NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.5% → 98.8%

Time: 19.3s

Alternatives: 20

Speedup: 1.7×

• 37.5% 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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{e^{\mathsf{fma}\left(x, eps\_m, x\right)}}}\\ \frac{e^{x \cdot \left(-1 + eps\_m\right)} + t\_0 \cdot t\_0}{2} \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (exp (fma x eps_m x))))))
   (/ (+ (exp (* x (+ -1.0 eps_m))) (* t_0 t_0)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 / sqrt(exp(fma(x, eps_m, x)));
	return (exp((x * (-1.0 + eps_m))) + (t_0 * t_0)) / 2.0;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 / sqrt(exp(fma(x, eps_m, x))))
	return Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + Float64(t_0 * t_0)) / 2.0)
end

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Simplified 75.0%

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

5. Taylor expanded in eps around inf 98.9%

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

   1. *-commutative 98.9%

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

   2. distribute-rgt-in 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. add-log-exp 98.9%

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

   5. pow-to-exp 98.9%

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

   6. add-sqr-sqrt 98.9%

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

   7. unpow-prod-down 98.9%

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

   8. +-commutative 98.9%

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

   9. *-commutative 98.9%

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

   10. fma-define 98.9%

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

   11. +-commutative 98.9%

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

   12. *-commutative 98.9%

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

   13. fma-define 98.9%

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

7. Applied egg-rr 98.9%

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

   1. unpow-1 98.9%

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

9. Applied egg-rr 98.9%

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

    1. unpow-1 98.9%

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

11. Applied egg-rr 98.9%

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

12. Final simplification 98.9%

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

Alternative 2: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(1 - eps\_m\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{e^{-x} + \frac{-1}{e^{x}}}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+157} \lor \neg \left(x \leq 7.2 \cdot 10^{+212}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + \frac{1}{1 + x \cdot \left(1 + eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-288)
   (/ (+ 1.0 (pow E (* x (- 1.0 eps_m)))) 2.0)
   (if (<= x 6.8e+68)
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (if (<= x 1.4e+77)
       (/ (/ (+ (exp (- x)) (/ -1.0 (exp x))) eps_m) 2.0)
       (if (or (<= x 7.5e+157) (not (<= x 7.2e+212)))
         (/
          (+ (exp (* x (+ -1.0 eps_m))) (/ 1.0 (+ 1.0 (* x (+ 1.0 eps_m)))))
          2.0)
         (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-288) {
		tmp = (1.0 + pow(((double) M_E), (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 6.8e+68) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else if (x <= 1.4e+77) {
		tmp = ((exp(-x) + (-1.0 / exp(x))) / eps_m) / 2.0;
	} else if ((x <= 7.5e+157) || !(x <= 7.2e+212)) {
		tmp = (exp((x * (-1.0 + eps_m))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-288) {
		tmp = (1.0 + Math.pow(Math.E, (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 6.8e+68) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else if (x <= 1.4e+77) {
		tmp = ((Math.exp(-x) + (-1.0 / Math.exp(x))) / eps_m) / 2.0;
	} else if ((x <= 7.5e+157) || !(x <= 7.2e+212)) {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -5e-288:
		tmp = (1.0 + math.pow(math.e, (x * (1.0 - eps_m)))) / 2.0
	elif x <= 6.8e+68:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	elif x <= 1.4e+77:
		tmp = ((math.exp(-x) + (-1.0 / math.exp(x))) / eps_m) / 2.0
	elif (x <= 7.5e+157) or not (x <= 7.2e+212):
		tmp = (math.exp((x * (-1.0 + eps_m))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -5e-288)
		tmp = Float64(Float64(1.0 + (exp(1) ^ Float64(x * Float64(1.0 - eps_m)))) / 2.0);
	elseif (x <= 6.8e+68)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	elseif (x <= 1.4e+77)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) + Float64(-1.0 / exp(x))) / eps_m) / 2.0);
	elseif ((x <= 7.5e+157) || !(x <= 7.2e+212))
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + Float64(1.0 / Float64(1.0 + Float64(x * Float64(1.0 + eps_m))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / 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 <= -5e-288)
		tmp = (1.0 + (2.71828182845904523536 ^ (x * (1.0 - eps_m)))) / 2.0;
	elseif (x <= 6.8e+68)
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	elseif (x <= 1.4e+77)
		tmp = ((exp(-x) + (-1.0 / exp(x))) / eps_m) / 2.0;
	elseif ((x <= 7.5e+157) || ~((x <= 7.2e+212)))
		tmp = (exp((x * (-1.0 + eps_m))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5e-288], N[(N[(1.0 + N[Power[E, N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.8e+68], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.4e+77], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] + N[(-1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 7.5e+157], N[Not[LessEqual[x, 7.2e+212]], $MachinePrecision]], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 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]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -5.00000000000000011e-288

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in x around 0 45.5%

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

   6. Taylor expanded in eps around inf 71.6%

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

      1. neg-mul-1 71.6%

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

      2. distribute-lft-neg-in 71.6%

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

   8. Simplified 71.6%

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

      1. *-un-lft-identity 71.6%

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

      2. exp-prod 71.6%

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

      3. add-sqr-sqrt 71.6%

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

      4. sqrt-unprod 68.8%

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

      5. sqr-neg 68.8%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 66.1%

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

   10. Applied egg-rr 66.1%

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

   if -5.00000000000000011e-288 < x < 6.8000000000000003e68

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 44.1%

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

   6. Taylor expanded in eps around inf 80.6%

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

      1. neg-mul-1 80.6%

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

      2. distribute-lft-neg-in 80.6%

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

   8. Simplified 80.6%

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

   9. Taylor expanded in eps around inf 81.0%

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

   if 6.8000000000000003e68 < x < 1.4e77

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 100.0%

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

   if 1.4e77 < x < 7.5e157 or 7.2e212 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 33.1%

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

   if 7.5e157 < x < 7.2e212

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 24.0%

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

   6. Taylor expanded in x around 0 78.5%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(1 - \varepsilon\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{e^{-x} + \frac{-1}{e^{x}}}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+157} \lor \neg \left(x \leq 7.2 \cdot 10^{+212}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \frac{1}{1 + x \cdot \left(1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Simplified 66.8%

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

5. Taylor expanded in eps around inf 98.9%

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

6. Final simplification 98.9%

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

Alternative 4: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Simplified 75.0%

