NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.7% → 98.9%

Time: 12.5s

Alternatives: 12

Speedup: 1.9×

• 39.3% 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: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 59.0%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in eps around inf 97.8%

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

   6. Taylor expanded in eps around 0 84.2%

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

      1. neg-mul-1 84.2%

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

   8. Simplified 84.2%

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

   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 76.5%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

      1. distribute-lft-neg-out 100.0%

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

      2. exp-neg 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 88.5%

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

Alternative 2: 98.9% accurate, 1.1× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.0)
   (/ (* 2.0 (exp (- 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 <= 1.0) {
		tmp = (2.0 * exp(-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 <= 1.0d0) then
        tmp = (2.0d0 * exp(-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 <= 1.0) {
		tmp = (2.0 * Math.exp(-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 <= 1.0:
		tmp = (2.0 * math.exp(-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 <= 1.0)
		tmp = Float64(Float64(2.0 * exp(Float64(-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 <= 1.0)
		tmp = (2.0 * exp(-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, 1.0], N[(N[(2.0 * N[Exp[(-x)], $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 < 1

   1. Initial program 59.0%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in eps around inf 97.8%

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

   6. Taylor expanded in eps around 0 84.2%

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

      1. neg-mul-1 84.2%

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

   8. Simplified 84.2%

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

   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 76.5%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
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 70.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in eps around inf 98.4%

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

6. Final simplification 98.4%

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

Alternative 4: 83.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-240}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 72000:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + \left(1 - x \cdot eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-240)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (if (<= x 72000.0) (/ (+ (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 <= -5e-240) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 72000.0) {
		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 <= (-5d-240)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 72000.0d0) 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 <= -5e-240) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 72000.0) {
		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 <= -5e-240:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif x <= 72000.0:
		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 <= -5e-240)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 72000.0)
		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 <= -5e-240)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 72000.0)
		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, -5e-240], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 72000.0], 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 < -5.0000000000000004e-240

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in x around 0 48.7%

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

   6. Taylor expanded in eps around inf 73.1%

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

      1. mul-1-neg 73.1%

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

      2. neg-mul-1 73.1%

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

      3. distribute-rgt-neg-in 73.1%

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

      4. distribute-neg-in 73.1%

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

      5. metadata-eval 73.1%

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

   8. Simplified 73.1%

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

   if -5.0000000000000004e-240 < x < 72000

   1. Initial program 48.3%

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

   3. Simplified 20.3%

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

   5. Taylor expanded in eps around inf 98.0%

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

   6. Taylor expanded in eps around inf 98.1%

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

      1. associate-*r* 98.1%

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

      2. neg-mul-1 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in eps around 0 89.2%

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

       1. neg-mul-1 89.2%

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

       2. unsub-neg 89.2%

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

   11. Simplified 89.2%

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

   12. Taylor expanded in eps around inf 89.9%

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

       1. *-commutative 98.8%

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

   14. Simplified 89.9%

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

   if 72000 < 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. Step-by-step derivation

      1. pow-exp 100.0%

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

      2. pow-neg 100.0%

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

      3. un-div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. sqr-neg 100.0%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 22.7%

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

      10. pow-exp 22.7%

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

      11. *-commutative 22.7%

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

      12. +-commutative 22.7%

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

      13. distribute-rgt-in 22.7%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 56.4%

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

      16. sqr-neg 56.4%

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

      17. sqrt-unprod 39.5%

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

      18. add-sqr-sqrt 39.5%

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

      19. *-un-lft-identity 39.5%

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

      20. fma-define 39.5%

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

      21. add-sqr-sqrt 0.0%

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

      22. sqrt-unprod 91.6%

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

      23. sqr-neg 91.6%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in eps around 0 62.0%

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

      1. rec-exp 62.0%

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

      2. div-sub 62.0%

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

      3. neg-mul-1 62.0%

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

      4. +-inverses 62.0%

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

   9. Simplified 62.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
   10. Step-by-step derivation

       1. metadata-eval 62.0%

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

   11. Applied egg-rr 62.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-240}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 72000:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 6000:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + \left(1 - x \cdot eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -550.0)
   (/ (* 2.0 (exp (- x))) 2.0)
   (if (<= x 6000.0) (/ (+ (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 <= -550.0) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else if (x <= 6000.0) {
		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 <= (-550.0d0)) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else if (x <= 6000.0d0) 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 <= -550.0) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else if (x <= 6000.0) {
		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 <= -550.0:
		tmp = (2.0 * math.exp(-x)) / 2.0
	elif x <= 6000.0:
		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 <= -550.0)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	elseif (x <= 6000.0)
		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 <= -550.0)
		tmp = (2.0 * exp(-x)) / 2.0;
	elseif (x <= 6000.0)
		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, -550.0], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6000.0], 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 < -550

