NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.5% → 98.9%

Time: 19.0s

Alternatives: 18

Speedup: 2.0×

• 38.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(-1 + eps\_m\right)\\ \mathbf{if}\;x \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;\frac{{e}^{t\_0} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t\_0} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ -1.0 eps_m))))
   (if (<= x 1.65e-9)
     (/ (+ (pow E t_0) (exp (* x (- eps_m)))) 2.0)
     (/ (+ (exp t_0) (exp (- x))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (-1.0 + eps_m);
	double tmp;
	if (x <= 1.65e-9) {
		tmp = (pow(((double) M_E), t_0) + exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (exp(t_0) + exp(-x)) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (-1.0 + eps_m);
	double tmp;
	if (x <= 1.65e-9) {
		tmp = (Math.pow(Math.E, t_0) + Math.exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (Math.exp(t_0) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (-1.0 + eps_m)
	tmp = 0
	if x <= 1.65e-9:
		tmp = (math.pow(math.e, t_0) + math.exp((x * -eps_m))) / 2.0
	else:
		tmp = (math.exp(t_0) + math.exp(-x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(-1.0 + eps_m))
	tmp = 0.0
	if (x <= 1.65e-9)
		tmp = Float64(Float64((exp(1) ^ t_0) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(exp(t_0) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (-1.0 + eps_m);
	tmp = 0.0;
	if (x <= 1.65e-9)
		tmp = ((2.71828182845904523536 ^ t_0) + exp((x * -eps_m))) / 2.0;
	else
		tmp = (exp(t_0) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.65e-9], N[(N[(N[Power[E, t$95$0], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[t$95$0], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65000000000000009e-9

   1. Initial program 64.5%

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

   3. Simplified 45.1%

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

   5. Taylor expanded in eps around inf 99.3%

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

      1. *-un-lft-identity 99.3%

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

      2. exp-prod 99.3%

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

      3. sub-neg 99.3%

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

      4. metadata-eval 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. exp-1-e 99.3%

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

   9. Simplified 99.3%

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

   10. Taylor expanded in eps around inf 99.3%

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

       1. *-commutative 99.3%

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

   12. Simplified 99.3%

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

   if 1.65000000000000009e-9 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 83.8%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;\frac{{e}^{\left(x \cdot \left(-1 + \varepsilon\right)\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(-eps\_m\right)}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-227}:\\ \;\;\;\;\frac{1 + t\_0}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+54}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + \left(1 - x \cdot eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+197}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+262} \lor \neg \left(x \leq 1.05 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (- eps_m)))))
   (if (<= x -1e-227)
     (/ (+ 1.0 t_0) 2.0)
     (if (<= x 9e+54)
       (/ (+ (exp (* x (+ -1.0 eps_m))) (- 1.0 (* x eps_m))) 2.0)
       (if (<= x 3.1e+197)
         (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)
         (if (or (<= x 6e+262) (not (<= x 1.05e+300)))
           (/ (* x eps_m) 2.0)
           (/ (+ t_0 (exp (- x))) 2.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * -eps_m));
	double tmp;
	if (x <= -1e-227) {
		tmp = (1.0 + t_0) / 2.0;
	} else if (x <= 9e+54) {
		tmp = (exp((x * (-1.0 + eps_m))) + (1.0 - (x * eps_m))) / 2.0;
	} else if (x <= 3.1e+197) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else if ((x <= 6e+262) || !(x <= 1.05e+300)) {
		tmp = (x * eps_m) / 2.0;
	} else {
		tmp = (t_0 + exp(-x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * -eps_m))
    if (x <= (-1d-227)) then
        tmp = (1.0d0 + t_0) / 2.0d0
    else if (x <= 9d+54) then
        tmp = (exp((x * ((-1.0d0) + eps_m))) + (1.0d0 - (x * eps_m))) / 2.0d0
    else if (x <= 3.1d+197) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    else if ((x <= 6d+262) .or. (.not. (x <= 1.05d+300))) then
        tmp = (x * eps_m) / 2.0d0
    else
        tmp = (t_0 + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * -eps_m));
	double tmp;
	if (x <= -1e-227) {
		tmp = (1.0 + t_0) / 2.0;
	} else if (x <= 9e+54) {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + (1.0 - (x * eps_m))) / 2.0;
	} else if (x <= 3.1e+197) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else if ((x <= 6e+262) || !(x <= 1.05e+300)) {
		tmp = (x * eps_m) / 2.0;
	} else {
		tmp = (t_0 + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * -eps_m))
	tmp = 0
	if x <= -1e-227:
		tmp = (1.0 + t_0) / 2.0
	elif x <= 9e+54:
		tmp = (math.exp((x * (-1.0 + eps_m))) + (1.0 - (x * eps_m))) / 2.0
	elif x <= 3.1e+197:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	elif (x <= 6e+262) or not (x <= 1.05e+300):
		tmp = (x * eps_m) / 2.0
	else:
		tmp = (t_0 + math.exp(-x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(-eps_m)))
	tmp = 0.0
	if (x <= -1e-227)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	elseif (x <= 9e+54)
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + Float64(1.0 - Float64(x * eps_m))) / 2.0);
	elseif (x <= 3.1e+197)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / eps_m))) / 2.0);
	elseif ((x <= 6e+262) || !(x <= 1.05e+300))
		tmp = Float64(Float64(x * eps_m) / 2.0);
	else
		tmp = Float64(Float64(t_0 + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * -eps_m));
	tmp = 0.0;
	if (x <= -1e-227)
		tmp = (1.0 + t_0) / 2.0;
	elseif (x <= 9e+54)
		tmp = (exp((x * (-1.0 + eps_m))) + (1.0 - (x * eps_m))) / 2.0;
	elseif (x <= 3.1e+197)
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	elseif ((x <= 6e+262) || ~((x <= 1.05e+300)))
		tmp = (x * eps_m) / 2.0;
	else
		tmp = (t_0 + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1e-227], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9e+54], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.1e+197], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 6e+262], N[Not[LessEqual[x, 1.05e+300]], $MachinePrecision]], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -9.99999999999999945e-228