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

5. Taylor expanded in eps around inf 98.9%

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

6. Final simplification 98.9%

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

Alternative 5: 84.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-284}:\\ \;\;\;\;\frac{1 + e^{x - x \cdot eps\_m}}{2}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+77} \lor \neg \left(x \leq 6.5 \cdot 10^{+157}\right) \land x \leq 2.05 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + \frac{1}{1 + x \cdot \left(1 + eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-284)
   (/ (+ 1.0 (exp (- x (* x eps_m)))) 2.0)
   (if (<= x 6.6e+67)
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (if (or (<= x 1.4e+77) (and (not (<= x 6.5e+157)) (<= x 2.05e+212)))
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)
       (/
        (+ (exp (* x (+ -1.0 eps_m))) (/ 1.0 (+ 1.0 (* x (+ 1.0 eps_m)))))
        2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-284) {
		tmp = (1.0 + exp((x - (x * eps_m)))) / 2.0;
	} else if (x <= 6.6e+67) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else if ((x <= 1.4e+77) || (!(x <= 6.5e+157) && (x <= 2.05e+212))) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) + (1.0 / (1.0 + (x * (1.0 + 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 <= (-2d-284)) then
        tmp = (1.0d0 + exp((x - (x * eps_m)))) / 2.0d0
    else if (x <= 6.6d+67) then
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    else if ((x <= 1.4d+77) .or. (.not. (x <= 6.5d+157)) .and. (x <= 2.05d+212)) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps_m))) + (1.0d0 / (1.0d0 + (x * (1.0d0 + 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 <= -2e-284) {
		tmp = (1.0 + Math.exp((x - (x * eps_m)))) / 2.0;
	} else if (x <= 6.6e+67) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else if ((x <= 1.4e+77) || (!(x <= 6.5e+157) && (x <= 2.05e+212))) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-284:
		tmp = (1.0 + math.exp((x - (x * eps_m)))) / 2.0
	elif x <= 6.6e+67:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	elif (x <= 1.4e+77) or (not (x <= 6.5e+157) and (x <= 2.05e+212)):
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-284)
		tmp = Float64(Float64(1.0 + exp(Float64(x - Float64(x * eps_m)))) / 2.0);
	elseif (x <= 6.6e+67)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	elseif ((x <= 1.4e+77) || (!(x <= 6.5e+157) && (x <= 2.05e+212)))
		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(exp(Float64(x * Float64(-1.0 + eps_m))) + Float64(1.0 / Float64(1.0 + Float64(x * Float64(1.0 + 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 <= -2e-284)
		tmp = (1.0 + exp((x - (x * eps_m)))) / 2.0;
	elseif (x <= 6.6e+67)
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	elseif ((x <= 1.4e+77) || (~((x <= 6.5e+157)) && (x <= 2.05e+212)))
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-284], N[(N[(1.0 + N[Exp[N[(x - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.6e+67], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.4e+77], And[N[Not[LessEqual[x, 6.5e+157]], $MachinePrecision], LessEqual[x, 2.05e+212]]], 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[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.00000000000000007e-284

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in x around 0 45.5%

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

   6. Taylor expanded in eps around inf 71.6%

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

      1. neg-mul-1 71.6%

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

      2. distribute-lft-neg-in 71.6%

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

   8. Simplified 71.6%

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

      1. add-sqr-sqrt 71.6%

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

      2. sqrt-unprod 68.8%

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

      3. sqr-neg 68.8%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 66.1%

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

      6. sub-neg 66.1%

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

      7. distribute-rgt-in 66.1%

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

      8. *-un-lft-identity 66.1%

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

   10. Applied egg-rr 66.1%

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

   if -2.00000000000000007e-284 < x < 6.6000000000000006e67

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 44.1%

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

   6. Taylor expanded in eps around inf 80.6%

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

      1. neg-mul-1 80.6%

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

      2. distribute-lft-neg-in 80.6%

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

   8. Simplified 80.6%

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

   9. Taylor expanded in eps around inf 81.0%

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

   if 6.6000000000000006e67 < x < 1.4e77 or 6.5e157 < x < 2.04999999999999995e212

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 16.2%

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

   6. Taylor expanded in x around 0 79.2%

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

   if 1.4e77 < x < 6.5e157 or 2.04999999999999995e212 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 33.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-284}:\\ \;\;\;\;\frac{1 + e^{x - x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+77} \lor \neg \left(x \leq 6.5 \cdot 10^{+157}\right) \land x \leq 2.05 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \frac{1}{1 + x \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(1 - eps\_m\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+77} \lor \neg \left(x \leq 3.7 \cdot 10^{+157}\right) \land x \leq 1.62 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + \frac{1}{1 + x \cdot \left(1 + eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-288)
   (/ (+ 1.0 (pow E (* x (- 1.0 eps_m)))) 2.0)
   (if (<= x 9e+68)
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (if (or (<= x 1.45e+77) (and (not (<= x 3.7e+157)) (<= x 1.62e+212)))
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)
       (/
        (+ (exp (* x (+ -1.0 eps_m))) (/ 1.0 (+ 1.0 (* x (+ 1.0 eps_m)))))
        2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-288) {
		tmp = (1.0 + pow(((double) M_E), (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 9e+68) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else if ((x <= 1.45e+77) || (!(x <= 3.7e+157) && (x <= 1.62e+212))) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-288) {
		tmp = (1.0 + Math.pow(Math.E, (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 9e+68) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else if ((x <= 1.45e+77) || (!(x <= 3.7e+157) && (x <= 1.62e+212))) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -5e-288:
		tmp = (1.0 + math.pow(math.e, (x * (1.0 - eps_m)))) / 2.0
	elif x <= 9e+68:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	elif (x <= 1.45e+77) or (not (x <= 3.7e+157) and (x <= 1.62e+212)):
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -5e-288)
		tmp = Float64(Float64(1.0 + (exp(1) ^ Float64(x * Float64(1.0 - eps_m)))) / 2.0);
	elseif (x <= 9e+68)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	elseif ((x <= 1.45e+77) || (!(x <= 3.7e+157) && (x <= 1.62e+212)))
		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(exp(Float64(x * Float64(-1.0 + eps_m))) + Float64(1.0 / Float64(1.0 + Float64(x * Float64(1.0 + 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 <= -5e-288)
		tmp = (1.0 + (2.71828182845904523536 ^ (x * (1.0 - eps_m)))) / 2.0;
	elseif (x <= 9e+68)
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	elseif ((x <= 1.45e+77) || (~((x <= 3.7e+157)) && (x <= 1.62e+212)))
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5e-288], N[(N[(1.0 + N[Power[E, N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9e+68], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.45e+77], And[N[Not[LessEqual[x, 3.7e+157]], $MachinePrecision], LessEqual[x, 1.62e+212]]], 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[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.00000000000000011e-288

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in x around 0 45.5%

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

   6. Taylor expanded in eps around inf 71.6%

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

      1. neg-mul-1 71.6%

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

      2. distribute-lft-neg-in 71.6%

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

   8. Simplified 71.6%

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

      1. *-un-lft-identity 71.6%

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

      2. exp-prod 71.6%

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

      3. add-sqr-sqrt 71.6%

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

      4. sqrt-unprod 68.8%

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

      5. sqr-neg 68.8%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 66.1%

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

   10. Applied egg-rr 66.1%

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

   if -5.00000000000000011e-288 < x < 9.0000000000000007e68

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 44.1%

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

   6. Taylor expanded in eps around inf 80.6%

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

      1. neg-mul-1 80.6%

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

      2. distribute-lft-neg-in 80.6%

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

   8. Simplified 80.6%

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

   9. Taylor expanded in eps around inf 81.0%

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

   if 9.0000000000000007e68 < x < 1.4500000000000001e77 or 3.6999999999999999e157 < x < 1.61999999999999994e212

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 16.2%

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

   6. Taylor expanded in x around 0 79.2%

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

   if 1.4500000000000001e77 < x < 3.6999999999999999e157 or 1.61999999999999994e212 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 33.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(1 - \varepsilon\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+77} \lor \neg \left(x \leq 3.7 \cdot 10^{+157}\right) \land x \leq 1.62 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \frac{1}{1 + x \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+77} \lor \neg \left(x \leq 1.65 \cdot 10^{+155}\right) \land x \leq 1.38 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-288)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 8.5e+68)
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (if (or (<= x 1.4e+77) (and (not (<= x 1.65e+155)) (<= x 1.38e+212)))
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)
       (/ (+ 1.0 (exp (* x (+ -1.0 eps_m)))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-288) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 8.5e+68) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else if ((x <= 1.4e+77) || (!(x <= 1.65e+155) && (x <= 1.38e+212))) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (-1.0 + 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 <= (-5d-288)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 8.5d+68) then
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    else if ((x <= 1.4d+77) .or. (.not. (x <= 1.65d+155)) .and. (x <= 1.38d+212)) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) + 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 <= -5e-288) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 8.5e+68) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else if ((x <= 1.4e+77) || (!(x <= 1.65e+155) && (x <= 1.38e+212))) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -5e-288:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 8.5e+68:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	elif (x <= 1.4e+77) or (not (x <= 1.65e+155) and (x <= 1.38e+212)):
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -5e-288)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 8.5e+68)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	elseif ((x <= 1.4e+77) || (!(x <= 1.65e+155) && (x <= 1.38e+212)))
		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(1.0 + exp(Float64(x * Float64(-1.0 + 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 <= -5e-288)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 8.5e+68)
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	elseif ((x <= 1.4e+77) || (~((x <= 1.65e+155)) && (x <= 1.38e+212)))
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5e-288], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.5e+68], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.4e+77], And[N[Not[LessEqual[x, 1.65e+155]], $MachinePrecision], LessEqual[x, 1.38e+212]]], 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[(1.0 + N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.00000000000000011e-288

