NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 99.9%

Time: 12.8s

Alternatives: 12

Speedup: 2.1×

• 38.1% 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: 73.8% 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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \mathbf{if}\;eps_m \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps_m\right)} + e^{x \cdot \left(-eps_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (+ x 1.0) (exp (- x)))))
   (if (<= eps_m 5e-19)
     (/ (+ t_0 t_0) 2.0)
     (/ (+ (exp (* x (+ -1.0 eps_m))) (exp (* x (- eps_m)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x + 1.0) * exp(-x);
	double tmp;
	if (eps_m <= 5e-19) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) + exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) * exp(-x)
    if (eps_m <= 5d-19) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps_m))) + exp((x * -eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (x + 1.0) * Math.exp(-x);
	double tmp;
	if (eps_m <= 5e-19) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + Math.exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x + 1.0) * math.exp(-x)
	tmp = 0
	if eps_m <= 5e-19:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) + math.exp((x * -eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x + 1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps_m <= 5e-19)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x + 1.0) * exp(-x);
	tmp = 0.0;
	if (eps_m <= 5e-19)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) + exp((x * -eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 5e-19], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 5.0000000000000004e-19

   1. Initial program 60.6%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in eps around 0 74.7%

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

   6. Simplified 75.8%

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

   if 5.0000000000000004e-19 < eps

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. sub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. mul-1-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. mul-1-neg 100.0%

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

      10. associate-*r* 100.0%

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

      11. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]

Alternative 2: 98.7% 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 72.0%

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

3. Simplified 72.0%

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

4. Taylor expanded in eps around inf 98.6%

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

5. Final simplification 98.6%

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

Alternative 3: 84.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{e^{x \cdot eps_m} + e^{x \cdot \left(-eps_m\right)}}{2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+234} \lor \neg \left(x \leq 3.7 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps_m}\right) - \left|-1 + \frac{-1}{eps_m}\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps_m\right)} - -1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 3.1e+39)
   (/ (+ (exp (* x eps_m)) (exp (* x (- eps_m)))) 2.0)
   (if (or (<= x 5.5e+234) (not (<= x 3.7e+259)))
     (/ (- (+ 1.0 (/ 1.0 eps_m)) (fabs (+ -1.0 (/ -1.0 eps_m)))) 2.0)
     (/ (- (exp (* x (+ -1.0 eps_m))) -1.0) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 3.1e+39) {
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	} else if ((x <= 5.5e+234) || !(x <= 3.7e+259)) {
		tmp = ((1.0 + (1.0 / eps_m)) - fabs((-1.0 + (-1.0 / eps_m)))) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) - -1.0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 3.1d+39) then
        tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0d0
    else if ((x <= 5.5d+234) .or. (.not. (x <= 3.7d+259))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - abs(((-1.0d0) + ((-1.0d0) / eps_m)))) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps_m))) - (-1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 3.1e+39) {
		tmp = (Math.exp((x * eps_m)) + Math.exp((x * -eps_m))) / 2.0;
	} else if ((x <= 5.5e+234) || !(x <= 3.7e+259)) {
		tmp = ((1.0 + (1.0 / eps_m)) - Math.abs((-1.0 + (-1.0 / eps_m)))) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) - -1.0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 3.1e+39:
		tmp = (math.exp((x * eps_m)) + math.exp((x * -eps_m))) / 2.0
	elif (x <= 5.5e+234) or not (x <= 3.7e+259):
		tmp = ((1.0 + (1.0 / eps_m)) - math.fabs((-1.0 + (-1.0 / eps_m)))) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) - -1.0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 3.1e+39)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif ((x <= 5.5e+234) || !(x <= 3.7e+259))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - abs(Float64(-1.0 + Float64(-1.0 / eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) - -1.0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 3.1e+39)
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	elseif ((x <= 5.5e+234) || ~((x <= 3.7e+259)))
		tmp = ((1.0 + (1.0 / eps_m)) - abs((-1.0 + (-1.0 / eps_m)))) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) - -1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 3.1e+39], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 5.5e+234], N[Not[LessEqual[x, 3.7e+259]], $MachinePrecision]], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[Abs[N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.1000000000000003e39

