NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.3% → 99.0%

Time: 21.1s

Alternatives: 14

Speedup: 2.0×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \frac{eps\_m \cdot \left(2 \cdot \frac{x + 1}{e^{x}}\right)}{eps\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \left(-1 + eps\_m\right)} + e^{eps\_m \cdot \left(-x\right)}\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2e-89)
   (* 0.5 (/ (* eps_m (* 2.0 (/ (+ x 1.0) (exp x)))) eps_m))
   (* 0.5 (+ (exp (* x (+ -1.0 eps_m))) (exp (* eps_m (- x)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2e-89) {
		tmp = 0.5 * ((eps_m * (2.0 * ((x + 1.0) / exp(x)))) / eps_m);
	} else {
		tmp = 0.5 * (exp((x * (-1.0 + eps_m))) + exp((eps_m * -x)));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 2d-89) then
        tmp = 0.5d0 * ((eps_m * (2.0d0 * ((x + 1.0d0) / exp(x)))) / eps_m)
    else
        tmp = 0.5d0 * (exp((x * ((-1.0d0) + eps_m))) + exp((eps_m * -x)))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2e-89) {
		tmp = 0.5 * ((eps_m * (2.0 * ((x + 1.0) / Math.exp(x)))) / eps_m);
	} else {
		tmp = 0.5 * (Math.exp((x * (-1.0 + eps_m))) + Math.exp((eps_m * -x)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 2e-89:
		tmp = 0.5 * ((eps_m * (2.0 * ((x + 1.0) / math.exp(x)))) / eps_m)
	else:
		tmp = 0.5 * (math.exp((x * (-1.0 + eps_m))) + math.exp((eps_m * -x)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 2e-89)
		tmp = Float64(0.5 * Float64(Float64(eps_m * Float64(2.0 * Float64(Float64(x + 1.0) / exp(x)))) / eps_m));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + exp(Float64(eps_m * Float64(-x)))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 2e-89)
		tmp = 0.5 * ((eps_m * (2.0 * ((x + 1.0) / exp(x)))) / eps_m);
	else
		tmp = 0.5 * (exp((x * (-1.0 + eps_m))) + exp((eps_m * -x)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 2e-89], N[(0.5 * N[(N[(eps$95$m * N[(2.0 * N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.00000000000000008e-89

   1. Initial program 59.3%

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

   3. Simplified 44.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 27.6%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot x} + \left(-1 \cdot e^{-1 \cdot x} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)\right)}{\varepsilon}} \cdot 0.5 \]
   6. Step-by-step derivation

   7. Simplified 68.6%

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot \left(2 \cdot \frac{x + 1}{e^{x}}\right) + 0}{\varepsilon}} \cdot 0.5 \]

   if 2.00000000000000008e-89 < eps

   1. Initial program 93.6%

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

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

      \[\leadsto \left(e^{-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}} + e^{x \cdot \left(\varepsilon - 1\right)}\right) \cdot 0.5 \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon \cdot \left(2 \cdot \frac{x + 1}{e^{x}}\right)}{\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \left(-1 + \varepsilon\right)} + e^{\varepsilon \cdot \left(-x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \left({\left({\left(e^{x \cdot \left(-1 - eps\_m\right)}\right)}^{3}\right)}^{0.3333333333333333} + e^{x \cdot \left(-1 + eps\_m\right)}\right) \cdot 0.5 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (*
  (+
   (pow (pow (exp (* x (- -1.0 eps_m))) 3.0) 0.3333333333333333)
   (exp (* x (+ -1.0 eps_m))))
  0.5))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (pow(pow(exp((x * (-1.0 - eps_m))), 3.0), 0.3333333333333333) + exp((x * (-1.0 + eps_m)))) * 0.5;
}

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))) ** 3.0d0) ** 0.3333333333333333d0) + exp((x * ((-1.0d0) + eps_m)))) * 0.5d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return (Math.pow(Math.pow(Math.exp((x * (-1.0 - eps_m))), 3.0), 0.3333333333333333) + Math.exp((x * (-1.0 + eps_m)))) * 0.5;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (math.pow(math.pow(math.exp((x * (-1.0 - eps_m))), 3.0), 0.3333333333333333) + math.exp((x * (-1.0 + eps_m)))) * 0.5

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(((exp(Float64(x * Float64(-1.0 - eps_m))) ^ 3.0) ^ 0.3333333333333333) + exp(Float64(x * Float64(-1.0 + eps_m)))) * 0.5)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (((exp((x * (-1.0 - eps_m))) ^ 3.0) ^ 0.3333333333333333) + exp((x * (-1.0 + eps_m)))) * 0.5;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[Power[N[Power[N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

3. Simplified 54.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
4. Add Preprocessing

5. Taylor expanded in eps around inf 99.2%

   \[\leadsto \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)} + e^{x \cdot \left(\varepsilon - 1\right)}\right)} \cdot 0.5 \]
6. Step-by-step derivation

   1. exp-prod 99.2%

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

7. Applied egg-rr 99.2%

   \[\leadsto \left(\color{blue}{{\left(e^{-1}\right)}^{\left(x \cdot \left(1 + \varepsilon\right)\right)}} + e^{x \cdot \left(\varepsilon - 1\right)}\right) \cdot 0.5 \]
8. Step-by-step derivation