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in x around 0 45.5%

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

   6. Taylor expanded in eps around inf 71.6%

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

      1. neg-mul-1 71.6%

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

      2. distribute-lft-neg-in 71.6%

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

   8. Simplified 71.6%

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

   9. Taylor expanded in eps around 0 77.0%

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

   if -5.00000000000000011e-288 < x < 8.49999999999999966e68

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 44.1%

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

   6. Taylor expanded in eps around inf 80.6%

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

      1. neg-mul-1 80.6%

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

      2. distribute-lft-neg-in 80.6%

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

   8. Simplified 80.6%

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

   9. Taylor expanded in eps around inf 81.0%

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

   if 8.49999999999999966e68 < x < 1.4e77 or 1.6499999999999999e155 < x < 1.38e212

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 16.2%

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

   6. Taylor expanded in x around 0 79.2%

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

   if 1.4e77 < x < 1.6499999999999999e155 or 1.38e212 < x

   1. Initial program 100.0%

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

   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 32.2%

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

   6. Taylor expanded in eps around inf 32.5%

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

      1. neg-mul-1 32.5%

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

      2. distribute-lft-neg-in 32.5%

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

   8. Simplified 32.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+77} \lor \neg \left(x \leq 1.65 \cdot 10^{+155}\right) \land x \leq 1.38 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-288}:\\ \;\;\;\;\frac{1 + e^{x - x \cdot eps\_m}}{2}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+77} \lor \neg \left(x \leq 3.6 \cdot 10^{+154}\right) \land x \leq 7.5 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4e-288)
   (/ (+ 1.0 (exp (- x (* x eps_m)))) 2.0)
   (if (<= x 4.4e+68)
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (if (or (<= x 1.85e+77) (and (not (<= x 3.6e+154)) (<= x 7.5e+212)))
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)
       (/ (+ 1.0 (exp (* x (+ -1.0 eps_m)))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -4e-288) {
		tmp = (1.0 + exp((x - (x * eps_m)))) / 2.0;
	} else if (x <= 4.4e+68) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else if ((x <= 1.85e+77) || (!(x <= 3.6e+154) && (x <= 7.5e+212))) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (-1.0 + 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 <= (-4d-288)) then
        tmp = (1.0d0 + exp((x - (x * eps_m)))) / 2.0d0
    else if (x <= 4.4d+68) then
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    else if ((x <= 1.85d+77) .or. (.not. (x <= 3.6d+154)) .and. (x <= 7.5d+212)) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) + 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 <= -4e-288) {
		tmp = (1.0 + Math.exp((x - (x * eps_m)))) / 2.0;
	} else if (x <= 4.4e+68) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else if ((x <= 1.85e+77) || (!(x <= 3.6e+154) && (x <= 7.5e+212))) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -4e-288:
		tmp = (1.0 + math.exp((x - (x * eps_m)))) / 2.0
	elif x <= 4.4e+68:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	elif (x <= 1.85e+77) or (not (x <= 3.6e+154) and (x <= 7.5e+212)):
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -4e-288)
		tmp = Float64(Float64(1.0 + exp(Float64(x - Float64(x * eps_m)))) / 2.0);
	elseif (x <= 4.4e+68)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	elseif ((x <= 1.85e+77) || (!(x <= 3.6e+154) && (x <= 7.5e+212)))
		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(1.0 + exp(Float64(x * Float64(-1.0 + 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 <= -4e-288)
		tmp = (1.0 + exp((x - (x * eps_m)))) / 2.0;
	elseif (x <= 4.4e+68)
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	elseif ((x <= 1.85e+77) || (~((x <= 3.6e+154)) && (x <= 7.5e+212)))
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -4e-288], N[(N[(1.0 + N[Exp[N[(x - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.4e+68], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.85e+77], And[N[Not[LessEqual[x, 3.6e+154]], $MachinePrecision], LessEqual[x, 7.5e+212]]], 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[(1.0 + N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.00000000000000023e-288

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in x around 0 45.5%

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

   6. Taylor expanded in eps around inf 71.6%

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

      1. neg-mul-1 71.6%

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

      2. distribute-lft-neg-in 71.6%

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

   8. Simplified 71.6%

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

      1. add-sqr-sqrt 71.6%

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

      2. sqrt-unprod 68.8%

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

      3. sqr-neg 68.8%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 66.1%

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

      6. sub-neg 66.1%

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

      7. distribute-rgt-in 66.1%

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

      8. *-un-lft-identity 66.1%

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

   10. Applied egg-rr 66.1%

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

   if -4.00000000000000023e-288 < x < 4.39999999999999974e68

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 44.1%

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

   6. Taylor expanded in eps around inf 80.6%

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

      1. neg-mul-1 80.6%

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

      2. distribute-lft-neg-in 80.6%

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

   8. Simplified 80.6%

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

   9. Taylor expanded in eps around inf 81.0%

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

   if 4.39999999999999974e68 < x < 1.84999999999999997e77 or 3.6000000000000001e154 < x < 7.5000000000000003e212

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 16.2%

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

   6. Taylor expanded in x around 0 79.2%

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

   if 1.84999999999999997e77 < x < 3.6000000000000001e154 or 7.5000000000000003e212 < x

   1. Initial program 100.0%

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

   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 32.2%

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

   6. Taylor expanded in eps around inf 32.5%

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

      1. neg-mul-1 32.5%

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

      2. distribute-lft-neg-in 32.5%

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

   8. Simplified 32.5%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-288}:\\ \;\;\;\;\frac{1 + e^{x - x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+77} \lor \neg \left(x \leq 3.6 \cdot 10^{+154}\right) \land x \leq 7.5 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-287}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+68} \lor \neg \left(x \leq 2 \cdot 10^{+77} \lor \neg \left(x \leq 3.35 \cdot 10^{+156}\right) \land x \leq 6.6 \cdot 10^{+212}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-287)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (or (<= x 4.8e+68)
           (not
            (or (<= x 2e+77) (and (not (<= x 3.35e+156)) (<= x 6.6e+212)))))
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-287) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if ((x <= 4.8e+68) || !((x <= 2e+77) || (!(x <= 3.35e+156) && (x <= 6.6e+212)))) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / 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 <= (-2d-287)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if ((x <= 4.8d+68) .or. (.not. (x <= 2d+77) .or. (.not. (x <= 3.35d+156)) .and. (x <= 6.6d+212))) then
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / 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 <= -2e-287) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if ((x <= 4.8e+68) || !((x <= 2e+77) || (!(x <= 3.35e+156) && (x <= 6.6e+212)))) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-287:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif (x <= 4.8e+68) or not ((x <= 2e+77) or (not (x <= 3.35e+156) and (x <= 6.6e+212))):
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-287)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif ((x <= 4.8e+68) || !((x <= 2e+77) || (!(x <= 3.35e+156) && (x <= 6.6e+212))))
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / 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 <= -2e-287)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif ((x <= 4.8e+68) || ~(((x <= 2e+77) || (~((x <= 3.35e+156)) && (x <= 6.6e+212)))))
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-287], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 4.8e+68], N[Not[Or[LessEqual[x, 2e+77], And[N[Not[LessEqual[x, 3.35e+156]], $MachinePrecision], LessEqual[x, 6.6e+212]]]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000004e-287

