NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.3% → 99.3%

Time: 23.1s

Alternatives: 16

Speedup: 1.8×

• 38.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 7 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + \frac{1}{{e}^{\left(x \cdot eps\_m\right)}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 7e-15)
   (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   (/ (+ (exp (* x eps_m)) (/ 1.0 (pow E (* x eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 7e-15) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (exp((x * eps_m)) + (1.0 / pow(((double) M_E), (x * eps_m)))) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 7e-15) {
		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.pow(Math.E, (x * eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 7e-15:
		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.pow(math.e, (x * eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 7e-15)
		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 * eps_m)) + Float64(1.0 / (exp(1) ^ Float64(x * eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 7e-15)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	else
		tmp = (exp((x * eps_m)) + (1.0 / (2.71828182845904523536 ^ (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, 7e-15], 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 * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[E, N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 7.0000000000000001e-15

   1. Initial program 58.4%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in eps around 0 29.2%

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

      1. associate-+r+ 71.1%

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

      2. mul-1-neg 71.1%

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

      3. sub-neg 71.1%

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

      4. +-inverses 71.1%

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

      5. associate-*r* 71.1%

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

      6. distribute-rgt-out 71.1%

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

      7. mul-1-neg 71.1%

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

   7. Simplified 71.1%

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

   if 7.0000000000000001e-15 < eps

   1. Initial program 100.0%

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

   3. Simplified 90.4%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. exp-1-e 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   15. Simplified 100.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 7 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \frac{1}{{e}^{\left(x \cdot \varepsilon\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot eps\_m}\\ \mathbf{if}\;eps\_m \leq 7 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + \frac{1}{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x eps_m))))
   (if (<= eps_m 7e-15)
     (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
     (/ (+ t_0 (/ 1.0 t_0)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * eps_m));
	double tmp;
	if (eps_m <= 7e-15) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (t_0 + (1.0 / t_0)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * eps_m))
    if (eps_m <= 7d-15) then
        tmp = ((eps_m * (exp(-x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else
        tmp = (t_0 + (1.0d0 / t_0)) / 2.0d0
    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));
	double tmp;
	if (eps_m <= 7e-15) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (t_0 + (1.0 / t_0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * eps_m))
	tmp = 0
	if eps_m <= 7e-15:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	else:
		tmp = (t_0 + (1.0 / t_0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * eps_m))
	tmp = 0.0
	if (eps_m <= 7e-15)
		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(t_0 + Float64(1.0 / t_0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * eps_m));
	tmp = 0.0;
	if (eps_m <= 7e-15)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	else
		tmp = (t_0 + (1.0 / t_0)) / 2.0;
	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 * eps$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eps$95$m, 7e-15], 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[(t$95$0 + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 7.0000000000000001e-15

   1. Initial program 58.4%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in eps around 0 29.2%

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

      1. associate-+r+ 71.1%

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

      2. mul-1-neg 71.1%

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

      3. sub-neg 71.1%

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

      4. +-inverses 71.1%

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

      5. associate-*r* 71.1%

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

      6. distribute-rgt-out 71.1%

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

      7. mul-1-neg 71.1%

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

   7. Simplified 71.1%

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

   if 7.0000000000000001e-15 < eps

   1. Initial program 100.0%

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

   3. Simplified 90.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. 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 eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 7 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \frac{1}{e^{x \cdot \varepsilon}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

Fortran

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return (Math.exp((x * (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.exp((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(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 = (exp((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[Exp[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 73.5%

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

3. Simplified 65.4%

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

5. Taylor expanded in eps around inf 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-282}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(1 - eps\_m\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+66}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+80} \lor \neg \left(x \leq 2 \cdot 10^{+261}\right):\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-282)
   (/ (+ 1.0 (pow E (* x (- 1.0 eps_m)))) 2.0)
   (if (<= x 1.12e+66)
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (if (or (<= x 2.1e+80) (not (<= x 2e+261)))
       (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
       (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-282) {
		tmp = (1.0 + pow(((double) M_E), (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.12e+66) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else if ((x <= 2.1e+80) || !(x <= 2e+261)) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-282) {
		tmp = (1.0 + Math.pow(Math.E, (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.12e+66) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else if ((x <= 2.1e+80) || !(x <= 2e+261)) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -5e-282:
		tmp = (1.0 + math.pow(math.e, (x * (1.0 - eps_m)))) / 2.0
	elif x <= 1.12e+66:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	elif (x <= 2.1e+80) or not (x <= 2e+261):
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -5e-282)
		tmp = Float64(Float64(1.0 + (exp(1) ^ Float64(x * Float64(1.0 - eps_m)))) / 2.0);
	elseif (x <= 1.12e+66)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	elseif ((x <= 2.1e+80) || !(x <= 2e+261))
		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(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -5e-282)
		tmp = (1.0 + (2.71828182845904523536 ^ (x * (1.0 - eps_m)))) / 2.0;
	elseif (x <= 1.12e+66)
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	elseif ((x <= 2.1e+80) || ~((x <= 2e+261)))
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5e-282], N[(N[(1.0 + N[Power[E, N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.12e+66], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 2.1e+80], N[Not[LessEqual[x, 2e+261]], $MachinePrecision]], 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[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.0000000000000001e-282

   1. Initial program 71.6%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in eps around inf 99.8%

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

   6. Taylor expanded in x around 0 68.9%

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

      1. sub-neg 68.9%

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

      2. metadata-eval 68.9%

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

      3. distribute-rgt-in 68.9%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 73.5%

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

      6. sqr-neg 73.5%

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

      7. sqrt-unprod 77.4%

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

      8. add-sqr-sqrt 77.4%

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

      9. neg-mul-1 77.4%

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

      10. *-un-lft-identity 77.4%

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

      11. distribute-rgt-in 77.4%

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

      12. +-commutative 77.4%

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

      13. *-commutative 77.4%

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

      14. distribute-rgt-neg-out 77.4%

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

      15. neg-sub0 77.4%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 66.9%

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

      18. sqr-neg 66.9%

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

      19. sqrt-unprod 68.8%

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

      20. add-sqr-sqrt 68.8%

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

      21. *-commutative 68.8%

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

      22. +-commutative 68.8%

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

      23. distribute-rgt-in 68.8%

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

   8. Applied egg-rr 77.4%

      \[\leadsto \frac{e^{\color{blue}{0 - x \cdot \left(\varepsilon + -1\right)}} + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 77.4%

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

      2. distribute-rgt-neg-in 77.4%

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

      3. +-commutative 77.4%

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

      4. distribute-neg-in 77.4%

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

      5. metadata-eval 77.4%

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

      6. sub-neg 77.4%

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

   10. Simplified 77.4%

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

       1. *-commutative 77.4%

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

       2. *-un-lft-identity 77.4%

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

       3. exp-prod 77.4%

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

       4. e-exp-1 77.4%

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

       5. *-commutative 77.4%

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

   12. Applied egg-rr 77.4%

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

   if -5.0000000000000001e-282 < x < 1.12e66

   1. Initial program 57.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in eps around inf 98.1%

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

   6. Taylor expanded in x around 0 84.0%

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

   7. Taylor expanded in eps around inf 84.7%

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

      1. *-commutative 97.9%

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

   9. Simplified 84.7%

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

   if 1.12e66 < x < 2.10000000000000001e80 or 1.9999999999999999e261 < 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 78.1%

