NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.1% → 99.8%

Time: 13.6s

Alternatives: 18

Speedup: 2.0×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% 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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps\_m \leq 0.0068:\\ \;\;\;\;\frac{t\_0 \cdot \left(1 + \left(x + 1\right)\right) + x \cdot t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 0.0068)
     (/ (+ (* t_0 (+ 1.0 (+ x 1.0))) (* x t_0)) 2.0)
     (/ (+ (exp (* x eps_m)) (exp (* x (- eps_m)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 0.0068) {
		tmp = ((t_0 * (1.0 + (x + 1.0))) + (x * t_0)) / 2.0;
	} else {
		tmp = (exp((x * 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 = exp(-x)
    if (eps_m <= 0.0068d0) then
        tmp = ((t_0 * (1.0d0 + (x + 1.0d0))) + (x * t_0)) / 2.0d0
    else
        tmp = (exp((x * 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 = Math.exp(-x);
	double tmp;
	if (eps_m <= 0.0068) {
		tmp = ((t_0 * (1.0 + (x + 1.0))) + (x * t_0)) / 2.0;
	} else {
		tmp = (Math.exp((x * eps_m)) + Math.exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 0.0068:
		tmp = ((t_0 * (1.0 + (x + 1.0))) + (x * t_0)) / 2.0
	else:
		tmp = (math.exp((x * eps_m)) + math.exp((x * -eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 0.0068)
		tmp = Float64(Float64(Float64(t_0 * Float64(1.0 + Float64(x + 1.0))) + Float64(x * t_0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * 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 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 0.0068)
		tmp = ((t_0 * (1.0 + (x + 1.0))) + (x * t_0)) / 2.0;
	else
		tmp = (exp((x * 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[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 0.0068], N[(N[(N[(t$95$0 * N[(1.0 + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $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 < 0.00679999999999999962

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in eps around 0 73.7%

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

      1. associate--r+ 73.7%

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

      2. associate-*r* 73.7%

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

      3. mul-1-neg 73.7%

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

      4. cancel-sign-sub 73.7%

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

      5. distribute-rgt1-in 73.7%

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

      6. distribute-rgt-out-- 74.2%

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

      7. mul-1-neg 74.2%

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

      8. mul-1-neg 74.2%

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

   7. Simplified 74.2%

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

   if 0.00679999999999999962 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in eps around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. neg-mul-1 100.0%

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

       3. distribute-rgt-neg-in 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0068:\\ \;\;\;\;\frac{e^{-x} \cdot \left(1 + \left(x + 1\right)\right) + x \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.0068:\\ \;\;\;\;\frac{x + \left(1 + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.0068)
   (/ (+ x (+ 1.0 (- 1.0 x))) 2.0)
   (/ (+ (exp (* x eps_m)) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.0068) {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	} else {
		tmp = (exp((x * 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) :: tmp
    if (eps_m <= 0.0068d0) then
        tmp = (x + (1.0d0 + (1.0d0 - x))) / 2.0d0
    else
        tmp = (exp((x * 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 tmp;
	if (eps_m <= 0.0068) {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	} else {
		tmp = (Math.exp((x * eps_m)) + Math.exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.0068:
		tmp = (x + (1.0 + (1.0 - x))) / 2.0
	else:
		tmp = (math.exp((x * eps_m)) + math.exp((x * -eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.0068)
		tmp = Float64(Float64(x + Float64(1.0 + Float64(1.0 - x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * 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)
	tmp = 0.0;
	if (eps_m <= 0.0068)
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	else
		tmp = (exp((x * 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_] := If[LessEqual[eps$95$m, 0.0068], N[(N[(x + N[(1.0 + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $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 < 0.00679999999999999962

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in x around 0 27.1%

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

      1. mul-1-neg 27.1%

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

      2. distribute-lft-neg-in 27.1%

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

   7. Simplified 27.1%

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

   8. Taylor expanded in x around 0 21.6%

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

   9. Taylor expanded in eps around 0 73.4%

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

       1. +-commutative 73.4%

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

       2. associate--l+ 73.4%

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

       3. neg-mul-1 73.4%

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

       4. distribute-lft-in 73.4%

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

       5. metadata-eval 73.4%

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

       6. neg-mul-1 73.4%

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

       7. associate-*r* 73.4%

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

       8. metadata-eval 73.4%

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

       9. *-lft-identity 73.4%

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

       10. +-commutative 73.4%

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

   11. Simplified 73.4%

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

   if 0.00679999999999999962 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in eps around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. neg-mul-1 100.0%

