NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.5% → 99.8%

Time: 22.1s

Alternatives: 17

Speedup: 1.8×

• 38.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 10^{-22}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1e-22)
   (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1e-22) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + 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 <= 1d-22) then
        tmp = ((eps_m * (exp(-x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + 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 <= 1e-22) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + 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 <= 1e-22:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + 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 <= 1e-22)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + 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 <= 1e-22)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + 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, 1e-22], N[(N[(N[(eps$95$m * N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $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 < 1e-22

   1. Initial program 63.9%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in eps around 0 29.8%

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

   7. Simplified 65.8%

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

   if 1e-22 < eps

   1. Initial program 100.0%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. rec-exp 100.0%

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

       2. *-commutative 100.0%

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

       3. distribute-rgt-neg-in 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-22}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64((exp(1) ^ Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / exp(Float64(x + Float64(x * eps_m))))) / 2.0)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = ((2.71828182845904523536 ^ (x * (eps_m + -1.0))) + (1.0 / exp((x + (x * eps_m))))) / 2.0;
end

Wolfram

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

Derivation

1. Initial program 73.1%

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

3. Simplified 65.9%

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

5. Taylor expanded in eps around inf 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. exp-prod 98.6%

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

   3. sub-neg 98.6%

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

   4. metadata-eval 98.6%

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

7. Applied egg-rr 98.6%

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

   1. exp-1-e 98.6%

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 3: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return ((1.0 / exp((x + (x * eps_m)))) + exp((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 = ((1.0d0 / exp((x + (x * eps_m)))) + exp((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 ((1.0 / Math.exp((x + (x * eps_m)))) + Math.exp((x * (eps_m + -1.0)))) / 2.0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

3. Simplified 65.9%

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

5. Taylor expanded in eps around inf 98.6%

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

6. Final simplification 98.6%

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

Alternative 4: 84.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-307}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x \cdot eps\_m}}}{2}\\ \mathbf{elif}\;x \leq 10^{+101} \lor \neg \left(x \leq 3.3 \cdot 10^{+198}\right) \land x \leq 5.6 \cdot 10^{+288}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-307)
   (/ (+ 1.0 (/ 1.0 (exp (* x eps_m)))) 2.0)
   (if (or (<= x 1e+101) (and (not (<= x 3.3e+198)) (<= x 5.6e+288)))
     (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-307) {
		tmp = (1.0 + (1.0 / exp((x * eps_m)))) / 2.0;
	} else if ((x <= 1e+101) || (!(x <= 3.3e+198) && (x <= 5.6e+288))) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= (-5d-307)) then
        tmp = (1.0d0 + (1.0d0 / exp((x * eps_m)))) / 2.0d0
    else if ((x <= 1d+101) .or. (.not. (x <= 3.3d+198)) .and. (x <= 5.6d+288)) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    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 <= -5e-307) {
		tmp = (1.0 + (1.0 / Math.exp((x * eps_m)))) / 2.0;
	} else if ((x <= 1e+101) || (!(x <= 3.3e+198) && (x <= 5.6e+288))) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -5e-307:
		tmp = (1.0 + (1.0 / math.exp((x * eps_m)))) / 2.0
	elif (x <= 1e+101) or (not (x <= 3.3e+198) and (x <= 5.6e+288)):
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -5e-307)
		tmp = Float64(Float64(1.0 + Float64(1.0 / exp(Float64(x * eps_m)))) / 2.0);
	elseif ((x <= 1e+101) || (!(x <= 3.3e+198) && (x <= 5.6e+288)))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -5e-307)
		tmp = (1.0 + (1.0 / exp((x * eps_m)))) / 2.0;
	elseif ((x <= 1e+101) || (~((x <= 3.3e+198)) && (x <= 5.6e+288)))
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5e-307], N[(N[(1.0 + N[(1.0 / N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1e+101], And[N[Not[LessEqual[x, 3.3e+198]], $MachinePrecision], LessEqual[x, 5.6e+288]]], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000014e-307

   1. Initial program 71.6%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in eps around inf 99.7%

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

   6. Taylor expanded in eps around inf 99.7%

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

      1. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around 0 67.2%

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

   if -5.00000000000000014e-307 < x < 9.9999999999999998e100 or 3.29999999999999994e198 < x < 5.59999999999999961e288

   1. Initial program 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in x around 0 33.4%

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

   6. Taylor expanded in eps around inf 62.0%

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

      1. *-commutative 62.0%

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

      2. associate-*r* 62.0%

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

      3. *-commutative 62.0%

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

      4. mul-1-neg 62.0%

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

      5. sub-neg 62.0%

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

      6. mul-1-neg 62.0%

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

      7. distribute-neg-in 62.0%

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

      8. metadata-eval 62.0%

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

      9. mul-1-neg 62.0%

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

      10. remove-double-neg 62.0%

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

      11. +-commutative 62.0%

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

   8. Simplified 62.0%

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

   if 9.9999999999999998e100 < x < 3.29999999999999994e198 or 5.59999999999999961e288 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 71.3%

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

      1. mul-1-neg 71.3%

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

      2. mul-1-neg 71.3%

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

      3. rec-exp 71.3%

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

      4. sub-neg 71.3%

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

      5. div-sub 71.3%

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

      6. mul-1-neg 71.3%

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

      7. rec-exp 71.3%

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

      8. +-inverses 71.3%

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

   7. Simplified 71.3%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-307}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 10^{+101} \lor \neg \left(x \leq 3.3 \cdot 10^{+198}\right) \land x \leq 5.6 \cdot 10^{+288}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-307}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+96} \lor \neg \left(x \leq 4.2 \cdot 10^{+201}\right) \land x \leq 2.8 \cdot 10^{+288}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-307)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (or (<= x 7e+96) (and (not (<= x 4.2e+201)) (<= x 2.8e+288)))
     (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-307) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if ((x <= 7e+96) || (!(x <= 4.2e+201) && (x <= 2.8e+288))) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= (-5d-307)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if ((x <= 7d+96) .or. (.not. (x <= 4.2d+201)) .and. (x <= 2.8d+288)) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    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 <= -5e-307) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if ((x <= 7e+96) || (!(x <= 4.2e+201) && (x <= 2.8e+288))) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -5e-307:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif (x <= 7e+96) or (not (x <= 4.2e+201) and (x <= 2.8e+288)):
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -5e-307)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif ((x <= 7e+96) || (!(x <= 4.2e+201) && (x <= 2.8e+288)))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -5e-307)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif ((x <= 7e+96) || (~((x <= 4.2e+201)) && (x <= 2.8e+288)))
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5e-307], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 7e+96], And[N[Not[LessEqual[x, 4.2e+201]], $MachinePrecision], LessEqual[x, 2.8e+288]]], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000014e-307

