.7%

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

      10. +-commutative 20.7%

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

   6. Simplified 20.7%

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

   7. Taylor expanded in eps around inf 20.7%

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

   9. Applied egg-rr 77.6%

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

   10. Taylor expanded in eps around inf 77.6%

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

   if 2.1499999999999999e279 < eps < 1.2199999999999999e294

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

      3. distribute-neg-in 3.1%

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

      4. metadata-eval 3.1%

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

      5. sub-neg 3.1%

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

      6. mul-1-neg 3.1%

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

      7. +-commutative 3.1%

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

      8. sub-neg 3.1%

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

      9. metadata-eval 3.1%

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

      10. +-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in eps around inf 3.1%

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

      1. distribute-lft-in 3.1%

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

      2. flip-+ 75.0%

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

      3. +-commutative 75.0%

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

      4. +-commutative 75.0%

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

      5. un-div-inv 75.0%

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

      6. +-commutative 75.0%

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

      7. un-div-inv 75.0%

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

      8. +-commutative 75.0%

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

      9. +-commutative 75.0%

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

      10. un-div-inv 75.0%

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

      11. +-commutative 75.0%

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

   9. Applied egg-rr 75.0%

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

   11. Simplified 75.0%

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

   if 1.2199999999999999e294 < eps

   1. Initial program 100.0%

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

   3. Simplified 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 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

      3. distribute-neg-in 3.1%

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

      4. metadata-eval 3.1%

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

      5. sub-neg 3.1%

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

      6. mul-1-neg 3.1%

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

      7. +-commutative 3.1%

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

      8. sub-neg 3.1%

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

      9. metadata-eval 3.1%

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

      10. +-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in eps around inf 3.1%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{+139}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.15 \cdot 10^{+279}:\\ \;\;\;\;\frac{2 + \frac{x \cdot {\left(\varepsilon + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}^{2}}{\varepsilon \cdot 2}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.22 \cdot 10^{+294}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{\left(\frac{1}{\varepsilon} - \varepsilon\right) \cdot \left(\left(-1 - \varepsilon\right) + \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{\left(-1 - \varepsilon\right) + \left(-1 + \frac{-1}{\varepsilon}\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 2 \cdot \left(x \cdot \varepsilon\right)}{2}\\ \end{array} \]

