NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.0% → 98.6%

Time: 13.0s

Alternatives: 22

Speedup: 1.9×

• 37.4% 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: 72.0% 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: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{e^{x \cdot \mathsf{expm1}\left(\log eps\_m\right)} + \frac{1}{e^{x + eps\_m \cdot x}}}{2} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (/ (+ (exp (* x (expm1 (log eps_m)))) (/ 1.0 (exp (+ x (* eps_m x))))) 2.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (exp((x * expm1(log(eps_m)))) + (1.0 / exp((x + (eps_m * x))))) / 2.0;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return (Math.exp((x * Math.expm1(Math.log(eps_m)))) + (1.0 / Math.exp((x + (eps_m * x))))) / 2.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (math.exp((x * math.expm1(math.log(eps_m)))) + (1.0 / math.exp((x + (eps_m * x))))) / 2.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(exp(Float64(x * expm1(log(eps_m)))) + Float64(1.0 / exp(Float64(x + Float64(eps_m * x))))) / 2.0)
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[Exp[N[(x * N[(Exp[N[Log[eps$95$m], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[N[(x + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 74.3%

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

3. Simplified 67.7%

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

5. Taylor expanded in eps around inf 99.5%

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

   1. add-exp-log 50.6%

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

   2. expm1-define 50.6%

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

7. Applied egg-rr 50.6%

   \[\leadsto \frac{e^{x \cdot \color{blue}{\mathsf{expm1}\left(\log \varepsilon\right)}} + \frac{1}{e^{x + \varepsilon \cdot x}}}{2} \]
8. Add Preprocessing

Alternative 2: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + \frac{1}{e^{eps\_m \cdot x}}}{2}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+275}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m - 1\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.3e+19)
   (/ (+ (exp (* x eps_m)) (/ 1.0 (exp (* eps_m x)))) 2.0)
   (if (<= x 2.9e+83)
     (/ (- (+ 1.0 (/ 1.0 eps_m)) (- (/ 1.0 eps_m) 1.0)) 2.0)
     (if (<= x 8.2e+275)
       (/ (+ (exp (* x (- eps_m 1.0))) 1.0) 2.0)
       (/ (/ (- (exp (* -1.0 x)) (/ 1.0 (exp x))) eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.3e+19) {
		tmp = (exp((x * eps_m)) + (1.0 / exp((eps_m * x)))) / 2.0;
	} else if (x <= 2.9e+83) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else if (x <= 8.2e+275) {
		tmp = (exp((x * (eps_m - 1.0))) + 1.0) / 2.0;
	} else {
		tmp = ((exp((-1.0 * x)) - (1.0 / exp(x))) / eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 1.3d+19) then
        tmp = (exp((x * eps_m)) + (1.0d0 / exp((eps_m * x)))) / 2.0d0
    else if (x <= 2.9d+83) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - ((1.0d0 / eps_m) - 1.0d0)) / 2.0d0
    else if (x <= 8.2d+275) then
        tmp = (exp((x * (eps_m - 1.0d0))) + 1.0d0) / 2.0d0
    else
        tmp = ((exp(((-1.0d0) * x)) - (1.0d0 / exp(x))) / eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.3e+19) {
		tmp = (Math.exp((x * eps_m)) + (1.0 / Math.exp((eps_m * x)))) / 2.0;
	} else if (x <= 2.9e+83) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else if (x <= 8.2e+275) {
		tmp = (Math.exp((x * (eps_m - 1.0))) + 1.0) / 2.0;
	} else {
		tmp = ((Math.exp((-1.0 * x)) - (1.0 / Math.exp(x))) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.3e+19:
		tmp = (math.exp((x * eps_m)) + (1.0 / math.exp((eps_m * x)))) / 2.0
	elif x <= 2.9e+83:
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0
	elif x <= 8.2e+275:
		tmp = (math.exp((x * (eps_m - 1.0))) + 1.0) / 2.0
	else:
		tmp = ((math.exp((-1.0 * x)) - (1.0 / math.exp(x))) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.3e+19)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + Float64(1.0 / exp(Float64(eps_m * x)))) / 2.0);
	elseif (x <= 2.9e+83)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	elseif (x <= 8.2e+275)
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m - 1.0))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(exp(Float64(-1.0 * x)) - Float64(1.0 / exp(x))) / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.3e+19)
		tmp = (exp((x * eps_m)) + (1.0 / exp((eps_m * x)))) / 2.0;
	elseif (x <= 2.9e+83)
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	elseif (x <= 8.2e+275)
		tmp = (exp((x * (eps_m - 1.0))) + 1.0) / 2.0;
	else
		tmp = ((exp((-1.0 * x)) - (1.0 / exp(x))) / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.3e+19], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.9e+83], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.2e+275], N[(N[(N[Exp[N[(x * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[Exp[N[(-1.0 * x), $MachinePrecision]], $MachinePrecision] - N[(1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.3e19

   1. Initial program 61.5%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in eps around inf 99.3%

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

      1. add-exp-log 48.2%

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

      2. expm1-define 48.2%

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

   7. Applied egg-rr 48.2%

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

   8. Taylor expanded in eps around inf 48.3%

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

   9. Taylor expanded in eps around inf 99.3%

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

   if 1.3e19 < x < 2.89999999999999999e83

   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 16.4%

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

   6. Taylor expanded in x around 0 72.3%

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

   if 2.89999999999999999e83 < x < 8.1999999999999994e275

   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 48.0%

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

   if 8.1999999999999994e275 < 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%

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

   5. Taylor expanded in eps around 0 80.3%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+18}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + \frac{1}{e^{eps\_m \cdot x}}}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m - 1\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 9e+18)
   (/ (+ (exp (* x eps_m)) (/ 1.0 (exp (* eps_m x)))) 2.0)
   (if (<= x 1.55e+81)
     (/ (- (+ 1.0 (/ 1.0 eps_m)) (- (/ 1.0 eps_m) 1.0)) 2.0)
     (/ (+ (exp (* x (- eps_m 1.0))) 1.0) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 9e+18) {
		tmp = (exp((x * eps_m)) + (1.0 / exp((eps_m * x)))) / 2.0;
	} else if (x <= 1.55e+81) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (exp((x * (eps_m - 1.0))) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 9d+18) then
        tmp = (exp((x * eps_m)) + (1.0d0 / exp((eps_m * x)))) / 2.0d0
    else if (x <= 1.55d+81) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - ((1.0d0 / eps_m) - 1.0d0)) / 2.0d0
    else
        tmp = (exp((x * (eps_m - 1.0d0))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 9e+18) {
		tmp = (Math.exp((x * eps_m)) + (1.0 / Math.exp((eps_m * x)))) / 2.0;
	} else if (x <= 1.55e+81) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m - 1.0))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 9e+18:
		tmp = (math.exp((x * eps_m)) + (1.0 / math.exp((eps_m * x)))) / 2.0
	elif x <= 1.55e+81:
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0
	else:
		tmp = (math.exp((x * (eps_m - 1.0))) + 1.0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 9e+18)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + Float64(1.0 / exp(Float64(eps_m * x)))) / 2.0);
	elseif (x <= 1.55e+81)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m - 1.0))) + 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 <= 9e+18)
		tmp = (exp((x * eps_m)) + (1.0 / exp((eps_m * x)))) / 2.0;
	elseif (x <= 1.55e+81)
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	else
		tmp = (exp((x * (eps_m - 1.0))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 9e+18], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.55e+81], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 9e18

   1. Initial program 61.5%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in eps around inf 99.3%

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

      1. add-exp-log 48.2%

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

      2. expm1-define 48.2%

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

   7. Applied egg-rr 48.2%

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

   8. Taylor expanded in eps around inf 48.3%

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

   9. Taylor expanded in eps around inf 99.3%

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

   if 9e18 < x < 1.55e81

   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 16.4%

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

   6. Taylor expanded in x around 0 72.3%

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

   if 1.55e81 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 42.5%

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

Alternative 4: 98.6% accurate, 1.1× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (/ (+ (exp (* x (- eps_m 1.0))) (/ 1.0 (exp (+ x (* eps_m x))))) 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 + (eps_m * x))))) / 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 + (eps_m * x))))) / 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 + (eps_m * x))))) / 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 + (eps_m * x))))) / 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(eps_m * x))))) / 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 + (eps_m * x))))) / 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[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 74.3%

