NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.6% → 98.8%

Time: 11.2s

Alternatives: 8

Speedup: 2.0×

• 38.0% of points produce a very large (infinite) output. You may want to add a precondition.

• Rival profile vector overflowed, profile may not be complete

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.6% 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: 98.8% 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 62.6%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in eps around inf 99.5%

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

   6. Taylor expanded in eps around 0 79.6%

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

      1. neg-mul-1 79.6%

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

   8. Simplified 79.6%

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

   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 82.8%

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

   5. Taylor expanded in eps around inf 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. distribute-lft-neg-out 100.0%

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

       3. *-commutative 100.0%

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

       4. cosh-undef 100.0%

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

   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. Add Preprocessing

Alternative 2: 98.8% 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 72.4%

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

3. Simplified 63.3%

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

5. Taylor expanded in eps around inf 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 73.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 46:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;eps\_m \leq 2.2 \cdot 10^{+115}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;eps\_m \leq 6.6 \cdot 10^{+125}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{x + eps\_m \cdot \left(x \cdot \left(eps\_m + -2\right)\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 46.0)
   (/ (* 2.0 (exp (- x))) 2.0)
   (if (<= eps_m 2.2e+115)
     (/ (+ 1.0 (exp x)) 2.0)
     (if (<= eps_m 6.6e+125)
       (+ 1.0 (* x (- (* x (+ 0.25 (* x -0.08333333333333333))) 0.5)))
       (/ (+ 2.0 (/ (+ x (* eps_m (* x (+ 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 <= 46.0) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else if (eps_m <= 2.2e+115) {
		tmp = (1.0 + exp(x)) / 2.0;
	} else if (eps_m <= 6.6e+125) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else {
		tmp = (2.0 + ((x + (eps_m * (x * (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 <= 46.0d0) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else if (eps_m <= 2.2d+115) then
        tmp = (1.0d0 + exp(x)) / 2.0d0
    else if (eps_m <= 6.6d+125) then
        tmp = 1.0d0 + (x * ((x * (0.25d0 + (x * (-0.08333333333333333d0)))) - 0.5d0))
    else
        tmp = (2.0d0 + ((x + (eps_m * (x * (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 <= 46.0) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else if (eps_m <= 2.2e+115) {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	} else if (eps_m <= 6.6e+125) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else {
		tmp = (2.0 + ((x + (eps_m * (x * (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 <= 46.0:
		tmp = (2.0 * math.exp(-x)) / 2.0
	elif eps_m <= 2.2e+115:
		tmp = (1.0 + math.exp(x)) / 2.0
	elif eps_m <= 6.6e+125:
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5))
	else:
		tmp = (2.0 + ((x + (eps_m * (x * (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 <= 46.0)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	elseif (eps_m <= 2.2e+115)
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	elseif (eps_m <= 6.6e+125)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.25 + Float64(x * -0.08333333333333333))) - 0.5)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(x + Float64(eps_m * Float64(x * 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 <= 46.0)
		tmp = (2.0 * exp(-x)) / 2.0;
	elseif (eps_m <= 2.2e+115)
		tmp = (1.0 + exp(x)) / 2.0;
	elseif (eps_m <= 6.6e+125)
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	else
		tmp = (2.0 + ((x + (eps_m * (x * (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, 46.0], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 2.2e+115], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 6.6e+125], N[(1.0 + N[(x * N[(N[(x * N[(0.25 + N[(x * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(x + N[(eps$95$m * N[(x * N[(eps$95$m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 46

   1. Initial program 62.8%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in eps around inf 99.5%

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

   6. Taylor expanded in eps around 0 79.7%

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

      1. neg-mul-1 79.7%

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

   8. Simplified 79.7%

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

   if 46 < eps < 2.2e115

   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 68.6%

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

   5. Taylor expanded in eps around inf 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 0 54.4%

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

       1. neg-mul-1 54.4%

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

   11. Simplified 54.4%

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

       1. frac-2neg 54.4%

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

       2. distribute-frac-neg 54.4%

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

       3. add-sqr-sqrt 35.3%

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

       4. sqrt-unprod 100.0%

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

       5. sqr-neg 100.0%

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

       6. sqrt-unprod 64.7%

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

       7. add-sqr-sqrt 82.9%

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

       8. metadata-eval 82.9%

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

   13. Applied egg-rr 82.9%

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

       1. distribute-neg-frac2 82.9%

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

       2. metadata-eval 82.9%

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

   15. Simplified 82.9%

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

   if 2.2e115 < eps < 6.60000000000000011e125

   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 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 0 100.0%