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

       3. distribute-rgt-neg-in 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0068:\\ \;\;\;\;\frac{x + \left(1 + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in eps around inf 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 85.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{x + \mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-251}:\\ \;\;\;\;\frac{\left(1 + x \cdot eps\_m\right) + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + \left(1 - x \cdot eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e+26)
   (/ (/ (+ x (expm1 (- x))) eps_m) 2.0)
   (if (<= x -2.3e-251)
     (/ (+ (+ 1.0 (* x eps_m)) (exp (* x (- eps_m)))) 2.0)
     (if (<= x 1.9e+22)
       (/ (+ (exp (* x eps_m)) (- 1.0 (* x eps_m))) 2.0)
       0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e+26) {
		tmp = ((x + expm1(-x)) / eps_m) / 2.0;
	} else if (x <= -2.3e-251) {
		tmp = ((1.0 + (x * eps_m)) + exp((x * -eps_m))) / 2.0;
	} else if (x <= 1.9e+22) {
		tmp = (exp((x * eps_m)) + (1.0 - (x * eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e+26) {
		tmp = ((x + Math.expm1(-x)) / eps_m) / 2.0;
	} else if (x <= -2.3e-251) {
		tmp = ((1.0 + (x * eps_m)) + Math.exp((x * -eps_m))) / 2.0;
	} else if (x <= 1.9e+22) {
		tmp = (Math.exp((x * eps_m)) + (1.0 - (x * eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e+26:
		tmp = ((x + math.expm1(-x)) / eps_m) / 2.0
	elif x <= -2.3e-251:
		tmp = ((1.0 + (x * eps_m)) + math.exp((x * -eps_m))) / 2.0
	elif x <= 1.9e+22:
		tmp = (math.exp((x * eps_m)) + (1.0 - (x * eps_m))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e+26)
		tmp = Float64(Float64(Float64(x + expm1(Float64(-x))) / eps_m) / 2.0);
	elseif (x <= -2.3e-251)
		tmp = Float64(Float64(Float64(1.0 + Float64(x * eps_m)) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 1.9e+22)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + Float64(1.0 - Float64(x * eps_m))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1e+26], N[(N[(N[(x + N[(Exp[(-x)] - 1), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -2.3e-251], N[(N[(N[(1.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.9e+22], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.00000000000000005e26

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 48.4%

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

   6. Taylor expanded in eps around 0 44.7%

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

      1. neg-mul-1 44.7%

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

      2. associate--r+ 44.7%

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

      3. neg-mul-1 44.7%

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

      4. expm1-undefine 44.7%

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

      5. sub-neg 44.7%

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

      6. remove-double-neg 44.7%

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

   8. Simplified 44.7%

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

   if -1.00000000000000005e26 < x < -2.30000000000000017e-251

   1. Initial program 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in eps around inf 98.2%

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

   6. Taylor expanded in eps around inf 98.2%

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

      1. *-commutative 98.2%

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

   8. Simplified 98.2%

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

   9. Taylor expanded in eps around inf 98.2%

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

       1. associate-*r* 98.2%

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

       2. neg-mul-1 98.2%

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

   11. Simplified 98.2%

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

   12. Taylor expanded in x around 0 84.9%

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

   if -2.30000000000000017e-251 < x < 1.9000000000000002e22

   1. Initial program 54.7%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 98.9%

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

      1. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in eps around inf 98.9%

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

       1. associate-*r* 98.9%

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

       2. neg-mul-1 98.9%

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

   11. Simplified 98.9%

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

   12. Taylor expanded in eps around 0 89.9%

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

   if 1.9000000000000002e22 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 14.7%

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

      1. mul-1-neg 14.7%

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

      2. distribute-lft-neg-in 14.7%

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

   7. Simplified 14.7%

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

   8. Taylor expanded in x around 0 22.9%

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

   9. Taylor expanded in eps around 0 18.6%

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

   11. Simplified 18.6%

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

   12. Taylor expanded in eps around 0 62.6%

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

       1. distribute-rgt1-in 62.6%

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

       2. metadata-eval 62.6%

          \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

       3. associate-*r/ 24.5%

          \[\leadsto \frac{\color{blue}{0 \cdot \frac{x}{\varepsilon}}}{2} \]

       4. mul0-lft 62.6%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   14. Simplified 62.6%

       \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{x + \mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-251}:\\ \;\;\;\;\frac{\left(1 + x \cdot \varepsilon\right) + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.115:\\ \;\;\;\;\frac{x + \left(1 + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot \left(\frac{e^{x \cdot \left(-1 + eps\_m\right)} + \left(x + \left(1 - x\right)\right)}{eps\_m} - x\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.115)
   (/ (+ x (+ 1.0 (- 1.0 x))) 2.0)
   (/
    (* eps_m (- (/ (+ (exp (* x (+ -1.0 eps_m))) (+ x (- 1.0 x))) eps_m) x))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.115) {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	} else {
		tmp = (eps_m * (((exp((x * (-1.0 + eps_m))) + (x + (1.0 - x))) / eps_m) - x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.115d0) then
        tmp = (x + (1.0d0 + (1.0d0 - x))) / 2.0d0
    else
        tmp = (eps_m * (((exp((x * ((-1.0d0) + eps_m))) + (x + (1.0d0 - x))) / eps_m) - x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.115) {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	} else {
		tmp = (eps_m * (((Math.exp((x * (-1.0 + eps_m))) + (x + (1.0 - x))) / eps_m) - x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.115:
		tmp = (x + (1.0 + (1.0 - x))) / 2.0
	else:
		tmp = (eps_m * (((math.exp((x * (-1.0 + eps_m))) + (x + (1.0 - x))) / eps_m) - x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.115)
		tmp = Float64(Float64(x + Float64(1.0 + Float64(1.0 - x))) / 2.0);
	else
		tmp = Float64(Float64(eps_m * Float64(Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + Float64(x + Float64(1.0 - x))) / eps_m) - x)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.115)
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	else
		tmp = (eps_m * (((exp((x * (-1.0 + eps_m))) + (x + (1.0 - x))) / eps_m) - x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.115], N[(N[(x + N[(1.0 + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(eps$95$m * N[(N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.115000000000000005

