NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.3% → 99.0%

Time: 18.7s

Alternatives: 14

Speedup: 1.8×

• 38.2% 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: 72.3% 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: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2e-89)
   (/ (+ (/ (+ x 1.0) (exp x)) (* (+ x 1.0) (exp (- x)))) 2.0)
   (/ (+ (exp (* x (+ -1.0 eps_m))) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2e-89) {
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) + exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 2d-89) then
        tmp = (((x + 1.0d0) / exp(x)) + ((x + 1.0d0) * exp(-x))) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps_m))) + exp((x * -eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2e-89) {
		tmp = (((x + 1.0) / Math.exp(x)) + ((x + 1.0) * Math.exp(-x))) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + Math.exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 2e-89:
		tmp = (((x + 1.0) / math.exp(x)) + ((x + 1.0) * math.exp(-x))) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) + math.exp((x * -eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 2e-89)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / exp(x)) + Float64(Float64(x + 1.0) * exp(Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 2e-89)
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) + exp((x * -eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 2e-89], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.00000000000000008e-89

   1. Initial program 59.3%

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

      1. fma-neg 59.3%

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

      2. /-rgt-identity 59.3%

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

      3. fma-neg 59.3%

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

      4. /-rgt-identity 59.3%

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

      5. distribute-rgt-neg-in 59.3%

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

      6. sub-neg 59.3%

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

      7. metadata-eval 59.3%

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

      8. distribute-rgt-neg-in 59.3%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in eps around 0 68.6%

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

   7. Simplified 68.6%

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

      1. exp-neg 68.6%

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

      2. un-div-inv 68.6%

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

   9. Applied egg-rr 68.6%

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

   if 2.00000000000000008e-89 < eps

   1. Initial program 93.6%

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

      1. fma-neg 93.6%

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

      2. /-rgt-identity 93.6%

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

      3. fma-neg 93.6%

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

      4. /-rgt-identity 93.6%

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

      5. distribute-rgt-neg-in 93.6%

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

      6. sub-neg 93.6%

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

      7. metadata-eval 93.6%

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

      8. distribute-rgt-neg-in 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. fma-neg 73.5%

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

   2. /-rgt-identity 73.5%

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

   3. fma-neg 73.5%

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

   4. /-rgt-identity 73.5%

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

   5. distribute-rgt-neg-in 73.5%

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

   6. sub-neg 73.5%

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

   7. metadata-eval 73.5%

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

   8. distribute-rgt-neg-in 73.5%

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

3. Simplified 73.5%

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

5. Taylor expanded in eps around inf 99.2%

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

   1. exp-prod 99.2%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 3: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. fma-neg 73.5%

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

   2. /-rgt-identity 73.5%

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

   3. fma-neg 73.5%

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

   4. /-rgt-identity 73.5%

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

   5. distribute-rgt-neg-in 73.5%

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

   6. sub-neg 73.5%

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

   7. metadata-eval 73.5%

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

   8. distribute-rgt-neg-in 73.5%

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

3. Simplified 73.5%

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

5. Taylor expanded in eps around inf 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 64.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.27:\\ \;\;\;\;\frac{\left(x \cdot eps\_m\right) \cdot \left(-1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq 580:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.3 \cdot 10^{+68} \lor \neg \left(x \leq 1.1 \cdot 10^{+87}\right) \land \left(x \leq 4.6 \cdot 10^{+129} \lor \neg \left(x \leq 1.65 \cdot 10^{+210}\right) \land x \leq 1.58 \cdot 10^{+261}\right):\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{eps\_m} + \left(\left(1 - \frac{-1}{eps\_m}\right) + 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.27)
   (/ (* (* x eps_m) (+ -1.0 (/ -1.0 eps_m))) 2.0)
   (if (<= x 580.0)
     1.0
     (if (or (<= x 8.3e+68)
             (and (not (<= x 1.1e+87))
                  (or (<= x 4.6e+129)
                      (and (not (<= x 1.65e+210)) (<= x 1.58e+261)))))
       (/ (/ (expm1 x) eps_m) 2.0)
       (/ (+ (/ -1.0 eps_m) (+ (- 1.0 (/ -1.0 eps_m)) 1.0)) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.27) {
		tmp = ((x * eps_m) * (-1.0 + (-1.0 / eps_m))) / 2.0;
	} else if (x <= 580.0) {
		tmp = 1.0;
	} else if ((x <= 8.3e+68) || (!(x <= 1.1e+87) && ((x <= 4.6e+129) || (!(x <= 1.65e+210) && (x <= 1.58e+261))))) {
		tmp = (expm1(x) / eps_m) / 2.0;
	} else {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.27) {
		tmp = ((x * eps_m) * (-1.0 + (-1.0 / eps_m))) / 2.0;
	} else if (x <= 580.0) {
		tmp = 1.0;
	} else if ((x <= 8.3e+68) || (!(x <= 1.1e+87) && ((x <= 4.6e+129) || (!(x <= 1.65e+210) && (x <= 1.58e+261))))) {
		tmp = (Math.expm1(x) / eps_m) / 2.0;
	} else {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.27:
		tmp = ((x * eps_m) * (-1.0 + (-1.0 / eps_m))) / 2.0
	elif x <= 580.0:
		tmp = 1.0
	elif (x <= 8.3e+68) or (not (x <= 1.1e+87) and ((x <= 4.6e+129) or (not (x <= 1.65e+210) and (x <= 1.58e+261)))):
		tmp = (math.expm1(x) / eps_m) / 2.0
	else:
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.27)
		tmp = Float64(Float64(Float64(x * eps_m) * Float64(-1.0 + Float64(-1.0 / eps_m))) / 2.0);
	elseif (x <= 580.0)
		tmp = 1.0;
	elseif ((x <= 8.3e+68) || (!(x <= 1.1e+87) && ((x <= 4.6e+129) || (!(x <= 1.65e+210) && (x <= 1.58e+261)))))
		tmp = Float64(Float64(expm1(x) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(Float64(-1.0 / eps_m) + Float64(Float64(1.0 - Float64(-1.0 / eps_m)) + 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.27], N[(N[(N[(x * eps$95$m), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 580.0], 1.0, If[Or[LessEqual[x, 8.3e+68], And[N[Not[LessEqual[x, 1.1e+87]], $MachinePrecision], Or[LessEqual[x, 4.6e+129], And[N[Not[LessEqual[x, 1.65e+210]], $MachinePrecision], LessEqual[x, 1.58e+261]]]]], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.27000000000000002

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 42.6%

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

   6. Taylor expanded in x around inf 13.2%

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

      1. mul-1-neg 13.2%

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

      2. *-commutative 13.2%

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

      3. associate-*l* 13.2%

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

      4. distribute-rgt-neg-in 13.2%

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

      5. distribute-neg-in 13.2%

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

      6. metadata-eval 13.2%

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

      7. distribute-neg-frac 13.2%

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

      8. metadata-eval 13.2%

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

   8. Simplified 13.2%

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

   9. Taylor expanded in eps around inf 13.2%

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

       1. neg-mul-1 13.2%

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

       2. distribute-lft-neg-in 13.2%

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

       3. *-commutative 13.2%

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

   11. Simplified 13.2%

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

       1. /-rgt-identity 13.2%

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

       2. clear-num 13.2%

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

       3. add-sqr-sqrt 0.1%

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

       4. sqrt-unprod 0.1%

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

       5. frac-times 0.1%

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

       6. metadata-eval 0.1%

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

       7. metadata-eval 0.1%

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

       8. frac-times 0.1%

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

       9. sqrt-unprod 0.0%

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

       10. add-sqr-sqrt 40.2%

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

       11. clear-num 40.2%

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

       12. div-inv 40.2%

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

       13. metadata-eval 40.2%

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

   13. Applied egg-rr 40.2%

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

       1. *-commutative 40.2%

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

       2. neg-mul-1 40.2%

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

   15. Simplified 40.2%

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

   if -0.27000000000000002 < x < 580

   1. Initial program 54.5%

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

      1. fma-neg 54.5%

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

      2. /-rgt-identity 54.5%

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

      3. fma-neg 54.5%

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

      4. /-rgt-identity 54.5%

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

      5. distribute-rgt-neg-in 54.5%

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

      6. sub-neg 54.5%

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

      7. metadata-eval 54.5%

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

      8. distribute-rgt-neg-in 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in x around 0 74.0%

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

   if 580 < x < 8.30000000000000041e68 or 1.1e87 < x < 4.59999999999999981e129 or 1.64999999999999997e210 < x < 1.5800000000000001e261