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

3. Simplified 67.7%

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

5. Taylor expanded in eps around inf 99.5%

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

Alternative 5: 92.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (exp((x * (eps_m - 1.0))) + (1.0 / exp((eps_m * x)))) / 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[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 74.3%

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

3. Simplified 67.7%

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

5. Taylor expanded in eps around inf 99.5%

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

6. Taylor expanded in eps around inf 88.2%

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

Alternative 6: 85.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{eps\_m \cdot x}\\ \mathbf{if}\;x \leq -6.4 \cdot 10^{-292}:\\ \;\;\;\;\frac{1 + \frac{1}{t\_0}}{2}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{eps\_m \cdot \left(\left(\frac{1}{eps\_m} + \frac{t\_0}{eps\_m}\right) - x\right)}{2}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m - 1\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* eps_m x))))
   (if (<= x -6.4e-292)
     (/ (+ 1.0 (/ 1.0 t_0)) 2.0)
     (if (<= x 8.8e+18)
       (/ (* eps_m (- (+ (/ 1.0 eps_m) (/ t_0 eps_m)) x)) 2.0)
       (if (<= x 7.2e+83)
         (/ (- (+ 1.0 (/ 1.0 eps_m)) (- (/ 1.0 eps_m) 1.0)) 2.0)
         (/ (+ (exp (* x (- eps_m 1.0))) 1.0) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((eps_m * x));
	double tmp;
	if (x <= -6.4e-292) {
		tmp = (1.0 + (1.0 / t_0)) / 2.0;
	} else if (x <= 8.8e+18) {
		tmp = (eps_m * (((1.0 / eps_m) + (t_0 / eps_m)) - x)) / 2.0;
	} else if (x <= 7.2e+83) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (exp((x * (eps_m - 1.0))) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((eps_m * x))
    if (x <= (-6.4d-292)) then
        tmp = (1.0d0 + (1.0d0 / t_0)) / 2.0d0
    else if (x <= 8.8d+18) then
        tmp = (eps_m * (((1.0d0 / eps_m) + (t_0 / eps_m)) - x)) / 2.0d0
    else if (x <= 7.2d+83) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - ((1.0d0 / eps_m) - 1.0d0)) / 2.0d0
    else
        tmp = (exp((x * (eps_m - 1.0d0))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((eps_m * x));
	double tmp;
	if (x <= -6.4e-292) {
		tmp = (1.0 + (1.0 / t_0)) / 2.0;
	} else if (x <= 8.8e+18) {
		tmp = (eps_m * (((1.0 / eps_m) + (t_0 / eps_m)) - x)) / 2.0;
	} else if (x <= 7.2e+83) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m - 1.0))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((eps_m * x))
	tmp = 0
	if x <= -6.4e-292:
		tmp = (1.0 + (1.0 / t_0)) / 2.0
	elif x <= 8.8e+18:
		tmp = (eps_m * (((1.0 / eps_m) + (t_0 / eps_m)) - x)) / 2.0
	elif x <= 7.2e+83:
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0
	else:
		tmp = (math.exp((x * (eps_m - 1.0))) + 1.0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(eps_m * x))
	tmp = 0.0
	if (x <= -6.4e-292)
		tmp = Float64(Float64(1.0 + Float64(1.0 / t_0)) / 2.0);
	elseif (x <= 8.8e+18)
		tmp = Float64(Float64(eps_m * Float64(Float64(Float64(1.0 / eps_m) + Float64(t_0 / eps_m)) - x)) / 2.0);
	elseif (x <= 7.2e+83)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m - 1.0))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((eps_m * x));
	tmp = 0.0;
	if (x <= -6.4e-292)
		tmp = (1.0 + (1.0 / t_0)) / 2.0;
	elseif (x <= 8.8e+18)
		tmp = (eps_m * (((1.0 / eps_m) + (t_0 / eps_m)) - x)) / 2.0;
	elseif (x <= 7.2e+83)
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	else
		tmp = (exp((x * (eps_m - 1.0))) + 1.0) / 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[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.4e-292], N[(N[(1.0 + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.8e+18], N[(N[(eps$95$m * N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(t$95$0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7.2e+83], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.4000000000000003e-292

   1. Initial program 66.3%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in eps around inf 99.5%

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

      1. add-exp-log 52.1%

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

      2. expm1-define 52.1%

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

   7. Applied egg-rr 52.1%

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

   8. Taylor expanded in eps around inf 52.1%

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

   9. Taylor expanded in x around 0 77.2%

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

   if -6.4000000000000003e-292 < x < 8.8e18

   1. Initial program 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in x around 0 40.3%

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

   6. Taylor expanded in eps around inf 84.0%

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

   7. Taylor expanded in eps around inf 84.7%

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

   if 8.8e18 < x < 7.1999999999999995e83

   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 16.4%

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

   6. Taylor expanded in x around 0 72.3%

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

   if 7.1999999999999995e83 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 42.5%

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

Alternative 7: 84.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{eps\_m \cdot x}}}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + 1}{2}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m - 1\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-281)
   (/ (+ 1.0 (/ 1.0 (exp (* eps_m x)))) 2.0)
   (if (<= x 1.5e+19)
     (/ (+ (exp (* x eps_m)) 1.0) 2.0)
     (if (<= x 4.8e+81)
       (/ (- (+ 1.0 (/ 1.0 eps_m)) (- (/ 1.0 eps_m) 1.0)) 2.0)
       (/ (+ (exp (* x (- eps_m 1.0))) 1.0) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-281) {
		tmp = (1.0 + (1.0 / exp((eps_m * x)))) / 2.0;
	} else if (x <= 1.5e+19) {
		tmp = (exp((x * eps_m)) + 1.0) / 2.0;
	} else if (x <= 4.8e+81) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (exp((x * (eps_m - 1.0))) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2d-281)) then
        tmp = (1.0d0 + (1.0d0 / exp((eps_m * x)))) / 2.0d0
    else if (x <= 1.5d+19) then
        tmp = (exp((x * eps_m)) + 1.0d0) / 2.0d0
    else if (x <= 4.8d+81) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - ((1.0d0 / eps_m) - 1.0d0)) / 2.0d0
    else
        tmp = (exp((x * (eps_m - 1.0d0))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-281) {
		tmp = (1.0 + (1.0 / Math.exp((eps_m * x)))) / 2.0;
	} else if (x <= 1.5e+19) {
		tmp = (Math.exp((x * eps_m)) + 1.0) / 2.0;
	} else if (x <= 4.8e+81) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m - 1.0))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-281:
		tmp = (1.0 + (1.0 / math.exp((eps_m * x)))) / 2.0
	elif x <= 1.5e+19:
		tmp = (math.exp((x * eps_m)) + 1.0) / 2.0
	elif x <= 4.8e+81:
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0
	else:
		tmp = (math.exp((x * (eps_m - 1.0))) + 1.0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-281)
		tmp = Float64(Float64(1.0 + Float64(1.0 / exp(Float64(eps_m * x)))) / 2.0);
	elseif (x <= 1.5e+19)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + 1.0) / 2.0);
	elseif (x <= 4.8e+81)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m - 1.0))) + 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 <= -2e-281)
		tmp = (1.0 + (1.0 / exp((eps_m * x)))) / 2.0;
	elseif (x <= 1.5e+19)
		tmp = (exp((x * eps_m)) + 1.0) / 2.0;
	elseif (x <= 4.8e+81)
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	else
		tmp = (exp((x * (eps_m - 1.0))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-281], N[(N[(1.0 + N[(1.0 / N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.5e+19], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.8e+81], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2e-281