   1. Initial program 74.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in eps around inf 98.9%

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

      1. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around 0 63.8%

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

   if -9.99999999999999945e-228 < x < 8.99999999999999968e54

   1. Initial program 63.2%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in eps around inf 99.7%

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

   6. Taylor expanded in eps around inf 95.6%

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

      1. *-commutative 95.6%

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

   8. Simplified 95.6%

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

   9. Taylor expanded in x around 0 83.8%

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

       1. mul-1-neg 83.8%

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

       2. *-commutative 83.8%

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

       3. unsub-neg 83.8%

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

   11. Simplified 83.8%

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

   if 8.99999999999999968e54 < x < 3.1e197

   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 18.3%

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

   6. Taylor expanded in x around 0 63.3%

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

   if 3.1e197 < x < 6.0000000000000001e262 or 1.05000000000000006e300 < 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 63.8%

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

   6. Taylor expanded in x around inf 63.8%

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

      1. associate-*r* 63.8%

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

      2. *-commutative 63.8%

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

      3. associate-*r* 63.8%

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

      4. sub-neg 63.8%

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

      5. neg-mul-1 63.8%

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

      6. associate-*r* 63.8%

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

      7. mul-1-neg 63.8%

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

      8. neg-sub0 63.8%

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

      9. distribute-rgt-in 63.8%

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

      10. *-lft-identity 63.8%

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

      11. neg-mul-1 63.8%

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

      12. cancel-sign-sub-inv 63.8%

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

      13. associate-+l- 63.8%

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

      14. neg-sub0 63.8%

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

      15. neg-mul-1 63.8%

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

      16. +-commutative 63.8%

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

      17. distribute-rgt-in 63.8%

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

   8. Simplified 63.8%

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

   9. Taylor expanded in eps around inf 63.9%

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

       1. *-commutative 63.9%

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

   11. Simplified 63.9%

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

   if 6.0000000000000001e262 < x < 1.05000000000000006e300

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 40.9%

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

      1. *-commutative 40.9%

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

   8. Simplified 40.9%

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

   9. Taylor expanded in eps around 0 40.9%

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

       1. neg-mul-1 40.9%

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

   11. Simplified 40.9%

       \[\leadsto \frac{e^{-1 \cdot \left(x \cdot \varepsilon\right)} + \color{blue}{e^{-x}}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-227}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+54}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+197}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+262} \lor \neg \left(x \leq 1.05 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 + eps\_m\right)}\\ \mathbf{if}\;x \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;\frac{t\_0 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ -1.0 eps_m)))))
   (if (<= x 1.65e-9)
     (/ (+ t_0 (exp (* x (- eps_m)))) 2.0)
     (/ (+ t_0 (exp (- x))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (-1.0 + eps_m)));
	double tmp;
	if (x <= 1.65e-9) {
		tmp = (t_0 + exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (t_0 + exp(-x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * ((-1.0d0) + eps_m)))
    if (x <= 1.65d-9) then
        tmp = (t_0 + exp((x * -eps_m))) / 2.0d0
    else
        tmp = (t_0 + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (-1.0 + eps_m)));
	double tmp;
	if (x <= 1.65e-9) {
		tmp = (t_0 + Math.exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (t_0 + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (-1.0 + eps_m)))
	tmp = 0
	if x <= 1.65e-9:
		tmp = (t_0 + math.exp((x * -eps_m))) / 2.0
	else:
		tmp = (t_0 + math.exp(-x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(-1.0 + eps_m)))
	tmp = 0.0
	if (x <= 1.65e-9)
		tmp = Float64(Float64(t_0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(t_0 + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (-1.0 + eps_m)));
	tmp = 0.0;
	if (x <= 1.65e-9)
		tmp = (t_0 + exp((x * -eps_m))) / 2.0;
	else
		tmp = (t_0 + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.65e-9], N[(N[(t$95$0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65000000000000009e-9

   1. Initial program 64.5%

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

   3. Simplified 45.1%

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

   5. Taylor expanded in eps around inf 99.3%

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

   6. Taylor expanded in eps around inf 99.3%

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

      1. *-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around inf 99.3%

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

       1. associate-*r* 99.3%

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

       2. neg-mul-1 99.3%

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

   11. Simplified 99.3%

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

   if 1.65000000000000009e-9 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 83.8%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(x \cdot \left(-1 + eps\_m\right)\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -9.8e-228)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (/ (+ (pow E (* x (+ -1.0 eps_m))) (exp (- x))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -9.8e-228) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (pow(((double) M_E), (x * (-1.0 + eps_m))) + exp(-x)) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -9.8e-228) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (Math.pow(Math.E, (x * (-1.0 + eps_m))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -9.8e-228:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	else:
		tmp = (math.pow(math.e, (x * (-1.0 + eps_m))) + math.exp(-x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -9.8e-228)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	else
		tmp = Float64(Float64((exp(1) ^ Float64(x * Float64(-1.0 + eps_m))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -9.8e-228)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	else
		tmp = ((2.71828182845904523536 ^ (x * (-1.0 + eps_m))) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -9.8e-228], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Power[E, N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.79999999999999976e-228

   1. Initial program 74.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in eps around inf 98.9%

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

      1. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around 0 63.8%

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

   if -9.79999999999999976e-228 < x

   1. Initial program 74.7%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in eps around inf 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. exp-prod 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. exp-1-e 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in eps around 0 86.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(x \cdot \left(-1 + \varepsilon\right)\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-221}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-221)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (/ (+ (exp (* x (+ -1.0 eps_m))) (exp (- x))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-221) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) + exp(-x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1d-221)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps_m))) + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-221) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e-221:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) + math.exp(-x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-221)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1e-221)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1e-221], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000002e-221