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 47.5%

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

   6. Taylor expanded in eps around 0 1.7%

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

      1. expm1-def 1.7%

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

      2. neg-mul-1 1.7%

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

   8. Simplified 1.7%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 1.5%

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

      3. div-inv 1.5%

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

      4. div-inv 1.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 46.0%

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

      7. sqr-neg 46.0%

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

      8. sqrt-unprod 46.0%

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

      9. add-sqr-sqrt 46.0%

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

   10. Applied egg-rr 46.0%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)} - 1}}{2} \]
   11. Step-by-step derivation

       1. expm1-def 46.0%

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

       2. expm1-log1p 46.2%

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

   12. Simplified 46.2%

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

   if 8.30000000000000041e68 < x < 1.1e87 or 4.59999999999999981e129 < x < 1.64999999999999997e210 or 1.5800000000000001e261 < 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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 24.1%

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

   6. Taylor expanded in x around 0 70.4%

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

      1. +-commutative 70.4%

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

      2. associate--l+ 70.4%

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

      3. metadata-eval 70.4%

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

      4. add-sqr-sqrt 26.0%

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

      5. sqrt-unprod 29.6%

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

      6. sqr-neg 29.6%

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

      7. sqrt-unprod 0.9%

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

      8. add-sqr-sqrt 2.6%

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

      9. frac-2neg 2.6%

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

      10. *-rgt-identity 2.6%

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

      11. +-commutative 2.6%

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

      12. metadata-eval 2.6%

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

      13. add-sqr-sqrt 1.6%

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

      14. sqrt-unprod 18.8%

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

      15. sqr-neg 18.8%

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

      16. sqrt-unprod 14.5%

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

      17. add-sqr-sqrt 70.4%

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

      18. frac-2neg 70.4%

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

   8. Applied egg-rr 70.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.27:\\ \;\;\;\;\frac{\left(x \cdot \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 580:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.3 \cdot 10^{+68} \lor \neg \left(x \leq 1.1 \cdot 10^{+87}\right) \land \left(x \leq 4.6 \cdot 10^{+129} \lor \neg \left(x \leq 1.65 \cdot 10^{+210}\right) \land x \leq 1.58 \cdot 10^{+261}\right):\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\varepsilon} + \left(\left(1 - \frac{-1}{\varepsilon}\right) + 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -560:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 680:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+68} \lor \neg \left(x \leq 2.6 \cdot 10^{+82}\right) \land \left(x \leq 1.26 \cdot 10^{+129} \lor \neg \left(x \leq 2.9 \cdot 10^{+206}\right) \land x \leq 3.75 \cdot 10^{+258}\right):\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{eps\_m} + \left(\left(1 - \frac{-1}{eps\_m}\right) + 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -560.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (<= x 680.0)
     1.0
     (if (or (<= x 1.32e+68)
             (and (not (<= x 2.6e+82))
                  (or (<= x 1.26e+129)
                      (and (not (<= x 2.9e+206)) (<= x 3.75e+258)))))
       (/ (/ (expm1 x) eps_m) 2.0)
       (/ (+ (/ -1.0 eps_m) (+ (- 1.0 (/ -1.0 eps_m)) 1.0)) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -560.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= 680.0) {
		tmp = 1.0;
	} else if ((x <= 1.32e+68) || (!(x <= 2.6e+82) && ((x <= 1.26e+129) || (!(x <= 2.9e+206) && (x <= 3.75e+258))))) {
		tmp = (expm1(x) / eps_m) / 2.0;
	} else {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -560.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= 680.0) {
		tmp = 1.0;
	} else if ((x <= 1.32e+68) || (!(x <= 2.6e+82) && ((x <= 1.26e+129) || (!(x <= 2.9e+206) && (x <= 3.75e+258))))) {
		tmp = (Math.expm1(x) / eps_m) / 2.0;
	} else {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -560.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= 680.0:
		tmp = 1.0
	elif (x <= 1.32e+68) or (not (x <= 2.6e+82) and ((x <= 1.26e+129) or (not (x <= 2.9e+206) and (x <= 3.75e+258)))):
		tmp = (math.expm1(x) / eps_m) / 2.0
	else:
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -560.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 680.0)
		tmp = 1.0;
	elseif ((x <= 1.32e+68) || (!(x <= 2.6e+82) && ((x <= 1.26e+129) || (!(x <= 2.9e+206) && (x <= 3.75e+258)))))
		tmp = Float64(Float64(expm1(x) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(Float64(-1.0 / eps_m) + Float64(Float64(1.0 - Float64(-1.0 / eps_m)) + 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -560.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 680.0], 1.0, If[Or[LessEqual[x, 1.32e+68], And[N[Not[LessEqual[x, 2.6e+82]], $MachinePrecision], Or[LessEqual[x, 1.26e+129], And[N[Not[LessEqual[x, 2.9e+206]], $MachinePrecision], LessEqual[x, 3.75e+258]]]]], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -560

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 34.5%

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

   6. Taylor expanded in eps around 0 67.6%

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

      1. expm1-def 67.6%

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

      2. neg-mul-1 67.6%

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

   8. Simplified 67.6%

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

   if -560 < x < 680

   1. Initial program 54.5%

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

      1. fma-neg 54.5%

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

      2. /-rgt-identity 54.5%

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

      3. fma-neg 54.5%

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

      4. /-rgt-identity 54.5%

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

      5. distribute-rgt-neg-in 54.5%

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

      6. sub-neg 54.5%

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

      7. metadata-eval 54.5%

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

      8. distribute-rgt-neg-in 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in x around 0 74.0%

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

   if 680 < x < 1.3200000000000001e68 or 2.5999999999999998e82 < x < 1.26e129 or 2.9e206 < x < 3.75000000000000016e258

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 47.5%

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

   6. Taylor expanded in eps around 0 1.7%

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

      1. expm1-def 1.7%

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

      2. neg-mul-1 1.7%

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

   8. Simplified 1.7%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 1.5%

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

      3. div-inv 1.5%

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

      4. div-inv 1.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 46.0%

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

      7. sqr-neg 46.0%

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

      8. sqrt-unprod 46.0%

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

      9. add-sqr-sqrt 46.0%

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

   10. Applied egg-rr 46.0%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)} - 1}}{2} \]
   11. Step-by-step derivation

       1. expm1-def 46.0%

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

       2. expm1-log1p 46.2%

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

   12. Simplified 46.2%

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

   if 1.3200000000000001e68 < x < 2.5999999999999998e82 or 1.26e129 < x < 2.9e206 or 3.75000000000000016e258 < 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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 24.1%

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

   6. Taylor expanded in x around 0 70.4%

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

      1. +-commutative 70.4%

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

      2. associate--l+ 70.4%

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

      3. metadata-eval 70.4%

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

      4. add-sqr-sqrt 26.0%

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

      5. sqrt-unprod 29.6%

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

      6. sqr-neg 29.6%

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

      7. sqrt-unprod 0.9%

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

      8. add-sqr-sqrt 2.6%

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

      9. frac-2neg 2.6%

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

      10. *-rgt-identity 2.6%

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

      11. +-commutative 2.6%

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

      12. metadata-eval 2.6%

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

      13. add-sqr-sqrt 1.6%

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

      14. sqrt-unprod 18.8%

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

      15. sqr-neg 18.8%

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

      16. sqrt-unprod 14.5%

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

      17. add-sqr-sqrt 70.4%

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

      18. frac-2neg 70.4%

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

   8. Applied egg-rr 70.4%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -560:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 680:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+68} \lor \neg \left(x \leq 2.6 \cdot 10^{+82}\right) \land \left(x \leq 1.26 \cdot 10^{+129} \lor \neg \left(x \leq 2.9 \cdot 10^{+206}\right) \land x \leq 3.75 \cdot 10^{+258}\right):\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\varepsilon} + \left(\left(1 - \frac{-1}{\varepsilon}\right) + 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -480:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{+69} \lor \neg \left(x \leq 1.4 \cdot 10^{+80}\right) \land \left(x \leq 2.8 \cdot 10^{+129} \lor \neg \left(x \leq 2.1 \cdot 10^{+206}\right) \land x \leq 4.65 \cdot 10^{+266}\right):\\ \;\;\;\;\frac{e^{x + x \cdot eps\_m} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{eps\_m} + \left(\left(1 - \frac{-1}{eps\_m}\right) + 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -480.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (or (<= x 6.7e+69)
           (and (not (<= x 1.4e+80))
                (or (<= x 2.8e+129)
                    (and (not (<= x 2.1e+206)) (<= x 4.65e+266)))))
     (/ (+ (exp (+ x (* x eps_m))) 1.0) 2.0)
     (/ (+ (/ -1.0 eps_m) (+ (- 1.0 (/ -1.0 eps_m)) 1.0)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -480.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if ((x <= 6.7e+69) || (!(x <= 1.4e+80) && ((x <= 2.8e+129) || (!(x <= 2.1e+206) && (x <= 4.65e+266))))) {
		tmp = (exp((x + (x * eps_m))) + 1.0) / 2.0;
	} else {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -480.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if ((x <= 6.7e+69) || (!(x <= 1.4e+80) && ((x <= 2.8e+129) || (!(x <= 2.1e+206) && (x <= 4.65e+266))))) {
		tmp = (Math.exp((x + (x * eps_m))) + 1.0) / 2.0;
	} else {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -480.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif (x <= 6.7e+69) or (not (x <= 1.4e+80) and ((x <= 2.8e+129) or (not (x <= 2.1e+206) and (x <= 4.65e+266)))):
		tmp = (math.exp((x + (x * eps_m))) + 1.0) / 2.0
	else:
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -480.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif ((x <= 6.7e+69) || (!(x <= 1.4e+80) && ((x <= 2.8e+129) || (!(x <= 2.1e+206) && (x <= 4.65e+266)))))
		tmp = Float64(Float64(exp(Float64(x + Float64(x * eps_m))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(-1.0 / eps_m) + Float64(Float64(1.0 - Float64(-1.0 / eps_m)) + 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -480.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 6.7e+69], And[N[Not[LessEqual[x, 1.4e+80]], $MachinePrecision], Or[LessEqual[x, 2.8e+129], And[N[Not[LessEqual[x, 2.1e+206]], $MachinePrecision], LessEqual[x, 4.65e+266]]]]], N[(N[(N[Exp[N[(x + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -480