Alternative 5: 73.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := -1 + \frac{-1}{\varepsilon}\\ t_2 := \left(-1 - \varepsilon\right) + t_1\\ \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{t_0 \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.15 \cdot 10^{+279}:\\ \;\;\;\;\frac{2 + \frac{x \cdot {\left(\varepsilon + \left(1 - \varepsilon\right) \cdot t_1\right)}^{2}}{\varepsilon \cdot 2}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{+293}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{\left(\frac{1}{\varepsilon} - \varepsilon\right) \cdot t_2}{t_2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 2 \cdot \left(x \cdot \varepsilon\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x)))
        (t_1 (+ -1.0 (/ -1.0 eps)))
        (t_2 (+ (- -1.0 eps) t_1)))
   (if (<= eps 1.0)
     (/ (* t_0 (+ 2.0 (* x 2.0))) 2.0)
     (if (<= eps 3.4e+139)
       (/ (+ t_0 1.0) 2.0)
       (if (<= eps 2.15e+279)
         (/
          (+ 2.0 (/ (* x (pow (+ eps (* (- 1.0 eps) t_1)) 2.0)) (* eps 2.0)))
          2.0)
         (if (<= eps 2.9e+293)
           (/ (+ 2.0 (* x (+ eps (/ (* (- (/ 1.0 eps) eps) t_2) t_2)))) 2.0)
           (/ (+ 2.0 (* 2.0 (* x eps))) 2.0)))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = exp(-x);
	double t_1 = -1.0 + (-1.0 / eps);
	double t_2 = (-1.0 - eps) + t_1;
	double tmp;
	if (eps <= 1.0) {
		tmp = (t_0 * (2.0 + (x * 2.0))) / 2.0;
	} else if (eps <= 3.4e+139) {
		tmp = (t_0 + 1.0) / 2.0;
	} else if (eps <= 2.15e+279) {
		tmp = (2.0 + ((x * pow((eps + ((1.0 - eps) * t_1)), 2.0)) / (eps * 2.0))) / 2.0;
	} else if (eps <= 2.9e+293) {
		tmp = (2.0 + (x * (eps + ((((1.0 / eps) - eps) * t_2) / t_2)))) / 2.0;
	} else {
		tmp = (2.0 + (2.0 * (x * eps))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = exp(-x)
    t_1 = (-1.0d0) + ((-1.0d0) / eps)
    t_2 = ((-1.0d0) - eps) + t_1
    if (eps <= 1.0d0) then
        tmp = (t_0 * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else if (eps <= 3.4d+139) then
        tmp = (t_0 + 1.0d0) / 2.0d0
    else if (eps <= 2.15d+279) then
        tmp = (2.0d0 + ((x * ((eps + ((1.0d0 - eps) * t_1)) ** 2.0d0)) / (eps * 2.0d0))) / 2.0d0
    else if (eps <= 2.9d+293) then
        tmp = (2.0d0 + (x * (eps + ((((1.0d0 / eps) - eps) * t_2) / t_2)))) / 2.0d0
    else
        tmp = (2.0d0 + (2.0d0 * (x * eps))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = Math.exp(-x);
	double t_1 = -1.0 + (-1.0 / eps);
	double t_2 = (-1.0 - eps) + t_1;
	double tmp;
	if (eps <= 1.0) {
		tmp = (t_0 * (2.0 + (x * 2.0))) / 2.0;
	} else if (eps <= 3.4e+139) {
		tmp = (t_0 + 1.0) / 2.0;
	} else if (eps <= 2.15e+279) {
		tmp = (2.0 + ((x * Math.pow((eps + ((1.0 - eps) * t_1)), 2.0)) / (eps * 2.0))) / 2.0;
	} else if (eps <= 2.9e+293) {
		tmp = (2.0 + (x * (eps + ((((1.0 / eps) - eps) * t_2) / t_2)))) / 2.0;
	} else {
		tmp = (2.0 + (2.0 * (x * eps))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = math.exp(-x)
	t_1 = -1.0 + (-1.0 / eps)
	t_2 = (-1.0 - eps) + t_1
	tmp = 0
	if eps <= 1.0:
		tmp = (t_0 * (2.0 + (x * 2.0))) / 2.0
	elif eps <= 3.4e+139:
		tmp = (t_0 + 1.0) / 2.0
	elif eps <= 2.15e+279:
		tmp = (2.0 + ((x * math.pow((eps + ((1.0 - eps) * t_1)), 2.0)) / (eps * 2.0))) / 2.0
	elif eps <= 2.9e+293:
		tmp = (2.0 + (x * (eps + ((((1.0 / eps) - eps) * t_2) / t_2)))) / 2.0
	else:
		tmp = (2.0 + (2.0 * (x * eps))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = exp(Float64(-x))
	t_1 = Float64(-1.0 + Float64(-1.0 / eps))
	t_2 = Float64(Float64(-1.0 - eps) + t_1)
	tmp = 0.0
	if (eps <= 1.0)
		tmp = Float64(Float64(t_0 * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	elseif (eps <= 3.4e+139)
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	elseif (eps <= 2.15e+279)
		tmp = Float64(Float64(2.0 + Float64(Float64(x * (Float64(eps + Float64(Float64(1.0 - eps) * t_1)) ^ 2.0)) / Float64(eps * 2.0))) / 2.0);
	elseif (eps <= 2.9e+293)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(eps + Float64(Float64(Float64(Float64(1.0 / eps) - eps) * t_2) / t_2)))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(2.0 * Float64(x * eps))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = exp(-x);
	t_1 = -1.0 + (-1.0 / eps);
	t_2 = (-1.0 - eps) + t_1;
	tmp = 0.0;
	if (eps <= 1.0)
		tmp = (t_0 * (2.0 + (x * 2.0))) / 2.0;
	elseif (eps <= 3.4e+139)
		tmp = (t_0 + 1.0) / 2.0;
	elseif (eps <= 2.15e+279)
		tmp = (2.0 + ((x * ((eps + ((1.0 - eps) * t_1)) ^ 2.0)) / (eps * 2.0))) / 2.0;
	elseif (eps <= 2.9e+293)
		tmp = (2.0 + (x * (eps + ((((1.0 / eps) - eps) * t_2) / t_2)))) / 2.0;
	else
		tmp = (2.0 + (2.0 * (x * eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-1.0 - eps), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[eps, 1.0], N[(N[(t$95$0 * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 3.4e+139], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 2.15e+279], N[(N[(2.0 + N[(N[(x * N[Power[N[(eps + N[(N[(1.0 - eps), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(eps * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 2.9e+293], N[(N[(2.0 + N[(x * N[(eps + N[(N[(N[(N[(1.0 / eps), $MachinePrecision] - eps), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(2.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if eps < 1

   1. Initial program 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in eps around 0 70.3%

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

      1. associate--r+ 70.2%

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

      2. associate-*r* 70.2%

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

      3. mul-1-neg 70.2%

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

      4. cancel-sign-sub 70.2%

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

      5. distribute-rgt1-in 70.2%

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

      6. distribute-rgt-out-- 70.2%

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

      7. mul-1-neg 70.2%

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

      8. mul-1-neg 70.2%

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

   6. Simplified 70.2%

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

   7. Taylor expanded in x around inf 70.3%

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

      1. associate-*r* 70.3%

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

      2. distribute-rgt-out 70.2%

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

   9. Simplified 70.2%

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

   if 1 < eps < 3.4000000000000002e139

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 72.2%

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

      1. mul-1-neg 72.2%

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

   10. Simplified 72.2%

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

   if 3.4000000000000002e139 < eps < 2.1499999999999999e279

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 20.7%

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

      1. associate-*r* 20.7%

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

      2. neg-mul-1 20.7%

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

      3. distribute-neg-in 20.7%

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

      4. metadata-eval 20.7%

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

      5. sub-neg 20.7%

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

      6. mul-1-neg 20.7%

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

      7. +-commutative 20.7%

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

      8. sub-neg 20.7%

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

      9. metadata-eval 20.7%

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

      10. +-commutative 20.7%

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

   6. Simplified 20.7%

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

   7. Taylor expanded in eps around inf 20.7%

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

   9. Applied egg-rr 77.6%

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

   10. Taylor expanded in eps around inf 77.6%

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

   if 2.1499999999999999e279 < eps < 2.89999999999999999e293

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

      3. distribute-neg-in 3.1%

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

      4. metadata-eval 3.1%

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

      5. sub-neg 3.1%

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

      6. mul-1-neg 3.1%

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

      7. +-commutative 3.1%

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

      8. sub-neg 3.1%

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

      9. metadata-eval 3.1%

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

      10. +-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in eps around inf 3.1%