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

      1. associate-+r+ 78.1%

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

      2. mul-1-neg 78.1%

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

      3. sub-neg 78.1%

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

      4. +-inverses 78.1%

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

      5. associate-*r* 78.1%

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

      6. distribute-rgt-out 78.1%

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

      7. mul-1-neg 78.1%

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

   7. Simplified 78.1%

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

   if 2.10000000000000001e80 < x < 1.9999999999999999e261

   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 41.0%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-282}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(1 - \varepsilon\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+66}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+80} \lor \neg \left(x \leq 2 \cdot 10^{+261}\right):\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{elif}\;x \leq 10^{+81}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{+266}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-280)
   (/ (+ 1.0 (exp (* x (- 1.0 eps_m)))) 2.0)
   (if (<= x 3.5e+70)
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (if (<= x 1e+81)
       0.0
       (if (<= x 3.15e+266) (/ (+ 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 <= -1e-280) {
		tmp = (1.0 + exp((x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 3.5e+70) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else if (x <= 1e+81) {
		tmp = 0.0;
	} else if (x <= 3.15e+266) {
		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 <= (-1d-280)) then
        tmp = (1.0d0 + exp((x * (1.0d0 - eps_m)))) / 2.0d0
    else if (x <= 3.5d+70) then
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    else if (x <= 1d+81) then
        tmp = 0.0d0
    else if (x <= 3.15d+266) 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 <= -1e-280) {
		tmp = (1.0 + Math.exp((x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 3.5e+70) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else if (x <= 1e+81) {
		tmp = 0.0;
	} else if (x <= 3.15e+266) {
		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 <= -1e-280:
		tmp = (1.0 + math.exp((x * (1.0 - eps_m)))) / 2.0
	elif x <= 3.5e+70:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	elif x <= 1e+81:
		tmp = 0.0
	elif x <= 3.15e+266:
		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 <= -1e-280)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(1.0 - eps_m)))) / 2.0);
	elseif (x <= 3.5e+70)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	elseif (x <= 1e+81)
		tmp = 0.0;
	elseif (x <= 3.15e+266)
		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 <= -1e-280)
		tmp = (1.0 + exp((x * (1.0 - eps_m)))) / 2.0;
	elseif (x <= 3.5e+70)
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	elseif (x <= 1e+81)
		tmp = 0.0;
	elseif (x <= 3.15e+266)
		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, -1e-280], N[(N[(1.0 + N[Exp[N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.5e+70], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+81], 0.0, If[LessEqual[x, 3.15e+266], 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 4 regimes

2. if x < -9.9999999999999996e-281

   1. Initial program 71.6%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in eps around inf 99.8%

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

   6. Taylor expanded in x around 0 68.9%

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

      1. sub-neg 68.9%

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

      2. metadata-eval 68.9%

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

      3. distribute-rgt-in 68.9%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 73.5%

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

      6. sqr-neg 73.5%

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

      7. sqrt-unprod 77.4%

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

      8. add-sqr-sqrt 77.4%

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

      9. neg-mul-1 77.4%

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

      10. *-un-lft-identity 77.4%

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

      11. distribute-rgt-in 77.4%

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

      12. +-commutative 77.4%

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

      13. *-commutative 77.4%

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

      14. distribute-rgt-neg-out 77.4%

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

      15. neg-sub0 77.4%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 66.9%

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

      18. sqr-neg 66.9%

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

      19. sqrt-unprod 68.8%

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

      20. add-sqr-sqrt 68.8%

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

      21. *-commutative 68.8%

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

      22. +-commutative 68.8%

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

      23. distribute-rgt-in 68.8%

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

   8. Applied egg-rr 77.4%

      \[\leadsto \frac{e^{\color{blue}{0 - x \cdot \left(\varepsilon + -1\right)}} + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 77.4%

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

      2. distribute-rgt-neg-in 77.4%

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

      3. +-commutative 77.4%

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

      4. distribute-neg-in 77.4%

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

      5. metadata-eval 77.4%

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

      6. sub-neg 77.4%

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

   10. Simplified 77.4%

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

   if -9.9999999999999996e-281 < x < 3.50000000000000002e70

   1. Initial program 57.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in eps around inf 98.1%

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

   6. Taylor expanded in x around 0 84.0%

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

   7. Taylor expanded in eps around inf 84.7%

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

      1. *-commutative 97.9%

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

   9. Simplified 84.7%

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

   if 3.50000000000000002e70 < x < 9.99999999999999921e80 or 3.15e266 < 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 78.1%

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

      1. mul-1-neg 78.1%

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

      2. mul-1-neg 78.1%

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

      3. rec-exp 78.1%

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

      4. sub-neg 78.1%

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

      5. div-sub 78.1%

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

      6. mul-1-neg 78.1%

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

      7. rec-exp 78.1%

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

      8. +-inverses 78.1%

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

   7. Simplified 78.1%

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

   if 9.99999999999999921e80 < x < 3.15e266

   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 41.0%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 10^{+81}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{+266}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(1 - eps\_m\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+66}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+85}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.65 \cdot 10^{+266}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-280)
   (/ (+ 1.0 (pow E (* x (- 1.0 eps_m)))) 2.0)
   (if (<= x 1.56e+66)
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (if (<= x 1.15e+85)
       0.0
       (if (<= x 4.65e+266) (/ (+ 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 <= -1e-280) {
		tmp = (1.0 + pow(((double) M_E), (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.56e+66) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else if (x <= 1.15e+85) {
		tmp = 0.0;
	} else if (x <= 4.65e+266) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-280) {
		tmp = (1.0 + Math.pow(Math.E, (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.56e+66) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else if (x <= 1.15e+85) {
		tmp = 0.0;
	} else if (x <= 4.65e+266) {
		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 <= -1e-280:
		tmp = (1.0 + math.pow(math.e, (x * (1.0 - eps_m)))) / 2.0
	elif x <= 1.56e+66:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	elif x <= 1.15e+85:
		tmp = 0.0
	elif x <= 4.65e+266:
		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 <= -1e-280)
		tmp = Float64(Float64(1.0 + (exp(1) ^ Float64(x * Float64(1.0 - eps_m)))) / 2.0);
	elseif (x <= 1.56e+66)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	elseif (x <= 1.15e+85)
		tmp = 0.0;
	elseif (x <= 4.65e+266)
		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 <= -1e-280)
		tmp = (1.0 + (2.71828182845904523536 ^ (x * (1.0 - eps_m)))) / 2.0;
	elseif (x <= 1.56e+66)
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	elseif (x <= 1.15e+85)
		tmp = 0.0;
	elseif (x <= 4.65e+266)
		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, -1e-280], N[(N[(1.0 + N[Power[E, N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.56e+66], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.15e+85], 0.0, If[LessEqual[x, 4.65e+266], 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 4 regimes