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 34.5%

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

   6. Taylor expanded in eps around 0 67.6%

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

      1. expm1-def 67.6%

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

      2. neg-mul-1 67.6%

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

   8. Simplified 67.6%

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

   if -480 < x < 6.7000000000000001e69 or 1.39999999999999992e80 < x < 2.79999999999999975e129 or 2.09999999999999987e206 < x < 4.65e266

   1. Initial program 63.5%

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

      1. fma-neg 63.5%

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

      2. /-rgt-identity 63.5%

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

      3. fma-neg 63.5%

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

      4. /-rgt-identity 63.5%

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

      5. distribute-rgt-neg-in 63.5%

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

      6. sub-neg 63.5%

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

      7. metadata-eval 63.5%

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

      8. distribute-rgt-neg-in 63.5%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in x around 0 44.2%

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

   6. Taylor expanded in eps around inf 79.6%

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

      1. associate-*r* 79.6%

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

      2. sub-neg 79.6%

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

      3. neg-mul-1 79.6%

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

      4. associate-*r* 79.6%

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

      5. associate-*r* 79.6%

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

      6. neg-mul-1 79.6%

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

      7. sub-neg 79.6%

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

      8. associate-*r* 79.6%

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

      9. neg-mul-1 79.6%

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

      10. distribute-rgt-neg-in 79.6%

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

   8. Simplified 79.6%

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

      1. add-sqr-sqrt 26.4%

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

      2. sqrt-unprod 81.5%

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

      3. sqr-neg 81.5%

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

      4. sqrt-unprod 58.8%

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

      5. add-sqr-sqrt 74.7%

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

      6. sub-neg 74.7%

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

      7. distribute-rgt-in 74.7%

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

      8. *-un-lft-identity 74.7%

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

      9. add-sqr-sqrt 39.7%

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

      10. sqrt-unprod 81.5%

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

      11. sqr-neg 81.5%

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

      12. sqrt-unprod 45.4%

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

      13. add-sqr-sqrt 79.6%

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

      14. *-commutative 79.6%

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

   10. Applied egg-rr 79.6%

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

   if 6.7000000000000001e69 < x < 1.39999999999999992e80 or 2.79999999999999975e129 < x < 2.09999999999999987e206 or 4.65e266 < 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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 24.1%

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

   6. Taylor expanded in x around 0 70.4%

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

      1. +-commutative 70.4%

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

      2. associate--l+ 70.4%

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

      3. metadata-eval 70.4%

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

      4. add-sqr-sqrt 26.0%

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

      5. sqrt-unprod 29.6%

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

      6. sqr-neg 29.6%

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

      7. sqrt-unprod 0.9%

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

      8. add-sqr-sqrt 2.6%

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

      9. frac-2neg 2.6%

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

      10. *-rgt-identity 2.6%

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

      11. +-commutative 2.6%

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

      12. metadata-eval 2.6%

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

      13. add-sqr-sqrt 1.6%

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

      14. sqrt-unprod 18.8%

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

      15. sqr-neg 18.8%

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

      16. sqrt-unprod 14.5%

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

      17. add-sqr-sqrt 70.4%

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

      18. frac-2neg 70.4%

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

   8. Applied egg-rr 70.4%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -480:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{+69} \lor \neg \left(x \leq 1.4 \cdot 10^{+80}\right) \land \left(x \leq 2.8 \cdot 10^{+129} \lor \neg \left(x \leq 2.1 \cdot 10^{+206}\right) \land x \leq 4.65 \cdot 10^{+266}\right):\\ \;\;\;\;\frac{e^{x + x \cdot \varepsilon} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\varepsilon} + \left(\left(1 - \frac{-1}{\varepsilon}\right) + 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-281}:\\ \;\;\;\;\frac{e^{x \cdot \left(1 - eps\_m\right)} + 1}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+69} \lor \neg \left(x \leq 8 \cdot 10^{+84}\right) \land \left(x \leq 9.5 \cdot 10^{+128} \lor \neg \left(x \leq 6 \cdot 10^{+211}\right) \land x \leq 1.22 \cdot 10^{+266}\right):\\ \;\;\;\;\frac{e^{x + x \cdot eps\_m} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{eps\_m} + \left(\left(1 - \frac{-1}{eps\_m}\right) + 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-281)
   (/ (+ (exp (* x (- 1.0 eps_m))) 1.0) 2.0)
   (if (or (<= x 1.85e+69)
           (and (not (<= x 8e+84))
                (or (<= x 9.5e+128)
                    (and (not (<= x 6e+211)) (<= x 1.22e+266)))))
     (/ (+ (exp (+ x (* x eps_m))) 1.0) 2.0)
     (/ (+ (/ -1.0 eps_m) (+ (- 1.0 (/ -1.0 eps_m)) 1.0)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-281) {
		tmp = (exp((x * (1.0 - eps_m))) + 1.0) / 2.0;
	} else if ((x <= 1.85e+69) || (!(x <= 8e+84) && ((x <= 9.5e+128) || (!(x <= 6e+211) && (x <= 1.22e+266))))) {
		tmp = (exp((x + (x * eps_m))) + 1.0) / 2.0;
	} else {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1d-281)) then
        tmp = (exp((x * (1.0d0 - eps_m))) + 1.0d0) / 2.0d0
    else if ((x <= 1.85d+69) .or. (.not. (x <= 8d+84)) .and. (x <= 9.5d+128) .or. (.not. (x <= 6d+211)) .and. (x <= 1.22d+266)) then
        tmp = (exp((x + (x * eps_m))) + 1.0d0) / 2.0d0
    else
        tmp = (((-1.0d0) / eps_m) + ((1.0d0 - ((-1.0d0) / eps_m)) + 1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-281) {
		tmp = (Math.exp((x * (1.0 - eps_m))) + 1.0) / 2.0;
	} else if ((x <= 1.85e+69) || (!(x <= 8e+84) && ((x <= 9.5e+128) || (!(x <= 6e+211) && (x <= 1.22e+266))))) {
		tmp = (Math.exp((x + (x * eps_m))) + 1.0) / 2.0;
	} else {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e-281:
		tmp = (math.exp((x * (1.0 - eps_m))) + 1.0) / 2.0
	elif (x <= 1.85e+69) or (not (x <= 8e+84) and ((x <= 9.5e+128) or (not (x <= 6e+211) and (x <= 1.22e+266)))):
		tmp = (math.exp((x + (x * eps_m))) + 1.0) / 2.0
	else:
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-281)
		tmp = Float64(Float64(exp(Float64(x * Float64(1.0 - eps_m))) + 1.0) / 2.0);
	elseif ((x <= 1.85e+69) || (!(x <= 8e+84) && ((x <= 9.5e+128) || (!(x <= 6e+211) && (x <= 1.22e+266)))))
		tmp = Float64(Float64(exp(Float64(x + Float64(x * eps_m))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(-1.0 / eps_m) + Float64(Float64(1.0 - Float64(-1.0 / eps_m)) + 1.0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1e-281)
		tmp = (exp((x * (1.0 - eps_m))) + 1.0) / 2.0;
	elseif ((x <= 1.85e+69) || (~((x <= 8e+84)) && ((x <= 9.5e+128) || (~((x <= 6e+211)) && (x <= 1.22e+266)))))
		tmp = (exp((x + (x * eps_m))) + 1.0) / 2.0;
	else
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1e-281], N[(N[(N[Exp[N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.85e+69], And[N[Not[LessEqual[x, 8e+84]], $MachinePrecision], Or[LessEqual[x, 9.5e+128], And[N[Not[LessEqual[x, 6e+211]], $MachinePrecision], LessEqual[x, 1.22e+266]]]]], N[(N[(N[Exp[N[(x + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e-281