   1. Initial program 71.6%

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

   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 46.9%

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

   6. Taylor expanded in eps around inf 74.9%

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

      1. *-commutative 74.9%

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

      2. associate-*r* 74.9%

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

      3. *-commutative 74.9%

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

      4. mul-1-neg 74.9%

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

      5. sub-neg 74.9%

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

      6. mul-1-neg 74.9%

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

      7. distribute-neg-in 74.9%

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

      8. metadata-eval 74.9%

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

      9. mul-1-neg 74.9%

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

      10. remove-double-neg 74.9%

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

      11. +-commutative 74.9%

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

   8. Simplified 74.9%

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

   9. Taylor expanded in eps around 0 79.1%

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

       1. mul-1-neg 79.1%

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

   11. Simplified 79.1%

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

   if -5.00000000000000014e-307 < x < 6.9999999999999998e96 or 4.1999999999999998e201 < x < 2.7999999999999998e288

   1. Initial program 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in x around 0 33.4%

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

   6. Taylor expanded in eps around inf 62.0%

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

      1. *-commutative 62.0%

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

      2. associate-*r* 62.0%

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

      3. *-commutative 62.0%

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

      4. mul-1-neg 62.0%

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

      5. sub-neg 62.0%

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

      6. mul-1-neg 62.0%

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

      7. distribute-neg-in 62.0%

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

      8. metadata-eval 62.0%

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

      9. mul-1-neg 62.0%

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

      10. remove-double-neg 62.0%

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

      11. +-commutative 62.0%

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

   8. Simplified 62.0%

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

   if 6.9999999999999998e96 < x < 4.1999999999999998e201 or 2.7999999999999998e288 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 71.3%

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

      1. mul-1-neg 71.3%

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

      2. mul-1-neg 71.3%

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

      3. rec-exp 71.3%

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

      4. sub-neg 71.3%

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

      5. div-sub 71.3%

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

      6. mul-1-neg 71.3%

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

      7. rec-exp 71.3%

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

      8. +-inverses 71.3%

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

   7. Simplified 71.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-307}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+96} \lor \neg \left(x \leq 4.2 \cdot 10^{+201}\right) \land x \leq 2.8 \cdot 10^{+288}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{-225}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x \cdot eps\_m}}}{2}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+99}:\\ \;\;\;\;\frac{t\_0 + \left(1 - x \cdot eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+205}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+286}:\\ \;\;\;\;\frac{1 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x -7.6e-225)
     (/ (+ 1.0 (/ 1.0 (exp (* x eps_m)))) 2.0)
     (if (<= x 2.05e+99)
       (/ (+ t_0 (- 1.0 (* x eps_m))) 2.0)
       (if (<= x 1.05e+205)
         0.0
         (if (<= x 3.6e+286) (/ (+ 1.0 t_0) 2.0) 0.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -7.6e-225) {
		tmp = (1.0 + (1.0 / exp((x * eps_m)))) / 2.0;
	} else if (x <= 2.05e+99) {
		tmp = (t_0 + (1.0 - (x * eps_m))) / 2.0;
	} else if (x <= 1.05e+205) {
		tmp = 0.0;
	} else if (x <= 3.6e+286) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * (eps_m + (-1.0d0))))
    if (x <= (-7.6d-225)) then
        tmp = (1.0d0 + (1.0d0 / exp((x * eps_m)))) / 2.0d0
    else if (x <= 2.05d+99) then
        tmp = (t_0 + (1.0d0 - (x * eps_m))) / 2.0d0
    else if (x <= 1.05d+205) then
        tmp = 0.0d0
    else if (x <= 3.6d+286) then
        tmp = (1.0d0 + t_0) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -7.6e-225) {
		tmp = (1.0 + (1.0 / Math.exp((x * eps_m)))) / 2.0;
	} else if (x <= 2.05e+99) {
		tmp = (t_0 + (1.0 - (x * eps_m))) / 2.0;
	} else if (x <= 1.05e+205) {
		tmp = 0.0;
	} else if (x <= 3.6e+286) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= -7.6e-225:
		tmp = (1.0 + (1.0 / math.exp((x * eps_m)))) / 2.0
	elif x <= 2.05e+99:
		tmp = (t_0 + (1.0 - (x * eps_m))) / 2.0
	elif x <= 1.05e+205:
		tmp = 0.0
	elif x <= 3.6e+286:
		tmp = (1.0 + t_0) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= -7.6e-225)
		tmp = Float64(Float64(1.0 + Float64(1.0 / exp(Float64(x * eps_m)))) / 2.0);
	elseif (x <= 2.05e+99)
		tmp = Float64(Float64(t_0 + Float64(1.0 - Float64(x * eps_m))) / 2.0);
	elseif (x <= 1.05e+205)
		tmp = 0.0;
	elseif (x <= 3.6e+286)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= -7.6e-225)
		tmp = (1.0 + (1.0 / exp((x * eps_m)))) / 2.0;
	elseif (x <= 2.05e+99)
		tmp = (t_0 + (1.0 - (x * eps_m))) / 2.0;
	elseif (x <= 1.05e+205)
		tmp = 0.0;
	elseif (x <= 3.6e+286)
		tmp = (1.0 + t_0) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -7.6e-225], N[(N[(1.0 + N[(1.0 / N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.05e+99], N[(N[(t$95$0 + N[(1.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.05e+205], 0.0, If[LessEqual[x, 3.6e+286], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.6000000000000005e-225