2. if x < -9.9999999999999996e-281

   1. Initial program 71.6%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in eps around inf 99.8%

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

   6. Taylor expanded in x around 0 68.9%

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

      1. sub-neg 68.9%

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

      2. metadata-eval 68.9%

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

      3. distribute-rgt-in 68.9%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 73.5%

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

      6. sqr-neg 73.5%

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

      7. sqrt-unprod 77.4%

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

      8. add-sqr-sqrt 77.4%

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

      9. neg-mul-1 77.4%

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

      10. *-un-lft-identity 77.4%

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

      11. distribute-rgt-in 77.4%

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

      12. +-commutative 77.4%

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

      13. *-commutative 77.4%

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

      14. distribute-rgt-neg-out 77.4%

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

      15. neg-sub0 77.4%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 66.9%

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

      18. sqr-neg 66.9%

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

      19. sqrt-unprod 68.8%

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

      20. add-sqr-sqrt 68.8%

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

      21. *-commutative 68.8%

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

      22. +-commutative 68.8%

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

      23. distribute-rgt-in 68.8%

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

   8. Applied egg-rr 77.4%

      \[\leadsto \frac{e^{\color{blue}{0 - x \cdot \left(\varepsilon + -1\right)}} + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 77.4%

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

      2. distribute-rgt-neg-in 77.4%

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

      3. +-commutative 77.4%

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

      4. distribute-neg-in 77.4%

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

      5. metadata-eval 77.4%

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

      6. sub-neg 77.4%

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

   10. Simplified 77.4%

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

       1. *-commutative 77.4%

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

       2. *-un-lft-identity 77.4%

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

       3. exp-prod 77.4%

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

       4. e-exp-1 77.4%

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

       5. *-commutative 77.4%

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

   12. Applied egg-rr 77.4%

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

   if -9.9999999999999996e-281 < x < 1.5599999999999999e66

   1. Initial program 57.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in eps around inf 98.1%

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

   6. Taylor expanded in x around 0 84.0%

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

   7. Taylor expanded in eps around inf 84.7%

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

      1. *-commutative 97.9%

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

   9. Simplified 84.7%

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

   if 1.5599999999999999e66 < x < 1.1499999999999999e85 or 4.65e266 < x

   1. Initial program 100.0%

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

   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 78.1%

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

      1. mul-1-neg 78.1%

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

      2. mul-1-neg 78.1%

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

      3. rec-exp 78.1%

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

      4. sub-neg 78.1%

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

      5. div-sub 78.1%

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

      6. mul-1-neg 78.1%

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

      7. rec-exp 78.1%

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

      8. +-inverses 78.1%

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

   7. Simplified 78.1%

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

   if 1.1499999999999999e85 < x < 4.65e266

   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 41.0%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(1 - \varepsilon\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+66}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+85}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.65 \cdot 10^{+266}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -450:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+70} \lor \neg \left(x \leq 1.2 \cdot 10^{+80}\right) \land x \leq 4.65 \cdot 10^{+266}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -450.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (or (<= x 4.9e+70) (and (not (<= x 1.2e+80)) (<= x 4.65e+266)))
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -450.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if ((x <= 4.9e+70) || (!(x <= 1.2e+80) && (x <= 4.65e+266))) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -450.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if ((x <= 4.9e+70) || (!(x <= 1.2e+80) && (x <= 4.65e+266))) {
		tmp = (1.0 + Math.exp((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 <= -450.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif (x <= 4.9e+70) or (not (x <= 1.2e+80) and (x <= 4.65e+266)):
		tmp = (1.0 + math.exp((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 <= -450.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif ((x <= 4.9e+70) || (!(x <= 1.2e+80) && (x <= 4.65e+266)))
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -450.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 4.9e+70], And[N[Not[LessEqual[x, 1.2e+80]], $MachinePrecision], LessEqual[x, 4.65e+266]]], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -450

   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 34.5%

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

   6. Taylor expanded in eps around 0 67.6%

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

      1. expm1-define 67.6%

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

      2. neg-mul-1 67.6%

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

   8. Simplified 67.6%

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

   if -450 < x < 4.90000000000000028e70 or 1.1999999999999999e80 < x < 4.65e266

   1. Initial program 66.7%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in eps around inf 99.0%

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

   6. Taylor expanded in x around 0 75.2%

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

   7. Taylor expanded in eps around inf 75.6%

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

      1. *-commutative 91.3%

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

   9. Simplified 75.6%

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

   if 4.90000000000000028e70 < x < 1.1999999999999999e80 or 4.65e266 < x

   1. Initial program 100.0%

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

   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 78.1%

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

      1. mul-1-neg 78.1%

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

      2. mul-1-neg 78.1%

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

      3. rec-exp 78.1%

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

      4. sub-neg 78.1%

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

      5. div-sub 78.1%

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

      6. mul-1-neg 78.1%

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

      7. rec-exp 78.1%

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

      8. +-inverses 78.1%

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

   7. Simplified 78.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -450:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+70} \lor \neg \left(x \leq 1.2 \cdot 10^{+80}\right) \land x \leq 4.65 \cdot 10^{+266}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+66} \lor \neg \left(x \leq 1.2 \cdot 10^{+80}\right) \land x \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-280)
   (/ (+ 1.0 (exp (* x (- 1.0 eps_m)))) 2.0)
   (if (or (<= x 1.25e+66) (and (not (<= x 1.2e+80)) (<= x 5e+260)))
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-280) {
		tmp = (1.0 + exp((x * (1.0 - eps_m)))) / 2.0;
	} else if ((x <= 1.25e+66) || (!(x <= 1.2e+80) && (x <= 5e+260))) {
		tmp = (1.0 + exp((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 <= (-1d-280)) then
        tmp = (1.0d0 + exp((x * (1.0d0 - eps_m)))) / 2.0d0
    else if ((x <= 1.25d+66) .or. (.not. (x <= 1.2d+80)) .and. (x <= 5d+260)) then
        tmp = (1.0d0 + exp((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 <= -1e-280) {
		tmp = (1.0 + Math.exp((x * (1.0 - eps_m)))) / 2.0;
	} else if ((x <= 1.25e+66) || (!(x <= 1.2e+80) && (x <= 5e+260))) {
		tmp = (1.0 + Math.exp((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 <= -1e-280:
		tmp = (1.0 + math.exp((x * (1.0 - eps_m)))) / 2.0
	elif (x <= 1.25e+66) or (not (x <= 1.2e+80) and (x <= 5e+260)):
		tmp = (1.0 + math.exp((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 <= -1e-280)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(1.0 - eps_m)))) / 2.0);
	elseif ((x <= 1.25e+66) || (!(x <= 1.2e+80) && (x <= 5e+260)))
		tmp = Float64(Float64(1.0 + exp(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 <= -1e-280)
		tmp = (1.0 + exp((x * (1.0 - eps_m)))) / 2.0;
	elseif ((x <= 1.25e+66) || (~((x <= 1.2e+80)) && (x <= 5e+260)))
		tmp = (1.0 + exp((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, -1e-280], N[(N[(1.0 + N[Exp[N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.25e+66], And[N[Not[LessEqual[x, 1.2e+80]], $MachinePrecision], LessEqual[x, 5e+260]]], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -9.9999999999999996e-281