   1. Initial program 74.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in eps around inf 98.9%

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

      1. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around 0 63.8%

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

   if -1.00000000000000002e-221 < x

   1. Initial program 74.7%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in eps around inf 99.8%

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

   6. Taylor expanded in eps around 0 86.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-221}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

3. Simplified 60.8%

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

5. Taylor expanded in eps around inf 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. exp-prod 99.5%

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

   3. sub-neg 99.5%

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

   4. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

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

   1. exp-1-e 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 7: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

3. Simplified 60.8%

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

5. Taylor expanded in eps around inf 99.5%

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

6. Final simplification 99.5%

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

Alternative 8: 84.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 - \frac{1}{eps\_m}\\ t_1 := 1 + \frac{1}{eps\_m}\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + \left(1 - x \cdot eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+196}:\\ \;\;\;\;\frac{t\_1 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 + e^{x \cdot eps\_m - x} \cdot t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ 1.0 eps_m))) (t_1 (+ 1.0 (/ 1.0 eps_m))))
   (if (<= x -9.8e-228)
     (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
     (if (<= x 9.5e+54)
       (/ (+ (exp (* x (+ -1.0 eps_m))) (- 1.0 (* x eps_m))) 2.0)
       (if (<= x 7e+196)
         (/ (+ t_1 t_0) 2.0)
         (/ (+ t_1 (* (exp (- (* x eps_m) x)) t_0)) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 - (1.0 / eps_m);
	double t_1 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -9.8e-228) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if (x <= 9.5e+54) {
		tmp = (exp((x * (-1.0 + eps_m))) + (1.0 - (x * eps_m))) / 2.0;
	} else if (x <= 7e+196) {
		tmp = (t_1 + t_0) / 2.0;
	} else {
		tmp = (t_1 + (exp(((x * eps_m) - x)) * t_0)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (1.0d0 / eps_m)
    t_1 = 1.0d0 + (1.0d0 / eps_m)
    if (x <= (-9.8d-228)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if (x <= 9.5d+54) then
        tmp = (exp((x * ((-1.0d0) + eps_m))) + (1.0d0 - (x * eps_m))) / 2.0d0
    else if (x <= 7d+196) then
        tmp = (t_1 + t_0) / 2.0d0
    else
        tmp = (t_1 + (exp(((x * eps_m) - x)) * t_0)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = 1.0 - (1.0 / eps_m);
	double t_1 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -9.8e-228) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if (x <= 9.5e+54) {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + (1.0 - (x * eps_m))) / 2.0;
	} else if (x <= 7e+196) {
		tmp = (t_1 + t_0) / 2.0;
	} else {
		tmp = (t_1 + (Math.exp(((x * eps_m) - x)) * t_0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 1.0 - (1.0 / eps_m)
	t_1 = 1.0 + (1.0 / eps_m)
	tmp = 0
	if x <= -9.8e-228:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif x <= 9.5e+54:
		tmp = (math.exp((x * (-1.0 + eps_m))) + (1.0 - (x * eps_m))) / 2.0
	elif x <= 7e+196:
		tmp = (t_1 + t_0) / 2.0
	else:
		tmp = (t_1 + (math.exp(((x * eps_m) - x)) * t_0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 - Float64(1.0 / eps_m))
	t_1 = Float64(1.0 + Float64(1.0 / eps_m))
	tmp = 0.0
	if (x <= -9.8e-228)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 9.5e+54)
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + Float64(1.0 - Float64(x * eps_m))) / 2.0);
	elseif (x <= 7e+196)
		tmp = Float64(Float64(t_1 + t_0) / 2.0);
	else
		tmp = Float64(Float64(t_1 + Float64(exp(Float64(Float64(x * eps_m) - x)) * t_0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 1.0 - (1.0 / eps_m);
	t_1 = 1.0 + (1.0 / eps_m);
	tmp = 0.0;
	if (x <= -9.8e-228)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif (x <= 9.5e+54)
		tmp = (exp((x * (-1.0 + eps_m))) + (1.0 - (x * eps_m))) / 2.0;
	elseif (x <= 7e+196)
		tmp = (t_1 + t_0) / 2.0;
	else
		tmp = (t_1 + (exp(((x * eps_m) - x)) * t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.8e-228], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.5e+54], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7e+196], N[(N[(t$95$1 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$1 + N[(N[Exp[N[(N[(x * eps$95$m), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.79999999999999976e-228

   1. Initial program 74.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in eps around inf 98.9%

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

      1. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around 0 63.8%

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

   if -9.79999999999999976e-228 < x < 9.4999999999999999e54

   1. Initial program 63.2%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in eps around inf 99.7%

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

   6. Taylor expanded in eps around inf 95.6%

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

      1. *-commutative 95.6%

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

   8. Simplified 95.6%

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

   9. Taylor expanded in x around 0 83.8%

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

       1. mul-1-neg 83.8%

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

       2. *-commutative 83.8%

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

       3. unsub-neg 83.8%

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

   11. Simplified 83.8%

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

   if 9.4999999999999999e54 < x < 6.9999999999999997e196

   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 18.3%

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

   6. Taylor expanded in x around 0 63.3%

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

   if 6.9999999999999997e196 < 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 21.0%

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

      1. *-commutative 21.0%

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

      2. +-commutative 21.0%

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

      3. distribute-rgt-in 21.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 45.1%