   1. Initial program 71.6%

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

      1. fma-neg 71.6%

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

      2. /-rgt-identity 71.6%

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

      3. fma-neg 71.6%

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

      4. /-rgt-identity 71.6%

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

      5. distribute-rgt-neg-in 71.6%

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

      6. sub-neg 71.6%

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

      7. metadata-eval 71.6%

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

      8. distribute-rgt-neg-in 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in x around 0 40.7%

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

   6. Taylor expanded in eps around inf 68.9%

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

      1. associate-*r* 68.9%

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

      2. sub-neg 68.9%

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

      3. neg-mul-1 68.9%

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

      4. associate-*r* 68.9%

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

      5. associate-*r* 68.9%

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

      6. neg-mul-1 68.9%

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

      7. sub-neg 68.9%

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

      8. associate-*r* 68.9%

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

      9. neg-mul-1 68.9%

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

      10. distribute-rgt-neg-in 68.9%

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

   8. Simplified 68.9%

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

      1. distribute-rgt-neg-out 68.9%

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

      2. add-sqr-sqrt 61.8%

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

      3. sqrt-unprod 97.2%

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

      4. sqr-neg 97.2%

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

      5. sqrt-unprod 37.4%

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

      6. add-sqr-sqrt 77.4%

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

      7. neg-sub0 77.4%

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

      8. associate--r- 77.4%

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

      9. metadata-eval 77.4%

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

   10. Applied egg-rr 77.4%

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

   if -1e-281 < x < 1.8499999999999999e69 or 8.00000000000000046e84 < x < 9.50000000000000014e128 or 6e211 < x < 1.22e266

   1. Initial program 67.2%

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

      1. fma-neg 67.2%

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

      2. /-rgt-identity 67.2%

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

      3. fma-neg 67.2%

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

      4. /-rgt-identity 67.2%

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

      5. distribute-rgt-neg-in 67.2%

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

      6. sub-neg 67.2%

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

      7. metadata-eval 67.2%

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

      8. distribute-rgt-neg-in 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in x around 0 44.3%

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

   6. Taylor expanded in eps around inf 75.7%

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

      1. associate-*r* 75.7%

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

      2. sub-neg 75.7%

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

      3. neg-mul-1 75.7%

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

      4. associate-*r* 75.7%

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

      5. associate-*r* 75.7%

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

      6. neg-mul-1 75.7%

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

      7. sub-neg 75.7%

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

      8. associate-*r* 75.7%

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

      9. neg-mul-1 75.7%

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

      10. distribute-rgt-neg-in 75.7%

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

   8. Simplified 75.7%

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

      1. add-sqr-sqrt 35.3%

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

      2. sqrt-unprod 89.0%

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

      3. sqr-neg 89.0%

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

      4. sqrt-unprod 57.9%

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

      5. add-sqr-sqrt 70.8%

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

      6. sub-neg 70.8%

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

      7. distribute-rgt-in 70.8%

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

      8. *-un-lft-identity 70.8%

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

      9. add-sqr-sqrt 40.8%

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

      10. sqrt-unprod 89.0%

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

      11. sqr-neg 89.0%

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

      12. sqrt-unprod 52.5%

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

      13. add-sqr-sqrt 75.6%

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

      14. *-commutative 75.6%

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

   10. Applied egg-rr 75.6%

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

   if 1.8499999999999999e69 < x < 8.00000000000000046e84 or 9.50000000000000014e128 < x < 6e211 or 1.22e266 < 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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 24.1%

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

   6. Taylor expanded in x around 0 70.4%

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

      1. +-commutative 70.4%

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

      2. associate--l+ 70.4%

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

      3. metadata-eval 70.4%

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

      4. add-sqr-sqrt 26.0%

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

      5. sqrt-unprod 29.6%

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

      6. sqr-neg 29.6%

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

      7. sqrt-unprod 0.9%

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

      8. add-sqr-sqrt 2.6%

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

      9. frac-2neg 2.6%

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

      10. *-rgt-identity 2.6%

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

      11. +-commutative 2.6%

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

      12. metadata-eval 2.6%

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

      13. add-sqr-sqrt 1.6%

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

      14. sqrt-unprod 18.8%

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

      15. sqr-neg 18.8%

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

      16. sqrt-unprod 14.5%

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

      17. add-sqr-sqrt 70.4%

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

      18. frac-2neg 70.4%

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

   8. Applied egg-rr 70.4%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-281}:\\ \;\;\;\;\frac{e^{x \cdot \left(1 - \varepsilon\right)} + 1}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+69} \lor \neg \left(x \leq 8 \cdot 10^{+84}\right) \land \left(x \leq 9.5 \cdot 10^{+128} \lor \neg \left(x \leq 6 \cdot 10^{+211}\right) \land x \leq 1.22 \cdot 10^{+266}\right):\\ \;\;\;\;\frac{e^{x + x \cdot \varepsilon} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\varepsilon} + \left(\left(1 - \frac{-1}{\varepsilon}\right) + 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-282}:\\ \;\;\;\;\frac{e^{x \cdot \left(1 - eps\_m\right)} + 1}{2}\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+68}:\\ \;\;\;\;\frac{e^{x + x \cdot eps\_m} + 1}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+83} \lor \neg \left(x \leq 9.6 \cdot 10^{+263}\right):\\ \;\;\;\;\frac{\frac{-1}{eps\_m} + \left(\left(1 - \frac{-1}{eps\_m}\right) + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.8e-282)
   (/ (+ (exp (* x (- 1.0 eps_m))) 1.0) 2.0)
   (if (<= x 1.42e+68)
     (/ (+ (exp (+ x (* x eps_m))) 1.0) 2.0)
     (if (or (<= x 7.5e+83) (not (<= x 9.6e+263)))
       (/ (+ (/ -1.0 eps_m) (+ (- 1.0 (/ -1.0 eps_m)) 1.0)) 2.0)
       (/ (+ (exp (* x (+ -1.0 eps_m))) 1.0) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.8e-282) {
		tmp = (exp((x * (1.0 - eps_m))) + 1.0) / 2.0;
	} else if (x <= 1.42e+68) {
		tmp = (exp((x + (x * eps_m))) + 1.0) / 2.0;
	} else if ((x <= 7.5e+83) || !(x <= 9.6e+263)) {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.8d-282)) then
        tmp = (exp((x * (1.0d0 - eps_m))) + 1.0d0) / 2.0d0
    else if (x <= 1.42d+68) then
        tmp = (exp((x + (x * eps_m))) + 1.0d0) / 2.0d0
    else if ((x <= 7.5d+83) .or. (.not. (x <= 9.6d+263))) then
        tmp = (((-1.0d0) / eps_m) + ((1.0d0 - ((-1.0d0) / eps_m)) + 1.0d0)) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps_m))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.8e-282) {
		tmp = (Math.exp((x * (1.0 - eps_m))) + 1.0) / 2.0;
	} else if (x <= 1.42e+68) {
		tmp = (Math.exp((x + (x * eps_m))) + 1.0) / 2.0;
	} else if ((x <= 7.5e+83) || !(x <= 9.6e+263)) {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.8e-282:
		tmp = (math.exp((x * (1.0 - eps_m))) + 1.0) / 2.0
	elif x <= 1.42e+68:
		tmp = (math.exp((x + (x * eps_m))) + 1.0) / 2.0
	elif (x <= 7.5e+83) or not (x <= 9.6e+263):
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) + 1.0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.8e-282)
		tmp = Float64(Float64(exp(Float64(x * Float64(1.0 - eps_m))) + 1.0) / 2.0);
	elseif (x <= 1.42e+68)
		tmp = Float64(Float64(exp(Float64(x + Float64(x * eps_m))) + 1.0) / 2.0);
	elseif ((x <= 7.5e+83) || !(x <= 9.6e+263))
		tmp = Float64(Float64(Float64(-1.0 / eps_m) + Float64(Float64(1.0 - Float64(-1.0 / eps_m)) + 1.0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.8e-282)
		tmp = (exp((x * (1.0 - eps_m))) + 1.0) / 2.0;
	elseif (x <= 1.42e+68)
		tmp = (exp((x + (x * eps_m))) + 1.0) / 2.0;
	elseif ((x <= 7.5e+83) || ~((x <= 9.6e+263)))
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.8e-282], N[(N[(N[Exp[N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.42e+68], N[(N[(N[Exp[N[(x + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 7.5e+83], N[Not[LessEqual[x, 9.6e+263]], $MachinePrecision]], N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.7999999999999999e-282

