NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.5% → 99.0%

Time: 11.5s

Alternatives: 10

Speedup: 2.0×

• 38.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -6.494161538886467e+298
  2. eps = 4.71022859007781e-192

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 2.0× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.0)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (* 2.0 (cosh (* 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 = (2.0 * cosh((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 = (2.0d0 * cosh((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 = (2.0 * Math.cosh((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 = (2.0 * math.cosh((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(2.0 * cosh(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.0)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (2.0 * cosh((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[(2.0 * N[Cosh[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 64.5%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in eps around inf 97.8%

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

   6. Taylor expanded in eps around 0 80.8%

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

      1. *-commutative 80.8%

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

      2. neg-mul-1 80.8%

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

   8. Simplified 80.8%

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

   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 81.9%

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

   5. Taylor expanded in eps around inf 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%

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

   11. Simplified 100.0%

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

       1. +-commutative 100.0%

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

       2. *-commutative 100.0%

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

       3. distribute-lft-neg-out 100.0%

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

       4. cosh-undef 100.0%

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

       5. *-commutative 100.0%

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

   13. Applied egg-rr 100.0%

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

4. Final simplification 85.2%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

3. Simplified 61.1%

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

5. Taylor expanded in eps around inf 98.3%

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

6. Final simplification 98.3%

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

Alternative 3: 72.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 7.2 \cdot 10^{+223}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 7.2e+223)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (exp (* x (- -1.0 eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 7.2e+223) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = exp((x * (-1.0 - eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 7.2d+223) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = exp((x * ((-1.0d0) - eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 7.2e+223) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = Math.exp((x * (-1.0 - eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 7.2e+223:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = math.exp((x * (-1.0 - eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 7.2e+223)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(exp(Float64(x * Float64(-1.0 - eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 7.2e+223)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = exp((x * (-1.0 - eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 7.2e+223], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 7.19999999999999982e223

   1. Initial program 70.8%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in eps around inf 98.2%

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

   6. Taylor expanded in eps around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. neg-mul-1 75.7%

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

   8. Simplified 75.7%

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

   if 7.19999999999999982e223 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 76.9%

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

   7. Taylor expanded in eps around inf 76.8%

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

      1. associate-*r* 76.8%

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

      2. +-commutative 76.8%

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

      3. mul-1-neg 76.8%

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

      4. distribute-lft-neg-in 76.8%

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

      5. neg-mul-1 76.8%

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

      6. +-commutative 76.8%

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

      7. *-lft-identity 76.8%

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

      8. metadata-eval 76.8%

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

      9. cancel-sign-sub-inv 76.8%

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

      10. mul-1-neg 76.8%

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

      11. cancel-sign-sub-inv 76.8%

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

      12. metadata-eval 76.8%

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

      13. *-lft-identity 76.8%

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

      14. +-commutative 76.8%

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

      15. distribute-rgt-neg-in 76.8%

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

      16. +-commutative 76.8%

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

      17. mul-1-neg 76.8%

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

      18. distribute-lft-in 76.8%

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

      19. metadata-eval 76.8%

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

      20. mul-1-neg 76.8%

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

      21. unsub-neg 76.8%

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

   9. Simplified 76.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 7.2 \cdot 10^{+223}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 7.8 \cdot 10^{+223}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 + eps\_m \cdot \left(-1 + eps\_m \cdot -2\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 7.8e+223)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (+ 2.0 (* x (/ (+ 1.0 (* eps_m (+ -1.0 (* eps_m -2.0)))) eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 7.8e+223) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 + (eps_m * (-1.0 + (eps_m * -2.0)))) / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 7.8d+223) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((1.0d0 + (eps_m * ((-1.0d0) + (eps_m * (-2.0d0))))) / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 7.8e+223) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 + (eps_m * (-1.0 + (eps_m * -2.0)))) / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 7.8e+223:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (2.0 + (x * ((1.0 + (eps_m * (-1.0 + (eps_m * -2.0)))) / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 7.8e+223)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 + Float64(eps_m * Float64(-1.0 + Float64(eps_m * -2.0)))) / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 7.8e+223)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (2.0 + (x * ((1.0 + (eps_m * (-1.0 + (eps_m * -2.0)))) / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 7.8e+223], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(1.0 + N[(eps$95$m * N[(-1.0 + N[(eps$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 7.7999999999999997e223