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in x around 0 45.5%

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

   6. Taylor expanded in eps around inf 71.6%

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

      1. neg-mul-1 71.6%

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

      2. distribute-lft-neg-in 71.6%

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

   8. Simplified 71.6%

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

   9. Taylor expanded in eps around 0 77.0%

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

   if -2.00000000000000004e-287 < x < 4.80000000000000016e68 or 1.99999999999999997e77 < x < 3.35e156 or 6.6e212 < x

   1. Initial program 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in x around 0 40.9%

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

   6. Taylor expanded in eps around inf 67.9%

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

      1. neg-mul-1 67.9%

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

      2. distribute-lft-neg-in 67.9%

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

   8. Simplified 67.9%

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

   9. Taylor expanded in eps around inf 68.1%

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

   if 4.80000000000000016e68 < x < 1.99999999999999997e77 or 3.35e156 < x < 6.6e212

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 16.2%

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

   6. Taylor expanded in x around 0 79.2%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-287}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+68} \lor \neg \left(x \leq 2 \cdot 10^{+77} \lor \neg \left(x \leq 3.35 \cdot 10^{+156}\right) \land x \leq 6.6 \cdot 10^{+212}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -210000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{-1}{eps\_m} + \left(1 - eps\_m\right) \cdot \left(-1 + \frac{1}{eps\_m}\right)\right) - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+213} \lor \neg \left(x \leq 5.2 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -210000000.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (<= x 1.12e-10)
     (/
      (+
       2.0
       (*
        x
        (- (+ (/ -1.0 eps_m) (* (- 1.0 eps_m) (+ -1.0 (/ 1.0 eps_m)))) eps_m)))
      2.0)
     (if (or (<= x 3.7e+213) (not (<= x 5.2e+303)))
       (/ (+ (+ 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 <= -210000000.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= 1.12e-10) {
		tmp = (2.0 + (x * (((-1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 + (1.0 / eps_m)))) - eps_m))) / 2.0;
	} else if ((x <= 3.7e+213) || !(x <= 5.2e+303)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -210000000.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= 1.12e-10) {
		tmp = (2.0 + (x * (((-1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 + (1.0 / eps_m)))) - eps_m))) / 2.0;
	} else if ((x <= 3.7e+213) || !(x <= 5.2e+303)) {
		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 <= -210000000.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= 1.12e-10:
		tmp = (2.0 + (x * (((-1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 + (1.0 / eps_m)))) - eps_m))) / 2.0
	elif (x <= 3.7e+213) or not (x <= 5.2e+303):
		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 <= -210000000.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 1.12e-10)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(-1.0 / eps_m) + Float64(Float64(1.0 - eps_m) * Float64(-1.0 + Float64(1.0 / eps_m)))) - eps_m))) / 2.0);
	elseif ((x <= 3.7e+213) || !(x <= 5.2e+303))
		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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -210000000.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.12e-10], N[(N[(2.0 + N[(x * N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3.7e+213], N[Not[LessEqual[x, 5.2e+303]], $MachinePrecision]], 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 < -2.1e8

   1. Initial program 100.0%

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

   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 55.5%

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

   6. Taylor expanded in eps around 0 45.9%

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

      1. expm1-define 45.9%

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

      2. mul-1-neg 45.9%

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

   8. Simplified 45.9%

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

   if -2.1e8 < x < 1.12e-10

   1. Initial program 52.9%

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

   3. Simplified 37.5%

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

   5. Taylor expanded in x around 0 70.7%

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

      1. +-commutative 70.7%

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

      2. *-un-lft-identity 70.7%

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

      3. fma-define 70.7%

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

      4. add-sqr-sqrt 28.9%

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

      5. sqrt-unprod 38.3%

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

      6. frac-times 37.6%

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

      7. metadata-eval 37.6%

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

      8. metadata-eval 37.6%

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

      9. frac-times 38.3%

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

      10. sqrt-unprod 20.8%

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

      11. add-sqr-sqrt 38.0%

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

      12. add-sqr-sqrt 17.2%

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

      13. sqrt-unprod 38.2%

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

      14. frac-times 37.5%

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

      15. metadata-eval 37.5%

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

      16. metadata-eval 37.5%

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

      17. frac-times 38.2%

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

      18. sqrt-unprod 29.3%

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

      19. add-sqr-sqrt 70.7%

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

      20. sub-neg 70.7%

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

      21. metadata-eval 70.7%

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

   7. Applied egg-rr 70.7%

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

      1. fma-undefine 70.7%

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

      2. *-lft-identity 70.7%

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

      3. +-commutative 70.7%

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

   9. Simplified 70.7%

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

   if 1.12e-10 < x < 3.69999999999999993e213 or 5.20000000000000024e303 < x

   1. Initial program 100.0%

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

   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 30.4%

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

   6. Taylor expanded in x around 0 50.8%

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

   if 3.69999999999999993e213 < x < 5.20000000000000024e303

   1. Initial program 100.0%

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

   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 46.5%

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

   6. Taylor expanded in x around inf 26.5%

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

      1. mul-1-neg 26.5%

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

      2. *-commutative 26.5%

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

      3. distribute-rgt-neg-in 26.5%

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

      4. distribute-rgt-neg-in 26.5%

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

      5. mul-1-neg 26.5%

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

      6. distribute-lft-in 26.5%

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

      7. metadata-eval 26.5%

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

      8. neg-mul-1 26.5%

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

      9. distribute-neg-frac 26.5%

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

      10. metadata-eval 26.5%

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

   8. Simplified 26.5%

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

   9. Taylor expanded in eps around inf 26.7%

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

       1. *-commutative 26.7%

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

   11. Simplified 26.7%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -210000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{-1}{\varepsilon} + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+213} \lor \neg \left(x \leq 5.2 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+214} \lor \neg \left(x \leq 2.1 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.12e-10)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (or (<= x 2.7e+214) (not (<= x 2.1e+303)))
     (/ (+ (+ 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 <= 1.12e-10) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if ((x <= 2.7e+214) || !(x <= 2.1e+303)) {
		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 <= 1.12d-10) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if ((x <= 2.7d+214) .or. (.not. (x <= 2.1d+303))) 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 <= 1.12e-10) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if ((x <= 2.7e+214) || !(x <= 2.1e+303)) {
		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 <= 1.12e-10:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif (x <= 2.7e+214) or not (x <= 2.1e+303):
		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 <= 1.12e-10)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif ((x <= 2.7e+214) || !(x <= 2.1e+303))
		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 <= 1.12e-10)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif ((x <= 2.7e+214) || ~((x <= 2.1e+303)))
		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, 1.12e-10], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 2.7e+214], N[Not[LessEqual[x, 2.1e+303]], $MachinePrecision]], 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 3 regimes