   1. add-cbrt-cube 99.2%

      \[\leadsto \left(\color{blue}{\sqrt[3]{\left({\left(e^{-1}\right)}^{\left(x \cdot \left(1 + \varepsilon\right)\right)} \cdot {\left(e^{-1}\right)}^{\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \cdot {\left(e^{-1}\right)}^{\left(x \cdot \left(1 + \varepsilon\right)\right)}}} + e^{x \cdot \left(\varepsilon - 1\right)}\right) \cdot 0.5 \]

   2. pow1/3 99.2%

      \[\leadsto \left(\color{blue}{{\left(\left({\left(e^{-1}\right)}^{\left(x \cdot \left(1 + \varepsilon\right)\right)} \cdot {\left(e^{-1}\right)}^{\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \cdot {\left(e^{-1}\right)}^{\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}^{0.3333333333333333}} + e^{x \cdot \left(\varepsilon - 1\right)}\right) \cdot 0.5 \]

   3. pow3 99.2%

      \[\leadsto \left({\color{blue}{\left({\left({\left(e^{-1}\right)}^{\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}^{3}\right)}}^{0.3333333333333333} + e^{x \cdot \left(\varepsilon - 1\right)}\right) \cdot 0.5 \]

   4. pow-exp 99.2%

      \[\leadsto \left({\left({\color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}}^{3}\right)}^{0.3333333333333333} + e^{x \cdot \left(\varepsilon - 1\right)}\right) \cdot 0.5 \]

   5. *-commutative 99.2%

      \[\leadsto \left({\left({\left(e^{-1 \cdot \color{blue}{\left(\left(1 + \varepsilon\right) \cdot x\right)}}\right)}^{3}\right)}^{0.3333333333333333} + e^{x \cdot \left(\varepsilon - 1\right)}\right) \cdot 0.5 \]

   6. associate-*r* 99.2%

      \[\leadsto \left({\left({\left(e^{\color{blue}{\left(-1 \cdot \left(1 + \varepsilon\right)\right) \cdot x}}\right)}^{3}\right)}^{0.3333333333333333} + e^{x \cdot \left(\varepsilon - 1\right)}\right) \cdot 0.5 \]

   7. distribute-lft-in 99.2%

      \[\leadsto \left({\left({\left(e^{\color{blue}{\left(-1 \cdot 1 + -1 \cdot \varepsilon\right)} \cdot x}\right)}^{3}\right)}^{0.3333333333333333} + e^{x \cdot \left(\varepsilon - 1\right)}\right) \cdot 0.5 \]

   8. metadata-eval 99.2%

      \[\leadsto \left({\left({\left(e^{\left(\color{blue}{-1} + -1 \cdot \varepsilon\right) \cdot x}\right)}^{3}\right)}^{0.3333333333333333} + e^{x \cdot \left(\varepsilon - 1\right)}\right) \cdot 0.5 \]

   9. neg-mul-1 99.2%

      \[\leadsto \left({\left({\left(e^{\left(-1 + \color{blue}{\left(-\varepsilon\right)}\right) \cdot x}\right)}^{3}\right)}^{0.3333333333333333} + e^{x \cdot \left(\varepsilon - 1\right)}\right) \cdot 0.5 \]

9. Applied egg-rr 99.2%

   \[\leadsto \left(\color{blue}{{\left({\left(e^{\left(-1 + \left(-\varepsilon\right)\right) \cdot x}\right)}^{3}\right)}^{0.3333333333333333}} + e^{x \cdot \left(\varepsilon - 1\right)}\right) \cdot 0.5 \]

10. Final simplification 99.2%

    \[\leadsto \left({\left({\left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)}^{3}\right)}^{0.3333333333333333} + e^{x \cdot \left(-1 + \varepsilon\right)}\right) \cdot 0.5 \]
11. Add Preprocessing

Alternative 3: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0.5 \cdot \left(e^{x \cdot \left(-1 + eps\_m\right)} + {\left(e^{-1}\right)}^{\left(x \cdot \left(eps\_m + 1\right)\right)}\right) \end{array} \]

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.5 * (exp((x * (-1.0 + eps_m))) + pow(exp(-1.0), (x * (eps_m + 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 = 0.5d0 * (exp((x * ((-1.0d0) + eps_m))) + (exp((-1.0d0)) ** (x * (eps_m + 1.0d0))))
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.5 * (Math.exp((x * (-1.0 + eps_m))) + Math.pow(Math.exp(-1.0), (x * (eps_m + 1.0))));
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.5 * (math.exp((x * (-1.0 + eps_m))) + math.pow(math.exp(-1.0), (x * (eps_m + 1.0))))

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(0.5 * Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + (exp(-1.0) ^ Float64(x * Float64(eps_m + 1.0)))))
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.5 * (exp((x * (-1.0 + eps_m))) + (exp(-1.0) ^ (x * (eps_m + 1.0))));
end