   1. Initial program 78.8%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in eps around inf 99.6%

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

   6. Taylor expanded in eps around inf 99.6%

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

      1. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 59.5%

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

   if -7.6000000000000005e-225 < x < 2.0499999999999999e99

   1. Initial program 61.2%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in eps around inf 97.6%

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

   6. Taylor expanded in eps around inf 89.5%

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

      1. *-commutative 89.5%

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

   8. Simplified 89.5%

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

   9. Taylor expanded in x around 0 74.1%

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

       1. mul-1-neg 74.1%

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

       2. *-commutative 74.1%

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

       3. unsub-neg 74.1%

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

       4. *-commutative 74.1%

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

   11. Simplified 74.1%

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

   if 2.0499999999999999e99 < x < 1.05e205 or 3.6e286 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 71.3%

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

      1. mul-1-neg 71.3%

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

      2. mul-1-neg 71.3%

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

      3. rec-exp 71.3%

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

      4. sub-neg 71.3%

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

      5. div-sub 71.3%

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

      6. mul-1-neg 71.3%

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

      7. rec-exp 71.3%

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

      8. +-inverses 71.3%

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

   7. Simplified 71.3%

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

   if 1.05e205 < x < 3.6e286

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 22.2%

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

   6. Taylor expanded in eps around inf 22.5%

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

      1. *-commutative 22.5%

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

      2. associate-*r* 22.5%

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

      3. *-commutative 22.5%

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

      4. mul-1-neg 22.5%

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

      5. sub-neg 22.5%

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

      6. mul-1-neg 22.5%

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

      7. distribute-neg-in 22.5%

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

      8. metadata-eval 22.5%

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

      9. mul-1-neg 22.5%

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

      10. remove-double-neg 22.5%

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

      11. +-commutative 22.5%

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

   8. Simplified 22.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-225}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+99}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+205}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+286}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 350.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 1.3e+19) (/ (+ 1.0 (exp x)) 2.0) 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 350.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 1.3e+19) {
		tmp = (1.0 + exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= 350.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 1.3d+19) then
        tmp = (1.0d0 + exp(x)) / 2.0d0
    else
        tmp = 0.0d0
    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 <= 350.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 1.3e+19) {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 350.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 1.3e+19:
		tmp = (1.0 + math.exp(x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 350.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 1.3e+19)
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 350.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 1.3e+19)
		tmp = (1.0 + exp(x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 350.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.3e+19], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 350

   1. Initial program 63.0%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in x around 0 43.7%

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

   6. Taylor expanded in eps around inf 79.0%

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

      1. *-commutative 79.0%

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

      2. associate-*r* 79.0%

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

      3. *-commutative 79.0%

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

      4. mul-1-neg 79.0%

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

      5. sub-neg 79.0%

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

      6. mul-1-neg 79.0%

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

      7. distribute-neg-in 79.0%

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

      8. metadata-eval 79.0%

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

      9. mul-1-neg 79.0%

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

      10. remove-double-neg 79.0%

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

      11. +-commutative 79.0%

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

   8. Simplified 79.0%

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

   9. Taylor expanded in eps around 0 76.8%

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

       1. mul-1-neg 76.8%

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

   11. Simplified 76.8%

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

   if 350 < x < 1.3e19

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 80.6%

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

   6. Taylor expanded in eps around inf 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-*r* 80.6%

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

      3. *-commutative 80.6%

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

      4. mul-1-neg 80.6%

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

      5. sub-neg 80.6%

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

      6. mul-1-neg 80.6%

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

      7. distribute-neg-in 80.6%

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

      8. metadata-eval 80.6%

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

      9. mul-1-neg 80.6%

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

      10. remove-double-neg 80.6%

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

      11. +-commutative 80.6%

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

   8. Simplified 80.6%

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

   9. Taylor expanded in eps around 0 3.1%

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

       1. mul-1-neg 3.1%

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

   11. Simplified 3.1%

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

       1. *-un-lft-identity 3.1%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 100.0%

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

       4. sqr-neg 100.0%

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

       5. sqrt-unprod 100.0%

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

       6. add-sqr-sqrt 100.0%

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

   13. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

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

   15. Simplified 100.0%

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

   if 1.3e19 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 54.6%

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

      1. mul-1-neg 54.6%

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

      2. mul-1-neg 54.6%

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

      3. rec-exp 54.6%

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

      4. sub-neg 54.6%

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

      5. div-sub 54.6%

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

      6. mul-1-neg 54.6%

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

      7. rec-exp 54.6%

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

      8. +-inverses 54.6%

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

   7. Simplified 54.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+194}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+282}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.5)
   (/ (+ 2.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666)))))) 2.0)
   (if (<= x 2e+19)
     (/ (+ 1.0 (exp x)) 2.0)
     (if (<= x 5e+194)
       0.0
       (if (<= x 1e+282) (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0) 0.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.5) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 2e+19) {
		tmp = (1.0 + exp(x)) / 2.0;
	} else if (x <= 5e+194) {
		tmp = 0.0;
	} else if (x <= 1e+282) {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= 2.5d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))) / 2.0d0
    else if (x <= 2d+19) then
        tmp = (1.0d0 + exp(x)) / 2.0d0
    else if (x <= 5d+194) then
        tmp = 0.0d0
    else if (x <= 1d+282) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 2.0d0
    else
        tmp = 0.0d0
    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 <= 2.5) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 2e+19) {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	} else if (x <= 5e+194) {
		tmp = 0.0;
	} else if (x <= 1e+282) {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.5:
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0
	elif x <= 2e+19:
		tmp = (1.0 + math.exp(x)) / 2.0
	elif x <= 5e+194:
		tmp = 0.0
	elif x <= 1e+282:
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.5)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666)))))) / 2.0);
	elseif (x <= 2e+19)
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	elseif (x <= 5e+194)
		tmp = 0.0;
	elseif (x <= 1e+282)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	elseif (x <= 2e+19)
		tmp = (1.0 + exp(x)) / 2.0;
	elseif (x <= 5e+194)
		tmp = 0.0;
	elseif (x <= 1e+282)
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.5], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+19], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+194], 0.0, If[LessEqual[x, 1e+282], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.5