2. if x < 1.12e-10

   1. Initial program 63.0%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in x around 0 44.0%

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

   6. Taylor expanded in eps around inf 78.9%

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

      1. neg-mul-1 78.9%

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

      2. distribute-lft-neg-in 78.9%

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

   8. Simplified 78.9%

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

   9. Taylor expanded in eps around 0 76.6%

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

   if 1.12e-10 < x < 2.70000000000000009e214 or 2.1e303 < x

   1. Initial program 100.0%

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

   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 30.4%

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

   6. Taylor expanded in x around 0 50.8%

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

   if 2.70000000000000009e214 < x < 2.1e303

   1. Initial program 100.0%

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

   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 46.5%

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

   6. Taylor expanded in x around inf 26.5%

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

      1. mul-1-neg 26.5%

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

      2. *-commutative 26.5%

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

      3. distribute-rgt-neg-in 26.5%

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

      4. distribute-rgt-neg-in 26.5%

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

      5. mul-1-neg 26.5%

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

      6. distribute-lft-in 26.5%

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

      7. metadata-eval 26.5%

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

      8. neg-mul-1 26.5%

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

      9. distribute-neg-frac 26.5%

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

      10. metadata-eval 26.5%

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

   8. Simplified 26.5%

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

   9. Taylor expanded in eps around inf 26.7%

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

       1. *-commutative 26.7%

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

   11. Simplified 26.7%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+214} \lor \neg \left(x \leq 2.1 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.2% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0008:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{eps\_m} + \frac{1}{eps\_m}\right) - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+213} \lor \neg \left(x \leq 1.35 \cdot 10^{+306}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.0008)
   (/ (+ 2.0 (* x (- (+ (/ 1.0 eps_m) (/ 1.0 eps_m)) eps_m))) 2.0)
   (if (<= x 1.12e-10)
     1.0
     (if (or (<= x 1.22e+213) (not (<= x 1.35e+306)))
       (/ (+ (+ 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 <= -0.0008) {
		tmp = (2.0 + (x * (((1.0 / eps_m) + (1.0 / eps_m)) - eps_m))) / 2.0;
	} else if (x <= 1.12e-10) {
		tmp = 1.0;
	} else if ((x <= 1.22e+213) || !(x <= 1.35e+306)) {
		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 <= (-0.0008d0)) then
        tmp = (2.0d0 + (x * (((1.0d0 / eps_m) + (1.0d0 / eps_m)) - eps_m))) / 2.0d0
    else if (x <= 1.12d-10) then
        tmp = 1.0d0
    else if ((x <= 1.22d+213) .or. (.not. (x <= 1.35d+306))) 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 <= -0.0008) {
		tmp = (2.0 + (x * (((1.0 / eps_m) + (1.0 / eps_m)) - eps_m))) / 2.0;
	} else if (x <= 1.12e-10) {
		tmp = 1.0;
	} else if ((x <= 1.22e+213) || !(x <= 1.35e+306)) {
		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 <= -0.0008:
		tmp = (2.0 + (x * (((1.0 / eps_m) + (1.0 / eps_m)) - eps_m))) / 2.0
	elif x <= 1.12e-10:
		tmp = 1.0
	elif (x <= 1.22e+213) or not (x <= 1.35e+306):
		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 <= -0.0008)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 / eps_m) + Float64(1.0 / eps_m)) - eps_m))) / 2.0);
	elseif (x <= 1.12e-10)
		tmp = 1.0;
	elseif ((x <= 1.22e+213) || !(x <= 1.35e+306))
		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 <= -0.0008)
		tmp = (2.0 + (x * (((1.0 / eps_m) + (1.0 / eps_m)) - eps_m))) / 2.0;
	elseif (x <= 1.12e-10)
		tmp = 1.0;
	elseif ((x <= 1.22e+213) || ~((x <= 1.35e+306)))
		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, -0.0008], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.12e-10], 1.0, If[Or[LessEqual[x, 1.22e+213], N[Not[LessEqual[x, 1.35e+306]], $MachinePrecision]], 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 < -8.00000000000000038e-4

   1. Initial program 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in x around 0 2.9%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 7.9%

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

      3. frac-times 7.9%

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

      4. metadata-eval 7.9%

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

      5. metadata-eval 7.9%

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

      6. frac-times 7.9%

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

      7. sqrt-unprod 1.7%

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

      8. add-sqr-sqrt 3.1%

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

      9. div-inv 3.1%

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

      10. mul-1-neg 3.1%

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

      11. sub-neg 3.1%

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

      12. add-sqr-sqrt 1.3%

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

      13. sqrt-unprod 3.0%

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

      14. frac-times 3.0%

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

      15. metadata-eval 3.0%

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

      16. metadata-eval 3.0%

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

      17. frac-times 3.0%

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

      18. sqrt-unprod 1.7%

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

      19. add-sqr-sqrt 3.0%

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

      20. sub-neg 3.0%

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

      21. metadata-eval 3.0%

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

      22. add-sqr-sqrt 1.3%

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

   7. Applied egg-rr 3.1%

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

   8. Taylor expanded in eps around 0 26.1%

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

   if -8.00000000000000038e-4 < x < 1.12e-10

   1. Initial program 53.4%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in x around 0 72.2%

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

   if 1.12e-10 < x < 1.2199999999999999e213 or 1.3499999999999999e306 < x

   1. Initial program 100.0%

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

   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 30.4%

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

   6. Taylor expanded in x around 0 50.8%

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

   if 1.2199999999999999e213 < x < 1.3499999999999999e306

   1. Initial program 100.0%

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

   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 46.5%

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

   6. Taylor expanded in x around inf 26.5%

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

      1. mul-1-neg 26.5%

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

      2. *-commutative 26.5%

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

      3. distribute-rgt-neg-in 26.5%

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

      4. distribute-rgt-neg-in 26.5%

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

      5. mul-1-neg 26.5%

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

      6. distribute-lft-in 26.5%

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

      7. metadata-eval 26.5%

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

      8. neg-mul-1 26.5%

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

      9. distribute-neg-frac 26.5%

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

      10. metadata-eval 26.5%

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

   8. Simplified 26.5%

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

   9. Taylor expanded in eps around inf 26.7%

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

       1. *-commutative 26.7%

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

   11. Simplified 26.7%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0008:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+213} \lor \neg \left(x \leq 1.35 \cdot 10^{+306}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0008:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{eps\_m} + \left(\frac{1}{eps\_m} - 2\right)\right) - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+212} \lor \neg \left(x \leq 3.8 \cdot 10^{+302}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.0008)
   (/ (+ 2.0 (* x (- (+ (/ 1.0 eps_m) (- (/ 1.0 eps_m) 2.0)) eps_m))) 2.0)
   (if (<= x 1.12e-10)
     1.0
     (if (or (<= x 2e+212) (not (<= x 3.8e+302)))
       (/ (+ (+ 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 <= -0.0008) {
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 / eps_m) - 2.0)) - eps_m))) / 2.0;
	} else if (x <= 1.12e-10) {
		tmp = 1.0;
	} else if ((x <= 2e+212) || !(x <= 3.8e+302)) {
		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 <= (-0.0008d0)) then
        tmp = (2.0d0 + (x * (((1.0d0 / eps_m) + ((1.0d0 / eps_m) - 2.0d0)) - eps_m))) / 2.0d0
    else if (x <= 1.12d-10) then
        tmp = 1.0d0
    else if ((x <= 2d+212) .or. (.not. (x <= 3.8d+302))) 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 <= -0.0008) {
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 / eps_m) - 2.0)) - eps_m))) / 2.0;
	} else if (x <= 1.12e-10) {
		tmp = 1.0;
	} else if ((x <= 2e+212) || !(x <= 3.8e+302)) {
		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 <= -0.0008:
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 / eps_m) - 2.0)) - eps_m))) / 2.0
	elif x <= 1.12e-10:
		tmp = 1.0
	elif (x <= 2e+212) or not (x <= 3.8e+302):
		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 <= -0.0008)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 / eps_m) - 2.0)) - eps_m))) / 2.0);
	elseif (x <= 1.12e-10)
		tmp = 1.0;
	elseif ((x <= 2e+212) || !(x <= 3.8e+302))
		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 <= -0.0008)
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 / eps_m) - 2.0)) - eps_m))) / 2.0;
	elseif (x <= 1.12e-10)
		tmp = 1.0;
	elseif ((x <= 2e+212) || ~((x <= 3.8e+302)))
		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, -0.0008], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 / eps$95$m), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.12e-10], 1.0, If[Or[LessEqual[x, 2e+212], N[Not[LessEqual[x, 3.8e+302]], $MachinePrecision]], 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 < -8.00000000000000038e-4

   1. Initial program 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in x around 0 2.9%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 7.9%