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

      1. distribute-lft-in 3.1%

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

      2. flip-+ 75.0%

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

      3. +-commutative 75.0%

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

      4. +-commutative 75.0%

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

      5. un-div-inv 75.0%

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

      6. +-commutative 75.0%

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

      7. un-div-inv 75.0%

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

      8. +-commutative 75.0%

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

      9. +-commutative 75.0%

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

      10. un-div-inv 75.0%

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

      11. +-commutative 75.0%

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

   9. Applied egg-rr 75.0%

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

   11. Simplified 75.0%

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

   if 2.89999999999999999e293 < eps

   1. Initial program 100.0%

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

   3. Simplified 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 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

      3. distribute-neg-in 3.1%

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

      4. metadata-eval 3.1%

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

      5. sub-neg 3.1%

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

      6. mul-1-neg 3.1%

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

      7. +-commutative 3.1%

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

      8. sub-neg 3.1%

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

      9. metadata-eval 3.1%

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

      10. +-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in eps around inf 3.1%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.15 \cdot 10^{+279}:\\ \;\;\;\;\frac{2 + \frac{x \cdot {\left(\varepsilon + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}^{2}}{\varepsilon \cdot 2}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{+293}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{\left(\frac{1}{\varepsilon} - \varepsilon\right) \cdot \left(\left(-1 - \varepsilon\right) + \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{\left(-1 - \varepsilon\right) + \left(-1 + \frac{-1}{\varepsilon}\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 2 \cdot \left(x \cdot \varepsilon\right)}{2}\\ \end{array} \]

Alternative 6: 71.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 11500:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+67} \lor \neg \left(x \leq 5 \cdot 10^{+90}\right) \land x \leq 3.6 \cdot 10^{+205}:\\ \;\;\;\;\frac{e^{x} \cdot \left(2 + \left(x + x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 11500.0)
   (/ (+ (exp (- x)) 1.0) 2.0)
   (if (or (<= x 4e+67) (and (not (<= x 5e+90)) (<= x 3.6e+205)))
     (/ (* (exp x) (+ 2.0 (+ x x))) 2.0)
     (/ (/ (* x 2.0) (exp x)) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 11500.0) {
		tmp = (exp(-x) + 1.0) / 2.0;
	} else if ((x <= 4e+67) || (!(x <= 5e+90) && (x <= 3.6e+205))) {
		tmp = (exp(x) * (2.0 + (x + x))) / 2.0;
	} else {
		tmp = ((x * 2.0) / exp(x)) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 11500.0d0) then
        tmp = (exp(-x) + 1.0d0) / 2.0d0
    else if ((x <= 4d+67) .or. (.not. (x <= 5d+90)) .and. (x <= 3.6d+205)) then
        tmp = (exp(x) * (2.0d0 + (x + x))) / 2.0d0
    else
        tmp = ((x * 2.0d0) / exp(x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 11500.0) {
		tmp = (Math.exp(-x) + 1.0) / 2.0;
	} else if ((x <= 4e+67) || (!(x <= 5e+90) && (x <= 3.6e+205))) {
		tmp = (Math.exp(x) * (2.0 + (x + x))) / 2.0;
	} else {
		tmp = ((x * 2.0) / Math.exp(x)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 11500.0:
		tmp = (math.exp(-x) + 1.0) / 2.0
	elif (x <= 4e+67) or (not (x <= 5e+90) and (x <= 3.6e+205)):
		tmp = (math.exp(x) * (2.0 + (x + x))) / 2.0
	else:
		tmp = ((x * 2.0) / math.exp(x)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 11500.0)
		tmp = Float64(Float64(exp(Float64(-x)) + 1.0) / 2.0);
	elseif ((x <= 4e+67) || (!(x <= 5e+90) && (x <= 3.6e+205)))
		tmp = Float64(Float64(exp(x) * Float64(2.0 + Float64(x + x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(x * 2.0) / exp(x)) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 11500.0)
		tmp = (exp(-x) + 1.0) / 2.0;
	elseif ((x <= 4e+67) || (~((x <= 5e+90)) && (x <= 3.6e+205)))
		tmp = (exp(x) * (2.0 + (x + x))) / 2.0;
	else
		tmp = ((x * 2.0) / exp(x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 11500.0], N[(N[(N[Exp[(-x)], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 4e+67], And[N[Not[LessEqual[x, 5e+90]], $MachinePrecision], LessEqual[x, 3.6e+205]]], N[(N[(N[Exp[x], $MachinePrecision] * N[(2.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 11500

   1. Initial program 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in eps around inf 99.5%

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

   5. Taylor expanded in eps around inf 99.0%

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

      1. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in eps around 0 80.9%

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

      1. mul-1-neg 80.9%

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

   10. Simplified 80.9%

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

   if 11500 < x < 3.99999999999999993e67 or 5.0000000000000004e90 < x < 3.60000000000000002e205

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 35.8%

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

      1. associate--r+ 35.8%

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

      2. associate-*r* 35.8%

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

      3. mul-1-neg 35.8%

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

      4. cancel-sign-sub 35.8%

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

      5. distribute-rgt1-in 35.8%

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

      6. distribute-rgt-out-- 35.8%

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

      7. mul-1-neg 35.8%

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

      8. mul-1-neg 35.8%

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

   6. Simplified 35.8%

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

      1. expm1-log1p-u 35.8%

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

      2. expm1-udef 35.8%

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

   8. Applied egg-rr 65.8%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(e^{x} \cdot \left(\left(x + 2\right) + x\right)\right)} - 1}}{2} \]
   9. Step-by-step derivation