   1. Initial program 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in eps around inf 98.1%

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

   5. Taylor expanded in eps around inf 97.1%

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

      1. *-commutative 97.1%

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

   7. Simplified 97.1%

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

   8. Taylor expanded in x around inf 97.1%

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

      1. associate-*r* 97.1%

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

      2. sub-neg 97.1%

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

      3. mul-1-neg 97.1%

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

      4. associate-*r* 97.1%

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

      5. associate-*r* 97.1%

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

      6. neg-mul-1 97.1%

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

      7. mul-1-neg 97.1%

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

      8. sub-neg 97.1%

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

      9. mul-1-neg 97.1%

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

      10. associate-*r* 97.1%

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

      11. mul-1-neg 97.1%

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

   10. Simplified 97.1%

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

   11. Taylor expanded in eps around inf 97.6%

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

       1. *-commutative 97.6%

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

   13. Simplified 97.6%

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

   if 3.1000000000000003e39 < x < 5.5e234 or 3.70000000000000015e259 < 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. Taylor expanded in x around 0 18.4%

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

   5. Taylor expanded in x around 0 65.1%

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

      1. sub-neg 65.1%

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

      2. add-sqr-sqrt 12.2%

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

      3. metadata-eval 12.2%

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

      4. sqrt-unprod 18.4%

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

      5. frac-times 13.6%

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

      6. metadata-eval 13.6%

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

      7. metadata-eval 13.6%

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

      8. frac-times 18.4%

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

      9. sqrt-unprod 1.5%

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

      10. add-sqr-sqrt 2.5%

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

      11. +-commutative 2.5%

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

      12. add-sqr-sqrt 1.0%

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

      13. sqrt-unprod 21.1%

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

      14. pow2 21.1%

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

   7. Applied egg-rr 21.1%

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

      1. unpow2 21.1%

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

      2. rem-sqrt-square 29.3%

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

   9. Simplified 29.3%

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

   if 5.5e234 < x < 3.70000000000000015e259

   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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 87.7%

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

      1. *-commutative 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in x around inf 87.7%

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

      1. associate-*r* 87.7%

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

      2. sub-neg 87.7%

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

      3. mul-1-neg 87.7%

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

      4. associate-*r* 87.7%

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

      5. associate-*r* 87.7%

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

      6. neg-mul-1 87.7%

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

      7. mul-1-neg 87.7%

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

      8. sub-neg 87.7%

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

      9. mul-1-neg 87.7%

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

      10. associate-*r* 87.7%

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

      11. mul-1-neg 87.7%

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

   10. Simplified 87.7%

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

   11. Taylor expanded in eps around 0 39.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+234} \lor \neg \left(x \leq 3.7 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left|-1 + \frac{-1}{\varepsilon}\right|}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} - -1}{2}\\ \end{array} \]

Alternative 4: 91.7% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 72.0%

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

3. Simplified 72.0%

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

4. Taylor expanded in eps around inf 98.6%

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

5. Taylor expanded in eps around inf 85.3%

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

   1. *-commutative 85.3%

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

7. Simplified 85.3%

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

8. Taylor expanded in x around inf 85.3%

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

   1. associate-*r* 85.3%

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

   2. sub-neg 85.3%

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

   3. mul-1-neg 85.3%

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

   4. associate-*r* 85.3%

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

   5. associate-*r* 85.3%

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

   6. neg-mul-1 85.3%

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

   7. mul-1-neg 85.3%

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

   8. sub-neg 85.3%

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

   9. mul-1-neg 85.3%

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

   10. associate-*r* 85.3%

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

   11. mul-1-neg 85.3%

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

10. Simplified 85.3%

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

11. Final simplification 85.3%

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

Alternative 5: 84.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-287}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps_m\right)}}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+40} \lor \neg \left(x \leq 3.1 \cdot 10^{+234}\right) \land x \leq 7.2 \cdot 10^{+259}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps_m\right)} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps_m}\right) - \left|-1 + \frac{-1}{eps_m}\right|}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-287)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (or (<= x 6.8e+40) (and (not (<= x 3.1e+234)) (<= x 7.2e+259)))
     (/ (- (exp (* x (+ -1.0 eps_m))) -1.0) 2.0)
     (/ (- (+ 1.0 (/ 1.0 eps_m)) (fabs (+ -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-287) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if ((x <= 6.8e+40) || (!(x <= 3.1e+234) && (x <= 7.2e+259))) {
		tmp = (exp((x * (-1.0 + eps_m))) - -1.0) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) - fabs((-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 <= (-5d-287)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if ((x <= 6.8d+40) .or. (.not. (x <= 3.1d+234)) .and. (x <= 7.2d+259)) then
        tmp = (exp((x * ((-1.0d0) + eps_m))) - (-1.0d0)) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - abs(((-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 <= -5e-287) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if ((x <= 6.8e+40) || (!(x <= 3.1e+234) && (x <= 7.2e+259))) {
		tmp = (Math.exp((x * (-1.0 + eps_m))) - -1.0) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) - Math.abs((-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-287:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif (x <= 6.8e+40) or (not (x <= 3.1e+234) and (x <= 7.2e+259)):
		tmp = (math.exp((x * (-1.0 + eps_m))) - -1.0) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) - math.fabs((-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-287)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif ((x <= 6.8e+40) || (!(x <= 3.1e+234) && (x <= 7.2e+259)))
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) - -1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - abs(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-287)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif ((x <= 6.8e+40) || (~((x <= 3.1e+234)) && (x <= 7.2e+259)))
		tmp = (exp((x * (-1.0 + eps_m))) - -1.0) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) - abs((-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-287], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 6.8e+40], And[N[Not[LessEqual[x, 3.1e+234]], $MachinePrecision], LessEqual[x, 7.2e+259]]], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[Abs[N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000025e-287