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in x around 0 43.9%

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

   6. Taylor expanded in eps around inf 79.3%

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

      1. *-commutative 79.3%

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

      2. associate-*r* 79.3%

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

      3. *-commutative 79.3%

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

      4. mul-1-neg 79.3%

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

      5. sub-neg 79.3%

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

      6. mul-1-neg 79.3%

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

      7. distribute-neg-in 79.3%

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

      8. metadata-eval 79.3%

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

      9. mul-1-neg 79.3%

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

      10. remove-double-neg 79.3%

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

      11. +-commutative 79.3%

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

   8. Simplified 79.3%

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

   9. Taylor expanded in eps around 0 77.1%

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

       1. mul-1-neg 77.1%

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

   11. Simplified 77.1%

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

   12. Taylor expanded in x around 0 70.4%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)}}{2} \]

   if 2.5 < x < 2e19

   1. Initial program 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in x around 0 67.4%

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

   6. Taylor expanded in eps around inf 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-*r* 69.2%

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

      3. *-commutative 69.2%

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

      4. mul-1-neg 69.2%

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

      5. sub-neg 69.2%

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

      6. mul-1-neg 69.2%

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

      7. distribute-neg-in 69.2%

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

      8. metadata-eval 69.2%

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

      9. mul-1-neg 69.2%

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

      10. remove-double-neg 69.2%

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

      11. +-commutative 69.2%

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

   8. Simplified 69.2%

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

   9. Taylor expanded in eps around 0 4.6%

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

       1. mul-1-neg 4.6%

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

   11. Simplified 4.6%

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

       1. *-un-lft-identity 4.6%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 85.0%

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

       4. sqr-neg 85.0%

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

       5. sqrt-unprod 85.0%

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

       6. add-sqr-sqrt 85.0%

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

   13. Applied egg-rr 85.0%

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

       1. *-lft-identity 85.0%

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

   15. Simplified 85.0%

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

   if 2e19 < x < 4.99999999999999989e194 or 1.00000000000000003e282 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 60.6%

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

      1. mul-1-neg 60.6%

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

      2. mul-1-neg 60.6%

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

      3. rec-exp 60.6%

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

      4. sub-neg 60.6%

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

      5. div-sub 60.6%

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

      6. mul-1-neg 60.6%

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

      7. rec-exp 60.6%

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

      8. +-inverses 60.6%

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

   7. Simplified 60.6%

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

   if 4.99999999999999989e194 < x < 1.00000000000000003e282

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 22.2%

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

   6. Taylor expanded in eps around inf 22.5%

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

      1. *-commutative 22.5%

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

      2. associate-*r* 22.5%

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

      3. *-commutative 22.5%

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

      4. mul-1-neg 22.5%

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

      5. sub-neg 22.5%

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

      6. mul-1-neg 22.5%

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

      7. distribute-neg-in 22.5%

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

      8. metadata-eval 22.5%

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

      9. mul-1-neg 22.5%

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

      10. remove-double-neg 22.5%

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

      11. +-commutative 22.5%

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

   8. Simplified 22.5%

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

   9. Taylor expanded in eps around 0 3.1%

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

       1. mul-1-neg 3.1%

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

   11. Simplified 3.1%

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

   12. Taylor expanded in x around 0 67.2%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(0.5 \cdot x - 1\right)}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+194}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+282}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.3% accurate, 8.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.52:\\ \;\;\;\;\frac{x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+209}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+283}:\\ \;\;\;\;\frac{x \cdot eps\_m + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.52)
   (/ (* x (- -1.0 eps_m)) 2.0)
   (if (<= x 3.4e+18)
     1.0
     (if (<= x 3.1e+209)
       0.0
       (if (<= x 2.5e+283) (/ (+ (* x eps_m) 2.0) 2.0) 0.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.52) {
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	} else if (x <= 3.4e+18) {
		tmp = 1.0;
	} else if (x <= 3.1e+209) {
		tmp = 0.0;
	} else if (x <= 2.5e+283) {
		tmp = ((x * eps_m) + 2.0) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= (-0.52d0)) then
        tmp = (x * ((-1.0d0) - eps_m)) / 2.0d0
    else if (x <= 3.4d+18) then
        tmp = 1.0d0
    else if (x <= 3.1d+209) then
        tmp = 0.0d0
    else if (x <= 2.5d+283) then
        tmp = ((x * eps_m) + 2.0d0) / 2.0d0
    else
        tmp = 0.0d0
    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 <= -0.52) {
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	} else if (x <= 3.4e+18) {
		tmp = 1.0;
	} else if (x <= 3.1e+209) {
		tmp = 0.0;
	} else if (x <= 2.5e+283) {
		tmp = ((x * eps_m) + 2.0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.52:
		tmp = (x * (-1.0 - eps_m)) / 2.0
	elif x <= 3.4e+18:
		tmp = 1.0
	elif x <= 3.1e+209:
		tmp = 0.0
	elif x <= 2.5e+283:
		tmp = ((x * eps_m) + 2.0) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.52)
		tmp = Float64(Float64(x * Float64(-1.0 - eps_m)) / 2.0);
	elseif (x <= 3.4e+18)
		tmp = 1.0;
	elseif (x <= 3.1e+209)
		tmp = 0.0;
	elseif (x <= 2.5e+283)
		tmp = Float64(Float64(Float64(x * eps_m) + 2.0) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.52)
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	elseif (x <= 3.4e+18)
		tmp = 1.0;
	elseif (x <= 3.1e+209)
		tmp = 0.0;
	elseif (x <= 2.5e+283)
		tmp = ((x * eps_m) + 2.0) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.52], N[(N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.4e+18], 1.0, If[LessEqual[x, 3.1e+209], 0.0, If[LessEqual[x, 2.5e+283], N[(N[(N[(x * eps$95$m), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.52000000000000002