Wolfram

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

Derivation

1. Initial program 73.5%

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

3. Simplified 54.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
4. Add Preprocessing

5. Taylor expanded in eps around inf 99.2%

   \[\leadsto \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)} + e^{x \cdot \left(\varepsilon - 1\right)}\right)} \cdot 0.5 \]
6. Step-by-step derivation

   1. exp-prod 99.2%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

   \[\leadsto 0.5 \cdot \left(e^{x \cdot \left(-1 + \varepsilon\right)} + {\left(e^{-1}\right)}^{\left(x \cdot \left(\varepsilon + 1\right)\right)}\right) \]
9. Add Preprocessing

Alternative 4: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0.5 \cdot \left(e^{x \cdot \left(-1 + eps\_m\right)} + e^{x \cdot \left(-1 - eps\_m\right)}\right) \end{array} \]

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.5 * (exp((x * (-1.0 + eps_m))) + exp((x * (-1.0 - eps_m))));
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.5d0 * (exp((x * ((-1.0d0) + eps_m))) + exp((x * ((-1.0d0) - eps_m))))
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.5 * (Math.exp((x * (-1.0 + eps_m))) + Math.exp((x * (-1.0 - eps_m))));
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.5 * (math.exp((x * (-1.0 + eps_m))) + math.exp((x * (-1.0 - eps_m))))

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(0.5 * Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + exp(Float64(x * Float64(-1.0 - eps_m)))))
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.5 * (exp((x * (-1.0 + eps_m))) + exp((x * (-1.0 - eps_m))));
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(0.5 * 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]), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

3. Simplified 54.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
4. Add Preprocessing

5. Taylor expanded in eps around inf 99.2%

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

6. Final simplification 99.2%

   \[\leadsto 0.5 \cdot \left(e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \]
7. Add Preprocessing

Alternative 5: 91.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-282}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{x \cdot \left(-1 + eps\_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{eps\_m} + \left(-1 + eps\_m\right) \cdot \left(1 + \frac{1}{eps\_m}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.7e-282)
   (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
   (if (<= x 8.4e+70)
     (* 0.5 (+ 1.0 (exp (* x (+ -1.0 eps_m)))))
     (/
      (* x (+ (/ 1.0 eps_m) (* (+ -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 <= -1.7e-282) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if (x <= 8.4e+70) {
		tmp = 0.5 * (1.0 + exp((x * (-1.0 + eps_m))));
	} else {
		tmp = (x * ((1.0 / eps_m) + ((-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 <= (-1.7d-282)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if (x <= 8.4d+70) then
        tmp = 0.5d0 * (1.0d0 + exp((x * ((-1.0d0) + eps_m))))
    else
        tmp = (x * ((1.0d0 / eps_m) + (((-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 <= -1.7e-282) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if (x <= 8.4e+70) {
		tmp = 0.5 * (1.0 + Math.exp((x * (-1.0 + eps_m))));
	} else {
		tmp = (x * ((1.0 / eps_m) + ((-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 <= -1.7e-282:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif x <= 8.4e+70:
		tmp = 0.5 * (1.0 + math.exp((x * (-1.0 + eps_m))))
	else:
		tmp = (x * ((1.0 / eps_m) + ((-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 <= -1.7e-282)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif (x <= 8.4e+70)
		tmp = Float64(0.5 * Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps_m)))));
	else
		tmp = Float64(Float64(x * Float64(Float64(1.0 / eps_m) + Float64(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 <= -1.7e-282)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif (x <= 8.4e+70)
		tmp = 0.5 * (1.0 + exp((x * (-1.0 + eps_m))));
	else
		tmp = (x * ((1.0 / eps_m) + ((-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, -1.7e-282], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.4e+70], N[(0.5 * N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 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]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.69999999999999999e-282

   1. Initial program 71.6%

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

   3. Taylor expanded in x around 0 49.2%

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

   4. Taylor expanded in eps around inf 77.4%

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

   5. Taylor expanded in eps around inf 77.6%

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

      1. *-commutative 99.8%

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

   7. Simplified 77.6%

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

   if -1.69999999999999999e-282 < x < 8.4000000000000003e70

   1. Initial program 57.3%

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

   3. Simplified 27.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 98.1%

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

   6. Taylor expanded in x around 0 84.0%

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

   if 8.4000000000000003e70 < 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. Add Preprocessing