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

      3. frac-times 7.9%

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

      4. metadata-eval 7.9%

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

      5. metadata-eval 7.9%

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

      6. frac-times 7.9%

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

      7. sqrt-unprod 1.7%

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

      8. add-sqr-sqrt 3.1%

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

      9. div-inv 3.1%

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

      10. mul-1-neg 3.1%

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

      11. sub-neg 3.1%

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

      12. add-sqr-sqrt 1.3%

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

      13. sqrt-unprod 3.0%

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

      14. frac-times 3.0%

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

      15. metadata-eval 3.0%

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

      16. metadata-eval 3.0%

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

      17. frac-times 3.0%

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

      18. sqrt-unprod 1.7%

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

      19. add-sqr-sqrt 3.0%

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

      20. sub-neg 3.0%

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

      21. metadata-eval 3.0%

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

      22. add-sqr-sqrt 1.3%

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

   7. Applied egg-rr 3.1%

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

   8. Taylor expanded in eps around 0 26.1%

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

   if -8.00000000000000038e-4 < x < 1.12e-10

   1. Initial program 53.4%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in x around 0 72.2%

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

   if 1.12e-10 < x < 1.9999999999999998e212 or 3.8000000000000004e302 < x

   1. Initial program 100.0%

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

   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 30.4%

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

   6. Taylor expanded in x around 0 50.8%

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

   if 1.9999999999999998e212 < x < 3.8000000000000004e302

   1. Initial program 100.0%

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

   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 46.5%

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

   6. Taylor expanded in x around inf 26.5%

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

      1. mul-1-neg 26.5%

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

      2. *-commutative 26.5%

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

      3. distribute-rgt-neg-in 26.5%

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

      4. distribute-rgt-neg-in 26.5%

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

      5. mul-1-neg 26.5%

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

      6. distribute-lft-in 26.5%

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

      7. metadata-eval 26.5%

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

      8. neg-mul-1 26.5%

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

      9. distribute-neg-frac 26.5%

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

      10. metadata-eval 26.5%

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

   8. Simplified 26.5%

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

   9. Taylor expanded in eps around inf 26.7%

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

       1. *-commutative 26.7%

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

   11. Simplified 26.7%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0008:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \left(\frac{1}{\varepsilon} - 2\right)\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+212} \lor \neg \left(x \leq 3.8 \cdot 10^{+302}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.2% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0008:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{eps\_m} + \left(\frac{1}{eps\_m} - 2\right)\right) - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{eps\_m} + \left(1 - eps\_m\right) \cdot \left(-1 - \frac{1}{eps\_m}\right)\right) - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+212} \lor \neg \left(x \leq 4.3 \cdot 10^{+306}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.0008)
   (/ (+ 2.0 (* x (- (+ (/ 1.0 eps_m) (- (/ 1.0 eps_m) 2.0)) eps_m))) 2.0)
   (if (<= x 1.12e-10)
     (/
      (+
       2.0
       (*
        x
        (- (+ (/ 1.0 eps_m) (* (- 1.0 eps_m) (- -1.0 (/ 1.0 eps_m)))) eps_m)))
      2.0)
     (if (or (<= x 3.6e+212) (not (<= x 4.3e+306)))
       (/ (+ (+ 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 <= -0.0008) {
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 / eps_m) - 2.0)) - eps_m))) / 2.0;
	} else if (x <= 1.12e-10) {
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 - (1.0 / eps_m)))) - eps_m))) / 2.0;
	} else if ((x <= 3.6e+212) || !(x <= 4.3e+306)) {
		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 <= (-0.0008d0)) then
        tmp = (2.0d0 + (x * (((1.0d0 / eps_m) + ((1.0d0 / eps_m) - 2.0d0)) - eps_m))) / 2.0d0
    else if (x <= 1.12d-10) then
        tmp = (2.0d0 + (x * (((1.0d0 / eps_m) + ((1.0d0 - eps_m) * ((-1.0d0) - (1.0d0 / eps_m)))) - eps_m))) / 2.0d0
    else if ((x <= 3.6d+212) .or. (.not. (x <= 4.3d+306))) 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 <= -0.0008) {
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 / eps_m) - 2.0)) - eps_m))) / 2.0;
	} else if (x <= 1.12e-10) {
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 - (1.0 / eps_m)))) - eps_m))) / 2.0;
	} else if ((x <= 3.6e+212) || !(x <= 4.3e+306)) {
		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 <= -0.0008:
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 / eps_m) - 2.0)) - eps_m))) / 2.0
	elif x <= 1.12e-10:
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 - (1.0 / eps_m)))) - eps_m))) / 2.0
	elif (x <= 3.6e+212) or not (x <= 4.3e+306):
		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 <= -0.0008)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 / eps_m) - 2.0)) - eps_m))) / 2.0);
	elseif (x <= 1.12e-10)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 - eps_m) * Float64(-1.0 - Float64(1.0 / eps_m)))) - eps_m))) / 2.0);
	elseif ((x <= 3.6e+212) || !(x <= 4.3e+306))
		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 <= -0.0008)
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 / eps_m) - 2.0)) - eps_m))) / 2.0;
	elseif (x <= 1.12e-10)
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 - (1.0 / eps_m)))) - eps_m))) / 2.0;
	elseif ((x <= 3.6e+212) || ~((x <= 4.3e+306)))
		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, -0.0008], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 / eps$95$m), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.12e-10], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(-1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3.6e+212], N[Not[LessEqual[x, 4.3e+306]], $MachinePrecision]], 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 < -8.00000000000000038e-4

   1. Initial program 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in x around 0 2.9%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 7.9%