   1. Initial program 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in eps around inf 97.4%

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

   6. Taylor expanded in eps around 0 95.0%

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

      1. neg-mul-1 95.0%

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

   8. Simplified 95.0%

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

   if -550 < x < 6e3

   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 24.8%

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

   5. Taylor expanded in eps around inf 97.9%

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

   6. Taylor expanded in eps around inf 98.0%

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

      1. associate-*r* 98.0%

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

      2. neg-mul-1 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in eps around 0 85.4%

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

       1. neg-mul-1 85.4%

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

       2. unsub-neg 85.4%

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

   11. Simplified 85.4%

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

   12. Taylor expanded in eps around inf 85.9%

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

       1. *-commutative 98.5%

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

   14. Simplified 85.9%

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

   if 6e3 < 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. Step-by-step derivation

      1. pow-exp 100.0%

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

      2. pow-neg 100.0%

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

      3. un-div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. sqr-neg 100.0%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 22.7%

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

      10. pow-exp 22.7%

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

      11. *-commutative 22.7%

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

      12. +-commutative 22.7%

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

      13. distribute-rgt-in 22.7%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 56.4%

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

      16. sqr-neg 56.4%

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

      17. sqrt-unprod 39.5%

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

      18. add-sqr-sqrt 39.5%

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

      19. *-un-lft-identity 39.5%

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

      20. fma-define 39.5%

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

      21. add-sqr-sqrt 0.0%

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

      22. sqrt-unprod 91.6%

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

      23. sqr-neg 91.6%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in eps around 0 62.0%

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

      1. rec-exp 62.0%

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

      2. div-sub 62.0%

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

      3. neg-mul-1 62.0%

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

      4. +-inverses 62.0%

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

   9. Simplified 62.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
   10. Step-by-step derivation

       1. metadata-eval 62.0%

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

   11. Applied egg-rr 62.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 6000:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-216}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 68000:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.1e-216)
   (/ (* 2.0 (exp (- x))) 2.0)
   (if (<= x 68000.0) (/ (+ 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 <= 1.1e-216) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else if (x <= 68000.0) {
		tmp = (1.0 + exp((x * (-1.0 + 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 <= 1.1d-216) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else if (x <= 68000.0d0) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) + 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 <= 1.1e-216) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else if (x <= 68000.0) {
		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 <= 1.1e-216:
		tmp = (2.0 * math.exp(-x)) / 2.0
	elif x <= 68000.0:
		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 <= 1.1e-216)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	elseif (x <= 68000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + 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 <= 1.1e-216)
		tmp = (2.0 * exp(-x)) / 2.0;
	elseif (x <= 68000.0)
		tmp = (1.0 + exp((x * (-1.0 + 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, 1.1e-216], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 68000.0], 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 < 1.09999999999999995e-216

   1. Initial program 65.4%

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

   3. Simplified 43.9%

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

   5. Taylor expanded in eps around inf 98.4%

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

   6. Taylor expanded in eps around 0 86.8%

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

      1. neg-mul-1 86.8%

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

   8. Simplified 86.8%

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

   if 1.09999999999999995e-216 < x < 68000

   1. Initial program 43.9%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in eps around inf 96.4%

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

   6. Taylor expanded in eps around inf 96.5%

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

      1. associate-*r* 96.5%

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

      2. neg-mul-1 96.5%

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

   8. Simplified 96.5%

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

      1. distribute-lft-neg-out 96.5%

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

      2. exp-neg 96.5%

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

   10. Applied egg-rr 96.5%

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

   11. Taylor expanded in eps around 0 81.2%

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

   if 68000 < 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. Step-by-step derivation

      1. pow-exp 100.0%

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

      2. pow-neg 100.0%

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

      3. un-div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. sqr-neg 100.0%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 22.7%

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

      10. pow-exp 22.7%

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

      11. *-commutative 22.7%

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

      12. +-commutative 22.7%

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

      13. distribute-rgt-in 22.7%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 56.4%

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

      16. sqr-neg 56.4%

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

      17. sqrt-unprod 39.5%

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

      18. add-sqr-sqrt 39.5%

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

      19. *-un-lft-identity 39.5%

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

      20. fma-define 39.5%

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

      21. add-sqr-sqrt 0.0%

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

      22. sqrt-unprod 91.6%

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

      23. sqr-neg 91.6%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in eps around 0 62.0%