   1. Initial program 60.7%

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

   3. Simplified 60.7%

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

   4. Taylor expanded in x around 0 44.8%

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

   5. Taylor expanded in eps around inf 81.8%

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

   6. Taylor expanded in eps around inf 82.4%

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

      1. *-commutative 97.7%

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

   8. Simplified 82.4%

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

   if -5.00000000000000025e-287 < x < 6.79999999999999977e40 or 3.0999999999999999e234 < x < 7.2000000000000006e259

   1. Initial program 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in eps around inf 99.0%

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

   5. Taylor expanded in eps around inf 95.5%

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

      1. *-commutative 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in x around inf 95.5%

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

      1. associate-*r* 95.5%

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

      2. sub-neg 95.5%

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

      3. mul-1-neg 95.5%

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

      4. associate-*r* 95.5%

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

      5. associate-*r* 95.5%

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

      6. neg-mul-1 95.5%

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

      7. mul-1-neg 95.5%

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

      8. sub-neg 95.5%

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

      9. mul-1-neg 95.5%

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

      10. associate-*r* 95.5%

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

      11. mul-1-neg 95.5%

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

   10. Simplified 95.5%

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

   11. Taylor expanded in eps around 0 74.8%

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

   if 6.79999999999999977e40 < x < 3.0999999999999999e234 or 7.2000000000000006e259 < 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. Taylor expanded in x around 0 18.4%

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

   5. Taylor expanded in x around 0 65.1%

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

      1. sub-neg 65.1%

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

      2. add-sqr-sqrt 12.2%

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

      3. metadata-eval 12.2%

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

      4. sqrt-unprod 18.4%

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

      5. frac-times 13.6%

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

      6. metadata-eval 13.6%

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

      7. metadata-eval 13.6%

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

      8. frac-times 18.4%

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

      9. sqrt-unprod 1.5%

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

      10. add-sqr-sqrt 2.5%

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

      11. +-commutative 2.5%

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

      12. add-sqr-sqrt 1.0%

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

      13. sqrt-unprod 21.1%

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

      14. pow2 21.1%

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

   7. Applied egg-rr 21.1%

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

      1. unpow2 21.1%

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

      2. rem-sqrt-square 29.3%

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

   9. Simplified 29.3%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-287}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+40} \lor \neg \left(x \leq 3.1 \cdot 10^{+234}\right) \land x \leq 7.2 \cdot 10^{+259}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left|-1 + \frac{-1}{\varepsilon}\right|}{2}\\ \end{array} \]