   3. Taylor expanded in x around 0 3.2%

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

   4. Taylor expanded in eps around 0 22.5%

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

   5. Taylor expanded in x around inf 67.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-282}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{x \cdot \left(-1 + \varepsilon\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{\varepsilon} + \left(-1 + \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-258}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{-x}\right)\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{x \cdot \left(-1 + eps\_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{eps\_m} + \left(-1 + eps\_m\right) \cdot \left(1 + \frac{1}{eps\_m}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.4e-258)
   (* 0.5 (+ 1.0 (exp (- x))))
   (if (<= x 5.1e+70)
     (* 0.5 (+ 1.0 (exp (* x (+ -1.0 eps_m)))))
     (/
      (* x (+ (/ 1.0 eps_m) (* (+ -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 <= 2.4e-258) {
		tmp = 0.5 * (1.0 + exp(-x));
	} else if (x <= 5.1e+70) {
		tmp = 0.5 * (1.0 + exp((x * (-1.0 + eps_m))));
	} else {
		tmp = (x * ((1.0 / eps_m) + ((-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 <= 2.4d-258) then
        tmp = 0.5d0 * (1.0d0 + exp(-x))
    else if (x <= 5.1d+70) then
        tmp = 0.5d0 * (1.0d0 + exp((x * ((-1.0d0) + eps_m))))
    else
        tmp = (x * ((1.0d0 / eps_m) + (((-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 <= 2.4e-258) {
		tmp = 0.5 * (1.0 + Math.exp(-x));
	} else if (x <= 5.1e+70) {
		tmp = 0.5 * (1.0 + Math.exp((x * (-1.0 + eps_m))));
	} else {
		tmp = (x * ((1.0 / eps_m) + ((-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 <= 2.4e-258:
		tmp = 0.5 * (1.0 + math.exp(-x))
	elif x <= 5.1e+70:
		tmp = 0.5 * (1.0 + math.exp((x * (-1.0 + eps_m))))
	else:
		tmp = (x * ((1.0 / eps_m) + ((-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 <= 2.4e-258)
		tmp = Float64(0.5 * Float64(1.0 + exp(Float64(-x))));
	elseif (x <= 5.1e+70)
		tmp = Float64(0.5 * Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps_m)))));
	else
		tmp = Float64(Float64(x * Float64(Float64(1.0 / eps_m) + Float64(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 <= 2.4e-258)
		tmp = 0.5 * (1.0 + exp(-x));
	elseif (x <= 5.1e+70)
		tmp = 0.5 * (1.0 + exp((x * (-1.0 + eps_m))));
	else
		tmp = (x * ((1.0 / eps_m) + ((-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, 2.4e-258], N[(0.5 * N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.1e+70], N[(0.5 * N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 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]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.4000000000000002e-258

   1. Initial program 66.8%

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

   3. Simplified 44.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 99.8%

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

   6. Taylor expanded in x around 0 73.5%

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

   7. Taylor expanded in eps around 0 83.4%

      \[\leadsto \color{blue}{\left(1 + e^{-1 \cdot x}\right)} \cdot 0.5 \]
   8. Step-by-step derivation

      1. mul-1-neg 83.4%

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

   9. Simplified 83.4%

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

   if 2.4000000000000002e-258 < x < 5.10000000000000014e70

   1. Initial program 61.3%

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

   3. Simplified 30.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 97.6%

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

   6. Taylor expanded in x around 0 80.4%

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

   if 5.10000000000000014e70 < 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. Add Preprocessing

   3. Taylor expanded in x around 0 3.2%

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

   4. Taylor expanded in eps around 0 22.5%

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

   5. Taylor expanded in x around inf 67.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-258}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{-x}\right)\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{x \cdot \left(-1 + \varepsilon\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{\varepsilon} + \left(-1 + \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 7:\\ \;\;\;\;0.5 \cdot \left(1 + e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{eps\_m} + \left(-1 + eps\_m\right) \cdot \left(1 + \frac{1}{eps\_m}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 7.0)
   (* 0.5 (+ 1.0 (exp (- x))))
   (/ (* x (+ (/ 1.0 eps_m) (* (+ -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 <= 7.0) {
		tmp = 0.5 * (1.0 + exp(-x));
	} else {
		tmp = (x * ((1.0 / eps_m) + ((-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 <= 7.0d0) then
        tmp = 0.5d0 * (1.0d0 + exp(-x))
    else
        tmp = (x * ((1.0d0 / eps_m) + (((-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 <= 7.0) {
		tmp = 0.5 * (1.0 + Math.exp(-x));
	} else {
		tmp = (x * ((1.0 / eps_m) + ((-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 <= 7.0:
		tmp = 0.5 * (1.0 + math.exp(-x))
	else:
		tmp = (x * ((1.0 / eps_m) + ((-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 <= 7.0)
		tmp = Float64(0.5 * Float64(1.0 + exp(Float64(-x))));
	else
		tmp = Float64(Float64(x * Float64(Float64(1.0 / eps_m) + Float64(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 <= 7.0)
		tmp = 0.5 * (1.0 + exp(-x));
	else
		tmp = (x * ((1.0 / eps_m) + ((-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, 7.0], N[(0.5 * N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 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]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7

   1. Initial program 62.5%

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

   3. Simplified 35.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 77.9%

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

   7. Taylor expanded in eps around 0 79.2%

      \[\leadsto \color{blue}{\left(1 + e^{-1 \cdot x}\right)} \cdot 0.5 \]
   8. Step-by-step derivation