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 45.7%

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

      1. associate-*r* 45.7%

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

      2. neg-mul-1 45.7%

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

   8. Simplified 45.7%

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

   9. Taylor expanded in x around inf 11.6%

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

       1. +-commutative 11.6%

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

       2. *-commutative 11.6%

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

       3. distribute-rgt1-in 11.6%

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

       4. distribute-lft-in 11.6%

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

       5. associate-*r* 11.6%

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

       6. neg-mul-1 11.6%

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

       7. distribute-rgt-in 11.6%

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

       8. unsub-neg 11.6%

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

   11. Simplified 11.6%

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

   if -0.52000000000000002 < x < 3.4e18

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in x around 0 69.6%

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

   if 3.4e18 < x < 3.1000000000000001e209 or 2.5000000000000002e283 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 60.6%

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

      1. mul-1-neg 60.6%

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

      2. mul-1-neg 60.6%

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

      3. rec-exp 60.6%

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

      4. sub-neg 60.6%

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

      5. div-sub 60.6%

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

      6. mul-1-neg 60.6%

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

      7. rec-exp 60.6%

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

      8. +-inverses 60.6%

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

   7. Simplified 60.6%

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

   if 3.1000000000000001e209 < x < 2.5000000000000002e283

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 22.2%

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

   6. Taylor expanded in x around 0 20.5%

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

      1. mul-1-neg 20.5%

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

      2. unsub-neg 20.5%

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

      3. associate-*r* 20.5%

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

   8. Simplified 20.5%

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

   9. Taylor expanded in eps around inf 20.9%

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

       1. associate-*r* 20.9%

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

       2. neg-mul-1 20.9%

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

   11. Simplified 20.9%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.52:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+209}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+283}:\\ \;\;\;\;\frac{x \cdot \varepsilon + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+18} \lor \neg \left(x \leq 10^{+198}\right) \land x \leq 10^{+288}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (or (<= x 3.4e+18) (and (not (<= x 1e+198)) (<= x 1e+288)))
   (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0)
   0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((x <= 3.4e+18) || (!(x <= 1e+198) && (x <= 1e+288))) {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= 3.4d+18) .or. (.not. (x <= 1d+198)) .and. (x <= 1d+288)) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 2.0d0
    else
        tmp = 0.0d0
    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 <= 3.4e+18) || (!(x <= 1e+198) && (x <= 1e+288))) {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if (x <= 3.4e+18) or (not (x <= 1e+198) and (x <= 1e+288)):
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if ((x <= 3.4e+18) || (!(x <= 1e+198) && (x <= 1e+288)))
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if ((x <= 3.4e+18) || (~((x <= 1e+198)) && (x <= 1e+288)))
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[Or[LessEqual[x, 3.4e+18], And[N[Not[LessEqual[x, 1e+198]], $MachinePrecision], LessEqual[x, 1e+288]]], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.4e18 or 1.00000000000000002e198 < x < 1e288

   1. Initial program 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in x around 0 43.0%

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

   6. Taylor expanded in eps around inf 74.9%

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

      1. *-commutative 74.9%

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

      2. associate-*r* 74.9%

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

      3. *-commutative 74.9%

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

      4. mul-1-neg 74.9%

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

      5. sub-neg 74.9%

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

      6. mul-1-neg 74.9%

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

      7. distribute-neg-in 74.9%

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

      8. metadata-eval 74.9%

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

      9. mul-1-neg 74.9%

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

      10. remove-double-neg 74.9%

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

      11. +-commutative 74.9%

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

   8. Simplified 74.9%

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

   9. Taylor expanded in eps around 0 69.6%

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

       1. mul-1-neg 69.6%

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

   11. Simplified 69.6%

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

   12. Taylor expanded in x around 0 65.1%

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

   if 3.4e18 < x < 1.00000000000000002e198 or 1e288 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 60.6%

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

      1. mul-1-neg 60.6%

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

      2. mul-1-neg 60.6%

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

      3. rec-exp 60.6%

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

      4. sub-neg 60.6%

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

      5. div-sub 60.6%

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

      6. mul-1-neg 60.6%

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

      7. rec-exp 60.6%

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

      8. +-inverses 60.6%

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

   7. Simplified 60.6%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+18} \lor \neg \left(x \leq 10^{+198}\right) \land x \leq 10^{+288}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+206}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+284}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.5)
   (/ (+ 2.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666)))))) 2.0)
   (if (<= x 1e+206)
     0.0
     (if (<= x 1.42e+284) (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.5) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 1e+206) {
		tmp = 0.0;
	} else if (x <= 1.42e+284) {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= 2.5d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))) / 2.0d0
    else if (x <= 1d+206) then
        tmp = 0.0d0
    else if (x <= 1.42d+284) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 2.0d0
    else
        tmp = 0.0d0
    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 <= 2.5) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= 1e+206) {
		tmp = 0.0;
	} else if (x <= 1.42e+284) {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.5:
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0
	elif x <= 1e+206:
		tmp = 0.0
	elif x <= 1.42e+284:
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.5)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666)))))) / 2.0);
	elseif (x <= 1e+206)
		tmp = 0.0;
	elseif (x <= 1.42e+284)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	elseif (x <= 1e+206)
		tmp = 0.0;
	elseif (x <= 1.42e+284)
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.5], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+206], 0.0, If[LessEqual[x, 1.42e+284], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.5