   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 58.2%

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

   5. Taylor expanded in eps around inf 99.5%

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

      1. add-exp-log 52.2%

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

      2. expm1-define 52.2%

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

   7. Applied egg-rr 52.2%

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

   8. Taylor expanded in eps around inf 52.2%

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

   9. Taylor expanded in x around 0 76.7%

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

   if -2e-281 < x < 1.5e19

   1. Initial program 55.3%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in eps around inf 99.2%

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

   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 84.6%

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

   if 1.5e19 < x < 4.79999999999999979e81

   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 16.4%

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

   6. Taylor expanded in x around 0 72.3%

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

   if 4.79999999999999979e81 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 42.5%

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

Alternative 8: 84.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{e^{x \cdot eps\_m} + 1}{2}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{eps\_m \cdot x}}}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ (exp (* x eps_m)) 1.0) 2.0)))
   (if (<= x -2e-281)
     (/ (+ 1.0 (/ 1.0 (exp (* eps_m x)))) 2.0)
     (if (<= x 1.3e+19)
       t_0
       (if (<= x 2.4e+80)
         (/ (- (+ 1.0 (/ 1.0 eps_m)) (- (/ 1.0 eps_m) 1.0)) 2.0)
         t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (exp((x * eps_m)) + 1.0) / 2.0;
	double tmp;
	if (x <= -2e-281) {
		tmp = (1.0 + (1.0 / exp((eps_m * x)))) / 2.0;
	} else if (x <= 1.3e+19) {
		tmp = t_0;
	} else if (x <= 2.4e+80) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (exp((x * eps_m)) + 1.0d0) / 2.0d0
    if (x <= (-2d-281)) then
        tmp = (1.0d0 + (1.0d0 / exp((eps_m * x)))) / 2.0d0
    else if (x <= 1.3d+19) then
        tmp = t_0
    else if (x <= 2.4d+80) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - ((1.0d0 / eps_m) - 1.0d0)) / 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (Math.exp((x * eps_m)) + 1.0) / 2.0;
	double tmp;
	if (x <= -2e-281) {
		tmp = (1.0 + (1.0 / Math.exp((eps_m * x)))) / 2.0;
	} else if (x <= 1.3e+19) {
		tmp = t_0;
	} else if (x <= 2.4e+80) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (math.exp((x * eps_m)) + 1.0) / 2.0
	tmp = 0
	if x <= -2e-281:
		tmp = (1.0 + (1.0 / math.exp((eps_m * x)))) / 2.0
	elif x <= 1.3e+19:
		tmp = t_0
	elif x <= 2.4e+80:
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(exp(Float64(x * eps_m)) + 1.0) / 2.0)
	tmp = 0.0
	if (x <= -2e-281)
		tmp = Float64(Float64(1.0 + Float64(1.0 / exp(Float64(eps_m * x)))) / 2.0);
	elseif (x <= 1.3e+19)
		tmp = t_0;
	elseif (x <= 2.4e+80)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (exp((x * eps_m)) + 1.0) / 2.0;
	tmp = 0.0;
	if (x <= -2e-281)
		tmp = (1.0 + (1.0 / exp((eps_m * x)))) / 2.0;
	elseif (x <= 1.3e+19)
		tmp = t_0;
	elseif (x <= 2.4e+80)
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	else
		tmp = t_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[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2e-281], N[(N[(1.0 + N[(1.0 / N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.3e+19], t$95$0, If[LessEqual[x, 2.4e+80], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e-281

   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 58.2%

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

   5. Taylor expanded in eps around inf 99.5%

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

      1. add-exp-log 52.2%

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

      2. expm1-define 52.2%

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

   7. Applied egg-rr 52.2%

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

   8. Taylor expanded in eps around inf 52.2%

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

   9. Taylor expanded in x around 0 76.7%

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

   if -2e-281 < x < 1.3e19 or 2.39999999999999979e80 < x

   1. Initial program 75.4%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in eps around inf 99.5%

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

   6. Taylor expanded in x around 0 65.3%

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

   7. Taylor expanded in eps around inf 65.5%

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

   if 1.3e19 < x < 2.39999999999999979e80

   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 16.4%

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

   6. Taylor expanded in x around 0 72.3%

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

Alternative 9: 77.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{e^{x \cdot eps\_m} + 1}{2}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;x \leq 1.58 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ (exp (* x eps_m)) 1.0) 2.0)))
   (if (<= x -1.0)
     (/ (+ (exp (- x)) 1.0) 2.0)
     (if (<= x 1.58e+19)
       t_0
       (if (<= x 1.6e+85)
         (/ (- (+ 1.0 (/ 1.0 eps_m)) (- (/ 1.0 eps_m) 1.0)) 2.0)
         t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (exp((x * eps_m)) + 1.0) / 2.0;
	double tmp;
	if (x <= -1.0) {
		tmp = (exp(-x) + 1.0) / 2.0;
	} else if (x <= 1.58e+19) {
		tmp = t_0;
	} else if (x <= 1.6e+85) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (exp((x * eps_m)) + 1.0d0) / 2.0d0
    if (x <= (-1.0d0)) then
        tmp = (exp(-x) + 1.0d0) / 2.0d0
    else if (x <= 1.58d+19) then
        tmp = t_0
    else if (x <= 1.6d+85) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - ((1.0d0 / eps_m) - 1.0d0)) / 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (Math.exp((x * eps_m)) + 1.0) / 2.0;
	double tmp;
	if (x <= -1.0) {
		tmp = (Math.exp(-x) + 1.0) / 2.0;
	} else if (x <= 1.58e+19) {
		tmp = t_0;
	} else if (x <= 1.6e+85) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (math.exp((x * eps_m)) + 1.0) / 2.0
	tmp = 0
	if x <= -1.0:
		tmp = (math.exp(-x) + 1.0) / 2.0
	elif x <= 1.58e+19:
		tmp = t_0
	elif x <= 1.6e+85:
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(exp(Float64(x * eps_m)) + 1.0) / 2.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(exp(Float64(-x)) + 1.0) / 2.0);
	elseif (x <= 1.58e+19)
		tmp = t_0;
	elseif (x <= 1.6e+85)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (exp((x * eps_m)) + 1.0) / 2.0;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (exp(-x) + 1.0) / 2.0;
	elseif (x <= 1.58e+19)
		tmp = t_0;
	elseif (x <= 1.6e+85)
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	else
		tmp = t_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[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(N[(N[Exp[(-x)], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.58e+19], t$95$0, If[LessEqual[x, 1.6e+85], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 48.3%

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

   7. Taylor expanded in eps around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. neg-mul-1 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -1 < x < 1.58e19 or 1.60000000000000009e85 < x