      1. mul-1-neg 79.2%

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

   9. Simplified 79.2%

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

   if 7 < 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. Add Preprocessing

   3. Taylor expanded in x around 0 3.1%

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

   4. Taylor expanded in eps around 0 19.6%

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

   5. Taylor expanded in x around inf 59.7%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7:\\ \;\;\;\;0.5 \cdot \left(1 + e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{\varepsilon} + \left(-1 + \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{eps\_m \cdot x}{-2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+185} \lor \neg \left(x \leq 1.25 \cdot 10^{+261}\right):\\ \;\;\;\;0.5 \cdot \frac{0}{eps\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{eps\_m \cdot x}}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (/ (* eps_m x) -2.0)
   (if (<= x 4.9e+33)
     1.0
     (if (or (<= x 5.2e+185) (not (<= x 1.25e+261)))
       (* 0.5 (/ 0.0 eps_m))
       (/ 1.0 (/ 2.0 (* eps_m x)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (eps_m * x) / -2.0;
	} else if (x <= 4.9e+33) {
		tmp = 1.0;
	} else if ((x <= 5.2e+185) || !(x <= 1.25e+261)) {
		tmp = 0.5 * (0.0 / eps_m);
	} else {
		tmp = 1.0 / (2.0 / (eps_m * x));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (eps_m * x) / (-2.0d0)
    else if (x <= 4.9d+33) then
        tmp = 1.0d0
    else if ((x <= 5.2d+185) .or. (.not. (x <= 1.25d+261))) then
        tmp = 0.5d0 * (0.0d0 / eps_m)
    else
        tmp = 1.0d0 / (2.0d0 / (eps_m * x))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (eps_m * x) / -2.0;
	} else if (x <= 4.9e+33) {
		tmp = 1.0;
	} else if ((x <= 5.2e+185) || !(x <= 1.25e+261)) {
		tmp = 0.5 * (0.0 / eps_m);
	} else {
		tmp = 1.0 / (2.0 / (eps_m * x));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (eps_m * x) / -2.0
	elif x <= 4.9e+33:
		tmp = 1.0
	elif (x <= 5.2e+185) or not (x <= 1.25e+261):
		tmp = 0.5 * (0.0 / eps_m)
	else:
		tmp = 1.0 / (2.0 / (eps_m * x))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(eps_m * x) / -2.0);
	elseif (x <= 4.9e+33)
		tmp = 1.0;
	elseif ((x <= 5.2e+185) || !(x <= 1.25e+261))
		tmp = Float64(0.5 * Float64(0.0 / eps_m));
	else
		tmp = Float64(1.0 / Float64(2.0 / Float64(eps_m * x)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (eps_m * x) / -2.0;
	elseif (x <= 4.9e+33)
		tmp = 1.0;
	elseif ((x <= 5.2e+185) || ~((x <= 1.25e+261)))
		tmp = 0.5 * (0.0 / eps_m);
	else
		tmp = 1.0 / (2.0 / (eps_m * x));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(eps$95$m * x), $MachinePrecision] / -2.0), $MachinePrecision], If[LessEqual[x, 4.9e+33], 1.0, If[Or[LessEqual[x, 5.2e+185], N[Not[LessEqual[x, 1.25e+261]], $MachinePrecision]], N[(0.5 * N[(0.0 / eps$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 / N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 3.1%

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

   4. Taylor expanded in eps around 0 13.2%

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

   5. Taylor expanded in eps around inf 13.2%

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

      1. *-commutative 13.2%

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

   7. Simplified 13.2%

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

      1. frac-2neg 13.2%

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

      2. distribute-rgt-neg-in 13.2%

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

      3. add-sqr-sqrt 13.2%

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

      4. sqrt-unprod 72.3%

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

      5. sqr-neg 72.3%

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

      6. sqrt-unprod 40.2%

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

      7. add-sqr-sqrt 40.2%

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

      8. *-commutative 40.2%

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

      9. metadata-eval 40.2%

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

   9. Applied egg-rr 40.2%

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

   if -1 < x < 4.90000000000000014e33

   1. Initial program 55.1%

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

   3. Simplified 22.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 73.1%

      \[\leadsto \color{blue}{2} \cdot 0.5 \]

   if 4.90000000000000014e33 < x < 5.20000000000000001e185 or 1.25e261 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 50.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. distribute-rgt1-in 50.8%

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

      2. metadata-eval 50.8%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{\varepsilon} \cdot 0.5 \]

      3. neg-mul-1 50.8%

         \[\leadsto \frac{0 \cdot e^{\color{blue}{-x}}}{\varepsilon} \cdot 0.5 \]

      4. mul0-lft 50.8%

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \cdot 0.5 \]

   7. Simplified 50.8%

      \[\leadsto \color{blue}{\frac{0}{\varepsilon}} \cdot 0.5 \]