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in x around 0 27.1%

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

      1. mul-1-neg 27.1%

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

      2. distribute-lft-neg-in 27.1%

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

   7. Simplified 27.1%

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

   8. Taylor expanded in x around 0 21.6%

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

   9. Taylor expanded in eps around 0 73.4%

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

       1. +-commutative 73.4%

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

       2. associate--l+ 73.4%

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

       3. neg-mul-1 73.4%

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

       4. distribute-lft-in 73.4%

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

       5. metadata-eval 73.4%

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

       6. neg-mul-1 73.4%

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

       7. associate-*r* 73.4%

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

       8. metadata-eval 73.4%

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

       9. *-lft-identity 73.4%

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

       10. +-commutative 73.4%

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

   11. Simplified 73.4%

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

   if 0.115000000000000005 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 72.8%

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

   6. Taylor expanded in eps around -inf 79.9%

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

      1. associate-*r* 79.9%

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

      2. neg-mul-1 79.9%

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

      3. sub-neg 79.9%

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

   8. Simplified 79.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.115:\\ \;\;\;\;\frac{x + \left(1 + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(x + \left(1 - x\right)\right)}{\varepsilon} - x\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-251}:\\ \;\;\;\;\frac{eps\_m \cdot \left(x + \frac{e^{x \cdot \left(-1 - eps\_m\right)} + \left(1 - x\right)}{eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+20}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + \left(1 - x \cdot eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2.3e-251)
   (/ (* eps_m (+ x (/ (+ (exp (* x (- -1.0 eps_m))) (- 1.0 x)) eps_m))) 2.0)
   (if (<= x 1.42e+20) (/ (+ (exp (* x eps_m)) (- 1.0 (* x eps_m))) 2.0) 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2.3e-251) {
		tmp = (eps_m * (x + ((exp((x * (-1.0 - eps_m))) + (1.0 - x)) / eps_m))) / 2.0;
	} else if (x <= 1.42e+20) {
		tmp = (exp((x * eps_m)) + (1.0 - (x * eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2.3d-251)) then
        tmp = (eps_m * (x + ((exp((x * ((-1.0d0) - eps_m))) + (1.0d0 - x)) / eps_m))) / 2.0d0
    else if (x <= 1.42d+20) then
        tmp = (exp((x * eps_m)) + (1.0d0 - (x * eps_m))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2.3e-251) {
		tmp = (eps_m * (x + ((Math.exp((x * (-1.0 - eps_m))) + (1.0 - x)) / eps_m))) / 2.0;
	} else if (x <= 1.42e+20) {
		tmp = (Math.exp((x * eps_m)) + (1.0 - (x * eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2.3e-251:
		tmp = (eps_m * (x + ((math.exp((x * (-1.0 - eps_m))) + (1.0 - x)) / eps_m))) / 2.0
	elif x <= 1.42e+20:
		tmp = (math.exp((x * eps_m)) + (1.0 - (x * eps_m))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2.3e-251)
		tmp = Float64(Float64(eps_m * Float64(x + Float64(Float64(exp(Float64(x * Float64(-1.0 - eps_m))) + Float64(1.0 - x)) / eps_m))) / 2.0);
	elseif (x <= 1.42e+20)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + Float64(1.0 - Float64(x * eps_m))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2.3e-251)
		tmp = (eps_m * (x + ((exp((x * (-1.0 - eps_m))) + (1.0 - x)) / eps_m))) / 2.0;
	elseif (x <= 1.42e+20)
		tmp = (exp((x * eps_m)) + (1.0 - (x * eps_m))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2.3e-251], N[(N[(eps$95$m * N[(x + N[(N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.42e+20], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.30000000000000017e-251

   1. Initial program 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 72.2%

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

      1. neg-mul-1 72.2%

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

      2. distribute-rgt-neg-in 72.2%

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

      3. sub-neg 72.2%

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

      4. neg-mul-1 72.2%

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

      5. distribute-neg-in 72.2%

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

      6. metadata-eval 72.2%

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

      7. neg-mul-1 72.2%

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

      8. remove-double-neg 72.2%

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

   8. Simplified 72.2%

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

   9. Taylor expanded in eps around -inf 84.0%

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

   11. Simplified 84.0%

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

   if -2.30000000000000017e-251 < x < 1.42e20

   1. Initial program 54.7%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 98.9%

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

      1. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in eps around inf 98.9%

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

       1. associate-*r* 98.9%

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

       2. neg-mul-1 98.9%

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

   11. Simplified 98.9%

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

   12. Taylor expanded in eps around 0 89.9%

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

   if 1.42e20 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 14.7%

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

      1. mul-1-neg 14.7%

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

      2. distribute-lft-neg-in 14.7%

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

   7. Simplified 14.7%

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

   8. Taylor expanded in x around 0 22.9%

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

   9. Taylor expanded in eps around 0 18.6%

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

   11. Simplified 18.6%

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

   12. Taylor expanded in eps around 0 62.6%

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

       1. distribute-rgt1-in 62.6%

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

       2. metadata-eval 62.6%

          \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

       3. associate-*r/ 24.5%

          \[\leadsto \frac{\color{blue}{0 \cdot \frac{x}{\varepsilon}}}{2} \]