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in x around 0 43.9%

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

   6. Taylor expanded in eps around inf 79.3%

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

      1. *-commutative 79.3%

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

      2. associate-*r* 79.3%

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

      3. *-commutative 79.3%

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

      4. mul-1-neg 79.3%

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

      5. sub-neg 79.3%

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

      6. mul-1-neg 79.3%

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

      7. distribute-neg-in 79.3%

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

      8. metadata-eval 79.3%

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

      9. mul-1-neg 79.3%

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

      10. remove-double-neg 79.3%

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

      11. +-commutative 79.3%

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

   8. Simplified 79.3%

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

   9. Taylor expanded in eps around 0 77.1%

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

       1. mul-1-neg 77.1%

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

   11. Simplified 77.1%

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

   12. Taylor expanded in x around 0 70.4%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)}}{2} \]

   if 2.5 < x < 1e206 or 1.42000000000000002e284 < x

   1. Initial program 98.3%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in eps around 0 54.3%

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

      1. mul-1-neg 54.3%

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

      2. mul-1-neg 54.3%

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

      3. rec-exp 54.3%

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

      4. sub-neg 54.3%

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

      5. div-sub 54.3%

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

      6. mul-1-neg 54.3%

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

      7. rec-exp 54.3%

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

      8. +-inverses 54.3%

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

   7. Simplified 54.3%

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

   if 1e206 < x < 1.42000000000000002e284

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 22.2%

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

   6. Taylor expanded in eps around inf 22.5%

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

      1. *-commutative 22.5%

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

      2. associate-*r* 22.5%

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

      3. *-commutative 22.5%

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

      4. mul-1-neg 22.5%

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

      5. sub-neg 22.5%

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

      6. mul-1-neg 22.5%

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

      7. distribute-neg-in 22.5%

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

      8. metadata-eval 22.5%

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

      9. mul-1-neg 22.5%

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

      10. remove-double-neg 22.5%

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

      11. +-commutative 22.5%

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

   8. Simplified 22.5%

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

   9. Taylor expanded in eps around 0 3.1%

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

       1. mul-1-neg 3.1%

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

   11. Simplified 3.1%

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

   12. Taylor expanded in x around 0 67.2%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+206}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+284}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.3% accurate, 9.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.62:\\ \;\;\;\;\frac{x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+211}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+284}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.62)
   (/ (* x (- -1.0 eps_m)) 2.0)
   (if (<= x 3.4e+18)
     1.0
     (if (<= x 5e+211) 0.0 (if (<= x 2.3e+284) (/ (* x eps_m) 2.0) 0.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.62) {
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	} else if (x <= 3.4e+18) {
		tmp = 1.0;
	} else if (x <= 5e+211) {
		tmp = 0.0;
	} else if (x <= 2.3e+284) {
		tmp = (x * eps_m) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= (-0.62d0)) then
        tmp = (x * ((-1.0d0) - eps_m)) / 2.0d0
    else if (x <= 3.4d+18) then
        tmp = 1.0d0
    else if (x <= 5d+211) then
        tmp = 0.0d0
    else if (x <= 2.3d+284) then
        tmp = (x * eps_m) / 2.0d0
    else
        tmp = 0.0d0
    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 <= -0.62) {
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	} else if (x <= 3.4e+18) {
		tmp = 1.0;
	} else if (x <= 5e+211) {
		tmp = 0.0;
	} else if (x <= 2.3e+284) {
		tmp = (x * eps_m) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.62:
		tmp = (x * (-1.0 - eps_m)) / 2.0
	elif x <= 3.4e+18:
		tmp = 1.0
	elif x <= 5e+211:
		tmp = 0.0
	elif x <= 2.3e+284:
		tmp = (x * eps_m) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.62)
		tmp = Float64(Float64(x * Float64(-1.0 - eps_m)) / 2.0);
	elseif (x <= 3.4e+18)
		tmp = 1.0;
	elseif (x <= 5e+211)
		tmp = 0.0;
	elseif (x <= 2.3e+284)
		tmp = Float64(Float64(x * eps_m) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.62)
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	elseif (x <= 3.4e+18)
		tmp = 1.0;
	elseif (x <= 5e+211)
		tmp = 0.0;
	elseif (x <= 2.3e+284)
		tmp = (x * eps_m) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.62], N[(N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.4e+18], 1.0, If[LessEqual[x, 5e+211], 0.0, If[LessEqual[x, 2.3e+284], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.619999999999999996

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 45.7%

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

      1. associate-*r* 45.7%

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

      2. neg-mul-1 45.7%

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

   8. Simplified 45.7%

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

   9. Taylor expanded in x around inf 11.6%

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

       1. +-commutative 11.6%

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

       2. *-commutative 11.6%

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

       3. distribute-rgt1-in 11.6%

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

       4. distribute-lft-in 11.6%

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

       5. associate-*r* 11.6%

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

       6. neg-mul-1 11.6%

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

       7. distribute-rgt-in 11.6%

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

       8. unsub-neg 11.6%

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

   11. Simplified 11.6%

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

   if -0.619999999999999996 < x < 3.4e18

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in x around 0 69.6%

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

   if 3.4e18 < x < 4.9999999999999995e211 or 2.29999999999999984e284 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 60.6%