      1. expm1-def 65.8%

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

      2. expm1-log1p 65.8%

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

      3. +-commutative 65.8%

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

      4. associate-+r+ 65.8%

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

   10. Simplified 65.8%

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

   if 3.99999999999999993e67 < x < 5.0000000000000004e90 or 3.60000000000000002e205 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 67.2%

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

      1. associate--r+ 67.2%

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

      2. associate-*r* 67.2%

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

      3. mul-1-neg 67.2%

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

      4. cancel-sign-sub 67.2%

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

      5. distribute-rgt1-in 67.2%

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

      6. distribute-rgt-out-- 67.2%

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

      7. mul-1-neg 67.2%

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

      8. mul-1-neg 67.2%

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

   6. Simplified 67.2%

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

   7. Taylor expanded in x around inf 67.2%

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

      1. *-commutative 67.2%

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

      2. associate-*l* 67.2%

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

      3. *-commutative 67.2%

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

   9. Simplified 67.2%

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

       1. associate-*r* 67.2%

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

       2. *-commutative 67.2%

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

       3. exp-neg 67.2%

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

       4. un-div-inv 67.2%

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

       5. *-commutative 67.2%

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

   11. Applied egg-rr 67.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 11500:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+67} \lor \neg \left(x \leq 5 \cdot 10^{+90}\right) \land x \leq 3.6 \cdot 10^{+205}:\\ \;\;\;\;\frac{e^{x} \cdot \left(2 + \left(x + x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{e^{x}}}{2}\\ \end{array} \]

Alternative 7: 71.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 11500:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+65} \lor \neg \left(x \leq 5 \cdot 10^{+90}\right) \land x \leq 5 \cdot 10^{+204}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 11500.0)
   (/ (+ (exp (- x)) 1.0) 2.0)
   (if (or (<= x 1.8e+65) (and (not (<= x 5e+90)) (<= x 5e+204)))
     (/ (* (exp x) (* x 2.0)) 2.0)
     (/ (/ (* x 2.0) (exp x)) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 11500.0) {
		tmp = (exp(-x) + 1.0) / 2.0;
	} else if ((x <= 1.8e+65) || (!(x <= 5e+90) && (x <= 5e+204))) {
		tmp = (exp(x) * (x * 2.0)) / 2.0;
	} else {
		tmp = ((x * 2.0) / exp(x)) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 11500.0d0) then
        tmp = (exp(-x) + 1.0d0) / 2.0d0
    else if ((x <= 1.8d+65) .or. (.not. (x <= 5d+90)) .and. (x <= 5d+204)) then
        tmp = (exp(x) * (x * 2.0d0)) / 2.0d0
    else
        tmp = ((x * 2.0d0) / exp(x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 11500.0) {
		tmp = (Math.exp(-x) + 1.0) / 2.0;
	} else if ((x <= 1.8e+65) || (!(x <= 5e+90) && (x <= 5e+204))) {
		tmp = (Math.exp(x) * (x * 2.0)) / 2.0;
	} else {
		tmp = ((x * 2.0) / Math.exp(x)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 11500.0:
		tmp = (math.exp(-x) + 1.0) / 2.0
	elif (x <= 1.8e+65) or (not (x <= 5e+90) and (x <= 5e+204)):
		tmp = (math.exp(x) * (x * 2.0)) / 2.0
	else:
		tmp = ((x * 2.0) / math.exp(x)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 11500.0)
		tmp = Float64(Float64(exp(Float64(-x)) + 1.0) / 2.0);
	elseif ((x <= 1.8e+65) || (!(x <= 5e+90) && (x <= 5e+204)))
		tmp = Float64(Float64(exp(x) * Float64(x * 2.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(x * 2.0) / exp(x)) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 11500.0)
		tmp = (exp(-x) + 1.0) / 2.0;
	elseif ((x <= 1.8e+65) || (~((x <= 5e+90)) && (x <= 5e+204)))
		tmp = (exp(x) * (x * 2.0)) / 2.0;
	else
		tmp = ((x * 2.0) / exp(x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 11500.0], N[(N[(N[Exp[(-x)], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.8e+65], And[N[Not[LessEqual[x, 5e+90]], $MachinePrecision], LessEqual[x, 5e+204]]], N[(N[(N[Exp[x], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 11500

   1. Initial program 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in eps around inf 99.5%

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

   5. Taylor expanded in eps around inf 99.0%

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

      1. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in eps around 0 80.9%

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

      1. mul-1-neg 80.9%

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

   10. Simplified 80.9%

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

   if 11500 < x < 1.79999999999999989e65 or 5.0000000000000004e90 < x < 5.00000000000000008e204

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 35.8%

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

      1. associate--r+ 35.8%

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

      2. associate-*r* 35.8%

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

      3. mul-1-neg 35.8%

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

      4. cancel-sign-sub 35.8%

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

      5. distribute-rgt1-in 35.8%

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

      6. distribute-rgt-out-- 35.8%

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

      7. mul-1-neg 35.8%

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

      8. mul-1-neg 35.8%

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

   6. Simplified 35.8%

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

   7. Taylor expanded in x around inf 35.8%

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

      1. *-commutative 35.8%

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

      2. associate-*l* 35.8%

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

      3. *-commutative 35.8%

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

   9. Simplified 35.8%

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

       1. expm1-log1p-u 35.8%

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

       2. expm1-udef 35.8%

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

       3. associate-*r* 35.8%

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

       4. *-commutative 35.8%

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

       5. associate-*l* 35.8%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 65.8%