       4. mul0-lft 62.6%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   14. Simplified 62.6%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-251}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 - x\right)}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+20}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -370:\\ \;\;\;\;\frac{\frac{x + \mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 3.45 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -370.0)
   (/ (/ (+ x (expm1 (- x))) eps_m) 2.0)
   (if (<= x 3.45e+22) (/ (+ 1.0 (exp (* x (+ -1.0 eps_m)))) 2.0) 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -370.0) {
		tmp = ((x + expm1(-x)) / eps_m) / 2.0;
	} else if (x <= 3.45e+22) {
		tmp = (1.0 + exp((x * (-1.0 + eps_m)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -370.0) {
		tmp = ((x + Math.expm1(-x)) / eps_m) / 2.0;
	} else if (x <= 3.45e+22) {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps_m)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -370.0:
		tmp = ((x + math.expm1(-x)) / eps_m) / 2.0
	elif x <= 3.45e+22:
		tmp = (1.0 + math.exp((x * (-1.0 + eps_m)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -370.0)
		tmp = Float64(Float64(Float64(x + expm1(Float64(-x))) / eps_m) / 2.0);
	elseif (x <= 3.45e+22)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps_m)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -370.0], N[(N[(N[(x + N[(Exp[(-x)] - 1), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.45e+22], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -370

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 46.4%

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

   6. Taylor expanded in eps around 0 45.2%

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

      1. neg-mul-1 45.2%

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

      2. associate--r+ 45.2%

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

      3. neg-mul-1 45.2%

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

      4. expm1-undefine 45.2%

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

      5. sub-neg 45.2%

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

      6. remove-double-neg 45.2%

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

   8. Simplified 45.2%

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

   if -370 < x < 3.4499999999999999e22

   1. Initial program 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 88.1%

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

   if 3.4499999999999999e22 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 14.7%

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

      1. mul-1-neg 14.7%

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

      2. distribute-lft-neg-in 14.7%

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

   7. Simplified 14.7%

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

   8. Taylor expanded in x around 0 22.9%

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

   9. Taylor expanded in eps around 0 18.6%

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

   11. Simplified 18.6%

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

   12. Taylor expanded in eps around 0 62.6%

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

       1. distribute-rgt1-in 62.6%

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

       2. metadata-eval 62.6%

          \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

       3. associate-*r/ 24.5%

          \[\leadsto \frac{\color{blue}{0 \cdot \frac{x}{\varepsilon}}}{2} \]

       4. mul0-lft 62.6%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   14. Simplified 62.6%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -370:\\ \;\;\;\;\frac{\frac{x + \mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 3.45 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.0068:\\ \;\;\;\;\frac{x + \left(1 + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} - \left(-1 + x \cdot \left(1 + eps\_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.0068)
   (/ (+ x (+ 1.0 (- 1.0 x))) 2.0)
   (/ (- (exp (* x eps_m)) (+ -1.0 (* x (+ 1.0 eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.0068) {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	} else {
		tmp = (exp((x * eps_m)) - (-1.0 + (x * (1.0 + eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.0068d0) then
        tmp = (x + (1.0d0 + (1.0d0 - x))) / 2.0d0
    else
        tmp = (exp((x * eps_m)) - ((-1.0d0) + (x * (1.0d0 + eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.0068) {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	} else {
		tmp = (Math.exp((x * eps_m)) - (-1.0 + (x * (1.0 + eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.0068:
		tmp = (x + (1.0 + (1.0 - x))) / 2.0
	else:
		tmp = (math.exp((x * eps_m)) - (-1.0 + (x * (1.0 + eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.0068)
		tmp = Float64(Float64(x + Float64(1.0 + Float64(1.0 - x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - Float64(-1.0 + Float64(x * Float64(1.0 + eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.0068)
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	else
		tmp = (exp((x * eps_m)) - (-1.0 + (x * (1.0 + eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.0068], N[(N[(x + N[(1.0 + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - N[(-1.0 + N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.00679999999999999962

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in x around 0 27.1%

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

      1. mul-1-neg 27.1%

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

      2. distribute-lft-neg-in 27.1%

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

   7. Simplified 27.1%

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

   8. Taylor expanded in x around 0 21.6%

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

   9. Taylor expanded in eps around 0 73.4%

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

       1. +-commutative 73.4%

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

       2. associate--l+ 73.4%

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

       3. neg-mul-1 73.4%

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

       4. distribute-lft-in 73.4%

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

       5. metadata-eval 73.4%

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

       6. neg-mul-1 73.4%

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

       7. associate-*r* 73.4%

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

       8. metadata-eval 73.4%

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

       9. *-lft-identity 73.4%

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

       10. +-commutative 73.4%

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

   11. Simplified 73.4%

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

   if 0.00679999999999999962 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 72.8%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0068:\\ \;\;\;\;\frac{x + \left(1 + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} - \left(-1 + x \cdot \left(1 + \varepsilon\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -460:\\ \;\;\;\;\frac{\frac{x + \mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -460.0)
   (/ (/ (+ x (expm1 (- x))) eps_m) 2.0)
   (if (<= x 5e+22) (/ (+ 1.0 (exp (* x eps_m))) 2.0) 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -460.0) {
		tmp = ((x + expm1(-x)) / eps_m) / 2.0;
	} else if (x <= 5e+22) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -460.0) {
		tmp = ((x + Math.expm1(-x)) / eps_m) / 2.0;
	} else if (x <= 5e+22) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -460.0:
		tmp = ((x + math.expm1(-x)) / eps_m) / 2.0
	elif x <= 5e+22:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -460.0)
		tmp = Float64(Float64(Float64(x + expm1(Float64(-x))) / eps_m) / 2.0);
	elseif (x <= 5e+22)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -460.0], N[(N[(N[(x + N[(Exp[(-x)] - 1), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+22], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -460