   1. Initial program 71.6%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in eps around inf 99.8%

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

   6. Taylor expanded in x around 0 68.9%

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

      1. sub-neg 68.9%

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

      2. metadata-eval 68.9%

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

      3. distribute-rgt-in 68.9%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 73.5%

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

      6. sqr-neg 73.5%

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

      7. sqrt-unprod 77.4%

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

      8. add-sqr-sqrt 77.4%

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

      9. neg-mul-1 77.4%

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

      10. *-un-lft-identity 77.4%

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

      11. distribute-rgt-in 77.4%

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

      12. +-commutative 77.4%

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

      13. *-commutative 77.4%

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

      14. distribute-rgt-neg-out 77.4%

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

      15. neg-sub0 77.4%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 66.9%

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

      18. sqr-neg 66.9%

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

      19. sqrt-unprod 68.8%

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

      20. add-sqr-sqrt 68.8%

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

      21. *-commutative 68.8%

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

      22. +-commutative 68.8%

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

      23. distribute-rgt-in 68.8%

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

   8. Applied egg-rr 77.4%

      \[\leadsto \frac{e^{\color{blue}{0 - x \cdot \left(\varepsilon + -1\right)}} + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 77.4%

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

      2. distribute-rgt-neg-in 77.4%

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

      3. +-commutative 77.4%

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

      4. distribute-neg-in 77.4%

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

      5. metadata-eval 77.4%

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

      6. sub-neg 77.4%

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

   10. Simplified 77.4%

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

   if -9.9999999999999996e-281 < x < 1.24999999999999998e66 or 1.1999999999999999e80 < x < 4.9999999999999996e260

   1. Initial program 71.4%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in x around 0 69.8%

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

   7. Taylor expanded in eps around inf 70.2%

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

      1. *-commutative 87.3%

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

   9. Simplified 70.2%

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

   if 1.24999999999999998e66 < x < 1.1999999999999999e80 or 4.9999999999999996e260 < 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 78.1%

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

      1. mul-1-neg 78.1%

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

      2. mul-1-neg 78.1%

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

      3. rec-exp 78.1%

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

      4. sub-neg 78.1%

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

      5. div-sub 78.1%

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

      6. mul-1-neg 78.1%

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

      7. rec-exp 78.1%

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

      8. +-inverses 78.1%

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

   7. Simplified 78.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-280}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+66} \lor \neg \left(x \leq 1.2 \cdot 10^{+80}\right) \land x \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \frac{x + eps\_m \cdot \left(x \cdot 0 - x \cdot eps\_m\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+61} \lor \neg \left(x \leq 3 \cdot 10^{+81}\right) \land x \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -3.2e-12)
   (/ (+ 2.0 (/ (+ x (* eps_m (- (* x 0.0) (* x eps_m)))) eps_m)) 2.0)
   (if (or (<= x 4e+61) (and (not (<= x 3e+81)) (<= x 2e+256)))
     (/ (+ 1.0 (exp x)) 2.0)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -3.2e-12) {
		tmp = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	} else if ((x <= 4e+61) || (!(x <= 3e+81) && (x <= 2e+256))) {
		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 <= (-3.2d-12)) then
        tmp = (2.0d0 + ((x + (eps_m * ((x * 0.0d0) - (x * eps_m)))) / eps_m)) / 2.0d0
    else if ((x <= 4d+61) .or. (.not. (x <= 3d+81)) .and. (x <= 2d+256)) 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 <= -3.2e-12) {
		tmp = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	} else if ((x <= 4e+61) || (!(x <= 3e+81) && (x <= 2e+256))) {
		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 <= -3.2e-12:
		tmp = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0
	elif (x <= 4e+61) or (not (x <= 3e+81) and (x <= 2e+256)):
		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 <= -3.2e-12)
		tmp = Float64(Float64(2.0 + Float64(Float64(x + Float64(eps_m * Float64(Float64(x * 0.0) - Float64(x * eps_m)))) / eps_m)) / 2.0);
	elseif ((x <= 4e+61) || (!(x <= 3e+81) && (x <= 2e+256)))
		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 <= -3.2e-12)
		tmp = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	elseif ((x <= 4e+61) || (~((x <= 3e+81)) && (x <= 2e+256)))
		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, -3.2e-12], N[(N[(2.0 + N[(N[(x + N[(eps$95$m * N[(N[(x * 0.0), $MachinePrecision] - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 4e+61], And[N[Not[LessEqual[x, 3e+81]], $MachinePrecision], LessEqual[x, 2e+256]]], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e-12

   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 34.5%

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

   6. Taylor expanded in x around 0 13.2%

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

      1. associate-*r* 13.2%

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

      2. neg-mul-1 13.2%

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

      3. *-commutative 13.2%

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

   8. Simplified 13.2%

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

      1. distribute-lft-in 13.2%

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

      2. *-rgt-identity 13.2%

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

      3. distribute-rgt-in 13.2%

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

      4. add-sqr-sqrt 13.2%

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

      5. sqrt-unprod 16.0%

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

      6. sqr-neg 16.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 40.2%

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

      9. un-div-inv 40.2%

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

      10. add-sqr-sqrt 40.2%

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

      11. sqrt-unprod 10.3%

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

      12. sqr-neg 10.3%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 40.2%

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

   10. Applied egg-rr 40.2%

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

   11. Taylor expanded in eps around 0 56.4%

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

       1. associate-+r+ 56.4%

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

       2. distribute-rgt1-in 56.4%

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

       3. metadata-eval 56.4%

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

       4. associate-*r* 56.4%

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

       5. neg-mul-1 56.4%

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

       6. *-commutative 56.4%

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

   13. Simplified 56.4%

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

   if -3.2000000000000001e-12 < x < 3.9999999999999998e61 or 2.99999999999999997e81 < x < 2.0000000000000001e256

   1. Initial program 66.7%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in eps around inf 99.0%

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

   6. Taylor expanded in x around 0 75.2%

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

      1. sub-neg 75.2%

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

      2. metadata-eval 75.2%

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

      3. distribute-rgt-in 75.2%

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

      4. add-sqr-sqrt 44.6%

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

      5. sqrt-unprod 70.9%

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

      6. sqr-neg 70.9%

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

      7. sqrt-unprod 29.1%

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

      8. add-sqr-sqrt 69.4%

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

      9. neg-mul-1 69.4%

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

      10. *-un-lft-identity 69.4%

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

      11. distribute-rgt-in 69.4%

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

      12. +-commutative 69.4%

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

      13. *-commutative 69.4%

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

      14. distribute-rgt-neg-out 69.4%

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

      15. neg-sub0 69.4%

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

      16. add-sqr-sqrt 40.3%

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

      17. sqrt-unprod 69.6%

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

      18. sqr-neg 69.6%

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

      19. sqrt-unprod 30.6%

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

      20. add-sqr-sqrt 75.1%

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

      21. *-commutative 75.1%

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

      22. +-commutative 75.1%

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

      23. distribute-rgt-in 75.1%

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

   8. Applied egg-rr 69.3%

      \[\leadsto \frac{e^{\color{blue}{0 - x \cdot \left(\varepsilon + -1\right)}} + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 69.3%