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

       8. sqr-neg 65.8%

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

       9. sqrt-unprod 65.8%

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

       10. add-sqr-sqrt 65.8%

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

   11. Applied egg-rr 65.8%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot e^{x}\right)\right)} - 1}}{2} \]
   12. Step-by-step derivation

       1. expm1-def 65.8%

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

       2. expm1-log1p 65.8%

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

       3. associate-*r* 65.8%

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

       4. *-commutative 65.8%

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

   13. Simplified 65.8%

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

   if 1.79999999999999989e65 < x < 5.0000000000000004e90 or 5.00000000000000008e204 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 67.2%

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

      1. associate--r+ 67.2%

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

      2. associate-*r* 67.2%

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

      3. mul-1-neg 67.2%

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

      4. cancel-sign-sub 67.2%

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

      5. distribute-rgt1-in 67.2%

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

      6. distribute-rgt-out-- 67.2%

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

      7. mul-1-neg 67.2%

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

      8. mul-1-neg 67.2%

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

   6. Simplified 67.2%

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

   7. Taylor expanded in x around inf 67.2%

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

      1. *-commutative 67.2%

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

      2. associate-*l* 67.2%

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

      3. *-commutative 67.2%

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

   9. Simplified 67.2%

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

       1. associate-*r* 67.2%

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

       2. *-commutative 67.2%

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

       3. exp-neg 67.2%

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

       4. un-div-inv 67.2%

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

       5. *-commutative 67.2%

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

   11. Applied egg-rr 67.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 11500:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+65} \lor \neg \left(x \leq 5 \cdot 10^{+90}\right) \land x \leq 5 \cdot 10^{+204}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{e^{x}}}{2}\\ \end{array} \]

Alternative 8: 73.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \left(-1 - \varepsilon\right) + \left(-1 + \frac{-1}{\varepsilon}\right)\\ \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{e^{x} \cdot \left(2 + \left(x + x\right)\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.1 \cdot 10^{+295}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{\left(\frac{1}{\varepsilon} - \varepsilon\right) \cdot t_0}{t_0}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 2 \cdot \left(x \cdot \varepsilon\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (- -1.0 eps) (+ -1.0 (/ -1.0 eps)))))
   (if (<= eps 1.0)
     (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
     (if (<= eps 5e+277)
       (/ (* (exp x) (+ 2.0 (+ x x))) 2.0)
       (if (<= eps 1.1e+295)
         (/ (+ 2.0 (* x (+ eps (/ (* (- (/ 1.0 eps) eps) t_0) t_0)))) 2.0)
         (/ (+ 2.0 (* 2.0 (* x eps))) 2.0))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (-1.0 - eps) + (-1.0 + (-1.0 / eps));
	double tmp;
	if (eps <= 1.0) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else if (eps <= 5e+277) {
		tmp = (exp(x) * (2.0 + (x + x))) / 2.0;
	} else if (eps <= 1.1e+295) {
		tmp = (2.0 + (x * (eps + ((((1.0 / eps) - eps) * t_0) / t_0)))) / 2.0;
	} else {
		tmp = (2.0 + (2.0 * (x * eps))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - eps) + ((-1.0d0) + ((-1.0d0) / eps))
    if (eps <= 1.0d0) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else if (eps <= 5d+277) then
        tmp = (exp(x) * (2.0d0 + (x + x))) / 2.0d0
    else if (eps <= 1.1d+295) then
        tmp = (2.0d0 + (x * (eps + ((((1.0d0 / eps) - eps) * t_0) / t_0)))) / 2.0d0
    else
        tmp = (2.0d0 + (2.0d0 * (x * eps))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = (-1.0 - eps) + (-1.0 + (-1.0 / eps));
	double tmp;
	if (eps <= 1.0) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else if (eps <= 5e+277) {
		tmp = (Math.exp(x) * (2.0 + (x + x))) / 2.0;
	} else if (eps <= 1.1e+295) {
		tmp = (2.0 + (x * (eps + ((((1.0 / eps) - eps) * t_0) / t_0)))) / 2.0;
	} else {
		tmp = (2.0 + (2.0 * (x * eps))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = (-1.0 - eps) + (-1.0 + (-1.0 / eps))
	tmp = 0
	if eps <= 1.0:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	elif eps <= 5e+277:
		tmp = (math.exp(x) * (2.0 + (x + x))) / 2.0
	elif eps <= 1.1e+295:
		tmp = (2.0 + (x * (eps + ((((1.0 / eps) - eps) * t_0) / t_0)))) / 2.0
	else:
		tmp = (2.0 + (2.0 * (x * eps))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(-1.0 - eps) + Float64(-1.0 + Float64(-1.0 / eps)))
	tmp = 0.0
	if (eps <= 1.0)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	elseif (eps <= 5e+277)
		tmp = Float64(Float64(exp(x) * Float64(2.0 + Float64(x + x))) / 2.0);
	elseif (eps <= 1.1e+295)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(eps + Float64(Float64(Float64(Float64(1.0 / eps) - eps) * t_0) / t_0)))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(2.0 * Float64(x * eps))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = (-1.0 - eps) + (-1.0 + (-1.0 / eps));
	tmp = 0.0;
	if (eps <= 1.0)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	elseif (eps <= 5e+277)
		tmp = (exp(x) * (2.0 + (x + x))) / 2.0;
	elseif (eps <= 1.1e+295)
		tmp = (2.0 + (x * (eps + ((((1.0 / eps) - eps) * t_0) / t_0)))) / 2.0;
	else
		tmp = (2.0 + (2.0 * (x * eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(N[(-1.0 - eps), $MachinePrecision] + N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, 1.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 5e+277], N[(N[(N[Exp[x], $MachinePrecision] * N[(2.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 1.1e+295], N[(N[(2.0 + N[(x * N[(eps + N[(N[(N[(N[(1.0 / eps), $MachinePrecision] - eps), $MachinePrecision] * t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(2.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 1