Alternative 6: 83.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps_m\right)}}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+47} \lor \neg \left(x \leq 3.6 \cdot 10^{+234}\right) \land x \leq 4 \cdot 10^{+259}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps_m\right)} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps_m}\right) + \left(\frac{-1}{eps_m} - -1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-288)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (or (<= x 1.1e+47) (and (not (<= x 3.6e+234)) (<= x 4e+259)))
     (/ (- (exp (* x (+ -1.0 eps_m))) -1.0) 2.0)
     (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- (/ -1.0 eps_m) -1.0)) 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 * -eps_m))) / 2.0;
	} else if ((x <= 1.1e+47) || (!(x <= 3.6e+234) && (x <= 4e+259))) {
		tmp = (exp((x * (-1.0 + eps_m))) - -1.0) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-5d-288)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if ((x <= 1.1d+47) .or. (.not. (x <= 3.6d+234)) .and. (x <= 4d+259)) then
        tmp = (exp((x * ((-1.0d0) + eps_m))) - (-1.0d0)) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (((-1.0d0) / eps_m) - (-1.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-288) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if ((x <= 1.1e+47) || (!(x <= 3.6e+234) && (x <= 4e+259))) {
		tmp = (Math.exp((x * (-1.0 + eps_m))) - -1.0) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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 * -eps_m))) / 2.0
	elif (x <= 1.1e+47) or (not (x <= 3.6e+234) and (x <= 4e+259)):
		tmp = (math.exp((x * (-1.0 + eps_m))) - -1.0) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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 * Float64(-eps_m)))) / 2.0);
	elseif ((x <= 1.1e+47) || (!(x <= 3.6e+234) && (x <= 4e+259)))
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) - -1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(Float64(-1.0 / eps_m) - -1.0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -5e-288)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif ((x <= 1.1e+47) || (~((x <= 3.6e+234)) && (x <= 4e+259)))
		tmp = (exp((x * (-1.0 + eps_m))) - -1.0) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5e-288], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.1e+47], And[N[Not[LessEqual[x, 3.6e+234]], $MachinePrecision], LessEqual[x, 4e+259]]], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000011e-288

   1. Initial program 60.7%

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

   3. Simplified 60.7%

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

   4. Taylor expanded in x around 0 44.8%

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

   5. Taylor expanded in eps around inf 81.8%

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

   6. Taylor expanded in eps around inf 82.4%

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

      1. *-commutative 97.7%

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

   8. Simplified 82.4%

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

   if -5.00000000000000011e-288 < x < 1.1e47 or 3.59999999999999999e234 < x < 4e259

   1. Initial program 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in eps around inf 99.0%

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

   5. Taylor expanded in eps around inf 94.4%

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

      1. *-commutative 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in x around inf 94.4%

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

      1. associate-*r* 94.4%

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

      2. sub-neg 94.4%

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

      3. mul-1-neg 94.4%

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

      4. associate-*r* 94.4%

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

      5. associate-*r* 94.4%

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

      6. neg-mul-1 94.4%

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

      7. mul-1-neg 94.4%

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

      8. sub-neg 94.4%

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

      9. mul-1-neg 94.4%

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

      10. associate-*r* 94.4%

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

      11. mul-1-neg 94.4%

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

   10. Simplified 94.4%

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

   11. Taylor expanded in eps around 0 74.0%

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

   if 1.1e47 < x < 3.59999999999999999e234 or 4e259 < 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. Taylor expanded in x around 0 18.7%

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

   5. Taylor expanded in x around 0 66.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+47} \lor \neg \left(x \leq 3.6 \cdot 10^{+234}\right) \land x \leq 4 \cdot 10^{+259}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \end{array} \]

Alternative 7: 76.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps_m\right)}}{2}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+234} \lor \neg \left(x \leq 3.7 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps_m}\right) + \left(\frac{-1}{eps_m} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 360.0)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (or (<= x 3.3e+234) (not (<= x 3.7e+259)))
     (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- (/ -1.0 eps_m) -1.0)) 2.0)
     (/ (* x eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 360.0) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if ((x <= 3.3e+234) || !(x <= 3.7e+259)) {
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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 <= 360.0d0) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if ((x <= 3.3d+234) .or. (.not. (x <= 3.7d+259))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (((-1.0d0) / eps_m) - (-1.0d0))) / 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 <= 360.0) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if ((x <= 3.3e+234) || !(x <= 3.7e+259)) {
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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 <= 360.0:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif (x <= 3.3e+234) or not (x <= 3.7e+259):
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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 <= 360.0)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif ((x <= 3.3e+234) || !(x <= 3.7e+259))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(Float64(-1.0 / eps_m) - -1.0)) / 2.0);
	else
		tmp = Float64(Float64(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 <= 360.0)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif ((x <= 3.3e+234) || ~((x <= 3.7e+259)))
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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, 360.0], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3.3e+234], N[Not[LessEqual[x, 3.7e+259]], $MachinePrecision]], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 360

   1. Initial program 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in x around 0 43.9%

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

   5. Taylor expanded in eps around inf 81.9%

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

   6. Taylor expanded in eps around inf 82.4%

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

      1. *-commutative 98.1%

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

   8. Simplified 82.4%

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

   if 360 < x < 3.3000000000000003e234 or 3.70000000000000015e259 < 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. Taylor expanded in x around 0 24.0%