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

      1. rec-exp 62.0%

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

      2. div-sub 62.0%

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

      3. neg-mul-1 62.0%

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

      4. +-inverses 62.0%

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

   9. Simplified 62.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
   10. Step-by-step derivation

       1. metadata-eval 62.0%

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

   11. Applied egg-rr 62.0%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-216}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 68000:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (2.0 * exp(-x)) / 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 = (2.0d0 * exp(-x)) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in eps around inf 98.4%

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

6. Taylor expanded in eps around 0 76.9%

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

   1. neg-mul-1 76.9%

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

8. Simplified 76.9%

   \[\leadsto \frac{\color{blue}{2 \cdot e^{-x}}}{2} \]
9. Add Preprocessing

Alternative 8: 62.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(\frac{2}{x} - -2\right)\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\frac{\left(x \cdot \left(\frac{2}{x} + -2\right)\right) \cdot t\_0}{t\_0}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (- (/ 2.0 x) -2.0))))
   (if (<= x 1.0) (/ (/ (* (* x (+ (/ 2.0 x) -2.0)) t_0) t_0) 2.0) 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * ((2.0 / x) - -2.0);
	double tmp;
	if (x <= 1.0) {
		tmp = (((x * ((2.0 / x) + -2.0)) * t_0) / t_0) / 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) :: t_0
    real(8) :: tmp
    t_0 = x * ((2.0d0 / x) - (-2.0d0))
    if (x <= 1.0d0) then
        tmp = (((x * ((2.0d0 / x) + (-2.0d0))) * t_0) / t_0) / 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 t_0 = x * ((2.0 / x) - -2.0);
	double tmp;
	if (x <= 1.0) {
		tmp = (((x * ((2.0 / x) + -2.0)) * t_0) / t_0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * ((2.0 / x) - -2.0)
	tmp = 0
	if x <= 1.0:
		tmp = (((x * ((2.0 / x) + -2.0)) * t_0) / t_0) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(N[(2.0 / x), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.0], N[(N[(N[(N[(x * N[(N[(2.0 / x), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 58.4%

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

   3. Simplified 38.9%

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

   5. Taylor expanded in eps around inf 97.8%

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

   6. Taylor expanded in x around 0 64.9%

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

      1. associate--l+ 64.3%

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

      2. sub-neg 64.3%

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

      3. distribute-lft-in 64.3%

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

      4. metadata-eval 64.3%

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

      5. neg-mul-1 64.3%

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

      6. unsub-neg 64.3%

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

      7. metadata-eval 64.3%

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

   8. Simplified 64.3%

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

   9. Taylor expanded in x around inf 64.8%

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

       1. sub-neg 64.8%

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

       2. associate-*r/ 64.8%

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

       3. metadata-eval 64.8%

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

       4. metadata-eval 64.8%

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

   11. Simplified 64.8%

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

       1. distribute-lft-in 64.8%

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

       2. flip-+ 74.8%

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

   13. Applied egg-rr 74.8%

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

       1. difference-of-squares 74.8%

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

       2. distribute-lft-in 74.8%

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

       3. distribute-lft-out-- 74.8%

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

       4. distribute-lft-out-- 74.8%

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

   15. Simplified 74.8%

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

   if 1 < 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. Step-by-step derivation

      1. pow-exp 100.0%

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

      2. pow-neg 100.0%

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

      3. un-div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. sqr-neg 100.0%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 22.1%

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

      10. pow-exp 22.1%

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

      11. *-commutative 22.1%

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

      12. +-commutative 22.1%

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

      13. distribute-rgt-in 22.1%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 57.6%

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

      16. sqr-neg 57.6%

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

      17. sqrt-unprod 41.2%

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

      18. add-sqr-sqrt 41.2%

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

      19. *-un-lft-identity 41.2%

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

      20. fma-define 41.2%

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

      21. add-sqr-sqrt 0.0%

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

      22. sqrt-unprod 91.8%

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

      23. sqr-neg 91.8%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in eps around 0 60.4%