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

      6. sqr-neg 45.1%

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

      7. sqrt-unprod 45.2%

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

      8. add-sqr-sqrt 45.2%

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

      9. *-un-lft-identity 45.2%

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

      10. unsub-neg 45.2%

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

   7. Applied egg-rr 45.2%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+196}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + e^{x \cdot \varepsilon - x} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-215}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + \left(1 - x \cdot eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+196}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-215)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (<= x 6.2e+54)
     (/ (+ (exp (* x (+ -1.0 eps_m))) (- 1.0 (* x eps_m))) 2.0)
     (if (<= x 7e+196)
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)
       (/ (* x eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-215) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if (x <= 6.2e+54) {
		tmp = (exp((x * (-1.0 + eps_m))) + (1.0 - (x * eps_m))) / 2.0;
	} else if (x <= 7e+196) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2d-215)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if (x <= 6.2d+54) then
        tmp = (exp((x * ((-1.0d0) + eps_m))) + (1.0d0 - (x * eps_m))) / 2.0d0
    else if (x <= 7d+196) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-215) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if (x <= 6.2e+54) {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + (1.0 - (x * eps_m))) / 2.0;
	} else if (x <= 7e+196) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-215:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif x <= 6.2e+54:
		tmp = (math.exp((x * (-1.0 + eps_m))) + (1.0 - (x * eps_m))) / 2.0
	elif x <= 7e+196:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-215)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 6.2e+54)
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + Float64(1.0 - Float64(x * eps_m))) / 2.0);
	elseif (x <= 7e+196)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2e-215)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif (x <= 6.2e+54)
		tmp = (exp((x * (-1.0 + eps_m))) + (1.0 - (x * eps_m))) / 2.0;
	elseif (x <= 7e+196)
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-215], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.2e+54], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7e+196], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.00000000000000008e-215

   1. Initial program 74.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in eps around inf 98.9%

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

      1. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around 0 63.8%

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

   if -2.00000000000000008e-215 < x < 6.1999999999999999e54

   1. Initial program 63.2%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in eps around inf 99.7%

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

   6. Taylor expanded in eps around inf 95.6%

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

      1. *-commutative 95.6%

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

   8. Simplified 95.6%

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

   9. Taylor expanded in x around 0 83.8%

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

       1. mul-1-neg 83.8%

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

       2. *-commutative 83.8%

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

       3. unsub-neg 83.8%

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

   11. Simplified 83.8%

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

   if 6.1999999999999999e54 < x < 6.9999999999999997e196

   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 18.3%

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

   6. Taylor expanded in x around 0 63.3%

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

   if 6.9999999999999997e196 < 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 44.3%

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

   6. Taylor expanded in x around inf 44.3%

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

      1. associate-*r* 44.3%

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

      2. *-commutative 44.3%

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

      3. associate-*r* 44.3%

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

      4. sub-neg 44.3%

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

      5. neg-mul-1 44.3%

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

      6. associate-*r* 44.3%

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

      7. mul-1-neg 44.3%

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

      8. neg-sub0 44.3%

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

      9. distribute-rgt-in 44.3%

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

      10. *-lft-identity 44.3%

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

      11. neg-mul-1 44.3%

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

      12. cancel-sign-sub-inv 44.3%

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

      13. associate-+l- 44.3%

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

      14. neg-sub0 44.3%

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

      15. neg-mul-1 44.3%

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

      16. +-commutative 44.3%

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

      17. distribute-rgt-in 44.3%

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

   8. Simplified 44.3%

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

   9. Taylor expanded in eps around inf 44.6%

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

       1. *-commutative 44.6%

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

   11. Simplified 44.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-215}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+196}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(2 + x \cdot eps\_m\right) - x}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+197}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 4.5e-51)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (<= x 8.5e+54)
     (/ (/ (- (* eps_m (+ 2.0 (* x eps_m))) x) eps_m) 2.0)
     (if (<= x 3.4e+197)
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)
       (/ (* x eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 4.5e-51) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if (x <= 8.5e+54) {
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	} else if (x <= 3.4e+197) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 4.5d-51) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if (x <= 8.5d+54) then
        tmp = (((eps_m * (2.0d0 + (x * eps_m))) - x) / eps_m) / 2.0d0
    else if (x <= 3.4d+197) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 4.5e-51) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if (x <= 8.5e+54) {
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	} else if (x <= 3.4e+197) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 4.5e-51:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif x <= 8.5e+54:
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0
	elif x <= 3.4e+197:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 4.5e-51)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 8.5e+54)
		tmp = Float64(Float64(Float64(Float64(eps_m * Float64(2.0 + Float64(x * eps_m))) - x) / eps_m) / 2.0);
	elseif (x <= 3.4e+197)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 4.5e-51)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif (x <= 8.5e+54)
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	elseif (x <= 3.4e+197)
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 4.5e-51], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.5e+54], N[(N[(N[(N[(eps$95$m * N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.4e+197], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.49999999999999974e-51

   1. Initial program 63.5%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in eps around inf 99.4%

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

   6. Taylor expanded in eps around inf 99.4%

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

      1. *-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in x around 0 77.4%

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

   if 4.49999999999999974e-51 < x < 8.4999999999999995e54

   1. Initial program 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in x around 0 27.7%

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

   6. Taylor expanded in x around 0 6.5%

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

   7. Taylor expanded in eps around 0 28.9%

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

   if 8.4999999999999995e54 < x < 3.40000000000000017e197

   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 18.3%

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

   6. Taylor expanded in x around 0 63.3%

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

   if 3.40000000000000017e197 < 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 44.3%