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

   5. Taylor expanded in x around 0 61.5%

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

   if 3.3000000000000003e234 < x < 3.70000000000000015e259

   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. Taylor expanded in x around 0 37.7%

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

   5. Taylor expanded in x around inf 37.7%

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

      1. mul-1-neg 37.7%

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

      2. associate-*r* 37.7%

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

      3. *-commutative 37.7%

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

      4. distribute-rgt-neg-in 37.7%

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

      5. distribute-rgt-neg-in 37.7%

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

      6. distribute-neg-in 37.7%

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

      7. metadata-eval 37.7%

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

      8. distribute-neg-frac 37.7%

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

      9. metadata-eval 37.7%

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

   7. Simplified 37.7%

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

   8. Taylor expanded in eps around inf 37.9%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+234} \lor \neg \left(x \leq 3.7 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]

Alternative 8: 69.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+234} \lor \neg \left(x \leq 9.8 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps_m}\right) + \left(\frac{-1}{eps_m} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 360.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (or (<= x 3.8e+234) (not (<= x 9.8e+259)))
     (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- (/ -1.0 eps_m) -1.0)) 2.0)
     (/ (* x eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 360.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if ((x <= 3.8e+234) || !(x <= 9.8e+259)) {
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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 <= 360.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if ((x <= 3.8d+234) .or. (.not. (x <= 9.8d+259))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (((-1.0d0) / eps_m) - (-1.0d0))) / 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 <= 360.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if ((x <= 3.8e+234) || !(x <= 9.8e+259)) {
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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 <= 360.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif (x <= 3.8e+234) or not (x <= 9.8e+259):
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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 <= 360.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif ((x <= 3.8e+234) || !(x <= 9.8e+259))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(Float64(-1.0 / eps_m) - -1.0)) / 2.0);
	else
		tmp = Float64(Float64(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 <= 360.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif ((x <= 3.8e+234) || ~((x <= 9.8e+259)))
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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, 360.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3.8e+234], N[Not[LessEqual[x, 9.8e+259]], $MachinePrecision]], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 360

   1. Initial program 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in eps around inf 98.1%

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

   5. Taylor expanded in eps around inf 98.1%

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

      1. *-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in eps around 0 75.7%

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

      1. neg-mul-1 75.7%

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

   10. Simplified 75.7%

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

   if 360 < x < 3.8e234 or 9.79999999999999958e259 < 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. Taylor expanded in x around 0 24.0%

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

   5. Taylor expanded in x around 0 61.5%

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

   if 3.8e234 < x < 9.79999999999999958e259

   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. Taylor expanded in x around 0 37.7%

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

   5. Taylor expanded in x around inf 37.7%

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

      1. mul-1-neg 37.7%

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

      2. associate-*r* 37.7%

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

      3. *-commutative 37.7%

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

      4. distribute-rgt-neg-in 37.7%

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

      5. distribute-rgt-neg-in 37.7%

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

      6. distribute-neg-in 37.7%

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

      7. metadata-eval 37.7%

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

      8. distribute-neg-frac 37.7%

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

      9. metadata-eval 37.7%

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

   7. Simplified 37.7%

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

   8. Taylor expanded in eps around inf 37.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+234} \lor \neg \left(x \leq 9.8 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]

Alternative 9: 63.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 165:\\ \;\;\;\;\frac{2 - x \cdot eps_m}{2}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+234} \lor \neg \left(x \leq 1.1 \cdot 10^{+260}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps_m}\right) + \left(\frac{-1}{eps_m} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 165.0)
   (/ (- 2.0 (* x eps_m)) 2.0)
   (if (or (<= x 6.5e+234) (not (<= x 1.1e+260)))
     (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- (/ -1.0 eps_m) -1.0)) 2.0)
     (/ (* x eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 165.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if ((x <= 6.5e+234) || !(x <= 1.1e+260)) {
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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 <= 165.0d0) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else if ((x <= 6.5d+234) .or. (.not. (x <= 1.1d+260))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (((-1.0d0) / eps_m) - (-1.0d0))) / 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 <= 165.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if ((x <= 6.5e+234) || !(x <= 1.1e+260)) {
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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 <= 165.0:
		tmp = (2.0 - (x * eps_m)) / 2.0
	elif (x <= 6.5e+234) or not (x <= 1.1e+260):
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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 <= 165.0)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	elseif ((x <= 6.5e+234) || !(x <= 1.1e+260))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(Float64(-1.0 / eps_m) - -1.0)) / 2.0);
	else
		tmp = Float64(Float64(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 <= 165.0)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	elseif ((x <= 6.5e+234) || ~((x <= 1.1e+260)))
		tmp = ((1.0 + (1.0 / eps_m)) + ((-1.0 / eps_m) - -1.0)) / 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, 165.0], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 6.5e+234], N[Not[LessEqual[x, 1.1e+260]], $MachinePrecision]], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 165