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

      2. distribute-rgt-neg-in 69.3%

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

      3. +-commutative 69.3%

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

      4. distribute-neg-in 69.3%

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

      5. metadata-eval 69.3%

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

      6. sub-neg 69.3%

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

   10. Simplified 69.3%

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

   11. Taylor expanded in eps around 0 72.4%

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

   if 3.9999999999999998e61 < x < 2.99999999999999997e81 or 2.0000000000000001e256 < 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 78.1%

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

      1. mul-1-neg 78.1%

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

      2. mul-1-neg 78.1%

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

      3. rec-exp 78.1%

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

      4. sub-neg 78.1%

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

      5. div-sub 78.1%

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

      6. mul-1-neg 78.1%

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

      7. rec-exp 78.1%

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

      8. +-inverses 78.1%

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

   7. Simplified 78.1%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+61} \lor \neg \left(x \leq 3 \cdot 10^{+81}\right) \land x \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -450:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+70} \lor \neg \left(x \leq 2 \cdot 10^{+80}\right) \land x \leq 4.2 \cdot 10^{+266}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -450.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (or (<= x 1.1e+70) (and (not (<= x 2e+80)) (<= x 4.2e+266)))
     (/ (+ 1.0 (exp x)) 2.0)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -450.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if ((x <= 1.1e+70) || (!(x <= 2e+80) && (x <= 4.2e+266))) {
		tmp = (1.0 + exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -450.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if ((x <= 1.1e+70) || (!(x <= 2e+80) && (x <= 4.2e+266))) {
		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 <= -450.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif (x <= 1.1e+70) or (not (x <= 2e+80) and (x <= 4.2e+266)):
		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 <= -450.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif ((x <= 1.1e+70) || (!(x <= 2e+80) && (x <= 4.2e+266)))
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -450.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.1e+70], And[N[Not[LessEqual[x, 2e+80]], $MachinePrecision], LessEqual[x, 4.2e+266]]], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -450

   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 34.5%

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

   6. Taylor expanded in eps around 0 67.6%

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

      1. expm1-define 67.6%

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

      2. neg-mul-1 67.6%

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

   8. Simplified 67.6%

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

   if -450 < x < 1.1e70 or 2e80 < x < 4.19999999999999996e266

   1. Initial program 66.7%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in eps around inf 99.0%

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

   6. Taylor expanded in x around 0 75.2%

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

      1. sub-neg 75.2%

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

      2. metadata-eval 75.2%

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

      3. distribute-rgt-in 75.2%

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

      4. add-sqr-sqrt 44.6%

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

      5. sqrt-unprod 70.9%

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

      6. sqr-neg 70.9%

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

      7. sqrt-unprod 29.1%

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

      8. add-sqr-sqrt 69.4%

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

      9. neg-mul-1 69.4%

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

      10. *-un-lft-identity 69.4%

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

      11. distribute-rgt-in 69.4%

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

      12. +-commutative 69.4%

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

      13. *-commutative 69.4%

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

      14. distribute-rgt-neg-out 69.4%

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

      15. neg-sub0 69.4%

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

      16. add-sqr-sqrt 40.3%

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

      17. sqrt-unprod 69.6%

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

      18. sqr-neg 69.6%

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

      19. sqrt-unprod 30.6%

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

      20. add-sqr-sqrt 75.1%

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

      21. *-commutative 75.1%

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

      22. +-commutative 75.1%

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

      23. distribute-rgt-in 75.1%

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

   8. Applied egg-rr 69.3%

      \[\leadsto \frac{e^{\color{blue}{0 - x \cdot \left(\varepsilon + -1\right)}} + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 69.3%

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

      2. distribute-rgt-neg-in 69.3%

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

      3. +-commutative 69.3%

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

      4. distribute-neg-in 69.3%

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

      5. metadata-eval 69.3%

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

      6. sub-neg 69.3%

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

   10. Simplified 69.3%

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

   11. Taylor expanded in eps around 0 72.4%

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

   if 1.1e70 < x < 2e80 or 4.19999999999999996e266 < 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 78.1%