   if 5.20000000000000001e185 < x < 1.25e261

   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. Add Preprocessing

   3. Taylor expanded in x around 0 3.1%

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

   4. Taylor expanded in eps around 0 30.1%

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

   5. Taylor expanded in eps around inf 30.0%

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

      1. *-commutative 30.0%

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

   7. Simplified 30.0%

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

      1. clear-num 30.0%

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

      2. *-commutative 30.0%

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

   9. Applied egg-rr 30.0%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\varepsilon \cdot x}{-2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+185} \lor \neg \left(x \leq 1.25 \cdot 10^{+261}\right):\\ \;\;\;\;0.5 \cdot \frac{0}{\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{\varepsilon \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.1% accurate, 9.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;\frac{eps\_m \cdot x}{-2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+182} \lor \neg \left(x \leq 3.5 \cdot 10^{+263}\right):\\ \;\;\;\;0.5 \cdot \frac{0}{eps\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.85)
   (/ (* eps_m x) -2.0)
   (if (<= x 4.9e+33)
     1.0
     (if (or (<= x 9.5e+182) (not (<= x 3.5e+263)))
       (* 0.5 (/ 0.0 eps_m))
       (/ (* eps_m x) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.85) {
		tmp = (eps_m * x) / -2.0;
	} else if (x <= 4.9e+33) {
		tmp = 1.0;
	} else if ((x <= 9.5e+182) || !(x <= 3.5e+263)) {
		tmp = 0.5 * (0.0 / eps_m);
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-0.85d0)) then
        tmp = (eps_m * x) / (-2.0d0)
    else if (x <= 4.9d+33) then
        tmp = 1.0d0
    else if ((x <= 9.5d+182) .or. (.not. (x <= 3.5d+263))) then
        tmp = 0.5d0 * (0.0d0 / eps_m)
    else
        tmp = (eps_m * x) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.85) {
		tmp = (eps_m * x) / -2.0;
	} else if (x <= 4.9e+33) {
		tmp = 1.0;
	} else if ((x <= 9.5e+182) || !(x <= 3.5e+263)) {
		tmp = 0.5 * (0.0 / eps_m);
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.85:
		tmp = (eps_m * x) / -2.0
	elif x <= 4.9e+33:
		tmp = 1.0
	elif (x <= 9.5e+182) or not (x <= 3.5e+263):
		tmp = 0.5 * (0.0 / eps_m)
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.85)
		tmp = Float64(Float64(eps_m * x) / -2.0);
	elseif (x <= 4.9e+33)
		tmp = 1.0;
	elseif ((x <= 9.5e+182) || !(x <= 3.5e+263))
		tmp = Float64(0.5 * Float64(0.0 / eps_m));
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.85)
		tmp = (eps_m * x) / -2.0;
	elseif (x <= 4.9e+33)
		tmp = 1.0;
	elseif ((x <= 9.5e+182) || ~((x <= 3.5e+263)))
		tmp = 0.5 * (0.0 / eps_m);
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.85], N[(N[(eps$95$m * x), $MachinePrecision] / -2.0), $MachinePrecision], If[LessEqual[x, 4.9e+33], 1.0, If[Or[LessEqual[x, 9.5e+182], N[Not[LessEqual[x, 3.5e+263]], $MachinePrecision]], N[(0.5 * N[(0.0 / eps$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.849999999999999978

   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. Add Preprocessing

   3. Taylor expanded in x around 0 3.1%

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

   4. Taylor expanded in eps around 0 13.2%

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

   5. Taylor expanded in eps around inf 13.2%

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

      1. *-commutative 13.2%

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

   7. Simplified 13.2%

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

      1. frac-2neg 13.2%

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

      2. distribute-rgt-neg-in 13.2%

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

      3. add-sqr-sqrt 13.2%

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

      4. sqrt-unprod 72.3%

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

      5. sqr-neg 72.3%

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

      6. sqrt-unprod 40.2%

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

      7. add-sqr-sqrt 40.2%

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

      8. *-commutative 40.2%

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

      9. metadata-eval 40.2%

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

   9. Applied egg-rr 40.2%

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

   if -0.849999999999999978 < x < 4.90000000000000014e33

   1. Initial program 55.1%

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

   3. Simplified 22.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 73.1%

      \[\leadsto \color{blue}{2} \cdot 0.5 \]

   if 4.90000000000000014e33 < x < 9.50000000000000002e182 or 3.49999999999999999e263 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 50.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. distribute-rgt1-in 50.8%

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

      2. metadata-eval 50.8%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{\varepsilon} \cdot 0.5 \]

      3. neg-mul-1 50.8%

         \[\leadsto \frac{0 \cdot e^{\color{blue}{-x}}}{\varepsilon} \cdot 0.5 \]

      4. mul0-lft 50.8%

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \cdot 0.5 \]

   7. Simplified 50.8%

      \[\leadsto \color{blue}{\frac{0}{\varepsilon}} \cdot 0.5 \]