   1. Initial program 67.9%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in eps around inf 99.4%

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

   6. Taylor expanded in x around 0 71.8%

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

   7. Taylor expanded in eps around inf 72.2%

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

   if 1.58e19 < x < 1.60000000000000009e85

   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 16.4%

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

   6. Taylor expanded in x around 0 72.3%

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

Alternative 10: 67.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{0.5 \cdot x + eps\_m \cdot \left(1 + -0.5 \cdot \left(eps\_m \cdot x\right)\right)}{eps\_m}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;1 + -0.5 \cdot {x}^{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + e^{x}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.8e-36)
   (/ (+ (* 0.5 x) (* eps_m (+ 1.0 (* -0.5 (* eps_m x))))) eps_m)
   (if (<= x 0.018)
     (+ 1.0 (* -0.5 (pow x 2.0)))
     (if (<= x 2.8e+83)
       (/ (- (+ 1.0 (/ 1.0 eps_m)) (- (/ 1.0 eps_m) 1.0)) 2.0)
       (* (+ 1.0 (exp x)) 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.8e-36) {
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	} else if (x <= 0.018) {
		tmp = 1.0 + (-0.5 * pow(x, 2.0));
	} else if (x <= 2.8e+83) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (1.0 + exp(x)) * 0.5;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.8d-36)) then
        tmp = ((0.5d0 * x) + (eps_m * (1.0d0 + ((-0.5d0) * (eps_m * x))))) / eps_m
    else if (x <= 0.018d0) then
        tmp = 1.0d0 + ((-0.5d0) * (x ** 2.0d0))
    else if (x <= 2.8d+83) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - ((1.0d0 / eps_m) - 1.0d0)) / 2.0d0
    else
        tmp = (1.0d0 + exp(x)) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.8e-36) {
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	} else if (x <= 0.018) {
		tmp = 1.0 + (-0.5 * Math.pow(x, 2.0));
	} else if (x <= 2.8e+83) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp(x)) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.8e-36:
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m
	elif x <= 0.018:
		tmp = 1.0 + (-0.5 * math.pow(x, 2.0))
	elif x <= 2.8e+83:
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0
	else:
		tmp = (1.0 + math.exp(x)) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.8e-36)
		tmp = Float64(Float64(Float64(0.5 * x) + Float64(eps_m * Float64(1.0 + Float64(-0.5 * Float64(eps_m * x))))) / eps_m);
	elseif (x <= 0.018)
		tmp = Float64(1.0 + Float64(-0.5 * (x ^ 2.0)));
	elseif (x <= 2.8e+83)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(x)) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.8e-36)
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	elseif (x <= 0.018)
		tmp = 1.0 + (-0.5 * (x ^ 2.0));
	elseif (x <= 2.8e+83)
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	else
		tmp = (1.0 + exp(x)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.8e-36], N[(N[(N[(0.5 * x), $MachinePrecision] + N[(eps$95$m * N[(1.0 + N[(-0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision], If[LessEqual[x, 0.018], N[(1.0 + N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e+83], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.80000000000000016e-36

   1. Initial program 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around 0 55.5%

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

   6. Taylor expanded in x around 0 28.5%

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

   7. Taylor expanded in eps around 0 41.1%

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

   if -1.80000000000000016e-36 < x < 0.0179999999999999986

   1. Initial program 50.4%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in eps around 0 79.4%

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

   6. Taylor expanded in x around 0 79.4%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]

   if 0.0179999999999999986 < x < 2.8e83

   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 18.7%

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

   6. Taylor expanded in x around 0 67.7%

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

   if 2.8e83 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 42.5%

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

   7. Taylor expanded in eps around 0 3.1%

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

      1. div-inv 3.1%

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

      2. +-commutative 3.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 63.1%

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

      5. *-commutative 63.1%

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

      6. *-commutative 63.1%

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

      7. swap-sqr 63.1%

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

      8. metadata-eval 63.1%

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

      9. *-un-lft-identity 63.1%

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

      10. sqrt-unprod 63.1%

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

      11. add-sqr-sqrt 63.1%

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

      12. metadata-eval 63.1%

          \[\leadsto \left(1 + e^{x}\right) \cdot \color{blue}{0.5} \]

   9. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\left(1 + e^{x}\right) \cdot 0.5} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{1}{eps\_m}\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{0.5 \cdot x + eps\_m \cdot \left(1 + -0.5 \cdot \left(eps\_m \cdot x\right)\right)}{eps\_m}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(t\_0 \cdot \left(eps\_m - 1\right) + \frac{1}{eps\_m}\right) - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{t\_0 - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + e^{x}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps_m))))
   (if (<= x -1.4e-36)
     (/ (+ (* 0.5 x) (* eps_m (+ 1.0 (* -0.5 (* eps_m x))))) eps_m)
     (if (<= x 0.018)
       (/ (+ 2.0 (* x (- (+ (* t_0 (- eps_m 1.0)) (/ 1.0 eps_m)) eps_m))) 2.0)
       (if (<= x 1.5e+81)
         (/ (- t_0 (- (/ 1.0 eps_m) 1.0)) 2.0)
         (* (+ 1.0 (exp x)) 0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -1.4e-36) {
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	} else if (x <= 0.018) {
		tmp = (2.0 + (x * (((t_0 * (eps_m - 1.0)) + (1.0 / eps_m)) - eps_m))) / 2.0;
	} else if (x <= 1.5e+81) {
		tmp = (t_0 - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (1.0 + exp(x)) * 0.5;
	}
	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 = 1.0d0 + (1.0d0 / eps_m)
    if (x <= (-1.4d-36)) then
        tmp = ((0.5d0 * x) + (eps_m * (1.0d0 + ((-0.5d0) * (eps_m * x))))) / eps_m
    else if (x <= 0.018d0) then
        tmp = (2.0d0 + (x * (((t_0 * (eps_m - 1.0d0)) + (1.0d0 / eps_m)) - eps_m))) / 2.0d0
    else if (x <= 1.5d+81) then
        tmp = (t_0 - ((1.0d0 / eps_m) - 1.0d0)) / 2.0d0
    else
        tmp = (1.0d0 + exp(x)) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -1.4e-36) {
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	} else if (x <= 0.018) {
		tmp = (2.0 + (x * (((t_0 * (eps_m - 1.0)) + (1.0 / eps_m)) - eps_m))) / 2.0;
	} else if (x <= 1.5e+81) {
		tmp = (t_0 - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp(x)) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 1.0 + (1.0 / eps_m)
	tmp = 0
	if x <= -1.4e-36:
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m
	elif x <= 0.018:
		tmp = (2.0 + (x * (((t_0 * (eps_m - 1.0)) + (1.0 / eps_m)) - eps_m))) / 2.0
	elif x <= 1.5e+81:
		tmp = (t_0 - ((1.0 / eps_m) - 1.0)) / 2.0
	else:
		tmp = (1.0 + math.exp(x)) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 + Float64(1.0 / eps_m))
	tmp = 0.0
	if (x <= -1.4e-36)
		tmp = Float64(Float64(Float64(0.5 * x) + Float64(eps_m * Float64(1.0 + Float64(-0.5 * Float64(eps_m * x))))) / eps_m);
	elseif (x <= 0.018)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(t_0 * Float64(eps_m - 1.0)) + Float64(1.0 / eps_m)) - eps_m))) / 2.0);
	elseif (x <= 1.5e+81)
		tmp = Float64(Float64(t_0 - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(x)) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 1.0 + (1.0 / eps_m);
	tmp = 0.0;
	if (x <= -1.4e-36)
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	elseif (x <= 0.018)
		tmp = (2.0 + (x * (((t_0 * (eps_m - 1.0)) + (1.0 / eps_m)) - eps_m))) / 2.0;
	elseif (x <= 1.5e+81)
		tmp = (t_0 - ((1.0 / eps_m) - 1.0)) / 2.0;
	else
		tmp = (1.0 + exp(x)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e-36], N[(N[(N[(0.5 * x), $MachinePrecision] + N[(eps$95$m * N[(1.0 + N[(-0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision], If[LessEqual[x, 0.018], N[(N[(2.0 + N[(x * N[(N[(N[(t$95$0 * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.5e+81], N[(N[(t$95$0 - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.4000000000000001e-36

   1. Initial program 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around 0 55.5%

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

   6. Taylor expanded in x around 0 28.5%

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

   7. Taylor expanded in eps around 0 41.1%

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

   if -1.4000000000000001e-36 < x < 0.0179999999999999986

   1. Initial program 50.4%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in x around 0 79.3%

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

   if 0.0179999999999999986 < x < 1.49999999999999999e81

   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 18.7%

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

   6. Taylor expanded in x around 0 67.7%

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

   if 1.49999999999999999e81 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 42.5%

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

   7. Taylor expanded in eps around 0 3.1%

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

      1. div-inv 3.1%

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

      2. +-commutative 3.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 63.1%

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

      5. *-commutative 63.1%

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

      6. *-commutative 63.1%

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

      7. swap-sqr 63.1%

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

      8. metadata-eval 63.1%

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

      9. *-un-lft-identity 63.1%

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

      10. sqrt-unprod 63.1%

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

      11. add-sqr-sqrt 63.1%

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

      12. metadata-eval 63.1%

          \[\leadsto \left(1 + e^{x}\right) \cdot \color{blue}{0.5} \]