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

   6. Taylor expanded in x around inf 44.3%

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

      1. associate-*r* 44.3%

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

      2. *-commutative 44.3%

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

      3. associate-*r* 44.3%

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

      4. sub-neg 44.3%

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

      5. neg-mul-1 44.3%

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

      6. associate-*r* 44.3%

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

      7. mul-1-neg 44.3%

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

      8. neg-sub0 44.3%

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

      9. distribute-rgt-in 44.3%

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

      10. *-lft-identity 44.3%

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

      11. neg-mul-1 44.3%

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

      12. cancel-sign-sub-inv 44.3%

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

      13. associate-+l- 44.3%

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

      14. neg-sub0 44.3%

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

      15. neg-mul-1 44.3%

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

      16. +-commutative 44.3%

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

      17. distribute-rgt-in 44.3%

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

   8. Simplified 44.3%

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

   9. Taylor expanded in eps around inf 44.6%

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

       1. *-commutative 44.6%

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

   11. Simplified 44.6%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 + x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+197}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(2 + x \cdot eps\_m\right) - x}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+196}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.2e-50)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 9e+54)
     (/ (/ (- (* eps_m (+ 2.0 (* x eps_m))) x) eps_m) 2.0)
     (if (<= x 9e+196)
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)
       (/ (* x eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.2e-50) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 9e+54) {
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	} else if (x <= 9e+196) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 1.2d-50) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 9d+54) then
        tmp = (((eps_m * (2.0d0 + (x * eps_m))) - x) / eps_m) / 2.0d0
    else if (x <= 9d+196) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.2e-50) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 9e+54) {
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	} else if (x <= 9e+196) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.2e-50:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 9e+54:
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0
	elif x <= 9e+196:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.2e-50)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 9e+54)
		tmp = Float64(Float64(Float64(Float64(eps_m * Float64(2.0 + Float64(x * eps_m))) - x) / eps_m) / 2.0);
	elseif (x <= 9e+196)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.2e-50)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 9e+54)
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	elseif (x <= 9e+196)
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.2e-50], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9e+54], N[(N[(N[(N[(eps$95$m * N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9e+196], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.20000000000000001e-50

   1. Initial program 63.5%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in eps around inf 99.4%

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

   6. Taylor expanded in eps around inf 99.4%

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

      1. *-commutative 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in eps around 0 81.3%

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

       1. neg-mul-1 81.3%

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

   11. Simplified 81.3%

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

   if 1.20000000000000001e-50 < x < 8.99999999999999968e54

   1. Initial program 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in x around 0 27.7%

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

   6. Taylor expanded in x around 0 6.5%

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

   7. Taylor expanded in eps around 0 28.9%

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

   if 8.99999999999999968e54 < x < 8.99999999999999956e196

   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 18.3%

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

   6. Taylor expanded in x around 0 63.3%

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

   if 8.99999999999999956e196 < 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 44.3%

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

   6. Taylor expanded in x around inf 44.3%

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

      1. associate-*r* 44.3%

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

      2. *-commutative 44.3%

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

      3. associate-*r* 44.3%

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

      4. sub-neg 44.3%

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

      5. neg-mul-1 44.3%

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

      6. associate-*r* 44.3%

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

      7. mul-1-neg 44.3%

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

      8. neg-sub0 44.3%

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

      9. distribute-rgt-in 44.3%

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

      10. *-lft-identity 44.3%

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

      11. neg-mul-1 44.3%

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

      12. cancel-sign-sub-inv 44.3%

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

      13. associate-+l- 44.3%

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

      14. neg-sub0 44.3%

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

      15. neg-mul-1 44.3%

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

      16. +-commutative 44.3%

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

      17. distribute-rgt-in 44.3%

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

   8. Simplified 44.3%

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

   9. Taylor expanded in eps around inf 44.6%

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

       1. *-commutative 44.6%

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

   11. Simplified 44.6%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 + x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+196}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.5% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-49}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + eps\_m\right) \cdot \left(-1 + \frac{1}{eps\_m}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(2 + x \cdot eps\_m\right) - x}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+197}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.25e-49)
   (/ (+ 2.0 (* x (* (+ 1.0 eps_m) (+ -1.0 (/ 1.0 eps_m))))) 2.0)
   (if (<= x 1.6e-50)
     1.0
     (if (<= x 6.5e+54)
       (/ (/ (- (* eps_m (+ 2.0 (* x eps_m))) x) eps_m) 2.0)
       (if (<= x 4e+197)
         (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)
         (/ (* x eps_m) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-49) {
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0;
	} else if (x <= 1.6e-50) {
		tmp = 1.0;
	} else if (x <= 6.5e+54) {
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	} else if (x <= 4e+197) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.25d-49)) then
        tmp = (2.0d0 + (x * ((1.0d0 + eps_m) * ((-1.0d0) + (1.0d0 / eps_m))))) / 2.0d0
    else if (x <= 1.6d-50) then
        tmp = 1.0d0
    else if (x <= 6.5d+54) then
        tmp = (((eps_m * (2.0d0 + (x * eps_m))) - x) / eps_m) / 2.0d0
    else if (x <= 4d+197) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-49) {
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0;
	} else if (x <= 1.6e-50) {
		tmp = 1.0;
	} else if (x <= 6.5e+54) {
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	} else if (x <= 4e+197) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.25e-49:
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0
	elif x <= 1.6e-50:
		tmp = 1.0
	elif x <= 6.5e+54:
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0
	elif x <= 4e+197:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.25e-49)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 + eps_m) * Float64(-1.0 + Float64(1.0 / eps_m))))) / 2.0);
	elseif (x <= 1.6e-50)
		tmp = 1.0;
	elseif (x <= 6.5e+54)
		tmp = Float64(Float64(Float64(Float64(eps_m * Float64(2.0 + Float64(x * eps_m))) - x) / eps_m) / 2.0);
	elseif (x <= 4e+197)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.25e-49)
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0;
	elseif (x <= 1.6e-50)
		tmp = 1.0;
	elseif (x <= 6.5e+54)
		tmp = (((eps_m * (2.0 + (x * eps_m))) - x) / eps_m) / 2.0;
	elseif (x <= 4e+197)
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.25e-49], N[(N[(2.0 + N[(x * N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.6e-50], 1.0, If[LessEqual[x, 6.5e+54], N[(N[(N[(N[(eps$95$m * N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4e+197], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.25e-49