   if 9.50000000000000002e182 < x < 3.49999999999999999e263

   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. Add Preprocessing

   3. Taylor expanded in x around 0 3.1%

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

   4. Taylor expanded in eps around 0 30.1%

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

   5. Taylor expanded in eps around inf 30.0%

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

      1. *-commutative 30.0%

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

   7. Simplified 30.0%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;\frac{\varepsilon \cdot x}{-2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+182} \lor \neg \left(x \leq 3.5 \cdot 10^{+263}\right):\\ \;\;\;\;0.5 \cdot \frac{0}{\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{x \cdot \left(-1 - eps\_m\right) + 2}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+184} \lor \neg \left(x \leq 3.5 \cdot 10^{+266}\right):\\ \;\;\;\;0.5 \cdot \frac{0}{eps\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{eps\_m \cdot x}}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.0)
   (/ (+ (* x (- -1.0 eps_m)) 2.0) 2.0)
   (if (or (<= x 1.4e+184) (not (<= x 3.5e+266)))
     (* 0.5 (/ 0.0 eps_m))
     (/ 1.0 (/ 2.0 (* eps_m x))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = ((x * (-1.0 - eps_m)) + 2.0) / 2.0;
	} else if ((x <= 1.4e+184) || !(x <= 3.5e+266)) {
		tmp = 0.5 * (0.0 / eps_m);
	} else {
		tmp = 1.0 / (2.0 / (eps_m * x));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = ((x * ((-1.0d0) - eps_m)) + 2.0d0) / 2.0d0
    else if ((x <= 1.4d+184) .or. (.not. (x <= 3.5d+266))) then
        tmp = 0.5d0 * (0.0d0 / eps_m)
    else
        tmp = 1.0d0 / (2.0d0 / (eps_m * x))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = ((x * (-1.0 - eps_m)) + 2.0) / 2.0;
	} else if ((x <= 1.4e+184) || !(x <= 3.5e+266)) {
		tmp = 0.5 * (0.0 / eps_m);
	} else {
		tmp = 1.0 / (2.0 / (eps_m * x));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.0:
		tmp = ((x * (-1.0 - eps_m)) + 2.0) / 2.0
	elif (x <= 1.4e+184) or not (x <= 3.5e+266):
		tmp = 0.5 * (0.0 / eps_m)
	else:
		tmp = 1.0 / (2.0 / (eps_m * x))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 - eps_m)) + 2.0) / 2.0);
	elseif ((x <= 1.4e+184) || !(x <= 3.5e+266))
		tmp = Float64(0.5 * Float64(0.0 / eps_m));
	else
		tmp = Float64(1.0 / Float64(2.0 / Float64(eps_m * x)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = ((x * (-1.0 - eps_m)) + 2.0) / 2.0;
	elseif ((x <= 1.4e+184) || ~((x <= 3.5e+266)))
		tmp = 0.5 * (0.0 / eps_m);
	else
		tmp = 1.0 / (2.0 / (eps_m * x));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.0], N[(N[(N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.4e+184], N[Not[LessEqual[x, 3.5e+266]], $MachinePrecision]], N[(0.5 * N[(0.0 / eps$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 / N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2

   1. Initial program 62.5%

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

   3. Taylor expanded in x around 0 44.5%

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

   4. Taylor expanded in eps around inf 81.0%

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

   5. Taylor expanded in x around 0 67.4%

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

      1. mul-1-neg 67.4%

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

      2. unsub-neg 67.4%

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

      3. +-commutative 67.4%

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

   7. Simplified 67.4%

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

   if 2 < x < 1.39999999999999995e184 or 3.50000000000000025e266 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 47.1%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. distribute-rgt1-in 47.1%

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

      2. metadata-eval 47.1%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{\varepsilon} \cdot 0.5 \]

      3. neg-mul-1 47.1%

         \[\leadsto \frac{0 \cdot e^{\color{blue}{-x}}}{\varepsilon} \cdot 0.5 \]

      4. mul0-lft 47.1%

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \cdot 0.5 \]

   7. Simplified 47.1%

      \[\leadsto \color{blue}{\frac{0}{\varepsilon}} \cdot 0.5 \]

   if 1.39999999999999995e184 < x < 3.50000000000000025e266

   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. Add Preprocessing

   3. Taylor expanded in x around 0 3.1%

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

   4. Taylor expanded in eps around 0 30.1%

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

   5. Taylor expanded in eps around inf 30.0%

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

      1. *-commutative 30.0%

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

   7. Simplified 30.0%

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

      1. clear-num 30.0%

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

      2. *-commutative 30.0%

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

   9. Applied egg-rr 30.0%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right) + 2}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+184} \lor \neg \left(x \leq 3.5 \cdot 10^{+266}\right):\\ \;\;\;\;0.5 \cdot \frac{0}{\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{\varepsilon \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.5% accurate, 10.3× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.12)
   (/ (+ (* x (- -1.0 eps_m)) 2.0) 2.0)
   (/ (* x (+ (/ 1.0 eps_m) (* (+ -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 <= 1.12) {
		tmp = ((x * (-1.0 - eps_m)) + 2.0) / 2.0;
	} else {
		tmp = (x * ((1.0 / eps_m) + ((-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 <= 1.12d0) then
        tmp = ((x * ((-1.0d0) - eps_m)) + 2.0d0) / 2.0d0
    else
        tmp = (x * ((1.0d0 / eps_m) + (((-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 <= 1.12) {
		tmp = ((x * (-1.0 - eps_m)) + 2.0) / 2.0;
	} else {
		tmp = (x * ((1.0 / eps_m) + ((-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 <= 1.12:
		tmp = ((x * (-1.0 - eps_m)) + 2.0) / 2.0
	else:
		tmp = (x * ((1.0 / eps_m) + ((-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 <= 1.12)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 - eps_m)) + 2.0) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(1.0 / eps_m) + Float64(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 <= 1.12)
		tmp = ((x * (-1.0 - eps_m)) + 2.0) / 2.0;
	else
		tmp = (x * ((1.0 / eps_m) + ((-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, 1.12], N[(N[(N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * 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]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1200000000000001

   1. Initial program 62.5%

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

   3. Taylor expanded in x around 0 44.5%

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

   4. Taylor expanded in eps around inf 81.0%

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

   5. Taylor expanded in x around 0 67.4%

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

      1. mul-1-neg 67.4%

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

      2. unsub-neg 67.4%

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

      3. +-commutative 67.4%

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

   7. Simplified 67.4%

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

   if 1.1200000000000001 < 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. Add Preprocessing