   1. Initial program 71.6%

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

      1. fma-neg 71.6%

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

      2. /-rgt-identity 71.6%

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

      3. fma-neg 71.6%

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

      4. /-rgt-identity 71.6%

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

      5. distribute-rgt-neg-in 71.6%

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

      6. sub-neg 71.6%

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

      7. metadata-eval 71.6%

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

      8. distribute-rgt-neg-in 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in x around 0 40.7%

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

   6. Taylor expanded in eps around inf 68.9%

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

      1. associate-*r* 68.9%

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

      2. sub-neg 68.9%

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

      3. neg-mul-1 68.9%

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

      4. associate-*r* 68.9%

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

      5. associate-*r* 68.9%

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

      6. neg-mul-1 68.9%

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

      7. sub-neg 68.9%

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

      8. associate-*r* 68.9%

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

      9. neg-mul-1 68.9%

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

      10. distribute-rgt-neg-in 68.9%

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

   8. Simplified 68.9%

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

      1. distribute-rgt-neg-out 68.9%

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

      2. add-sqr-sqrt 61.8%

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

      3. sqrt-unprod 97.2%

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

      4. sqr-neg 97.2%

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

      5. sqrt-unprod 37.4%

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

      6. add-sqr-sqrt 77.4%

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

      7. neg-sub0 77.4%

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

      8. associate--r- 77.4%

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

      9. metadata-eval 77.4%

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

   10. Applied egg-rr 77.4%

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

   if -1.7999999999999999e-282 < x < 1.41999999999999998e68

   1. Initial program 57.3%

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

      1. fma-neg 57.3%

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

      2. /-rgt-identity 57.3%

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

      3. fma-neg 57.3%

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

      4. /-rgt-identity 57.3%

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

      5. distribute-rgt-neg-in 57.3%

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

      6. sub-neg 57.3%

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

      7. metadata-eval 57.3%

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

      8. distribute-rgt-neg-in 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in x around 0 43.2%

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

   6. Taylor expanded in eps around inf 84.0%

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

      1. associate-*r* 84.0%

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

      2. sub-neg 84.0%

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

      3. neg-mul-1 84.0%

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

      4. associate-*r* 84.0%

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

      5. associate-*r* 84.0%

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

      6. neg-mul-1 84.0%

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

      7. sub-neg 84.0%

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

      8. associate-*r* 84.0%

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

      9. neg-mul-1 84.0%

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

      10. distribute-rgt-neg-in 84.0%

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

   8. Simplified 84.0%

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

      1. add-sqr-sqrt 32.0%

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

      2. sqrt-unprod 91.0%

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

      3. sqr-neg 91.0%

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

      4. sqrt-unprod 64.5%

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

      5. add-sqr-sqrt 80.9%

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

      6. sub-neg 80.9%

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

      7. distribute-rgt-in 80.9%

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

      8. *-un-lft-identity 80.9%

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

      9. add-sqr-sqrt 42.3%

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

      10. sqrt-unprod 91.0%

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

      11. sqr-neg 91.0%

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

      12. sqrt-unprod 54.2%

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

      13. add-sqr-sqrt 84.0%

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

      14. *-commutative 84.0%

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

   10. Applied egg-rr 84.0%

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

   if 1.41999999999999998e68 < x < 7.49999999999999989e83 or 9.6000000000000002e263 < 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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 18.7%

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

   6. Taylor expanded in x around 0 78.5%

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

      1. +-commutative 78.5%

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

      2. associate--l+ 78.5%

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

      3. metadata-eval 78.5%

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

      4. add-sqr-sqrt 28.7%

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

      5. sqrt-unprod 35.0%

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

      6. sqr-neg 35.0%

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

      7. sqrt-unprod 1.0%

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

      8. add-sqr-sqrt 2.6%

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

      9. frac-2neg 2.6%

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

      10. *-rgt-identity 2.6%

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

      11. +-commutative 2.6%

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

      12. metadata-eval 2.6%

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

      13. add-sqr-sqrt 1.6%

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

      14. sqrt-unprod 24.1%

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

      15. sqr-neg 24.1%

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

      16. sqrt-unprod 17.2%

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

      17. add-sqr-sqrt 78.5%

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

      18. frac-2neg 78.5%

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

   8. Applied egg-rr 78.5%

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

   if 7.49999999999999989e83 < x < 9.6000000000000002e263

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 40.7%

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

   6. Taylor expanded in eps around inf 41.0%

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

      1. associate-*r* 41.0%

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

      2. sub-neg 41.0%

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

      3. neg-mul-1 41.0%

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

      4. associate-*r* 41.0%

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

      5. associate-*r* 41.0%

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

      6. neg-mul-1 41.0%

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

      7. sub-neg 41.0%

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

      8. associate-*r* 41.0%

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

      9. neg-mul-1 41.0%

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

      10. distribute-rgt-neg-in 41.0%

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

   8. Simplified 41.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-282}:\\ \;\;\;\;\frac{e^{x \cdot \left(1 - \varepsilon\right)} + 1}{2}\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+68}:\\ \;\;\;\;\frac{e^{x + x \cdot \varepsilon} + 1}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+83} \lor \neg \left(x \leq 9.6 \cdot 10^{+263}\right):\\ \;\;\;\;\frac{\frac{-1}{\varepsilon} + \left(\left(1 - \frac{-1}{\varepsilon}\right) + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-282}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \frac{1}{\frac{-1}{eps\_m}}\right)} + 1}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+71}:\\ \;\;\;\;\frac{e^{x + x \cdot eps\_m} + 1}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+80} \lor \neg \left(x \leq 2.36 \cdot 10^{+266}\right):\\ \;\;\;\;\frac{\frac{-1}{eps\_m} + \left(\left(1 - \frac{-1}{eps\_m}\right) + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.7e-282)
   (/ (+ (exp (* x (+ -1.0 (/ 1.0 (/ -1.0 eps_m))))) 1.0) 2.0)
   (if (<= x 2e+71)
     (/ (+ (exp (+ x (* x eps_m))) 1.0) 2.0)
     (if (or (<= x 4.6e+80) (not (<= x 2.36e+266)))
       (/ (+ (/ -1.0 eps_m) (+ (- 1.0 (/ -1.0 eps_m)) 1.0)) 2.0)
       (/ (+ (exp (* x (+ -1.0 eps_m))) 1.0) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.7e-282) {
		tmp = (exp((x * (-1.0 + (1.0 / (-1.0 / eps_m))))) + 1.0) / 2.0;
	} else if (x <= 2e+71) {
		tmp = (exp((x + (x * eps_m))) + 1.0) / 2.0;
	} else if ((x <= 4.6e+80) || !(x <= 2.36e+266)) {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.7d-282)) then
        tmp = (exp((x * ((-1.0d0) + (1.0d0 / ((-1.0d0) / eps_m))))) + 1.0d0) / 2.0d0
    else if (x <= 2d+71) then
        tmp = (exp((x + (x * eps_m))) + 1.0d0) / 2.0d0
    else if ((x <= 4.6d+80) .or. (.not. (x <= 2.36d+266))) then
        tmp = (((-1.0d0) / eps_m) + ((1.0d0 - ((-1.0d0) / eps_m)) + 1.0d0)) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps_m))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.7e-282) {
		tmp = (Math.exp((x * (-1.0 + (1.0 / (-1.0 / eps_m))))) + 1.0) / 2.0;
	} else if (x <= 2e+71) {
		tmp = (Math.exp((x + (x * eps_m))) + 1.0) / 2.0;
	} else if ((x <= 4.6e+80) || !(x <= 2.36e+266)) {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.7e-282:
		tmp = (math.exp((x * (-1.0 + (1.0 / (-1.0 / eps_m))))) + 1.0) / 2.0
	elif x <= 2e+71:
		tmp = (math.exp((x + (x * eps_m))) + 1.0) / 2.0
	elif (x <= 4.6e+80) or not (x <= 2.36e+266):
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) + 1.0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.7e-282)
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + Float64(1.0 / Float64(-1.0 / eps_m))))) + 1.0) / 2.0);
	elseif (x <= 2e+71)
		tmp = Float64(Float64(exp(Float64(x + Float64(x * eps_m))) + 1.0) / 2.0);
	elseif ((x <= 4.6e+80) || !(x <= 2.36e+266))
		tmp = Float64(Float64(Float64(-1.0 / eps_m) + Float64(Float64(1.0 - Float64(-1.0 / eps_m)) + 1.0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.7e-282)
		tmp = (exp((x * (-1.0 + (1.0 / (-1.0 / eps_m))))) + 1.0) / 2.0;
	elseif (x <= 2e+71)
		tmp = (exp((x + (x * eps_m))) + 1.0) / 2.0;
	elseif ((x <= 4.6e+80) || ~((x <= 2.36e+266)))
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.7e-282], N[(N[(N[Exp[N[(x * N[(-1.0 + N[(1.0 / N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+71], N[(N[(N[Exp[N[(x + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 4.6e+80], N[Not[LessEqual[x, 2.36e+266]], $MachinePrecision]], N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.69999999999999999e-282