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

      1. mul-1-neg 78.1%

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

      2. mul-1-neg 78.1%

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

      3. rec-exp 78.1%

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

      4. sub-neg 78.1%

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

      5. div-sub 78.1%

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

      6. mul-1-neg 78.1%

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

      7. rec-exp 78.1%

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

      8. +-inverses 78.1%

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

   7. Simplified 78.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -450:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+70} \lor \neg \left(x \leq 2 \cdot 10^{+80}\right) \land x \leq 4.2 \cdot 10^{+266}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{\frac{eps\_m \cdot \left(x \cdot eps\_m + 2\right) - x}{eps\_m}}{2}\\ \mathbf{if}\;x \leq -0.00039:\\ \;\;\;\;\frac{2 + \frac{x + eps\_m \cdot \left(x \cdot 0 - x \cdot eps\_m\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 0.076:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+128}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+183}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+265}:\\ \;\;\;\;\frac{eps\_m \cdot \left(x + \frac{2}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (/ (- (* eps_m (+ (* x eps_m) 2.0)) x) eps_m) 2.0)))
   (if (<= x -0.00039)
     (/ (+ 2.0 (/ (+ x (* eps_m (- (* x 0.0) (* x eps_m)))) eps_m)) 2.0)
     (if (<= x 0.076)
       1.0
       (if (<= x 3.5e+65)
         t_0
         (if (<= x 2.3e+82)
           0.0
           (if (<= x 5.2e+128)
             t_0
             (if (<= x 8.5e+183)
               0.0
               (if (<= x 6.8e+265)
                 (/ (* eps_m (+ x (/ 2.0 eps_m))) 2.0)
                 0.0)))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double tmp;
	if (x <= -0.00039) {
		tmp = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	} else if (x <= 0.076) {
		tmp = 1.0;
	} else if (x <= 3.5e+65) {
		tmp = t_0;
	} else if (x <= 2.3e+82) {
		tmp = 0.0;
	} else if (x <= 5.2e+128) {
		tmp = t_0;
	} else if (x <= 8.5e+183) {
		tmp = 0.0;
	} else if (x <= 6.8e+265) {
		tmp = (eps_m * (x + (2.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (((eps_m * ((x * eps_m) + 2.0d0)) - x) / eps_m) / 2.0d0
    if (x <= (-0.00039d0)) then
        tmp = (2.0d0 + ((x + (eps_m * ((x * 0.0d0) - (x * eps_m)))) / eps_m)) / 2.0d0
    else if (x <= 0.076d0) then
        tmp = 1.0d0
    else if (x <= 3.5d+65) then
        tmp = t_0
    else if (x <= 2.3d+82) then
        tmp = 0.0d0
    else if (x <= 5.2d+128) then
        tmp = t_0
    else if (x <= 8.5d+183) then
        tmp = 0.0d0
    else if (x <= 6.8d+265) then
        tmp = (eps_m * (x + (2.0d0 / 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 t_0 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double tmp;
	if (x <= -0.00039) {
		tmp = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	} else if (x <= 0.076) {
		tmp = 1.0;
	} else if (x <= 3.5e+65) {
		tmp = t_0;
	} else if (x <= 2.3e+82) {
		tmp = 0.0;
	} else if (x <= 5.2e+128) {
		tmp = t_0;
	} else if (x <= 8.5e+183) {
		tmp = 0.0;
	} else if (x <= 6.8e+265) {
		tmp = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0
	tmp = 0
	if x <= -0.00039:
		tmp = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0
	elif x <= 0.076:
		tmp = 1.0
	elif x <= 3.5e+65:
		tmp = t_0
	elif x <= 2.3e+82:
		tmp = 0.0
	elif x <= 5.2e+128:
		tmp = t_0
	elif x <= 8.5e+183:
		tmp = 0.0
	elif x <= 6.8e+265:
		tmp = (eps_m * (x + (2.0 / eps_m))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(Float64(eps_m * Float64(Float64(x * eps_m) + 2.0)) - x) / eps_m) / 2.0)
	tmp = 0.0
	if (x <= -0.00039)
		tmp = Float64(Float64(2.0 + Float64(Float64(x + Float64(eps_m * Float64(Float64(x * 0.0) - Float64(x * eps_m)))) / eps_m)) / 2.0);
	elseif (x <= 0.076)
		tmp = 1.0;
	elseif (x <= 3.5e+65)
		tmp = t_0;
	elseif (x <= 2.3e+82)
		tmp = 0.0;
	elseif (x <= 5.2e+128)
		tmp = t_0;
	elseif (x <= 8.5e+183)
		tmp = 0.0;
	elseif (x <= 6.8e+265)
		tmp = Float64(Float64(eps_m * Float64(x + Float64(2.0 / eps_m))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	tmp = 0.0;
	if (x <= -0.00039)
		tmp = (2.0 + ((x + (eps_m * ((x * 0.0) - (x * eps_m)))) / eps_m)) / 2.0;
	elseif (x <= 0.076)
		tmp = 1.0;
	elseif (x <= 3.5e+65)
		tmp = t_0;
	elseif (x <= 2.3e+82)
		tmp = 0.0;
	elseif (x <= 5.2e+128)
		tmp = t_0;
	elseif (x <= 8.5e+183)
		tmp = 0.0;
	elseif (x <= 6.8e+265)
		tmp = (eps_m * (x + (2.0 / 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_] := Block[{t$95$0 = N[(N[(N[(N[(eps$95$m * N[(N[(x * eps$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.00039], N[(N[(2.0 + N[(N[(x + N[(eps$95$m * N[(N[(x * 0.0), $MachinePrecision] - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.076], 1.0, If[LessEqual[x, 3.5e+65], t$95$0, If[LessEqual[x, 2.3e+82], 0.0, If[LessEqual[x, 5.2e+128], t$95$0, If[LessEqual[x, 8.5e+183], 0.0, If[LessEqual[x, 6.8e+265], N[(N[(eps$95$m * N[(x + N[(2.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -3.89999999999999993e-4

   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 34.5%

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

   6. Taylor expanded in x around 0 13.2%

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

      1. associate-*r* 13.2%

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

      2. neg-mul-1 13.2%

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

      3. *-commutative 13.2%

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

   8. Simplified 13.2%

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

      1. distribute-lft-in 13.2%

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

      2. *-rgt-identity 13.2%

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

      3. distribute-rgt-in 13.2%

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

      4. add-sqr-sqrt 13.2%

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

      5. sqrt-unprod 16.0%

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

      6. sqr-neg 16.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 40.2%

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

      9. un-div-inv 40.2%

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

      10. add-sqr-sqrt 40.2%

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

      11. sqrt-unprod 10.3%

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

      12. sqr-neg 10.3%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 40.2%

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

   10. Applied egg-rr 40.2%

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

   11. Taylor expanded in eps around 0 56.4%

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

       1. associate-+r+ 56.4%

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

       2. distribute-rgt1-in 56.4%

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

       3. metadata-eval 56.4%

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

       4. associate-*r* 56.4%

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

       5. neg-mul-1 56.4%

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

       6. *-commutative 56.4%

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

   13. Simplified 56.4%

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

   if -3.89999999999999993e-4 < x < 0.0759999999999999981

   1. Initial program 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in x around 0 75.0%

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

   if 0.0759999999999999981 < x < 3.5000000000000001e65 or 2.29999999999999988e82 < x < 5.2e128

   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 51.4%

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

   6. Taylor expanded in x around 0 14.8%

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

      1. associate-*r* 14.8%

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

      2. neg-mul-1 14.8%

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

      3. *-commutative 14.8%

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

   8. Simplified 14.8%

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

   9. Taylor expanded in eps around 0 30.4%

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

   if 3.5000000000000001e65 < x < 2.29999999999999988e82 or 5.2e128 < x < 8.5000000000000004e183 or 6.79999999999999981e265 < 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 73.8%