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 46.4%

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

   6. Taylor expanded in eps around 0 45.2%

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

      1. neg-mul-1 45.2%

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

      2. associate--r+ 45.2%

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

      3. neg-mul-1 45.2%

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

      4. expm1-undefine 45.2%

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

      5. sub-neg 45.2%

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

      6. remove-double-neg 45.2%

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

   8. Simplified 45.2%

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

   if -460 < x < 4.9999999999999996e22

   1. Initial program 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 99.3%

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

      1. *-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around 0 88.1%

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

   if 4.9999999999999996e22 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 14.7%

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

      1. mul-1-neg 14.7%

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

      2. distribute-lft-neg-in 14.7%

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

   7. Simplified 14.7%

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

   8. Taylor expanded in x around 0 22.9%

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

   9. Taylor expanded in eps around 0 18.6%

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

   11. Simplified 18.6%

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

   12. Taylor expanded in eps around 0 62.6%

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

       1. distribute-rgt1-in 62.6%

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

       2. metadata-eval 62.6%

          \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

       3. associate-*r/ 24.5%

          \[\leadsto \frac{\color{blue}{0 \cdot \frac{x}{\varepsilon}}}{2} \]

       4. mul0-lft 62.6%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   14. Simplified 62.6%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -460:\\ \;\;\;\;\frac{\frac{x + \mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \frac{\left(1 - x\right) + eps\_m \cdot \left(-1 + x \cdot eps\_m\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.3e-11)
   (/
    (-
     (+ 1.0 (/ 1.0 eps_m))
     (/ (+ (- 1.0 x) (* eps_m (+ -1.0 (* x eps_m)))) eps_m))
    2.0)
   (if (<= x 5e+22) (/ (+ 1.0 (exp (* x eps_m))) 2.0) 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.3e-11) {
		tmp = ((1.0 + (1.0 / eps_m)) - (((1.0 - x) + (eps_m * (-1.0 + (x * eps_m)))) / eps_m)) / 2.0;
	} else if (x <= 5e+22) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.3d-11)) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - (((1.0d0 - x) + (eps_m * ((-1.0d0) + (x * eps_m)))) / eps_m)) / 2.0d0
    else if (x <= 5d+22) then
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.3e-11) {
		tmp = ((1.0 + (1.0 / eps_m)) - (((1.0 - x) + (eps_m * (-1.0 + (x * eps_m)))) / eps_m)) / 2.0;
	} else if (x <= 5e+22) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.3e-11:
		tmp = ((1.0 + (1.0 / eps_m)) - (((1.0 - x) + (eps_m * (-1.0 + (x * eps_m)))) / eps_m)) / 2.0
	elif x <= 5e+22:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.3e-11)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(Float64(Float64(1.0 - x) + Float64(eps_m * Float64(-1.0 + Float64(x * eps_m)))) / eps_m)) / 2.0);
	elseif (x <= 5e+22)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.3e-11)
		tmp = ((1.0 + (1.0 / eps_m)) - (((1.0 - x) + (eps_m * (-1.0 + (x * eps_m)))) / eps_m)) / 2.0;
	elseif (x <= 5e+22)
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.3e-11], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 - x), $MachinePrecision] + N[(eps$95$m * N[(-1.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+22], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3e-11

   1. Initial program 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 56.0%

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

      1. mul-1-neg 56.0%

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

      2. distribute-lft-neg-in 56.0%

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

   7. Simplified 56.0%

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

   8. Taylor expanded in x around 0 0.8%

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

   9. Taylor expanded in eps around 0 7.3%

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

   11. Simplified 7.3%

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

   12. Taylor expanded in x around 0 32.6%

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

   if -1.3e-11 < x < 4.9999999999999996e22

   1. Initial program 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 99.3%

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

      1. *-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around 0 88.6%

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

   if 4.9999999999999996e22 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 14.7%

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

      1. mul-1-neg 14.7%

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

      2. distribute-lft-neg-in 14.7%

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

   7. Simplified 14.7%

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

   8. Taylor expanded in x around 0 22.9%

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

   9. Taylor expanded in eps around 0 18.6%

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

   11. Simplified 18.6%

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

   12. Taylor expanded in eps around 0 62.6%

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

       1. distribute-rgt1-in 62.6%

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

       2. metadata-eval 62.6%

          \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

       3. associate-*r/ 24.5%

          \[\leadsto \frac{\color{blue}{0 \cdot \frac{x}{\varepsilon}}}{2} \]

       4. mul0-lft 62.6%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   14. Simplified 62.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \frac{\left(1 - x\right) + \varepsilon \cdot \left(-1 + x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.0068:\\ \;\;\;\;\frac{x + \left(1 - \left(-1 + x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + \left(1 - x \cdot eps\_m\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.0068)
   (/ (+ x (- 1.0 (+ -1.0 x))) 2.0)
   (/ (+ (exp (* x eps_m)) (- 1.0 (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.0068) {
		tmp = (x + (1.0 - (-1.0 + x))) / 2.0;
	} else {
		tmp = (exp((x * eps_m)) + (1.0 - (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 (eps_m <= 0.0068d0) then
        tmp = (x + (1.0d0 - ((-1.0d0) + x))) / 2.0d0
    else
        tmp = (exp((x * eps_m)) + (1.0d0 - (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 (eps_m <= 0.0068) {
		tmp = (x + (1.0 - (-1.0 + x))) / 2.0;
	} else {
		tmp = (Math.exp((x * eps_m)) + (1.0 - (x * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.0068:
		tmp = (x + (1.0 - (-1.0 + x))) / 2.0
	else:
		tmp = (math.exp((x * eps_m)) + (1.0 - (x * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.0068)
		tmp = Float64(Float64(x + Float64(1.0 - Float64(-1.0 + x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + Float64(1.0 - 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 (eps_m <= 0.0068)
		tmp = (x + (1.0 - (-1.0 + x))) / 2.0;
	else
		tmp = (exp((x * eps_m)) + (1.0 - (x * eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.0068], N[(N[(x + N[(1.0 - N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.00679999999999999962