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

       1. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0 100.0%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.25 + -0.08333333333333333 \cdot x\right) - 0.5\right)} \]

   if 6.60000000000000011e125 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 43.3%

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

      1. *-commutative 43.3%

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

      2. +-commutative 43.3%

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

      3. distribute-rgt-in 43.3%

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

      4. add-sqr-sqrt 33.8%

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

      5. sqrt-unprod 84.9%

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

      6. sqr-neg 84.9%

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

      7. sqrt-unprod 64.3%

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

      8. add-sqr-sqrt 70.7%

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

      9. *-un-lft-identity 70.7%

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

      10. unsub-neg 70.7%

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

   7. Applied egg-rr 70.7%

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

   8. Taylor expanded in x around 0 41.1%

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

      1. mul-1-neg 41.1%

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

      2. distribute-rgt-neg-in 41.1%

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

      3. sub-neg 41.1%

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

      4. metadata-eval 41.1%

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

      5. sub-neg 41.1%

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

      6. metadata-eval 41.1%

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

      7. distribute-rgt-neg-in 41.1%

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

      8. +-commutative 41.1%

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

      9. neg-mul-1 41.1%

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

      10. +-commutative 41.1%

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

      11. distribute-rgt-in 41.1%

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

      12. metadata-eval 41.1%

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

      13. associate-*l/ 41.1%

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

      14. metadata-eval 41.1%

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

   10. Simplified 41.1%

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

   11. Taylor expanded in eps around 0 63.5%

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

       1. distribute-rgt-out 63.5%

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

   13. Simplified 63.5%

       \[\leadsto \frac{2 + \color{blue}{\frac{x + \varepsilon \cdot \left(x \cdot \left(-2 + \varepsilon\right)\right)}{\varepsilon}}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 46:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.2 \cdot 10^{+115}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 6.6 \cdot 10^{+125}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{x + \varepsilon \cdot \left(x \cdot \left(\varepsilon + -2\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-82}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-82)
   (+ 1.0 (* x (- (* x (+ 0.25 (* x -0.08333333333333333))) 0.5)))
   (/ (+ 1.0 (exp x)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-82) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else {
		tmp = (1.0 + exp(x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-5d-82)) then
        tmp = 1.0d0 + (x * ((x * (0.25d0 + (x * (-0.08333333333333333d0)))) - 0.5d0))
    else
        tmp = (1.0d0 + exp(x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-82) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -5e-82:
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5))
	else:
		tmp = (1.0 + math.exp(x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -5e-82)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.25 + Float64(x * -0.08333333333333333))) - 0.5)));
	else
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -5e-82)
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	else
		tmp = (1.0 + exp(x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5e-82], N[(1.0 + N[(x * N[(N[(x * N[(0.25 + N[(x * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-82

   1. Initial program 73.7%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around inf 98.8%

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

      1. associate-*r* 98.8%

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

      2. neg-mul-1 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in eps around 0 88.7%

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

       1. neg-mul-1 88.7%

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

   11. Simplified 88.7%

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

   12. Taylor expanded in x around 0 80.8%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.25 + -0.08333333333333333 \cdot x\right) - 0.5\right)} \]

   if -4.9999999999999998e-82 < x

   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 61.2%

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

   5. Taylor expanded in eps around inf 99.8%

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

   6. Taylor expanded in eps around inf 84.4%

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

      1. associate-*r* 84.4%

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

      2. neg-mul-1 84.4%

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

   8. Simplified 84.4%

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

   9. Taylor expanded in eps around 0 45.0%

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

       1. neg-mul-1 45.0%

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

   11. Simplified 45.0%

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

       1. frac-2neg 45.0%

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

       2. distribute-frac-neg 45.0%

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

       3. add-sqr-sqrt 21.5%

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

       4. sqrt-unprod 66.6%

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

       5. sqr-neg 66.6%

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

       6. sqrt-unprod 45.1%

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

       7. add-sqr-sqrt 66.6%

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

       8. metadata-eval 66.6%

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

   13. Applied egg-rr 66.6%

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

       1. distribute-neg-frac2 66.6%

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

       2. metadata-eval 66.6%

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

   15. Simplified 66.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-82}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.2% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4500 \lor \neg \left(x \leq 3.8 \cdot 10^{+146}\right):\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (or (<= x 4500.0) (not (<= x 3.8e+146)))
   (+ 1.0 (* x (- (* x 0.25) 0.5)))
   0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((x <= 4500.0) || !(x <= 3.8e+146)) {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	} 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 <= 4500.0d0) .or. (.not. (x <= 3.8d+146))) then
        tmp = 1.0d0 + (x * ((x * 0.25d0) - 0.5d0))
    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 <= 4500.0) || !(x <= 3.8e+146)) {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if (x <= 4500.0) or not (x <= 3.8e+146):
		tmp = 1.0 + (x * ((x * 0.25) - 0.5))
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if ((x <= 4500.0) || !(x <= 3.8e+146))
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * 0.25) - 0.5)));
	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 <= 4500.0) || ~((x <= 3.8e+146)))
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[Or[LessEqual[x, 4500.0], N[Not[LessEqual[x, 3.8e+146]], $MachinePrecision]], N[(1.0 + N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4500 or 3.79999999999999979e146 < x