   1. Initial program 70.8%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in eps around inf 98.2%

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

   6. Taylor expanded in eps around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. neg-mul-1 75.7%

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

   8. Simplified 75.7%

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

   if 7.7999999999999997e223 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

      3. +-commutative 3.1%

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

      4. sub-neg 3.1%

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

      5. metadata-eval 3.1%

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

      6. +-commutative 3.1%

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

   8. Simplified 3.1%

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

      1. distribute-rgt-in 3.1%

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

      2. *-un-lft-identity 3.1%

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

      3. inv-pow 3.1%

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

      4. pow-plus 3.1%

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

      5. metadata-eval 3.1%

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

      6. metadata-eval 3.1%

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

      7. add-sqr-sqrt 3.0%

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

      8. sqrt-unprod 23.5%

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

      9. sqr-neg 23.5%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 42.8%

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

      12. neg-sub0 42.8%

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

   10. Applied egg-rr 42.8%

       \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(0 - \left(\varepsilon + 1\right)\right)} - \left(-\left(\varepsilon + 1\right)\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2} \]
   11. Step-by-step derivation

       1. neg-sub0 42.8%

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

       2. +-commutative 42.8%

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

       3. distribute-neg-in 42.8%

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

       4. metadata-eval 42.8%

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

       5. unsub-neg 42.8%

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

   12. Simplified 42.8%

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

   13. Taylor expanded in eps around 0 76.5%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 7.8 \cdot 10^{+223}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 + \varepsilon \cdot \left(-1 + \varepsilon \cdot -2\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \frac{x + eps\_m \cdot \left(-2 \cdot \left(x \cdot eps\_m\right) - x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -7e-12)
   (/ (+ 2.0 (/ (+ x (* eps_m (- (* -2.0 (* x eps_m)) x))) eps_m)) 2.0)
   (if (<= x 480.0) 1.0 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -7e-12) {
		tmp = (2.0 + ((x + (eps_m * ((-2.0 * (x * eps_m)) - x))) / eps_m)) / 2.0;
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-7d-12)) then
        tmp = (2.0d0 + ((x + (eps_m * (((-2.0d0) * (x * eps_m)) - x))) / eps_m)) / 2.0d0
    else if (x <= 480.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -7e-12) {
		tmp = (2.0 + ((x + (eps_m * ((-2.0 * (x * eps_m)) - x))) / eps_m)) / 2.0;
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -7e-12:
		tmp = (2.0 + ((x + (eps_m * ((-2.0 * (x * eps_m)) - x))) / eps_m)) / 2.0
	elif x <= 480.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -7e-12)
		tmp = Float64(Float64(2.0 + Float64(Float64(x + Float64(eps_m * Float64(Float64(-2.0 * Float64(x * eps_m)) - x))) / eps_m)) / 2.0);
	elseif (x <= 480.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -7e-12)
		tmp = (2.0 + ((x + (eps_m * ((-2.0 * (x * eps_m)) - x))) / eps_m)) / 2.0;
	elseif (x <= 480.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -7e-12], N[(N[(2.0 + N[(N[(x + N[(eps$95$m * N[(N[(-2.0 * N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 480.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -7.0000000000000001e-12

   1. Initial program 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in eps around inf 91.3%

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

   6. Taylor expanded in x around 0 2.9%

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

      1. associate-*r* 2.9%

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

      2. neg-mul-1 2.9%

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

      3. +-commutative 2.9%

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

      4. sub-neg 2.9%

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

      5. metadata-eval 2.9%

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

      6. +-commutative 2.9%

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

   8. Simplified 2.9%

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

      1. distribute-rgt-in 2.9%

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

      2. *-un-lft-identity 2.9%

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

      3. inv-pow 2.9%

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

      4. pow-plus 2.9%

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

      5. metadata-eval 2.9%

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

      6. metadata-eval 2.9%

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

      7. add-sqr-sqrt 3.3%

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

      8. sqrt-unprod 0.4%

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

      9. sqr-neg 0.4%

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

      10. sqrt-unprod 0.1%

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

      11. add-sqr-sqrt 24.7%

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

      12. neg-sub0 24.7%

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

   10. Applied egg-rr 24.7%

       \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(0 - \left(\varepsilon + 1\right)\right)} - \left(-\left(\varepsilon + 1\right)\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2} \]
   11. Step-by-step derivation