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

      1. mul-1-neg 60.6%

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

      2. mul-1-neg 60.6%

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

      3. rec-exp 60.6%

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

      4. sub-neg 60.6%

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

      5. div-sub 60.6%

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

      6. mul-1-neg 60.6%

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

      7. rec-exp 60.6%

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

      8. +-inverses 60.6%

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

   7. Simplified 60.6%

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

   if 4.9999999999999995e211 < x < 2.29999999999999984e284

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 27.2%

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

      1. +-commutative 27.2%

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

      2. +-commutative 27.2%

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

      3. associate-+l+ 27.2%

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

      4. mul-1-neg 27.2%

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

      5. distribute-rgt-neg-in 27.2%

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

      6. *-commutative 27.2%

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

      7. distribute-rgt-neg-in 27.2%

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

      8. neg-mul-1 27.2%

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

      9. distribute-rgt-in 27.2%

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

      10. metadata-eval 27.2%

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

      11. associate-*l/ 27.2%

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

      12. metadata-eval 27.2%

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

   7. Simplified 27.2%

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

   8. Taylor expanded in eps around inf 67.5%

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

   10. Simplified 67.5%

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

   11. Taylor expanded in eps around inf 20.9%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.62:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+211}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+284}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.0% accurate, 9.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot eps\_m}{-2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+197}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+287}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (/ (* x eps_m) (- 2.0))
   (if (<= x 3.4e+18)
     1.0
     (if (<= x 3e+197) 0.0 (if (<= x 3.5e+287) (/ (* x eps_m) 2.0) 0.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) / -2.0;
	} else if (x <= 3.4e+18) {
		tmp = 1.0;
	} else if (x <= 3e+197) {
		tmp = 0.0;
	} else if (x <= 3.5e+287) {
		tmp = (x * eps_m) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (x * eps_m) / -2.0d0
    else if (x <= 3.4d+18) then
        tmp = 1.0d0
    else if (x <= 3d+197) then
        tmp = 0.0d0
    else if (x <= 3.5d+287) then
        tmp = (x * eps_m) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) / -2.0;
	} else if (x <= 3.4e+18) {
		tmp = 1.0;
	} else if (x <= 3e+197) {
		tmp = 0.0;
	} else if (x <= 3.5e+287) {
		tmp = (x * eps_m) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (x * eps_m) / -2.0
	elif x <= 3.4e+18:
		tmp = 1.0
	elif x <= 3e+197:
		tmp = 0.0
	elif x <= 3.5e+287:
		tmp = (x * eps_m) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * eps_m) / Float64(-2.0));
	elseif (x <= 3.4e+18)
		tmp = 1.0;
	elseif (x <= 3e+197)
		tmp = 0.0;
	elseif (x <= 3.5e+287)
		tmp = Float64(Float64(x * eps_m) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * eps_m) / -2.0;
	elseif (x <= 3.4e+18)
		tmp = 1.0;
	elseif (x <= 3e+197)
		tmp = 0.0;
	elseif (x <= 3.5e+287)
		tmp = (x * eps_m) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(x * eps$95$m), $MachinePrecision] / (-2.0)), $MachinePrecision], If[LessEqual[x, 3.4e+18], 1.0, If[LessEqual[x, 3e+197], 0.0, If[LessEqual[x, 3.5e+287], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 45.7%

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

      1. associate-*r* 45.7%

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

      2. neg-mul-1 45.7%

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

   8. Simplified 45.7%

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

   9. Taylor expanded in x around 0 11.6%

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

   10. Taylor expanded in eps around inf 11.6%

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

       1. associate-*r* 11.6%

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

       2. neg-mul-1 11.6%

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

       3. *-commutative 11.6%

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

   12. Simplified 11.6%

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

   if -1 < x < 3.4e18

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in x around 0 69.6%

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

   if 3.4e18 < x < 3.0000000000000002e197 or 3.49999999999999976e287 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 60.6%

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

      1. mul-1-neg 60.6%

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

      2. mul-1-neg 60.6%

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

      3. rec-exp 60.6%

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

      4. sub-neg 60.6%

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

      5. div-sub 60.6%

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

      6. mul-1-neg 60.6%

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

      7. rec-exp 60.6%

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

      8. +-inverses 60.6%

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

   7. Simplified 60.6%

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

   if 3.0000000000000002e197 < x < 3.49999999999999976e287

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 27.2%

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

      1. +-commutative 27.2%

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

      2. +-commutative 27.2%

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

      3. associate-+l+ 27.2%

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

      4. mul-1-neg 27.2%

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

      5. distribute-rgt-neg-in 27.2%

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

      6. *-commutative 27.2%

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

      7. distribute-rgt-neg-in 27.2%

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

      8. neg-mul-1 27.2%

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

      9. distribute-rgt-in 27.2%

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

      10. metadata-eval 27.2%

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

      11. associate-*l/ 27.2%

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

      12. metadata-eval 27.2%

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

   7. Simplified 27.2%

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

   8. Taylor expanded in eps around inf 67.5%

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

   10. Simplified 67.5%

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

   11. Taylor expanded in eps around inf 20.9%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \varepsilon}{-2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+197}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+287}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+195}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+288}:\\ \;\;\;\;\frac{x \cdot eps\_m + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.0)
   (/ (+ 2.0 (* x (- -1.0 eps_m))) 2.0)
   (if (<= x 1e+195)
     0.0
     (if (<= x 4.2e+288) (/ (+ (* x eps_m) 2.0) 2.0) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1e+195) {
		tmp = 0.0;
	} else if (x <= 4.2e+288) {
		tmp = ((x * eps_m) + 2.0) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= 2.0d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps_m))) / 2.0d0
    else if (x <= 1d+195) then
        tmp = 0.0d0
    else if (x <= 4.2d+288) then
        tmp = ((x * eps_m) + 2.0d0) / 2.0d0
    else
        tmp = 0.0d0
    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 <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1e+195) {
		tmp = 0.0;
	} else if (x <= 4.2e+288) {
		tmp = ((x * eps_m) + 2.0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0
	elif x <= 1e+195:
		tmp = 0.0
	elif x <= 4.2e+288:
		tmp = ((x * eps_m) + 2.0) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 1e+195)
		tmp = 0.0;
	elseif (x <= 4.2e+288)
		tmp = Float64(Float64(Float64(x * eps_m) + 2.0) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	elseif (x <= 1e+195)
		tmp = 0.0;
	elseif (x <= 4.2e+288)
		tmp = ((x * eps_m) + 2.0) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.0], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+195], 0.0, If[LessEqual[x, 4.2e+288], N[(N[(N[(x * eps$95$m), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 2