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in x around 0 27.1%

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

      1. mul-1-neg 27.1%

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

      2. distribute-lft-neg-in 27.1%

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

   7. Simplified 27.1%

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

   8. Taylor expanded in x around 0 21.6%

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

   9. Taylor expanded in eps around 0 73.4%

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

       1. +-commutative 73.4%

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

       2. associate--l+ 73.4%

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

       3. neg-mul-1 73.4%

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

       4. distribute-lft-in 73.4%

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

       5. metadata-eval 73.4%

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

       6. neg-mul-1 73.4%

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

       7. associate-*r* 73.4%

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

       8. metadata-eval 73.4%

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

       9. *-lft-identity 73.4%

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

       10. +-commutative 73.4%

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

   11. Simplified 73.4%

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

   if 0.00679999999999999962 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in eps around 0 72.8%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0068:\\ \;\;\;\;\frac{x + \left(1 - \left(-1 + x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.4% accurate, 8.1× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.2)
   (/ (+ x (+ 1.0 (- 1.0 x))) 2.0)
   (/
    (-
     (/ (+ 1.0 (- eps_m x)) eps_m)
     (/ (+ (- 1.0 x) (* eps_m (+ -1.0 (* x eps_m)))) eps_m))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.2) {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	} else {
		tmp = (((1.0 + (eps_m - x)) / eps_m) - (((1.0 - x) + (eps_m * (-1.0 + (x * eps_m)))) / 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 (eps_m <= 1.2d0) then
        tmp = (x + (1.0d0 + (1.0d0 - x))) / 2.0d0
    else
        tmp = (((1.0d0 + (eps_m - x)) / eps_m) - (((1.0d0 - x) + (eps_m * ((-1.0d0) + (x * eps_m)))) / 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 (eps_m <= 1.2) {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	} else {
		tmp = (((1.0 + (eps_m - x)) / eps_m) - (((1.0 - x) + (eps_m * (-1.0 + (x * eps_m)))) / eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.2:
		tmp = (x + (1.0 + (1.0 - x))) / 2.0
	else:
		tmp = (((1.0 + (eps_m - x)) / eps_m) - (((1.0 - x) + (eps_m * (-1.0 + (x * eps_m)))) / eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.2)
		tmp = Float64(Float64(x + Float64(1.0 + Float64(1.0 - x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(eps_m - x)) / eps_m) - Float64(Float64(Float64(1.0 - x) + Float64(eps_m * Float64(-1.0 + Float64(x * eps_m)))) / eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.2)
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	else
		tmp = (((1.0 + (eps_m - x)) / eps_m) - (((1.0 - x) + (eps_m * (-1.0 + (x * eps_m)))) / eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1.19999999999999996

   1. Initial program 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in x around 0 27.5%

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

      1. mul-1-neg 27.5%

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

      2. distribute-lft-neg-in 27.5%

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

   7. Simplified 27.5%

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

   8. Taylor expanded in x around 0 22.0%

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

   9. Taylor expanded in eps around 0 73.5%

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

       1. +-commutative 73.5%

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

       2. associate--l+ 73.5%

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

       3. neg-mul-1 73.5%

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

       4. distribute-lft-in 73.5%

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

       5. metadata-eval 73.5%

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

       6. neg-mul-1 73.5%

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

       7. associate-*r* 73.5%

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

       8. metadata-eval 73.5%

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

       9. *-lft-identity 73.5%

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

       10. +-commutative 73.5%

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

   11. Simplified 73.5%

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

   if 1.19999999999999996 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 48.7%

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

      1. mul-1-neg 48.7%

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

      2. distribute-lft-neg-in 48.7%

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

   7. Simplified 48.7%

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

   8. Taylor expanded in x around 0 20.0%

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

   9. Taylor expanded in eps around 0 26.8%

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

   11. Simplified 26.8%

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

   12. Taylor expanded in eps around 0 42.3%

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

       1. neg-mul-1 42.3%

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

       2. +-commutative 42.3%

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

       3. unsub-neg 42.3%

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

       4. distribute-lft-in 42.3%

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

       5. distribute-rgt1-in 42.3%

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

       6. metadata-eval 42.3%

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

       7. mul0-lft 42.3%

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

       8. distribute-lft-in 42.3%

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

       9. metadata-eval 42.3%

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

       10. *-rgt-identity 42.3%

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

   14. Simplified 42.3%

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

4. Final simplification 65.1%

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

Alternative 13: 69.4% accurate, 8.7× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) - (((1.0 - x) + (eps_m * (-1.0 + (x * eps_m)))) / 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 (eps_m <= 1.0d0) then
        tmp = (x + (1.0d0 + (1.0d0 - x))) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - (((1.0d0 - x) + (eps_m * ((-1.0d0) + (x * eps_m)))) / 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 (eps_m <= 1.0) {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) - (((1.0 - x) + (eps_m * (-1.0 + (x * eps_m)))) / eps_m)) / 2.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in x around 0 27.5%