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 40.9%

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

   6. Taylor expanded in x around 0 25.2%

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

      1. +-commutative 25.2%

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

      2. sub-neg 25.2%

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

      3. metadata-eval 25.2%

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

      4. +-commutative 25.2%

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

   8. Simplified 25.2%

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

   if -1.25e-49 < x < 1.6e-50

   1. Initial program 52.7%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in x around 0 81.4%

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

   if 1.6e-50 < x < 6.5e54

   1. Initial program 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in x around 0 27.7%

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

   6. Taylor expanded in x around 0 6.5%

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

   7. Taylor expanded in eps around 0 28.9%

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

   if 6.5e54 < x < 3.9999999999999998e197

   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 18.3%

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

   6. Taylor expanded in x around 0 63.3%

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

   if 3.9999999999999998e197 < 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 44.3%

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

   6. Taylor expanded in x around inf 44.3%

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

      1. associate-*r* 44.3%

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

      2. *-commutative 44.3%

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

      3. associate-*r* 44.3%

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

      4. sub-neg 44.3%

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

      5. neg-mul-1 44.3%

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

      6. associate-*r* 44.3%

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

      7. mul-1-neg 44.3%

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

      8. neg-sub0 44.3%

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

      9. distribute-rgt-in 44.3%

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

      10. *-lft-identity 44.3%

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

      11. neg-mul-1 44.3%

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

      12. cancel-sign-sub-inv 44.3%

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

      13. associate-+l- 44.3%

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

      14. neg-sub0 44.3%

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

      15. neg-mul-1 44.3%

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

      16. +-commutative 44.3%

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

      17. distribute-rgt-in 44.3%

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

   8. Simplified 44.3%

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

   9. Taylor expanded in eps around inf 44.6%

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

       1. *-commutative 44.6%

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

   11. Simplified 44.6%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-49}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 + x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+197}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-49}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+197}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.25e-49)
   (* (* x eps_m) -0.5)
   (if (<= x 1.65e-9)
     1.0
     (if (<= x 1.9e+197)
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)
       (/ (* x eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-49) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 1.65e-9) {
		tmp = 1.0;
	} else if (x <= 1.9e+197) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.25d-49)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 1.65d-9) then
        tmp = 1.0d0
    else if (x <= 1.9d+197) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-49) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 1.65e-9) {
		tmp = 1.0;
	} else if (x <= 1.9e+197) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.25e-49:
		tmp = (x * eps_m) * -0.5
	elif x <= 1.65e-9:
		tmp = 1.0
	elif x <= 1.9e+197:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.25e-49)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 1.65e-9)
		tmp = 1.0;
	elseif (x <= 1.9e+197)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.25e-49)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 1.65e-9)
		tmp = 1.0;
	elseif (x <= 1.9e+197)
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.25e-49], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 1.65e-9], 1.0, If[LessEqual[x, 1.9e+197], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.25e-49

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 43.6%

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

   6. Taylor expanded in x around inf 29.1%

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

      1. associate-*r* 29.1%

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

      2. *-commutative 29.1%

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

      3. associate-*r* 29.1%

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

      4. sub-neg 29.1%

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

      5. neg-mul-1 29.1%

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

      6. associate-*r* 29.1%

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

      7. mul-1-neg 29.1%

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

      8. neg-sub0 29.1%

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

      9. distribute-rgt-in 29.1%

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

      10. *-lft-identity 29.1%

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

      11. neg-mul-1 29.1%

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

      12. cancel-sign-sub-inv 29.1%

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

      13. associate-+l- 29.1%

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

      14. neg-sub0 29.1%

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

      15. neg-mul-1 29.1%

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

      16. +-commutative 29.1%

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

      17. distribute-rgt-in 29.1%

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

   8. Simplified 29.1%

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

   9. Taylor expanded in eps around inf 29.0%

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

       1. *-commutative 29.0%

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

   11. Simplified 29.0%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
   12. Step-by-step derivation

       1. frac-2neg 29.0%

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

       2. mul-1-neg 29.0%

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

       3. div-inv 29.0%

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

       4. add-sqr-sqrt 0.1%

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

       5. sqrt-unprod 0.2%

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

       6. mul-1-neg 0.2%

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

       7. mul-1-neg 0.2%

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

       8. sqr-neg 0.2%

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

       9. sqrt-unprod 0.2%

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

       10. add-sqr-sqrt 25.3%

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

       11. metadata-eval 25.3%

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

       12. metadata-eval 25.3%

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

   13. Applied egg-rr 25.3%

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

   if -1.25e-49 < x < 1.65000000000000009e-9

   1. Initial program 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around 0 76.3%

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

   if 1.65000000000000009e-9 < x < 1.9000000000000001e197

   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 18.0%

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

   6. Taylor expanded in x around 0 50.7%

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

   if 1.9000000000000001e197 < 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 44.3%