       1. neg-sub0 24.7%

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

       2. +-commutative 24.7%

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

       3. distribute-neg-in 24.7%

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

       4. metadata-eval 24.7%

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

       5. unsub-neg 24.7%

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

   12. Simplified 24.7%

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

   13. Taylor expanded in eps around 0 30.7%

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

   if -7.0000000000000001e-12 < x < 480

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in eps around inf 51.6%

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

   6. Taylor expanded in x around 0 54.8%

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

      1. associate-*r* 54.8%

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

      2. neg-mul-1 54.8%

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

      3. +-commutative 54.8%

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

      4. sub-neg 54.8%

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

      5. metadata-eval 54.8%

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

      6. +-commutative 54.8%

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

   8. Simplified 54.8%

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

   9. Taylor expanded in eps around inf 79.2%

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

   10. Taylor expanded in x around 0 79.5%

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

   if 480 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 53.4%

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

   6. Taylor expanded in x around 0 53.4%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(-2 \cdot \left(x \cdot \varepsilon\right) - x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 + eps\_m \cdot \left(-1 + eps\_m \cdot -2\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -6.5e-12)
   (/ (+ 2.0 (* x (/ (+ 1.0 (* eps_m (+ -1.0 (* eps_m -2.0)))) eps_m))) 2.0)
   (if (<= x 500.0) 1.0 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -6.5e-12) {
		tmp = (2.0 + (x * ((1.0 + (eps_m * (-1.0 + (eps_m * -2.0)))) / eps_m))) / 2.0;
	} else if (x <= 500.0) {
		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 <= (-6.5d-12)) then
        tmp = (2.0d0 + (x * ((1.0d0 + (eps_m * ((-1.0d0) + (eps_m * (-2.0d0))))) / eps_m))) / 2.0d0
    else if (x <= 500.0d0) 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 <= -6.5e-12) {
		tmp = (2.0 + (x * ((1.0 + (eps_m * (-1.0 + (eps_m * -2.0)))) / eps_m))) / 2.0;
	} else if (x <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -6.5e-12:
		tmp = (2.0 + (x * ((1.0 + (eps_m * (-1.0 + (eps_m * -2.0)))) / eps_m))) / 2.0
	elif x <= 500.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -6.5e-12)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 + Float64(eps_m * Float64(-1.0 + Float64(eps_m * -2.0)))) / eps_m))) / 2.0);
	elseif (x <= 500.0)
		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 <= -6.5e-12)
		tmp = (2.0 + (x * ((1.0 + (eps_m * (-1.0 + (eps_m * -2.0)))) / eps_m))) / 2.0;
	elseif (x <= 500.0)
		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, -6.5e-12], N[(N[(2.0 + N[(x * N[(N[(1.0 + N[(eps$95$m * N[(-1.0 + N[(eps$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 500.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5000000000000002e-12

   1. Initial program 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in eps around inf 91.3%

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

   6. Taylor expanded in x around 0 2.9%

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

      1. associate-*r* 2.9%

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

      2. neg-mul-1 2.9%

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

      3. +-commutative 2.9%

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

      4. sub-neg 2.9%

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

      5. metadata-eval 2.9%

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

      6. +-commutative 2.9%

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

   8. Simplified 2.9%

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

      1. distribute-rgt-in 2.9%

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

      2. *-un-lft-identity 2.9%

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

      3. inv-pow 2.9%

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

      4. pow-plus 2.9%

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

      5. metadata-eval 2.9%

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

      6. metadata-eval 2.9%

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

      7. add-sqr-sqrt 3.3%

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

      8. sqrt-unprod 0.4%

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

      9. sqr-neg 0.4%

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

      10. sqrt-unprod 0.1%

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

      11. add-sqr-sqrt 24.7%

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

      12. neg-sub0 24.7%

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

   10. Applied egg-rr 24.7%

       \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(0 - \left(\varepsilon + 1\right)\right)} - \left(-\left(\varepsilon + 1\right)\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2} \]
   11. Step-by-step derivation