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

      3. frac-times 7.9%

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

      4. metadata-eval 7.9%

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

      5. metadata-eval 7.9%

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

      6. frac-times 7.9%

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

      7. sqrt-unprod 1.7%

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

      8. add-sqr-sqrt 3.1%

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

      9. div-inv 3.1%

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

      10. mul-1-neg 3.1%

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

      11. sub-neg 3.1%

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

      12. add-sqr-sqrt 1.3%

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

      13. sqrt-unprod 3.0%

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

      14. frac-times 3.0%

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

      15. metadata-eval 3.0%

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

      16. metadata-eval 3.0%

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

      17. frac-times 3.0%

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

      18. sqrt-unprod 1.7%

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

      19. add-sqr-sqrt 3.0%

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

      20. sub-neg 3.0%

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

      21. metadata-eval 3.0%

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

      22. add-sqr-sqrt 1.3%

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

   7. Applied egg-rr 3.1%

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

   8. Taylor expanded in eps around 0 26.1%

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

   if -8.00000000000000038e-4 < x < 1.12e-10

   1. Initial program 53.4%

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

   3. Simplified 37.6%

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

   5. Taylor expanded in x around 0 72.2%

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

   if 1.12e-10 < x < 3.6e212 or 4.2999999999999998e306 < x

   1. Initial program 100.0%

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

   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 30.4%

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

   6. Taylor expanded in x around 0 50.8%

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

   if 3.6e212 < x < 4.2999999999999998e306

   1. Initial program 100.0%

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

   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 46.5%

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

   6. Taylor expanded in x around inf 26.5%

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

      1. mul-1-neg 26.5%

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

      2. *-commutative 26.5%

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

      3. distribute-rgt-neg-in 26.5%

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

      4. distribute-rgt-neg-in 26.5%

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

      5. mul-1-neg 26.5%

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

      6. distribute-lft-in 26.5%

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

      7. metadata-eval 26.5%

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

      8. neg-mul-1 26.5%

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

      9. distribute-neg-frac 26.5%

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

      10. metadata-eval 26.5%

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

   8. Simplified 26.5%

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

   9. Taylor expanded in eps around inf 26.7%

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

       1. *-commutative 26.7%

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

   11. Simplified 26.7%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0008:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \left(\frac{1}{\varepsilon} - 2\right)\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \left(1 - \varepsilon\right) \cdot \left(-1 - \frac{1}{\varepsilon}\right)\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+212} \lor \neg \left(x \leq 4.3 \cdot 10^{+306}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.2% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0008:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{eps\_m} + \left(\frac{1}{eps\_m} - 2\right)\right) - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{-1}{eps\_m} + \left(1 - eps\_m\right) \cdot \left(-1 + \frac{1}{eps\_m}\right)\right) - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+212} \lor \neg \left(x \leq 1.06 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.0008)
   (/ (+ 2.0 (* x (- (+ (/ 1.0 eps_m) (- (/ 1.0 eps_m) 2.0)) eps_m))) 2.0)
   (if (<= x 1.12e-10)
     (/
      (+
       2.0
       (*
        x
        (- (+ (/ -1.0 eps_m) (* (- 1.0 eps_m) (+ -1.0 (/ 1.0 eps_m)))) eps_m)))
      2.0)
     (if (or (<= x 1.4e+212) (not (<= x 1.06e+303)))
       (/ (+ (+ 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 <= -0.0008) {
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 / eps_m) - 2.0)) - eps_m))) / 2.0;
	} else if (x <= 1.12e-10) {
		tmp = (2.0 + (x * (((-1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 + (1.0 / eps_m)))) - eps_m))) / 2.0;
	} else if ((x <= 1.4e+212) || !(x <= 1.06e+303)) {
		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 <= (-0.0008d0)) then
        tmp = (2.0d0 + (x * (((1.0d0 / eps_m) + ((1.0d0 / eps_m) - 2.0d0)) - eps_m))) / 2.0d0
    else if (x <= 1.12d-10) then
        tmp = (2.0d0 + (x * ((((-1.0d0) / eps_m) + ((1.0d0 - eps_m) * ((-1.0d0) + (1.0d0 / eps_m)))) - eps_m))) / 2.0d0
    else if ((x <= 1.4d+212) .or. (.not. (x <= 1.06d+303))) 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 <= -0.0008) {
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 / eps_m) - 2.0)) - eps_m))) / 2.0;
	} else if (x <= 1.12e-10) {
		tmp = (2.0 + (x * (((-1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 + (1.0 / eps_m)))) - eps_m))) / 2.0;
	} else if ((x <= 1.4e+212) || !(x <= 1.06e+303)) {
		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 <= -0.0008:
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 / eps_m) - 2.0)) - eps_m))) / 2.0
	elif x <= 1.12e-10:
		tmp = (2.0 + (x * (((-1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 + (1.0 / eps_m)))) - eps_m))) / 2.0
	elif (x <= 1.4e+212) or not (x <= 1.06e+303):
		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 <= -0.0008)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 / eps_m) - 2.0)) - eps_m))) / 2.0);
	elseif (x <= 1.12e-10)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(-1.0 / eps_m) + Float64(Float64(1.0 - eps_m) * Float64(-1.0 + Float64(1.0 / eps_m)))) - eps_m))) / 2.0);
	elseif ((x <= 1.4e+212) || !(x <= 1.06e+303))
		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 <= -0.0008)
		tmp = (2.0 + (x * (((1.0 / eps_m) + ((1.0 / eps_m) - 2.0)) - eps_m))) / 2.0;
	elseif (x <= 1.12e-10)
		tmp = (2.0 + (x * (((-1.0 / eps_m) + ((1.0 - eps_m) * (-1.0 + (1.0 / eps_m)))) - eps_m))) / 2.0;
	elseif ((x <= 1.4e+212) || ~((x <= 1.06e+303)))
		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, -0.0008], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 / eps$95$m), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.12e-10], N[(N[(2.0 + N[(x * N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.4e+212], N[Not[LessEqual[x, 1.06e+303]], $MachinePrecision]], 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 < -8.00000000000000038e-4

   1. Initial program 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in x around 0 2.9%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-unprod 7.9%