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

      1. mul-1-neg 73.8%

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

      2. mul-1-neg 73.8%

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

      3. rec-exp 73.8%

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

      4. sub-neg 73.8%

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

      5. div-sub 73.8%

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

      6. mul-1-neg 73.8%

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

      7. rec-exp 73.8%

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

      8. +-inverses 73.8%

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

   7. Simplified 73.8%

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

   if 8.5000000000000004e183 < x < 6.79999999999999981e265

   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 39.8%

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

   6. Taylor expanded in x around 0 29.7%

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

      1. associate-*r* 29.7%

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

      2. neg-mul-1 29.7%

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

      3. *-commutative 29.7%

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

   8. Simplified 29.7%

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

   9. Taylor expanded in eps around inf 30.0%

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

       1. associate-*r/ 30.0%

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

       2. metadata-eval 30.0%

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

   11. Simplified 30.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00039:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot 0 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 0.076:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+183}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+265}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{\frac{eps\_m \cdot \left(x \cdot eps\_m + 2\right) - x}{eps\_m}}{2}\\ \mathbf{if}\;x \leq 2.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+81}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+185}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+258}:\\ \;\;\;\;\frac{eps\_m \cdot \left(x + \frac{2}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (/ (- (* eps_m (+ (* x eps_m) 2.0)) x) eps_m) 2.0)))
   (if (<= x 2.1e-15)
     (/ (- 2.0 (* x eps_m)) 2.0)
     (if (<= x 5.8e+70)
       t_0
       (if (<= x 7.2e+81)
         0.0
         (if (<= x 2.3e+129)
           t_0
           (if (<= x 4.5e+185)
             0.0
             (if (<= x 1.02e+258)
               (/ (* eps_m (+ x (/ 2.0 eps_m))) 2.0)
               0.0))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double tmp;
	if (x <= 2.1e-15) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= 5.8e+70) {
		tmp = t_0;
	} else if (x <= 7.2e+81) {
		tmp = 0.0;
	} else if (x <= 2.3e+129) {
		tmp = t_0;
	} else if (x <= 4.5e+185) {
		tmp = 0.0;
	} else if (x <= 1.02e+258) {
		tmp = (eps_m * (x + (2.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (((eps_m * ((x * eps_m) + 2.0d0)) - x) / eps_m) / 2.0d0
    if (x <= 2.1d-15) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else if (x <= 5.8d+70) then
        tmp = t_0
    else if (x <= 7.2d+81) then
        tmp = 0.0d0
    else if (x <= 2.3d+129) then
        tmp = t_0
    else if (x <= 4.5d+185) then
        tmp = 0.0d0
    else if (x <= 1.02d+258) then
        tmp = (eps_m * (x + (2.0d0 / 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 t_0 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	double tmp;
	if (x <= 2.1e-15) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= 5.8e+70) {
		tmp = t_0;
	} else if (x <= 7.2e+81) {
		tmp = 0.0;
	} else if (x <= 2.3e+129) {
		tmp = t_0;
	} else if (x <= 4.5e+185) {
		tmp = 0.0;
	} else if (x <= 1.02e+258) {
		tmp = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0
	tmp = 0
	if x <= 2.1e-15:
		tmp = (2.0 - (x * eps_m)) / 2.0
	elif x <= 5.8e+70:
		tmp = t_0
	elif x <= 7.2e+81:
		tmp = 0.0
	elif x <= 2.3e+129:
		tmp = t_0
	elif x <= 4.5e+185:
		tmp = 0.0
	elif x <= 1.02e+258:
		tmp = (eps_m * (x + (2.0 / eps_m))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(Float64(eps_m * Float64(Float64(x * eps_m) + 2.0)) - x) / eps_m) / 2.0)
	tmp = 0.0
	if (x <= 2.1e-15)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	elseif (x <= 5.8e+70)
		tmp = t_0;
	elseif (x <= 7.2e+81)
		tmp = 0.0;
	elseif (x <= 2.3e+129)
		tmp = t_0;
	elseif (x <= 4.5e+185)
		tmp = 0.0;
	elseif (x <= 1.02e+258)
		tmp = Float64(Float64(eps_m * Float64(x + Float64(2.0 / eps_m))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (((eps_m * ((x * eps_m) + 2.0)) - x) / eps_m) / 2.0;
	tmp = 0.0;
	if (x <= 2.1e-15)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	elseif (x <= 5.8e+70)
		tmp = t_0;
	elseif (x <= 7.2e+81)
		tmp = 0.0;
	elseif (x <= 2.3e+129)
		tmp = t_0;
	elseif (x <= 4.5e+185)
		tmp = 0.0;
	elseif (x <= 1.02e+258)
		tmp = (eps_m * (x + (2.0 / 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_] := Block[{t$95$0 = N[(N[(N[(N[(eps$95$m * N[(N[(x * eps$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, 2.1e-15], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.8e+70], t$95$0, If[LessEqual[x, 7.2e+81], 0.0, If[LessEqual[x, 2.3e+129], t$95$0, If[LessEqual[x, 4.5e+185], 0.0, If[LessEqual[x, 1.02e+258], N[(N[(eps$95$m * N[(x + N[(2.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.09999999999999981e-15

   1. Initial program 62.9%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in x around 0 42.3%

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

   6. Taylor expanded in x around 0 43.0%

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

      1. associate-*r* 43.0%

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

      2. neg-mul-1 43.0%

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

      3. *-commutative 43.0%

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

   8. Simplified 43.0%

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

      1. add-sqr-sqrt 22.2%

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

      2. sqrt-unprod 48.7%

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

      3. sqr-neg 48.7%

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

      4. sqrt-unprod 20.7%

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

      5. add-sqr-sqrt 48.1%

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

      6. distribute-rgt-in 48.1%

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

      7. *-un-lft-identity 48.1%

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

      8. distribute-rgt-in 48.1%

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

   10. Applied egg-rr 48.1%

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

       1. associate-*l/ 48.1%

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

       2. *-lft-identity 48.1%

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

       3. distribute-rgt-out 48.1%

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

       4. +-commutative 48.1%

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

       5. associate-+r- 48.1%

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

       6. div-sub 48.1%

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

       7. *-inverses 48.1%

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

       8. associate-+l- 48.1%

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

       9. metadata-eval 48.1%

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

   12. Simplified 48.1%

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

   13. Taylor expanded in eps around inf 69.0%

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

       1. associate-*r* 69.0%

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

       2. neg-mul-1 69.0%

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

       3. *-commutative 69.0%

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

   15. Simplified 69.0%

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

   if 2.09999999999999981e-15 < x < 5.7999999999999997e70 or 7.20000000000000011e81 < x < 2.2999999999999999e129

   1. Initial program 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in x around 0 45.0%

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

   6. Taylor expanded in x around 0 12.3%

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

      1. associate-*r* 12.3%

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

      2. neg-mul-1 12.3%

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

      3. *-commutative 12.3%

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

   8. Simplified 12.3%

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

   9. Taylor expanded in eps around 0 24.7%

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

   if 5.7999999999999997e70 < x < 7.20000000000000011e81 or 2.2999999999999999e129 < x < 4.5000000000000002e185 or 1.0200000000000001e258 < 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 73.8%