   1. Initial program 71.6%

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

      1. fma-neg 71.6%

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

      2. /-rgt-identity 71.6%

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

      3. fma-neg 71.6%

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

      4. /-rgt-identity 71.6%

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

      5. distribute-rgt-neg-in 71.6%

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

      6. sub-neg 71.6%

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

      7. metadata-eval 71.6%

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

      8. distribute-rgt-neg-in 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in x around 0 40.7%

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

   6. Taylor expanded in eps around inf 68.9%

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

      1. associate-*r* 68.9%

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

      2. sub-neg 68.9%

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

      3. neg-mul-1 68.9%

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

      4. associate-*r* 68.9%

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

      5. associate-*r* 68.9%

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

      6. neg-mul-1 68.9%

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

      7. sub-neg 68.9%

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

      8. associate-*r* 68.9%

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

      9. neg-mul-1 68.9%

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

      10. distribute-rgt-neg-in 68.9%

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

   8. Simplified 68.9%

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

      1. /-rgt-identity 68.9%

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

      2. clear-num 68.9%

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

      3. add-sqr-sqrt 22.0%

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

      4. sqrt-unprod 45.4%

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

      5. frac-times 45.4%

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

      6. metadata-eval 45.4%

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

      7. metadata-eval 45.4%

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

      8. frac-times 45.4%

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

      9. sqrt-unprod 25.1%

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

      10. add-sqr-sqrt 77.4%

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

   10. Applied egg-rr 77.4%

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

   if -1.69999999999999999e-282 < x < 2.0000000000000001e71

   1. Initial program 57.3%

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

      1. fma-neg 57.3%

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

      2. /-rgt-identity 57.3%

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

      3. fma-neg 57.3%

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

      4. /-rgt-identity 57.3%

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

      5. distribute-rgt-neg-in 57.3%

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

      6. sub-neg 57.3%

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

      7. metadata-eval 57.3%

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

      8. distribute-rgt-neg-in 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in x around 0 43.2%

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

   6. Taylor expanded in eps around inf 84.0%

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

      1. associate-*r* 84.0%

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

      2. sub-neg 84.0%

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

      3. neg-mul-1 84.0%

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

      4. associate-*r* 84.0%

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

      5. associate-*r* 84.0%

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

      6. neg-mul-1 84.0%

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

      7. sub-neg 84.0%

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

      8. associate-*r* 84.0%

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

      9. neg-mul-1 84.0%

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

      10. distribute-rgt-neg-in 84.0%

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

   8. Simplified 84.0%

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

      1. add-sqr-sqrt 32.0%

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

      2. sqrt-unprod 91.0%

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

      3. sqr-neg 91.0%

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

      4. sqrt-unprod 64.5%

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

      5. add-sqr-sqrt 80.9%

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

      6. sub-neg 80.9%

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

      7. distribute-rgt-in 80.9%

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

      8. *-un-lft-identity 80.9%

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

      9. add-sqr-sqrt 42.3%

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

      10. sqrt-unprod 91.0%

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

      11. sqr-neg 91.0%

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

      12. sqrt-unprod 54.2%

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

      13. add-sqr-sqrt 84.0%

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

      14. *-commutative 84.0%

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

   10. Applied egg-rr 84.0%

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

   if 2.0000000000000001e71 < x < 4.60000000000000008e80 or 2.36e266 < 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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 18.7%

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

   6. Taylor expanded in x around 0 78.5%

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

      1. +-commutative 78.5%

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

      2. associate--l+ 78.5%

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

      3. metadata-eval 78.5%

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

      4. add-sqr-sqrt 28.7%

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

      5. sqrt-unprod 35.0%

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

      6. sqr-neg 35.0%

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

      7. sqrt-unprod 1.0%

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

      8. add-sqr-sqrt 2.6%

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

      9. frac-2neg 2.6%

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

      10. *-rgt-identity 2.6%

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

      11. +-commutative 2.6%

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

      12. metadata-eval 2.6%

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

      13. add-sqr-sqrt 1.6%

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

      14. sqrt-unprod 24.1%

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

      15. sqr-neg 24.1%

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

      16. sqrt-unprod 17.2%

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

      17. add-sqr-sqrt 78.5%

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

      18. frac-2neg 78.5%

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

   8. Applied egg-rr 78.5%

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

   if 4.60000000000000008e80 < x < 2.36e266

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 40.7%

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

   6. Taylor expanded in eps around inf 41.0%

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

      1. associate-*r* 41.0%

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

      2. sub-neg 41.0%

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

      3. neg-mul-1 41.0%

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

      4. associate-*r* 41.0%

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

      5. associate-*r* 41.0%

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

      6. neg-mul-1 41.0%

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

      7. sub-neg 41.0%

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

      8. associate-*r* 41.0%

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

      9. neg-mul-1 41.0%

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

      10. distribute-rgt-neg-in 41.0%

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

   8. Simplified 41.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-282}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \frac{1}{\frac{-1}{\varepsilon}}\right)} + 1}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+71}:\\ \;\;\;\;\frac{e^{x + x \cdot \varepsilon} + 1}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+80} \lor \neg \left(x \leq 2.36 \cdot 10^{+266}\right):\\ \;\;\;\;\frac{\frac{-1}{\varepsilon} + \left(\left(1 - \frac{-1}{\varepsilon}\right) + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.2% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.27:\\ \;\;\;\;\frac{\left(x \cdot eps\_m\right) \cdot \left(-1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq 17:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+212} \lor \neg \left(x \leq 2.25 \cdot 10^{+265}\right):\\ \;\;\;\;\frac{\frac{-1}{eps\_m} + \left(\left(1 - \frac{-1}{eps\_m}\right) + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.27)
   (/ (* (* x eps_m) (+ -1.0 (/ -1.0 eps_m))) 2.0)
   (if (<= x 17.0)
     1.0
     (if (or (<= x 2.35e+212) (not (<= x 2.25e+265)))
       (/ (+ (/ -1.0 eps_m) (+ (- 1.0 (/ -1.0 eps_m)) 1.0)) 2.0)
       (/ (* x eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.27) {
		tmp = ((x * eps_m) * (-1.0 + (-1.0 / eps_m))) / 2.0;
	} else if (x <= 17.0) {
		tmp = 1.0;
	} else if ((x <= 2.35e+212) || !(x <= 2.25e+265)) {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-0.27d0)) then
        tmp = ((x * eps_m) * ((-1.0d0) + ((-1.0d0) / eps_m))) / 2.0d0
    else if (x <= 17.0d0) then
        tmp = 1.0d0
    else if ((x <= 2.35d+212) .or. (.not. (x <= 2.25d+265))) then
        tmp = (((-1.0d0) / eps_m) + ((1.0d0 - ((-1.0d0) / eps_m)) + 1.0d0)) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.27) {
		tmp = ((x * eps_m) * (-1.0 + (-1.0 / eps_m))) / 2.0;
	} else if (x <= 17.0) {
		tmp = 1.0;
	} else if ((x <= 2.35e+212) || !(x <= 2.25e+265)) {
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.27:
		tmp = ((x * eps_m) * (-1.0 + (-1.0 / eps_m))) / 2.0
	elif x <= 17.0:
		tmp = 1.0
	elif (x <= 2.35e+212) or not (x <= 2.25e+265):
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.27)
		tmp = Float64(Float64(Float64(x * eps_m) * Float64(-1.0 + Float64(-1.0 / eps_m))) / 2.0);
	elseif (x <= 17.0)
		tmp = 1.0;
	elseif ((x <= 2.35e+212) || !(x <= 2.25e+265))
		tmp = Float64(Float64(Float64(-1.0 / eps_m) + Float64(Float64(1.0 - Float64(-1.0 / eps_m)) + 1.0)) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.27)
		tmp = ((x * eps_m) * (-1.0 + (-1.0 / eps_m))) / 2.0;
	elseif (x <= 17.0)
		tmp = 1.0;
	elseif ((x <= 2.35e+212) || ~((x <= 2.25e+265)))
		tmp = ((-1.0 / eps_m) + ((1.0 - (-1.0 / eps_m)) + 1.0)) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.27], N[(N[(N[(x * eps$95$m), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 17.0], 1.0, If[Or[LessEqual[x, 2.35e+212], N[Not[LessEqual[x, 2.25e+265]], $MachinePrecision]], N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.27000000000000002