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

   6. Taylor expanded in x around inf 44.3%

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

      1. associate-*r* 44.3%

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

      2. *-commutative 44.3%

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

      3. associate-*r* 44.3%

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

      4. sub-neg 44.3%

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

      5. neg-mul-1 44.3%

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

      6. associate-*r* 44.3%

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

      7. mul-1-neg 44.3%

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

      8. neg-sub0 44.3%

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

      9. distribute-rgt-in 44.3%

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

      10. *-lft-identity 44.3%

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

      11. neg-mul-1 44.3%

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

      12. cancel-sign-sub-inv 44.3%

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

      13. associate-+l- 44.3%

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

      14. neg-sub0 44.3%

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

      15. neg-mul-1 44.3%

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

      16. +-commutative 44.3%

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

      17. distribute-rgt-in 44.3%

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

   8. Simplified 44.3%

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

   9. Taylor expanded in eps around inf 44.6%

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

       1. *-commutative 44.6%

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

   11. Simplified 44.6%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-49}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+197}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-49}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + eps\_m\right) \cdot \left(-1 + \frac{1}{eps\_m}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+197}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.25e-49)
   (/ (+ 2.0 (* x (* (+ 1.0 eps_m) (+ -1.0 (/ 1.0 eps_m))))) 2.0)
   (if (<= x 1.65e-9)
     1.0
     (if (<= x 1e+197)
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)
       (/ (* x eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-49) {
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0;
	} else if (x <= 1.65e-9) {
		tmp = 1.0;
	} else if (x <= 1e+197) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.25d-49)) then
        tmp = (2.0d0 + (x * ((1.0d0 + eps_m) * ((-1.0d0) + (1.0d0 / eps_m))))) / 2.0d0
    else if (x <= 1.65d-9) then
        tmp = 1.0d0
    else if (x <= 1d+197) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-49) {
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0;
	} else if (x <= 1.65e-9) {
		tmp = 1.0;
	} else if (x <= 1e+197) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.25e-49:
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0
	elif x <= 1.65e-9:
		tmp = 1.0
	elif x <= 1e+197:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.25e-49)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 + eps_m) * Float64(-1.0 + Float64(1.0 / eps_m))))) / 2.0);
	elseif (x <= 1.65e-9)
		tmp = 1.0;
	elseif (x <= 1e+197)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.25e-49)
		tmp = (2.0 + (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m))))) / 2.0;
	elseif (x <= 1.65e-9)
		tmp = 1.0;
	elseif (x <= 1e+197)
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.25e-49], N[(N[(2.0 + N[(x * N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.65e-9], 1.0, If[LessEqual[x, 1e+197], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.25e-49

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 40.9%

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

   6. Taylor expanded in x around 0 25.2%

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

      1. +-commutative 25.2%

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

      2. sub-neg 25.2%

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

      3. metadata-eval 25.2%

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

      4. +-commutative 25.2%

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

   8. Simplified 25.2%

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

   if -1.25e-49 < x < 1.65000000000000009e-9

   1. Initial program 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around 0 76.3%

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

   if 1.65000000000000009e-9 < x < 9.9999999999999995e196

   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 18.0%

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

   6. Taylor expanded in x around 0 50.7%

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

   if 9.9999999999999995e196 < 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 44.3%

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

   6. Taylor expanded in x around inf 44.3%

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

      1. associate-*r* 44.3%

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

      2. *-commutative 44.3%

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

      3. associate-*r* 44.3%

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

      4. sub-neg 44.3%

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

      5. neg-mul-1 44.3%

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

      6. associate-*r* 44.3%

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

      7. mul-1-neg 44.3%

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

      8. neg-sub0 44.3%

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

      9. distribute-rgt-in 44.3%

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

      10. *-lft-identity 44.3%

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

      11. neg-mul-1 44.3%

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

      12. cancel-sign-sub-inv 44.3%

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

      13. associate-+l- 44.3%

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

      14. neg-sub0 44.3%

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

      15. neg-mul-1 44.3%

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

      16. +-commutative 44.3%

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

      17. distribute-rgt-in 44.3%

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

   8. Simplified 44.3%

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

   9. Taylor expanded in eps around inf 44.6%

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

       1. *-commutative 44.6%

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

   11. Simplified 44.6%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-49}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+197}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.8% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-49}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.25e-49)
   (* (* x eps_m) -0.5)
   (if (<= x 1.65e-9) 1.0 (/ (* x eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-49) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 1.65e-9) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.25d-49)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 1.65d-9) then
        tmp = 1.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-49) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 1.65e-9) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.25e-49:
		tmp = (x * eps_m) * -0.5
	elif x <= 1.65e-9:
		tmp = 1.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.25e-49)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 1.65e-9)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.25e-49)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 1.65e-9)
		tmp = 1.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.25e-49], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 1.65e-9], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25e-49

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 43.6%

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

   6. Taylor expanded in x around inf 29.1%

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

      1. associate-*r* 29.1%

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

      2. *-commutative 29.1%

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

      3. associate-*r* 29.1%

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

      4. sub-neg 29.1%

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

      5. neg-mul-1 29.1%

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

      6. associate-*r* 29.1%

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

      7. mul-1-neg 29.1%

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

      8. neg-sub0 29.1%

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

      9. distribute-rgt-in 29.1%

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

      10. *-lft-identity 29.1%

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

      11. neg-mul-1 29.1%

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

      12. cancel-sign-sub-inv 29.1%

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

      13. associate-+l- 29.1%

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

      14. neg-sub0 29.1%

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

      15. neg-mul-1 29.1%

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

      16. +-commutative 29.1%

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

      17. distribute-rgt-in 29.1%

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

   8. Simplified 29.1%

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

   9. Taylor expanded in eps around inf 29.0%

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

       1. *-commutative 29.0%

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

   11. Simplified 29.0%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
   12. Step-by-step derivation

       1. frac-2neg 29.0%

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

       2. mul-1-neg 29.0%

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

       3. div-inv 29.0%

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

       4. add-sqr-sqrt 0.1%

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

       5. sqrt-unprod 0.2%

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

       6. mul-1-neg 0.2%

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

       7. mul-1-neg 0.2%

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

       8. sqr-neg 0.2%

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

       9. sqrt-unprod 0.2%

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

       10. add-sqr-sqrt 25.3%

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

       11. metadata-eval 25.3%

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

       12. metadata-eval 25.3%

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

   13. Applied egg-rr 25.3%

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

   if -1.25e-49 < x < 1.65000000000000009e-9

   1. Initial program 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around 0 76.3%

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

   if 1.65000000000000009e-9 < 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 21.4%