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

      1. mul-1-neg 27.5%

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

      2. distribute-lft-neg-in 27.5%

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

   7. Simplified 27.5%

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

   8. Taylor expanded in x around 0 22.0%

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

   9. Taylor expanded in eps around 0 73.5%

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

       1. +-commutative 73.5%

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

       2. associate--l+ 73.5%

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

       3. neg-mul-1 73.5%

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

       4. distribute-lft-in 73.5%

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

       5. metadata-eval 73.5%

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

       6. neg-mul-1 73.5%

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

       7. associate-*r* 73.5%

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

       8. metadata-eval 73.5%

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

       9. *-lft-identity 73.5%

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

       10. +-commutative 73.5%

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

   11. Simplified 73.5%

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

   if 1 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 48.7%

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

      1. mul-1-neg 48.7%

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

      2. distribute-lft-neg-in 48.7%

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

   7. Simplified 48.7%

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

   8. Taylor expanded in x around 0 20.0%

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

   9. Taylor expanded in eps around 0 26.8%

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

   11. Simplified 26.8%

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

   12. Taylor expanded in x around 0 42.3%

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

4. Final simplification 65.1%

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

Alternative 14: 63.3% accurate, 12.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + eps\_m\right) \cdot \left(-1 + \frac{1}{eps\_m}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 + \left(1 - x\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.3e-11)
   (/ (* x (* (+ 1.0 eps_m) (+ -1.0 (/ 1.0 eps_m)))) 2.0)
   (/ (+ x (+ 1.0 (- 1.0 x))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.3e-11) {
		tmp = (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m)))) / 2.0;
	} else {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.3d-11)) then
        tmp = (x * ((1.0d0 + eps_m) * ((-1.0d0) + (1.0d0 / eps_m)))) / 2.0d0
    else
        tmp = (x + (1.0d0 + (1.0d0 - x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.3e-11) {
		tmp = (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m)))) / 2.0;
	} else {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.3e-11:
		tmp = (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m)))) / 2.0
	else:
		tmp = (x + (1.0 + (1.0 - x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.3e-11)
		tmp = Float64(Float64(x * Float64(Float64(1.0 + eps_m) * Float64(-1.0 + Float64(1.0 / eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(x + Float64(1.0 + Float64(1.0 - x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.3e-11)
		tmp = (x * ((1.0 + eps_m) * (-1.0 + (1.0 / eps_m)))) / 2.0;
	else
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.3e-11], N[(N[(x * N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x + N[(1.0 + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e-11

   1. Initial program 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 46.7%

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

   6. Taylor expanded in x around inf 26.3%

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

      1. +-commutative 26.3%

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

      2. sub-neg 26.3%

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

      3. metadata-eval 26.3%

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

      4. +-commutative 26.3%

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

   8. Simplified 26.3%

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

   if -1.3e-11 < x

   1. Initial program 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in x around 0 28.4%

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

      1. mul-1-neg 28.4%

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

      2. distribute-lft-neg-in 28.4%

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

   7. Simplified 28.4%

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

   8. Taylor expanded in x around 0 25.8%

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

   9. Taylor expanded in eps around 0 71.2%

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

       1. +-commutative 71.2%

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

       2. associate--l+ 71.2%

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

       3. neg-mul-1 71.2%

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

       4. distribute-lft-in 71.2%

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

       5. metadata-eval 71.2%

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

       6. neg-mul-1 71.2%

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

       7. associate-*r* 71.2%

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

       8. metadata-eval 71.2%

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

       9. *-lft-identity 71.2%

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

       10. +-commutative 71.2%

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

   11. Simplified 71.2%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 + \left(1 - x\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.3% accurate, 16.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;eps\_m \cdot \left(x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 + \left(1 - x\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.3e-11) (* eps_m (* x -0.5)) (/ (+ x (+ 1.0 (- 1.0 x))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.3e-11) {
		tmp = eps_m * (x * -0.5);
	} else {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.3d-11)) then
        tmp = eps_m * (x * (-0.5d0))
    else
        tmp = (x + (1.0d0 + (1.0d0 - x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.3e-11) {
		tmp = eps_m * (x * -0.5);
	} else {
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.3e-11:
		tmp = eps_m * (x * -0.5)
	else:
		tmp = (x + (1.0 + (1.0 - x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.3e-11)
		tmp = Float64(eps_m * Float64(x * -0.5));
	else
		tmp = Float64(Float64(x + Float64(1.0 + Float64(1.0 - x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.3e-11)
		tmp = eps_m * (x * -0.5);
	else
		tmp = (x + (1.0 + (1.0 - x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.3e-11], N[(eps$95$m * N[(x * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(1.0 + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e-11