   1. Initial program 63.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in eps around inf 98.6%

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

   6. Taylor expanded in x around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. neg-mul-1 76.6%

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

   8. Simplified 76.6%

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

   9. Taylor expanded in x around 0 56.8%

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

       1. associate-+r+ 56.8%

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

       2. distribute-lft-neg-out 56.8%

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

       3. unsub-neg 56.8%

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

       4. metadata-eval 56.8%

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

       5. +-commutative 56.8%

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

   11. Applied egg-rr 56.8%

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

   if 2 < x < 9.99999999999999977e194 or 4.19999999999999998e288 < x

   1. Initial program 98.3%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in eps around 0 54.3%

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

      1. mul-1-neg 54.3%

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

      2. mul-1-neg 54.3%

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

      3. rec-exp 54.3%

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

      4. sub-neg 54.3%

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

      5. div-sub 54.3%

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

      6. mul-1-neg 54.3%

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

      7. rec-exp 54.3%

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

      8. +-inverses 54.3%

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

   7. Simplified 54.3%

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

   if 9.99999999999999977e194 < x < 4.19999999999999998e288

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 22.2%

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

   6. Taylor expanded in x around 0 20.5%

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

      1. mul-1-neg 20.5%

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

      2. unsub-neg 20.5%

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

      3. associate-*r* 20.5%

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

   8. Simplified 20.5%

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

   9. Taylor expanded in eps around inf 20.9%

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

       1. associate-*r* 20.9%

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

       2. neg-mul-1 20.9%

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

   11. Simplified 20.9%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+195}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+288}:\\ \;\;\;\;\frac{x \cdot \varepsilon + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.6% accurate, 11.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+202}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+288}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 3.4e+18)
   1.0
   (if (<= x 2.5e+202) 0.0 (if (<= x 3.7e+288) (/ (* x eps_m) 2.0) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 3.4e+18) {
		tmp = 1.0;
	} else if (x <= 2.5e+202) {
		tmp = 0.0;
	} else if (x <= 3.7e+288) {
		tmp = (x * eps_m) / 2.0;
	} else {
		tmp = 0.0;
	}
	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 <= 3.4d+18) then
        tmp = 1.0d0
    else if (x <= 2.5d+202) then
        tmp = 0.0d0
    else if (x <= 3.7d+288) then
        tmp = (x * eps_m) / 2.0d0
    else
        tmp = 0.0d0
    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 <= 3.4e+18) {
		tmp = 1.0;
	} else if (x <= 2.5e+202) {
		tmp = 0.0;
	} else if (x <= 3.7e+288) {
		tmp = (x * eps_m) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 3.4e+18:
		tmp = 1.0
	elif x <= 2.5e+202:
		tmp = 0.0
	elif x <= 3.7e+288:
		tmp = (x * eps_m) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 3.4e+18)
		tmp = 1.0;
	elseif (x <= 2.5e+202)
		tmp = 0.0;
	elseif (x <= 3.7e+288)
		tmp = Float64(Float64(x * eps_m) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 3.4e+18)
		tmp = 1.0;
	elseif (x <= 2.5e+202)
		tmp = 0.0;
	elseif (x <= 3.7e+288)
		tmp = (x * eps_m) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 3.4e+18], 1.0, If[LessEqual[x, 2.5e+202], 0.0, If[LessEqual[x, 3.7e+288], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.4e18

   1. Initial program 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in x around 0 55.3%

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

   if 3.4e18 < x < 2.5e202 or 3.6999999999999998e288 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 60.6%

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

      1. mul-1-neg 60.6%

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

      2. mul-1-neg 60.6%

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

      3. rec-exp 60.6%

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

      4. sub-neg 60.6%

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

      5. div-sub 60.6%

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

      6. mul-1-neg 60.6%

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

      7. rec-exp 60.6%

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

      8. +-inverses 60.6%

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

   7. Simplified 60.6%

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

   if 2.5e202 < x < 3.6999999999999998e288

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 27.2%

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

      1. +-commutative 27.2%

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

      2. +-commutative 27.2%

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

      3. associate-+l+ 27.2%

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

      4. mul-1-neg 27.2%

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

      5. distribute-rgt-neg-in 27.2%

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

      6. *-commutative 27.2%

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

      7. distribute-rgt-neg-in 27.2%

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

      8. neg-mul-1 27.2%

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

      9. distribute-rgt-in 27.2%

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

      10. metadata-eval 27.2%

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

      11. associate-*l/ 27.2%

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

      12. metadata-eval 27.2%

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

   7. Simplified 27.2%

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

   8. Taylor expanded in eps around inf 67.5%

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

   10. Simplified 67.5%

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

   11. Taylor expanded in eps around inf 20.9%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+202}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+288}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.3% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 3.4e+18) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 3.4e+18) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	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 <= 3.4d+18) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    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 <= 3.4e+18) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 3.4e+18:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 3.4e+18)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 3.4e+18)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 3.4e+18], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.4e18

   1. Initial program 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in x around 0 55.3%

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

   if 3.4e18 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 54.6%

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

      1. mul-1-neg 54.6%

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

      2. mul-1-neg 54.6%

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

      3. rec-exp 54.6%

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

      4. sub-neg 54.6%

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

      5. div-sub 54.6%

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

      6. mul-1-neg 54.6%

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

      7. rec-exp 54.6%

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

      8. +-inverses 54.6%

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

   7. Simplified 54.6%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 16.1% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

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

Fortran

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 0.0
end

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 73.1%

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

3. Simplified 62.1%

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

5. Taylor expanded in eps around 0 15.4%

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

   1. mul-1-neg 15.4%

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

   2. mul-1-neg 15.4%

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

   3. rec-exp 15.4%

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

   4. sub-neg 15.4%

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

   5. div-sub 15.4%

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

   6. mul-1-neg 15.4%

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

   7. rec-exp 15.4%

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

   8. +-inverses 15.7%

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

7. Simplified 15.7%

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

8. Final simplification 15.7%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(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))