   9. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\left(1 + e^{x}\right) \cdot 0.5} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 70.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 0.018:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + e^{x}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 0.018)
   (/ (+ (exp (- x)) 1.0) 2.0)
   (if (<= x 7.8e+80)
     (/ (- (+ 1.0 (/ 1.0 eps_m)) (- (/ 1.0 eps_m) 1.0)) 2.0)
     (* (+ 1.0 (exp x)) 0.5))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.018) {
		tmp = (exp(-x) + 1.0) / 2.0;
	} else if (x <= 7.8e+80) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (1.0 + exp(x)) * 0.5;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 0.018d0) then
        tmp = (exp(-x) + 1.0d0) / 2.0d0
    else if (x <= 7.8d+80) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - ((1.0d0 / eps_m) - 1.0d0)) / 2.0d0
    else
        tmp = (1.0d0 + exp(x)) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.018) {
		tmp = (Math.exp(-x) + 1.0) / 2.0;
	} else if (x <= 7.8e+80) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp(x)) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 0.018:
		tmp = (math.exp(-x) + 1.0) / 2.0
	elif x <= 7.8e+80:
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0
	else:
		tmp = (1.0 + math.exp(x)) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 0.018)
		tmp = Float64(Float64(exp(Float64(-x)) + 1.0) / 2.0);
	elseif (x <= 7.8e+80)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(x)) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 0.018)
		tmp = (exp(-x) + 1.0) / 2.0;
	elseif (x <= 7.8e+80)
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	else
		tmp = (1.0 + exp(x)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 0.018], N[(N[(N[Exp[(-x)], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7.8e+80], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0179999999999999986

   1. Initial program 60.8%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in eps around inf 99.3%

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

   6. Taylor expanded in x around 0 79.5%

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

   7. Taylor expanded in eps around 0 79.8%

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

      1. *-commutative 79.8%

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

      2. neg-mul-1 79.8%

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

   9. Applied egg-rr 79.8%

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

   if 0.0179999999999999986 < x < 7.79999999999999998e80

   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 18.7%

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

   6. Taylor expanded in x around 0 67.7%

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

   if 7.79999999999999998e80 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 42.5%

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

   7. Taylor expanded in eps around 0 3.1%

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

      1. div-inv 3.1%

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

      2. +-commutative 3.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 63.1%

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

      5. *-commutative 63.1%

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

      6. *-commutative 63.1%

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

      7. swap-sqr 63.1%

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

      8. metadata-eval 63.1%

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

      9. *-un-lft-identity 63.1%

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

      10. sqrt-unprod 63.1%

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

      11. add-sqr-sqrt 63.1%

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

      12. metadata-eval 63.1%

          \[\leadsto \left(1 + e^{x}\right) \cdot \color{blue}{0.5} \]

   9. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\left(1 + e^{x}\right) \cdot 0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 66.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{1}{eps\_m}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{0.5 \cdot x + eps\_m \cdot \left(1 + -0.5 \cdot \left(eps\_m \cdot x\right)\right)}{eps\_m}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(t\_0 \cdot \left(eps\_m - 1\right) + \frac{1}{eps\_m}\right) - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+182}:\\ \;\;\;\;\frac{t\_0 - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.25 \cdot x - 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps_m))))
   (if (<= x -3.4e-37)
     (/ (+ (* 0.5 x) (* eps_m (+ 1.0 (* -0.5 (* eps_m x))))) eps_m)
     (if (<= x 0.018)
       (/ (+ 2.0 (* x (- (+ (* t_0 (- eps_m 1.0)) (/ 1.0 eps_m)) eps_m))) 2.0)
       (if (<= x 2.3e+182)
         (/ (- t_0 (- (/ 1.0 eps_m) 1.0)) 2.0)
         (+ 1.0 (* x (- (* 0.25 x) 0.5))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -3.4e-37) {
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	} else if (x <= 0.018) {
		tmp = (2.0 + (x * (((t_0 * (eps_m - 1.0)) + (1.0 / eps_m)) - eps_m))) / 2.0;
	} else if (x <= 2.3e+182) {
		tmp = (t_0 - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	}
	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 = 1.0d0 + (1.0d0 / eps_m)
    if (x <= (-3.4d-37)) then
        tmp = ((0.5d0 * x) + (eps_m * (1.0d0 + ((-0.5d0) * (eps_m * x))))) / eps_m
    else if (x <= 0.018d0) then
        tmp = (2.0d0 + (x * (((t_0 * (eps_m - 1.0d0)) + (1.0d0 / eps_m)) - eps_m))) / 2.0d0
    else if (x <= 2.3d+182) then
        tmp = (t_0 - ((1.0d0 / eps_m) - 1.0d0)) / 2.0d0
    else
        tmp = 1.0d0 + (x * ((0.25d0 * x) - 0.5d0))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -3.4e-37) {
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	} else if (x <= 0.018) {
		tmp = (2.0 + (x * (((t_0 * (eps_m - 1.0)) + (1.0 / eps_m)) - eps_m))) / 2.0;
	} else if (x <= 2.3e+182) {
		tmp = (t_0 - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 1.0 + (1.0 / eps_m)
	tmp = 0
	if x <= -3.4e-37:
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m
	elif x <= 0.018:
		tmp = (2.0 + (x * (((t_0 * (eps_m - 1.0)) + (1.0 / eps_m)) - eps_m))) / 2.0
	elif x <= 2.3e+182:
		tmp = (t_0 - ((1.0 / eps_m) - 1.0)) / 2.0
	else:
		tmp = 1.0 + (x * ((0.25 * x) - 0.5))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 + Float64(1.0 / eps_m))
	tmp = 0.0
	if (x <= -3.4e-37)
		tmp = Float64(Float64(Float64(0.5 * x) + Float64(eps_m * Float64(1.0 + Float64(-0.5 * Float64(eps_m * x))))) / eps_m);
	elseif (x <= 0.018)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(t_0 * Float64(eps_m - 1.0)) + Float64(1.0 / eps_m)) - eps_m))) / 2.0);
	elseif (x <= 2.3e+182)
		tmp = Float64(Float64(t_0 - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(0.25 * x) - 0.5)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 1.0 + (1.0 / eps_m);
	tmp = 0.0;
	if (x <= -3.4e-37)
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	elseif (x <= 0.018)
		tmp = (2.0 + (x * (((t_0 * (eps_m - 1.0)) + (1.0 / eps_m)) - eps_m))) / 2.0;
	elseif (x <= 2.3e+182)
		tmp = (t_0 - ((1.0 / eps_m) - 1.0)) / 2.0;
	else
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e-37], N[(N[(N[(0.5 * x), $MachinePrecision] + N[(eps$95$m * N[(1.0 + N[(-0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision], If[LessEqual[x, 0.018], N[(N[(2.0 + N[(x * N[(N[(N[(t$95$0 * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.3e+182], N[(N[(t$95$0 - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 + N[(x * N[(N[(0.25 * x), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.40000000000000018e-37