   1. Initial program 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in eps around 0 70.3%

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

      1. associate--r+ 70.2%

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

      2. associate-*r* 70.2%

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

      3. mul-1-neg 70.2%

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

      4. cancel-sign-sub 70.2%

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

      5. distribute-rgt1-in 70.2%

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

      6. distribute-rgt-out-- 70.2%

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

      7. mul-1-neg 70.2%

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

      8. mul-1-neg 70.2%

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

   6. Simplified 70.2%

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

   7. Taylor expanded in x around inf 70.3%

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

      1. associate-*r* 70.3%

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

      2. distribute-rgt-out 70.2%

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

   9. Simplified 70.2%

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

   if 1 < eps < 4.99999999999999982e277

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 37.8%

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

      1. associate--r+ 37.8%

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

      2. associate-*r* 37.8%

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

      3. mul-1-neg 37.8%

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

      4. cancel-sign-sub 37.8%

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

      5. distribute-rgt1-in 37.8%

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

      6. distribute-rgt-out-- 37.8%

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

      7. mul-1-neg 37.8%

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

      8. mul-1-neg 37.8%

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

   6. Simplified 37.8%

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

      1. expm1-log1p-u 36.9%

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

      2. expm1-udef 36.9%

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

   8. Applied egg-rr 64.8%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(e^{x} \cdot \left(\left(x + 2\right) + x\right)\right)} - 1}}{2} \]
   9. Step-by-step derivation

      1. expm1-def 64.8%

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

      2. expm1-log1p 65.7%

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

      3. +-commutative 65.7%

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

      4. associate-+r+ 65.7%

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

   10. Simplified 65.7%

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

   if 4.99999999999999982e277 < eps < 1.0999999999999999e295

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

      3. distribute-neg-in 3.1%

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

      4. metadata-eval 3.1%

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

      5. sub-neg 3.1%

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

      6. mul-1-neg 3.1%

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

      7. +-commutative 3.1%

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

      8. sub-neg 3.1%

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

      9. metadata-eval 3.1%

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

      10. +-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in eps around inf 3.1%

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

      1. distribute-lft-in 3.1%

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

      2. flip-+ 75.0%

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

      3. +-commutative 75.0%

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

      4. +-commutative 75.0%

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

      5. un-div-inv 75.0%

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

      6. +-commutative 75.0%

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

      7. un-div-inv 75.0%

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

      8. +-commutative 75.0%

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

      9. +-commutative 75.0%

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

      10. un-div-inv 75.0%

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

      11. +-commutative 75.0%

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

   9. Applied egg-rr 75.0%

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

   11. Simplified 75.0%

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

   if 1.0999999999999999e295 < eps

   1. Initial program 100.0%

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

   3. Simplified 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 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

      3. distribute-neg-in 3.1%

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

      4. metadata-eval 3.1%

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

      5. sub-neg 3.1%

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

      6. mul-1-neg 3.1%

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

      7. +-commutative 3.1%

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

      8. sub-neg 3.1%

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

      9. metadata-eval 3.1%

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

      10. +-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in eps around inf 3.1%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{e^{x} \cdot \left(2 + \left(x + x\right)\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.1 \cdot 10^{+295}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{\left(\frac{1}{\varepsilon} - \varepsilon\right) \cdot \left(\left(-1 - \varepsilon\right) + \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{\left(-1 - \varepsilon\right) + \left(-1 + \frac{-1}{\varepsilon}\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 2 \cdot \left(x \cdot \varepsilon\right)}{2}\\ \end{array} \]

Alternative 9: 70.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 11500:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x \cdot 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 11500.0) (/ (+ (exp (- x)) 1.0) 2.0) (/ (* (exp x) (* x 2.0)) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 11500.0) {
		tmp = (exp(-x) + 1.0) / 2.0;
	} else {
		tmp = (exp(x) * (x * 2.0)) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 11500.0d0) then
        tmp = (exp(-x) + 1.0d0) / 2.0d0
    else
        tmp = (exp(x) * (x * 2.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 11500.0) {
		tmp = (Math.exp(-x) + 1.0) / 2.0;
	} else {
		tmp = (Math.exp(x) * (x * 2.0)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 11500.0:
		tmp = (math.exp(-x) + 1.0) / 2.0
	else:
		tmp = (math.exp(x) * (x * 2.0)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 11500.0)
		tmp = Float64(Float64(exp(Float64(-x)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(exp(x) * Float64(x * 2.0)) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 11500.0)
		tmp = (exp(-x) + 1.0) / 2.0;
	else
		tmp = (exp(x) * (x * 2.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 11500.0], N[(N[(N[Exp[(-x)], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[x], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 11500