   1. Initial program 66.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in eps around inf 99.5%

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

   6. Taylor expanded in eps around inf 94.5%

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

      1. associate-*r* 94.5%

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

      2. neg-mul-1 94.5%

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

   8. Simplified 94.5%

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

   9. Taylor expanded in eps around 0 63.6%

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

       1. neg-mul-1 63.6%

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

   11. Simplified 63.6%

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

   12. Taylor expanded in x around 0 67.8%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]

   if 4500 < x < 3.79999999999999979e146

   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 55.3%

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

   6. Taylor expanded in x around 0 55.3%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4500 \lor \neg \left(x \leq 3.8 \cdot 10^{+146}\right):\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+146}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.5)
   (+ 1.0 (* x (- (* x (+ 0.25 (* x -0.08333333333333333))) 0.5)))
   (if (<= x 3.8e+146) 0.0 (+ 1.0 (* x (- (* x 0.25) 0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.5) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else if (x <= 3.8e+146) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	}
	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 <= 2.5d0) then
        tmp = 1.0d0 + (x * ((x * (0.25d0 + (x * (-0.08333333333333333d0)))) - 0.5d0))
    else if (x <= 3.8d+146) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 + (x * ((x * 0.25d0) - 0.5d0))
    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 <= 2.5) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else if (x <= 3.8e+146) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.5:
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5))
	elif x <= 3.8e+146:
		tmp = 0.0
	else:
		tmp = 1.0 + (x * ((x * 0.25) - 0.5))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.5)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.25 + Float64(x * -0.08333333333333333))) - 0.5)));
	elseif (x <= 3.8e+146)
		tmp = 0.0;
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * 0.25) - 0.5)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	elseif (x <= 3.8e+146)
		tmp = 0.0;
	else
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.5], N[(1.0 + N[(x * N[(N[(x * N[(0.25 + N[(x * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e+146], 0.0, N[(1.0 + N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.5

   1. Initial program 57.4%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in eps around inf 99.4%

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

   6. Taylor expanded in eps around inf 99.4%

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

      1. associate-*r* 99.4%

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

      2. neg-mul-1 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in eps around 0 80.4%

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

       1. neg-mul-1 80.4%

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

   11. Simplified 80.4%

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

   12. Taylor expanded in x around 0 78.1%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.25 + -0.08333333333333333 \cdot x\right) - 0.5\right)} \]

   if 2.5 < x < 3.79999999999999979e146

   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 54.1%

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

   6. Taylor expanded in x around 0 54.1%

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

   if 3.79999999999999979e146 < 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 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 76.0%

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

      1. associate-*r* 76.0%

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

      2. neg-mul-1 76.0%

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

   8. Simplified 76.0%

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

   9. Taylor expanded in eps around 0 3.1%

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

       1. neg-mul-1 3.1%

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

   11. Simplified 3.1%

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

   12. Taylor expanded in x around 0 54.5%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+146}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.6% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 4500.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 4500.0) {
		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 <= 4500.0d0) 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 <= 4500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 4500.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 4500.0)
		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 <= 4500.0)
		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, 4500.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4500

   1. Initial program 57.6%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in eps around inf 99.4%

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

   6. Taylor expanded in eps around inf 99.4%

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

      1. associate-*r* 99.4%

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

      2. neg-mul-1 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in x around 0 63.9%

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

       1. distribute-rgt1-in 63.9%

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

       2. metadata-eval 63.9%

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

       3. mul0-lft 63.9%

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

       4. metadata-eval 63.9%

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

       5. *-commutative 63.9%

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

       6. neg-mul-1 63.9%

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

       7. unsub-neg 63.9%

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

   11. Simplified 63.9%

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

   12. Taylor expanded in x around 0 64.1%

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

   if 4500 < 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 49.1%

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

   6. Taylor expanded in x around 0 49.1%

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

Alternative 8: 15.8% 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 72.4%

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

3. Simplified 63.3%

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

5. Taylor expanded in eps around 0 18.6%

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

6. Taylor expanded in x around 0 18.7%

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

Reproduce

herbie shell --seed 2024155 
(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))