   1. Initial program 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around 0 55.5%

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

   6. Taylor expanded in x around 0 28.5%

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

   7. Taylor expanded in eps around 0 41.1%

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

   if -3.40000000000000018e-37 < x < 0.0179999999999999986

   1. Initial program 50.4%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in x around 0 79.3%

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

   if 0.0179999999999999986 < x < 2.3e182

   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 31.0%

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

   6. Taylor expanded in x around 0 49.5%

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

   if 2.3e182 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 41.9%

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

   7. Taylor expanded in eps around 0 3.1%

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

   8. Taylor expanded in x around 0 58.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 67.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-36}:\\ \;\;\;\;\frac{0.5 \cdot x + eps\_m \cdot \left(1 + -0.5 \cdot \left(eps\_m \cdot x\right)\right)}{eps\_m}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+182}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.25 \cdot x - 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.25e-36)
   (/ (+ (* 0.5 x) (* eps_m (+ 1.0 (* -0.5 (* eps_m x))))) eps_m)
   (if (<= x 0.018)
     1.0
     (if (<= x 3.6e+182)
       (/ (- (+ 1.0 (/ 1.0 eps_m)) (- (/ 1.0 eps_m) 1.0)) 2.0)
       (+ 1.0 (* x (- (* 0.25 x) 0.5)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-36) {
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	} else if (x <= 0.018) {
		tmp = 1.0;
	} else if (x <= 3.6e+182) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.25d-36)) then
        tmp = ((0.5d0 * x) + (eps_m * (1.0d0 + ((-0.5d0) * (eps_m * x))))) / eps_m
    else if (x <= 0.018d0) then
        tmp = 1.0d0
    else if (x <= 3.6d+182) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - ((1.0d0 / eps_m) - 1.0d0)) / 2.0d0
    else
        tmp = 1.0d0 + (x * ((0.25d0 * x) - 0.5d0))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-36) {
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	} else if (x <= 0.018) {
		tmp = 1.0;
	} else if (x <= 3.6e+182) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.25e-36:
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m
	elif x <= 0.018:
		tmp = 1.0
	elif x <= 3.6e+182:
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0
	else:
		tmp = 1.0 + (x * ((0.25 * x) - 0.5))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.25e-36)
		tmp = Float64(Float64(Float64(0.5 * x) + Float64(eps_m * Float64(1.0 + Float64(-0.5 * Float64(eps_m * x))))) / eps_m);
	elseif (x <= 0.018)
		tmp = 1.0;
	elseif (x <= 3.6e+182)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(0.25 * x) - 0.5)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.25e-36)
		tmp = ((0.5 * x) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	elseif (x <= 0.018)
		tmp = 1.0;
	elseif (x <= 3.6e+182)
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	else
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.25e-36], N[(N[(N[(0.5 * x), $MachinePrecision] + N[(eps$95$m * N[(1.0 + N[(-0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision], If[LessEqual[x, 0.018], 1.0, If[LessEqual[x, 3.6e+182], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 + N[(x * N[(N[(0.25 * x), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.25000000000000001e-36

   1. Initial program 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around 0 55.5%

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

   6. Taylor expanded in x around 0 28.5%

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

   7. Taylor expanded in eps around 0 41.1%

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

   if -1.25000000000000001e-36 < x < 0.0179999999999999986

   1. Initial program 50.4%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in x around 0 79.3%

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

   if 0.0179999999999999986 < x < 3.6e182

   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 31.0%

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

   6. Taylor expanded in x around 0 49.5%

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

   if 3.6e182 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 41.9%

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

   7. Taylor expanded in eps around 0 3.1%

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

   8. Taylor expanded in x around 0 58.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 66.3% accurate, 9.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 0.018:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + -0.08333333333333333 \cdot x\right) - 0.5\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+182}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.25 \cdot x - 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 0.018)
   (+ 1.0 (* x (- (* x (+ 0.25 (* -0.08333333333333333 x))) 0.5)))
   (if (<= x 7.2e+182)
     (/ (- (+ 1.0 (/ 1.0 eps_m)) (- (/ 1.0 eps_m) 1.0)) 2.0)
     (+ 1.0 (* x (- (* 0.25 x) 0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.018) {
		tmp = 1.0 + (x * ((x * (0.25 + (-0.08333333333333333 * x))) - 0.5));
	} else if (x <= 7.2e+182) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 0.018d0) then
        tmp = 1.0d0 + (x * ((x * (0.25d0 + ((-0.08333333333333333d0) * x))) - 0.5d0))
    else if (x <= 7.2d+182) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - ((1.0d0 / eps_m) - 1.0d0)) / 2.0d0
    else
        tmp = 1.0d0 + (x * ((0.25d0 * x) - 0.5d0))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.018) {
		tmp = 1.0 + (x * ((x * (0.25 + (-0.08333333333333333 * x))) - 0.5));
	} else if (x <= 7.2e+182) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 0.018:
		tmp = 1.0 + (x * ((x * (0.25 + (-0.08333333333333333 * x))) - 0.5))
	elif x <= 7.2e+182:
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0
	else:
		tmp = 1.0 + (x * ((0.25 * x) - 0.5))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 0.018)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.25 + Float64(-0.08333333333333333 * x))) - 0.5)));
	elseif (x <= 7.2e+182)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(0.25 * x) - 0.5)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 0.018)
		tmp = 1.0 + (x * ((x * (0.25 + (-0.08333333333333333 * x))) - 0.5));
	elseif (x <= 7.2e+182)
		tmp = ((1.0 + (1.0 / eps_m)) - ((1.0 / eps_m) - 1.0)) / 2.0;
	else
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 0.018], N[(1.0 + N[(x * N[(N[(x * N[(0.25 + N[(-0.08333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e+182], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 + N[(x * N[(N[(0.25 * x), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0179999999999999986

   1. Initial program 60.8%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in eps around inf 99.3%

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

   6. Taylor expanded in x around 0 79.5%

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

   7. Taylor expanded in eps around 0 79.8%

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

   8. Taylor expanded in x around 0 74.3%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.25 + -0.08333333333333333 \cdot x\right) - 0.5\right)} \]

   if 0.0179999999999999986 < x < 7.2e182

   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 31.0%

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

   6. Taylor expanded in x around 0 49.5%

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

   if 7.2e182 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 41.9%

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

   7. Taylor expanded in eps around 0 3.1%

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

   8. Taylor expanded in x around 0 58.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 62.0% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-53}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + -0.08333333333333333 \cdot x\right) - 0.5\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+160}:\\ \;\;\;\;\frac{eps\_m \cdot \left(\frac{2}{eps\_m} + x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.25 \cdot x - 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 3e-53)
   (+ 1.0 (* x (- (* x (+ 0.25 (* -0.08333333333333333 x))) 0.5)))
   (if (<= x 1.9e+160)
     (/ (* eps_m (+ (/ 2.0 eps_m) x)) 2.0)
     (+ 1.0 (* x (- (* 0.25 x) 0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 3e-53) {
		tmp = 1.0 + (x * ((x * (0.25 + (-0.08333333333333333 * x))) - 0.5));
	} else if (x <= 1.9e+160) {
		tmp = (eps_m * ((2.0 / eps_m) + x)) / 2.0;
	} else {
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	}
	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 <= 3d-53) then
        tmp = 1.0d0 + (x * ((x * (0.25d0 + ((-0.08333333333333333d0) * x))) - 0.5d0))
    else if (x <= 1.9d+160) then
        tmp = (eps_m * ((2.0d0 / eps_m) + x)) / 2.0d0
    else
        tmp = 1.0d0 + (x * ((0.25d0 * x) - 0.5d0))
    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 <= 3e-53) {
		tmp = 1.0 + (x * ((x * (0.25 + (-0.08333333333333333 * x))) - 0.5));
	} else if (x <= 1.9e+160) {
		tmp = (eps_m * ((2.0 / eps_m) + x)) / 2.0;
	} else {
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 3e-53:
		tmp = 1.0 + (x * ((x * (0.25 + (-0.08333333333333333 * x))) - 0.5))
	elif x <= 1.9e+160:
		tmp = (eps_m * ((2.0 / eps_m) + x)) / 2.0
	else:
		tmp = 1.0 + (x * ((0.25 * x) - 0.5))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 3e-53)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.25 + Float64(-0.08333333333333333 * x))) - 0.5)));
	elseif (x <= 1.9e+160)
		tmp = Float64(Float64(eps_m * Float64(Float64(2.0 / eps_m) + x)) / 2.0);
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(0.25 * x) - 0.5)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 3e-53)
		tmp = 1.0 + (x * ((x * (0.25 + (-0.08333333333333333 * x))) - 0.5));
	elseif (x <= 1.9e+160)
		tmp = (eps_m * ((2.0 / eps_m) + x)) / 2.0;
	else
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 3e-53], N[(1.0 + N[(x * N[(N[(x * N[(0.25 + N[(-0.08333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e+160], N[(N[(eps$95$m * N[(N[(2.0 / eps$95$m), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 + N[(x * N[(N[(0.25 * x), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.0000000000000002e-53