   1. Initial program 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in eps around inf 99.5%

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

   5. Taylor expanded in eps around inf 99.0%

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

      1. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in eps around 0 80.9%

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

      1. mul-1-neg 80.9%

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

   10. Simplified 80.9%

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

   if 11500 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 49.6%

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

      1. associate--r+ 49.6%

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

      2. associate-*r* 49.6%

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

      3. mul-1-neg 49.6%

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

      4. cancel-sign-sub 49.6%

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

      5. distribute-rgt1-in 49.6%

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

      6. distribute-rgt-out-- 49.6%

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

      7. mul-1-neg 49.6%

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

      8. mul-1-neg 49.6%

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

   6. Simplified 49.6%

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

   7. Taylor expanded in x around inf 49.6%

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

      1. *-commutative 49.6%

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

      2. associate-*l* 49.6%

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

      3. *-commutative 49.6%

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

   9. Simplified 49.6%

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

       1. expm1-log1p-u 49.6%

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

       2. expm1-udef 49.6%

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

       3. associate-*r* 49.6%

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

       4. *-commutative 49.6%

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

       5. associate-*l* 49.6%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 52.0%

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

       8. sqr-neg 52.0%

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

       9. sqrt-unprod 52.0%

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

       10. add-sqr-sqrt 52.0%

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

   11. Applied egg-rr 52.0%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot e^{x}\right)\right)} - 1}}{2} \]
   12. Step-by-step derivation

       1. expm1-def 52.0%

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

       2. expm1-log1p 52.0%

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

       3. associate-*r* 52.0%

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

       4. *-commutative 52.0%

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

   13. Simplified 52.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 11500:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x \cdot 2\right)}{2}\\ \end{array} \]

Alternative 10: 64.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 7.9 \cdot 10^{-140}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(-1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 7.9e-140)
   (/ (+ (exp (- x)) 1.0) 2.0)
   (/ (+ 2.0 (* x (+ (/ 1.0 eps) (* (+ -1.0 eps) (+ (/ 1.0 eps) 1.0))))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 7.9e-140) {
		tmp = (exp(-x) + 1.0) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps) + ((-1.0 + eps) * ((1.0 / eps) + 1.0))))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 7.9d-140) then
        tmp = (exp(-x) + 1.0d0) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((1.0d0 / eps) + (((-1.0d0) + eps) * ((1.0d0 / eps) + 1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 7.9e-140) {
		tmp = (Math.exp(-x) + 1.0) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps) + ((-1.0 + eps) * ((1.0 / eps) + 1.0))))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 7.9e-140:
		tmp = (math.exp(-x) + 1.0) / 2.0
	else:
		tmp = (2.0 + (x * ((1.0 / eps) + ((-1.0 + eps) * ((1.0 / eps) + 1.0))))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 7.9e-140)
		tmp = Float64(Float64(exp(Float64(-x)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(-1.0 + eps) * Float64(Float64(1.0 / eps) + 1.0))))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 7.9e-140)
		tmp = (exp(-x) + 1.0) / 2.0;
	else
		tmp = (2.0 + (x * ((1.0 / eps) + ((-1.0 + eps) * ((1.0 / eps) + 1.0))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 7.9e-140], N[(N[(N[Exp[(-x)], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(-1.0 + eps), $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.89999999999999966e-140

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in eps around inf 99.3%

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

   5. Taylor expanded in eps around inf 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in eps around 0 86.8%

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

      1. mul-1-neg 86.8%

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

   10. Simplified 86.8%

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

   if 7.89999999999999966e-140 < x

   1. Initial program 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in x around 0 25.6%

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

      1. associate-*r* 25.6%

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

      2. neg-mul-1 25.6%

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

      3. distribute-neg-in 25.6%

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

      4. metadata-eval 25.6%

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

      5. sub-neg 25.6%

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

      6. mul-1-neg 25.6%

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

      7. +-commutative 25.6%

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

      8. sub-neg 25.6%

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

      9. metadata-eval 25.6%

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

      10. +-commutative 25.6%

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

   6. Simplified 25.6%

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

   7. Taylor expanded in eps around 0 32.6%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.9 \cdot 10^{-140}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(-1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)\right)}{2}\\ \end{array} \]

Alternative 11: 58.6% accurate, 10.8× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-279}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(-1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-279)
   (/ (- 2.0 (* x eps)) 2.0)
   (/ (+ 2.0 (* x (+ (/ 1.0 eps) (* (+ -1.0 eps) (+ (/ 1.0 eps) 1.0))))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -2e-279) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps) + ((-1.0 + eps) * ((1.0 / eps) + 1.0))))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2d-279)) then
        tmp = (2.0d0 - (x * eps)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((1.0d0 / eps) + (((-1.0d0) + eps) * ((1.0d0 / eps) + 1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -2e-279) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps) + ((-1.0 + eps) * ((1.0 / eps) + 1.0))))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -2e-279:
		tmp = (2.0 - (x * eps)) / 2.0
	else:
		tmp = (2.0 + (x * ((1.0 / eps) + ((-1.0 + eps) * ((1.0 / eps) + 1.0))))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -2e-279)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(-1.0 + eps) * Float64(Float64(1.0 / eps) + 1.0))))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2e-279)
		tmp = (2.0 - (x * eps)) / 2.0;
	else
		tmp = (2.0 + (x * ((1.0 / eps) + ((-1.0 + eps) * ((1.0 / eps) + 1.0))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -2e-279], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(-1.0 + eps), $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000011e-279