   1. Initial program 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 46.7%

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

   6. Taylor expanded in x around inf 26.3%

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

      1. +-commutative 26.3%

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

      2. sub-neg 26.3%

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

      3. metadata-eval 26.3%

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

      4. +-commutative 26.3%

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

   8. Simplified 26.3%

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

   9. Taylor expanded in eps around inf 26.3%

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

       1. *-commutative 26.3%

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

       2. *-commutative 26.3%

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

       3. *-commutative 26.3%

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

       4. associate-*r* 26.3%

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

   11. Simplified 26.3%

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

   if -1.3e-11 < x

   1. Initial program 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in x around 0 28.4%

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

      1. mul-1-neg 28.4%

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

      2. distribute-lft-neg-in 28.4%

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

   7. Simplified 28.4%

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

   8. Taylor expanded in x around 0 25.8%

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

   9. Taylor expanded in eps around 0 71.2%

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

       1. +-commutative 71.2%

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

       2. associate--l+ 71.2%

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

       3. neg-mul-1 71.2%

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

       4. distribute-lft-in 71.2%

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

       5. metadata-eval 71.2%

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

       6. neg-mul-1 71.2%

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

       7. associate-*r* 71.2%

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

       8. metadata-eval 71.2%

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

       9. *-lft-identity 71.2%

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

       10. +-commutative 71.2%

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

   11. Simplified 71.2%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 + \left(1 - x\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.0% accurate, 20.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;eps\_m \cdot \left(x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.3e-11) (* eps_m (* x -0.5)) (if (<= x 480.0) 1.0 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.3e-11) {
		tmp = eps_m * (x * -0.5);
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.3d-11)) then
        tmp = eps_m * (x * (-0.5d0))
    else if (x <= 480.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.3e-11) {
		tmp = eps_m * (x * -0.5);
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.3e-11:
		tmp = eps_m * (x * -0.5)
	elif x <= 480.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.3e-11)
		tmp = Float64(eps_m * Float64(x * -0.5));
	elseif (x <= 480.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.3e-11)
		tmp = eps_m * (x * -0.5);
	elseif (x <= 480.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.3e-11], N[(eps$95$m * N[(x * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 480.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3e-11

   1. Initial program 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0 46.7%

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

   6. Taylor expanded in x around inf 26.3%

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

      1. +-commutative 26.3%

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

      2. sub-neg 26.3%

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

      3. metadata-eval 26.3%

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

      4. +-commutative 26.3%

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

   8. Simplified 26.3%

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

   9. Taylor expanded in eps around inf 26.3%

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

       1. *-commutative 26.3%

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

       2. *-commutative 26.3%

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

       3. *-commutative 26.3%

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

       4. associate-*r* 26.3%

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

   11. Simplified 26.3%

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

   if -1.3e-11 < x < 480

   1. Initial program 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 85.7%

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

      1. neg-mul-1 85.7%

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

      2. distribute-rgt-neg-in 85.7%

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

      3. sub-neg 85.7%

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

      4. neg-mul-1 85.7%

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

      5. distribute-neg-in 85.7%

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

      6. metadata-eval 85.7%

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

      7. neg-mul-1 85.7%

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

      8. remove-double-neg 85.7%

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

   8. Simplified 85.7%

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

   9. Taylor expanded in x around 0 76.9%

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

   10. Taylor expanded in x around 0 77.7%

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

   if 480 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 15.5%

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

      1. mul-1-neg 15.5%

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

      2. distribute-lft-neg-in 15.5%

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

   7. Simplified 15.5%

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

   8. Taylor expanded in x around 0 23.1%

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

   9. Taylor expanded in eps around 0 19.0%

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

   11. Simplified 19.0%

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

   12. Taylor expanded in eps around 0 60.6%

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

       1. distribute-rgt1-in 60.6%

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

       2. metadata-eval 60.6%

          \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

       3. associate-*r/ 24.6%

          \[\leadsto \frac{\color{blue}{0 \cdot \frac{x}{\varepsilon}}}{2} \]

       4. mul0-lft 60.6%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   14. Simplified 60.6%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.5% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 620:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 620.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 620.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	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 <= 620.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    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 <= 620.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 620.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 620.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 620.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 620.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 620

   1. Initial program 61.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in eps around inf 99.4%

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

   6. Taylor expanded in x around 0 78.0%

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

      1. neg-mul-1 78.0%

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

      2. distribute-rgt-neg-in 78.0%

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

      3. sub-neg 78.0%

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

      4. neg-mul-1 78.0%

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

      5. distribute-neg-in 78.0%

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

      6. metadata-eval 78.0%

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

      7. neg-mul-1 78.0%

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

      8. remove-double-neg 78.0%

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

   8. Simplified 78.0%

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

   9. Taylor expanded in x around 0 66.3%

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

   10. Taylor expanded in x around 0 59.6%

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

   if 620 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 15.5%

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

      1. mul-1-neg 15.5%

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

      2. distribute-lft-neg-in 15.5%

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

   7. Simplified 15.5%

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

   8. Taylor expanded in x around 0 23.1%

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

   9. Taylor expanded in eps around 0 19.0%

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

   11. Simplified 19.0%

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

   12. Taylor expanded in eps around 0 60.6%

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

       1. distribute-rgt1-in 60.6%

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

       2. metadata-eval 60.6%

          \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

       3. associate-*r/ 24.6%

          \[\leadsto \frac{\color{blue}{0 \cdot \frac{x}{\varepsilon}}}{2} \]

       4. mul0-lft 60.6%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   14. Simplified 60.6%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 620:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.6% 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 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in eps around inf 99.6%

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

6. Taylor expanded in x around 0 59.8%

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

   1. neg-mul-1 59.8%

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

   2. distribute-rgt-neg-in 59.8%

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

   3. sub-neg 59.8%

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

   4. neg-mul-1 59.8%

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

   5. distribute-neg-in 59.8%

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

   6. metadata-eval 59.8%

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

   7. neg-mul-1 59.8%

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

   8. remove-double-neg 59.8%

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

8. Simplified 59.8%

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

9. Taylor expanded in x around 0 49.6%

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

10. Taylor expanded in x around 0 43.0%

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

Reproduce

herbie shell --seed 2024130 
(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))