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 42.6%

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

   6. Taylor expanded in x around inf 13.2%

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

      1. mul-1-neg 13.2%

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

      2. *-commutative 13.2%

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

      3. associate-*l* 13.2%

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

      4. distribute-rgt-neg-in 13.2%

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

      5. distribute-neg-in 13.2%

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

      6. metadata-eval 13.2%

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

      7. distribute-neg-frac 13.2%

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

      8. metadata-eval 13.2%

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

   8. Simplified 13.2%

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

   9. Taylor expanded in eps around inf 13.2%

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

       1. neg-mul-1 13.2%

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

       2. distribute-lft-neg-in 13.2%

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

       3. *-commutative 13.2%

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

   11. Simplified 13.2%

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

       1. /-rgt-identity 13.2%

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

       2. clear-num 13.2%

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

       3. add-sqr-sqrt 0.1%

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

       4. sqrt-unprod 0.1%

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

       5. frac-times 0.1%

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

       6. metadata-eval 0.1%

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

       7. metadata-eval 0.1%

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

       8. frac-times 0.1%

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

       9. sqrt-unprod 0.0%

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

       10. add-sqr-sqrt 40.2%

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

       11. clear-num 40.2%

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

       12. div-inv 40.2%

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

       13. metadata-eval 40.2%

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

   13. Applied egg-rr 40.2%

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

       1. *-commutative 40.2%

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

       2. neg-mul-1 40.2%

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

   15. Simplified 40.2%

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

   if -0.27000000000000002 < x < 17

   1. Initial program 53.8%

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

      1. fma-neg 53.9%

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

      2. /-rgt-identity 53.9%

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

      3. fma-neg 53.8%

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

      4. /-rgt-identity 53.8%

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

      5. distribute-rgt-neg-in 53.8%

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

      6. sub-neg 53.8%

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

      7. metadata-eval 53.8%

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

      8. distribute-rgt-neg-in 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in x around 0 75.0%

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

   if 17 < x < 2.34999999999999996e212 or 2.24999999999999993e265 < 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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 35.0%

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

   6. Taylor expanded in x around 0 48.3%

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

      1. +-commutative 48.3%

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

      2. associate--l+ 48.3%

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

      3. metadata-eval 48.3%

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

      4. add-sqr-sqrt 17.9%

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

      5. sqrt-unprod 20.6%

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

      6. sqr-neg 20.6%

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

      7. sqrt-unprod 1.1%

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

      8. add-sqr-sqrt 2.8%

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

      9. frac-2neg 2.8%

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

      10. *-rgt-identity 2.8%

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

      11. +-commutative 2.8%

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

      12. metadata-eval 2.8%

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

      13. add-sqr-sqrt 1.7%

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

      14. sqrt-unprod 14.1%

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

      15. sqr-neg 14.1%

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

      16. sqrt-unprod 9.2%

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

      17. add-sqr-sqrt 48.3%

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

      18. frac-2neg 48.3%

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

   8. Applied egg-rr 48.3%

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

   if 2.34999999999999996e212 < x < 2.24999999999999993e265

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 34.6%

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

   6. Taylor expanded in x around inf 27.9%

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

      1. mul-1-neg 27.9%

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

      2. *-commutative 27.9%

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

      3. associate-*l* 27.9%

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

      4. distribute-rgt-neg-in 27.9%

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

      5. distribute-neg-in 27.9%

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

      6. metadata-eval 27.9%

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

      7. distribute-neg-frac 27.9%

         \[\leadsto \frac{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \color{blue}{\frac{-1}{\varepsilon}}\right)}{2} \]

      8. metadata-eval 27.9%

         \[\leadsto \frac{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \frac{\color{blue}{-1}}{\varepsilon}\right)}{2} \]

   8. Simplified 27.9%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}}{2} \]

   9. Taylor expanded in eps around inf 28.1%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   10. Step-by-step derivation

       1. *-commutative 28.1%

          \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   11. Simplified 28.1%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.27:\\ \;\;\;\;\frac{\left(x \cdot \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 17:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+212} \lor \neg \left(x \leq 2.25 \cdot 10^{+265}\right):\\ \;\;\;\;\frac{\frac{-1}{\varepsilon} + \left(\left(1 - \frac{-1}{\varepsilon}\right) + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.0% accurate, 14.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.48:\\ \;\;\;\;\frac{\left(x \cdot eps\_m\right) \cdot \left(-1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.48)
   (/ (* (* x eps_m) (+ -1.0 (/ -1.0 eps_m))) 2.0)
   (/ (+ 2.0 (* x eps_m)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.48) {
		tmp = ((x * eps_m) * (-1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-0.48d0)) then
        tmp = ((x * eps_m) * ((-1.0d0) + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (2.0d0 + (x * eps_m)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.48) {
		tmp = ((x * eps_m) * (-1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.48:
		tmp = ((x * eps_m) * (-1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (2.0 + (x * eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.48)
		tmp = Float64(Float64(Float64(x * eps_m) * Float64(-1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.48)
		tmp = ((x * eps_m) * (-1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (2.0 + (x * eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.48], N[(N[(N[(x * eps$95$m), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.47999999999999998

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 100.0%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 42.6%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   6. Taylor expanded in x around inf 13.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 13.2%

         \[\leadsto \frac{\color{blue}{-x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}}{2} \]

      2. *-commutative 13.2%

         \[\leadsto \frac{-x \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

      3. associate-*l* 13.2%

         \[\leadsto \frac{-\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)}}{2} \]

      4. distribute-rgt-neg-in 13.2%

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-\left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

      5. distribute-neg-in 13.2%

         \[\leadsto \frac{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \color{blue}{\left(\left(-1\right) + \left(-\frac{1}{\varepsilon}\right)\right)}}{2} \]

      6. metadata-eval 13.2%

         \[\leadsto \frac{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(\color{blue}{-1} + \left(-\frac{1}{\varepsilon}\right)\right)}{2} \]

      7. distribute-neg-frac 13.2%

         \[\leadsto \frac{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \color{blue}{\frac{-1}{\varepsilon}}\right)}{2} \]

      8. metadata-eval 13.2%

         \[\leadsto \frac{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \frac{\color{blue}{-1}}{\varepsilon}\right)}{2} \]

   8. Simplified 13.2%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}}{2} \]

   9. Taylor expanded in eps around inf 13.2%

      \[\leadsto \frac{\color{blue}{\left(-1 \cdot \left(\varepsilon \cdot x\right)\right)} \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]
   10. Step-by-step derivation

       1. neg-mul-1 13.2%

          \[\leadsto \frac{\color{blue}{\left(-\varepsilon \cdot x\right)} \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       2. distribute-lft-neg-in 13.2%

          \[\leadsto \frac{\color{blue}{\left(\left(-\varepsilon\right) \cdot x\right)} \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       3. *-commutative 13.2%

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(-\varepsilon\right)\right)} \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

   11. Simplified 13.2%

       \[\leadsto \frac{\color{blue}{\left(x \cdot \left(-\varepsilon\right)\right)} \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]
   12. Step-by-step derivation

       1. /-rgt-identity 13.2%

          \[\leadsto \frac{\left(x \cdot \left(-\color{blue}{\frac{\varepsilon}{1}}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       2. clear-num 13.2%

          \[\leadsto \frac{\left(x \cdot \left(-\color{blue}{\frac{1}{\frac{1}{\varepsilon}}}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       3. add-sqr-sqrt 0.1%

          \[\leadsto \frac{\left(x \cdot \left(-\frac{1}{\color{blue}{\sqrt{\frac{1}{\varepsilon}} \cdot \sqrt{\frac{1}{\varepsilon}}}}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       4. sqrt-unprod 0.1%

          \[\leadsto \frac{\left(x \cdot \left(-\frac{1}{\color{blue}{\sqrt{\frac{1}{\varepsilon} \cdot \frac{1}{\varepsilon}}}}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       5. frac-times 0.1%

          \[\leadsto \frac{\left(x \cdot \left(-\frac{1}{\sqrt{\color{blue}{\frac{1 \cdot 1}{\varepsilon \cdot \varepsilon}}}}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       6. metadata-eval 0.1%

          \[\leadsto \frac{\left(x \cdot \left(-\frac{1}{\sqrt{\frac{\color{blue}{1}}{\varepsilon \cdot \varepsilon}}}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       7. metadata-eval 0.1%

          \[\leadsto \frac{\left(x \cdot \left(-\frac{1}{\sqrt{\frac{\color{blue}{-1 \cdot -1}}{\varepsilon \cdot \varepsilon}}}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       8. frac-times 0.1%

          \[\leadsto \frac{\left(x \cdot \left(-\frac{1}{\sqrt{\color{blue}{\frac{-1}{\varepsilon} \cdot \frac{-1}{\varepsilon}}}}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       9. sqrt-unprod 0.0%

          \[\leadsto \frac{\left(x \cdot \left(-\frac{1}{\color{blue}{\sqrt{\frac{-1}{\varepsilon}} \cdot \sqrt{\frac{-1}{\varepsilon}}}}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       10. add-sqr-sqrt 40.2%

           \[\leadsto \frac{\left(x \cdot \left(-\frac{1}{\color{blue}{\frac{-1}{\varepsilon}}}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       11. clear-num 40.2%

           \[\leadsto \frac{\left(x \cdot \left(-\color{blue}{\frac{\varepsilon}{-1}}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       12. div-inv 40.2%

           \[\leadsto \frac{\left(x \cdot \left(-\color{blue}{\varepsilon \cdot \frac{1}{-1}}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       13. metadata-eval 40.2%

           \[\leadsto \frac{\left(x \cdot \left(-\varepsilon \cdot \color{blue}{-1}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