   1. Initial program 65.3%

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

   3. Simplified 65.3%

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

   4. Taylor expanded in x around 0 54.3%

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

      1. associate-*r* 54.3%

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

      2. neg-mul-1 54.3%

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

      3. distribute-neg-in 54.3%

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

      4. metadata-eval 54.3%

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

      5. sub-neg 54.3%

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

      6. mul-1-neg 54.3%

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

      7. +-commutative 54.3%

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

      8. sub-neg 54.3%

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

      9. metadata-eval 54.3%

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

      10. +-commutative 54.3%

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

   6. Simplified 54.3%

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

   7. Taylor expanded in eps around 0 57.1%

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

   8. Taylor expanded in eps around 0 57.1%

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

      1. mul-1-neg 57.1%

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

   10. Simplified 57.1%

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

   if -2.00000000000000011e-279 < x

   1. Initial program 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in x around 0 42.4%

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

      1. associate-*r* 42.4%

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

      2. neg-mul-1 42.4%

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

      3. distribute-neg-in 42.4%

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

      4. metadata-eval 42.4%

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

      5. sub-neg 42.4%

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

      6. mul-1-neg 42.4%

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

      7. +-commutative 42.4%

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

      8. sub-neg 42.4%

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

      9. metadata-eval 42.4%

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

      10. +-commutative 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in eps around 0 47.3%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-279}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(-1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)\right)}{2}\\ \end{array} \]

Alternative 12: 58.6% accurate, 20.5× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-279}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 2 \cdot \left(x \cdot \varepsilon\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-279)
   (/ (- 2.0 (* x eps)) 2.0)
   (/ (+ 2.0 (* 2.0 (* x eps))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -2e-279) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = (2.0 + (2.0 * (x * eps))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2d-279)) then
        tmp = (2.0d0 - (x * eps)) / 2.0d0
    else
        tmp = (2.0d0 + (2.0d0 * (x * eps))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -2e-279) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = (2.0 + (2.0 * (x * eps))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -2e-279:
		tmp = (2.0 - (x * eps)) / 2.0
	else:
		tmp = (2.0 + (2.0 * (x * eps))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -2e-279)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(2.0 * Float64(x * eps))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2e-279)
		tmp = (2.0 - (x * eps)) / 2.0;
	else
		tmp = (2.0 + (2.0 * (x * eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -2e-279], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(2.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000011e-279

   1. Initial program 65.3%

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

   3. Simplified 65.3%

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

   4. Taylor expanded in x around 0 54.3%

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

      1. associate-*r* 54.3%

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

      2. neg-mul-1 54.3%

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

      3. distribute-neg-in 54.3%

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

      4. metadata-eval 54.3%

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

      5. sub-neg 54.3%

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

      6. mul-1-neg 54.3%

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

      7. +-commutative 54.3%

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

      8. sub-neg 54.3%

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

      9. metadata-eval 54.3%

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

      10. +-commutative 54.3%

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

   6. Simplified 54.3%

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

   7. Taylor expanded in eps around 0 57.1%

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

   8. Taylor expanded in eps around 0 57.1%

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

      1. mul-1-neg 57.1%

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

   10. Simplified 57.1%

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

   if -2.00000000000000011e-279 < x

   1. Initial program 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in x around 0 42.4%

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

      1. associate-*r* 42.4%

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

      2. neg-mul-1 42.4%

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

      3. distribute-neg-in 42.4%

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

      4. metadata-eval 42.4%

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

      5. sub-neg 42.4%

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

      6. mul-1-neg 42.4%

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

      7. +-commutative 42.4%

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

      8. sub-neg 42.4%

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

      9. metadata-eval 42.4%

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

      10. +-commutative 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in eps around inf 27.9%

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

   9. Applied egg-rr 31.8%

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

   10. Taylor expanded in eps around inf 47.2%

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

       1. *-commutative 47.2%

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

   12. Simplified 47.2%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-279}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + 2 \cdot \left(x \cdot \varepsilon\right)}{2}\\ \end{array} \]

Alternative 13: 51.1% accurate, 32.4× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \frac{2 - x \cdot \varepsilon}{2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

3. Simplified 73.5%

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

4. Taylor expanded in x around 0 46.2%

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

   1. associate-*r* 46.2%

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

   2. neg-mul-1 46.2%

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

   3. distribute-neg-in 46.2%

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

   4. metadata-eval 46.2%

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

   5. sub-neg 46.2%

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

   6. mul-1-neg 46.2%

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

   7. +-commutative 46.2%

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

   8. sub-neg 46.2%

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

   9. metadata-eval 46.2%

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

   10. +-commutative 46.2%

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

6. Simplified 46.2%

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

7. Taylor expanded in eps around 0 52.0%

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

8. Taylor expanded in eps around 0 52.0%

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

   1. mul-1-neg 52.0%

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

10. Simplified 52.0%

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

11. Final simplification 52.0%

    \[\leadsto \frac{2 - x \cdot \varepsilon}{2} \]

Alternative 14: 45.0% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ 1 \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

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

Python

eps = abs(eps)
def code(x, eps):
	return 1.0

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := 1.0

Derivation

1. Initial program 73.5%

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

3. Simplified 73.5%

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

4. Taylor expanded in eps around 0 61.4%

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

   1. associate--r+ 61.4%

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

   2. associate-*r* 61.4%

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

   3. mul-1-neg 61.4%

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

   4. cancel-sign-sub 61.4%

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

   5. distribute-rgt1-in 61.4%

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

   6. distribute-rgt-out-- 61.4%

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

   7. mul-1-neg 61.4%

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

   8. mul-1-neg 61.4%

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

6. Simplified 61.4%

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

7. Taylor expanded in x around 0 45.2%

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

   1. mul-1-neg 45.2%

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

   2. unsub-neg 45.2%

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

9. Simplified 45.2%

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

10. Taylor expanded in x around 0 46.2%

    \[\leadsto \color{blue}{1} \]

11. Final simplification 46.2%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023306 
(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))