   1. Initial program 62.7%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in eps around inf 99.7%

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

   6. Taylor expanded in x around 0 79.5%

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

   7. Taylor expanded in eps around 0 81.1%

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

   8. Taylor expanded in x around 0 75.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.25 + -0.08333333333333333 \cdot x\right) - 0.5\right)} \]

   if 3.0000000000000002e-53 < x < 1.90000000000000006e160

   1. Initial program 85.6%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in x around 0 31.5%

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

   6. Taylor expanded in eps around inf 48.8%

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

   7. Taylor expanded in x around 0 24.0%

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

      1. sub-neg 24.0%

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

      2. neg-mul-1 24.0%

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

      3. *-commutative 24.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 20.7%

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

      6. *-commutative 20.7%

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

      7. *-commutative 20.7%

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

      8. swap-sqr 20.7%

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

      9. metadata-eval 20.7%

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

      10. *-un-lft-identity 20.7%

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

      11. sqrt-unprod 20.7%

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

      12. add-sqr-sqrt 20.7%

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

   9. Applied egg-rr 20.7%

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

   if 1.90000000000000006e160 < 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%

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

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

   7. Taylor expanded in eps around 0 3.1%

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

   8. Taylor expanded in x around 0 57.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 59.9% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-292}:\\ \;\;\;\;1 + -0.5 \cdot \left(eps\_m \cdot x\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+160}:\\ \;\;\;\;\frac{eps\_m \cdot \left(\frac{2}{eps\_m} + x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.25 \cdot x - 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4.3e-292)
   (+ 1.0 (* -0.5 (* eps_m x)))
   (if (<= x 1.9e+160)
     (/ (* eps_m (+ (/ 2.0 eps_m) x)) 2.0)
     (+ 1.0 (* x (- (* 0.25 x) 0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -4.3e-292) {
		tmp = 1.0 + (-0.5 * (eps_m * x));
	} else if (x <= 1.9e+160) {
		tmp = (eps_m * ((2.0 / eps_m) + x)) / 2.0;
	} else {
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	}
	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 <= (-4.3d-292)) then
        tmp = 1.0d0 + ((-0.5d0) * (eps_m * x))
    else if (x <= 1.9d+160) then
        tmp = (eps_m * ((2.0d0 / eps_m) + x)) / 2.0d0
    else
        tmp = 1.0d0 + (x * ((0.25d0 * x) - 0.5d0))
    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 <= -4.3e-292) {
		tmp = 1.0 + (-0.5 * (eps_m * x));
	} else if (x <= 1.9e+160) {
		tmp = (eps_m * ((2.0 / eps_m) + x)) / 2.0;
	} else {
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -4.3e-292:
		tmp = 1.0 + (-0.5 * (eps_m * x))
	elif x <= 1.9e+160:
		tmp = (eps_m * ((2.0 / eps_m) + x)) / 2.0
	else:
		tmp = 1.0 + (x * ((0.25 * x) - 0.5))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -4.3e-292)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(eps_m * x)));
	elseif (x <= 1.9e+160)
		tmp = Float64(Float64(eps_m * Float64(Float64(2.0 / eps_m) + x)) / 2.0);
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(0.25 * x) - 0.5)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -4.3e-292)
		tmp = 1.0 + (-0.5 * (eps_m * x));
	elseif (x <= 1.9e+160)
		tmp = (eps_m * ((2.0 / eps_m) + x)) / 2.0;
	else
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -4.3e-292], N[(1.0 + N[(-0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e+160], N[(N[(eps$95$m * N[(N[(2.0 / eps$95$m), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 + N[(x * N[(N[(0.25 * x), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e-292

   1. Initial program 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in x around 0 44.0%

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

   6. Taylor expanded in eps around inf 85.4%

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

   7. Taylor expanded in x around 0 62.5%

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

   8. Taylor expanded in eps around 0 62.6%

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

   if -4.3e-292 < x < 1.90000000000000006e160

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in x around 0 38.5%

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

   6. Taylor expanded in eps around inf 69.3%

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

   7. Taylor expanded in x around 0 52.0%

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

      1. sub-neg 52.0%

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

      2. neg-mul-1 52.0%

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

      3. *-commutative 52.0%

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

      4. add-sqr-sqrt 1.7%

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

      5. sqrt-unprod 50.5%

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

      6. *-commutative 50.5%

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

      7. *-commutative 50.5%

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

      8. swap-sqr 50.5%

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

      9. metadata-eval 50.5%

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

      10. *-un-lft-identity 50.5%

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

      11. sqrt-unprod 48.7%

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

      12. add-sqr-sqrt 50.5%

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

   9. Applied egg-rr 50.5%

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

   if 1.90000000000000006e160 < 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%

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

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

   7. Taylor expanded in eps around 0 3.1%

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

   8. Taylor expanded in x around 0 57.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 59.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 0.018:\\ \;\;\;\;1 + -0.5 \cdot \left(eps\_m \cdot x\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+160}:\\ \;\;\;\;-0.5 \cdot \left(-eps\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.25 \cdot x - 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 0.018)
   (+ 1.0 (* -0.5 (* eps_m x)))
   (if (<= x 1.9e+160)
     (* -0.5 (- (* eps_m x)))
     (+ 1.0 (* x (- (* 0.25 x) 0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.018) {
		tmp = 1.0 + (-0.5 * (eps_m * x));
	} else if (x <= 1.9e+160) {
		tmp = -0.5 * -(eps_m * x);
	} else {
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 0.018d0) then
        tmp = 1.0d0 + ((-0.5d0) * (eps_m * x))
    else if (x <= 1.9d+160) then
        tmp = (-0.5d0) * -(eps_m * x)
    else
        tmp = 1.0d0 + (x * ((0.25d0 * x) - 0.5d0))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.018) {
		tmp = 1.0 + (-0.5 * (eps_m * x));
	} else if (x <= 1.9e+160) {
		tmp = -0.5 * -(eps_m * x);
	} else {
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 0.018:
		tmp = 1.0 + (-0.5 * (eps_m * x))
	elif x <= 1.9e+160:
		tmp = -0.5 * -(eps_m * x)
	else:
		tmp = 1.0 + (x * ((0.25 * x) - 0.5))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 0.018)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(eps_m * x)));
	elseif (x <= 1.9e+160)
		tmp = Float64(-0.5 * Float64(-Float64(eps_m * x)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(0.25 * x) - 0.5)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 0.018)
		tmp = 1.0 + (-0.5 * (eps_m * x));
	elseif (x <= 1.9e+160)
		tmp = -0.5 * -(eps_m * x);
	else
		tmp = 1.0 + (x * ((0.25 * x) - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 0.018], N[(1.0 + N[(-0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e+160], N[(-0.5 * (-N[(eps$95$m * x), $MachinePrecision])), $MachinePrecision], N[(1.0 + N[(x * N[(N[(0.25 * x), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0179999999999999986

   1. Initial program 60.8%

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

   3. Simplified 60.8%

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

   5. Taylor expanded in x around 0 42.5%

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

   6. Taylor expanded in eps around inf 85.7%

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

   7. Taylor expanded in x around 0 68.0%

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

   8. Taylor expanded in eps around 0 68.1%

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

   if 0.0179999999999999986 < x < 1.90000000000000006e160

   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 35.0%

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

   6. Taylor expanded in eps around inf 12.2%

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

   7. Taylor expanded in eps around 0 12.2%

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

      1. add-sqr-sqrt 12.2%

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

      2. sqrt-unprod 12.2%

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

      3. *-un-lft-identity 12.2%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\color{blue}{1 \cdot \left(x \cdot x\right)}}\right) \]