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

      1. rec-exp 60.4%

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

      2. div-sub 60.4%

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

      3. neg-mul-1 60.4%

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

      4. +-inverses 60.4%

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

   9. Simplified 60.4%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
   10. Step-by-step derivation

       1. metadata-eval 60.4%

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

   11. Applied egg-rr 60.4%

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

Alternative 9: 61.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(1 + eps\_m\right)\right) \cdot \left(-1 + \frac{1}{eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -9.5e-34)
   (/ (+ 2.0 (* (* x (+ 1.0 eps_m)) (+ -1.0 (/ 1.0 eps_m)))) 2.0)
   (if (<= x 520.0) 1.0 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -9.5e-34) {
		tmp = (2.0 + ((x * (1.0 + eps_m)) * (-1.0 + (1.0 / eps_m)))) / 2.0;
	} else if (x <= 520.0) {
		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 <= (-9.5d-34)) then
        tmp = (2.0d0 + ((x * (1.0d0 + eps_m)) * ((-1.0d0) + (1.0d0 / eps_m)))) / 2.0d0
    else if (x <= 520.0d0) 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 <= -9.5e-34) {
		tmp = (2.0 + ((x * (1.0 + eps_m)) * (-1.0 + (1.0 / eps_m)))) / 2.0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -9.5e-34:
		tmp = (2.0 + ((x * (1.0 + eps_m)) * (-1.0 + (1.0 / eps_m)))) / 2.0
	elif x <= 520.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -9.5e-34)
		tmp = Float64(Float64(2.0 + Float64(Float64(x * Float64(1.0 + eps_m)) * Float64(-1.0 + Float64(1.0 / eps_m)))) / 2.0);
	elseif (x <= 520.0)
		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 <= -9.5e-34)
		tmp = (2.0 + ((x * (1.0 + eps_m)) * (-1.0 + (1.0 / eps_m)))) / 2.0;
	elseif (x <= 520.0)
		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, -9.5e-34], N[(N[(2.0 + N[(N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 520.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -9.49999999999999985e-34

   1. Initial program 95.6%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in x around 0 57.0%

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

   6. Taylor expanded in x around 0 24.6%

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

      1. associate-*r* 24.6%

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

      2. +-commutative 24.6%

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

      3. sub-neg 24.6%

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

      4. metadata-eval 24.6%

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

      5. +-commutative 24.6%

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

   8. Simplified 24.6%

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

   if -9.49999999999999985e-34 < x < 520

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in x around 0 83.6%

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

   if 520 < 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. Step-by-step derivation

      1. pow-exp 100.0%

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

      2. pow-neg 100.0%

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

      3. un-div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. sqr-neg 100.0%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 22.7%

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

      10. pow-exp 22.7%

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

      11. *-commutative 22.7%

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

      12. +-commutative 22.7%

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

      13. distribute-rgt-in 22.7%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 56.4%

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

      16. sqr-neg 56.4%

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

      17. sqrt-unprod 39.5%

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

      18. add-sqr-sqrt 39.5%

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

      19. *-un-lft-identity 39.5%

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

      20. fma-define 39.5%

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

      21. add-sqr-sqrt 0.0%

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

      22. sqrt-unprod 91.6%

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

      23. sqr-neg 91.6%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in eps around 0 62.0%

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

      1. rec-exp 62.0%

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

      2. div-sub 62.0%

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

      3. neg-mul-1 62.0%

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

      4. +-inverses 62.0%

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

   9. Simplified 62.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
   10. Step-by-step derivation

       1. metadata-eval 62.0%

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

   11. Applied egg-rr 62.0%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.1% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 2.0) (/ (- 2.0 x) 2.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 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 <= 2.0d0) then
        tmp = (2.0d0 - x) / 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 <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 - x) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 - x) / 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 <= 2.0)
		tmp = (2.0 - x) / 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, 2.0], N[(N[(2.0 - x), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 58.4%

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

   3. Simplified 38.9%

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

   5. Taylor expanded in eps around inf 97.8%

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

   6. Taylor expanded in eps around inf 97.8%

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

      1. associate-*r* 97.8%

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

      2. neg-mul-1 97.8%

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

   8. Simplified 97.8%

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

   9. Taylor expanded in x around 0 65.0%

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

       1. distribute-rgt1-in 65.0%

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

       2. metadata-eval 65.0%

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

       3. mul0-lft 65.0%

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

       4. metadata-eval 65.0%

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

       5. *-commutative 65.0%

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

       6. neg-mul-1 65.0%

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

   11. Simplified 65.0%

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

   if 2 < 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. Step-by-step derivation

      1. pow-exp 100.0%

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

      2. pow-neg 100.0%

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

      3. un-div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. sqr-neg 100.0%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 22.1%

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

      10. pow-exp 22.1%

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

      11. *-commutative 22.1%

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

      12. +-commutative 22.1%

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

      13. distribute-rgt-in 22.1%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 57.6%