   1. Initial program 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in x around 0 43.9%

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

   5. Taylor expanded in x around 0 50.3%

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

   6. Taylor expanded in eps around inf 66.7%

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

      1. associate-*r* 66.7%

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

      2. mul-1-neg 66.7%

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

   8. Simplified 66.7%

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

   if 165 < x < 6.4999999999999995e234 or 1.10000000000000006e260 < 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. Taylor expanded in x around 0 24.0%

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

   5. Taylor expanded in x around 0 61.5%

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

   if 6.4999999999999995e234 < x < 1.10000000000000006e260

   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. Taylor expanded in x around 0 37.7%

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

   5. Taylor expanded in x around inf 37.7%

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

      1. mul-1-neg 37.7%

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

      2. associate-*r* 37.7%

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

      3. *-commutative 37.7%

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

      4. distribute-rgt-neg-in 37.7%

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

      5. distribute-rgt-neg-in 37.7%

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

      6. distribute-neg-in 37.7%

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

      7. metadata-eval 37.7%

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

      8. distribute-neg-frac 37.7%

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

      9. metadata-eval 37.7%

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

   7. Simplified 37.7%

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

   8. Taylor expanded in eps around inf 37.9%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 165:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+234} \lor \neg \left(x \leq 1.1 \cdot 10^{+260}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]

Alternative 10: 57.6% accurate, 25.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 21:\\ \;\;\;\;\frac{2 - x \cdot eps_m}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 21.0) (/ (- 2.0 (* x 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 <= 21.0) {
		tmp = (2.0 - (x * 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 <= 21.0d0) then
        tmp = (2.0d0 - (x * 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 <= 21.0) {
		tmp = (2.0 - (x * 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 <= 21.0:
		tmp = (2.0 - (x * 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 <= 21.0)
		tmp = Float64(Float64(2.0 - Float64(x * 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 <= 21.0)
		tmp = (2.0 - (x * 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, 21.0], N[(N[(2.0 - N[(x * eps$95$m), $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 < 21

   1. Initial program 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in x around 0 43.9%

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

   5. Taylor expanded in x around 0 50.3%

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

   6. Taylor expanded in eps around inf 66.7%

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

      1. associate-*r* 66.7%

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

      2. mul-1-neg 66.7%

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

   8. Simplified 66.7%

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

   if 21 < 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. Taylor expanded in x around 0 25.0%

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

   5. Taylor expanded in x around inf 11.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]
   6. 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. associate-*r* 11.9%

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

      3. *-commutative 11.9%

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

      4. distribute-rgt-neg-in 11.9%

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

      5. distribute-rgt-neg-in 11.9%

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

      6. distribute-neg-in 11.9%

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

      7. metadata-eval 11.9%

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

      8. distribute-neg-frac 11.9%

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

      9. metadata-eval 11.9%

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

   7. Simplified 11.9%

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

   8. Taylor expanded in eps around inf 12.7%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 21:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]

Alternative 11: 50.9% accurate, 32.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 42:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 42.0) 1.0 (/ (* x eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 42.0) {
		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 <= 42.0d0) 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 <= 42.0) {
		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 <= 42.0:
		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 <= 42.0)
		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 <= 42.0)
		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, 42.0], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 42

   1. Initial program 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in x around 0 63.3%

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

   if 42 < 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. Taylor expanded in x around 0 25.0%

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

   5. Taylor expanded in x around inf 11.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right)\right)}}{2} \]
   6. 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. associate-*r* 11.9%

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

      3. *-commutative 11.9%

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

      4. distribute-rgt-neg-in 11.9%

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

      5. distribute-rgt-neg-in 11.9%

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

      6. distribute-neg-in 11.9%

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

      7. metadata-eval 11.9%

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

      8. distribute-neg-frac 11.9%

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

      9. metadata-eval 11.9%

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

   7. Simplified 11.9%

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

   8. Taylor expanded in eps around inf 12.7%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 42:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]

Alternative 12: 43.8% 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 72.0%

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

3. Simplified 72.0%

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

4. Taylor expanded in x around 0 45.4%

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

5. Final simplification 45.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023322 
(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))