   3. Taylor expanded in x around 0 3.1%

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

   4. Taylor expanded in eps around 0 19.6%

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

   5. Taylor expanded in x around inf 59.7%

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

4. Final simplification 65.1%

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

Alternative 12: 63.2% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.47:\\ \;\;\;\;\frac{eps\_m \cdot x}{-2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{0}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.47)
   (/ (* eps_m x) -2.0)
   (if (<= x 4.9e+33) 1.0 (* 0.5 (/ 0.0 eps_m)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.47) {
		tmp = (eps_m * x) / -2.0;
	} else if (x <= 4.9e+33) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (0.0 / eps_m);
	}
	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 <= (-0.47d0)) then
        tmp = (eps_m * x) / (-2.0d0)
    else if (x <= 4.9d+33) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (0.0d0 / eps_m)
    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 <= -0.47) {
		tmp = (eps_m * x) / -2.0;
	} else if (x <= 4.9e+33) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (0.0 / eps_m);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.47:
		tmp = (eps_m * x) / -2.0
	elif x <= 4.9e+33:
		tmp = 1.0
	else:
		tmp = 0.5 * (0.0 / eps_m)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.47)
		tmp = Float64(Float64(eps_m * x) / -2.0);
	elseif (x <= 4.9e+33)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(0.0 / eps_m));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.47)
		tmp = (eps_m * x) / -2.0;
	elseif (x <= 4.9e+33)
		tmp = 1.0;
	else
		tmp = 0.5 * (0.0 / eps_m);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.47], N[(N[(eps$95$m * x), $MachinePrecision] / -2.0), $MachinePrecision], If[LessEqual[x, 4.9e+33], 1.0, N[(0.5 * N[(0.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.46999999999999997

   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. Add Preprocessing

   3. Taylor expanded in x around 0 3.1%

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

   4. Taylor expanded in eps around 0 13.2%

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

   5. Taylor expanded in eps around inf 13.2%

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

      1. *-commutative 13.2%

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

   7. Simplified 13.2%

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

      1. frac-2neg 13.2%

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

      2. distribute-rgt-neg-in 13.2%

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

      3. add-sqr-sqrt 13.2%

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

      4. sqrt-unprod 72.3%

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

      5. sqr-neg 72.3%

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

      6. sqrt-unprod 40.2%

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

      7. add-sqr-sqrt 40.2%

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

      8. *-commutative 40.2%

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

      9. metadata-eval 40.2%

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

   9. Applied egg-rr 40.2%

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

   if -0.46999999999999997 < x < 4.90000000000000014e33

   1. Initial program 55.1%

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

   3. Simplified 22.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 73.1%

      \[\leadsto \color{blue}{2} \cdot 0.5 \]

   if 4.90000000000000014e33 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 44.5%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. distribute-rgt1-in 44.5%

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

      2. metadata-eval 44.5%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{\varepsilon} \cdot 0.5 \]

      3. neg-mul-1 44.5%

         \[\leadsto \frac{0 \cdot e^{\color{blue}{-x}}}{\varepsilon} \cdot 0.5 \]

      4. mul0-lft 44.5%

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \cdot 0.5 \]

   7. Simplified 44.5%

      \[\leadsto \color{blue}{\frac{0}{\varepsilon}} \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.47:\\ \;\;\;\;\frac{\varepsilon \cdot x}{-2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{0}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.0% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{0}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 4.9e+33) 1.0 (* 0.5 (/ 0.0 eps_m))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 4.9e+33) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (0.0 / eps_m);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 4.9d+33) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (0.0d0 / eps_m)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 4.9e+33) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (0.0 / eps_m);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 4.9e+33:
		tmp = 1.0
	else:
		tmp = 0.5 * (0.0 / eps_m)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 4.9e+33)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(0.0 / eps_m));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 4.9e+33)
		tmp = 1.0;
	else
		tmp = 0.5 * (0.0 / eps_m);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 4.9e+33], 1.0, N[(0.5 * N[(0.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.90000000000000014e33

   1. Initial program 63.3%

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

   3. Simplified 36.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 60.2%

      \[\leadsto \color{blue}{2} \cdot 0.5 \]

   if 4.90000000000000014e33 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 44.5%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. distribute-rgt1-in 44.5%

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

      2. metadata-eval 44.5%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{\varepsilon} \cdot 0.5 \]

      3. neg-mul-1 44.5%

         \[\leadsto \frac{0 \cdot e^{\color{blue}{-x}}}{\varepsilon} \cdot 0.5 \]

      4. mul0-lft 44.5%

         \[\leadsto \frac{\color{blue}{0}}{\varepsilon} \cdot 0.5 \]

   7. Simplified 44.5%

      \[\leadsto \color{blue}{\frac{0}{\varepsilon}} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{0}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.5% 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 73.5%

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

3. Simplified 54.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, {\left(e^{-1 - \varepsilon}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot 0.5} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 44.4%

   \[\leadsto \color{blue}{2} \cdot 0.5 \]

6. Final simplification 44.4%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

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