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

      3. frac-times 7.9%

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

      4. metadata-eval 7.9%

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

      5. metadata-eval 7.9%

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

      6. frac-times 7.9%

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

      7. sqrt-unprod 1.7%

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

      8. add-sqr-sqrt 3.1%

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

      9. div-inv 3.1%

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

      10. mul-1-neg 3.1%

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

      11. sub-neg 3.1%

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

      12. add-sqr-sqrt 1.3%

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

      13. sqrt-unprod 3.0%

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

      14. frac-times 3.0%

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

      15. metadata-eval 3.0%

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

      16. metadata-eval 3.0%

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

      17. frac-times 3.0%

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

      18. sqrt-unprod 1.7%

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

      19. add-sqr-sqrt 3.0%

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

      20. sub-neg 3.0%

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

      21. metadata-eval 3.0%

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

      22. add-sqr-sqrt 1.3%

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

   7. Applied egg-rr 3.1%

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

   8. Taylor expanded in eps around 0 26.1%

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

   if -8.00000000000000038e-4 < x < 1.12e-10

   1. Initial program 53.4%

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

   3. Simplified 37.6%

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

   5. Taylor expanded in x around 0 72.2%

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

      1. +-commutative 72.2%

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

      2. *-un-lft-identity 72.2%

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

      3. fma-define 72.2%

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

      4. add-sqr-sqrt 29.5%

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

      5. sqrt-unprod 37.6%

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

      6. frac-times 36.9%

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

      7. metadata-eval 36.9%

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

      8. metadata-eval 36.9%

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

      9. frac-times 37.6%

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

      10. sqrt-unprod 21.2%

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

      11. add-sqr-sqrt 38.8%

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

      12. add-sqr-sqrt 17.6%

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

      13. sqrt-unprod 39.1%

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

      14. frac-times 38.3%

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

      15. metadata-eval 38.3%

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

      16. metadata-eval 38.3%

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

      17. frac-times 39.1%

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

      18. sqrt-unprod 30.0%

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

      19. add-sqr-sqrt 72.2%

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

      20. sub-neg 72.2%

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

      21. metadata-eval 72.2%

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

   7. Applied egg-rr 72.2%

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

      1. fma-undefine 72.2%

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

      2. *-lft-identity 72.2%

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

      3. +-commutative 72.2%

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

   9. Simplified 72.2%

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

   if 1.12e-10 < x < 1.39999999999999999e212 or 1.06e303 < x

   1. Initial program 100.0%

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

   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 30.4%

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

   6. Taylor expanded in x around 0 50.8%

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

   if 1.39999999999999999e212 < x < 1.06e303

   1. Initial program 100.0%

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

   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 46.5%

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

   6. Taylor expanded in x around inf 26.5%

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

      1. mul-1-neg 26.5%

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

      2. *-commutative 26.5%

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

      3. distribute-rgt-neg-in 26.5%

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

      4. distribute-rgt-neg-in 26.5%

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

      5. mul-1-neg 26.5%

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

      6. distribute-lft-in 26.5%

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

      7. metadata-eval 26.5%

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

      8. neg-mul-1 26.5%

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

      9. distribute-neg-frac 26.5%

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

      10. metadata-eval 26.5%

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

   8. Simplified 26.5%

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

   9. Taylor expanded in eps around inf 26.7%

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

       1. *-commutative 26.7%

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

   11. Simplified 26.7%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0008:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \left(\frac{1}{\varepsilon} - 2\right)\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{-1}{\varepsilon} + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+212} \lor \neg \left(x \leq 1.06 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + x \cdot -2}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+212} \lor \neg \left(x \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.12e-10)
   (/ (+ 2.0 (* x -2.0)) 2.0)
   (if (or (<= x 7e+212) (not (<= x 2e+306)))
     (/ (+ (+ 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 <= 1.12e-10) {
		tmp = (2.0 + (x * -2.0)) / 2.0;
	} else if ((x <= 7e+212) || !(x <= 2e+306)) {
		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 <= 1.12d-10) then
        tmp = (2.0d0 + (x * (-2.0d0))) / 2.0d0
    else if ((x <= 7d+212) .or. (.not. (x <= 2d+306))) 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 <= 1.12e-10) {
		tmp = (2.0 + (x * -2.0)) / 2.0;
	} else if ((x <= 7e+212) || !(x <= 2e+306)) {
		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 <= 1.12e-10:
		tmp = (2.0 + (x * -2.0)) / 2.0
	elif (x <= 7e+212) or not (x <= 2e+306):
		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 <= 1.12e-10)
		tmp = Float64(Float64(2.0 + Float64(x * -2.0)) / 2.0);
	elseif ((x <= 7e+212) || !(x <= 2e+306))
		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 <= 1.12e-10)
		tmp = (2.0 + (x * -2.0)) / 2.0;
	elseif ((x <= 7e+212) || ~((x <= 2e+306)))
		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, 1.12e-10], N[(N[(2.0 + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 7e+212], N[Not[LessEqual[x, 2e+306]], $MachinePrecision]], 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 3 regimes

2. if x < 1.12e-10

   1. Initial program 63.0%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in x around 0 56.2%

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

      1. add-sqr-sqrt 23.0%

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

      2. sqrt-unprod 30.8%

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

      3. frac-times 30.2%

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

      4. metadata-eval 30.2%

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

      5. metadata-eval 30.2%

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

      6. frac-times 30.8%

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

      7. sqrt-unprod 16.7%

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

      8. add-sqr-sqrt 30.5%

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

      9. div-inv 30.5%

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

      10. mul-1-neg 30.5%

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

      11. sub-neg 30.5%

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

      12. add-sqr-sqrt 13.9%

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

      13. sqrt-unprod 30.7%

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

      14. frac-times 30.2%

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

      15. metadata-eval 30.2%

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

      16. metadata-eval 30.2%

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

      17. frac-times 30.7%

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

      18. sqrt-unprod 23.4%

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

      19. add-sqr-sqrt 56.2%

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

      20. sub-neg 56.2%

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

      21. metadata-eval 56.2%

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

      22. add-sqr-sqrt 23.0%

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

   7. Applied egg-rr 30.6%

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

   8. Taylor expanded in eps around inf 56.3%

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

   if 1.12e-10 < x < 6.99999999999999974e212 or 2.00000000000000003e306 < x

   1. Initial program 100.0%

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

   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 30.4%

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

   6. Taylor expanded in x around 0 50.8%

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

   if 6.99999999999999974e212 < x < 2.00000000000000003e306

   1. Initial program 100.0%

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

   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 46.5%

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

   6. Taylor expanded in x around inf 26.5%

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

      1. mul-1-neg 26.5%

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

      2. *-commutative 26.5%

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

      3. distribute-rgt-neg-in 26.5%

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

      4. distribute-rgt-neg-in 26.5%

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

      5. mul-1-neg 26.5%

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

      6. distribute-lft-in 26.5%

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

      7. metadata-eval 26.5%

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

      8. neg-mul-1 26.5%

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

      9. distribute-neg-frac 26.5%

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

      10. metadata-eval 26.5%

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

   8. Simplified 26.5%

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

   9. Taylor expanded in eps around inf 26.7%

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

       1. *-commutative 26.7%

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

   11. Simplified 26.7%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + x \cdot -2}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+212} \lor \neg \left(x \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 50.6% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + x \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.12e-10) (/ (+ 2.0 (* x -2.0)) 2.0) (/ (* x eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.12e-10) {
		tmp = (2.0 + (x * -2.0)) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 1.12d-10) then
        tmp = (2.0d0 + (x * (-2.0d0))) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.12e-10) {
		tmp = (2.0 + (x * -2.0)) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.12e-10:
		tmp = (2.0 + (x * -2.0)) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.12e-10)
		tmp = Float64(Float64(2.0 + Float64(x * -2.0)) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.12e-10)
		tmp = (2.0 + (x * -2.0)) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.12e-10], N[(N[(2.0 + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.12e-10

   1. Initial program 63.0%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in x around 0 56.2%

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

      1. add-sqr-sqrt 23.0%

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

      2. sqrt-unprod 30.8%

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

      3. frac-times 30.2%

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

      4. metadata-eval 30.2%

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

      5. metadata-eval 30.2%

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

      6. frac-times 30.8%

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

      7. sqrt-unprod 16.7%

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

      8. add-sqr-sqrt 30.5%

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

      9. div-inv 30.5%

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

      10. mul-1-neg 30.5%

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

      11. sub-neg 30.5%

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

      12. add-sqr-sqrt 13.9%

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

      13. sqrt-unprod 30.7%

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

      14. frac-times 30.2%

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

      15. metadata-eval 30.2%

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

      16. metadata-eval 30.2%

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

      17. frac-times 30.7%

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

      18. sqrt-unprod 23.4%

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

      19. add-sqr-sqrt 56.2%

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

      20. sub-neg 56.2%

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

      21. metadata-eval 56.2%

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

      22. add-sqr-sqrt 23.0%

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

   7. Applied egg-rr 30.6%

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

   8. Taylor expanded in eps around inf 56.3%

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

   if 1.12e-10 < x

   1. Initial program 100.0%

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

   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 26.3%

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

   6. Taylor expanded in x around inf 11.9%

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

      1. mul-1-neg 11.9%

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

      2. *-commutative 11.9%

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

      3. distribute-rgt-neg-in 11.9%

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

      4. distribute-rgt-neg-in 11.9%

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

      5. mul-1-neg 11.9%

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

      6. distribute-lft-in 11.9%

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

      7. metadata-eval 11.9%

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

      8. neg-mul-1 11.9%

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

      9. distribute-neg-frac 11.9%

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

      10. metadata-eval 11.9%

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

   8. Simplified 11.9%

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

   9. Taylor expanded in eps around inf 12.6%

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

       1. *-commutative 12.6%

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

   11. Simplified 12.6%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + x \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.5% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.12e-10) 1.0 (/ (* x eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.12e-10) {
		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 <= 1.12d-10) 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 <= 1.12e-10) {
		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 <= 1.12e-10:
		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 <= 1.12e-10)
		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 <= 1.12e-10)
		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, 1.12e-10], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.12e-10

   1. Initial program 63.0%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in x around 0 56.2%

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

   if 1.12e-10 < x

   1. Initial program 100.0%

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

   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 26.3%

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

   6. Taylor expanded in x around inf 11.9%

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

      1. mul-1-neg 11.9%

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

      2. *-commutative 11.9%

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

      3. distribute-rgt-neg-in 11.9%

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

      4. distribute-rgt-neg-in 11.9%

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

      5. mul-1-neg 11.9%

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

      6. distribute-lft-in 11.9%

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

      7. metadata-eval 11.9%

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

      8. neg-mul-1 11.9%

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

      9. distribute-neg-frac 11.9%

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

      10. metadata-eval 11.9%

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

   8. Simplified 11.9%

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

   9. Taylor expanded in eps around inf 12.6%

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

       1. *-commutative 12.6%

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

   11. Simplified 12.6%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 49.5% accurate, 25.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Simplified 75.0%

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

5. Taylor expanded in x around 0 40.1%

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

6. Taylor expanded in eps around inf 63.8%

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

   1. neg-mul-1 63.8%

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

   2. distribute-lft-neg-in 63.8%

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

8. Simplified 63.8%

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

9. Taylor expanded in x around 0 45.7%

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

10. Final simplification 45.7%

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

Alternative 20: 44.1% 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 75.0%

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

3. Simplified 75.0%

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

5. Taylor expanded in x around 0 39.0%

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

6. Final simplification 39.0%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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))