       1. neg-sub0 24.7%

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

       2. +-commutative 24.7%

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

       3. distribute-neg-in 24.7%

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

       4. metadata-eval 24.7%

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

       5. unsub-neg 24.7%

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

   12. Simplified 24.7%

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

   13. Taylor expanded in eps around 0 26.7%

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

   if -6.5000000000000002e-12 < x < 500

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in eps around inf 51.6%

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

   6. Taylor expanded in x around 0 54.8%

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

      1. associate-*r* 54.8%

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

      2. neg-mul-1 54.8%

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

      3. +-commutative 54.8%

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

      4. sub-neg 54.8%

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

      5. metadata-eval 54.8%

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

      6. +-commutative 54.8%

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

   8. Simplified 54.8%

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

   9. Taylor expanded in eps around inf 79.2%

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

   10. Taylor expanded in x around 0 79.5%

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

   if 500 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 53.4%

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

   6. Taylor expanded in x around 0 53.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 + \varepsilon \cdot \left(-1 + \varepsilon \cdot -2\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.0% accurate, 16.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 12.4:\\ \;\;\;\;\frac{2 + x \cdot \left(eps\_m \cdot -2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 12.4) (/ (+ 2.0 (* x (* eps_m -2.0))) 2.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 12.4) {
		tmp = (2.0 + (x * (eps_m * -2.0))) / 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) :: tmp
    if (x <= 12.4d0) then
        tmp = (2.0d0 + (x * (eps_m * (-2.0d0)))) / 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 <= 12.4) {
		tmp = (2.0 + (x * (eps_m * -2.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 12.4:
		tmp = (2.0 + (x * (eps_m * -2.0))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 12.4)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(eps_m * -2.0))) / 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 <= 12.4)
		tmp = (2.0 + (x * (eps_m * -2.0))) / 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, 12.4], N[(N[(2.0 + N[(x * N[(eps$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 12.4000000000000004

   1. Initial program 61.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in eps around inf 61.4%

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

   6. Taylor expanded in x around 0 41.9%

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

      1. associate-*r* 41.9%

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

      2. neg-mul-1 41.9%

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

      3. +-commutative 41.9%

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

      4. sub-neg 41.9%

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

      5. metadata-eval 41.9%

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

      6. +-commutative 41.9%

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

   8. Simplified 41.9%

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

      1. distribute-rgt-in 41.9%

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

      2. *-un-lft-identity 41.9%

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

      3. inv-pow 41.9%

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

      4. pow-plus 41.9%

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

      5. metadata-eval 41.9%

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

      6. metadata-eval 41.9%

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

      7. add-sqr-sqrt 29.1%

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

      8. sqrt-unprod 41.5%

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

      9. sqr-neg 41.5%

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

      10. sqrt-unprod 12.3%

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

      11. add-sqr-sqrt 46.5%

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

      12. neg-sub0 46.5%

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

   10. Applied egg-rr 46.5%

       \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(0 - \left(\varepsilon + 1\right)\right)} - \left(-\left(\varepsilon + 1\right)\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2} \]
   11. Step-by-step derivation

       1. neg-sub0 46.5%

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

       2. +-commutative 46.5%

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

       3. distribute-neg-in 46.5%

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

       4. metadata-eval 46.5%

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

       5. unsub-neg 46.5%

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

   12. Simplified 46.5%

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

   13. Taylor expanded in eps around inf 65.1%

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

       1. *-commutative 65.1%

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

   15. Simplified 65.1%

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

   if 12.4000000000000004 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 52.8%

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

   6. Taylor expanded in x around 0 52.8%

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

Alternative 8: 57.1% 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 61.4%

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

   3. Simplified 44.9%

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

   5. Taylor expanded in eps around inf 97.6%

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

   6. Taylor expanded in eps around inf 97.6%

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

      1. associate-*r* 97.6%

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

      2. neg-mul-1 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in x around 0 60.7%

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

       1. *-commutative 60.7%

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

       2. distribute-rgt1-in 60.7%

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

       3. metadata-eval 60.7%

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

       4. mul0-lft 60.7%

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

       5. metadata-eval 60.7%

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

       6. mul-1-neg 60.7%

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

       7. unsub-neg 60.7%

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

   11. Simplified 60.7%

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

   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%

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

   5. Taylor expanded in eps around 0 52.8%

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

   6. Taylor expanded in x around 0 52.8%

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

Alternative 9: 57.1% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 550.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 550.0) {
		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 <= 550.0d0) 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 <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 550.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 550.0)
		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 <= 550.0)
		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, 550.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 550

   1. Initial program 61.6%

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

   3. Simplified 61.6%

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

   5. Taylor expanded in eps around inf 61.6%

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

   6. Taylor expanded in x around 0 41.6%

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

      1. associate-*r* 41.6%

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

      2. neg-mul-1 41.6%

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

      3. +-commutative 41.6%

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

      4. sub-neg 41.6%

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

      5. metadata-eval 41.6%

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

      6. +-commutative 41.6%

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

   8. Simplified 41.6%

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

   9. Taylor expanded in eps around inf 59.4%

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

   10. Taylor expanded in x around 0 60.1%

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

   if 550 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 53.4%

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

   6. Taylor expanded in x around 0 53.4%

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

Alternative 10: 16.0% 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 72.7%

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

3. Simplified 61.1%

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

5. Taylor expanded in eps around 0 17.0%

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

6. Taylor expanded in x around 0 17.2%

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

Reproduce

herbie shell --seed 2024177 
(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))