   13. Applied egg-rr 40.2%

       \[\leadsto \frac{\left(x \cdot \left(-\color{blue}{\varepsilon \cdot -1}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]
   14. Step-by-step derivation

       1. *-commutative 40.2%

          \[\leadsto \frac{\left(x \cdot \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

       2. neg-mul-1 40.2%

          \[\leadsto \frac{\left(x \cdot \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

   15. Simplified 40.2%

       \[\leadsto \frac{\left(x \cdot \left(-\color{blue}{\left(-\varepsilon\right)}\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2} \]

   if -0.47999999999999998 < x

   1. Initial program 69.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 69.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 69.5%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 69.4%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 69.4%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 69.4%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 69.4%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 69.4%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 69.4%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Simplified 69.4%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 40.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot \color{blue}{1}}{2} \]

   6. Taylor expanded in x around 0 38.4%

      \[\leadsto \frac{\color{blue}{2 + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 38.4%

         \[\leadsto \frac{2 + \color{blue}{\left(-x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]

      2. *-commutative 38.4%

         \[\leadsto \frac{2 + \left(-x \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}\right)}{2} \]

      3. associate-*l* 38.4%

         \[\leadsto \frac{2 + \left(-\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)}\right)}{2} \]

      4. distribute-rgt-neg-in 38.4%

         \[\leadsto \frac{2 + \color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-\left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

      5. distribute-neg-in 38.4%

         \[\leadsto \frac{2 + \left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \color{blue}{\left(\left(-1\right) + \left(-\frac{1}{\varepsilon}\right)\right)}}{2} \]

      6. metadata-eval 38.4%

         \[\leadsto \frac{2 + \left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(\color{blue}{-1} + \left(-\frac{1}{\varepsilon}\right)\right)}{2} \]

      7. distribute-neg-frac 38.4%

         \[\leadsto \frac{2 + \left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \color{blue}{\frac{-1}{\varepsilon}}\right)}{2} \]

      8. metadata-eval 38.4%

         \[\leadsto \frac{2 + \left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \frac{\color{blue}{-1}}{\varepsilon}\right)}{2} \]

   8. Simplified 38.4%

      \[\leadsto \frac{\color{blue}{2 + \left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}}{2} \]

   9. Taylor expanded in eps around inf 55.9%

      \[\leadsto \frac{2 + \color{blue}{\varepsilon \cdot x}}{2} \]
   10. Step-by-step derivation

       1. *-commutative 8.9%

          \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   11. Simplified 55.9%

       \[\leadsto \frac{2 + \color{blue}{x \cdot \varepsilon}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.48:\\ \;\;\;\;\frac{\left(x \cdot \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.4% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 15.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 15.2) 1.0 (/ (* x eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 15.2) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 15.2d0) then
        tmp = 1.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 15.2) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 15.2:
		tmp = 1.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 15.2)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 15.2)
		tmp = 1.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 15.2], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 15.199999999999999

   1. Initial program 62.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. fma-neg 62.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 62.5%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 62.5%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 62.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 62.5%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 62.5%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 62.5%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 62.5%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Simplified 62.5%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.5%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 15.199999999999999 < 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

      1. fma-neg 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. /-rgt-identity 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

      3. fma-neg 100.0%

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      4. /-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      5. distribute-rgt-neg-in 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      6. sub-neg 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      8. distribute-rgt-neg-in 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 31.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   6. Taylor expanded in x around inf 19.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 19.0%

         \[\leadsto \frac{\color{blue}{-x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)}}{2} \]

      2. *-commutative 19.0%

         \[\leadsto \frac{-x \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

      3. associate-*l* 19.0%

         \[\leadsto \frac{-\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)}}{2} \]

      4. distribute-rgt-neg-in 19.0%

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-\left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

      5. distribute-neg-in 19.0%

         \[\leadsto \frac{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \color{blue}{\left(\left(-1\right) + \left(-\frac{1}{\varepsilon}\right)\right)}}{2} \]

      6. metadata-eval 19.0%

         \[\leadsto \frac{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(\color{blue}{-1} + \left(-\frac{1}{\varepsilon}\right)\right)}{2} \]

      7. distribute-neg-frac 19.0%

         \[\leadsto \frac{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \color{blue}{\frac{-1}{\varepsilon}}\right)}{2} \]

      8. metadata-eval 19.0%

         \[\leadsto \frac{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \frac{\color{blue}{-1}}{\varepsilon}\right)}{2} \]

   8. Simplified 19.0%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}}{2} \]

   9. Taylor expanded in eps around inf 19.6%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   10. Step-by-step derivation

       1. *-commutative 19.6%

          \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   11. Simplified 19.6%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 15.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.5% accurate, 32.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{2 + x \cdot eps\_m}{2} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (/ (+ 2.0 (* x eps_m)) 2.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (2.0 + (x * eps_m)) / 2.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = (2.0d0 + (x * eps_m)) / 2.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return (2.0 + (x * eps_m)) / 2.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (2.0 + (x * eps_m)) / 2.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (2.0 + (x * eps_m)) / 2.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Step-by-step derivation

   1. fma-neg 73.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

   2. /-rgt-identity 73.5%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

   3. fma-neg 73.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   4. /-rgt-identity 73.5%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   5. distribute-rgt-neg-in 73.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   6. sub-neg 73.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   7. metadata-eval 73.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   8. distribute-rgt-neg-in 73.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

3. Simplified 73.5%

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 40.1%

   \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot \color{blue}{1}}{2} \]

6. Taylor expanded in x around 0 35.1%

   \[\leadsto \frac{\color{blue}{2 + -1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]
7. Step-by-step derivation

   1. mul-1-neg 35.1%

      \[\leadsto \frac{2 + \color{blue}{\left(-x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]

   2. *-commutative 35.1%

      \[\leadsto \frac{2 + \left(-x \cdot \color{blue}{\left(\left(1 - \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}\right)}{2} \]

   3. associate-*l* 35.1%

      \[\leadsto \frac{2 + \left(-\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)}\right)}{2} \]

   4. distribute-rgt-neg-in 35.1%

      \[\leadsto \frac{2 + \color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-\left(1 + \frac{1}{\varepsilon}\right)\right)}}{2} \]

   5. distribute-neg-in 35.1%

      \[\leadsto \frac{2 + \left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \color{blue}{\left(\left(-1\right) + \left(-\frac{1}{\varepsilon}\right)\right)}}{2} \]

   6. metadata-eval 35.1%

      \[\leadsto \frac{2 + \left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(\color{blue}{-1} + \left(-\frac{1}{\varepsilon}\right)\right)}{2} \]

   7. distribute-neg-frac 35.1%

      \[\leadsto \frac{2 + \left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \color{blue}{\frac{-1}{\varepsilon}}\right)}{2} \]

   8. metadata-eval 35.1%

      \[\leadsto \frac{2 + \left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \frac{\color{blue}{-1}}{\varepsilon}\right)}{2} \]

8. Simplified 35.1%

   \[\leadsto \frac{\color{blue}{2 + \left(x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}}{2} \]

9. Taylor expanded in eps around inf 50.2%

   \[\leadsto \frac{2 + \color{blue}{\varepsilon \cdot x}}{2} \]
10. Step-by-step derivation

    1. *-commutative 9.5%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

11. Simplified 50.2%

    \[\leadsto \frac{2 + \color{blue}{x \cdot \varepsilon}}{2} \]

12. Final simplification 50.2%

    \[\leadsto \frac{2 + x \cdot \varepsilon}{2} \]
13. Add Preprocessing

Alternative 14: 44.5% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 1.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 1.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 1.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 73.5%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Step-by-step derivation

   1. fma-neg 73.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

   2. /-rgt-identity 73.5%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1}}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2} \]

   3. fma-neg 73.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{1} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   4. /-rgt-identity 73.5%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   5. distribute-rgt-neg-in 73.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   6. sub-neg 73.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   7. metadata-eval 73.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   8. distribute-rgt-neg-in 73.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

3. Simplified 73.5%

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 44.4%

   \[\leadsto \frac{\color{blue}{2}}{2} \]

6. Final simplification 44.4%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))