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

      1. mul-1-neg 73.8%

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

      2. mul-1-neg 73.8%

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

      3. rec-exp 73.8%

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

      4. sub-neg 73.8%

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

      5. div-sub 73.8%

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

      6. mul-1-neg 73.8%

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

      7. rec-exp 73.8%

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

      8. +-inverses 73.8%

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

   7. Simplified 73.8%

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

   if 4.5000000000000002e185 < x < 1.0200000000000001e258

   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 39.8%

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

   6. Taylor expanded in x around 0 29.7%

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

      1. associate-*r* 29.7%

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

      2. neg-mul-1 29.7%

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

      3. *-commutative 29.7%

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

   8. Simplified 29.7%

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

   9. Taylor expanded in eps around inf 30.0%

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

       1. associate-*r/ 30.0%

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

       2. metadata-eval 30.0%

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

   11. Simplified 30.0%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+81}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(x \cdot \varepsilon + 2\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+185}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+258}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 17:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+182}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+266}:\\ \;\;\;\;\frac{eps\_m \cdot \left(x + \frac{2}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 17.0)
   (/ (- 2.0 (* x eps_m)) 2.0)
   (if (<= x 9e+182)
     0.0
     (if (<= x 2.2e+266) (/ (* eps_m (+ x (/ 2.0 eps_m))) 2.0) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 17.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= 9e+182) {
		tmp = 0.0;
	} else if (x <= 2.2e+266) {
		tmp = (eps_m * (x + (2.0 / 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 <= 17.0d0) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else if (x <= 9d+182) then
        tmp = 0.0d0
    else if (x <= 2.2d+266) then
        tmp = (eps_m * (x + (2.0d0 / 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 <= 17.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= 9e+182) {
		tmp = 0.0;
	} else if (x <= 2.2e+266) {
		tmp = (eps_m * (x + (2.0 / 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 <= 17.0:
		tmp = (2.0 - (x * eps_m)) / 2.0
	elif x <= 9e+182:
		tmp = 0.0
	elif x <= 2.2e+266:
		tmp = (eps_m * (x + (2.0 / 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 <= 17.0)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	elseif (x <= 9e+182)
		tmp = 0.0;
	elseif (x <= 2.2e+266)
		tmp = Float64(Float64(eps_m * Float64(x + Float64(2.0 / 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 <= 17.0)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	elseif (x <= 9e+182)
		tmp = 0.0;
	elseif (x <= 2.2e+266)
		tmp = (eps_m * (x + (2.0 / 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, 17.0], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9e+182], 0.0, If[LessEqual[x, 2.2e+266], N[(N[(eps$95$m * N[(x + N[(2.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 17

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in x around 0 41.6%

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

   6. Taylor expanded in x around 0 41.7%

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

      1. associate-*r* 41.7%

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

      2. neg-mul-1 41.7%

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

      3. *-commutative 41.7%

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

   8. Simplified 41.7%

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

      1. add-sqr-sqrt 21.5%

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

      2. sqrt-unprod 47.2%

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

      3. sqr-neg 47.2%

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

      4. sqrt-unprod 20.2%

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

      5. add-sqr-sqrt 46.6%

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

      6. distribute-rgt-in 46.6%

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

      7. *-un-lft-identity 46.6%

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

      8. distribute-rgt-in 46.6%

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

   10. Applied egg-rr 46.6%

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

       1. associate-*l/ 46.6%

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

       2. *-lft-identity 46.6%

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

       3. distribute-rgt-out 46.6%

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

       4. +-commutative 46.6%

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

       5. associate-+r- 46.6%

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

       6. div-sub 46.6%

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

       7. *-inverses 46.6%

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

       8. associate-+l- 46.6%

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

       9. metadata-eval 46.6%

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

   12. Simplified 46.6%

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

   13. Taylor expanded in eps around inf 67.9%

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

       1. associate-*r* 67.9%

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

       2. neg-mul-1 67.9%

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

       3. *-commutative 67.9%

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

   15. Simplified 67.9%

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

   if 17 < x < 9.00000000000000058e182 or 2.1999999999999999e266 < 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 47.1%

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

      1. mul-1-neg 47.1%

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

      2. mul-1-neg 47.1%

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

      3. rec-exp 47.1%

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

      4. sub-neg 47.1%

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

      5. div-sub 47.1%

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

      6. mul-1-neg 47.1%

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

      7. rec-exp 47.1%

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

      8. +-inverses 47.1%

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

   7. Simplified 47.1%

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

   if 9.00000000000000058e182 < x < 2.1999999999999999e266

   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 39.8%

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

   6. Taylor expanded in x around 0 29.7%

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

      1. associate-*r* 29.7%

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

      2. neg-mul-1 29.7%

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

      3. *-commutative 29.7%

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

   8. Simplified 29.7%

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

   9. Taylor expanded in eps around inf 30.0%

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

       1. associate-*r/ 30.0%

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

       2. metadata-eval 30.0%

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

   11. Simplified 30.0%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 17:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+182}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+266}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.2% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 17:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 17.0) (/ (- 2.0 (* x eps_m)) 2.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 17.0) {
		tmp = (2.0 - (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 <= 17.0d0) then
        tmp = (2.0d0 - (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 <= 17.0) {
		tmp = (2.0 - (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 <= 17.0:
		tmp = (2.0 - (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 <= 17.0)
		tmp = Float64(Float64(2.0 - 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 <= 17.0)
		tmp = (2.0 - (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, 17.0], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 17

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in x around 0 41.6%

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

   6. Taylor expanded in x around 0 41.7%

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

      1. associate-*r* 41.7%

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

      2. neg-mul-1 41.7%

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

      3. *-commutative 41.7%

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

   8. Simplified 41.7%

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

      1. add-sqr-sqrt 21.5%

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

      2. sqrt-unprod 47.2%

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

      3. sqr-neg 47.2%

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

      4. sqrt-unprod 20.2%

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

      5. add-sqr-sqrt 46.6%

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

      6. distribute-rgt-in 46.6%

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

      7. *-un-lft-identity 46.6%

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

      8. distribute-rgt-in 46.6%

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

   10. Applied egg-rr 46.6%

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

       1. associate-*l/ 46.6%

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

       2. *-lft-identity 46.6%

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

       3. distribute-rgt-out 46.6%

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

       4. +-commutative 46.6%

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

       5. associate-+r- 46.6%

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

       6. div-sub 46.6%

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

       7. *-inverses 46.6%

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

       8. associate-+l- 46.6%

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

       9. metadata-eval 46.6%

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

   12. Simplified 46.6%

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

   13. Taylor expanded in eps around inf 67.9%

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

       1. associate-*r* 67.9%

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

       2. neg-mul-1 67.9%

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

       3. *-commutative 67.9%

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

   15. Simplified 67.9%

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

   if 17 < 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 42.3%

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

      1. mul-1-neg 42.3%

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

      2. mul-1-neg 42.3%

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

      3. rec-exp 42.3%

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

      4. sub-neg 42.3%

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

      5. div-sub 42.3%

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

      6. mul-1-neg 42.3%

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

      7. rec-exp 42.3%

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

      8. +-inverses 42.3%

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

   7. Simplified 42.3%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 17:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.0% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 4.9e+33) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 4.9e+33) {
		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 <= 4.9d+33) 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 <= 4.9e+33) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 4.9e+33:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 4.9e+33)
		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 <= 4.9e+33)
		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, 4.9e+33], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.90000000000000014e33

   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 60.2%

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

   if 4.90000000000000014e33 < 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 44.5%

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

      1. mul-1-neg 44.5%

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

      2. mul-1-neg 44.5%

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

      3. rec-exp 44.5%

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

      4. sub-neg 44.5%

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

      5. div-sub 44.5%

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

      6. mul-1-neg 44.5%

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

      7. rec-exp 44.5%

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

      8. +-inverses 44.5%

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

   7. Simplified 44.5%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 15.6% 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 73.5%

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

3. Simplified 61.9%

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

5. Taylor expanded in eps around 0 14.0%

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

   1. mul-1-neg 14.0%

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

   2. mul-1-neg 14.0%

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

   3. rec-exp 13.9%

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

   4. sub-neg 13.9%

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

   5. div-sub 13.9%

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

   6. mul-1-neg 13.9%

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

   7. rec-exp 14.0%

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

   8. +-inverses 14.2%

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

7. Simplified 14.2%

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

8. Final simplification 14.2%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))