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

      16. sqr-neg 57.6%

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

      17. sqrt-unprod 41.2%

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

      18. add-sqr-sqrt 41.2%

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

      19. *-un-lft-identity 41.2%

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

      20. fma-define 41.2%

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

      21. add-sqr-sqrt 0.0%

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

      22. sqrt-unprod 91.8%

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

      23. sqr-neg 91.8%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in eps around 0 60.4%

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

      1. rec-exp 60.4%

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

      2. div-sub 60.4%

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

      3. neg-mul-1 60.4%

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

      4. +-inverses 60.4%

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

   9. Simplified 60.4%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
   10. Step-by-step derivation

       1. metadata-eval 60.4%

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

   11. Applied egg-rr 60.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.0% accurate, 28.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 58.4%

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

   3. Simplified 38.9%

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

   5. Taylor expanded in eps around inf 97.8%

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

   6. Taylor expanded in x around 0 64.9%

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

      1. associate--l+ 64.3%

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

      2. sub-neg 64.3%

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

      3. distribute-lft-in 64.3%

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

      4. metadata-eval 64.3%

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

      5. neg-mul-1 64.3%

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

      6. unsub-neg 64.3%

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

      7. metadata-eval 64.3%

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

   8. Simplified 64.3%

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

   9. Taylor expanded in x around inf 64.8%

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

       1. sub-neg 64.8%

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

       2. associate-*r/ 64.8%

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

       3. metadata-eval 64.8%

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

       4. metadata-eval 64.8%

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

   11. Simplified 64.8%

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

   12. Taylor expanded in x around 0 64.9%

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

       1. neg-mul-1 64.9%

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

       2. unsub-neg 64.9%

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

   14. Simplified 64.9%

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

   if 1 < 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. Step-by-step derivation

      1. pow-exp 100.0%

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

      2. pow-neg 100.0%

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

      3. un-div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. sqr-neg 100.0%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 22.1%

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

      10. pow-exp 22.1%

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

      11. *-commutative 22.1%

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

      12. +-commutative 22.1%

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

      13. distribute-rgt-in 22.1%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 57.6%

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

      16. sqr-neg 57.6%

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

      17. sqrt-unprod 41.2%

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

      18. add-sqr-sqrt 41.2%

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

      19. *-un-lft-identity 41.2%

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

      20. fma-define 41.2%

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

      21. add-sqr-sqrt 0.0%

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

      22. sqrt-unprod 91.8%

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

      23. sqr-neg 91.8%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in eps around 0 60.4%

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

      1. rec-exp 60.4%

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

      2. div-sub 60.4%

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

      3. neg-mul-1 60.4%

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

      4. +-inverses 60.4%

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

   9. Simplified 60.4%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
   10. Step-by-step derivation

       1. metadata-eval 60.4%

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

   11. Applied egg-rr 60.4%

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

Alternative 12: 15.7% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 0.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 70.1%

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

3. Simplified 70.1%

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

   1. pow-exp 56.1%

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

   2. pow-neg 56.1%

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

   3. un-div-inv 56.1%

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

   4. +-commutative 56.1%

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

   5. add-sqr-sqrt 34.9%

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

   6. sqrt-unprod 53.8%

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

   7. sqr-neg 53.8%

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

   8. sqrt-unprod 9.3%

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

   9. add-sqr-sqrt 19.4%

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

   10. pow-exp 33.4%

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

   11. *-commutative 33.4%

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

   12. +-commutative 33.4%

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

   13. distribute-rgt-in 33.4%

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

   14. add-sqr-sqrt 14.8%

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

   15. sqrt-unprod 45.8%

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

   16. sqr-neg 45.8%

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

   17. sqrt-unprod 27.0%

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

   18. add-sqr-sqrt 53.5%

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

   19. *-un-lft-identity 53.5%

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

   20. fma-define 53.5%

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

   21. add-sqr-sqrt 26.6%

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

   22. sqrt-unprod 63.2%

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

   23. sqr-neg 63.2%

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

6. Applied egg-rr 70.1%

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

7. Taylor expanded in eps around 0 18.6%

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

   1. rec-exp 18.6%

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

   2. div-sub 18.6%

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

   3. neg-mul-1 18.6%

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

   4. +-inverses 18.8%

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

9. Simplified 18.8%

   \[\leadsto \frac{\color{blue}{0}}{2} \]
10. Step-by-step derivation

    1. metadata-eval 18.8%

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

11. Applied egg-rr 18.8%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(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))