      4. metadata-eval 12.2%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\color{blue}{\left(-1 \cdot -1\right)} \cdot \left(x \cdot x\right)}\right) \]

      5. swap-sqr 12.2%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\color{blue}{\left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}}\right) \]

      6. *-commutative 12.2%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\color{blue}{\left(x \cdot -1\right)} \cdot \left(-1 \cdot x\right)}\right) \]

      7. *-commutative 12.2%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\left(x \cdot -1\right) \cdot \color{blue}{\left(x \cdot -1\right)}}\right) \]

      8. sqrt-unprod 0.0%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \color{blue}{\left(\sqrt{x \cdot -1} \cdot \sqrt{x \cdot -1}\right)}\right) \]

      9. add-sqr-sqrt 8.1%

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

      10. *-commutative 8.1%

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

      11. neg-mul-1 8.1%

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

      12. distribute-rgt-neg-in 8.1%

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

   9. Applied egg-rr 8.1%

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

   if 1.90000000000000006e160 < 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%

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

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

   7. Taylor expanded in eps around 0 3.1%

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

   8. Taylor expanded in x around 0 57.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 58.7% accurate, 14.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-0.5 \cdot \left(eps\_m \cdot x\right)\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(-eps\_m \cdot x\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (* -0.5 (* eps_m x))
   (if (<= x 0.018) 1.0 (* -0.5 (- (* eps_m x))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = -0.5 * (eps_m * x);
	} else if (x <= 0.018) {
		tmp = 1.0;
	} else {
		tmp = -0.5 * -(eps_m * x);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-0.5d0) * (eps_m * x)
    else if (x <= 0.018d0) then
        tmp = 1.0d0
    else
        tmp = (-0.5d0) * -(eps_m * x)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = -0.5 * (eps_m * x);
	} else if (x <= 0.018) {
		tmp = 1.0;
	} else {
		tmp = -0.5 * -(eps_m * x);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = -0.5 * (eps_m * x)
	elif x <= 0.018:
		tmp = 1.0
	else:
		tmp = -0.5 * -(eps_m * x)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-0.5 * Float64(eps_m * x));
	elseif (x <= 0.018)
		tmp = 1.0;
	else
		tmp = Float64(-0.5 * Float64(-Float64(eps_m * x)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -0.5 * (eps_m * x);
	elseif (x <= 0.018)
		tmp = 1.0;
	else
		tmp = -0.5 * -(eps_m * x);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(-0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.018], 1.0, N[(-0.5 * (-N[(eps$95$m * x), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 54.5%

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

   6. Taylor expanded in eps around inf 34.6%

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

   7. Taylor expanded in eps around 0 34.6%

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

   if -1 < x < 0.0179999999999999986

   1. Initial program 52.3%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in x around 0 76.0%

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

   if 0.0179999999999999986 < 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%

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

   5. Taylor expanded in x around 0 29.9%

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

   6. Taylor expanded in eps around inf 12.4%

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

   7. Taylor expanded in eps around 0 12.4%

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

      1. add-sqr-sqrt 12.4%

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

      2. sqrt-unprod 15.3%

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

      3. *-un-lft-identity 15.3%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\color{blue}{1 \cdot \left(x \cdot x\right)}}\right) \]

      4. metadata-eval 15.3%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\color{blue}{\left(-1 \cdot -1\right)} \cdot \left(x \cdot x\right)}\right) \]

      5. swap-sqr 15.3%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\color{blue}{\left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}}\right) \]

      6. *-commutative 15.3%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\color{blue}{\left(x \cdot -1\right)} \cdot \left(-1 \cdot x\right)}\right) \]

      7. *-commutative 15.3%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\left(x \cdot -1\right) \cdot \color{blue}{\left(x \cdot -1\right)}}\right) \]

      8. sqrt-unprod 0.0%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \color{blue}{\left(\sqrt{x \cdot -1} \cdot \sqrt{x \cdot -1}\right)}\right) \]

      9. add-sqr-sqrt 18.6%

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

      10. *-commutative 18.6%

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

      11. neg-mul-1 18.6%

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

      12. distribute-rgt-neg-in 18.6%

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

   9. Applied egg-rr 18.6%

      \[\leadsto -0.5 \cdot \color{blue}{\left(-\varepsilon \cdot x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 58.3% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 0.018:\\ \;\;\;\;1 + -0.5 \cdot \left(eps\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(-eps\_m \cdot x\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 0.018) (+ 1.0 (* -0.5 (* eps_m x))) (* -0.5 (- (* eps_m x)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.018) {
		tmp = 1.0 + (-0.5 * (eps_m * x));
	} else {
		tmp = -0.5 * -(eps_m * x);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 0.018d0) then
        tmp = 1.0d0 + ((-0.5d0) * (eps_m * x))
    else
        tmp = (-0.5d0) * -(eps_m * x)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.018) {
		tmp = 1.0 + (-0.5 * (eps_m * x));
	} else {
		tmp = -0.5 * -(eps_m * x);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 0.018:
		tmp = 1.0 + (-0.5 * (eps_m * x))
	else:
		tmp = -0.5 * -(eps_m * x)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 0.018)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(eps_m * x)));
	else
		tmp = Float64(-0.5 * Float64(-Float64(eps_m * x)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 0.018)
		tmp = 1.0 + (-0.5 * (eps_m * x));
	else
		tmp = -0.5 * -(eps_m * x);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 0.018], N[(1.0 + N[(-0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * (-N[(eps$95$m * x), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0179999999999999986

   1. Initial program 60.8%

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

   3. Simplified 60.8%

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

   5. Taylor expanded in x around 0 42.5%

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

   6. Taylor expanded in eps around inf 85.7%

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

   7. Taylor expanded in x around 0 68.0%

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

   8. Taylor expanded in eps around 0 68.1%

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

   if 0.0179999999999999986 < 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%

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

   5. Taylor expanded in x around 0 29.9%

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

   6. Taylor expanded in eps around inf 12.4%

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

   7. Taylor expanded in eps around 0 12.4%

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

      1. add-sqr-sqrt 12.4%

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

      2. sqrt-unprod 15.3%

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

      3. *-un-lft-identity 15.3%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\color{blue}{1 \cdot \left(x \cdot x\right)}}\right) \]

      4. metadata-eval 15.3%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\color{blue}{\left(-1 \cdot -1\right)} \cdot \left(x \cdot x\right)}\right) \]

      5. swap-sqr 15.3%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\color{blue}{\left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}}\right) \]

      6. *-commutative 15.3%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\color{blue}{\left(x \cdot -1\right)} \cdot \left(-1 \cdot x\right)}\right) \]

      7. *-commutative 15.3%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \sqrt{\left(x \cdot -1\right) \cdot \color{blue}{\left(x \cdot -1\right)}}\right) \]

      8. sqrt-unprod 0.0%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \color{blue}{\left(\sqrt{x \cdot -1} \cdot \sqrt{x \cdot -1}\right)}\right) \]

      9. add-sqr-sqrt 18.6%

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

      10. *-commutative 18.6%

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

      11. neg-mul-1 18.6%

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

      12. distribute-rgt-neg-in 18.6%

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

   9. Applied egg-rr 18.6%

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

Alternative 21: 52.0% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-0.5 \cdot \left(eps\_m \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0) (* -0.5 (* eps_m x)) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = -0.5 * (eps_m * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = -0.5 * (eps_m * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = -0.5 * (eps_m * x)
	else:
		tmp = 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-0.5 * Float64(eps_m * x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -0.5 * (eps_m * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(-0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 54.5%

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

   6. Taylor expanded in eps around inf 34.6%

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

   7. Taylor expanded in eps around 0 34.6%

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

   if -1 < x

   1. Initial program 70.8%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in x around 0 47.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 44.9% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

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

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 1.0d0
end function

Java

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

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 74.3%

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

3. Simplified 67.7%

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

5. Taylor expanded in x around 0 42.4%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024116 -o generate:simplify
(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))