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

   6. Taylor expanded in x around inf 14.6%

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

      1. associate-*r* 14.6%

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

      2. *-commutative 14.6%

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

      3. associate-*r* 14.6%

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

      4. sub-neg 14.6%

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

      5. neg-mul-1 14.6%

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

      6. associate-*r* 14.6%

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

      7. mul-1-neg 14.6%

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

      8. neg-sub0 14.6%

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

      9. distribute-rgt-in 14.6%

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

      10. *-lft-identity 14.6%

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

      11. neg-mul-1 14.6%

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

      12. cancel-sign-sub-inv 14.6%

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

      13. associate-+l- 14.6%

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

      14. neg-sub0 14.6%

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

      15. neg-mul-1 14.6%

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

      16. +-commutative 14.6%

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

      17. distribute-rgt-in 14.6%

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

   8. Simplified 14.6%

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

   9. Taylor expanded in eps around inf 15.5%

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

       1. *-commutative 15.5%

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

   11. Simplified 15.5%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-49}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.5% accurate, 16.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-49}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot \left(x + \frac{2}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.25e-49)
   (* (* x eps_m) -0.5)
   (/ (* eps_m (+ x (/ 2.0 eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-49) {
		tmp = (x * eps_m) * -0.5;
	} else {
		tmp = (eps_m * (x + (2.0 / 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 <= (-1.25d-49)) then
        tmp = (x * eps_m) * (-0.5d0)
    else
        tmp = (eps_m * (x + (2.0d0 / 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 <= -1.25e-49) {
		tmp = (x * eps_m) * -0.5;
	} else {
		tmp = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.25e-49:
		tmp = (x * eps_m) * -0.5
	else:
		tmp = (eps_m * (x + (2.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.25e-49)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	else
		tmp = Float64(Float64(eps_m * Float64(x + Float64(2.0 / 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 <= -1.25e-49)
		tmp = (x * eps_m) * -0.5;
	else
		tmp = (eps_m * (x + (2.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.25e-49], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(eps$95$m * N[(x + N[(2.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e-49

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 43.6%

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

   6. Taylor expanded in x around inf 29.1%

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

      1. associate-*r* 29.1%

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

      2. *-commutative 29.1%

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

      3. associate-*r* 29.1%

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

      4. sub-neg 29.1%

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

      5. neg-mul-1 29.1%

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

      6. associate-*r* 29.1%

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

      7. mul-1-neg 29.1%

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

      8. neg-sub0 29.1%

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

      9. distribute-rgt-in 29.1%

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

      10. *-lft-identity 29.1%

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

      11. neg-mul-1 29.1%

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

      12. cancel-sign-sub-inv 29.1%

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

      13. associate-+l- 29.1%

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

      14. neg-sub0 29.1%

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

      15. neg-mul-1 29.1%

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

      16. +-commutative 29.1%

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

      17. distribute-rgt-in 29.1%

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

   8. Simplified 29.1%

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

   9. Taylor expanded in eps around inf 29.0%

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

       1. *-commutative 29.0%

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

   11. Simplified 29.0%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
   12. Step-by-step derivation

       1. frac-2neg 29.0%

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

       2. mul-1-neg 29.0%

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

       3. div-inv 29.0%

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

       4. add-sqr-sqrt 0.1%

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

       5. sqrt-unprod 0.2%

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

       6. mul-1-neg 0.2%

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

       7. mul-1-neg 0.2%

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

       8. sqr-neg 0.2%

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

       9. sqrt-unprod 0.2%

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

       10. add-sqr-sqrt 25.3%

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

       11. metadata-eval 25.3%

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

       12. metadata-eval 25.3%

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

   13. Applied egg-rr 25.3%

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

   if -1.25e-49 < x

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in x around 0 35.3%

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

   6. Taylor expanded in x around 0 25.1%

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

   7. Taylor expanded in eps around inf 54.9%

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

      1. associate-*r/ 54.9%

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

      2. metadata-eval 54.9%

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

   9. Simplified 54.9%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-49}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.8% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-49}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.25e-49) (* (* x eps_m) -0.5) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.25e-49) {
		tmp = (x * eps_m) * -0.5;
	} 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 <= (-1.25d-49)) then
        tmp = (x * eps_m) * (-0.5d0)
    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 <= -1.25e-49) {
		tmp = (x * eps_m) * -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.25e-49:
		tmp = (x * eps_m) * -0.5
	else:
		tmp = 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.25e-49)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	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 <= -1.25e-49)
		tmp = (x * eps_m) * -0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.25e-49], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e-49

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 43.6%

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

   6. Taylor expanded in x around inf 29.1%

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

      1. associate-*r* 29.1%

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

      2. *-commutative 29.1%

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

      3. associate-*r* 29.1%

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

      4. sub-neg 29.1%

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

      5. neg-mul-1 29.1%

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

      6. associate-*r* 29.1%

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

      7. mul-1-neg 29.1%

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

      8. neg-sub0 29.1%

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

      9. distribute-rgt-in 29.1%

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

      10. *-lft-identity 29.1%

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

      11. neg-mul-1 29.1%

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

      12. cancel-sign-sub-inv 29.1%

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

      13. associate-+l- 29.1%

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

      14. neg-sub0 29.1%

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

      15. neg-mul-1 29.1%

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

      16. +-commutative 29.1%

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

      17. distribute-rgt-in 29.1%

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

   8. Simplified 29.1%

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

   9. Taylor expanded in eps around inf 29.0%

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

       1. *-commutative 29.0%

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

   11. Simplified 29.0%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
   12. Step-by-step derivation

       1. frac-2neg 29.0%

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

       2. mul-1-neg 29.0%

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

       3. div-inv 29.0%

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

       4. add-sqr-sqrt 0.1%

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

       5. sqrt-unprod 0.2%

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

       6. mul-1-neg 0.2%

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

       7. mul-1-neg 0.2%

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

       8. sqr-neg 0.2%

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

       9. sqrt-unprod 0.2%

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

       10. add-sqr-sqrt 25.3%

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

       11. metadata-eval 25.3%

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

       12. metadata-eval 25.3%

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

   13. Applied egg-rr 25.3%

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

   if -1.25e-49 < x

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in x around 0 51.5%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-49}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.4% 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 74.6%

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

3. Simplified 74.6%

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

5. Taylor expanded in x around 0 43.7%

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

6. Final simplification 43.7%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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))